Combined heat and mass transfer during bubble absorption in binary solutions

~ ) Pergamon Int. J. Heat Mass Transfer. Vol. 40, No. 3, pp. 589~603, 1997

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights l~-'served

0017-9310/97 $15.00+0.00

PII : S0017-9310(96)00118-4

Combined heat and mass transfer during bubble absorption in binary solutions

T. L. MERRILL Carrier Corporation, Carrier Parkway, P.O. Box 4808, Syracuse, NY 13221, U.S.A.

and

H. PEREZ-BLANCO Penn State University, 338 Reber Bldg, State College, PA 16801, U.S.A.

(Received 22 June 1995 and in final form 29 March 1996)

Abstract--Bubble injection is an effective method for vapor absorption. This analytical study describes bubble behavior from inception to collapse as it translates inside a subcooled liquid binary solution. A finite difference method is employed to solve the governing equations and their associated boundary conditions. Special consideration is given to the complex interface mass and heat transport process. Subcooled liquid temperature and concentration fields surrounding a collapsing bubble are described. Bubble hea.t and mass transport variables are presented. Bubble diameter and mass are also described over

the bubble life span. Copyright © 1996 Elsevier Science Ltd.

'1. I N T R O D U C T I O N

Although our problem deals with the absorption process in absorption machines for heating or cooling, as in Ref. [1], the solution may be of interest to those engaged in the broader problem of absorption of bubbles with subslantial thermal effects. Typical absorbers employ fallirLg films to carry out the absorption process. However,. by breaking up the vapor into small bubbles and injecting them into the liquid, the interfacial area per unit volume of vapor is increased and greater liquid mix:ing occurs at the vapor-liquid interface, resulting in compact absorbers. This study theor- etically analyzes the behavior of bubbles injected into subcooled liquid binary solutions, from inception to collapse. The two components involved in this study are ammonia and water.

Bubble behavior has been studied extensively. Typi- cally, studies involving bubbles have fallen into four categories : fluid mechanics, mass transfer, heat transfer, and coupled heat and mass transfer surrounding bubbles.

Levich [2], Chat) [3], and Moore [4] each developed similar perturbed potential flow solutions for liquid flow past sphericaL1 fluid spheres. Bhaga and Weber [5] confirmed the Levich-Chao-Moore solution for bubble Reynolds numbers as high as 265. For bubble Reynolds exceeding 265, Ref. [5] suggests that the Levich--Chao--Moore solution for liquid flow past spherical fluid bubbles is reasonable.

Numerous studies have been undertaken to describe bubble behavior under conditions of pure mass transfer, with no thermal effect. For this problem empirical

correlations for the mass transfer coefficient have been developed: Griffith [6], Leonard and Houghton [7], Calderbank and Lochiel [8], Johnson et al. [9], and Azbel [I0]. In general these correlations depend on the mass transfer Prclet number, Pem, defined as the product of the Reynolds and Schmidt numbers [11]. Analytical solutions for mass transfer rates based on potential flow around a fluid sphere bubble exist [12, 13].

Early attempts to understand heat transfer surrounding bubbles involved pure substances: either condensation or evaporation of vapor bubbles. Bubble growth in a superheated liquid was treated analytically by Plesset and Zwick [14] for a stagnant liquid. With additional contributions from Forster and Zuber [15] and Plesset and Zwick [16], it was established that the interface temperature rapidly approached the saturation temperature corresponding to the external liquid pressure and that the rate of growth of the bubble was controlled by the liquid side heat transfer. Florschuetz and Chao [17] focused on the relative importance of inertial and heat transfer effects, as well as the effect of noncondensibles on bubble collapse rates. More recent heat transfer studies have focused on bubble condensation in subcooled single component liquids [18-20].

Bubble behavior involving combined phase change and absorption has not been studied extensively. However, combined heat and mass transfer problems for liquid droplets have been studied [21, 22]. Several studies for falling films also exist [23-27].

