Effects of Reduced Gravity on Heat Transfer

Effects of Reduced Gravity on Heat Transfer ROBERT SIEGEL Lewis Research Center National Aeronautics and Space Administration Cleveland. Ohio I . Importance of Studies at Reduced Gravity . . . . . . A . Space Applications . . . . . . . . . . . . B. Gravity as an Independent Parameter . . . . . . C . Elimination of Free Convection . . . . . . . . D . Present Objective . . . . . . . . . . . I1 . Experimental Production of Reduced Gravity . . . . . A . Drop Tower . . . . . . . . . . . . . B. Airplane Trajectory . . . . . . . . . . . C . Rockets and Satellites . . . . . . . . . . . D . Magnetic Forces . . . . . . . . . . . . 111. Free Convection . . . . . . . . . . . . . A . Fluid Flow in Reduced Gravity . . . . . . . . B . Heat Transfer . . . . . . . . . . . . . C . Transient Development Times for Boundary Layer . . IV . Pool Boiling . . . . . . . . . . . . . . . A . Nucleate-Pool-Boiling Heat Transfer . . . . . . . B . Critical Heat Flux for Pool Boiling . . . . . . . C . Transition Region for Pool Boiling . . . . . . . D . Minimum Heat Flux between Transition Boiling and Film Boiling . . . . . . . . . . . . . . . E . Film-Boiling Heat Transfer . . . . . . . . . F. Dynamics of Vapor Bubbles in Saturated Nucleate Boiling . G . Bubble Dynamics in Subcooled Pool Boiling . . . . H . Vapor Patterns for Film Boiling in a Saturated Liquid . . . . . . . . . . . . V . Forced Convection Boiling A . Reduced Gravity Effect on Two-Phase Flow . . . . B . Two-Phase Heat Transfer . . . . . . . . . . C . Designs Involving Substitute Body Forces . . . . . VI . Condensation without Forced Flow . . . . . . . . A . Laminar Film Condensation on a Vertical Surface . . . B . Laminar-to-Turbulent Transition and Turbulent Flow . C . Transient Time to Establish Laminar Condensate Film .

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144 144 144 145 145 146 146 150 150 150 151 152 154 156 158 158 166 171 171 173 174 194 198 200 200 201 205 206 207 208 209

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ROBERT SIEGEL VII. Forced Flow Condensation . . . . . . A. Flow Behavior in Low Gravity . . . . B. Pressure Drop . . . . . . . . . C . Vapor-Liquid Interface . . . . . . D. Noncondensable Gas . . . . . . . VIII. Combustion . . . . . . . . . . . A. Candle Flame . . . . . . . . . B. Fuel Droplets . . . . . . . . . C. Solid Fuels . . . . . . . . . . IX. Summary and Areas for Further Investigation . Nomenclature . . . . . . . . . . References . . . . . . . . . . .

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I. Importance of Studies at Reduced Gravity

T h e study of heat-transfer processes in reduced and zero gravity is of interest for reasons of both practical and basic research importance.

A. SPACE APPLICATIONS One of the fascinating aspects of flight in space is the weightless condition experienced by the material within a vehicle that derives its acceleration solely from the force of the local gravitational field. In this instance, the inertial force will exactly oppose the gravitational force, and the contained material will experience a zero-gravity condition relative to the vehicle. If not tied down, the material will “free float” within the vehicle. This would be the case in an orbiting satellite, space capsule, space station, or any freely coasting device in space. In space, a vehicle that has a small spin or is undergoing a small acceleration will have a gravity field equal to a fraction of that on earth. This would also be true for a device on the surface of the moon. As a consequence, the design of systems for space applications has made it necessary to consider low gravity effects with regard to factors such as fluid orientation, influence on the human body, and the present subject of heat transfer.

B. GRAVITY AS AN INDEPENDENT PARAMETER T h e gravity field is one of the independent parameters in many theoretical solutions and experimental correlations. When experiments are performed throughout a range of gravity fields, a means is provided for further evaluating the applicability of assumptions and theoretical equations employed in an analysis. Experimental correlations based on observations at earth

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gravity can be tested to see whether they present the correct gravity dependence. Low gravity tests also provide a means by which gravity-dependent phenomena can be isolated from those that are independent of gravity, and this can further our basic understanding. For example, when studying bubble dynamics in nucleate pool boiling, the buoyancy force can be diminished or removed by low gravity testing. T h e remaining dynamic, drag, and surface tension forces can then be studied more clearly.

C. ELIMINATION OF FREE CONVECTION Low gravity tests can be used to remove unwanted secondary effects resulting from free convection. For example, if laminar-flow heat-transfer tests could be performed at zero gravity in an orbiting laboratory, larger temperature differences and lower flow rates than on earth could be utilized without free convection becoming of major influence. This could lead to additional substantiations of the laminar flow solutions and the momentum and energy equations from which they are derived. One of the difficulties in measuring the thermal conductivity of gases in a parallel-plate-type apparatus is the heat transferred between the plates by free convection. This unwanted component of heat flow could be eliminated by an orbital-type test. The use of very low gravity can provide a convenient change of scale by eliminating free convection. For example, the study of individual fuel droplet combustion in a very fine spray is made difficult by the small size of the drops. The drops are so small that negligible free convection develops around them when they are burned at earth gravity. For convenience, larger drops can be studied provided that the free convection which would develop around the drops in an earth environment is eliminated by utilizing low gravity tests.

D. PRESENT OBJECTIVE Although the gravity parameter appeared in many theoretical and experimental heat-transfer correlations throughout the development of the science of heat transfer, little had been done to experimentally verify the dependence at reduced gravities until the middle 1950’s. Previous to that time, some limited research had been done at high gravities-specifically, with reference to heat transfer in rotating machinery. It was the design of space devices, however, that sparked the interest in low gravity conditions and led to a rapid expansion of low gravity experimentation. This paper will review and summarize low gravity information up to about November 1966. Only heat transfer will be considered. Aspects of low

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gravity hydrodynamics or fluid orientation will be mentioned only when they are especially pertinent to the heat-transfer behavior. (A bibliography and brief review of hydrodynamic aspects is given by Habip (I).) The heattransfer subjects discussed are free convection, pool and forced flow boiling, condensation in stationary vapor and in forced flow, and combustion. Experimental and theoretical results will be compared when possible. When experimental results are not available, some of the theoretical correlations will be interpreted with regard to reductions in gravity. 11. Experimental Production of Reduced Gravity

T o experimentally study reduced and very low gravity behavior, a facility is needed that will provide a specified gravity reduction with reasonable convenience. Some of these facilities have been reviewed by Unterberg and Congelliere ( l a ) , and some pertinent characteristics will be considered here.

A. DROPTOWER Short periods of reduced gravity can be obtained with earth-bound equipment. A mass m that has a downward acceleration a can be regarded as having an upward inertial force ma. This force is in opposition to the earth gravity force; hence, when moving in the frame of reference of the accelerating mass, the effective acceleration is g = g, - a. Reduced gravity fields with magnitudes between 1 and Og, can be obtained by letting an experiment accelerate downward with an acceleration in the range from 0 to lg,. A body in perfectly free fall with no resisting force provides a zerogravity environment in the reference frame of the falling body. Therefore, to obtain reduced gravity by this means requires a structure of some type where the experiment can be hoisted and dropped under controlled conditions. A typical example of a free-fall drop tower is given in Fig. 1. The experiment could also be dropped from a natural elevation. For conditions approaching true zero gravity, air drag must be compensated for or eliminated. The drop tower could be enclosed so that it can be evacuated. Since this may not be feasible in most instances, the relative effect of air drag for a given falling package can be reduced by increasing the package weight. This, however, complicates stopping the package at the end of the fall. Air drag can be essentially eliminated by having the experiment freely falling within a larger falling container (2, 3) (Fig. 2). The outer container is acted upon by the air drag and falls slightly more slowly than the inner container. The experimental package could also be pulled downward mechanically to com-

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h

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&

FIG.1. Schematic of 85-ft drop tower, NASA Lewis Research Center (2).

pensate for air drag (4).T o obtain a range of reduced gravities, the drop tower can be counterweighted as shown in Fig. 3 ( 5 ) or a braking rocket could be attached to the package. The disadvantage of using drop towers is that they provide a relatively short test time. For a zero-gravity test, the test time t as a function of drop tower height s is given by s = *g,P

A 1-sec test requires a height of 16.1 ft, a 3-sec test 144.9 ft, and a 5-sec test

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402.5 ft. A drop tower facility at the NASA Lewis Research Center consists of a 550-ft shaft (in the ground) that will provide a test time of -5 sec. The shaft can be evacuated to eliminate air drag. The test package can also be projected upward from the bottom of the shaft and thereby double the test time to 10 sec. Longer test times magnify extremely the difficulty in bringing the package to a stop without destruction at the end of the fall. Since this review is concerned with heat transfer, space will not be taken to dwell on the engineering difficulties involved in the operation of drop towers. Rather the purpose here is to emphasize that drop tower test times are limited by

-..

Music-wi re support

Wire-release mechanism 1

LBase rounded to reduce a i r drag (a) Before test drop, experiment at top of drag shield.

shield

II II II

~ L d Ld

Ld

lDeceleration spikes

(b) During test drop, experiment moves downward within drag shield.

(c) Just before drag shield is decelerated.

Frc. 2. Schematic showing position of experiment package within drag shield (a) before, (b) during, and (c) at end of test drop (2).

practical considerations to a range of several seconds in duration. This is a short time when considering thermal equilibrium requirements for most heat-transfer experiments. When interpreting drop tower experimental results, the fact that transient conditions may be involved should be kept in mind. Fortunately, some processes appear to have reasonably short transient times. For example, in boiling studies the bubble generation is very rapid and the bubbles appear to adjust quickly to the change from normal to reduced gravity. However, even for boiling there is some question as to whether the thermal layer of superheated fluid adjacent to the heated surface can adjust rapidly enough to the lower gravity condition. The

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FIG.3 . Counterweighted drop tower used for boiling experiment (5). Drop height, 12.5 ft.

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thermal-layer thickness is probably partially governed by free convection, which has a slower time response than the bubble generation.

B. AIRPLANE TRAJECTORY T o obtain longer test times at zero or fractional gravities an airplane can be used. For zero gravity the airplane flies in a Keplerian ballistic arc so that the experiment free floats within the cabin. By this means times of 20 to 30 sec can be obtained. The gravity level is low, but there are large initial disturbances incurred from the variation in acceleration as the airplane first dives and then turns upward into the zero-gravity flight path. For a large experimental package it is impractical to have it free float. With the package tied down it is possible to control the low gravity to within k O.O2g, with an experienced pilot (6, 6a). This gravity level is an adequate approximation to Oge for the many cases where other forces are present in sufficient magnitude so that a small gravity force has negligible influence. When the experiment is fastened to the airplane, fractional gravities can also be obtained, but again there is the difficulty of the pilot to control the desired gravity within a close tolerance. C. ROCKETS AND SATELLITES Longer durations of very close to zero gravity can be obtained by having the experiment on a rocket flight, in a satellite, or in a manned space capsule (7, 8). Such tests, however, become quite costly. Nevertheless, these tests may be the only way to finally determine the true long-term performance in some devices. 'The experiments must be designed to withstand the high launch acceleration, and the space vehicle has to be stabilized against tumbling or spinning. A planned orbital boiling test is described by Kirkpatrick (9).T h e feasibility of several orbital heat transfer and fluid mechanics experiments is examined by Nein and Arnett (9a).

D. MAGNETIC FORCES A technique that provides a range of gravity fields and can operate for long durations is the use of a magnetic field with a fluid that has magnetic properties. T h e fluid is placed in a magnetic gradient such as in the core of a solenoidal magnet, and the magnetic force can be adjusted to counteract all or part of the gravitational body force. Some difficulties are encountered in obtaining magnetic fields that provide their force in a perfectly unidirectional manner because of end effects in a finite-sized magnet. Also, the liquid and vapor phases of the fluid, as would be present in a boiling or condensation experiment, are affected differently by the magnetic field, and hence there is not a perfectly uniform reduced gravity simulation thoughout the two-phase mixture.

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Another means of utilizing magnetic forces is an electromagnetic technique discussed by Kirk (10, 11).This technique arises from the electromagnetic principle that an electric current density will interact with a component of a magnetic field transverse to it and yield a body force normal to the current-magnetic field plane. The experiment could consist of a container of liquid metal with electrodes on two sides so that a current is passed horizontally through the fluid. The container is then placed between the poles of a magnet so that the magnetic field lines are horizontal and normal to the current flux. If the electric current is in the proper direction, a force will be produced in opposition to the earth gravity field and hence can be used to obtain reduced gravity conditions. This method has a number of disadvantages such as the production of joule heating in the fluid, and the fluid motion tending to distort the magnetic flux lines. The disadvantages are discussed in detail by Kirk (11).One serious difficulty that would be encountered with this method in a two-phase system such as pool boiling is that the presence of vapor voids distorts the electric field. The resulting electromagnetic body force is then nonuniform. Also, the vapor is not influenced by the electromagnetic force in the same manner as the liquid so there is not a uniform reduced gravity simulation throughout the twophase mixture. 111. Free Convection

Gravity or a similar body force field appears in the analytical predictions and experimental correlations for free convection as a natural consequence of the buoyancy force being the defined driving potential for the flow. The important dimensionless group in free convection is the Rayleigh number Ra, which is equal to the product of the Grashof G r and Prandtl Pr numbers. The Rayleigh number contains a characteristic dimension L of the system that can be the height of a vertical plate, diameter of a heated wire, or thickness of the convective boundary layer. For a surface at a specified temperature T, , the Rayleigh number is given by

where throughout the present discussion the body force will be limited to that resulting from the gravitational field g. For a surface dissipating a specified heat flux q, a modified Rayleigh number is utilized where in the dimensional analysis T, - Tb is replaced by gL/k to give

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I n free convection the magnitude of the Rayleigh number plays an important role in determining the threshold of convective motion, the limit where boundary-layer theory applies, and the transition from laminar to turbulent flow: it is also directly related to the Nusselt number. T h e gravitational field appears to the first power in Ra and Ra*; hence, when considering a gravity reduction it may be thought of as a proportionate reduction in Rayleigh number.

FLOWIN REDUCED GRAVITY A. FLUID 1. Threshold of Convective Motion Since a gravity reduction can result in low Rayleigh numbers, consideration must be given to the lower limit of free convective motion. T h e onset of free convection is a stability problem :extensive discussions of it are given by Ostrach (12) and Stuart (13), and a brief summary is given by Grober et al. (14). T h e stability is sensitive to the orientation of the heated layer. For a fluid layer confined between two horizontal plates and heated from below, Jeffreys (15)and Low (16) computed a critical Ra of 1706 based on the thickness of the layer. Above this value flow will start. This was confirmed by Schmidt and Saunders (17) and Malkus (Zd), among others, for a layer of water. Asa numerical illustration, if water at an average temperature of 100°F is contained between two horizontal plates 0.1 ft apart and the lower plate is 100°F warmer than the upper, a reduction of gravity field to 3.2 x 10-sge would cause the fluid motion to cease. If the upper surface of the layer is not bounded by a solid surface but is merely a free surface, the critical Ra is reduced to 1101 [Low (16) obtained 11081 according to the theory of Pellew and Southwell (19). For a vertical fluid layer the situation is not as clear as the horizontal case, and the onset of convection depends on the geometry enclosed between the vertical surfaces. A rectangular enclosure having two isothermal vertical walls at different temperatures, of height H , spaced L apart, and enclosed at the top and bottom by insulated plates has been studied by Eckert and Carlson (20). They found for air that the regime of pure conduction ended at RaL = 500H/L, which agreed fairly well with the analysis of Batchelor (21). A similar experiment by Emery and Chu (22)for water and oil (Pr = 30,000) and HIL = 10 and 20 indicated that the regime with conduction only is encountered for RaL < lo3.As a result of the direct dependence on Kayleigh number, the onset of the convective regime will be delayed in direct proportion to the magnitude of g as the gravity field is reduced. 2. Boundary-Layer 7’hrory When the Rayleigh number is sufficiently large so that fluid motion is present, a boundary layer may be established on the surface, depending on

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the surface orientation. Consider a vertical surface at constant temperature with laminar free convection. The boundary-layer thickness at a distance x from the leading edge is given by Eckert and Drake (23) from an integral theory as -6 _- 3.93(0.952 + Pr)114 1 (3) X Pr1/4 (Ep With all quantities held constant except g, Eq. (3) indicates that the boundary-layer thickness depends on g-1/4so that the thickness becomes very large as gravity approaches zero. For the usual boundary-layer analysis, the convective layer along the wall is assumed thin relative to the characteristic dimensions of the system. Hence if the gravity field is small, care should be taken to examine the Rayleigh number to determine whether boundarylayer-type correlations can be utilized. This will be discussed later relative to the expressions for heat transfer. T o give a numerical example, if Ra, is lo4 and Pr = 1, Eq. (3) yields S/x = 0.465 and it is doubtful that the thinboundary-layer assumptions could apply. Of course even at earth gravity a low Ra, region is encountered near the leading edge of the plate, but generally it occupies a short length and thus is not very significant except for a body of small size. In low gravity this “leading-edge region” becomes proportionately larger. 3. Boundary-Layer Transition Transition from laminar to turbulent flow depends on the Rayleigh number and for a vertical plate at uniform temperature occurs in the vicinity of Ra,= lo9. For a plate with uniform heat flux the Ra,* at transition is about 10” (24).There is no reason to believe that the basic mechanisms of fluid stability and transition should be influenced by gravity, and consequently these transition Rayleigh numbers should apply in the reduced gravity range. As a consequence of thegx3 in Ra, , as gravity is reduced for a plate at uniform temperature the physical location of transition is shifted to a larger x in relation to gp1l3.

