Vaporization of a spinning fuel droplet

Twenty-Fourth Symposium (International) on Combustion/The Combustion Institute, 1992/pp. 1483-1491

VAPORIZATION

OF

A SPINNING

FUEL

DROPLET

DAVID LOZINSKI AND MOSHE MATALON Dept. of Engineering Sciences and Applied Mathematics McCormick School of Engineering and Applied Science Northwestern University Evanston, 1L 60208 USA

The influence of rotation on droplet vaporization is investigated theoretically. A single, spherical liquid fuel droplet, spinning at a constant angular velocit~ about its own axis is considered. The analysis assumes that the rotational Reynolds number is small and that quasisteady conditions prevail in the gas phase. Thus, the swirling flow field and the accompanying heat transfer process are obtained as modifications of the classical solution of a stationary vaporizing droplet. It is found that the induced secondary flow, inwards towards the droplet poles and outwards from its equator, enhances the vaporization rate and shortens the droplet lifetime. An explicit expression is obtained for the correction to the evaporation rate and consequently a modification to the d2-1aw is derived. For most practical cases, the dependence of the evaporation rate on the rotational Reynolds number, Rer, and on the Prandtl number, Pr, is approximately of the form (Rer)4 Pr 7/5. Finally, an explicit expression for the torque exerted on the evaporating droplet compared to that exerted on a solid sphere is derived.

Introduction A liquid fuel injected into a combustion chamber is often disintegrated into numerous small droplets by means of atomizers. Either because of the atomizer design and/or because of the highly turbulent ambient gas, the droplets may acquire an angular velocity about their own axis. In dilute sprays, droplet collisions are infrequent and the behavior of individual droplets becomes of primary interest. This study examines the effect of rotation on the behavior of an individual droplet. The focus in this paper is on the vaporization process. Indeed, when the liquid fuel is dispersed into very fine drops, it will vaporize completely, and mix with the oxidizer in a preheat zone. The resultant is a typical premixed flame, where the fuel supply depends on the rate of vaporization of the droplets. For larger droplets, combustion takes the form of an envelope diffusion flame; this will be discussed in a subsequent paper. Measurements on the heat transfer from a rotating solid sphere 1 and the mass transfer rate to a rotating solid sphere 2 have been previously reported. The only study on evaporatin~g droplets, however, is that of Pearlman and Sohrab. They have experimentally examined the effects of rigid body rotation on the rate of vaporization of heptane droplets in a stagnant environment. Their results indicate that rotation enhances the rate of droplet evaporation and thus shortens droplet lifetimes.

This paper investigates theoretically the influence of rotation on droplet vaporization. The analysis is carried out in the low Reynolds number regime, where the process is diffusion controlled. A complete description of the flow field and of the heat transfer process is given, as well as the modification of the d2-1aw for spinning droplets. The assumption is well within the limits of the experiment3 reported above; and indeed, the present results are consistent with their observations.

Problem Formulation A single, spherical, liquid fuel droplet is vaporizing in a hot gaseous environment of temperature T=. The droplet is spinning at a constant angular velocity g2 about its own axis, but the relative translational velocity of the droplet with respect to the ambient gas is negligible. As a result, the velocity field remains axisymmetric, as do the processes of heat and mass transport. It is also assumed that the liquid surface tension is large enough that the droplet maintains its spherical shape. For simplicity, the liquid within the droplet is assumed to be incompressible and held at a constant and uniform temperature 7"~.< T,. As a consequence, apart from the rotational motion, heat and mass transport in the liquid phase are completely neglected and only the gas phase is considered. Owing to the disparity between the liquid and gas

1483

1484

DROPLETS AND SPRAYS

densities, quasi-steady conditions are attained in the gas phase. That is, the surface of the droplet recedes very slowly while an O(1) mass flux of fuel vapor leaves the droplet. The gas phase processes are thus described by the steady conservation equations. The surface regression rate is then determined by an overall mass conservation of the droplet. A fixed spherical coordinate system is used, centered at the centroid of the droplet. The radial distance is denoted by r, ~b is the collatitude and O the azimuthal angle. The components of the velocity 6 are then denoted by (u, v, w) respectively. The axis ~b = 0 is aligned with the axis of rotation. Axisymmetry implies 0/00 = 0. In this fixed frame, the flow within the droplet interior corresponds to a rigid body rotation: d)=Orsin~b,