Absorption of ammonia vapor bubbles into subcooled solutions with low ammonia concentrations

589

590 T.L. MERRILL and H. PEREZ-BLANCO

NOMENCLATURE

a bubble radius [m] 6 Cp specific heat [J kg-1 K- l ] A Bim mass transfer Biot number, hma/D~jot 0 Dij general binary diffusivity coefficient

[m 2 s -l] ~c dr bubble diameter [m] v DAB binary diffusivity coefficient of t9

ammonia in water [m 2 s -q ~b DBA binary diffusivity coefficient of water

in ammonia [m 2 s - q hfg heat of vaporization and dilution ~,

[kJ kg- 1] (~ mass transfer coefficient [m s-1] ~o0 heat transfer coefficient [W m -2 K -I] tangential position col radial position thermal conductivity [W m -1 K -1] bubble mass [kg] co o~ pressure [kPa] heat transfer P6clet number, Utdo/r. mass transfer P6clet number, Utde/Dij V heat transfer mesh P6clet number, ur Ar/~

Pemesh.m mass transfer mesh P6clet number, ur Ar/D~j

Pe*~,h critical mesh P6clet number equal to two

Pr Prandtl number, v/x Re Reynolds number, Utde/v r radial coordinate [m] S_c Schmidt number, v/D~j Sh average Sherwood number, hmde/D~i A T temperature [°C or K] B t time [s] f u velocity [m s-~] ff U far-field velocity [m s-~] g Ut bubble terminal velocity [m s-J] i v velocity component normal to an itf

interface [m s -1] j z coordinate in the direction of the 1

bubble ascent [m]. n r t V 0

hm h i

j

k mbub P Pe eem eemesh

Greek symbols ct distribution function (see Fig. 6) F nondimensional parameter (see

equation (11))

boundary layer thickness [m] difference spherical-polar angle, measured from the forward stagnation point (°) thermal diffusivity [m 2 s-l] kinematic viscosity [m 2 s-1] mass concentration [kg m -3] nondimensional general scalar variable or velocity potential function [m: s -1] stream function [m 3 s-1] liquid mass fraction liquid mass fraction initial condition (see Fig. 6) liquid mass fraction of the continuous region outside the fluid sphere (see Fig. 6) far-field liquid mass fraction of the continuous region outside the fluid sphere (see Fig. 6) for all.

Superscripts * nondimensional or local transfer

coefficient time derivative

n finite difference time level average.

Subscripts ammonia water liquid far-field gas or vapor initial or theta location interface radial location liquid normal direction radial direction or relative thermal vapor theta direction or dependent on the theta location.

requires a complete formulation of a combined heat and mass transfer problem. Both components can be present in both phases at any time. Liquid-vapor phase equilibrium may not exist for both components all of the time. The liquid-vapor interfacial temperature is a consequence of the magnitude and direction of mass fluxes and heats of vaporization and dilution. Vaporization of water into the vapor phase

influences the total amount of heat released by the bubble during bubble life span. Ultimately model results may serve as a design tool for the design of bubble absorbers for cooling/heating absorption machines.

After formulating the mathematical bubble absorption problem, the numerical technique adopted for solution is briefly explained. Since the governing equa-

Heat and mass transfer during bubble absorption 591

tions are coupled at a moving interface, uncommon problems exist. Hence, updating the computational grid and the values corresponding to each node as the bubble shrinks (or expands) is required. Convective effects arising from the radial velocities inherent with bubble collapse require a spatial treatment of the discretized equations. Numerical solutions show subcooled liquid temperature and concentration fields surrounding a collapsing bubble. Bubble heat and mass transport variables are compared to semi- empirical correlations. Direct comparisons between predicted and actual bubble behavior were not conducted.

2. PROBLEM FORMULATION

We describe briefly in this section what is known about the velocky field around collapsing bubbles, and the governing equations for our combined heat and mass transfer problem. The boundary conditions, the kernel of the model, are described in detail. Our approach consisted of adapting the known velocity field to our probh;m, whereby a numerical solution of the governing equations could be found.

Consider Fig. 1, with the spherical-polar coordinate system and the control volume surrounding the bubble under inw,'stigation. Symmetry about the vertical axis is assumed. To facilitate the formulation and solution of the problem, the control volume and the bubble coordinate system are fixed to the bubble center of mass. The origin of the z-axis coincides with the release point of the bubble. Tfr(z) represents the far-field temperature field along the z-coordinate.