4. Hayleigh Numbers Encountered in Low-Gravity Applications As discussed by Chin et al. (24), it is of interest to explore the range of Rayleigh numbers that would be encountered in reduced gravity applications to provide some practical judgment for the flow regimes to be expected. This will be done for three fluids: air, water, and liquid hydrogen. Free convection heat transfer is often of importance in space applications with reference to heating of fuel inside storage tanks. Here the wall heat flux is specified as imposed by solar radiation. For this reason the modified

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Rayleigh number Ra," will be considered. Figure 4 shows Ra," as a function ofg/g, for a length dimension x of 1ft and for a wall flux q of 1Btu/(hr)(sq. ft). T h e liquid-hydrogen properties were taken at saturation conditions for atmospheric pressure and are as follows: cp = 2.3 Btu/(lb)("R), p = 4.43 lb/ cu. ft, k = O.O684Btu/(hr)(ft)("R),p = 320 x lb/(ft)(hr),v = 2.01 x 1OW sq. ft/sec, and p = 0.0158/"R. Figure 4 shows that for a length of 1 ft the

i: x

Im x

Liquid hydrogen at saturation

/

Fraction of Earth gravity, g/ge

FIG.4. Magnitude of modified Rayleigh numher as function of reduced gravity. Heat transfer per unit time and area, q , 1 Rtu/(hr)(sq. ft); length, s,1 ft; all fluids at atmospheric pressure.

modified Kayleigh number can still be fairly high for liquid hydrogen even when considering gravity fields as low as 1O-'ge. Larger surfaces would give much larger Ra" because of the fourth-power dependence on length. €3. HEATTRANSFER Since the local Nusselt number depends on the local Rayleigh number and

Ra,ydepends on the productgx3(or similarly hY* depends ongx'), the effect

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of an increased or decreased gravity field on the Nusselt number can be simulated at earth gravity by a change in scale of the apparatus. This was considered for example by Schmidt (25) where the high gravity field g, in a rotating machine was simulated by scaling up the experimental equipment by a factor of (g,/ge)’’3when using an isothermal surface at earth gravity, Information concerning the effect of low gravity on free convection Nusselt numbers can thus be obtained by looking at an apparatus of reduced size. For example to simulate a reduced gravity of 1OP6ge, the apparatus can be scaled down by a factor of 100 to yield a geometrically similar boundary-layer formation in earth gravity. This follows from Eq. (3) where, for a given Prandtl number, S/x depends only on Ra,. Attention is then directed to experiments that have been performed with test sections having small dimensions such as fine wires or short vertical plates. For short vertical plates in air, data and a curve for natural convection are given by McAdams (26). T h e test results indicate that for lo4 < RaL < lo9,where the flow is laminar, the average Nusselt number over the surface varies as RaL’4. This one-quarter variation is the dependency predicted theoretically by utilizing the boundary-layer assumptions. For RaL < lo4,the NUL decreases less rapidly than Ra94, which indicates that the boundary-layer assumptions no longer apply and that the boundary-layer is becoming too thick compared with the characteristic length of the heated surface. The same behavior is observed for free convection from horizontal cylinders (26, p. 176) where below a Rayleigh number of lo4 based on the cylinder diameter the experimental data deviates from the boundary-layer prediction. Hence the limit can be assigned: if at low gravity fields RaL falls below lo4, boundary-layer theory should no longer be applied for computing the Nusselt number. T o examine the ranges of values where boundary-layer theory will not apply, consider a constant-temperature surface of characteristic dimension L and let [g$Pr/u2]L3(T, - Tb)(g/g,) = lo4. The quantity in square brackets has a specific value for any particular fluid under consideration. For a characteristic length L and a temperature difference T, - T b , the fractional gravity field can be determined below which boundary-layer theory no longer applies. Typical results are shown in Fig. 5 . For water, if T, - Tb= 100”F, and a surface of 1 ft characteristic size is considered, then gig, has to be below almost lo-’ for boundary-layer theory not to apply. If, however, the surface dimension is much smaller, say 0.01 ft, then for gig, below about 0.1 boundary-layer theory cannot be utilized. When RaI, is above lo4 it would be assumed, in the absence of any extensive low gravity information to the contrary, that the conventional boundary-layer free convection analyses and correlations would apply as given in texts such as McAdams (26) and Jakob (27). There are differences in interpretation, however. For example, in turbulent free convection from

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a vertical plate at constant temperature, Jakob (27, p. 530) provides the expression valid for los < RaL < l o i 2 : -

NU,

= &L/k= 0.129(RaL)'i3

(4)

Since RaL contains the factor L3, Eq. (4)shows that the heat-transfer coefficient & is independent of L. Hence, if RaL is varied within the range of applicability by varying L, the & value is unaffected. However, if RaL is varied by changingg, then h will vary a ~ g ' / ~ .

When RaL < lo4 and boundary-layer theory does not apply, the data correlations can be used as given by McAdams (26, pp. 173, 176) or Senftleben (28). An analysis is also available by Mahony (29).

C. TRANSIENT DEVELOPMENT TIMES FOR BOUNDARY LAYER In reduced gravity, the thermal boundary layers are of greater thickness than in earth gravity, and consequently a longer time is required for them to develop after a transient change in thermal conditions. Transient times have been discussed relative to cryogenic storage tanks for space applications

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by Schwartz and Adelberg (30,32).The development of the free convective pattern in very low gravity can be fairly slow. During the transient process the externally applied thermal boundary conditions may continue to change, or the conditions within the fluid may be altered, for example, by an outflow from the tank. Hence, in some practical applications a steady condition may never be achieved, so it is necessary to deal with the transient process. Unfortunately, free convection circulation can depend on the geometry of the heated surfaces and it is therefore difficult to formulate general statements. Some insight can be obtained, however, by considering a simplified case such as the vertical flat plate. Transient free convection from a vertical plate in laminar flow was analyzed by Siege1 (32).Experimental results by Goldstein and Eckert (33) demonstrated that the analysis was reasonable. These results were for a plate that had negligible heat capacity. When appreciable capacity within the surface is involved, the transient times are longer, so the results given here provide a lower limit for the transient times. In discussing these results it is assumed that the Rayleigh numbers at reduced gravity are still large enough so that the thin-boundary-layer assumptions apply. Consider a position at height x on a vertical plate of total length L.The plate and the surrounding fluid are initially isothermal. Then the plate has a step in either temperature or heat flux suddenly imposed on it. The analysis shows that up to a certain time the position x dissipates heat only by pure conduction into the fluid. The end of the conduction transient occurs at time t, or at dimensionless time T, . Then there is an adjustment where the thermal layer established by only conduction changes into the steady-state free convection layer. The adjustment is completed and steady state is achieved at the end of time t, or T, . Hence, for T < T, ,the transient conduction solutions for stationary media can be utilized to obtain the temperature distribution in the fluid and the associated heat transfer. For T > T,, the steady-state convection equations are employed. The expressions for the transient conduction and steady-state times are given as follows : For a step in wall temperature : T~ =

or

+[l.SO(l.S + Pr)'12+ 2.48(0.6 + Pr)'i2](RaLPr)-'/2(x/L)1/2 ( 5 4

+ 2.48(0.6 + Pr)'/'] [gp(Tzc Tb)]-1/2~1/2 (5b) = +[5.24(0.952 + Pr)'I2 + 7.10(0.377 + Pr)112] (RaLPr)-112(x/L)112(6a)

t, = $[130(1.5 + Pr)'12 T~

-

or t, = )[5.24(0.952

+ Pr)'I2 + 7.10(0.377 + Pr)112][g/3(T'c- Tb)]-112x1/2 (6b)

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For a step in wall heat flux: T, =

1.97(1 + Pr)2’5( RaL” Pr)-*i5 ( x / L ) ~ ’ ~

4.78(0.8 + Pr)2i5(RaLXPr)-’” ( x / L ) ~ ”

(7)

(8) Thus for a surface at constant temperature the transient times vary asg-‘I2, while for uniform wall heat flux the variation is as g-*l5. As a numerical example, the time is calculated to establish the free convection pattern in air at 70°F for the first foot of a plate that has been suddenly raised from 70 to 270°F. T h e Rayleigh number at earth gravity, computed from Eq. ( l ) , is about 2 x lo8 so that the laminar analysis can be utilized. Then from Eq. (6b) the steady-state time t, in earth gravity is about 2 sec. If the gravityfield is reduced to 10-4g,, the Rayleigh number is 2 x lo4 so that boundary-layer theory still applies. The steady-state time becomes 200 sec, or more than 3 min, for the boundary layer to become established. Further numerical examples are given by Schwartz and Adelberg (30, 34, who also discuss the transient development of a turbulent layer. T~ =

IV. Po01 Boiling Until about 1956 there had been practically no consideration given to a systematic study of the effect of gravity field on pool boiling. Pool boiling provides very high heat-transfer coefficients and hence can have useful applications in the high-performance heat-transfer devices employed in space applications. There are several factors of importance that must be studied in reduced gravity. It is necessary to know how the nucleate-boiling heat-transfer coefficient is influenced as gravity is reduced. T h e upper limit of the nucleate boiling flux (critical heat flux) is certainly very important for cooling surfaces where there is an imposed heating that must be dissipated. T h e upper limit is also important in the boiloff of cryogens from storage tanks where film boiling may be desirable as it will reduce the heat leakage. T h e behavior of the vapor formed during boiling is also of interest ; for example, in a liquid-cooled and moderated nuclear system, the distribution of voids in the moderator is important in determining the control of the reactor. In this section various factors in pool boiling will be treated individually with both theoretical and experimental results provided. .4. XCCLEATE-POOL-BOILING HEATTRANSFER

1. 7‘h~oql, for Kidpate Boiling ‘1‘0 obtain a possible indication of the gravity effects to he expected in nucleate pool boiling, two semitheoretical correlations can be considered as

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proposed by Rohsenow (34)and by Forster and Zuber [see Westwater (35), p. 19)]. T h e correlation of Rohsenow takes the velocity of a vapor bubble at the instant of breakoff from the surface as being the most meaningful velocity in the heat-transfer mechanism. Since this velocity depends at least in part on the buoyancy force of the bubble, the heat-transfer coefficient would be expected to be a function of gravity. T h e final result of the correlation, however, explicitly contains gravity to only a small power :

In the derivation of Forster and Zuber (35),it is postulated that the radial growth velocity of the bubbles while they are still close to the surface is the significant velocity governing the turbulent motion induced by the bubble action. As a result, this heat-transfer correlation appears to be independent of the gravity field. T h e gravity dependence indicated by these and other similar correlations must be viewed with considerable caution. The correlations were derived by utilizing physical models and bubble behavior characteristics obtained from boiling at earth gravity. Hence there could be other gravity dependencies implicitly contained in the derivations that have not been adequately accounted for.

2. Experimental Results In Table I are listed the experiments where nucleate boiling has been studied in reduced gravity. Several different fluids have been utilized such as water, liquid hydrogen, and ethyl alcohol. A variety of test section geometries were used: horizontal flat plates and wires, vertical wires, and spheres. These test sections were made from various materials such as platinum, lead, and copper. Thus, the experiments represent a diverse selection of geometry, surface material, and fluid combinations. Most of the experiments were performed by using drop towers of moderate height and hence were limited to only 1 or 2 sec at low gravity. Three experiments were conducted in airplanes that provided up to 17 sec of low gravity time. One experiment utilized a magnetic field to reduce the effective gravity on the fluid and provide results under steady-state conditions. The findings of these experiments will now be considered so that a conclusion can be made as to the behavior of nucleate boiling in reduced gravity. Our interest will be directed toward the variation of wall heat flux Q/A as a function of T,, - T,,, , which is a common way of displaying nucleate boiling

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data. In some preliminary nucleate boiling experiments in saturated water (37), it was found that for a fixed QIA no shift in T, - Tsatwas noted when the boiler was placed in near zero gravity for 0.75 sec. The instrumentation TABLE I REDUCED GRAVITY NUCLEATE POOLBOILINGEXPERIMENTS Authxe

Siegel and Usiskin

Ref.

Reduced gravity facilitv

(36) Drop tower

Usiskin and (37) Drop tower Siegel 9 ft high; counterweighied

Sherley

(38) Drop tower

Merte and Clark

(39) Drop tower 32 ft high;

Liquid Distilled water

Liquid condition Saturated

Cidelter Siege1 a d Keshoek

(5)

Test section material

Beaker

GimS

Horlzontal and Nichrome vertical ribbons: 0.006 in thick, 0.125 in. wide, 0.75 in long Horizontal wire: Platinum 0.0455-in. dlam, 2.5 in long Horizontal ribNickel ban: 0.010 in. thick, 0.2 in. wide, 2. 5 in. long Horizontal thin Lead

Distilled water

Saturated

Liquid hydrogen

Saturated

Liquid nitrogen

Saturated

Sphere: 1- and K-in. dlam

Copper

Distilled water

Saturated

Horizontal wire: 0.020-in. diam

Platinum

Airplane counterweighted ( 4 0 ) Drop tower 55 fl high

Test section immetrv

Drop tower Distiiled Saturated 12.5 It high water ethyl aleoh&. 6043 aqueoussucrose

film: 2-sq. in. area

Platinum Horizontal and vertical wires: 0.0197-indiam, 1.5 in. Long

Means of heating

Gravity range,

Test duration, see

e.

Hot plate -0 underneath Alternatingcurrent electricity

0.7

-0- 1 Directcurrent electrictty

0.75 for

zero gravity

Direct-0 1 current electricity 0 15 0.01 1 1.4 for .Heat storage of sphere zero test section gravIiy

-

-

Direct-0.01 Current electricity 0.014 Mrectcurrent electricity

1.85

0.9

MiUtiO"

Liquid nitrogen

Saturated

Sphere: K - and I-in. diam

Copper

Water

Saturated, becoming subcooled during test

Probably AlternatingStainlese current steel electricity

Schwvtz ( 4 3 ) Airplane and Mannes

Distilled water

Saturated

Rexand Knight

(44)

Ballistic

Propane (C,H. I

Vertical U-tube immersion heater: I- 1/16 in. Long, 5/16 in diam Horizontal ribbon: 2.15 in long.O.25 in. wlde.0.005 in thick Spherical tank: 25.4-em dlim

Papell and Faber

(45)

Magnet

Colloid of magnetic iron oxide in normal henane

Steel Saturated (pressure gradually inereasingj Horiaontal rtbChrome1 Saturated b n : 1/16 in. wide, 1 i n Long

Lewis e l 01. ( 4 1 ) Drop tower 31 It high; counterweighted Hedgepeth ( 4 2 ) Airplane and Z u a

missile

-

Heat storage C0.002of sphere 1 test section

NickelDirectchromium current iron alloy electricity

0

0.03 - 1

-

zero gravity

9 - 11

Direct(4.5x rvrrent 10-1 electricity Alternatingcurrent electricity

1.4 for

0-1

8

- 10

220

Steady state

of Usiskin and Siegel (37)was not able to detect a temperature shift of less than 6"F, but the indication was that gravity had little effect as long as nucleate boiling was sustained. Several more recent experiments summarized

REDUCED GRAVITY ON HEATTRANSFER

161

in the following paragraphs have utilized temperature measuring equipment of greater sensitivity. The data of Sherley (38) are shown in Fig. 6 where Q/A is plotted as a function of T, - T,,, for boiling of saturated liquid hydrogen. The data points include drop tower tests of 1-sec duration and airplane tests lasting 15 sec. There was a statistical scatter for both the 1 and Og, groups of data

FIG.6. Effect of gravity reduction on nucleate boiling curve for liquid hydrogen [Sherley (3811.

so a least-squares statistical line was fitted through each set. This revealed a very small shift in the data as a consequence of the gravity reduction. For zero gravity the T, - T,,, values were less than 0S"F smaller than at earth gravity. The data of Merte and Clark (39) for another cryogenic fluid, liquid nitrogen, displays a similar behavior. As shown in Fig. 7, the data in the nucleate region for lg, and 0.01-0.03ge are scattered within 1 or 2°F of each other. Forg/g, = 0.20 [(39),Fig. 101the data reveal a shift of about 1.5"F values than given by the near 0 and lg, results. toward larger T, - T,,,

ROBERTSIEGEL

162

For water, the data of Clodfelter (40) for a horizontal wire are given in Table 11. For a gravity reduction from 1 to O.O1ge the wall temperature decreased approximately 4"F, which indicates a higher heat-transfer coefficient ( h = (Q/A)/(Tro - TSat)) in the reduced gravity range. Table 111 from Siege1 and Keshock (5)includes data similar to those of Clodfelter (40) Fraction of Earth

inch 1/2

94, 0.01-0.03

1

0.01-0.03

1

0.20

1

0.33 0. 60

1

105

8 6

10"

Sphere

2

1 4

6

1/2

1

1

1

2 4 6 8 10' Temperature difference, ,T - TSat, "F

8 10'

2

4

6 8 lo3

FIG.7. Nucleate and transition boiling at earth and near zero gravity and film boiling at earth and fractional gravities for liquid nitrogen [Mertr and Clark (39)].

for a horizontal wire test section in water, and in this case T, - T,,, also decreased 2-4°F when gravity was reduced to 0.014ge. For a vertical test wire orientation, however, the temperature shift was in the opposite direction although of about the same magnitude as the horizontal case. The data of Schwartz and Mannes (43)for water were taken in an airplane providing a low gravity duration of 8 to 10 sec. The results are shown in

163

REDUCEDGRAVITY ON HEATTRANSFER TABLE I1 SHIFTI N SURFACE TEMPERATURE AS A RESULTOF GRAVITY REDUCTION FOR NUCLEATE BOILING OF SATURATED WATER& Heat flux Q/A Btu/(hr)(sq. ft) Fraction of earth gravity gig, = 1

1.28 x 4.36 x 5.22 x 6.09 x 6.38 x 6.87 x ~

Shift in surface temperature

Fraction of earth gravity g/ge= 0.01

1.27 x 4.33 x 5.20 x 6.09 x 6.35 x 6.82 x

104 104 104 104 104 104

T,,.(g/g, = 1) - T,,.(g/g, = 0.01) "F

3.5 4.4 4.3 3.0 4.6 4.1

104 104 104 104 104 104

~~

" From Clodfelter (40) TABLE 111 HEAT-TRANSFER DATA FOR NUCLEATE BOILING FROM AN ELECTRICALLY HEATED \TIRE" Horizontal wire Earth gravity, 1 ge Heat flux, Temp. Q/A difference,

Vertical wlre

Reduced gravity, 0.014 ge Heat flux, Temp. 4/A, difference,

Earth gravity, 1 4e Heat flux, Temp. Q/A difference, - B ~ n a t , ~ t u / ( h r ) ~w ;Teat, F (ss.ft) F

Reduced gravity, 0.014

Heat flux,

Q/A WU/(~)

gp

Temp. difference,

r,. sat,

~lu/b)

t ;Tmat,

Distilled water

30,300 50,100 90,700

15.3 16.1 18. 5

30,200 50,000 90,600

11.3 14.5 16. 5

29,100 48,700 62,800 69,400

11.3 14.2 15.9 11.3

29,100 48,900 62,900 69,700

13.8 11.6 19. 5 21.8

Ethyl alcohol

28,700 48,300 62,500

32.2 33.1 33.1

28,700 48,200 62,500

30.2 31.3 31.9

28,300 47,800

27. 1 30.1

28,300 47,800

28. 2 31.2

37.3

30,000 49,900 64,700 92,100

37.3 44.9 50.3 58.4

31,000 51,200 66,200 99,800

64.3 64. 1 79.0 87.1

31,000 51,400 66,400 94,000

65.8 61. 1 83.9 94. 5

(ss. ft)

60% by

weight aqueoussucrose solution a

30,000

49,900 64,700 92,100

F

44.9 50. 3 58.4

R U / ~ )

(scr.

ft)

~w

(6s.

It)

F

From Siege1 and Keshock ( 5 ) .

Fig. 8 and again reveal an insensitivity of the nucleate-boiling curve to the gravity reduction. Table I11 includes data for some additional fluids: ethyl alcohol and 60",, by weight aqueous-sucrose solution. For horizontal test wires, the gravity reduction caused a reduction in Tzc- T,,, of 2°F for alcohol, but had no effect for the sucrose solution. For vertical wires, T,, - T,,, increased somewhat when gravity was reduced-the increase for sucrose solution

ROBERTSIEGEL

164

being larger than for alcohol. However, the sucrose solution also required a substantially larger T, - T,,, for a given Q/A as compared with the alcohol. Figure 9 shows the data of Papell and Faber (45) who used a magnetic field to produce low gravity in normal heptane containing a colloidal suspension of magnetic particles. When this technique was used, boiling could be observed under steady-state conditions rather than for the short times available with drop towers. T h e results show a decrease in T, - T,,, of up to about 5°F when the gravity was changed from earth to near zero gravity.

loo

2

4 6 8 10' 2 4 Temperature difference, T, - lsat, "F

6 8 10'

FIG.8. Saturated nucleate pool boiling of water at 1 atm in earth and low gravity [Schwartz and Mannes ( 4 3 ) ] .