~=~3=0

(1)

= g~Ptr~sinZ6/2 + f)o

(2)

where I)1 is the liquid density. The time-dependent additive function #o is determined by balancing the normal stress components at the liquid/gas interface, as remarked later. This interior solution is only needed to appropriately determine the boundary conditions at the droplet surface. For nondimensionalization, the initial droplet radius, a, is used as a unit of length. Temperature and density are scaled with their values in the far field, T= and p=. The characteristic velocity is taken to be the diffusion velocity, K/a, where K = A/ p=Cp is the thermal diffusivity with cn the specific heat at constant pressure and A the thermal conductivity. The unit of time is (a2/K)(pl/p~), where the density ratio has been introduced in the scaling in keeping with the quasi-steady approximation. 4 The fuel mass fraction Y is normalized with respect to its value within the droplet. Since the resulting gas velocities are typically small compared to the speed of sound, the process is nearly isobaric. Consequently, the small pressure deviation from the ambient pressure, P=, is non-dimensionalized with respect to (K/a)2p~ which is of the order of a representative Mach number squared. The dimensionless conservation equations of mass, momentum and energy in the gas phase, supplemented by the equation of state, become: 7 . (p6) = 0

(3)

pg- V6 = - V p + er (7z6 + 7(V" 6)/3)

(4)

pS. VT = V2T

(5)

/96" VY = Le-lV~Y

(6)

pT = 1.

glected as a consequence of assuming P=/pz ~ 1, all variables depend on time, t, parametrically. The Prandtl number, Pr = becp/A, where be is the viscosity. The Lewis number, Le = K/D=, where D is the mass diffusivity. The following boundary conditions are imposed as r-'-~ ~ 6---->0,

p--->0,

T->l,

Y--->0,

p---->l.

At the droplet surface r = rs(O,

OT/Or= Lpu,

T = Ts

Le-lOY/Or = -pu(1 - Y) w = ~orssin~b, v = 0, which are statements of mass and energy balance across the surface, and of no relative motion of the two fluids at the surface. In these expressions, T~ = Ts/T=, L = L/cpT| where f~ is the latent heat of vaporization, and ag2

~o = K/a

Tangential Velocity Radial Diffusion Velocity

is the reciprocal of a Rossby number. The balance of tangential stress components at the droplet surface is of no additional consequence except for the determination of the O(p=/pl) correction to the flow field in the droplet interior. Assuming that an appropriately defined Weber number, representing a ratio of the dynamic to the capillary force, is small (as when the surface tension is high), the balance of the normal stress components yields that the droplet maintains its spherical shape, and also determines ~o in Eq. (2). The above conditions are sufficient to determine the flow field for any given re(t). An overall mass conservation across the droplet yields

d(47rr3/3) = 2~r dt

[r2pU]r=rsincbd~b.

(8)

The right hand side is the total mass leaving the droplet surface per unit time; it is also the net mass flow rate across a spherical shell of radius R > r.~ when evaluated at r = R instead, It is convenient to introduce the radial mass flux m = pu so that Eq. (8) may be written as

(7)

Note that although time derivatives have been ne-

d(r~) dt -

r~ 1 fo ~ [rem]r=r~sin~bd~b.

(9)

VAPORIZATION OF A SPINNING FUEL DROPLET The Flow Field Surrounding the Droplet

p r ~ l 0 (reOtbl~

In the following, the parameter to will be considered small; i.e., the tangential velocity resulting from rotation is smaller in magnitude than the radial flow induced by the vaporization process. For example, in the experiment of Pearlman and Sohrab, 3 a droplet of ~1 mm in diameter was used with an angular velocity of 36 rps; this yields to 0.1. For most practical fuel sprays, however, the mean droplet size is much smaller, approximately 100-200 /zm in diameter, which justifies the assumption of to ~ 1 for a wide range of angular velocities. The solution of Eqs. (3)-(7) is then sought as an asymptotic series in powers of to. All variables, with the exception of w, are expanded in the form u = Uo + to2uz + to4U4 + . . . . whereas the azimuthal velocity has the form w = toWl + to3w3 + ... This parity in the powers of to is to be expected from symmetry arguments. Substituting into the governing equations yields a system of problems at each power of to. In these problems, the equation for w is uncoupled from the remaining equations and can be solved after the other variables have been determined. To leading order, the classical solution of a spherically symmetric stationary vaporizing droplet is recovered: mo = M o / r 2,