In addition to :;ymmetry about a vertical axis, the following assumptions were made :

(1) A spherical[ bubble shape is maintained from injection to collapse, with no surface oscillations. This assumption is warranted by the experimental work of Wittke and Chao [28] and Johnson et al. [9], for bubbles with diameters smaller than 4 ram. With

. . . . . . . . }it ......-*.

lmbblc e m a ~ I

Fig. 1. Bubble dynamics problem coordinate system and control volume.

bubbles of this size surface tension forces dominate over pressure and viscous forces that would tend to deform the bubble.

(2) The pressure on both sides of the bubble surface is constant and equal. Surface tension effects are neglected ; this is a suitable assumption except when the bubble diameter is below 1% of its original value. Since the bubble collapse time is assumed to be short, the change in the external static pressure is negligible.

(3) No interfacial resistance to heat or mass transfer.

(4) The vapor concentration and temperature fields are uniform, since diffusion in the vapor bubble is large compared to diffusion in the liquid phase. Assuming a lumped concentration field in the bubble is reasonable considering the mass transfer Biot number (hma/Do~t) is below the critical value of 0.2. (p. 63 of Ref. [29]).

(5) Inertial effects associated with liquid motion into the bubble are negligible, as per the criteria given by Ref. [17].

(6) Outside the bubble, the flow is irrotational, with a radial inward velocity dependent on the absorption rate. The bubble moves upwards as its terminal velocity, which changes with bubble volume and density, and no separation occurs due to slip condition on the bubble surface (p. 127 of Ref. [29]). The flow field is described with more detail below.

Bridging upon the L e v i c h ~ h a o - M o o r e velocity field previously discussed, Wittke [30] described the potential velocity field about a moving, collapsing bubble. The radial and tangential velocity components are

U r =

and

a~ 1 a¢ ar r: sin 0 ~0

/ a3\ a 2 t ~

- U k l - 7) c°s°+ 7

(1)

uo . . . . . U 1 + sin 0, r a0 r sin 0 0r

(2)

where d represents the radius time derivative, and the term a2a in equation (1) reflects the effect of a collapsing vapor-liquid interface. Implicit in equation (1) is the condition that the liquid adjacent to the interface and the interface itself move at the same velocity a. Clearly, if mass transfer is occurring across the interface, the liquid velocity and interface velocity need not be equal. Hence, equation (1) must be revised. Our revision is given together with the interfacial boundary conditions, following the introduction of the governing equations below.

The energy equation for the fluid surrounding the bubble is given by :


OT OT pCpU 0 aT pcp ~ @ pCpU r ~r ÷ - - r --00

r z Or kr2 ksm 0 (3) -- r 2 sin 0

which under the assumption of constant properties, and the following thermal boundary layer approxi- mation shown to be valid for most of the bubble residence time [31] :

/ O a 2 ~ /1 0 1 0 2 \ >> k + r f f ~ ) (4) ~20r+r~r2) r O-O

reduces to

aT aT uo aT v [1 3 [" 2 aT'~l 0 - ~ + U ' ~ r -t r a0 Pr ~ 7 3 r k r ~-r)J" (5)

Introducing the following definitions of non-dimensional variables,

Ur UO r* -- r Ur *=- ~ t ( t ) , 1,l~ -~ Ut(t----~, a( t ) '

Vt (t) t T(r, O, t) t * - - - a ( t ) ' and T*o Ti (6)

and dropping the asterisk (*), the final dimensionless form of the energy equation is

aT 2 [I a (r2a_T'll (uaT+uo_~ a t - P c ~SOr \ ~ - r } J - \ ar r . (7)

Note that the heat transfer Prclet number, Pe, the product of the Re and Pr numbers, controls the influ- ence of diffusion on the unsteady temperature profile inside the thermal boundary layer. As the bubble nears complete collapse both the bubble radius and the terminal velocity approach zero. Hence, the heat transfer P6clet number also approaches zero, and diffusion dominates over convection when the bubble is near collapse.

The mass fraction (co) of ammonia or water surrounding the bubble responds to the conservation equation :

Oco aco Uoaco 1 a ( 23co) - - +Ur-~-- r + -- Dor at r aO r ~ &

o( .