T h e data of Hedgepeth and Zara (42) revealed some interesting transient effects in airplane tests of up to 17 sec. When the experimental package was fixed to the frame of the airplane, the reduced gravity fields were i. 0.03ge. This was sufficient to keep the water mixed throughout the test and the fluid pressure and temperature remained close to equilibrium at saturation conditions. The test section temperature rose about 2°F during the low gravity period. This is consistent with the data for avertical surface by Siege1and Keshock (5)(see Table 111).When the experiment was free floating in the airplane, however, so that the gravity field was reduced to about O.O1ge,the bulk temperature remained constant but the pressure increased, thereby causing

REDUCED GRAVITY ON HEATTRANSFER

165

the bulk fluid to become subcooled. The increase in pressure was thought to be caused by a stratification of a warm layer of fluid adjacent to the vapor space in the boiler. During this transient process T, - T,,, decreased about 2.5"F.

FIG.9. Gravitational effects on nucleate pool-boiling heat transfer in magnetic iron oxide-normal heptane colloid. Saturation temperature, 205°F [Papell and Faber ( 4 5 ) ] .

An experiment using a ballistic missile is given by Rex and Knight (44) for boiling propane in a heated closed spherical tank. As in the free floating tests of Hedgepeth and Zara (42),the tank pressure increased during the flight so that the nucleate boiling was undergoing a slow transient. The

166

ROBERTSIEGEL

pressure rise may have had a tendency to suppress bubble formation. The temperature difference between the tank wall and the liquid was about 4.5"F at the very low gravity field during the test so it appears that nucleate boiling was maintained. There were no rapid excursions in tank wall temperature that would be indicative of a film-boiling condition. Rex and Knight (44) conclude that for the temperature differences that existed during the very low gravity test the wall heat fluxes dissipated were about one-third the values for lg,, and that the heat transfer in nucleate boiling is therefore adversely influenced by a very low gravity environment. Since the test period was long compared with the results from drop tower and airplane tests, this is an important finding. However, this writer would not consider the findings conclusive without further long-term tests. A close examination of the work of Rex and Knight (44) raised some questions. The T, - T,,, values were a few centigrade degrees in magnitude and were obtained from telemetered data of temperature values that were an order-of-magnitude larger. Thus a small telemetry error could lead to correspondingly much different values in T, - T,,,. Another more worrisome factor was that the lg, data used for comparison with the low gravity values were not taken with the same boiler under the same conditions of the heating surface as the low gravity tests. Rather, the lg, data were taken from Cichelli and Bonilla (46). The curve of Q/Aas a function of T, - T,,, is generally sensitive to the heated surface conditions. Hence there is some question as to whether the differences measured were a result of gravity effects or were caused by using two entirely different boilers to obtain the low gravity and lg, data. Also, the lg, data shown in Rex and Knight (44)[as taken from Cichelli and Bonilla (46)] are very sketchy in the small T, - T,,, range of the low gravity tests, so they do not provide as good a set of lg, reference data as would be desirable. T h e need for additional long-duration tests at very low gravity levels is evident. They would reveal whether the insensitivity of nucleate boiling to gravity, as found in short-duration tests, would be substantiated.

H. CRITICAL HEATFLUX FOR POOLBOILING The experimental results discussed in the previous section have shown within the frame of their limited scope that nucleate pool boiling is practically insensitive to gravity reductions. This is subject to the restriction that, when gravity is reduced, nucleate boiling is sustained and transition to another form of boiling does not occur. The heat flux imposed must not be high enough so that it exceeds the peak nucleate boiling flux for the low gravity condition being considered. This brings us to the consideration of how the peak nucleate boiling flux depends on gravity. This is a very important factor when nucleate pool boiling is to be applied in low gravity situations.

REDUCEDGRAVITY ON HEAT TRANSFER

167

1. Theory for Critical Heat Flux The theories and correlations of Kutateladze (47),Zuber (48),Noyes (49), and others have indicated that the critical heat flux depends on g1I4.For example, the relation derived for pool boiling from a horizontal surface takes the form (48)

If it is assumed that the fluid properties are not influenced by gravity reductions (they might be influenced by a change in hydrostatic pressure head when gravity becomes small), the ratio can be written as

This indicates that as the gravity field approaches zero it is no longer possible to sustain nucleate boiling as the peak nucleate boiling flux will approach zero. Experimental results will be examined to judge whether the dependency in Eq. (11) is valid. I t should be pointed out that Eq. (11) may be an oversimplification, as the theory is based mainly on a horizontal infinite flat-plate model. As discussed by Lienhard and Watanabe (50),there may exist an additional geometric effect so that Qc should be written as

where the functional dependence f would depend on the geometry involved. This could lead to a more complex gravity dependence than in Eq. (11). 2, Experimental Critical Heat Flux Behavior For the reduced gravity range, boiling experiments that provide information on the upper limit of nucleate boiling are summarized in Table IV. Experimental data are given in Figs. 10-12. The early experiments by Steinle (51) and Siege1 and Usiskin (36) indicated that for conditions close to zero gravity the critical heat flux would be substantially decreased from the values at earth gravity. When looking at the data points in Figs. 10 and 11 it is seen that there is a vertical spread in the data at each gravity for each series of tests. When a comparison is made with theory, the lowest data points should be particularly considered. The test durations are short and a period of time is required for the vapor pattern to be formed at the critical flux in order to obtain a transition from nucleate to film boiling. T o provide the necessary

ROBERTSIEGEL

168

vapor buildup in a test of short duration, it would be expected that the measured critical heat flux might often be greater than the critical flux that would be measured in a test of long duration. Data for horizontal surfaces and horizontal wires obtained using drop towers and airplanes are shown in Fig. 10. The measurements of Sherley TABLE IV CRITICAL HEATFLUXEXPERIMENTS IN POOLBOILING Authors Steinle

Ref.

ReduCsd gravity facility

( 5 1 ) Drop tower 9 ft high

tower Usiskin and (36. BOD 37) 9 i high; Siege1 counterweighted Merte and (39) Drop tower 32 ft high; Clark counterweighted (38) Drop tower Sherley

Liquid

Freon 114

Distilled water

Liquid condition Atmospheric pressure, probably saturated Saturated, atmospheric pressure

Test

kleansol heating

Gravity, Test range, duration,

-

see

R-

Horizontal Platinum Direct0 0.75 wire: 0.0015current in diam, electrlcity 1.281 in Long -0 - 1 0.75 for Horizontal Platinum Directwire: 0.0453current zero in. diam, electricity gravity 2.5 in long Heat storage 0.01 - 1.4 for Sphere: 1- and Copper % -in. diam of sphere 1 zero test sectlo" gravity

-

Saturated, atmospheric pressure

Liquid hydrogen

Saturaied, atmospheric pressure Saturated

Horizontal thin film: 2 sp. in. area

Diatllled water

Saturated, atmospheric pressure

Horizontalwire: Platinum Dlrect0.01 0.020-in. diam; current horizontal ribelectricity bons: %and % In. wide Horizontal wire: <0.01 0,020-in. d a m

Horizontalwire: Platinum Direct-0.015- 0.9 sec 0.020-in diam, current 1 for zero electricity gravity 1.5 in long Yertical wire: 0.020-in diam, 1.5 and 9.0 in

Airplane ( 4 0 ) Drop tower 55 ft high

Siege1 and Hoarell

(52) Drop tower

Saturated, atmospheric pressure

Lewtsef 01. ( 4 1 ) Drop tower 31 I t high; counterweighted Papell and ( 4 5 ) Magnet Faber

Saturated, 1,3,5-atmospheres of pressure Saturated, Colloid of mngnetic tron atmospheric oxide in narpressure mal beptane -Atmaspheric Liquid pressure oxygen

Airplane

Didilled 12.5 ft high; water ethyl counteralcohdl,608 weighted by weight aqueouseucrose solution

(53) Magnet

Test

Liquid nitrogen

Clodfalter

Lyon e f al.

gr:yR :pal

Liquid nitrogen

Lead

Directcurrent electricity

-

0

1

0

15 1.85

3.5 - 7

long

where: 1-, %-, Copper and )&-in diam

Heat dorage <0.002- 1.4 for of sphere 1 zero test section gravity

Horizontal rib- Chromei Alternatingban: 1/16 in. current electricity wide, 1 in Long Horizontal flat surface: 0.75in. diam

--

0 1

steady state

Platinum Axial heat -0.03- 1 Steady conduction state throwhcopper cylinder

(38)showed that close to Og, the critical flux had decreased to the range of 0.5 to 0.7 times the value at lg,. The tests of Clodfelter (40) which were at fields less than 0.01geyielded critical heat fluxes as low as 0.15 times the lg, value. The tests by Sherley (38) and Clodfelter (40) were confined to low gravities between 0 and O.Olg,. The precise values of the gravities were unknown ; for the airplane tests extremely close to zero gravity would be expected but large initial disturbances in the fluid may be present. The data

REDUCEDGRAVITY ON HEATTRANSFER

169

of Usiskin and Siegel (37)and Siegel and Howell (52)extend over a range of gravities and hence can be compared with the theoretical one-quarter power gravity dependence. As shown in Fig. 10 the data for water from Usiskin and Siegel (37) are located a little above the one-quarter-power line while those from Siegel and Howell (52) are somewhat below the line in the intermediate gravity range. The data €or ethyl alcohol follow the one-quarter power relation quite well. Hence for the range O.Olg, < g < lg, the onequarter power dependence appears to be a reasonable engineering approximation. For gravities lower than O.Olg, insufficient information is available to make a definitive conclusion. Although the critical heat flux was reduced considerably when zero gravity was approached, it still cannot be ascertained

c

/-

A 0.020-Inch-diam wire A 114-Inch ribbon

-$Zt

"1

10-2

1 2

Sherley(38)

0

Clodfelter (airplane) (40) Siegel and Howell (52) Water Ethyl alcohol

0 h

of zero gravity, g/ge = 0

1

V

, I Clodfelter (drop tower) 140)

A 118-Inch ribbon

2

l0-d'

0 Usiskin and Siegel 137)

I

l

l

I l l

1

4 6 8 10-1 2 Fraction of Earth gravity, g1ge

1

I

4

I

I 6

I l l -

8 1 8

FIG.10. Reduced gravity critical heat flux data for horizontal surfaces and horizontal wires obtained with drop towers and airplanes.

whether the critical flux is really zero at zero gravity. Tests of longer duration with carefully measured values of the very low gravity fields are still needed to extend the lower end of the logarithmic plot in Fig. 10. Figure 11 shows data for vertical wires and spheres obtained in drop tower tests of up to 1.4-sec duration. The data for liquid nitrogen (39) follow the one-quarter power line quite well, and so do the lower data points for water and sucrose solution given by Siegel and Howell (52). The data for ethyl alcohol, however, show a definite deviation above the theory as gravity becomes less than about 0. lge. The data in Fig. 12 were obtained by a different experimental means than those in the previous two figures. Boiling fluids were used that are influenced by a magnetic field. Hence when the fluid is placed in a uniform magnetic gradient it is possible to counteract all or part of the gravitational body force. This method has the advantage, as discussed earlier, that steady-state tests

ROBERTSIEGEL

170

Reference

10-2

4

2

6

8 10-1

2

Fraction of Earth gravity, 919,

4

818

6

FIG. 11. Reduced gravity critical heat flux data for vertical wires and spheres obtained with drop towers.

can be made. As shown in Fig. 12, the data are farther above the one-quarter power line than most of those obtained with drop tower or airplane tests. It was found by Lyon et al. (53) that if the magnetic force were increased sufficiently to provide a negative gravity (presumably adequate to lift the loo

A"

L

0

0

0

Reference

0 Papell and Faber (45)

2 Limit of zero gravity, 919,

10-1

,/-

2

=

0

0

Lyon et al. (53)

4 6 8 10-1 2 Fraction of Earth gravity, 919,

4

6

8 1

P

FIG. 12. Reduced gravity critical heat flux data for horizontal surfaces obtained by using magnetic body forces.

liquid from the surface), then when gravity was about - 0 . 0 3 ~the ~ critical heat flux decreased to very close to zero. Lewis et al. (41), provide a few data points giving the critical heat flux variation with gravity at pressures of 3 and 5 atm. T h e information is limited so that it would be speculative to try to make a definite conclusion

REDUCEDGRAVITY ON HEATTRANSFER

171

from it. Further experimentation is required on the gravitational effects at elevated pressures.

REGIONFOR POOLBOILING C. TRANSITION The experiments by Merte and Clark (39)and Lewis et al. (41)have yielded information in the transition region between the nucleate and film regimes for liquid nitrogen boiling from a sphere. A typical set of their data is shown in Fig. 7. The few points at near-zero gravity that fall in the transition region agree within a reasonable scatter with the transitional data at earth gravity. Based on this limited information it appears that the & / Adependence on T, - T,,, in the transition region is insensitive to gravity reductions, and in this respect is similar to the nucleate portion of the boiling curve.

D. MINIMUM HEATFLUX BETWEEN TRANSITION BOILINGA N D FILM BOILING The boiling curve of &/A as a function of T, - T,,, passes through a minimum Q/A in going from the transition region to the film-boiling region. A relation for this minimum derived by Berenson (54) is

which predicts a variation of ( & / A ) m i n as gravity to the one-quarter power. Equation (13) was derived for a horizontal flat surface. Data is provided for the minimum flux by Merte and Clark (39)and Lewis et al. (41)for boiling of saturated liquid nitrogen at atmospheric pressure from a 1-in.-diam sphere. Numerical values are listed in Table V and are plotted in Fig. 13. If the fact is considered that the experimental data is for a sphere, while the theory is for a horizontal plate, the agreement is remarkably good and the one-quarter dependence on gravity appears to be substantiated. For some test section geometries the one-quarter power dependence may not be valid. This wasdiscussed by Lienhard and Watanabe(5O)and Lienhard (55). It was found that for many geometric configurations other than a flat plate the ( [and perhaps also the critical heat flux as pointed out in Eq. (IZ)] can be obtained by multiplying the flat-plate expression by a geometric scale factor f(R’) : where

ROBERTSIEGEL

172

TABLE V MINIMUM HEATFLUX BETWEEN TRANSITION AND FILM BOILINGREGIME^

Fraction of earth gravity

Predicted from Eq. (13)

Experimental

1.o 0.6 0.33 0.2

2100 1850 1590 1400

1700-2100 1550 1300-1400 1300

0.03 0.01

875

0.003 0.001

491 374

gig,

a

__

Minimum heat flux (Q/A),,, Btu/(hr)(sq. ft)

666

870-1100 180-530

Lewis et al. (41)

2

4

2 4 6 810-l Fraction of Earth gravity, 919,

6 810"

2

4

6 810°

FIG.13. Minimum film boiling heat flux for a 1-inch-diam sphere in saturated liqL id nitrogen at atmospheric pressure [Lewis et al. (41)].

It is noted that R' contains a gravity factor in addition to the characteristic length L of the heated surface. Then instead of the simple type of relation as in Eq. (11) there results

REDUCED GRAVITY ON HEATTRANSFER

173

In (50)the factor f(R') for horizontal cylinders was derived as

f ( R ' ) = (2 + R'-2)'/2[R'2'3/(R'2 + 0.5)]3/4

(16) Equation (16)is valid for R' less than about 2 and the characteristic dimension L in R is the cylinder radius. Thus it is found that the gravity dependence in Eq. (15) can be quite complex. Evidently for the l-inch sphere discussed in the previous paragraph the geometric factor is not significant so that the flat-plate expression applied. E. FILM-BOILING HEATTRANSFER

1. Theoretical Relations T h e heat transfer through the vapor film in film boiling depends on whether the film is laminar or turbulent. There have been many analyses of film boiling including various surface orientations and types of boundarylayer assumptions. No attempt will be made to review them here. The present discussion is only intended to deal with the gravity dependence, and a few analytical expressions will serve to indicate this. For a laminar vapor film on a horizontal cylinder of diameter L, the filmboiling heat-transfer coefficient given by Bromley (56) is

where

Equation (17)indicates that as gravity is reduced the heat-transfer coefficient decreases in proportion to the one-quarter power of the gravity field. When the vapor film becomes turbulent as discussed by Hsu and Westwater (57) (and in the discussion at the end of their paper by Bankoff), the exponent increases and a two-fifths to one-half power variation is more applicable. An exponent of was proposed for a sphere by Frederking and Clark (58) who gave the relation analogous to Bromley's correlation

+

Since this is an average heat-transfer coefficient, it contains contributions from both the laminar and turbulent regimes.

2. Experimental Results Figure 7 shows some typical film-boiling data as given by Merte and Clark (39)and Lewis et al. (41). At a fixed T,, - T,,, there is a decrease in

174 174

ROBERTSIEGEL SIEGEL ROBERT

Q/A as gravity is reduced so that the heat-transfer coefficient decreases with gravity as expected. Figure 14 shows a correlation of the data as compared with the laminar and turbulent theories. T h e data, which extend over a range of reduced gravities from 0.17 to lg,, follow the one-third power variation quite well. Later in this review, in the section on bubble dynamics, some photographs will be shown of film boiling from horizontal and vertical wires. T h e pictures illustrate how important surface tension is in determining the configuration of the vapor film. This is especially true as gravity becomes very low as then Qld 2-

,213 '4

'ii

Frederking et al. (58)correlation:

GL-0. 14(Ra')1'3-,

103 8= 6: 4-

-

2-

4D 6-

'LBromley (56)type correlation for laminar flow:

2101

I " 1 1 1 1 1 1

I

fiI111111

KL 0. 62(Ra')114

I ' 1 1 1 1 1 1 1

1

' 1 1 i 1 1 1 1

'

1ll'Ll.L

FIG.14. Comparison of fractional gravity film boiling data for liquid nitrogen with correlations by Frederking and Clark (58) and Bromley (56). Gravity range, 0.17
surface tension begins to become the dominating force. At very low gravities the surface tension will tend to make the vapor film form into large vapor masses of a spherical shape, and the correlations derived on the basis of boundary-layer theory may no longer apply.

F. DYNAMICS OF VAPOR BUBBLESIN SATURATED NUCLEATE BOILING Another aspect of reduced gravity boilingthat isinfluenced by thechangein buoyancy is the dynamic action of the vapor bubbles being produced. Some of the nucleate-boiling heat-transfer theories are based on considerations of the actions of individual bubbles, and hence the characteristics of bubbles in reduced gravity must be studied. T h e quantities that will be

REDUCEDGRAVITY ON HEATTRANSFER

I75

discussed are bubble growth, the forces acting during growth, the size of bubbles at departure from the heated surface, and the rise of bubbles through the liquid. After the characteristics of single bubbles are presented, the behavior at high heat fluxes where many bubbles are present will be briefly considered. Siege1 and Keshock (59, 60) studied bubble dynamics in saturated pool boiling from a flat horizontal surface 2+ inches in diameter with a heated area in the center inch in diameter. This provided an area sufficiently large so that the symmetric growth of the bubbles would not be hampered. T h e surface had a very low roughness which limited the number of nucleation sites, and when a low heat flux was used, only a few sites were active. Single bubbles could then be studied without mutual interference effects. All of the bubbles discussed here were obtained in drop tower tests of approximately I-sec duration.