vo = O

To = (Ts - L) + Le M~176 YO "~ 1 -

uo = mo/po,

e -LeM~

Po = 1/To

(lO)

with Mo = r~o ln(1 + B) where B is the so-called transfer number given by B = (1 - T~)/L. Eq. (9) can be integrated to produce the familiar ar2-1aw, ~o= 1-

Kt,

K----21n(I+B),

(11)

1485

"

Mo 0 r 3 Or (rib1) = 0

(14)

whose general solution is ~.)I(T') = cl~'e-M~ -

q- c2[-2r + 2(Mo/Pr)

(Mo/Pr)er-1].

(15)

After satisfying the boundary conditions (13), one finds ff~l(r) = r~of(r)/f(r~o) f(r) = 2re -M~

(16)

- 2 r + 2(M0/Pr)

(17)

- (Mo/Pr)~r -~.

Note that tbl decays as r -2 at large r. To higher orders, the problems consist of a system of linear partial differential equations that lend themselves naturally to solutions expressed in terms of the Legendre polynomials Pn(cos~b) and their derivatives with respect to ~b, P" (cos~b). At O(to2), the problem is being driven by the nonhomogeneous terms arising from the centrifugal accelerations w ~ / r and w~ cot~b/r in the momentum equations. The gb dependence of these terms is a combination of the two Legendre modes P0 ~- 1 and Pz =- (3 cos2~b - 1)/2 only. As a result, the solution is expressed in terms of these two modes; the remaining terms vanish identically. Thus u2 = a2(r) + a~(r)e~(cos~)

T2 = :tz(r) + ~2(r)e2(cos6) p2 = ~ ( r ) + ~(r)P2(cos4,) v~ = ~2(r)/'~(cos4,).

according to which the square of the droplet radius (or diameter) decreases linearly in time. 5 The problem for Wl is now (~.~)

Mo 0Wl +

= er

--~ \ 0r

{ ~ 0_ ( re 0wl~ + _ _ 1 O ( sind~ ~ ) 0r \ Or / r e sint~ a6

wl = r~osin~b at r = rso, wl~0

asr~oo.

Substituting into the governing equations, the system corresponding to the P0-mode (identified by the

(13)

The boundary conditions suggest seeking a solution of the form w 1 = I b l ( r ) sin~b. Substituting into Eq. (12) gives

Wl

}

(12)

re s-~nnZt,b

- ) decouples from that of the P2-mode, and can be solved independently. Furthermore, the equation for 1?2 decouples from the system, and can be solved a posteriori. Its solution is similar to that of the temperature field, and will not be further discussed in this paper. It is important to note that, when applying the Legendre expansions to equation (9), only the Po-

DROPLETS AND SPRAYS

1486

mode can contribute to the net flux away from the droplet surface. With any other Legendre mode, the integral on the right hand side of (9) vanishes, owing to the orthogonality of Legendre polynomials. The equations corresponding to the P0-mode,

a(r2~)/ar = 0 -of- +2- - r ~OTo e-

mOor

o~

(18)

l a l f r2 a ~ \ = 0

Or

~r\

-~-r/

(19)

Ouoa~+ -a~

mo - - + Or

Or

_pr(1

Or 0 (r~0fie/

\ ~ a-r \

_~)2

Or i - 2

t~ = 3 Po - f- '

As r ~ 0% all variables tend to 0. Although linear, the system (21)-(25) consists of coupled equations with highly complicated coefficients, and must therefore be solved numerically. The collocation software for boundary-value ODE's, COLSYS, 6 is used. The equations are written as a system of six first order equations for 172, 17~, Tz, ~,z , u2 and (4Pr u~ - 31~z). The conditions at infinity were applied at a sufficiently large r. The results are summarized in Fig. 1 where profiles of fiz, 172 and i"2 have been plotted showing their dependence on the parameters, L, Ts and Pr. The solid curves in the figure correspond to L = T~ = 0.1 and Pr = 0.7, and can be used as a reference to

(20) 0.1

are merely a perturbation of the leading order problem. When the boundary conditions are imposed, it is easily verified that they yield no correction to the flow field; hence lh2 = 1~2 = T2 = 172 = 0. There is only a pressure modification resulting from the balance of the centrifugal forces, 192 = - ( 2 / 3 )

lb~/r

...--'-'----_..