1 Dis sm 0 -ffff (8) + r 2 sin~ 0-0

Since the summation of ammonia and water mass fractions equals one, the gradient of one species equals the negative gradient of the other species. Equation (8) can be expressed in an analogous form to equation (7), with :

co, = co(r, 0, t) (9) coi

The final nondimensional form is (the asterisks have been dropped) :

aco+u@ + - - Tr at r aO ReSc

(10)

A nondimensional scalar variable q~ and a characteristic nondimensional variable F can be used to express equation (7) and (10) in a general nondimensional form :

as uo as 2 [ i 0 {r2° l N ÷ur + r -O0-FE r2 ar k Or]J ( I I)

where ~b may equal T/T~ or co/coi and F may equal RePr or ReSc.

To complete the mathematical formulation, boundary conditions must be specified on three boundaries of the liquid field, see Fig. 1 : at the interface, far from the interface or far-field, and along the symmetry axis. Although the symmetry and the far-field boundary conditions are conceptually straightforward, the interface boundary conditions require a more in-depth explanation: these interface boundary conditions couple the governing equations.

Figure 2 outlines conceptually our approach to a well-posed problem. The following initial conditions were adopted :

at t = 0 , T - T i and co=co i V0 and r

(12)

where the subscript i represents the initial value at time equal to zero.

At time greater than zero, far from the bubble interface, the liquid surrounding a bubble inside an absorber becomes subcooled and therefore equilibrium can not be assumed. Instead we make the assumption that the concentration does not vary appreciably far from the bubble interface :

&o for t > 0 , as r ~ m , ~ - r ~ 0 , V0. (13)

A far-field condition for temperature is of the Dirichlet type,

for t > 0 , as r ~ o o , T = Tff(z), V0

(14)

where the temperature may be constant or can be easily determined from experimental data [32]. For both the temperature and mass fraction fields, the symmetry condition is of the Neumann type :

for t > 0 , for r>~a, 0 = 0 or It,

aT aco 0. (15)

30 a0

A moving interface, across which one component is condensing wherea.s the other is evaporating must meet three criteria, see Fig. 3: mass conservation, energy conservation, and vapor-liquid equilibrium.


Governing Unsteady [ Partial Differential Equations

/

t.nporat r 1

\ Boundary Conditions ]

syrnrnetry, at 0 = 0.0 and x . )

mass fraction]

- - ' ( : fa r field, r - - - > % o )

~ temperature ] mass fraction]

(bubble interface, r = a )

mperature ~)

quilibrium [ mass balance and energy balance

"~ mass fraction ) Fig. 2. Conditions necessary for a well-posed problem.

In Fig. 3 the double arrows illustrate the complex counter-diffusion phenomenon occurring during binary solution bubble collapse, where one species may be entering or leaving the bubble interface depending upon tile interface conditions ([33] and p. 445 of Ref. [34]).

Mass conservation leads to an interesting modi- fication of equation (1) to account for convective effects during bubble collapse. Instead of assuming

phase boundary

a ( t ) ' ~ ~ n d ~ / ~ - qeonduetion

- I Fig. 3. Energy balance at the bubble interface.

that the liquid velocity at the interface equals t~, as was the case in deriving equation (1), the liquid velocity must be determined by a mass balance at the interface :

pv(v~., --~t) = p, (v,.n - 4 ) . (16)

Solving for the liquid normal velocity at the interface, and neglecting the vapor velocity as compared to the interface velocity [35, 36], we find that

o = Ur ---- 0 ( 1 - - 17,

Equation (17) shows that the interface and liquid radial velocity may indeed differ from each other, and hence the velocity potential leading to equation (1) must be modified, to give :

~ ) COS

(18)

The radial velocity thus correctly accounts for convective effects close to the interface [31]. Also,


conservation of mass of each species must hold at the interface, leading to the following:

pA.~ @~,° - a) = - p . ~ G 7 [,~, + p ' ` @' '° - ~)

and

(19)

- = - - S ; i , , f - p . ( v , , . - a )

(20)

where A stands for ammonia and B for water. It should be noted that equation (16) is the sum of equations (19) and (20) since DAB = DBA in binary mixtures (p. 502 of Ref. [37]). The coupled heat and mass transfer nature is reflected in the energy balance at the vapor-liquid interface :

6~f.OA I 7 --pDAB---~7--r [ +pA(V,,.--a)|hA.f, Iitf J

+I -flDBA ~(DBt~r itf+flB(Ol'n--Ll)lhB'fg

= - k , d~rTi~f. (21)

Conduction into the vapor phase has been neglected, consistent with our assumptions. Likewise, Ref. [36] has shown that it is reasonable to neglect two additional energy terms; the kinetic energy of both the vapor and the liquid, and the work done against viscous stresses. At first glance the change in the radial velocity (equations (1) and (18)) may seem negligible due to the small density ratio, Pjpl. However, an order of magnitude comparison between the diffusive terms and the convective terms in equation (21) reveals that the seemingly small change in the convective term is significant [31].