1. Nucleation Cycle and Coalescence of Successive Bubbles in Reduced Gravity As the gravity field is reduced, the detached bubbles begin to rise very slowly because of decreased buoyancy. This leads to a bubble coalescence mechanism that is much less frequently observed during earth gravity boiling. After a bubble departs and begins to move upward, if its rise velocity is small, the next bubble growing at the surface will collide with the rising bubble because of the rapid rate at which the diameter of the attached bubble increases during the early stages of growth. The succeeding bubbles formed at the nucleation site that contact the detached bubble and merge with it are thereby pulled from the surface before they can grow very large. This is illustrated by the sequence of photographs in Figs. 15(a) and 15(b). Several bubbles will rapidly feed into the larger bubble until it finally rises out of range. Then the next bubble will grow in an undisturbed manner. A bubble column in the low-gravity boiling regime is thus characterized by a distinctive cyclical behavior. An undisturbed bubble will grow to its final size and detach in a normal manner. Then several small bubbles will merge into it before the detached bubble can rise very far away from the surface. T h e large bubble thus serves as a temporary vapor reservoir near the surface and absorbs new bubbles while they are relatively very small. The bubble frequency at the surface is quite high when the small bubbles are merging into the larger one. This could greatly increase the turbulence induced near the surface and aid in promotinga high heat-transfer coefficient. Hence, this portion of the bubble cycle could play a significant role in reduced-gravity nucleate-pool-boiling heat transfer. At higher heat fluxes there will be many nucleation sites and a bubble will generally not have the opportunity to grow to completion without interference from adjacent

176

ROBERTSIECEL

FIG.lS(a). Comparison of bubbles growing in saturated water at atmospheric pressure for earth gravity and 6.1 04 of earth gravity. Time is measured from onset of growth for each - T,,,, 17°F bubble; heat flux, Q/A, 17,700 Btu/(hr)(sq. ft); temperature difference, T," [Siegel and Keshock (59)].

REDUCEDGRAVITY ON HEATTRANSFER

177

FIG.15(b). Continuation of the growth of the reduced gravity (0.061 g,) bubble in Fig. 15(a). It shows the merging of successive bubbles with the undisturbed bubble in Fig. 15(a).

bubbles. In this instance, the boiling process could be largely dependent on the small bubbles that form and rapidly merge with a vapor mass remaining near the heated surface. 2. Bubble-Growth Rates

When investigating the boiling process by examining the details of bubble dynamics, a factor of fundamental importance is the rate of growth of bubbles while they are attached to the surface.

178

ROBERTSIEGEL

a . Theoretical Relations. There have been many bubble-growth relations derived in the literature, and only a few of them will be discussed here to examine their gravity dependence. Fritz and Ende (61) considered bubble growth in an infinite uniformly superheated liquid. The heat conduction into the bubble was determined by having the temperature profile in the liquid adjacent to the bubble boundary equal to that for unsteady heat conduction in a slab. Their analysis resulted in the equation

Plesset and Zwick (62) included the influence of liquid inertia and accounted for the effect of the spherical shape of the boundary on the temperature profile, rather than using the temperature distribution in a plane slab as done by Fritz and Ende. This gave the same form as Eq. (19) except with an additional 2/3 multiplying the right side. Forster and Zuber (63)obtained 2 the right the same form as Eq. (19) except with an additional ~ r / multiplying side. Zuber (64) considered growth in a nonuniform temperature field and introduced a correction factor for sphericity to obtain

All of these expressions were derived for growth in an infinite medium away from solid surfaces. Surface tension, viscosity, and inertia were not considered to be important. Equation (19) indicates .a steady increase of diameter with the square root of time, while Eq. (20) predicts that a maximum diameter will be reached. The notable fact for our present purposes is that during bubble growth these expressions indicate that the functional dependency of bubble diameter on time is independent of gravity. 6. Experimental Results. Figures 16-18 show typical bubble-growth measurements for bubbles growing in liquids at saturation temperature and atmospheric pressure conditions. Figure 16 is for water, while Fig. 17 is for a 60°4 by weight aqueous-sucrose solution. The growth curves do not exhibit any definite trend with gravity, the differences between the curves being within the range of variations encountered between different bubbles in tests at a fixed gravity. This finding is in accord with the theory. The curves for water extend to much longer times as gravity is reduced because the bubble departure size in this fluid increases as gravity is reduced. This will be discussed in the next section. In Fig. 18, data for water are compared with the theoretical growth relations. The Fritz-Ende relation offers the best general agreement over the entire range of data. However, if the data are grouped into initial ( t < 0.02

REDUCEDGRAVITY ON HEATTRANSFER

179

Time, t, sec

FIG.16. Growth of typical single bubbles in saturated water at atmospheric pressure for seven different gravity fields. Heat flux, Q/A,10,900Btu/(hr)(sq.ft); temperature difference, T,,- T,,, , 11.1"F [Siegel and Keshock ( 5 9 ) ] .

,0002

,0004

,001

,002

,004

Time. 1.

sec

.01

.02

.c

FIG.17. Growth of typical single bubbles in 600: by weight aqueous-sucrose solution at saturation temperature and atmospheric pressure for five gravity fields. Heat flux, Q / A , 20,500 Btu/(hr)(sq.ft); temperaturedifference,T,$,- T,,, ,30.1°F [Keshockand Siegel(6O)l.

ROBERTSIEGEL

180

sec) and final ( t >0.02 sec) growth periods, the Fritz-Ende relation does not indicate the observed diameter variation during the final growth period. The data indicate D t3'* compared with t1I2 from theory. In the initial growth period, the time exponents were observed to range from 0.5 to 0.8. In the final growth period at low gravity, the bubbles in water become so large that they may extend out of the superheated thermal layer adjacent to the surface. In this instance the model used in deriving the Fritz-Ende relation N

FIG.18. Comparison of theoretical predictions with bubble growth data in reduced gravity for saturated water at atmospheric pressure. Heat flux, Q / A ,10,900 Btu/(hr)(sq. ft) ; temperature difference, T, - T,,,, 11.1"F[Siegel and Keshock ( 5 9 ) ] .

[(Eq. (19)]would not seem reasonable. In the final stage of growth, vaporization may actually occur from only around the cylindrical stem at the bubble base. If it is assumed that the heat transfer through the stem area is constant, since the base diameter is fairly constant as will be shown in a later figure, then

(E')

pUh --

=

const

or D t1I3.This is close to the three-eights power variation indicated by the data in Fig. 18. N

REDUCEDGRAVITY ON HEATTRANSFER

181

3 . Diameter of Bubbles at Departure a . Theoretical Relations. Several theoretical relations have been proposed for predicting the size of bubbles at departure from a horizontal surface, and two of these will now be reviewed so that the gravity dependence can be examined. T h e best known is the Fritz (65) equation

where the contact angle 8 is in degrees. A relation by Zuber (48)is

These relations indicate that the bubble departure diameter will increase with gravity reduction as g-"* or g-Il3, respectively, provided none of the

0 Water aq ueous-s ucrose

8

c

6 10-2

2

I

4

1

6

I

I

8 10-1

10

2

Fraction of Earth gravity, 919,

4

I

6

I

I

8 loo

FIG. 19. Effect of reduced gravity on diameters of bubbles in saturated liquids at instant of detachment from heated surface [Siegel and Keshock (59, 6 0 ) ] .

other factors in the equations, such as 8, depend on gravity. T h e Fritz relation [Eq. (22)] was derived by assuming that buoyancy alone is the force that overcomes the surface tension at the bubble base and pulls the bubble from the surface. Hence the negative one-half power function of gravity depends on the absence of appreciable dynamic forces to initiate bubble detachment. b. Experimental Results. Siegel and Keshock (59, 60) measured average bubble diameters in reduced gravity at departure from the heated surface and normalized the values with respect to the average departure size in earth gravity. The results are shown in Fig. 19 for distilled water and a 60y0by

182

ROBERTSIEGEL

weight aqueous-sucrose solution. T h e contact angle [B in Eq. (241 was found by Siegel and Keshock (59) to be independent of gravity, and hence Eq. (22) predicts D d ( g ) / D d ( l g e= ) (g/ge)-’lz. T h e data for water does have a negative one-half power slope for (gig,) < 0.1 ; hence, the large bubbles in this gravity range appear to be governed only by buoyancy and surface tension forces as this is the foundation for Eq. (22). For 0.1 < gige < 1, the slope for water in Fig. 19 tends more toward a negative one-third power variation, which indicates that additional forces are influencing bubble departure. For the 60n,, sucrose solution it is surprising that the departure diameter has hardly any variation with gravity. This indicates that buoyancy has practically no role in bubble departure for this fluid. T o examine why the bubble departure diameters have such different gravity dependencies in the two fluids, the forces acting on the bubbles were investigated.

4. Forces Acting on Bubbles during Bubble GiTowtA ‘The forces acting on bubbles during their growth in earth and reduced gravity have been computed by Keshock and Siegel (60) for saturated conditions and by Cochran et al. (66) for subcooled conditions. Since there are some differences in these two references in the way the forces were analyzed, the force expressions will now be reviewed. A typical bubble is shown in Fig. 20. Forces opposing bubble detachment will be taken as negative. T h e surface tension holding the bubble to the surface is given by Fs= -.rrUb 0 sin B (24) ‘The drag force was derived approximately by Keshock and Siegel (60) as

Fd = -(7~/16)bpLDdD/dt

(25)

where D is an equivalent spherical diameter for the bubble, and a value of

45 was dsed for b, which is a constant in the drag coefficient. A similar expression was used by Cochran et al. (66) with b = 48. T h e net buoyancy force is equal to the integral over the bubble surface of the vertical component of the liquid hydrostatic pressure force minus the downward-acting weight of the vapor in the bubble. For an unattached bubble this yields simply

When a bubble is attached to the surface, however, the buoyancy force must be modified to account for the fact that the base area does not have the

REDUCED GRAVITY ON HEATTRANSFER

183

ordinary liquid pressure acting underneath it. A correction to account for this was applied directly (60) to Eq. (26) to yield

T h e correction term does not contain gravity and will continue to aid bubble detachment as gravity is reduced. Equation (27) has the disadvantage that bubble contact angle 0 is needed, and for a growing bubble this cannot be measured with good accuracy. Also, the last term in Eq. (27) was derived by using some simplifying assumptions about the radii of curvature at the

FIG.20. Typical vapor bubble on heated surface.

bubble base. Since this term does not include gravity, there is some question as to whether it really should be included in the buoyancy force. Perhaps a better way of considering the buoyancy force is given by Cochran et al. (66). Here the buoyancy involves only the part of the bubble that has liquid pressure acting on both the upper and lower portions of the liquid-vapor interface. This constitutes only the unshaded volume in Fig. 20 and gives the net buoyancy as

The last term arises from the weight of the vapor in the bubble, and because of its gravity dependence is included in the buoyancy term. To account for

ROBERTSIEGEL

184

I

t

0

,*-

2

Time, sec (solidcurvesl

I

.02

I

.04

lime, sec (dashed curves)

I

.O 6

I

.08

FIG.21a. FIG.21. Variation of diameter, contact angle, and base diameter with time for bubble growth in saturated water. Heat transferred from solid surface to boiling liquid, 10,900 Btu/(hr)(sq. ft); temperature difference, T,, - T,,, , 11.1 "F. (a) Gravity fields, 1.O and 0.229ge. (b) Gravity fields, 0.061 and O.014ge. [Keshock and Siege1 (60).]

the pressure forces on the shaded volume V bin Fig. 20, the pressure force is utilized on the bubble interface area lying above the bubble base area. T h e pressure inside the bubble is higher than that outside so a net upward force is provided. Assuming the top of the shaded volume to be spherical with a radius of curvature R,, and p o and p i not to vary over this area, provides the pressure force

REDUCED GRAVITY ON HEATTRANSFER

185

.16 .12

.08 .04 0

.28 .24

.20

-32

.I6

.I2 .08

I

0

.d,

.Ib

.I

.2

20 2 5 Time, sec (solid curves) .115

J

.3

-.16

0.061 ,014

--O--

.04

0

- .24,

Fraction of Earth gravity,

I

.4

1

.5

Time, sec (dashed curves)

- .08

.& 1

.6

.is 1

.7

.40 O ~

1

.8

FIG.21b.

Since R, can be measured with good accuracy, the use of Eqs. (28) and (29) in preference to Eq. (27) has some advantage. Equation (29) emphasizes that part of the pressure force acting on the bubble, as given by F p , is independent of the gravity field. There are inertial forces developed during bubble growth primarily as a result of the liquid surrounding the bubble being placed into motion. The inertial force of the apparent liquid mass surrounding the bubble is given by Keshock and Siege1 (60) as

ROBERTSIEGEL

186

This equation represents an attempt to directly compute the liquid inertial force, but it involves a number of simplifying assumptions. Another approach, as given by Cochran et al. (66), utilizes the equation of motion stating that the sum of forces acting on the vapor in the bubble accelerates the center of mass of the vapor. This yields the dynamic force as

Fdy=

d (mv),,p,r - Fd - Fs - I i b - Fp dt bubble ~

Equation (31) provides the liquid dynamic force from the other bubble forces and hence is not as direct a calculation as is Eq. (30). Some typical .-2

0 8 -.

FIG.22. Variation of diameter, contact angle, and base diameter with time for bubble growth in saturated 60% aqueous-sucrose solution. Gravity fields, 1.0 and O.126ge;heat flux, Q / A ,20,500 Btu/(hr)(sq. ft); temperature difference, T,,,- T,,,,30.1”F [Keshock and Siege1 ( 6 0 ) ] .

Time. sec lrolid curvesl

0

.008

Time.

.016

.024

6

.032

rec ldashed curvesl

results will now be presented where the foregoing equations have been applied to observed bubble behavior in reduced gravity. T h e experimentally measured bubble dimensions, base diameter D,, contact angle 8, and equivalent spherical diameter D,are given for typical bubbles during growth in earth and reduced gravity for water in Fig. 21 and for sucrose solution in Fig. 22. T h e forces throughout bubble growth were computed from Eqs. (24), (25), (27), and (30), and they are shown in Figs. 23 and 24. T h e drag force was generally found to be negligible.

REDUCEDGRAVITY ON HEATTRANSFER

187

Fraction of

-.4

,

o

.ooe

,004

I

I

0

I

.02

.01

.012

.o16

I

I

I

I

I

I

.04 .05 sec (dashed curvesl

.03

Time,

.a

.024

.020

Time, SR (solid curves)

.O?

.06

(al 7 xlo-6

I

1

-I 0

I

0

I

I

.05

.I0

I

Time.

1 .I0

I

.20

I

.20

.I 5 SK

I .30

Time,

I

.30

I

.35

(solid curvesl

I .40

SK

I

.25 I SO

(dashed curves1

I

.60

I

.70

.

3 I

.80

(b)

FIG.23. Inertial, buoyancy, and surface-tension forces for bubbles growing in saturated water. Heat transferred from solid surface to boiling liquid, 10,900 Btu/(hr) (sq. ft); temperature difference, T,,, T,,, , 11.1"F. (a) Gravity fields, 1.O and 0.229ge; (b) gravity fields, 0.061 and O.014ge (60). ~

188

ROBERTSIEGEL

Figure 23(a) shows the forces for two typical bubbles growingin water, one at earth gravity and the other at a reduced gravity field of 0.229ge. T h e magnitudes of the forces are quite similar for the two bubbles, and the only effect of the gravity reduction is an increase in the total growth time. Because of the rapid initial growth of either bubble, the liquid inertial force reaches its maximum early in the growth period. By the time the inertial force reaches its maximum, however, the bubble base diameter has increased sufficiently to produce a surface-tension force that is somewhat largerthan the inertial force. Hence, the maximum liquid inertial force is insufficient to tear the bubble away from the surface. The inertial force then decreases while the buoyancy force continues to increase. T h e latter eventually surpasses the surface-tension force so that the bubble must detach. Since a finite time is required for the bubble base to form a neck and finally break loose, the bubble continues to grow, and at departure the buoyancy exceeds the surface-tension force. Figure 23(b) shows the forces on bubbles in water for the much lower gravity fields of 0.061 and 0.014ge. T h e total growth times are much longer than those of Fig. 23(a), and consequently the peak in the liquid inertial force occurs very early relative to the total growth period. As growth continues, the inertia decreases and the bubble base continues to spread so that the surface-tension force becomes quite large. T h e increase in buoyancy is slow because of the low gravity field, but as the bubble becomes quite large, the buoyancy comes into balance with the surface-tension force and departure occurs. Hence, for these bubbles in the very low gravity range, the departure is dependent on an equilibrium of buoyancy and surface-tension forces. It is reasonable that the departure diameter in this range should depend ong-112as was shown in Fig. 19, since this gravity dependence is predicted by a balance of surface-tension and buoyancy forces as in the Fritz equation, Eq. (22). Figure 24 shows the forces computed for two typical bubbles growing in 60°:, aqueous-sucrose solution, the solid lines being for a bubble in earth gravity while the dashed lines are for 0.126ge. The two sets of curves are quite similar to each other. One of the most important features of these curves is that the inertial forces are large compared with the surface-tension forces. This is a result of the larger growth rates characteristic of bubbles in sucrose solution. For these bubbles, the large inertial force soon overcomes the surface-tension force and hence initiates bubble detachment. This force unbalance occurs when the buoyancy force is still small. Consequently, the departure process is dominated by inertia. As gravity is further reduced below the values shown in Fig. 24, the buoyancy becomes smaller and is even less important in influencing departure. Hence, the departure of the rapidly growing bubbles observed in sucrose solution appears to be governed principally by inertial and surface-tension forces and should not exhibit a gravity dependence. This is in accord with the results in Fig. 19. Of course,

REDUCED GRAVITY ON HEAT TRANSFER

189

the removal of the detached bubbles from the vicinity of the heated surface is still gravity dependent. I t must not be inferred that all bubbles growing in water, for example, would be of the gravity-dependent type discussed here. If a particular 12TI0-5

3x10-6

--I -4

-3 -2 --I

-0

-2

-

-4

-

-6

-

-0

0

I

0

-

Fraction of Earth gravity.

- -I

1.0 0.126

- -2

----

- -3 .002

.004

.006

.000

DIO

.012

.014

Time, I& (solid curves)

I

.004

I

.000

I

.012

I

.016

1

.020

Time, rec (dashed curvesl

I

.024

I

.020

--4

.

6

I

.032

FIG.24. Inertial, buoyancy, surface-tension, and drag forces for bubbles growing in 60% aqueous-sucrose solution for 1.0 and O.126ge gravity fields. Heat flux, Q/A,20,500 Btu/(hr) (sq. ft); temperature difference, T,, - Tsa,, 30.1"F (60).

nucleation site emitted rapidly growing bubbles, then these would most likely exhibit a more significant inertial influence. It has been reported by Rohsenow (67) that for subcooled boiling, bubbles have sometimes been propelled away from the surface before condensing even for a horizontal surface facing downward. Usually, however, the bubbles grow and collapse while remaining either attached or very close to the surface. The differences in bubble behavior may result from the relative

190

ROBERTSIEGEL

magnitudes of the inertial, buoyancy, and surface-tension forces as discussed here. Some information on subcooled bubbles will be given a little later.

5. Rise of Detached Bubbles through Liquid Very limited experimental information in reduced gravity is available concerning the rise of boiling bubbles through the liquid after the bubbles detach from the heated surface. In Siegel and Keshock (59)boiling bubbles in saturated water were followed for approximately 1 inch of their rise away from the surface. At this height, the bubbles appeared to have reached a constant-rising velocity except for the lowest gravity tested, 0.014ge. An equation by Harmathy (68) for moderately distorted ellipsoidal bubbles predicts the rise velocity as (32) so that the steady velocity should decrease as gravity to the one-quarter power. The experimental data shown in Fig. 25 indicated that this power variation was correct.

