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Mass (or heat) transfer from an oscillating cylinder
Chemical Engineering Science, 1964, Vol. 19, pp. 793-800. Pergamon Press Ltd., Oxford.
Printed
in Great Britain.
Mass (or heat) transfer from an oscillating cylinder G.
J. JAMESON
Department of ChemicalEngineering, Universityof Cambridge (Received 13 December 1963; in revisedform 26 April 1964) Abstract-An equation has been derived for predicting the rate of mass (or heat) transfer from a cylinder oscillating with simple harmonic motion in an infinite body of fluid. It is shown that in certain circumstances the mechanism of the transfer is similar to that of heat transfer from a hot body in a fluctuating stream, in that the bulk of the transfer takes place in a steady streaming flow. The equation has been compared with experimental results obtained in the system benzoic acid/ glycerol-water solution, and shown to be in reasonable agreement. The mass transfer coefficients obtained with vibration were up to 28 times higher than those due to free convection alone. INTRODUCTION
THE effect of vibration on heat transfer from cylinders has been extensively investigated in recent years. In some cases, the solid was oscillated in an essentially stagnant fluid; in others, an oscillation was imparted to the fluid in contact with the body. The field in which most interest has been displayed is that in which a sound wave is passed through the fluid (generally air) surrounding a hot wire or cylinder, or through a fluid which is passing through a heated tube. Thus, KUBANSKII[l, 21 has studied the effect of stationary sound fields on free convection from an electrically heated horizontal cylinder in air, with the direction of oscillation parallel to the axis of the cylinder. The measured increase in the heat transfer coefficient due to the vibration was approximately 75 per cent. When a cross flow of air corresponding to Reynolds numbers between 1450 and 1770 was passed over the horizontal cylinder, KUBANSKII [3] was able to increase or reduce the heat transfer coefficient by moving the cylinder relative to the nodes of the stationary sound field, which was perpendicular to both the axis of the cylinder and the direction of the air flow. The maximum improvement in the observed coefficient, relative to that with no vibration, was about 50 per cent. Heat transfer from a wire vibrating at subsonic frequencies in stationary air was investigated by LEMLICH[4]. He presented a film model for the
heat transfer process which is applicable on a gross scale, but is rather unsatisfactory for the region close to the wire, where the flow configuration is an important factor. Similar work has been reported by ANANTANARAYANANand RAMACHANDRAN [5],
VAN DER HEGGE ZIJNEN [6],
LEMLICHand LEVY [7]
and FAND and KAYE [8]. Studies on the effect of vibrations on heat transfer to liquids have been relatively few. MARTINELLI and BOELTER[9] used a heated $ in. dia. horizontal tube oscillating in water at frequencies up to 40 cycles/set. The theoretical basis these workers gave to their work is oversimplified, however, and the validity of their experimental results has since been questioned [4]. In an excellent experimental paper, DEAVER, PENNEYand JEFFERSON [lo] considered heat transfer from an oscillating horizontal wire to water. They were able to correlate their results, with some success, by the simple expedient of forming dimensionless groups similar to those used in correlations for combined free and forced convection in steady flow past a cylinder. Detailed literature surveys on the subject have been given by LEMLICH[ll], and FAND and KAYE [8]. Despite the volume of literature on this topic, it seems that little progress has been made toward providing a satisfactory theoretical background for the experiments reported. It is clear that the general process of heat transfer to or from a body oscillating in a fluid may occur either by free convection, by
t Presentaddress:CaliforniaResearchCorporation,Richmond,California,U.S.A. 793
G. J. J-N forced convection, or by a combination of the two. A complicating factor however, is the compressibility of the fluid concerned. Thus, the equations governing the transfer of heat to gases will probably be different from those which apply to liquids, especially at high Mach numbers. Further, if the fluid is compressible, the equations may be different depending on whether the fluid oscillates about a fixed cylinder, or the cylinder oscillates in a stagnant fluid. The purpose of this present paper is to present a theoretical analysis of one aspect of this complex problem: mass or heat transfer by forced convection alone from a cylinder oscillating in an incompressible fluid. An equation based on an approximate boundary layer analysis of SCHLICHTING[13] is developed for predicting heat or mass transfer rates under these conditions. The equation, expressed in dimensionless form, has been verified experimentally by observing the rate of mass transfer from a benzoic acid cylinder oscillating in a glycerol-water solution.
of the axis of oscillation. The existence of this flow was demonstrated practically by SCHLICHTING,and has also been shown to exist in compressible fluids [14, 151. The velocity distribution derived by SCHLICH~NG may be represented (to the second order approximation) by the following stream function: * = $1 -t *z =-
2v 112 U&)F,($e’“’ n
0
+
FIG. 1 Curvilinear co-ordinate
0
-dxX n x CF2dv) + ~22(r)e2’“‘l
(1)
U,,(x), the amplitude of the periodic potential flow velocity, is 2nA sin (x/R) for a circular cylinder oscillating along a diameter with simple harmonic motion. The coordinate system is shown in Fit. 1. The terms F,, FzO, and Fz2 are functions of the dimensionless parameter 9 = y(n/2v)1’2 : F,
system.