0.0

dr.

Consequently, there is no correction to the overall vaporization rate at this order, and r~2 = 0. The equations for the Pz-mode are lO

- - - (rZrh2) - 6po172 = 0

/ / " ,,,,,," ~j ,:

-0.1

0fi2 mo Or

- -

Or m2

O1~2 + +

r Pr

ar

+ 3-~ fiz + -r --Or O~)2

~2

Pz

r

r

--+mo--+--+Pr

mo Or

8 + mo

ai"z Or

02

i , ,,,,'

[

[

1 aft2 3r Or

4 02fiz

8 off2

3 ar e

3 r Or

2 =

3r

[

OzOe

2 dO2

Or"~

r Or

~ 8 ] = _ poib~ o, az 3r

2

Or \

rh2 = Pofi2 + uolo2,

6

T~

Po~P2 + Tolo2 = 0.

7~2 = 0,

02 = 0.

0.1 0.2

0.2 0.1

0.7 0.7

0.1 0.1

0.08 0.7 0.1 0.6

3

4

5

6

7

8

9

I

10

(22) o,o

(23) -o.1

',

~

u2

.."

(24)

(25)

The conditions at the droplet surface r = r~o (recalling that rsz = O) are OT2/Or = Lr/~z,

Pr 0.7

o.1

-0.2 = 0

0.1

r

OTo Or 1 O (r2M'21

- ....

L

0.1

I;1111~11,111.11111111,11111111LIIIIIIII

I

OUo _ +

----

(21)

r Or - -

',,',:

Ts

(26)

-0.3

II,lllllllllllllllllllllllllllllllillllll[ll

2

3

4

5

6

7

8

I

9

10

r FIG, 1. Velocity and temperature profiles for varying values of the parameters T,, L and Pr.

1487

VAPORIZATION OF A SPINNING FUEL DROPLET identify the effect of varying any one of the parameters. Note that decreasing any of these values causes an increase in the induced velocity and temperature. This change is mostly affected by varying L and Pr. Varying T~ causes relatively minor changes; for example, doubling T~ produces a change of only 10%. The streamlines of the perturbed flow field, which must be superimposed on the radial flow emanating from the droplet, are shown in Fig. 2. The velocity field far from the droplet resembles the secondary flow that develops for a rotating impermeable sphere at low Reynolds number.7 As a result, heat is convected towards the droplet at the poles and away from the droplet at the equator. This causes an excess/reduction in the vaporization rate at the pole/ equator respectively. There is, however, no overall effect at this order.

mo c3~4 + -0:1"o - ~h4 + -rhz - - -0i"z + 6pof)2Tz - Or Or 5 Or 5r

(2s)

=Po-

The boundary conditions, when expressed at r~ = rso, become

T4 + r~4aTo/ar = 0

(29)

cgmo aT4 OeTo Lrh4 + L - Or - r~4 = - Or - + - ~ rs4

(30)

together with 7"4 ~ 0 as r ~ ~. Writing rh4 = M4/r 2, rh4 decays as r ~ + as it should. Integrating Eq. (28) and applying the boundary conditions yields, in addition to determining :]'4, an expression for M4:

Vaporization Rate The effect of rotation on the overall rate of vaporization appears at O(to4). The mass flow rate takes the form m4 = rh4 + rh4 P2(cos~b) + rn4P4(cos~b), which involves the Legendre polynomial/4- However, as commented on earlier, it is only the Pomode that contributes to the net mass flux in Eq. (9). The relevant equations are therefore

O(r2r'n4)/ar = 0

(27)

M 4 ~-~

-M~176 + B)L I'~.4

x [x2rh27~.~+ 6xpoOfl'e]dx + Mo ~'. r~o

(31)

The time dependence may be easily isolated from the integral in this expression, so that M4 = M~4(1 - Kt)9/2 + (K/2)rs4

(32)

where M4~ is the value of M4 when r~o = 1 (i.e. at t = 0). To O(t04), equation (9) becomes

d(r2~rs4)/ dt = - M 4.