Fraction of Earth gravity, g/g,

FIG.25. Effect of gravity field on velocity of freely rising bubbles for boiling water at saturation temperature and atmospheric pressure (59). Earth gravity, = 11.8 in./sec.

The drag coefficient for a rising bubble depends on the bubble Reynolds number. Harmathy (68) states that Eq. (32) is limited to a Reynolds number greater than 500 and hence will not apply for a low Reynolds number range where viscous forces are more significant. Some low Reynolds number information is given by Keshock and Siegel (60) for bubbles rising in 60” ,~by weight aqueous-sucrose solution, which is much more viscous than water. In this instance the duration of the drop tower experiments was insufficient

REDUCED GRAVITY ON HEAT TRANSFER

191

for the bubbles to achieve a steady-rise velocity. However, the motion of the bubbles accelerating for a short distance away from the surface demonstrated a considerable reduction in rise velocity resulting from decreasing the gravity field.

6. Some Effects of Higher Heat Fluxes T h e previous discussion of bubble dynamics has been limited to single bubbles that were observed at low heat fluxes. At low gravity, for higher fluxes the increased vapor formed coalesces into larger masses as it is not readily removed from the vicinity of the heated surface. This is illustrated in Figs. 26 and 27, which show nucleate boiling from horizontal and vertical wires at earth gravity and O.014ge.The heat flux is about 50,000 Btu/(hr)(sq. ft), which is still fairly low for boiling in earth gravity. However at O.014ge, the critical flux is only about 35% of that at lg, (using the relation Qc g1’4) so that the heat flux in Figs. 26 and 27 is relatively much closer to the critical heat flux when considering the low gravity. The photographs illustrate how after 0.5 sec in reduced gravity the average size of the vapor masses in the liquid has become much larger. Nucleate boiling is still present since the vapor masses have not grown around the wire. Schwartz and Mannes (43) studied the vapor formation for nucleate boiling in low gravity for heat fluxes from 8000 to 30,000 Btu/(hr)(sq. ft). T h e volumes of the bubbles were measured from photographs and the latent heat transported by the bubbles was then computed. This is given in Fig. 28 where the latent heat transport divided by the total heat transfer is given as a function of the total heat transfer. As gravity is reduced, the latent heat transport increases. I t is recalled from Fig. 8, however, which is also taken from Schwartz and Mannes (43),that the total nucleate-boiling heat flux was practically uninfluenced by a gravity reduction. This means that, although the latent heat transport increased as gravity was reduced, other means of heat transfer in the complex nucleate-boiling phenomenon must have been similarly decreased. Siege1 and Usiskin (36) and Clodfelter (40) show photographs for water at higher heat fluxes that are in the nucleate range at earth gravity but are above the critical heat flux in the reduced gravity tested. In this instance, the boiler in reduced gravity is quickly filled with large bubbles that envelop the heated wire or ribbon and cause a burnout of the electrically heated test section. This usually occurs at a local position along the test section, rather than the test section being completely blanketed with a vapor film. McGrew et al. (69) conducted an experiment to study bubble dynamics at zero gravity by a simulation of nucleate boiling. A porous rubber membrane was placed between a chamber of nitrogen gas and a pool of water, and N

FIG.26. Nucleate boiling for three fluids at saturation conditions from horizontal electrically heated wire (5). (a) Earth gravity. (b) After 0.50 sec in 0.014 earth gravity.

REDUCED GRAVITY ON HEATTRANSFER

193

Bu

x

ROBERTSIEGEL

194

the gas flowed through the membrane to produce streams of bubbles in the water. The apparatus was placed in a 100-ft drop tower yielding 23 sec of zero gravity. The gas flow was initiated as the free fall started. At first small bubbles grew at a number of positions on the membrane. Then neighboring bubbles coalesced into larger bubbles and eventually large bubbles formed that did not move away from the membrane. Thus, the surface tended to become covered with gas. The action was very similar to that of boiling bubbles in water and alcohol shown in Fig. 26. -Fraction

: 0

0

of Earth gravity, 919, 1

0.3

0.25 0.2 0.14

0.1

0.057 0.03-OM L

i2

-z

L

c ...

4-

-

a

B

d

c

F

2-

c L

5

c

1 0 4 r 8-

-

6-

-

4 0

.4 .6 .a Ratio of latent to total heat transfer

.2

1 I

FIG.28. Latent heat transport as function of total heat flux in lg, and low gravity fields [Schwartz and Mannes ( 4 3 ) ] .

G. BUBBLEDYNAMICS IN SUBCOOLED POOLBOILING The pool boiling discussion up to this point has been restricted to fluids

at or very near the saturation condition. There are a few reports on the influence of having the liquid subcooled, and these will now be discussed.

REDUCED GRAVITY ON HEATTRANSFER

195

The work of Cochran et al. (66) contains low gravity experimental data, while that of Rehm (70) interprets data at earth gravity to predict what would be expected when gravity is reduced. The tests by Cochran et al. (66) were carried out with distilled water at a pressure of 1 atm. A horizontal chrome1 strip 0.005 inch thick, 0.25 inch wide, and 0.50 inch long was heated with direct current. The strip was insulated on the back side so that boiling occurred only from the top surface. T h e boiler was mounted in an 85-ft drop tower equipped with a drag shield to reduce air friction. This yielded 2.3 sec at a gravity field less than 10-5ge. The boiler was also operated at earth gravity at the same heat flux, 28,900 Btu/(hr)(sq. ft), which was the value used throughout all the tests. T h e subcoolings tested were 5 , 10, 15, 25, and 40"F,and a few tests were also made close to the saturation temperature. For the low gravity test near saturation, a vapor mass formed above the surface and served as a collector for the vapor being generated at the surface. As time progressed, the vapor mass covered the heater and burnout seemed imminent. This behavior is similar to the results discussed in the previous section dealing with the saturated condition.

1. Bubble Growth Figure 29 shows typical bubble volumes as a function of time from the beginning of growth until the bubbles form a neck and break loose from the lWx10-8

Subcooling,

mc

##

0°

--* --*J.

"F

4

-------_______ --\

\

Fraction of Earth gravity, 919,

----14

Ti me, mil I iseconds

I

16

I

18

< 10-5 1

I

x)

I

22

I

FIG.29. Effect of gravity reduction on growth of bubbles for subcooled boiling of water at atmospheric pressure [Cochran et al. ( 6 6 ) ] .Heat transferred from solid surface to boiling liquid, 28,900 Btu/(hr)(sq. ft).

surface. At a large subcooling of about 38.5"F, the gravity reduction to less than 10-5g, had no appreciable influence on the bubble growth. At a smaller subcooling of about 5°F the gravity had little influence during the early

196

ROBERTSIEGEL

growth period. The significant influence was that bubble detachment was delayed and the maximum size of the bubble was increased somewhat. An average of several bubbles also showed that for small subcoolings the maximum size of the bubbles increased when gravity was reduced. This is consistent with the data for saturated water discussed previously.

2. Forces Acting on Bubbles Forces computed from Eqs. (24), (28), (29), and (31) are given in Fig. 30 for bubbles with large and small subcooling. The dynamic force in Fig. 30 should not be directly compared with the inertial force in Figs. 23 and 24 because of the different approaches used in computing the forces as discussed earlier in connection with (30) and (31). Figure 30(a) shows that for large subcooling there is little influence of the gravity reduction on the bubble forces. The buoyancy at lg, is already small because with large subcooling the bubbles condense before they can become very large. As a consequence, removing the buoyancy by going to zero gravity has little influence on the bubbles. Hence if subcoolings on the order of 30°F are maintained, it would be expected (based on the limited available data) that the nucleate-boiling heat transfer would be independent of gravity. Figure 30(b) shows the effect of gravity reduction for a smaller subcooling of about 5°F. At earth gravity the dynamic force is small; in fact, it becomes negative before departure and helps keep the bubble attached to the surface. The buoyancy in earth gravity becomes appreciable during the latter half of the bubble lifetime. At zero gravity the times involved are approximately doubled. The dynamic force becomes much larger than at earth gravity and must have a significant influence on bubble detachment. From Eqs. (29) and (24) the ratio of pressure to surface tension force is

For a bubble that is a truncated sphere, sin 8 = D,/2R, and FJ FS = - 1, which shows that for this geometry these forces would be equal in magnitude. A truncated spherical bubble shape would be expected for conditions of static equilibrium at zero gravity. The curves for pressure and surfacetension forces in Fig. 30 are similar in shape; their deviation from each other for zero gravity is evidently a result of the bubble distortion during growth. The findings by Cochran et al. (66) tend to agree with those of Kehm (70) who studied the bubble forces in subcooled boiling of water at earth gravity. Kehm found that as saturation was approached the removal force was due almost completely to gravity. This was not true for subcooled conditions, where the buoyancy force became of much less importance. For a horizontal

REDUCEDGRAVITY ON HEAT TRANSFER Fraction

I

1

I

2

1

3

I

(of

4

I

197

Sub-

I

5

6

Fraction Sub. of Earth CDolir OF gravity,

44 5.1

1

-

-1201 0

I

4

I

8

I

I

12 16 lime, millirexmdr

I

20

I

24

I

(b) FIG.30. Effect of gravity on bubble forces in water boiling at atmospheric pressure (66). Heat transferred from solid surface to boiling liquid, 28,900 Btu/(hr)(sq. ft). (a) Large subcooling. (b) Small subcooling.

plate facing upward, the ratio of the buoyant to the total removal force near bubble separation decreased from 0.8 at saturation to 0.15 at a subcooling of 30°F. Hence at zero gravity a subcooled condition is desirable when boiling is being used solely for cooling without the need for net vapor production.

198

ROBERTSIEGEL

FIG.31. Film boiling of saturated ethyl alcohol from electrically heated wire at earth and three reduced gravities (5). (a) Horizontal wire. (b) Vertical wire.

H. VAPORPATTERNS FOR FILMBOILING IN

A

SATURATED LIQUID

Some quantitative data giving the gravitational effect on film-boiling heat transfer were considered previously. Photographic results for filmboiling vapor patterns are given here for horizontal and vertical wires in earth and three reduced gravity fields. T h e fluid is ethyl alcohol and is at its saturation temperature at atmospheric pressure. The photographs in Figs. 31(a) and (b) indicate that as gravity is reduced there is a general increase in the size of the periodically spaced vapor masses along the wire. The vertical wire is especially interesting because the vapor does not rise in the form of a smooth boundary layer of increasing thickness with height. Rather, the circumferential component of the surface tension pulls the vapor into a series of regularly spaced vapor enlargements rising along the wire. When

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199

FIG.31b.

gravity becomes low, the influence of buoyancy and hence wire orientation is greatly reduced, and the vapor configuration around the vertical wire achieves an appearance similar to that for the horizontal wire. T h e principal difference is that for a vertical wire the entire vapor configuration moves axially along the wire. Additional information on the instability of a vapor layer surrounding a horizontal cylinder is given by Lienhard and Wong (71) and Siege1 and Keshock (5).

200

ROBERTSIEGEL V. Forced Convection Boiling

For low gravity applications in a system where there is a net vapor generation, a difficulty, arising as a result of the low buoyancy force, is the separation of the vapor from the liquid. This difficulty can be overcome by utilizing a forced-flow-type boiler in order to provide a substitute body force. If the pressure and drag forces exerted by the moving liquid and vapor are substantially larger than the buoyancy force, then the system performance should be independent of a gravity reduction. In this section two-phase flow boiling will be considered with reference to low gravity features. The limited experimental information on the effect of gravity on two-phase flow regimes and heat transfer will be reviewed and discussed.

A. REDUCED GRAVITY EFFECT ON TWO-PHASE FLOW T o obtain an indication of the importance of the body force in two-phase flow, the most simple case that can be considered is where no heat flow is present. An experiment of this type was carried out by Evans (72) who circulated a mixture of air and water through a $-in.-diam clear plastic tube 18inches long. The apparatus was flown in a zero-gravity airplane trajectory, which provided test durations of up to 15 sec. For some of the tests the apparatus was free-floating in the airplane, while for other tests it was tied to the aircraft frame and was subject to random accelerations of +0.02ge. The two different test conditions yielded the same results, so evidently the small gravitational body force of &0.02gewas negligible compared with the forces induced by the forced flow. Detailed descriptions of the flow patterns are given by Evans (72) as a function of percent air in the water and average velocity of the bubbles. T h e results reveal how the gravity field and tube orientation influence the distribution of bubbles over the tube cross section. For flow in a horizontal tube with approximately 30% by volume of air, the bubbles at earth gravity were mostly in the upper portion of the tube cross section with the smaller bubbles clustering near the wall. With a small volume of air the bubbles were quite small and were distributed a little more into the lower part of the tube cross section than for the higher air volume case. For either upward or downward vertical flow at earth gravity, the bubbles were distributed quite uniformly over the tube cross section except very close to the wall where the bubble population was somewhat diminished. In zero gravity the bubble distribution over the tube cross section was found to be very similar to the vertical case at earth gravity. This points out the possibility of utilizing a vertical-tube earth-bound experiment to simulate zero-gravity conditions. A second series of adiabatic visualization experiments were carried out by using twisted ribbons or coiled wires inserted into the tubes. These produced

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induced radial accelerations ranging up to 4ge. Swirling flow at earth gravity with coiled wire inserts in a vertical tube closely resembled swirling flow at Og,. T h e swirling increased the tendency of the air to coalesce in the central portion of the tube cross section. For swirling flow in a horizontal tube at lg,, the air masses still had a tendency to accumulate in the upper portion of the tube for the range of experimental conditions that were employed.

B. TWO-PHASE HEATTRANSFER What can be concluded from the adiabatic flow patterns concerning the two-phase flow behavior in low gravity with heat transfer ? Also, can tests at earth gravity be utilized to provide information applicable to the zero-gravity condition ? One difference fromadiabatic flow is that for the heat-transfer case there is bubble production at the wall. This is illustrated in a photographic investigation by Hsu and Graham (73) for a vertical tube at earth gravity. Hence the patterns given by Evans (72) have to be modified by postulating the generation of bubbles at the wall. Since in zero gravity the bubble generation would be symmetric around the tube periphery, it is reasonable that heat-transfer tests at earth gravity to simulate the zero-gravity case should be performed with a vertical tube orientation. Although a vertical-flow test at earth gravity will provide proper bubble symmetry over the flow cross section, there is still an effect of buoyancy along the tube length. If the flow is upward, the buoyancy force provided by the earth gravity field is aiding the bubbles in moving with the flow direction, Alternately, a test at earth gravity can be performed in downflow. The buoyancy force on the bubbles is then in opposition to the flow drag and there is an effective -lg, field with regard to the flow direction. In zero gravity the vapor bubbles leaving the wall should be dragged up to the speed of the liquid and then should move with the liquid. Hence in considering the simulation of zero gravity by earth-bound experiments, the interaction of the drag and buoyancy forces must be evaluated. Some estimates of the forces in forced convection boiling are discussed by Adelberg (74, 75). Ratios comparing drag, buoyancy, surface-tension, and inertial forces are formulated. There is so little heat-transfer data available on forced convection boiling in reduced gravity that at present generalized conclusions as to the relative importance of the various forces cannot be made. As an illustration consider one possible way to inquire into the gravity influence, at least for cases where there is not a high percentage of vapor in the liquid. This is by examining the terminal velocity u, for a bubble rising through a pool of liquid, as is given by expressions such as Eq. (32). For downflow in a vertical tube at earth gravity, the buoyancy

202

ROBERTSIEGEL

force can cause a bubble to move in opposition to the liquid with a velocity u , , ~relative to the liquid velocity. Thus if the liquid downflow velocity were u,,,, the bubble would remain at a fixed position along the tube length. If the liquid velocity were several times u,,,, the buoyancy effect should become negligible and the downflow case would perform the same as upflow. Some experimental work that approximately demonstrates the previous discussion is given by Simoneau and Simon (76). Boiling was studied from a heated wall in a vertical channel with liquid nitrogen in either upflow or downflow. The pressure was 35 psia and the bulk liquid was 6 to 10°F subcooled. For the range of bubble sizes encountered in nucleate boiling, the terminal rise velocity in a stationary liquid pool was computed to be in the range 0.53 < u , , ~< 0.75 ft/sec (76a). For a low flow velocity of 0.86 ft/sec, the downflow condition produced an appreciable vapor accumulation in the channel. At a higher velocity of 2.6 ft/sec, the downflow case had only slightly more vapor accumulation near the channel outlet than the upflow case. The behavior of the vapor patterns indicates that a flow velocity several times u, would be sufficient to cause buoyancy forces acting in the flow direction or opposite to it to have little influence on the vapor motion. Similar results were also found for film boiling. Some critical heat flux experiments to determine the influence of a positive or negative earth gravity relative to the flow direction have been carried out by Papell et al. (77). Liquid nitrogen was pumped in either upflow or downflow through a vertical heated tube 0.505 inch in diameter. Pressures ranged from 50 to 240 psia, the inlet velocity of the liquid from 0.5 to 11.0 ft/sec, and the inlet subcooling from 12 to 51°F. The critical heat flux was defined as the flux required to initiate an excursion of increasing temperature at any local position along the tube wall. Figure 32 shows a typical set of data for the critical heat flux as a function of liquid inlet velocity in upflow and downflow. In upflow the excursion of increasing wall temperature always started at the outlet end of the tube. Presumably this is caused by both the flow velocity and buoyancy force driving thevapor accumulation toward the upper end of the tube. The highest vapor accumulation would be reached at the tube outlet and tend to blanket the wall with vapor. When the inlet liquid velocity for upflow exceeds about 6.5 ft/sec for the particular conditions in Fig. 32, the upflow velocity begins to exert an effect on the critical flux. Below this velocity the influence of buoyancy must be appreciable. Turning now to the data for downflow, at very low velocities it was found that the critical flux was first reached at the inlet (upper end) of the tube. Evidently the flow velocity was insufficient to drive the bubbles downward, and consequently the vapor still accumulated near the top of the tube. For velocities higher than 2 to 3 ft/sec, the critical flux position shifted to the outlet (lower end) of the tube.

REDUCED GRAVITY ON HEATTRANSFER

203

Beyond a certainvelocity, about 6.5 ft/sec in the case of Fig. 32, the upflow and downflow data merge together so that there is no longer an influence of the buoyancy orientation relative to the flow direction. Since the reduced gravity range O,< gig, ,< 1lies between the lg, conditions tested, the merging of the curves indicates that above this inlet flow velocity the same performance should be obtained in reduced gravity as in the earth gravity tests. Results of the same nature as shown in Fig. 32 were obtained for a range

‘1

7 Upflow \

.20c u c

.-

-a -

Flow .16-

:‘

Inlet

Burnout independent of flow direction

xc

Z

c

‘Downflow

.08Inlet

m U .c ._ L

V Burnout at inlet

0 Burnout at outlet

V

@ 0

I

2

4 6 8 Liquid inlet velocity, ftlsec

10

FIG.32. Typical data for critical heat flux as function of inlet velocity for upflow and downflow of liquid nitrogen in O.SOS-in.-diamtube. Inlet pressure, 75 psia; inlet subcooling, 19°F [Papell et al. (77)].

of subcoolings and pressures. T h e velocities at which the effect of orientation vanishes are summarized in Fig. 33. T h e dotted lines are each for a different inlet pressure, and they show that for a given pressure the inlet velocity necessary to overcome the influence of gravity on the critical heat flux depends strongly on the inlet subcooling of the liquid nitrogen. When the pressure or subcooling is increased, the liquid velocities necessary to make gravity unimportant become smaller. This would be expected since an increase in pressure or subcooling should decrease the vapor volume thereby decreasing the influence of buoyancy on the critical heat flux.