Problem closure is completed with the condition for vapor-liquid equilibrium at interface :

(-Ditf = f(Titf, P). (22)

3. NUMI:ItlCAL SOLUTION

The numerical method chosen for the solution of equation (11) and the associated boundary conditions is briefly described here. Ad hoc techniques called for by the unique nature of the problem are also discussed.

The bubble absorption problem, defined in the previous section as an unsteady, two-dimensional, moving boundary/phase change problem, was solved with finite differences. The application of finite difference techniques to boundary-layer partial differential equations, like equation (11), is well understood (Chap. 7 of Ref. [38]). When the time derivative is approximated by forward differencing or backward differ-

encing and the spatial derivatives are approximated by central differencing, first-order accuracy in time and second-order accuracy in space is achieved. A fully implicit technique was chosen for two reasons: fully implicit techniques are unconditionally stable, lacking constraints on choices of time step, At, or grid spacing, Ar, and A0, and fully implicit techniques have been proven effective for solving a wide range of transport problems.

With boundary layer finite difference approxi- mations, where (O/Or >> 0/d0), the order of approxi- mation in the streamwise direction can be reduced from second to first (p. 363 of Ref. [38]). As a result, the implicit finite difference method creates a tri- diagonal system of algebraic equations, allowing the implementation of the Thomas algorithm (p. 457 of Ref. [39]). With the computational grid of Fig. 4, the general governing equation (11) can be discretized to obtain algebraic equations of the form:

_ n 4~i.j+ 1 - 4~ia-, At + ~(ij) 2 Ar

"~- l~O(i,j) Oin, f , n+ , n+ , n+

0 A0 r " # ArL s+ ,/2 t a r

- rJ- ' /2 t- Ar-4~2f-" +O(At, AO, Ar~). (23)

The overall accuracy of this finite difference equation is O(At, A0, Ar 2) [311.

Because the problem posed in this study involves phase change and a moving boundary condition, two ad hoe methods are necessary to obtain a reasonable solution. First, the moving boundary can make convection dominant over diffusion in the radial direction. As a result, central differencing leads to coefficient matrices that are not diagonally dominant, which in turn leads to unstable Thomas algorithm solutions [40, 41]. To address this problem, a mesh P~clet number, Pem~h is used to assign weighted central and upwinding differences (p. 345 of Ref. [38], and [42]). The mesh P6clet number is calculated with a characteristic length equal to the spacing between consecutivej values and a characteristic velocity equal to the radial velocity, which coincides with the direction of dominant diffusion. For example, the radial convection term in the generalized scalar equation (11), in the case where the radial velocity, u,, is positive may be expressed as :

Pc* \ an+l .h,+, + 1 -- mclhlip't tlPi'J - - ttli 'J-- ' ~ J " , o a ) ~ , (24)

where Pe*~h is the critical mesh P6clet number equal to two (p. 344 Ref. [38]).

Second, an adaptive computational grid must be

Heat and mass transfer during bubble absorption

• I

I

,/I _ ~ . . . . c o m p u t a t i o n a l

. . . . . ' , - - ' m o l e c u l e

Y • ,

" :c i I I I - - r - - r - - T - - 3 - -

¢ • ¢ . i / i i / .

, . ' ~ l I .'_--t

I e , ' ~ : ~__. !

/ . . " " . . . . . • . . . . - I J - " l

Ibubble I " - . _ ', , --*," l i n t e r f a c e l . . . . . ! . . . . I ' -" '

I i:imax I I j=jmax I

liquid I flow field I

Fig. 4. Finite difference computational grid for the liquid flow field.