(33)

Substituting Eq. (32) into the right hand side and integrating using r~4(0) = 0, yields the correction to the evolution of the droplet radius, 1 ( _ ~ ) [(1 - Kt)5 - 1 ]

DROP~~ FIG. 2. Streamlines, in the radial plane, for the induced flow field.

(34)

The constant M ~ is calculated from Eq. (31) using the O(w 2) numerical solution, and is always found to be positive. Thus, tile correction to the droplet radius is negative at all times. The dependence of the evaporation rate on Pr can be identified by writing M -=- Mo + (Rer)4pr4M4 + ...

(35)

where to = RerPr and Rer = a2g2p=/I,t is the rotational Reynolds number. There is still an implicit dependence on Pr in M ~ which can be isolated by rewriting M ~ = (K/2)Pr~-4M * so that

DROPLETS AND SPRAYS

1488

M = M o { l + (Rer)4pr"M*[ll~0~l Kt)5-

']} oO15M,

T,~.

(36)

The complete dependence on Pr is a power law with an exponent tr, which can be estimated by plotting the exact expression pr4M~4against Pr for various :Is and L as shown in Fig. 3. In a typical combustion chamber, where T~ -- 2300 K and ~P8 is near the boiling temperature, the dimensionless Ts ranges between 0.1 and 0.2. The dimensionless latent heat tends to be smaller for heavier fuels;s L - 0.08 for the heavier fuels such as Kerosene and n-Decane, L ~ 0.1 for n-heptane or n-hexane; and L -- 0.3 for the lighter fuels such as methanol and ethanol. In all these cases and others that were computed (e.g. Ts as low as 0.05), the exponent a varies only slightly with L and Ts and is estimated at 7/5 +0.1. Thus the various curves are closely approximated by (K/2)M*Pr7/5 with M* depending only on L and T~. The constant M* is a measure of the effect of rotation on the evaporation process. The dependence of M* on the transfer number B is shown in Fig. 4 for selected values of Ts and a representative Pr = 0.7. Rotation therefore has a more significant effect on the vaporization process the smaller L and T8 are. The time history of the droplet radius is, correct to 0(0)4),

(r,) 2 = (1 - Kt) -

0.01o :

4 oi P r M4 : A 0.008

0.006

(Re,)4pr~M*[I

-

Kt)5]/5

(1 -

T,L

(37)

A//

B C D

.1 .1 .1 .2

,08 .1 .2 .08

1.31 1.48 1.44 1.28

E

F

.2 .1

.1 .3

1.29 1.46

C H

.2 .2

.2 .3

1.34 1.37

0.004

/

/ /

/

// /

/

/

0.000

5

/

// /

F G

0.002

0.2

0.4

15

FIG. 4. The factor M*, which measures the effect of rotation on the vaporization process, versus the transfer number B for varying values of T,. which is the modification of the de-law in the presence of swirl. The experiment carried out by Pearlman and Sohrab 3 shows that at low rates of rotation, the dependence of the square of the droplet diameter on time is nearly linear during the first quarter of the droplet lifetime. Furthermore, these authors concluded, based on additional observations, that at higher angular velocities the dependence is no longer linear. Both observations are indeed consistent with the time dependence seen in the modified law (37). Finally, the droplet lifetime can be found by evaluating equation (37) with rs = 0, such that (38)

The total torque exerted on the evaporating droplet can be calculated (in dimensional form) from ~"= a 2

0.0

~

Torque E

0.000

10

In summary, rotation enhances the rate of vaporization of the droplet, increases the regression rate of the surface, and shortens the droplet lifetime.