ROBERTSIEGEL

204

Another illustration of the influence of gravity was found by Macbeth

(78).He gathered together data for boiling in forced axial flow through heated rod bundles at earth gravity. The majority of the data was for water at 1000 psia. The data for vertical axial upflow through the bundles were found to form a separate group from those for horizontal axial flow. T h e horizontal flow burnout heat fluxes were as much as SO:/, below those for the vertical upflow condition. Evidently in the horizontal case there was a stratification of vapor that would promote a burnout condition among the upper tubes in

1

Inlet static pressure, psia Constant inlet temperatur Constant inlet pressure

d

740

4

c

Buoyancy independent region

v1 m

I-

't 0

1

2

I

4

6

Liquid inlet velocity, ftlsec

a

1

FIG.33. Liquid inlet velocity at which critical heat flux becomes independent of upward or downward flow direction for liquid nitrogen in O.SOS-in.-diam tube (77).

the bundle. At very low gravity this stratification would not occur, and the critical heat flux would be expected to correspond more to the range of data for the vertical orientation. Some near-zero-gravity two-phase heat-transfer tests by Papell (79) revealed an instability effect. Distilled water at 50 psia and subcooled 175 to 198°F was pumped through an electrically heated stainless-steel tube that was 0.311 inch in diam. The flow rate was 0.20 lb/sec and the heat addition 1.40 Btu/(sec)(sq. in.). For higher flow rates the instability was not observed, and it also could not be induced at earth gravity. The apparatus was mounted in an airplane that flew a zero-gravity trajectory and produced

REDUCED GRAVITY ON HEATTRANSFER

205

test times of 15 sec at 0 to 0.02ge. The instability was induced as follows: After the apparatus had been in zero gravity for 3 sec, the flow was stopped for about 1sec. During the flow interruption the wall temperatures increased about 200°F. When the flow was restarted, the wall temperatures decreased to a lower level than had existed previous to the interruption even though none of the test conditions had been changed. The increase in heat-transfer coefficient was about 16%. During the flow interruption there is an increase in the amount of vapor within the tube. It is postulated that the presence of more vapor changes an originally highly subcooled bubbly flow to a more violent slug-type flow which then persists for the remainder of the test. In earth gravity the buoyancy would help separate the vapor back out of the liquid and prevent the slug flow pattern from developing. T h e zero-gravity performance of a flow loop containing an evaporator and condenser was investigated by Feldmanis (79u,b). Water was pumped through a straight electrically heated evaporator tube [some photographic results with swirl devices in the tubes are also included in (79u)l and the resulting two-phase mixture passed into a tapered condenser. Zero-gravity airplane flights were used to free float the apparatus and provide 8 to 19 seconds of zerog, testing time. Water entered the 0.305-inch i.d. evaporator tube with a velocity of about 0.09 ft/sec and was only partially evaporated; the fluid left as a mixture of liquid and vapor slugs with a velocity of 72 ft/sec. Prior to entering the zero-gravity flight path, the electrical power was turned on and the boiling and condensing processes established. Two significant findings in zero gravity were found with regard to system pressures ; detailed heat transfer measurements were not taken. One finding was an increase in system pressure during the zero-gravity condition. The second was the damping of pressure oscillations that existed in the flow loop during earth gravity operation. After about five seconds in zero gravity the loop operation became quite stable and the pressure oscillations almost entirely disappeared. The reasons for this behavior are not yet clearly defined.

C. DESIGNSINVOLVING SUBSTITUTE BODYFORCES Instead of utilizing drag forces produced by forced flow in a straight tube, there are alternate means of providing a substitute body force to obtain proper performance at low gravities. An informative discussion of some of these ideas is given by Ginwala et ul. (80). One approach is to utilize centrifugal forces to provide an artificial gravity field. Twisted ribbons or coiled wires can be inserted into the boiler tubes to provide a swirling action. This was mentioned earlier in connection with the work of Evans (72) who observed swirling flow patterns for air-water mixtures at earth and zero gravities. Extensive work has been done with twisted inserts in tubes as a

206

ROBERTSIEGEL

means of improving boiling performance. An example is the work of Gambill and Greene (81)and Gambill et al. (82),where swirl was utilized to increase the burnout heat flux. Since the inserts can produce effective radial accelerations of several or more earth gravities, a satisfactory performance should result in reduced and zero gravity. Instead of using internal inserts to swirl the flow, the whole boiler can be rotated, although this would probably not be as convenient for a space application. A number of investigations utilizing centrifuges have demonstrated the nature of boiling in high effective gravity fields and the accompanying improvement in the critical heat flux. No attempt will be made to review the extensive work in this area; some references (83-88) are provided for the interested reader. Another approach that can be employed where net vapor generation is required for a power generation cycle is to use flash evaporation by employing an expansion engine or by having a sudden pressure drop occur on the heated liquid. This method of vapor production can eliminate the need of a body force to separate the vapor from the liquid. Instead of utilizing a body force in boiling, there is the possibility of controlling the liquid by surface forces that are independent of gravity. The liquid can be pumped to the heated surface by the use of capillary forces. A wick-type material of either fibrous or metal mesh would be placed adjacent to the surface to keep it in contact with the liquid. Some tests at earth gravity with wick materials have been carried out by Ginwala et al. (80) and Allingham and McEntire (89). In the latter, it was found that at low heat fluxes the wicking material covering the surface aided bubble nucleation and improved the heat transfer as compared with ordinary pool boiling. At higher heat fluxes, however, the boiling was hindered as the incoming liquid motion was interferred with by the escaping vapor. If the wicking is wrapped around the heater in such a way that an open path for vapor removal is provided, high critical heat fluxes can be obtained. Values up to 1.2 x lo6 Btu/(hr)(sq. ft) were attained for water by Costello and Frea (90). VI. Condensation without Forced Flow

The discussion of condensation will begin with situations where there is no forced-flow pumping pressure applied to move either the liquid or the vapor so that the fluid motion is produced only by the gravity field. As would be expected, for these conditions low heat transfer is obtained when the gravity field is substantially reduced. T o mitigate this difficulty, a forced flow can be utilized to move the liquid and vapor over the surface. In the following discussion some aspects will first be presented of condensation

REDUCED GRAVITY ON HEATTRANSFER

207

when only the gravity force is present. Then in the succeeding section some results will be given with forced flow, such as in a condenser for a space power plant.

A. LAMINAR FILMCONDENSATION ON A VERTICAL SURFACE When condensing is occurring on a stationary surface in a quiescent vapor, the process is gravity dependent as it is the gravitational force that produces the liquid flow along the surface. For example, consider the classic

FIG.34. Laminar condensate film on vertical flat plate.

case of laminar film condensation on a vertical flat plate as illustrated in Fig. 34. The thickness of the liquid film flowing down the plate is (91)

Equation (34) shows that the film thickness at a fixed x position on the surface increases with decreased gravity in proportion to g-ll4. Hence at very low gravities the liquid layer would become extremely thick. Since the heat transfer depends inversely on the film thickness, within the simplifying assumptions of the Nusselt condensation theory the local heat-transfer coefficient is given by (91)

208

ROBERTSIEGEL

The Nusselt theory and more refined boundary-layer theories of film condensation have generally utilized boundary-layer-type assumptions that require the liquid layer to be thin relative to the length along the surface. These assumptions can be violated for the thick layers produced in low gravity. When the boundary-layer assumptions are valid, Eq. (35) shows that h, decreases asg’I4leading to low heat transfer as gravity becomes very small. The integrated average heat-transfer coefficient over the length of the surface also has this gravity dependence.

B. LAMINAR-TO-TURBULENT TRANSITION AND TURBULENT FLOW For condensation on a vertical flat plate as in Fig. 34, the transition from a

laminar to a turbulent condensate film depends on the Reynolds number of the film which can be defined as

Reg = CSlv (36) For laminar flow the mean film velocity C depends on gl/’, while 6 depends on g-’I4. Thus the film Reynolds number in the laminar range depends on g1/4.As discussed by Grober et al. (14, p. 330) this Reynolds number can be written (after slight modification) as

The numerical value of the transition Reynolds number is somewhat indefinite, and for purposes of the present discussion an average value of 350 will be utilized. Then Eq. (37) can be rearranged to give at transition

As an illustration, if saturated steam at a 1-atm pressure is condensing on a wall at 62”F, then using average film properties yields (T,,, - T,)x(g/ge)li3z 820 ft O F Since (T,,, - T,) = 150°F in this illustration, ~ ( g / g , )=~5.5 / ~ ft. At earth gravity a laminar condensate film would be maintained for 5.5. ft of condensing surface. In a reduced gravity field of O.OOlg,, the length having laminar flow would be increased to 55 ft. When the boundary layer becomes turbulent, the heat-transfer coefficient is more strongly dependent on gravity than for the laminar case. For a turbulent condensate layer on a vertical flat plate of height L , the integrated average heat-transfer coefficient is given by Grober et al. (14, p. 342)

REDUCED GRAVITY ON HEAT TRANSFER

209

This equation indicates a dependence on gravity to the one-half power rather than the one-quarter power as in laminar flow. C. TRANSIENT TIME TO ESTABLISH LAMINAR CONDENSATE FILM Consider the time that would be required to fully establish the film condensation layer for a typical situation-laminar condensation on a vertical flat plate. A vertical plate is suspended in a large expanse of quiescent pure vapor at its saturation temperature Tsatas in Fig. 34. Then the plate temperature, which is initially also at T,,, , is instantaneously dropped to a lower temperature T , and condensation commences. It is assumed that film condensation occurs throughout the transient process. Sparrow and Siege1 (92) give the time for a steady-state condensation layer to be established at a position x along the plate :

As gravity is reduced, the transient time increases in proportion to g-’I2. As an example consider saturated water vapor at 1 atm, 212”F, condensing on a plate whose temperature is suddenly dropped to 100°F. For these conditions the time required to achieve steady state is about 0.63 sec at x = 6 inches, and 0.9 sec at x = 12 inches. If gravity is reduced to O.Olg,, the time at x = 12 inches increases to 9 sec, and for 10-6gethe steady-state time becomes 900 sec or 15 min. This of course assumes that the laminar boundary-layer theory still applies for the thicker condensation layers encountered is 0.22 at low gravities. Since the layer thickness at x = 12 inches for 1OW6ge inch as computed from Eq. (34), this seems to be a reasonable assumption for this example. VII. Forced Flow Condensation

An interest in forced flow condensation in reduced gravity stems from the design of electric generating plants for space use. A system that shows some promise is a Rankine cycle turbogenerator unit that utilizes a liquid metal as the working fluid. A forced flow condenser would be employed so that gravity would not be required to collect the condensate, and the system would be able to operate properly in the zero-gravity space environment. In addition to requiring heat-transfer information for the condenser design, two-phase pressure drop data is needed. Condensation occurs in the lowpressure portion of the Rankine power cycle. It is restricted to a low-pressure drop allowable from the turbine exit pressure to the pressure required at the suction side of the pump recirculating the condensate to the boiler.

ROBERT SIEGEL

210

A. FLOW BEHAVIORI N Low GRAVITY Some flow visualization tests have been carried out for mercury condensing in horizontal tubes. Cummings et al. (93)[a brief discussion is also given by Trusela and Clodfelter (94)]employed a 12-inch-long glass tube with a 0.196-inch i.d. An airplane flying a ballistic parabola was utilized to obtain 12 sec at zero gravity. Albers and Macosko (95)and Albers and Namkoong

-

Flow direction

FIG.35. Flow configurations at interface location for 1 and zero g, mercury nonwetting condensation (#-in. 0.d. glass tube.) (a) lg,; vapor mass flow rate, 0.031 Ib/sec. (b) Zero g,; vapor mass flow rate, 0.035 Ib/sec. [Albers and Narnkoong (96)].

(96) observed condensation of mercury in constant-diameter horizontal glass tubes with 0.27-, 0.40-, and 0.49-inch i.d. and a tube length of 84 inches. The vapor flow rate ranged from 0.03 to 0.05 lb/sec, and the tubes were convectively cooled, thus providing a uniform heat flux boundary condition. An airplane was used to provide test times of 10 to 15 sec. In all

REDUCED GRAVITY ON HEATTRANSFER

21 1

of these tests the mercury did not wet the tube wall. Motion pictures taken with high-speed cameras revealed the details of the condensation process. An illustration of the differences in the flow at 1 and Og, is shown in Fig. 35. At earth gravity, and particularly when the vapor velocities were low, the mercury drops would run diagonally down along the tube wall resulting in a liquid accumulation in the bottom portion of the tube cross section. At Og, there tended to be uniform distributions of the drops over the cross section of the vapor stream and around the wall circumference. In general, the drops were observed to form on the tube surface and travel along the wall; they increased in size as they coalesced with other drops along their path and in some cases were swept into the vapor stream. At high vapor inlet velocities of 250 ft/sec the entrained drops became very small and a mist flow resulted. As an aid in the analysis of dropwise condensation, a detailed photographic study was made by Sturas et al. (96a)of the geometry of individual mercury drops in 0, 1, 1.5, and 2ge environments.

B. PRESSURE DROP One of the important objectives in the condensation experiments was to find whether the differences in flow for the 0 and lg, conditions would influence the pressure drop in the condenser tube. Detailed pressure drop information was obtained by Albers and Macosko (95, 97) and Albers and Namkoong (96) for both constant-diameter and tapered stainless-steel tubes, and the findings will now be discussed. The constant-diameter tube had an internal diameter of 0.31 1 inch and was 87 inches long. The tapered tube was 84 inches long and had internal diameters of 0.40 inch at the inlet and 0.15 inch at the exit. The effect of gravity on the static pressure drop from the beginning of the condensing length to a local position along the tubepo -pi is given in Fig. 36. Near the entrance of the condenser there are high vapor velocities that produce large friction losses. This caused the value of p, - p, to increase within the first half of the condensing length. In the last half of the condensing length, the pressure rise derived from momentum recovery exceeded the frictional loss so thatp, -pi decreased. The important finding for the present discussion was that the change from the 1 to Oge environment had no discernible influence on the static pressure drop distribution throughout the tube length (see Fig. 36). The over-all static pressure drop from the start of the condensing length to the vapor-liquid interface at the end of condensation is given in Fig. 37 as a function of condensing length for a typical mass flow rate. Although the majority of the Og, data fall slightly above the lg, values, there is little difference for practical purposes between the 0 and lg, conditions. The values of

212

ROBERTSIEGEL

the over-all static pressure drop ranged from 0.2 to 2.2 psi for the straight tube, and from a 0.9-psi pressure rise to a 0.1-psi drop in the tapered tube for the range of flow rates and condensing lengths tested. From these measurements it appears that the change in distribution of mercury drops within the tube, caused by a reduction from earth to near Fraction ot Earth gravity,

.-m

V c c VI

0

12 24 36 48 60 Distance from condensing tube inlet I , inches

1

(b) FIG.36. Effect of gravity on local static pressure drop (96).Vapor mass flow rate, 0.038 Ib/sec. (a) Constant diameter tube (0.31-in. i.d.). (b) Tapered tube (0.40-in. inlet i.d. with taper ratio, 0.036 in./ft).

zero gravity, has no significant influence on the pressure drop during condensation. Hence there may be no need to resort to low gravity tests when pressure drop data for condensing mercury is needed for a space vehicle application. I t must be carefully noted that this finding is based on a range of flow and heating variables that is rather limited when considering the wide range of flow conditions that are possible in a two-phase forced-flow situation. For conditions considerably different from those discussed here

REDUCED GRAVITY ON HEATTRANSFER

213

there might be a significant effect of gravity reduction. Since the tests have all been for mercury, it would be desirable to conduct similar tests with fluids that wet the tube wall.

I 1.61

48

Fraction of Earth gravity,

gh,

0

0

52

I

56

I

60

I 64

Condensing length, {, inches

I 68

72

(b) FIG.37. Effect of gravity on over-all static pressure difference for entire condensing length (96).Vapor mass flow rate, 0.028 lb/sec. (a) Constant diameter tube (0.31-in. i d . ) . (b) Tapered tube (0.40-in. inlet i.d. with taper ratio, 0.036 in./ft).

C. VAPOR-LIQUID INTERFACE Another factor of importance in low gravity forced flow condensing of nonwetting mercury is whether the vapor-liquid interface that is established across the tube as shown in Fig. 35(b) will remain stable. This is discussed from a theoretical viewpoint by Lancet et al. (98). Consider two adjacent fluid layers having different densities. If a body force is oriented from the more-dense toward the less-dense fluid, the interface will tend to become unstable. T h e interfacial disturbances that have the greatest tendency to amplify have a wavelength (the critical wavelength) given by the Taylor instability theory

Equation (41) is interpreted with respect to the interface in a tube to imply that the interface will be stable (i.e., the liquid will not spill out from a vertical tube) if the condenser tube diameter is less than one-half the critical wavelength. A more detailed derivation applying the stability theory to a round tube geometry yielded the same functional form as Eq. (41) but with a lower constant of 1.84 instead of 2n. These relations indicate that the

214

ROBERTSIEGEL

maximum stable tube diameter increases as gravity is reduced according to

g-1/2*

As discussed by Lancet et al. (98) [see also Denington et al. (98a)], it has been found experimentally at earth gravity that the range of critical tube diameters for mercury is up to 0.168 inch, depending on the tube surface finish and adhesion between mercury and the surface. This is in agreement with the experimental results of Reitz (99), where the maximum tube diameter for stability at lg, was 0.15 inch. In a reduced gravity field of, for example, O.05ge, the critical diameter would increase by about 4.5 times so that it would be about 0.75 inch. For low gravity testing the tube used by Reitz (99)was 0.196 inch in diameter and was found to have a stable interface in zero gravity. The interface experienced small oscillations resulting from impacts of condensate drops, but did not exhibit any large-scale unstable motion (93).It should be remarked that there are some restrictions in the Taylor theory ;for example, the body force is normal to the interface between the two fluids. A horizontal tube [see Fig. 35(a)] in a finite gravity field can have a long interface (not normal to the gravity vector) where the condensate layer in the lower portion of the tube cross section gradually thickens along the tube length. T h e interface can be especially long if the liquid wets the tube wall very well. In the Taylor theory thereis no mass addition at the interface. In condensation, new condensate is continually being supplied at the interface and there is liquid flowing away from it. This may influence the stability, especially for high rates of condensation where the flows would be large.

D. NONCONDENSABLE GAS

An indication of the behavior of noncondensable gas in the forced flow condensation process was obtained by Lancet et al. (98). Condensation at earth gravity was observed in a below-critical-diameter tube that was tilted at a few different angles away from the horizontal. The vapor-liquid interface was held stable by surface tension. Noncondensables accumulated in the vapor region upstream of the interface. In the region of high noncondensable concentration, little condensation occurred. When the noncondensables occupied several diameters of the tube length, the mercury drops were slowed by viscous damping in the gas. The drops rapidly coalesced and eventually bridged the tube diameter before the liquid could touch the vapor-liquid interface. This entrained a slug of noncondensables and at the same time formed a new interface, thereby starting the process again. The same behavior was observed at zero gravity by Cummings et crl. (93). In zero gravity the flow was more evenly distributed over the tube cross section. This permitted the noncondensables to be more easily trapped by the liquid and carried away by the liquid flow.