595

employed to handle the moving liquid-vapor interface. At each new time step the computational grid is adapted for each new bubble diameter. Once a new bubble radius is calculated at the new time step, the overall solution method steps in time by assigning node values based on the previous grid at the nth time level to the newly ~;enerated grid at the n + 1 time level.

4. RESULTS A N D DISCUSSION

To assess the relative accuracy of the model, initial efforts focused on comparing the results of the numerical solution to the results of existing analytical solutions. As mentioned in the Introduction, no complete previous solution to this problem exists, and hence our comparisons were done for cases somewhat sim- pler than the subject of the paper. To make the first evaluation, a pure mass diffusion problem was chosen. The exact solution for pure mass diffusion in a stagnant fluid surrounding a stationary solid sphere with constant surface concentration is available in Sher- wood et aL (pp. 69 and 70 in Ref. [11]). A comparison between the predictions of our model and Sherwood's solution is shown in Fig. 5, which displays mass fraction vs radial distance.

Next, a more complex problem involving simultaneous heat and mass transfer from a moving, well mixed fluid sphere to a liquid was solved. An analytical solution wa,; possible because the temperature and concentration fields are not coupled at the interface. The solution offered by Chao [13], assumes a potential flow field around a sphere of constant radius. Mass fraction at va:rious 0 positions is plotted vs radial distance in Fig. 6. The agreement is reasonable if one considers that Chao's solution assumes r << a, result-

ing in velocity fields which are simplified. These simplifications result in a diffusion term which differs slightly from the diffusion term shown in equation (11). Furthermore, these simplifications are not entirely valid in the trail of the sphere, where the boundary layers, 6 ~ a. The calculated total mass transferred differed by 1% at low Peru around 6680, increasing to a 3% difference for Pem around 66 800. The solution for the temperature field is not presented here, because it is identical to the concentration field.

With the model reasonably validated, a calculation of the temperature and concentration distributions surrounding a collapsing bubble were made. For the conditions shown in Table 1, an initial bubble diameter of 3.71 mm, and the temperature gradient along the z-direction equal to 45°C m - ' (compatible with our experimental observations [32]), the temperature and concentration distributions were calculated.

A study of convergence under these conditions resulted in the adoption of 850 radial nodes, or a Ar equal to 9/~m. In the theta direction, A0 was 9 °, arising from 20 intervals spaced over the computational domain of 180 °. The time step, At required for solution convergence, was 0.1 ms.

Figure 7(a) shows a contour plot of the nondimensional mass fraction field (o9/~oi) at a time equal to 0.22 s during the collapse of a binary bubble, whereas Fig. 7(b) displays the nondimensional temperature field (T/TD at the same time. As expected, the boundary layers are thin relative to the bubble radius over most of the bubble; the largest temperature and concentration gradients occur near the forward stagnation point. The temperature field contour plot differs from the mass fraction contour plot because the far-field boundary conditions are different.


@ @

8

I @

8

@ 8 I

i r/) r/) O

O o ~

r~

I wll,~

0.2:5

0.20

0.15

~ | | | | | | l | | | l I ~ U | | | | H | | | I l U l O l i l a i l l l l i | i | H | | n | l l i l i i k g l l i l l i | i n ~ | g B | i l I n l l i i l i | K N i l l l i i l i i l | S ] i ~ i i l i l i l l i G ~ !

- - O - - n u m e r i c a l so ln . theta = 86 - - exac t so ln . theta = 86 - - ~ - - n u m e r i c a l so ln . theta = 113 - - O - - exac t so ln . t h e m = 113 ..... • .... n u m e r i c a l so ln . theta = 140 - e - . - e x a c t so ln . t h e m = 140

- G - - n u m e r i c a l so ln . t h e m = 167

exac t so ln . t h e m = 167

-86 "113

167140 0.10 1

0.05 -

0 .00 -

0 .00 0.05 0 .10 0.15 0 .20 0.25 0 .30 0.35 0 .40

r /a - 1 Fig. 6. Chao [13] solution comparison: Peru = 6680, Re = 365, Sc = 18.3, Ar = 86.6 pm, A0 = 4.5 °,

At = 0.1 ms, and evaluated at time = 0.1 s.

whether or no t the se t ransfer coefficients were reasonable, average t ransfer coefficients over the bubb le life span were calculated and compared to a semi-empirical corre la t ion for heat and mass t ransfer coefficients found on p. 184 o f Ref. [10] :