2” 1/Z U,(x) dU,(x) n
FLUID DYNAMICS An analysis of the equations of motion around a cylinder oscillating in an incompressible fluid has been given by SCHLICHTING[12, 131. A method of successive approximations was used, based on boundary layer theory. It was shown that as well as the periodic flow pattern one would expect, a steady secondary flow is induced by the interaction of viscous and inertial effects in the boundary layer. The secondary flow streams in from the bulk of the fluid near the plane of the equator of the cylinder, passes over the surface, and leaves in the direction
+
=
-
(cq2
(1
-
,-(1+04)
+ q
(2)
13 3rl F 20 = -8 - T - &eSzq - se-” cos q - e-v sinq-fqe-“sinq F
22
_
(I + ‘) ,-(l+i)n,'Z + 2 4J2
2
,-(l+jjq
(3)
(1 + i) (4)
- 4J2
The term in II/ which is independent of time (the term associated with Fzo) gives rise to the steady streaming flow over the cylinder. It is a requirement of the boundary layer method used to calculate $ that the vibrational Reynolds number 2nAR/v and the length ratio R/A should both be large. Since $2 is of order A/R times til, the magnitude of the steady streaming flow is considerably less than the maximum value of the major oscillating flow component. The flow around an oscillating cylinder has also been investigated by HOLTSMARK et al. [21]. Where SCHLICHTINGsolved the boundary layer equations by successive approximations, these authors started from the Navier-Stokes equations. Their solutions are correspondingly more accurate, but since it is
794
Mass (or heat) transfer from an oscillating cylinder
not possible to express them in a simple general form, they are rather inconvenient to use in the present problem. Thus, when required, the flow profiles given by SCHLICHTING’Smethod will be used.
Since the velocity gradient may be assumed constant for any particular 8, the velocity may be written u = By, where /? is the velocity gradient, and we may define a stream function per unit length of cylinder as u dy = 38~~
i=so’
MASS (OR HEAT) TRANSFER
The mechanism that will be assumed to dominate transport from the oscillating cylinder is that of transfer into the steady streaming flow. This will only be valid when the time of an oscillation is very small when compared with a characteristic diffusion time of the system, which we take to be ~3~10, where 6 is the boundary layer thickness. Since 6 is of order (~/n)‘/~, this means that the steady flow mechanism will apply when the Schmidt number v/D is large. Now the equation for mass transfer into the two-dimensional steady boundary layer is:
If the thickness of the imagined boundary layer in which all the mass transfer occurs is A, $a = $fiA2 and 1+4= I,~~Y’, where y/Y = A. Thus, if $ is constant, so also is Y. Equation (6) may then be written
To transform this equation into a tractable form, we let dd
R
de
PA3
A=-
U?Z+$=D$ OY with boundary conditions c = 0 at x = 0 and at y = 00, and c = c* at y = 0. (The concentration in the oncoming stream is taken as zero for convenience.) Since we are already restricted to the case of large Schmidt number, or, in other words, a system where the diffusional boundary layer is considerably “thinner” than the velocity boundary layer, we may simplify this equation considerably. Very close to the body, the velocity component u will be negligible so that the term 2,&/8y may be neglected. The simplified equation is then
where 4 is a function of 8 only, equal to zero at 8 = 0. The differential equation then becomes
y&,=D($)+
(7)
with boundary conditions Y=O:
c=c*
Y=l:
c=o
+=o:
c=o
where c* is the saturation concentration of the solute. This equation may be solved by the introduction ac a2C UY-=DT (5) . , of a new variable s = Y/@/3 (Ref. [17]), since ax af substitution reduces equation (7) to the readily Also, in the layer very close to the body, the soluble form velocity gradient au/ay may be assumed constant. The method now used follows that of DAVIDSON d2c 2 and CULLEN [16], who were concerned with the solution of a slightly soluble gas in a film of liquid with boundary conditions s = 0, c = c* ; s = 00, flowing over a sphere, as a method of determining diffusion coefficients. By carrying out a mass c = 0. The solution of the equation is balance along a streamline of value $, they derived m e-s’3/9D ds’ an approximately equivalent form of equation (5),
~+gg=o
c
u(g),=RD($),
-=
(6) 795
c*
s s s
me-s”J9D
0
&
G. J. JAMFSON
The integral may be evaluated by gamma functions, to give a value of 1.198. Then, from (8), D2/3nr/2A2/3 k = o.746 ~2/3,,1/6
or we may write _
dc z
=
C*e-~'3/9D
me-s'",9DdSI s 0
The denominator may be expressed in terms of the gamma function I(+). Evaluation at c = 0 yields 2t3c*
(9D) s=.
=
-
3DF(3)
For calculation of the mass transfer rate, we need the concentration gradient at the wall, which is readily shown to be
We may define an overall mass transfer coefficient as
(8) Using the relations we find
d#de
= R/j3A3, Jla = @A2,
or rearranging in dimensionless form, Sh = 0.746 Re”2Sc”3(A/R)“6
(11)
The analogous heat transfer equation Nu = 0.746 Re’/’ Pr1/3(A/R)‘/6 would be equally valid providing the same assumptions held, i.e., that the Prandtl number was large, and that flow conditions were such that a boundary layer solution would apply. We have already remarked that for the boundary layer solution to apply, Re and R/A should both be large, since it is only in this case that the series solution given in equation (1) can be convergent. A further condition which must be met in the present analysis is that the radius of curvature of the cylinder should be large compared with the boundary layer thickness, since it is only under these circumstances that the curvilinear coordinate system may be used. This condition will be satisfied if nR’/v is large, but since nR2/v is equivalent (disregarding constants) to the product of Re and R/A, both of which are already assumed large, this is not really a further restriction. EXPERIMENTAL
where
The method used to experimentally investigate mass transfer from an oscillating cylinder was the dissolution of benzoic acid in a glycerol-water solution. This method was chosen because the benzoic acid is relatively insoluble in the liquid, so that the effects of natural convection are minimized. Furthermore, benzoic acid is readily formed into spheres or cylinders, and the process may be followed simply by measuring the loss in weight of the body after a known time. The cylinder was formed by first pressing benzoic acid (A.R. grade) in a spherical steel die, preheated to lOo”C, at 5 tons/in2. This produced spheres of about 3 cm dia., from which cylinders were turned. The cylinders formed were 2.1 cm long, 1.10 cm dia. They were mounted between perspex cylinders of the same diameter and length, in order to minimise end effects.