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215

VIII. Combustion

Some combustion processes depend on gravity because free convection accounts for the circulation of some of the reactants into the flame zone. A. CANDLE FLAME A common example of gravity-dependent combustion is a candle flame. Here free convection carries most of the oxygen into the flame. I n zero gravity the question is whether diffusion alone would suffice to carry the combustion products in and out of the flame zone to sustain the reaction. Experiments on burning a candle in various gravity fields were carried out by Hall (100). A white paraffin candle and a camera were mounted in a box that could be free floated in an airplane during a zero-gravity trajectory. Test results were also taken during the 2ge turn entering into the zerogravity flight path. Thirty-five tests were made, each lasting from 25 to 28 sec. Two atmospheres were used, air at 14.7 psia and oxygen at 5 psia. During all the tests the candle was found to burn continuously. At 2ge the flame heightened and narrowed as compared with lg, but remained yellow in color. In zero gravity the flame became spherical and had a lightblue color. Thus, at least for the experimental durations considered, the combustion in a candle flame can be sustained in the absence of free convection.

B. FUELDROPLETS The burning of fuel droplets in reduced and zero gravity has been studied by Kumagai and Isoda (101-103). In combustion of a finely divided fuel spray the fuel droplets are so small that the influence of natural convection around each drop is negligible. Consequently, the flame surrounding an individual droplet is essentially spherical. One way to study droplet combustion would be to observe the actual very fine droplets, but this has the difficulty of dealing with the small size. An alternative procedure is to use droplets of larger size and eliminate the free convection, which would occur for them at earth gravity, by studying the combustion in a low gravity field. The latter of these alternatives will be considered here. The resulting experimental information can be compared with analyses that assume the combustion is spherically symmetric about the droplet. Kumagai (101) and Kumagai and Isoda (102) placed a chamber in a counterweighted drop tower. The chamber contained a vertical silica filament from which a fuel droplet was suspended. T h e droplet diameter range was from 1 to 1.5 mm. The droplet was ignited by an electric spark

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216

at the beginning of the low gravity period. One photograph was taken during each test as the falling chamber passed a stationary camera 0.1 to 0.4 sec after ignition. Direct photographs were utilized to observe the flame boundary, and schlieren photographs were taken to reveal the hot-air zone. T h e fuels were ethyl alcohol and n-heptane. In earth gravity the flame had an oval shape and there was an upward flow of the hot air around the droplet.

.2

0

I

.1

t

.2

I

.3

I 1 1 I .4 .5 .6 .7 Fraction of Earth gravity, 919,

I

.8

I

.9

FIG.38. Evaporation constant and shape of luminous flame for burning droplets as function of gravity for two fuels [Kumagai and Isoda (IOZ)].

In zero gravity the combustion zone became spherically symmetric around the droplet. The dimensions of the flame and hot-air zones are given by Kumagai and Isoda (102)along with photographs at six different gravities. In Fig. 38 the outlines of typical flame zones are shown to illustrate the change toward a symmetric shape as gravity is reduced. The decrease of droplet size with time during combustion was expressed in terms of an evaporation coefficient K,, . This coefficient relates the rate of change of droplet mass to the droplet diameter. It was found experi-

REDUCED GRAVITY ON HEAT TRANSFER

217

mentally that there was a direct proportionality between these two quantities so that -dmf/dt = K,, 7rp (D/4) (42) Since the droplets are assumed spherical, Eq. (42) can be integrated to yield DO2 - D2 = K,, t (43) where Do is the initial size of the droplet. As shown in Fig. 38, the evaporation coefficient was found to decrease with gravity; for example, for n-heptane the value of K,, at zero gravity is about one-half that at earth gravity. A shortcoming of the experiments by Kumagai (101) and Kumagai and Isoda (102)was that the combustion was observed only at one instant during each test. T o observe the behavior of the combustion process as a function of time, another series of experiments was conducted (103). Here a movie camera was placed in free fall along with the combustion chamber and burning was observed in zero gravity for about a 1-sec duration. The fuels were n-heptane, ethyl alcohol, and benzene. Ignition was made just after the falling platform was released. The luminous flame boundary was completely spherical and concentric with the droplets. For benzene the luminosity of the flame obscured the view of the droplet so that droplet measurements could not be made. For n-heptane and ethyl alcohol, however, the burning rate was again found to obey Eq. (42). The K,, values at zero gravity for n-heptane and ethyl alcohol were 0.60 and 0.46 mm2/sec as compared with the corresponding higher values of 0.84 and 0.70 mm2/sec at earth gravity. These values are a little different from those in Fig. 38, but again reveal how the lack of free convection decreases the rate at which the droplets are consumed. The hot-air zone outside the flame expanded continuously with time during the zero-gravity period. The diameter of the luminous spherical flame increased at first, and then it decreased as the burning droplet became small.

C. SOLIDFUELS The combustion of several solid fuels in zero gravity was studied by Kimzey (104). An airplane was used to provide test durations up to 12 seconds. Some of the fuels tested were styrene, white foam rubber, paraffin, neoprene, and polyurethane. These were burned in pure oxygen, air, and oxygen-nitrogen mixtures at pressures from 5 to 14.7 psia. The fuel was ignited during weightlessness with an electrically heated nichrome wire, and motion pictures were taken with both color and infrared film. Ignition during weightlessness was found to be essentially unchanged from that in earth gravity, but the burning rate was considerably reduced in

218

ROBERTSIEGEL

zero gravity. For example, in 5-psia oxygen the burning rate of a tubular length of polyurethane was reduced from 0.6 to 0.08 inch/sec by a change from 1 to Og,. T h e reduction in burning rate is consistent with the decrease in evaporation coefficient discussed in the previous section. The flame behavior, however, for the several materials tested, was different from the continuous burning reported for a candle (100). Since one of the materials tested by Kimzey (104) was paraffin it is not clear why this difference was found. Soon after ignition the flame in (104) was found to reach a maximum size and brilliance; then, the flame receded and darkened. This was thought to be caused by an insufficient rate of diffusion of oxygen into the flame (the fact that the candle in (100) continued to burn indicates that the method of fuel supply, i.e., the use of a wick, may be significant). The flame darkening occurred within a few seconds after ignition. Often, the flame would appear to be extinguished as the zero-gravity period proceeded ;however, the flame would resume if gravity were restored even momentarily. The tests were of insufficient duration to determine whether the flame might eventually be completely extinguished in zero gravity. IX. Summary and Areas for Further Investigation

T o the harried reader who is endeavoring to cope with the explosive production of scientific literature is dedicated this capsule summary of some of the significant findings covered in this review. Within the summary, some areas for further research will also be brought to attention. Studies in reduced gravity are important (1) for specific applications in space devices, (2) to determine the validity of the gravity dependence in theoretical and experimental correlations, and (3) to remove unwanted effects of free convection in certain experiments. Experimental tests have been conducted in the following types of facilities (listed in order of providing increased durations of testing time) : (1) drop tower, (2) airplane flying ballistic trajectory, (3) rocket suborbital flight, (4) manned space capsule, (5) satellite (tests being planned), and (6) use of a magnetic field in conjunction with fluid having magnetic properties. In the area of free convection there is a lack of data at reduced gravities, so this provides a subject for future experimentation. A difficulty is that free convection boundary layers have a relatively slow time response so that tests cannot be conducted with convenient facilities such as drop towers or airplane flights. From analysis and correlations of lg, data some conclusions are the following : 1. Free convection depends on the Rayleigh number, which contains gravity to the first power. In laminar flow the free-convection heat-transfer

REDUCED GRAVITY ON HEATTRANSFER

219

coefficient should vary a ~ g ' /while ~ , for turbulent flow the exponent increases and tends toward g2lSas the Rayleigh number continues to increase. to g1/3 2. For laminar flow the boundary-layer thickness depends ong-'l4; hence, at very low gravities the thickness can become large enough so that thinboundary-layer theory no longer applies. 3. If a vertical plate initially at the same temperature as its surroundings has its temperature suddenly raised, the time required for steady state to be achieved for laminar flow depends on g-'/'. As a result, the transient times can become very long in low gravity. For pool boiling of a saturated liquid the following results have been found :

1. For nucleate boiling, both analysis and experiment indicate that the relation between temperature difference (T, - Tsst)and wall heat flux is insensitive to gravity. 2. T h e peak nucleate boiling (critical) heat flux was found experimentally to vary reasonably well as g1/4,which is in agreement with theory. 3. T h e relation between T, - T,,, and wall heat flux in the transition region between nucleate and film boiling appears from limited data to be insensitive to gravity reductions. 4. T h e minimum heat flux value where transition boiling changes to film boiling depends on g1/4. 5. In laminar film boiling the heat-transfer coefficientdepends ~ n g ' ' ~For . a turbulent vapor film the exponent increases and may be as large as 3 to $. The experimental evidence to substantiate these conclusions is mostly from tests of short duration. Additional experimental work is needed with longer testing times. This is especially true at very low gravities ;for example, it has not been conclusively demonstrated that the critical heat flux does go to zero as exactly zero gravity is approached. Longer test times would assure that the thermal layer near the surface has sufficient time to adjust to the reduced-gravity conditions. Photographic studies of reduced-gravity pool boiling have shown the following for saturated conditions :

1. Vapor accumulations tend to linger near the surface and collect new bubbles being formed thereby helping to remove them from the surface. 2. T h e dependency of bubble diameter on time during bubble growth is insensitive to gravity as shown by both theory and experiment. 3. The diameter of single bubbles at departure depends on g-'l2 for bubbles that grow slowly. For these bubbles, buoyancy is the force causing bubble departure.

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4. For rapidly growing bubbles, departure is governed by inertial forces and the departure size becomes insensitive to gravity. 5. There is a pressure force acting on the top portion of the bubble that lies above the area of the bubble base. This force aids bubble departure and does not depend on gravity (it is also present for subcooled conditions.) 6. The velocity at which detached bubbles rise through the liquid appears from limited data to decrease with gravity as predicted by theory. For subcooled conditions, the bubble formation' in water becomes independent of gravity when the subcooling is about 30°F. Again it is emphasized that these conclusions are based on limited tests of short duration and that longer testing times are a need for future work. For reduced gravity forced-convection boiling, little experimental information is available :

1. Isothermal two-phase flow visualization tests with air-water mixtures have shown that the bubble distribution over the cross section in a vertical tube at earth gravity is very similar to that in zero gravity. This points out the usefulness of the vertical orientation to simulate low gravity conditions. 2. Critical heat flux tests with upflow and downflow in an earth-bound vertical-tube experiment can define a fluid velocity range above which the critical flux is insensitive to flow orientation with respect to the gravity vector. Above this velocity range, performance should be insensitive to gravity reductions. For condensation without forced flow, no experimental results were found in the literature; hence, this would be another fruitful area for future work :

1. From laminar film condensation theory for a vertical plate at constant temperature, the heat-transfer coefficient depends on g114.For a turbulent condensate layer the exponential dependence increases to about g'l'. 2. The transient time to establish a laminar condensate layer on a vertical plate varies as g-'12 for a sudden reduction in plate temperature. For forced flow condensation in tubes, low gravity tests have been performed with nonwetting mercury :

1. The two-phase pressure drop was insensitive to gravity reductions for the conditions tested. 2. An indication of the stability of the vapor-liquid interface could be found from the Taylor type of instability theory. 3. Low gravity conditions can aid in the trapping of noncondensable gas.

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Further experimentation is needed to expand the range of condensing rates and to test fluids that wet the tube wall. A few combustion tests have been performed in reduced and zero gravity :

1. At zero gravity a candle continued to burn and the flame became

spherical with a light-blue color. 2. Reduced gravity decreased the rate of burning for fuel droplets. T h e luminous flame and hot-gas zones around a droplet became more spherical as gravity was reduced. 3. For several solid fuels the burning rate was substantially decreased by a change from earth to zero gravity. After ignition in zero gravity the flame reached a maximum size and brilliance ;then the flame receded and darkened.

NOMENCLATURE surface area acceleration relative to a fixed frame of reference constant in drag coefficient coefficient in boiling correlation, Eq. (9) specific heat at constant pressure bubble diameter [or droplet diameter in Eqs. (42) and (43)] bubble base diameter bubble departure diameter initial droplet size force buoyancy force defined by Eq. (28) buoyancy force defined by Eq. (27) drag force defined by Eq. (25) dynamic force defined by Eq. (31) inertial force defined by Eq. (30) pressure force defined by Eq. (29) surface-tension force defined by Eq. (24) functional dependence Grashof number based on length L gravity field of heat-transfer system being studied earth gravity field (also used in some equations to designate the conversion between Ib mass and lb force) effective gravity field caused by rotation

height of vertical surface heat-transfer coefficient local heat-transfer coefficient at position x average heat-transfer coefficient thermal conductivity characteristic length total condensing length mass average Nusselt number, i L / k Prandtl number, c p p / k static pressure pressure inside bubble static pressure at local position along tube static pressure at beginning of condensing length, or pressure outside bubble heat transfer per unit time critical heat transfer rate in boiling heat transfer per unit time and area heat flus from vapor in bubble to liquid radius of curvature modified characteristic length, Eq. (14) Rayleigh number based on length L , [gP(T,, - TdL3/v21Pr Rayleigh number based on length x, rgP(T,"- T b w / v 2 I p r

ROBERTSIEGEL

222 RaL* Ra,' Re, S

T Tb

t

urn

-U V Vb 2,

X

a

B Ycr

s

modified Rayleigh number based on length L , (g,G&"/kv2)Pr modified Rayleigh number based on length x, (g/?qx"/kv2)Pr Reynolds number based on length 6 height of drop tower temperature fluid bulk temperature time terminal velocity of bubble rising value in in a pool of liquid; earth gravity average velocity total volume of bubble portion of bubble volume lying above bubble base velocity length coordinate thermal diffusivity coefficient of volume expansion critical wavelength, Eq. (41) boundary-layer thickness or thickness of condensate layer

e bubble contact angle Ke v

evaporation coefficient

h latent heat of vaporization ; A' = h BcJT,,, - T,)

+

CL V

P U

T

absolute fluid viscosity kinematic fluid viscosity fluid density surface tension dimensionless time, t a / U

SUBSCRIPTS conduction regime (except in Q,) fuel 1 liquid phase (except in p,) min minimum flux between transition and film boiling steady state (except in F,) S sat saturation condition t top of bubble u vapor phase vf vapor film wall W C

f

REFERENCES l a . W. Unterberg and J. T. Congelliere, Zero gravity problems in space powerplants: A status survey. A R S (Am. Rocket SOC.) J . 32, 862 (1962). 1. I,. M. Habip, On the mechanics of liquids in subgravity, Astrotiuuticu Actu 11,401 (1965). 2. D . A. Petrash, R. C. Nussle, and E. W. Otto, Effect of contact angle and tank geometry on the configuration of the liquid-vapor interface during weightlessness. N A S A Tech. Note T N D-2075, October 1963. 3. M. Delattre, Installation d'Essais en impesanteur de L'0.N.E.R.A. 5th European Space Symp., Munich, Germany, July 1965, ONERA T.P. No. 258 (1965). 4. E. T . Benedikt and R. Lepper, Experimental production of a zero or near-zero gravity environment, in "Weightlessness-Physical Phenomena and Biological Effects" (E. T. Benedikt, ed.), p. 56. Plenum, New York, 1961. 5. R. Siege1 and E. G. Keshock, Nucleate and film boiling in reduced gravity from horizontal and vertical wires. N A S A Tech. Rept. TR-R-216, February 1965. 6a. J. W. Useller, J . H. Enders, and F. W. Haise, Jr., Use of aircraft for zero-gravity environment, N A S A Tech. Note T N D-3380, May 1966. 6. C . C. Crabs, AJ-2 zero-gravity flight facility. N A S A Tech. Note T N D-2838, Appendix A, June 1965. 7. R. R. Nunamaker, E. L. Corpas, and J. G. McArdle, Weightlessness experiments with liquid hydrogen in aerobee sounding rockets : uniform radiant heat addition-Flight 3. NASA T M X-872, December 1963 (confidential).

REDUCEDGRAVITY ON HEATTRANSFER

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8. D. A. Petrash, R. C. Nussle, and E. W. Otto, Effect of the acceleration disturbances encountered in the MA-7 spacecraft on the liquid-vapor interface in a baWed tank during weightlessness. N A S A Tech. Note T N D-1577, January 1963. 9. M . E. Kirkpatrick, A study of space propulsion system experiments utilizing environmental research satellites. AFRPL-TR-65-73 (DDC No. AD464581) T R W Space Techno]. Labs., p. 60, 1965. 9a. M. E. Nein and C. D. Arnett, Program plan for earth-orbital low g heat transfer and fluid mechanics experiments, NASA T M X-53395, February 1966. 10. D. A. Kirk, Means of creating simulated variable gravity fields for the study of free convective heat transfer, in Proc. 1st Space Vehicle Thermal Atmospheric Control Symp. February 1963, ASD-TDR-63-260, April 1963. 11. D. A. Kirk, T h e effect of gravity on free convection heat transfer, I : The feasibility of using an electromagnetic body force. WADD T R 60-303, August 1960. 12. S. Ostrach, Laminar flows with body forces, in “Theory of Laminar Flows,” (F. K. Moore, ed.), Vol. IV: High Speed Aerodynamics and Jet Propulsion, Princeton Univ. Press, Princeton, New Jersey, 1964. 13. J. T. Stuart, Hydrodynamic stability, in “Laminar Boundary Layers” (L. Rosenhead, ed.), Chapter 9, p. 492. Oxford Univ. Press (Clarendon) London and New York, 1963. 14. H. Grober, S. Erk, and U. Grigull, “Fundamentals of Heat Transfer,” p. 313 (transl. by J. R. Moszyenski). McGraw-Hill, New York, 1961. 15. H. Jeffreys, Some cases of instabilityin fluid motion. Proc. Roy. Sac. A118,195 (1928). 16. A. R. Low, On the criterion for stability of a layer of viscous fluid heated from below. Proc. Roy. Sac. A125, 180 (1929). 17. R. J. Schmidt and 0. A. Saunders, On the motion of a fluid heated from below. Proc. Roy. SOC.A165, 216 (1938). 18. W. V. R. Malkus, Discrete transitions in turbulent convection. Proc. Roy. Sac. A225, 185 (1954). 19. A. Pellew and R. V. Southwell, On maintained convective motion in a fluid heated from below. Proc. Roy. Sac. A176, 312 (1940). 20. E. R. G. Eckert and W. 0. Carlson, Natural convection in an air layer enclosed between two vertical plates with different temperatures. Intern. J. Heat Mass Transfer 2, 106 (1961). 21. G. K. Batchelor, Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quart. Appl. Math. 12, 209 (1954). 22. A. Emery and N. C. Chu, Heat transfer across vertical layers. J. Heat Transfer 87,110 (1965). 23. E. R. G. Eckert and R. M. Drake, Jr., “Heat and Mass Transfer,” 2nd ed., p. 314. McGraw-Hill, New York, 1959. 24. J. H. Chin, J. 0. Donaldson, L. W. Gallagher, E. Y. Harper, S. E. Hurd, and H. M. Satterlee, Analytical and experimental study of liquid orientation and stratification in standard and reduced gravity fields. Rept. 2-05-64-1, Lockheed Missiles and Space Co., July 1964. 25. E. H. W. Schmidt, Heat transmission by natural convection at high centrifugal acceleration in water-cooled gas-turbine blades. Proc. Gen. Discussion Heat Transfer ASME-IME, London, 1951, p. 361. Inst. Mech. Engrs, London, 1951. 26. W. H. McAdams, “Heat Transmission,” 3rd ed., p. 173. McGraw-Hill, New York, 1954. 27. M. Jakob, “Heat Transfer,” Val. 1. Wiley, New York, 1949. 28. H. Senftleben, Die Warmeabgabe von Korpern verschiedener Form in Flussigkeiten und Gasen bei freier Stromung. 2. Angew. Phys. 3, 361 (1951).