S'--h = I. 13" Pe~/2 (29)

and

]~--u = 1.13" Pe ~/2. (30)

Our model predicted an average mass transfer coefficient of 1.1554 x 10 -3 m s -1 and the semi-empirical correlation predicted a value of 1.1447 x 10 -3 m s - l . F o r the heat t ransfer coefficient the numerical model predicted art average value 16021 W m -2 K - I and the semi-empirical corre la t ion predicted a value of 16 783 W m -2 K-1 . In addi t ion, Brucker and Spar- row [18] have verified experimental ly average heat

t ransfer coefficients on the order of 104 W m -2 K - ~ for sa tura ted s team bubbles in t roduced into subcooled water envi ronments , where the initial diameters of these bubbles were abou t 3 mm.

Figure 11 shows the nond imens iona l bubble diameter and mass dur ing the bubble life span. Three sep- arate regions of bubb le collapses are illustrated. Dur- ing the initial 0.06 s, the d iameter remains relatively unchanged, where the a m o u n t of mass leaving the bubble matches the a m o u n t of mass enter ing the bubble. F r o m 0.06 to nearly 0.2 s the rate of bubb le collapses increases and then increases even fur ther beyond 0.2 s. The model was te rminated with the bubble mass was approximate ly 1% of its original mass and 24% of its original diameter. This relat ion- ship between the final bubble mass and final bubble d iameter arises f rom the fact tha t the bubb le mass is p ropor t iona l to the d iameter cubed.

598

la)

(b)

3.5

3.0

2.5

2.0 r / a

1.5

1.0

0.5

0.0

3.5

T. L. MERRILL and H. PEREZ-BLANCO

to* = to(r,0,t)/tai, so l id l ines ~* = t~]a2Ut, dot ted l ines

t . ~ o ........

1.011111 ~",,~1 •.. "~ . . . . t ,"i" I.~16

-2 -1 0 1 2 r / a

3.0

2.5

T* = T(r ,0 , t ) /Ti , so l id l ines ~* = ~/a2Ut, do t ted l ines

2.0 r / a

1.5

1.0

0.5

0.0 -2 -1 0 1 2

r / a Fig. 7. (a) Streamlines and mass fraction contours for flow past a collapsing bubble: Pem = 59 735 and time = 0.22 s, ( . . . . ) streamline, and (--) mass fraction, (b) Streamlines and temperature contours for flow

past a collapsing bubble Pe = 5611 and time = 0.22 s. ( . . . . ) streamline, and (--) temperature.

5. CONCLUSION

A unique model has been developed which reasonably predicts bubble dynamics in binary subcooled solutions. Our study extends bubble research to the formulation and solution of the combined heat and mass transfer problem surrounding collapsing vapor

bubbles during absorption. Whereas the conservation equations readily lead to a formulation of the problem, the treatment of the interfacial conditions required close examination. Correctly determining the radial velocity at the moving vapor- l iquid interface allowed the proper interface boundary conditions to be defined. The problem was then numerically solved

Heat and mass transfer during bubble absorp t ion 599

(a)

(b)

@

J .9.0

. m

1 . 0 7 -

1 . 0 6

1.05

1.04 -

1.03 -

1.02 -

~L.01 -

1~.00

\

\ t

i\\ X

0.99 -

I I I

- - ¢ - - 0 = 36 degrees

---m-- 0 ffi 72 degrees

--A.--. 0 = 108 degrees ~--

- - v - . 0 = 144 degrees _ _

f I 1

0.000 0.013 0.025 0.038 0.050 0.063 0.075 0.088 0.100

r / a - 1

1 . 2 5

" " 1 . 2 0

1 . 1 0 .

I " ~ 1 . 0 0

-" 0 = 36 degrees

---~-- 0 = 72 degrees

- - '~-" 0 = 108 degrees

- ~ , - - O = 144 degrees

I I

\ "v\

0.95

0.000 0.013 0.025 0.038 0.050 0.063 0.075 0.088 0.100

r / a - 1 Fig. 8. Mass fraction profile for flow past a collapsing bubble : (a) Peru = 66800 and time = 0.11 s. (b)

Mass fraction profile for flow past a collapsing bubble: Peru = 40000 and time = 0.33 s.

using a finite difference method which employed mesh P6clet numbers and an adaptive spatial grid to account for the moving vapor-liquid interface. After the solution method accuracy was verified with simple mass transport problems, the complete coupled heat and mass transfer problem for bubbles collapsing in binary solutions was attempted.