/

/

o.oo

tt = K - i l l - (Rer)4pr"M*/5].

e/ /

/ /

o oo

0.6

0.8

#

(27rrssin~b)(-O're)~ sin~bd~b

(39)

1.0

Pr

FIc. 3. The dependence of PPM ~ on the Prandti number for various L and T,. The overall dependence is of the from Pr"; these curves are used to approximate cr.

where tr,~ = pd2(cgw/Or - w/r) is the frictional force per unit area of the sphere. To leading order,

_

= _1 1

r,of'(r,o)]

Zo

3

[

f(r~o) J

(40)

1489

VAPORIZATION OF A SPINNING FUEL DROPLET where ~'o = 8V/z(arso)312 is the moment exerted on a solid sphere 9 of radius argo. The right hand side of equation (40) is always less than one and approaches one when K ~ 0 (i.e. no vaporization), as it should. The moment on a vaporizing droplet is thus smaller than that on a sphere of the same radius, by a factor that depends on the parameter Pr/K as shown in Fig. 5. In the limit K -+ ~, the torque is 2/3 of its value in the absence of evaporation.

1.0

~/~o

J

0.9

08

0.7

Conclusion A spinning droplet induces a secondary flow inwards towards its poles and outwards from the equator. As a result, additional heat is convected from the ambient to increase the droplet vaporization rate near the poles while reducing the rate near the equator. The overall effect serves to enhance the vaporization rate, and to shorten the droplet lifetime. For most practical cases, the correction of the vaporization rate, due to rotation, is proportional to (Rer)4prT/SM*, where M* depends on the ratio of the surface to the ambient temperature Ts/ T= and on the transfer number B = cp(T| - T~)/ L. The effect of rotation is found to be more significant for heavier fuels, which usually possess a relatively lower latent heat/,; and for smaller temperature ratios, for example when T= is higher, though in this case the Reynolds number is reduced because of the lower gas density. An explicit expression for the evolution of the droplet radius, which is a modification of the d
0

1

2

3

pc/i~( i +8)

FIG. 5. The torque exerted on the evaporating droplet plotted against the parameter Pr/ln(1 + B).

Acknowledgement This work has been partially supported by the NSF under grant number DMS-9104029. D. L. acknowledges support from NSERC Canada.

REFERENCES 1. EASTOI', T. D.: Int. J. Heat Mass Transfer 16, 1954 (1973). 2. NOOaDSH, M. P. AND RoTrE, ]. W.: Chem. Eng. Sci. 23, 657 (1968). 3. PEARLMAN, H. G. AND SOHRAB, S. H.: Combust. Sci. Tech. 76, 321 (1991). 4. MATALON, M. AND LAW, C. K.: Combust Flame 50, 219 (1983). 5. WILLmMS, F. A.: Combustion Theory, 2nd Ed., p. 52, Benjamin Cummings, 1985. 6. ASCHEm U., CRmSTUNSEN]. AND RUSSELL, R. D.: ACM Trans. Math. Softw- 7, 223 (1981). 7. BICKLEY, W. G.: Phil. Mag. 25, 746 (1938). 8. FRISTOM, R. M. AND WESTENBERG, A. A.: Flame Structure, McGraw-Hill, 1965. 9. LaNDaU, L. D. AND LIFSHII"Z E. M.: Fluid Mechanics, 2nd Ed., p. 65 Pergamon Press, 1987.

COMMENTS Roger H. Rangel, University of California at Irvine, USA. How would the internal circulation generated by the variable surface tension compare with the one you would expect from a relative transla-

tional velociD' between the gas and the drop? Could you provide an estimate of their respective strengths based on dimensionless parameters such as Reynolds numbers?