For the velocity gradient j$, we now go to equation (l), since
with only the steady part of II/ being considered. Thus,
Substituting in [9], R [nA” (:)“‘I do = (2$A)3’2 2R
1’2Jysinli2
20 de
796
Mass (or heat) transfer from an oscillatingcylinder
A sketch of the apparatus for oscillating the cylinder is shown in Fig. 2. The oscillation was produced by the crank-connecting rod system in which the length of the connecting rod was sufficient to produce essentially simple harmonic motion of
FIG.2 the
Diagram of apparatus
cylinder. The amplitude could be varied by altering the position of the crank, which was held in a milled slot. The crankshaft was belt driven from a Carter variable speed drive (with power input from a 4 h.p. motor), the variable speed unit being mounted on a separate frame from the oscillating cylinder mechanism, to minimise extraneous vibration. Speed measurements were by stroboscope, or by a Smiths tachometer at low speeds. The stroke (= 2x amplitude) was measured by a travelling microscope. Experiments were run at three levels, where A/R was respectively 0.102, O-198, and 0.298.
Prior to the actual experiments, the cylinder was placed in a saturated solution of benzoic acid in water, and then put under vacuum. This removed most of the trapped air in the interstices of the body, the voids being filled with liquid when atmospheric pressure was restored. Each cylinder was brought to constant weight by repeated immersions, and then kept under saturated benzoic acid solution. No difficulty was experienced in measuring the loss in weight of the cylinder during the course of the experiments. By careful turning, a high polish could be produced on the surface of the benzoic acid, allowing surface moisture to be removed by gently wiping with an absorbent towel. The weight readings were reproducible to within f0.0005 g; dissolution times were such as to give chahges in weight of at least 0.01 g. The liquid used in the experiments was a glycerolwater solution, maintained at a temperature of 23 k 0.1°C.. The density of the solution was 1.237 g/cm3, and the kinematic viscosity 1.55 cm”/ set (from the Tables of MINER and DALTON [18]), both at 23°C. The solubility and diffusivity data were taken from values given by WINT [19]. For densities of glycerol-water solutions of 1.149, 1.198, and 1.233 g/cm3, he gave values for the saturation concentration of benzoic acid of 8.3, 12.9 and 20.4 g/l of solvent, and diffusivities of 1.009 x 10e6, 0.281 x lo-‘j, and 0.0598 x 10V6 cm2/sec, respectively. The values used in present calculations were : c* = 22.5 g/l, and D = O-048 x 10e6 cm’/sec, so that the Schmidt number was 32.3 x 106. An experiment was carried out using the same glycerol-water mixture to determine the magnitude of the Sherwood number for free convection alone from the benzoic acid cylinder. The value found was Sh = 109. This may be compared with the range of Sherwood numbers found in the forced convection experiments, where the minimum was 275, and the maximum 2850. The natural convection value is therefore considerably smaller than those due to forced convection over most of the Reynolds number range investigated. RESULTSAND DISCUSSION
The results are shown (plotted in dimensionless form) in FIG. 3, together with the theoretical line 797
G.
J. JAMFSON
0.1
,
I.0
100
Re
FIG. 3
Comparison
of experimental
from equation (11). The experimental points are in fair agreement with the predicted, but they are, in every case, too high. One probable cause of this discrepancy is the neglect of the term u &/L+ in writing equation (5). While this term is very small over most of the cylinder, it is very large where the flow impinges on the surface, and where it departs. Continuity requires that in these areas, u approaches zero, while o approaches the main stream velocity. A second source of error lies in the boundary layer analysis of SCHLICHTING. Detailed calculations of HOLTSMARK et al. [21] have shown that the magnitude of the steady streaming flow is somewhat underestimated by the boundary layer theory, indicating that the velocity gradient given is also probably too low. A low value of the velocity gradient p would cause equation (11) to predict conservative results, as would neglect of the term v a~/+. The results of LEMLICH [4], and of DEAVER, PENNEY and JEFFERSON[lo] cannot be used to check equation (1 l), since in their experiments, the ratio AIR was considerably greater than unity, so that
results and theoretical prediction.
the present theory is not applicable. Nevertheless, DEAVER et al. found that at high Reynolds numbers, where the forced convection mechanism predominates, the Nusselt number was proportional to Re1’2Pr1i3, with a considerable scatter in the experimental data. They have since found that by inclusion of the dimensionless group AIR in the correlation, the scatter can be considerably reduced [20]. FAND and KAYE [8] studied the transfer of heat to air from a horizontal cylinder $ in. in dia., which was placed in a horizontal sound field normal to the axis of the cylinder. The wavelength of the disturbance was, in most of their experiments, large compared with the diameter of the cylinder, so that it would be safe to assume that any change in the pressure at the front of the cylinder would be instantaneously transmitted to the rear. The authors chose a rather idiosyncratic method of expressing their results, by plotting the observed heat transfer coefficient as a function of the temperature difference AT, at various values of the “sound pressure level” (SPL), which is merely a function of the peak velocity nA. Their results
798
Mass (or heat) transfer from an oscillating cylinder
showed that at low values of SPL, there was little change in the value of the heat transfer coefficient h, compared with the values obtained for free convection alone. At the highest values of SPL, however, there was an increase in h of a factor of approximately three. Values of SPL ranged from 138 to 151 dB, corresponding to values of nA between 1.79 and 8.16 ft/sec respectively. Now in these experiments, Pr = 0.7 approximately, so that the present theory could not be expected to hold. Nevertheless, some typical results have been plotted in Fig. 3, to see how much in error the predicted values would be. The results shown are taken from Fig. 10 of FAND and KAYE’Spaper, since these results showed the highest increase in the heat transfer rate when compared with the coefficient due to natural convection alone. Results were taken at three values of the “sound pressure level”, and it is observed that at a substantially constant Re, there is quite a range of values of Nu/Pr1/3(_4/R)“6. This is because the Nusselt number shows a considerable dependence on the temperature difference AT, indicating that free convection was still an important factor. Free convection is at a minimum however, when AT is smallest, i.e., for the lowest values of Nu/Pr1/3 (A/R)1’6, where AT was about 20 F”. Using these points as being representative of forced convection alone, one sees that they are rather lower than the values predicted by the “large Pr” theory. The experimental results show that the mass transfer coefficient with vibration was up to 28 times greater than without vibration. This factor is considerably higher than those reported by other workers [l-8] for heat transfer, a reflection of the rather low free convection mass transfer rates in the benzoic acid-glycerol solution system. This suggests that any practical use of vibrations to increase mass or heat transfer rates is likely to be most effective where natural convection rates are lowest, e.g., in solidliquid extraction.