224

ROBERTSIEGEL

29. J. J. Mahony, Heat transfer at small Grashof numbers. Proc. Roy. Sac. A238, 412 (1957). 30. S. H. Schwartz and M. Adelberg, Some thermal aspects of a contained fluid in a reduced gravity environment. Douglas Missile and Space Systems Div. Paper No. 3394, Symp. Fluid Mechs. Heat Transfer under Low Gravitational Conditions. USAF Office of Sci. Res. and Lockheed Missiles and Space Co., Palo Alto, California, 1965. 31. M. Adelberg and S. H. Schwartz, Heat transfer domains for fluids in a variable gravity field with some applications to storage of cryogens in space. Douglas Missile and Space Systems Div. Paper No. 3516, Cryogenic Eng. Conf. Houston, Texas, August 1965. 32. R. Siegel, Transient free convection from a vertical flat plate. Trans. A S M E 80, 347 (1958). 33. R. J. Goldstein and E. R. G. Eckert, The steady and transient free convection boundary layer on a uniformly heated vertical plate. Intern. J . Heat Mass Transfer 1,208 (1960). 34. W. M. Rohsenow, A method of correlating heat-transfer data for surface boiling of liquids. Trans. A S M E 74, 969 (1952). 35. J. W. Westwater, Boiling of liquids. Advan. Chem. Eng. 1, 1 (1956). 36. R. Siegel and C. Usiskin, A photographic study of boiling in the absence of gravity. J . Heat Transfer 81, 230 (1959). 37. C. M. Usiskin and R. Siegel, An experimental study of boiling in reduced and zero gravity fields. J . Heat Transfer 83, 243 (1961). 38. J. E. Sherley, Nucleate boiling heat-transfer data for liquid hydrogen at standard and zero gravity, in “Advances in Cryogenic Engineering,” (K. D. Timmerhaus, ed.), Vol. 8, p. 495. Plenum, New York, 1963. 39. H. Merte, Jr. and J. A. Clark, Boiling heat transfer with cryogenic fluids at standard, fractional, and near-zero gravity. J . Heat Transfer 86, 351 (1964). 40. R. G. Clodfelter, Low-gravity pool-boiling heat transfer. APL-TDR-64-19 (DDC No. AD-437803), 1964. 41. E. W. Lewis, H. Merte, Jr., and J. A. Clark, Heat transfer at zero gravity. AZChE55th Natl. Meeting, Houston, Texas, February 1965. Preprint 24e. 42. L. M. Hedgepeth and E. A. Zara, Zero gravity pool boiling. ASD-TDR-63-706 (DDC No. AD-431810), September 1963. 43. S. H. Schwartz and R. L. Mannes, A study of nucleate boiling with water in a low gravity field. Submitted for publication. 44. J. Rex and B. A. Knight, An experimental assessment of the heat transfer properties of propane in a near-zero gravity environment. Roy. Aircraft Estab. Tech. Note No. Space 69, August 1964. 45. S . S. Papell and 0. C. Faber, Jr., Zero and reduced gravity simulation on a magnetic colloid pool-boiling system. N A S A Tech. Note T N D-3288. February 1966. 46. M. T. Cichelli and C. F. Bonilla, Heat transfer to liquids boiling under pressure. Trans. A m . Inst. Chem. Engrs. 41, 755 (1945). 47. S. S. Kutateladze, A hydrodynamic theory of changes in the boiling process under free convection conditions. Zzv. Akad. Nauk S S S R , Otd. Tekhn. Nauk No. 4, 529 (1951) (transl. available as AEC-TR-1441). 48. N. Zuber, “Hydrodynamic Aspects of Boiling Heat Transfer,” Ph.D. Thesis, Univ. of California, Los Angeles, California, 1959 (also available as AECU-4439). 49. R. C. Noyes, An experimental study of sodium pool boiling heat transfer. .I. Heat Transfer 85, 125 (1963). 50. J . H. Lienhard and K. Watanabe, On correlating the peak and minimum boiling heat fluxes with pressure and heater configuration. ,I. Heat Transfer 88, 94 (1966).

REDUCED GRAVITY ON HEATTRANSFER

225

51. H. F. Steinle, An experimental study of the transition from nucleate to film boiling under zero-gravity conditions, in “Weightlessness-Physical Phenomena and Biological Effects” (E. T. Benedikt, ed.), p. 111. Plenum, New York, 1961 [see alternately Proc. 1960 Heat Transfer and Fluid Mechanics Institute (D. M. Mason, W. C. Reynolds, and W. G. Vincenti, eds.), p. 208. Stanford Univ. Press, Stanford, California, 19601. 52. R. Siegel and J. R. Howell, Critical heat flux for saturated pool boiling from horizontal and vertical wires in reduced gravity. NASA Tech. Note T N D-3123, December 1965. 53. D. N. Lyon, M. C. Jones, G. L. Ritter, C. Chiladakis, and P. G. Kosky, Peaknucleate boiling fluxes for liquid oxygen on a flat horizontal platinum surface at buoyancies corresponding to accelerations between -0.03 and lg,, A.Z.Ch.E. ( A m . Znst. Chem. Engrs.) J . 11, 773 (1965). 54. P. J. Berenson, Film-boiling heat transfer from a horizontal surface. J . Heat Transfer 83, 351 (1961). 55. J. H. Lienhard, Interacting effects of geometry and gravity upon the extreme boiling heat fluxes. Submitted for publication. 56. L. A. Bromley, Effect of heat capacity of condensate. Ind. Eng. Chem. 44, 2966 (195 2). 57. Y. Y. Hsu and J. W. Westwater, Approximate theory for film boiling on vertical surfaces. Chem. Eng. Progr. Symp. Ser. 56, No. 30, 15 (1960). 58. T. H. K. Frederking and J. A. Clark, Natural convection film boiling on a sphere, in “Advances in Cryogenic Engineering” (K. D. Timmerhaus, ed.), Vol. 8, p. 501. Plenum, New York, 1963. 59. R. Siegel and E. G. Keshock, Effects of reduced gravity on nucleate boiling bubble dynamics in saturated water. AZChE ( A m . Znst. Chem. Engrs.) J . 10, 509 (1964). 60. E. G. Keshock and R. Siegel, Forces acting on bubbles in nucleate boiling under normal and reduced gravity conditions. N A S A Tech. Note T N D-2299, August 1964. 61. W. Fritz and W. Ende, The vaporization process according to cinematographic pictures of vapor bubbles. Z . Physik. 37, 391 (1936). 62. M. S. Plesset and S. A. Zwick, The growth of vapor bubbles in superheated liquids. 1. Appl. Phys. 25,493 (1954). 63. H. K. Forster and N. Zuber, Growth of a vapor bubble in a superheated liquid. .I. Appl. Phys. 25, 474 (1954). 64. N. Zuber, T h e Dynamics of vapor bubbles in nonuniform temperature fields. Intern. J . Heat Mass Transfer 2, 83 (1961). 65. W. Fritz, Maximum volume of vapor bubbles. Z . Physik. 36, 379 (1935). 66. T. H. Cochran, J. C. Aydelott, and T. C. Frysinger, The effect of subcooling and gravity level on boiling in the discrete bubble region. N A S A Tech. Note T N D-3449, September 1966. 67. W. M. Rohsenow, Heat transfer with boiling in “Modern Developments in Heat Transfer” (W. E. Ibele, ed.), p. 111. Academic Press, New York, 1963. 6 8 . T. Z. Harmathy, Velocity of large drops and bubbles in media of infinite or restricted extent. AZChE ( A m . Inst. Chem. Engrs.) ,I. 6, 281 (1961). 69. J. L. McGrew, D. W. Murphy, and R. V. Bailey, Boiling heat transfer in a zero gravity environment. S A E - A S M E Air Transport and Space Meeting, New York, April 1964. SAE Paper 862C. 70. T. R. Rehm, Subcooled boiling in a negligible gravity field. Denver Res. Inst. Rep. DRI-2227, 1965. 71. J. H. Lienhard and P. T. Y. Wong, The dominant unstable wavelength and minimum heat flux during film boiling on a horizontal cylinder. J . Heat Transfer 86, 220 (1964).

226

ROBERTSIEGEL

72. D. G. Evans, Visual study of swirling and nonswirling two-phase two-component flow at 1 and 0 gravity. NASA T M X-725, August 1963. 73. Y. Y.Hsu and R. W. Graham, A visual study of two-phase flow in a vertical tube with heat addition. NASA Tech. Note T N D-1564, January 1963. 74. M. Adelberg, Effect of gravity upon nucleate boiling, in “Physical and Biological Phenomena in a Weightless State,” Advances in the Astronautical Sciences, (E. T. Benedikt and R. W. Halliburton, eds.), VoI. 14, p. 196. Western Periodicals, North Hollywood, California, 1963. 75. M. Adelberg, Boiling, condensation and convection in a gravitational field. A I C h E 55th N a t l . Meeting, S y m p . Effects Zero Gravity on Fluid Dynamics and Heat Transfer Houston, Texas, February 1965, Pt. 11, Paper 24d. 76. R. J. Simoneau and F. F. Simon, A visual study of velocity and buoyancy effects on boiling nitrogen. N A S A Tech. Note T N D-3354, September 1966. 76a. R. J. Simoneau and F. F. Simon, Personal communication, 1966. 77. S. S. Papell, R. J. Simoneau, and D. D. Brown, Buoyancy effects on the critical heat flux of forced convective boiling in vertical flow. N A S A Tech. Note T N 1)-3672, October 1966. 78. R. V. Macbeth, Burn-out analysis, 5: Examination of published world data for rod bundles. AEEW-R-358, 1964. 79. S. S. Papell, An instability effect on two-phase heat transfer for subcooled water flowing under conditions of zero gravity. A m . Rocket SOC.Space Power Systems Conf. Santa Monica, California, September 1962. Paper 2548-62. 79a. C. J. Feldmanis, Performance of boiling and condensing equipment under simulated outer space conditions, ASD-TDR-63-862, November 1963. 79b. C. J. Feldmanis, Pressure and temperature changes in closed loop forced convection boiling and condensing processes under zero gravity conditions. Proc. Inst. Enwiron. Sci. 1966 Ann. Tech. M t g . , Sun Diego, California, April 1966. 80. K. Ginwala, T. Blatt, W. Jansen, G. Khabbaz, and S. Shenkman, Engineering study of vapor cycle cooling equipment for zero-gravity environment. WADD T R 60-776. Northern Res. Engr. Corp., January 1961. 81. W. R. Gambill and N. D. Greene, Boiling burnout with water in vortex flow. Chem. Eng. Progr. 54, 68 (1958). 82. W. R. Gambill, R. D. Bundy, and R. W. Wansbrough, Heat transfer burnout, and pressure drop for water in swirl flow through tubes with internal twisted tapes. ORNL-2911, April 1960. 83. H. J. Ivey, Acceleration and the critical heat flux in pool boiling heat transfer. Proc. Znst. Mech. Engrs. (London) 177,15 (1963). 84. V. I . Morozkin, A. N. Amenitskii, and I. T. Alad’ev, An experimental investigation of the effect of acceleration on the boiling crisis in subcooled water. High Temp. (English Trans.) 2, 104 (1964). 85. C. P. Costello and J. M. Adams, Burnout heat fluxes in pool boilingat high accelerations, in “International Developmentsin Heat Transfer,” Proc. Heat Transfer Conf., Boulder, Colorado, and London, 1961-1962, p. 255. Am. SOC.Mech. Engrs. New York, 1963. 86. R. W. Graham, R. C. Hendricks, and R. C . Ehlers, Analytical and experimental study of pool heating of liquid hydrogen over a range of accelerations. NASA Tech. Note T N D-1883, February 1964. 87. M. L. Pomerantz, Film boiling on a horizontal tube in increased gravity fields. ./. Heat Transfer 86, 2 1 3 (1964). 88. H. Merte, Jr. and J. A. Clark, Pool boiling in an accelerating system. ./. Heat Transfer 83, 233 (1961).

REDUCED GRAVITY ON HEATTRANSFER

227

89. W. D. Allingham and J. A. McEntire, Determination of boiling film coefficient for a Heated horizontal tube in water-saturated wick material. J . Heat Transfer 83, 71 (1961). 90. C. P. Costello and W. J. Frea, The roles of capillary wicking and surface deposits in the attainment of high pool boiling burnout heat fluxes. AZChE ( A m . Inst. Chem. Engrs.) J . 10, 393 (1964). 91. W. M. Rohsenow and H. Y. Choi, “Heat, Mass, and Momentum Transfer,” p. 239. Prentice-Hall, Englewood Cliffs, New Jersey, 1961. 92. E. M. Sparrow and R. Siegel, Transient film condensation. ,I. Appl. Mech. 26, 120 (1959). 93. R. L. Cummings, P. E. Grevstad, and J. G. Reitz, Orbital force field boiling and condensing experiment, in “Weightlessness-Physical Phenomena and Biological Effects” (E. T. Benedikt, ed.), p. 121. Plenum, New York, 1961. 94. R. A. Trusela and R. G. Clodfelter, Heat transfer problems of space vehicle power systems. S A E Natl. Aeron. Meeting, New York, April 1960. Paper No. 154c. 95. J. A. Albers and R. P. Macosko, Experimental pressure-drop investigation of nonwetting, condensing flow of mercury vapor in a constant-diameter tube in 1-G and zero-gravity environments. N A S A Tech. Note T N D-2838, June 1965. 96. J. A. Albers and D. Namkoong, Jr., Single-tube mercury vapor condensing characteristics in one G and zero G environments. AZAA Rankine Cycle Space Power System Specialists Conf., Cleveland, Ohio, October 1965, AEC C O N F 65 1026, 1966. 96a. J. I. Sturas, C. C. Crabs, and S. H. Gorland, Photographic study of mercury droplet parameters including effects of gravity, N A S A Tech. Note T N D-3705, November 1966. 97. J. A. Albers and R. P. Macosko, Condensation pressure drop of nonwetting mercury in a uniformly tapered tube in 1-G and zero-gravity flight environments. N A S A Tech. Note T N D-3185, January 1966. 98. R. T. Lancet, P. Abramson, and R. P. Forslund, T h e fluid mechanics of condensing mercury in a low-gravity environment. Symp. Fluid Mech. Heat Transfer Under Low Gradational Conditions. USAF Office of Sci. Res. and Lockheed Missiles and Space Co., Palo Alto, California, 1965. 98a. R. J. Denington, A. Koestel, A. V. Saule, R. I. Shure, G. T. Stevens, and R. B. Taylor, Space radiator study. ASD-TDR-61-697 (DDC No. AD-424419), October 1963. 99. J. G. Reitz, Zero gravity mercury condensing research. Aerospace Eng. 19, 18 (1960). 100. A. L. Hall, Observations on the burning of a candle at zero gravity. School of Aviation Med., Rept. No. 5 (DDC No. AD-436897), February 1964. 101. S. Kumagai, Combustion of fuel droplets in a falling chamber with special reference to the effect of natural convection. Jet Propulsion 26, 786 (1956). 102. S. Kumagai and H. Isoda, Combustion of fuel droplets in a falling chamber. 6th Symp. Combustion, p. 726. Reinhold, New York, 1957. 103. H. Isoda and S. Kumagai, New aspects of droplet combustion. 7th Symp. Combustion, London, Oxford 1958, p. 523. Butterworths, London, 1959. 104. J. H. Kimzey, Flammability during weightlessness, Proc. Inst. Environ. Sci. 1966 Ann. Tech. Mtg., San Diego, Cal$ornia, April 1966, p. 433.

ADDITIONAL REFERENCES J. C. Aydelott, E. L. Corpas, and R. P. Gruber, Comparison of pressure rise in a hydrogen dewar for homogeneous, normal-gravity quiescent, and zero-gravity conditions-Flight 9. NASA T M X-1052, February 1965 (confidential).

ROBERTSIEGEL K. L . Abdalla, R. A. Flage, and R. G. Jackson, Zero-gravity performance of ullage control

surface with liquid hydrogen while subjected to unsymmetrical radiant heating. NASA T M X-1001, August 1964 (confidential). J. C. Aydelott, E. L. Corpas, and R. P. Gruber, Comparison of pressure rise in a hydrogen dewar for homogeneous, normal-gravity quiescent, and zero-gravity conditions-Flight 7. NASA T M X-1006, September 1964 (confidential). K. L. Abdalla, T. C. Frysinger, and C. R. Andracchio, Pressure-rise characteristics for a liquid-hydrogen dewar for homogeneous, normal-gravity quiescent, and zerogravity tests. NASA T M X-1134, September 1965 (confidential). M . Adelberg and S. H. Schwartz, Scaling of fluids for studying the effect of gravity upon Inst. Environ. Sci. 1966 Ann. Tech. Mtg., Sun Diego, Califortiia, nucleate boiling, PYOC. April 1966. C. S. Boyd, An investigation of boiling water in a porous material as a means of separating liquid and vapour in zero gravity, M.S. Thesis, Air Force Inst. of Tech., WrightPatterson AFB, Ohio, March 1965 ( D D C No. AD-618022). R. D. Cess, Laminar-film condensation on a flat plate in the absence of a body force, %. Angew. Math. Phys. 11, 426 (1960). J. T. Congelliere, W.Unterberg, and E. B. Zwick, The zero G flow loop: steady-flow, zerogravity simulation for investigation of two-phase phenomena, in “Physical and Biological Phenomena in a Weightless State,” Advances in the Astronautical Sciences, (E. T. Benedikt and R. W. Halliburton, eds.), Vol. 14, p. 223. Western Periodicals, North Hollywood, California, 1963. R . L . Hammel, J. M. Robinson, and H. T. Sliff, Environmental research satellites for space propulsion systems experiments, Quart. Progr. Repts. AFRPL-TR-66-64, AFRPL-TR66-178, T R W Systems, March and June 1966. R. H. Knoll, G. R. Smolak, and R. P. Nunamaker, Weightlessness experiments with liquid hydrogen in aerobee sounding rockets: uniform radiant heat addition-flight 1, KASA T M X-484, June 1962 (confidential). P. P. Mader, G. V. Colombo, and E. S. Mills, Summary report of tests conducted to determine effects of atmospheric compositions and gravity on burning properties, Ilouglas Missile and Space Systems Div. Rept. SM-48862, October 1965. J. G . McArdle, R. C. Dillon, and D. A. Altmos, if’eightlessness experiments with liquid hydrogen in aerobee sounding rockets: uniform radiant heat addition-flight 2, NASA T M X-718, December 1962 (confidential). R . M. Singer, T h e control of condensation heat transfer rates using an electromagnetic field, Argonne National Lab., ANL-6861, July 1964. E. A. Zara, Boiling heat transfer in zero gravity, R T D Technol. Briefs, Headquarters Res. and Tech. Div., Bolling AFB, D. C. 11, January 1964.