DROPLETS AND SPRAYS

1490

Author's Reply. The internal circulation generated by a variable surface tension is not described in the written version of the paper, though it was briefly discussed in the presentation; further details will appear in a forthcoming publication I. The present paper assumes that the droplet is held at a constant temperature. The more accurate requirement to be imposed at the droplet surface is the Clansius-Clapeyron condition relating the droplet surface temperature to the vapor pressure. As a consequence, there would be variations in temperature along the droplet surface and hence a variable surface tension that produces an internal circulation. This thermocapillary motion is characterized by a Marangoni number M

=

y/~t12

where y = - d ~ / d T with ~r the surface tension and /xt the viscosity of the liquid. The non-radial external gas phase motion in the presence of rotation may also cause an internal circulation due to viscous stresses. Its strength depends on the ratio of the two Reynolds numbers which characterize the liquid and gas phases or ,~ = (u,,/m)(~,Jo~12)

Where /~ is the viscosity of the gaseous mixture and vc is the characteristic velocity in the gas phase. Note that the two parameters M and 6 are independent of one another so that their relative importanee depends on the problem at hand. In the present problem, u~ - a12 so that 6 is typically small owing to the small viscosity ratio. As a consequence, the thermocapillary effect is more important. When there is also a relative translational velocity between the droplet and the gas, characterized by uc, the parameter 6 might no longer be small. The internal circulation in this ease will be affected or dominated by viscous stress effects. REFERENCE 1. LOZINSKI, D. AND MATATLON, M.: Tbermocapillary Motion in a Spinning Vaporizing Droplet, to appear.

Kenneth K. Kuo, Pennsylvania State University, USA. Very interesting paper. I would like to mention two effects which exist in the experiments of Pearlman and Sohrab but are absent in your modeling work. One effect is due to heat conduction from the spinning rod to the liquid droplet. The other effect is associated with liquid motion along the rod under spinning condition. Therefore, special cautions must be made in order to compare theoretical results with expermental data.

Author's Reply. We agree. These two effects may cause some quantitative differences between theoretical predictions and experimental data.

Dr. J. Bellan, Jet Propulsion Laboratory, USA. Your analysis assumes that the drop spins at a constant angular velocity. In fact the angular velocity will decrease as the drop evaporates. This will certainly influence the impact of your findings in terms of enhancement of the evaporation rate. Can you please comment on this? This point is particularly important since it seems that even with a constantant angular velocity, the enhancement (over the non-spinning case) in the evaporation rate is very small. A table of 12 and correspnding per cent enhancement of evaporation rate would be most helpful.

Author's Reply. Our analysis assumes that the droplet spins at a constant angular velocity in accord with the experimental conditions of Pearlman and Sohrab. Of course, in practical systems, the angular velocity of the droplet will decrease in time as a result of viscous damping. A balance of angular momentum, accounting for the angular momentum of the droplet and that imparted to the gas, yields the following estimate for the decay in angular velocity as a function of time 15Pr - -

12

=

120(1

-

(r/~'o).

Kt) K

Recall that z/~'0 is the torque ratio expressed in equation (40), and O0 is the initial angular velocity. The exponent in this expression is of order unity, suggesting that the time scale for the decay in angular velocity is comparable to the droplet lifetime. Thus, the analysis earl be easily modified to accomodate variations in angular velocity by replacing O with the above expression. Indeed, the enhancement in the evaporation rate is predicted by this work to be relatively small in magnitude, approximately as (Rer)4. This prediction does not change when variations of O are accounted for, and the magnitude of the effect will not vary significantly for much of the droplet lifetime. An analytical expression for the enhancement of the evaporation rate as a result of rotation has been derived; see equation (36). This expression depends on such parameters as the Reynolds number, the Prandtl number, the heat of vaporization, and the droplet and ambient temperatures. The per cent enhancement of the evaporation rate can therefore

VAPORIZATION OF A SPINNING F U E L DROPLET be easily determined from this expression for given system conditions.

1. GOkalp, CNRS-LCSR, France. The conclusion of this work which shows that the effect of rotation is more significant for heavier fuels is consistent with our finding concerning the turbulence effect on vaporization (G6kalp et al., Combustion and Flame, July 1992). It is thus important to estimate, in given flow conditions, the relative effects of the mean flow (the Fr6ssling effect), turbulence and rotation, on the

1491

vaporization rate of single droplets, and to establish a phase diagram for possible vaporization regimes. Could you give an indicaton of the relative weight of the rotation effect compared to other convection effects.

Author's Reply. The effect of rotation is characterized by the rotational Reynolds number Rer = a2pl2/ixg, where #~ and p are the viscosity and density of the ambient gas. This should be compared to the effective Reynolds n u m b e r characterizing other convective effects.