the surface of the cylinder, but by a steady streaming flow induced by the oscillations. Providing Re and SC are large, and A/R is small, the equation which describes the mass transfer rate is Sh = 0.746 Re1i2Sc’13(A/R)116 This equation has been shown to be in reasonable agreement with experimental data, giving rather conservative predictions. There is no information available to test the analogous heat transfer equation, but at low Prandtl numbers, the equation has been shown to overestimate the heat losses. Experimental mass transfer rates with vibration were up to 28 times the free convection mass transfer rate, suggesting that vibrations could be used to increase mass transfer coefficients in operations such as solid-liquid extraction. Acknowle&ement-The author wishes to express his gratitude to DR J. FT DAVIDSONfor much helpful guidance and discussion.
NOTATION A C C*
D : P: R Re Sd Sh SPL ; u V x
Y F 8,: e
CONCLUSION It has been shown that under certain conditions, mass transfer from a cylinder oscillating with simple harmonic motion in a large body of fluid is dominated not by the periodic velocity components near D
799
Amplitude of oscillation Concentration of solute Saturation concentration of solute Diffusion coefficient Heat transfer coefficient Mass transfer coefficient Circular frequency Prandtl number Radius of cylinder Reynolds number, 2nA/v Y/Q* Schmidt number, v/D Sherwood number, 2kRID Sound pressure level Time Temperature Velocity component parallel to solid surface Velocity component normal to solid surface Distance parallel to solid surface Distance normal to solid surface YIA Velocity gradient Gamma function Boundary layer thickness Angular displacement from axis of oscillation Kinematic viscosity Liquid density Function of 6 Stream function Value of $I at outer edge of boundary layer
G. J. JAMESON
REFERENCES KUBANSKII P. N., J. Tech. Phys. (Moscow) 1952 22 585. KULIANSK~P. N., Dokl. Acad. Nauk, 1952 82 585. KUBANSKII P. N., J. Tech. Phys. (Moscow) 1952 22 593. [4] LEMLICH R., Industr. Engng Chem. 1955 411175. [5] ANANTANARAYANAN R. and RAMACHANDRANA., Trans. Amer. Sot. mech. Engrs 1958 88 1426. [6] VAN DER HEGGEZIJNEN B. G., Appl. Sci. Res. 1958 Al 205. [7] LEMLICH R. and LEW M. R., Amer. Inst. Chem. Engrs J. 1961 7 241. [S] FANLI R. M. and KAYE J., Trans. Amer. Sot. mech. Engrs 1961 83 C 133. [9] MARTINELLI R. C. and BOELTER L. M. K., Proc. Fifth Intern. Congr. Appl. Mech. 1938 578. [lo] DEAVER F. K., PENNEY W. R. and JEFFERSON T. B.. Trans. Amer. Sot. mech. Enars 1962 84 C 251. iiij LEMLICH R., them. Engng 1961 68 171. WI SCHLICHTING H., Boundary Layer Theory (4th ed.) Chap. 11. McGraw-Hill, New York 1960. [131 SCHLICHTINGH., Phys. 2. 1932 33 327.
[l] [2] [3]
wl WESTG. D., Proc. Phys. Sot. Land. 1951 B64 483. 1151 RANEY W. P., C~RELLI J. C. and WESTERVELT P. J., J. Acoust. Sot. America 1954 26 1006. 1161 DAVIDSON J. F. and CULLEN E. J., Trans. Inst. Chem. Engrs 1957 35 51. 1171 MICKLEY H. S., SHERWOOD T. K. and REED C. E., Applied Mathematics in Chemical Engineering (2nd ed.) McGrawHill, New York 1957. t181 MINER C. S. and DALTON N. N., Glycerol. Reinhold, New York 1953. [I91 WINT A., Ph.D. THESIS, University of Cambridge, 1960. PO1 JEFFERSON T. B., private communication. PII HOL~MARK J., JOHNSEN I., SIKKELAND T. and SKAVLEM S., J. Acoust. Sot. America 1954 26 26.
Resume-L’auteur a Ctabli une equation qui prevoit la vitesse d&change massique ou thermique entre un cylindre oscillant d’un mouvement harmonique simple et un milieu infini de fluide. 11 montre que, dans certaines circonstances, le mecanisme du transfert est semblable a celui de l’echange de chaleur entre un corps chaud et un 6coulement fluctuant, en ce que la majeure partie de T&change se produit dans un flot en 6coulement permanent. L’auteur a compare i’equation avec les resultats experimentaux obtenus dans le systeme acide benzolque/solution glyctrine-eau et a montre qu’il y a un accord raisonnable. Les coefficients d&change massique obtenus avec vibration sont jusqu’a 28 fois plus forts que ceux dus a la convection seule.
800
Printed
in Great Britain.
Mass (or heat) transfer from an oscillating cylinder G.
J. JAMESON
Department of ChemicalEngineering, Universityof Cambridge (Received 13 December 1963; in revisedform 26 April 1964) Abstract-An equation has been derived for predicting the rate of mass (or heat) transfer from a cylinder oscillating with simple harmonic motion in an infinite body of fluid. It is shown that in certain circumstances the mechanism of the transfer is similar to that of heat transfer from a hot body in a fluctuating stream, in that the bulk of the transfer takes place in a steady streaming flow. The equation has been compared with experimental results obtained in the system benzoic acid/ glycerol-water solution, and shown to be in reasonable agreement. The mass transfer coefficients obtained with vibration were up to 28 times higher than those due to free convection alone. INTRODUCTION
THE effect of vibration on heat transfer from cylinders has been extensively investigated in recent years. In some cases, the solid was oscillated in an essentially stagnant fluid; in others, an oscillation was imparted to the fluid in contact with the body. The field in which most interest has been displayed is that in which a sound wave is passed through the fluid (generally air) surrounding a hot wire or cylinder, or through a fluid which is passing through a heated tube. Thus, KUBANSKII[l, 21 has studied the effect of stationary sound fields on free convection from an electrically heated horizontal cylinder in air, with the direction of oscillation parallel to the axis of the cylinder. The measured increase in the heat transfer coefficient due to the vibration was approximately 75 per cent. When a cross flow of air corresponding to Reynolds numbers between 1450 and 1770 was passed over the horizontal cylinder, KUBANSKII [3] was able to increase or reduce the heat transfer coefficient by moving the cylinder relative to the nodes of the stationary sound field, which was perpendicular to both the axis of the cylinder and the direction of the air flow. The maximum improvement in the observed coefficient, relative to that with no vibration, was about 50 per cent. Heat transfer from a wire vibrating at subsonic frequencies in stationary air was investigated by LEMLICH[4]. He presented a film model for the
heat transfer process which is applicable on a gross scale, but is rather unsatisfactory for the region close to the wire, where the flow configuration is an important factor. Similar work has been reported by ANANTANARAYANANand RAMACHANDRAN [5],
VAN DER HEGGE ZIJNEN [6],
LEMLICHand LEVY [7]
and FAND and KAYE [8]. Studies on the effect of vibrations on heat transfer to liquids have been relatively few. MARTINELLI and BOELTER[9] used a heated $ in. dia. horizontal tube oscillating in water at frequencies up to 40 cycles/set. The theoretical basis these workers gave to their work is oversimplified, however, and the validity of their experimental results has since been questioned [4]. In an excellent experimental paper, DEAVER, PENNEYand JEFFERSON [lo] considered heat transfer from an oscillating horizontal wire to water. They were able to correlate their results, with some success, by the simple expedient of forming dimensionless groups similar to those used in correlations for combined free and forced convection in steady flow past a cylinder. Detailed literature surveys on the subject have been given by LEMLICH[ll], and FAND and KAYE [8]. Despite the volume of literature on this topic, it seems that little progress has been made toward providing a satisfactory theoretical background for the experiments reported. It is clear that the general process of heat transfer to or from a body oscillating in a fluid may occur either by free convection, by
t Presentaddress:CaliforniaResearchCorporation,Richmond,California,U.S.A. 793
G. J. J-N forced convection, or by a combination of the two. A complicating factor however, is the compressibility of the fluid concerned. Thus, the equations governing the transfer of heat to gases will probably be different from those which apply to liquids, especially at high Mach numbers. Further, if the fluid is compressible, the equations may be different depending on whether the fluid oscillates about a fixed cylinder, or the cylinder oscillates in a stagnant fluid. The purpose of this present paper is to present a theoretical analysis of one aspect of this complex problem: mass or heat transfer by forced convection alone from a cylinder oscillating in an incompressible fluid. An equation based on an approximate boundary layer analysis of SCHLICHTING[13] is developed for predicting heat or mass transfer rates under these conditions. The equation, expressed in dimensionless form, has been verified experimentally by observing the rate of mass transfer from a benzoic acid cylinder oscillating in a glycerol-water solution.
of the axis of oscillation. The existence of this flow was demonstrated practically by SCHLICHTING,and has also been shown to exist in compressible fluids [14, 151. The velocity distribution derived by SCHLICH~NG may be represented (to the second order approximation) by the following stream function: * = $1 -t *z =-
2v 112 U&)F,($e’“’ n
0
+
FIG. 1 Curvilinear co-ordinate
0
-dxX n x CF2dv) + ~22(r)e2’“‘l
(1)
U,,(x), the amplitude of the periodic potential flow velocity, is 2nA sin (x/R) for a circular cylinder oscillating along a diameter with simple harmonic motion. The coordinate system is shown in Fit. 1. The terms F,, FzO, and Fz2 are functions of the dimensionless parameter 9 = y(n/2v)1’2 : F,
system.
2” 1/Z U,(x) dU,(x) n
FLUID DYNAMICS An analysis of the equations of motion around a cylinder oscillating in an incompressible fluid has been given by SCHLICHTING[12, 131. A method of successive approximations was used, based on boundary layer theory. It was shown that as well as the periodic flow pattern one would expect, a steady secondary flow is induced by the interaction of viscous and inertial effects in the boundary layer. The secondary flow streams in from the bulk of the fluid near the plane of the equator of the cylinder, passes over the surface, and leaves in the direction
+
=
-
(cq2
(1
-
,-(1+04)
+ q
(2)
13 3rl F 20 = -8 - T - &eSzq - se-” cos q - e-v sinq-fqe-“sinq F
22
_
(I + ‘) ,-(l+i)n,'Z + 2 4J2
2
,-(l+jjq
(3)
(1 + i) (4)
- 4J2
The term in II/ which is independent of time (the term associated with Fzo) gives rise to the steady streaming flow over the cylinder. It is a requirement of the boundary layer method used to calculate $ that the vibrational Reynolds number 2nAR/v and the length ratio R/A should both be large. Since $2 is of order A/R times til, the magnitude of the steady streaming flow is considerably less than the maximum value of the major oscillating flow component. The flow around an oscillating cylinder has also been investigated by HOLTSMARK et al. [21]. Where SCHLICHTINGsolved the boundary layer equations by successive approximations, these authors started from the Navier-Stokes equations. Their solutions are correspondingly more accurate, but since it is
794
Mass (or heat) transfer from an oscillating cylinder
not possible to express them in a simple general form, they are rather inconvenient to use in the present problem. Thus, when required, the flow profiles given by SCHLICHTING’Smethod will be used.
Since the velocity gradient may be assumed constant for any particular 8, the velocity may be written u = By, where /? is the velocity gradient, and we may define a stream function per unit length of cylinder as u dy = 38~~
i=so’
MASS (OR HEAT) TRANSFER
The mechanism that will be assumed to dominate transport from the oscillating cylinder is that of transfer into the steady streaming flow. This will only be valid when the time of an oscillation is very small when compared with a characteristic diffusion time of the system, which we take to be ~3~10, where 6 is the boundary layer thickness. Since 6 is of order (~/n)‘/~, this means that the steady flow mechanism will apply when the Schmidt number v/D is large. Now the equation for mass transfer into the two-dimensional steady boundary layer is:
If the thickness of the imagined boundary layer in which all the mass transfer occurs is A, $a = $fiA2 and 1+4= I,~~Y’, where y/Y = A. Thus, if $ is constant, so also is Y. Equation (6) may then be written
To transform this equation into a tractable form, we let dd
R
de
PA3
A=-
U?Z+$=D$ OY with boundary conditions c = 0 at x = 0 and at y = 00, and c = c* at y = 0. (The concentration in the oncoming stream is taken as zero for convenience.) Since we are already restricted to the case of large Schmidt number, or, in other words, a system where the diffusional boundary layer is considerably “thinner” than the velocity boundary layer, we may simplify this equation considerably. Very close to the body, the velocity component u will be negligible so that the term 2,&/8y may be neglected. The simplified equation is then
where 4 is a function of 8 only, equal to zero at 8 = 0. The differential equation then becomes
y&,=D($)+
(7)
with boundary conditions Y=O:
c=c*
Y=l:
c=o
+=o:
c=o
where c* is the saturation concentration of the solute. This equation may be solved by the introduction ac a2C UY-=DT (5) . , of a new variable s = Y/@/3 (Ref. [17]), since ax af substitution reduces equation (7) to the readily Also, in the layer very close to the body, the soluble form velocity gradient au/ay may be assumed constant. The method now used follows that of DAVIDSON d2c 2 and CULLEN [16], who were concerned with the solution of a slightly soluble gas in a film of liquid with boundary conditions s = 0, c = c* ; s = 00, flowing over a sphere, as a method of determining diffusion coefficients. By carrying out a mass c = 0. The solution of the equation is balance along a streamline of value $, they derived m e-s’3/9D ds’ an approximately equivalent form of equation (5),
~+gg=o
c
u(g),=RD($),
-=
(6) 795
c*
s s s
me-s”J9D
0
&
G. J. JAMFSON
The integral may be evaluated by gamma functions, to give a value of 1.198. Then, from (8), D2/3nr/2A2/3 k = o.746 ~2/3,,1/6
or we may write _
dc z
=
C*e-~'3/9D
me-s'",9DdSI s 0
The denominator may be expressed in terms of the gamma function I(+). Evaluation at c = 0 yields 2t3c*
(9D) s=.
=
-
3DF(3)
For calculation of the mass transfer rate, we need the concentration gradient at the wall, which is readily shown to be
We may define an overall mass transfer coefficient as
(8) Using the relations we find
d#de
= R/j3A3, Jla = @A2,
or rearranging in dimensionless form, Sh = 0.746 Re”2Sc”3(A/R)“6
(11)
The analogous heat transfer equation Nu = 0.746 Re’/’ Pr1/3(A/R)‘/6 would be equally valid providing the same assumptions held, i.e., that the Prandtl number was large, and that flow conditions were such that a boundary layer solution would apply. We have already remarked that for the boundary layer solution to apply, Re and R/A should both be large, since it is only in this case that the series solution given in equation (1) can be convergent. A further condition which must be met in the present analysis is that the radius of curvature of the cylinder should be large compared with the boundary layer thickness, since it is only under these circumstances that the curvilinear coordinate system may be used. This condition will be satisfied if nR’/v is large, but since nR2/v is equivalent (disregarding constants) to the product of Re and R/A, both of which are already assumed large, this is not really a further restriction. EXPERIMENTAL
where
The method used to experimentally investigate mass transfer from an oscillating cylinder was the dissolution of benzoic acid in a glycerol-water solution. This method was chosen because the benzoic acid is relatively insoluble in the liquid, so that the effects of natural convection are minimized. Furthermore, benzoic acid is readily formed into spheres or cylinders, and the process may be followed simply by measuring the loss in weight of the body after a known time. The cylinder was formed by first pressing benzoic acid (A.R. grade) in a spherical steel die, preheated to lOo”C, at 5 tons/in2. This produced spheres of about 3 cm dia., from which cylinders were turned. The cylinders formed were 2.1 cm long, 1.10 cm dia. They were mounted between perspex cylinders of the same diameter and length, in order to minimise end effects.
For the velocity gradient j$, we now go to equation (l), since
with only the steady part of II/ being considered. Thus,
Substituting in [9], R [nA” (:)“‘I do = (2$A)3’2 2R
1’2Jysinli2
20 de
796
Mass (or heat) transfer from an oscillatingcylinder
A sketch of the apparatus for oscillating the cylinder is shown in Fig. 2. The oscillation was produced by the crank-connecting rod system in which the length of the connecting rod was sufficient to produce essentially simple harmonic motion of
FIG.2 the
Diagram of apparatus
cylinder. The amplitude could be varied by altering the position of the crank, which was held in a milled slot. The crankshaft was belt driven from a Carter variable speed drive (with power input from a 4 h.p. motor), the variable speed unit being mounted on a separate frame from the oscillating cylinder mechanism, to minimise extraneous vibration. Speed measurements were by stroboscope, or by a Smiths tachometer at low speeds. The stroke (= 2x amplitude) was measured by a travelling microscope. Experiments were run at three levels, where A/R was respectively 0.102, O-198, and 0.298.
Prior to the actual experiments, the cylinder was placed in a saturated solution of benzoic acid in water, and then put under vacuum. This removed most of the trapped air in the interstices of the body, the voids being filled with liquid when atmospheric pressure was restored. Each cylinder was brought to constant weight by repeated immersions, and then kept under saturated benzoic acid solution. No difficulty was experienced in measuring the loss in weight of the cylinder during the course of the experiments. By careful turning, a high polish could be produced on the surface of the benzoic acid, allowing surface moisture to be removed by gently wiping with an absorbent towel. The weight readings were reproducible to within f0.0005 g; dissolution times were such as to give chahges in weight of at least 0.01 g. The liquid used in the experiments was a glycerolwater solution, maintained at a temperature of 23 k 0.1°C.. The density of the solution was 1.237 g/cm3, and the kinematic viscosity 1.55 cm”/ set (from the Tables of MINER and DALTON [18]), both at 23°C. The solubility and diffusivity data were taken from values given by WINT [19]. For densities of glycerol-water solutions of 1.149, 1.198, and 1.233 g/cm3, he gave values for the saturation concentration of benzoic acid of 8.3, 12.9 and 20.4 g/l of solvent, and diffusivities of 1.009 x 10e6, 0.281 x lo-‘j, and 0.0598 x 10V6 cm2/sec, respectively. The values used in present calculations were : c* = 22.5 g/l, and D = O-048 x 10e6 cm’/sec, so that the Schmidt number was 32.3 x 106. An experiment was carried out using the same glycerol-water mixture to determine the magnitude of the Sherwood number for free convection alone from the benzoic acid cylinder. The value found was Sh = 109. This may be compared with the range of Sherwood numbers found in the forced convection experiments, where the minimum was 275, and the maximum 2850. The natural convection value is therefore considerably smaller than those due to forced convection over most of the Reynolds number range investigated. RESULTSAND DISCUSSION
The results are shown (plotted in dimensionless form) in FIG. 3, together with the theoretical line 797
G.
J. JAMFSON
0.1
,
I.0
100
Re
FIG. 3
Comparison
of experimental
from equation (11). The experimental points are in fair agreement with the predicted, but they are, in every case, too high. One probable cause of this discrepancy is the neglect of the term u &/L+ in writing equation (5). While this term is very small over most of the cylinder, it is very large where the flow impinges on the surface, and where it departs. Continuity requires that in these areas, u approaches zero, while o approaches the main stream velocity. A second source of error lies in the boundary layer analysis of SCHLICHTING. Detailed calculations of HOLTSMARK et al. [21] have shown that the magnitude of the steady streaming flow is somewhat underestimated by the boundary layer theory, indicating that the velocity gradient given is also probably too low. A low value of the velocity gradient p would cause equation (11) to predict conservative results, as would neglect of the term v a~/+. The results of LEMLICH [4], and of DEAVER, PENNEY and JEFFERSON[lo] cannot be used to check equation (1 l), since in their experiments, the ratio AIR was considerably greater than unity, so that
results and theoretical prediction.
the present theory is not applicable. Nevertheless, DEAVER et al. found that at high Reynolds numbers, where the forced convection mechanism predominates, the Nusselt number was proportional to Re1’2Pr1i3, with a considerable scatter in the experimental data. They have since found that by inclusion of the dimensionless group AIR in the correlation, the scatter can be considerably reduced [20]. FAND and KAYE [8] studied the transfer of heat to air from a horizontal cylinder $ in. in dia., which was placed in a horizontal sound field normal to the axis of the cylinder. The wavelength of the disturbance was, in most of their experiments, large compared with the diameter of the cylinder, so that it would be safe to assume that any change in the pressure at the front of the cylinder would be instantaneously transmitted to the rear. The authors chose a rather idiosyncratic method of expressing their results, by plotting the observed heat transfer coefficient as a function of the temperature difference AT, at various values of the “sound pressure level” (SPL), which is merely a function of the peak velocity nA. Their results
798
Mass (or heat) transfer from an oscillating cylinder
showed that at low values of SPL, there was little change in the value of the heat transfer coefficient h, compared with the values obtained for free convection alone. At the highest values of SPL, however, there was an increase in h of a factor of approximately three. Values of SPL ranged from 138 to 151 dB, corresponding to values of nA between 1.79 and 8.16 ft/sec respectively. Now in these experiments, Pr = 0.7 approximately, so that the present theory could not be expected to hold. Nevertheless, some typical results have been plotted in Fig. 3, to see how much in error the predicted values would be. The results shown are taken from Fig. 10 of FAND and KAYE’Spaper, since these results showed the highest increase in the heat transfer rate when compared with the coefficient due to natural convection alone. Results were taken at three values of the “sound pressure level”, and it is observed that at a substantially constant Re, there is quite a range of values of Nu/Pr1/3(_4/R)“6. This is because the Nusselt number shows a considerable dependence on the temperature difference AT, indicating that free convection was still an important factor. Free convection is at a minimum however, when AT is smallest, i.e., for the lowest values of Nu/Pr1/3 (A/R)1’6, where AT was about 20 F”. Using these points as being representative of forced convection alone, one sees that they are rather lower than the values predicted by the “large Pr” theory. The experimental results show that the mass transfer coefficient with vibration was up to 28 times greater than without vibration. This factor is considerably higher than those reported by other workers [l-8] for heat transfer, a reflection of the rather low free convection mass transfer rates in the benzoic acid-glycerol solution system. This suggests that any practical use of vibrations to increase mass or heat transfer rates is likely to be most effective where natural convection rates are lowest, e.g., in solidliquid extraction.
the surface of the cylinder, but by a steady streaming flow induced by the oscillations. Providing Re and SC are large, and A/R is small, the equation which describes the mass transfer rate is Sh = 0.746 Re1i2Sc’13(A/R)116 This equation has been shown to be in reasonable agreement with experimental data, giving rather conservative predictions. There is no information available to test the analogous heat transfer equation, but at low Prandtl numbers, the equation has been shown to overestimate the heat losses. Experimental mass transfer rates with vibration were up to 28 times the free convection mass transfer rate, suggesting that vibrations could be used to increase mass transfer coefficients in operations such as solid-liquid extraction. Acknowle&ement-The author wishes to express his gratitude to DR J. FT DAVIDSONfor much helpful guidance and discussion.
NOTATION A C C*
D : P: R Re Sd Sh SPL ; u V x
Y F 8,: e
CONCLUSION It has been shown that under certain conditions, mass transfer from a cylinder oscillating with simple harmonic motion in a large body of fluid is dominated not by the periodic velocity components near D
799
Amplitude of oscillation Concentration of solute Saturation concentration of solute Diffusion coefficient Heat transfer coefficient Mass transfer coefficient Circular frequency Prandtl number Radius of cylinder Reynolds number, 2nA/v Y/Q* Schmidt number, v/D Sherwood number, 2kRID Sound pressure level Time Temperature Velocity component parallel to solid surface Velocity component normal to solid surface Distance parallel to solid surface Distance normal to solid surface YIA Velocity gradient Gamma function Boundary layer thickness Angular displacement from axis of oscillation Kinematic viscosity Liquid density Function of 6 Stream function Value of $I at outer edge of boundary layer
G. J. JAMESON
REFERENCES KUBANSKII P. N., J. Tech. Phys. (Moscow) 1952 22 585. KULIANSK~P. N., Dokl. Acad. Nauk, 1952 82 585. KUBANSKII P. N., J. Tech. Phys. (Moscow) 1952 22 593. [4] LEMLICH R., Industr. Engng Chem. 1955 411175. [5] ANANTANARAYANAN R. and RAMACHANDRANA., Trans. Amer. Sot. mech. Engrs 1958 88 1426. [6] VAN DER HEGGEZIJNEN B. G., Appl. Sci. Res. 1958 Al 205. [7] LEMLICH R. and LEW M. R., Amer. Inst. Chem. Engrs J. 1961 7 241. [S] FANLI R. M. and KAYE J., Trans. Amer. Sot. mech. Engrs 1961 83 C 133. [9] MARTINELLI R. C. and BOELTER L. M. K., Proc. Fifth Intern. Congr. Appl. Mech. 1938 578. [lo] DEAVER F. K., PENNEY W. R. and JEFFERSON T. B.. Trans. Amer. Sot. mech. Enars 1962 84 C 251. iiij LEMLICH R., them. Engng 1961 68 171. WI SCHLICHTING H., Boundary Layer Theory (4th ed.) Chap. 11. McGraw-Hill, New York 1960. [131 SCHLICHTINGH., Phys. 2. 1932 33 327.
[l] [2] [3]
wl WESTG. D., Proc. Phys. Sot. Land. 1951 B64 483. 1151 RANEY W. P., C~RELLI J. C. and WESTERVELT P. J., J. Acoust. Sot. America 1954 26 1006. 1161 DAVIDSON J. F. and CULLEN E. J., Trans. Inst. Chem. Engrs 1957 35 51. 1171 MICKLEY H. S., SHERWOOD T. K. and REED C. E., Applied Mathematics in Chemical Engineering (2nd ed.) McGrawHill, New York 1957. t181 MINER C. S. and DALTON N. N., Glycerol. Reinhold, New York 1953. [I91 WINT A., Ph.D. THESIS, University of Cambridge, 1960. PO1 JEFFERSON T. B., private communication. PII HOL~MARK J., JOHNSEN I., SIKKELAND T. and SKAVLEM S., J. Acoust. Sot. America 1954 26 26.
Resume-L’auteur a Ctabli une equation qui prevoit la vitesse d&change massique ou thermique entre un cylindre oscillant d’un mouvement harmonique simple et un milieu infini de fluide. 11 montre que, dans certaines circonstances, le mecanisme du transfert est semblable a celui de l’echange de chaleur entre un corps chaud et un 6coulement fluctuant, en ce que la majeure partie de T&change se produit dans un flot en 6coulement permanent. L’auteur a compare i’equation avec les resultats experimentaux obtenus dans le systeme acide benzolque/solution glyctrine-eau et a montre qu’il y a un accord raisonnable. Les coefficients d&change massique obtenus avec vibration sont jusqu’a 28 fois plus forts que ceux dus a la convection seule.
800