Assessing the model accuracy proved difficult. Because of the limited research involving combined heat and mass transfer surrounding bubbles, no comparisons between actual bubble behavior data and model predictions were conducted. However, the calculated concentration and temperature fields surrounding a collapsing bubble were reasonable. Fur-

600 T . L . M E R R I L L and H. P E R E Z - B L A N C O

(a)

(b)

0.9895 I I t n

: Offi36degrees .[t~t A" ~ 0.9890 ~, ---4-- 0 = 72 degrees

0.9885 i ~ --'--" 0 = 108 degrees ' ~ -'~-" e--144igree ~

0.9880 ~-. "~ "v'v" ~ ' I " ~ ' ~ , v- r '~'4' v-~-~- ~ . , v , ~ ,~ i

• ~ _ ~ ~ ,L j . . . . i~.~l ~ A o.7o 0.9865 _1,.,, ***'*~

0.9860 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

r/a-1 0.965

0.964

0.963

o o 0.962

i 0.961

0.960 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

I t t : O=36degrees [ I I

[ . - e -- 72 degrees I ~ 1 e = / U ' 1--~-" e--108 degrees I / / ~ , r

-- 0 = 144 degrees [

r/a-1 Fig. 9. Tempera tu re profi le for f low pas t a co l laps ing bubb l e : (a) Pe = 6720 and t ime = 0.11 s. (b)

Tempera tu r e profi le for f low pas t a co l laps ing bubble Pe = 3750 and t ime = 0.33 s.


(a)

(b)

3(t000

~ 25000

20,000

o 15000 o O

10000

-~ 5000 O o

0

V - A k

m m

'"4 • i". I "

I I I I I I I I -- " t ime = 0.076 sec.

t ime = O. 152 sec. - - - ~ - t ime = 0.228 sec.

- V - - t ime = 0.304 sec.

4 ' t ime = 0.380 sec. ~ i M m m n m m n m m m ~ ~ m m m m ~ m m m m m m m m m m ~ w ~ m m ~ r l m m m m m m m m m m m . m m m m m m m m m m w l m m m m ~ a m m L J m m m m m m m ~ ~ m m m m m m m m m m m m m m m ~ ~ m ~ m m r m m m m m m m m m m w ~ m m ~ m m m m

m m m m m m m m ~ m L ~ m ~ m m m m m m m m m m m ~ ~ m u m m ~ = ~ ~ m m m m m ~ ~ I m m m m w k ~ i m m m L ~ . m ~ m ~ ~ m m ~ m ~ m m ~ ~ m m ~ ~ ~ ~ ~ m m ~ M m m m m m m m ~ [ z ~ m ~ ~ . m m m m m m m m m m ~ . m ~ i ~ m m m m m m m m m m m m ~ ~ -

0 1 n/2 2 3 4

Theta location, 0 (radians)

0.0025 Theta location, 0 (radians) i I I I I I I I I I

! ~ ' ---0-- time -- 0.076 sec.

0.0020 r ~ ~ I t - -al --- ~ e = 0.152 sec. " " " 6 " 1 ~ - t i m e = 0 . 2 2 8 SeC.

I " J x, , - - ~ - - - t i m e = 0 . 3 0 4 s e c .

0.(1015 - m , p '~ , --4~"" t ~ e = 0 . 3 8 s e c .

r ~- w. . ~ \ _

0 . 0 0 1 0 r " " - ] "" ~ " ] ~ ' ' ~ ' d ~ - ~ ~ • ~

I l l -- ,..j ,~ i k '~ ..~,~\

0.11005 ~ ?W "~ m [ . \

k \ - 0.0000 •

0 I v,./2 2 3 4

Theta location, 0 (radians) Fig. 10. (a) Local heat transfer vs theta location for various times during bubble collapse. (b) Local mass

transfer coefficient vs theta location for various times during bubble collapse.


1.0 ,4 ,,,,-4hi .~ -.. w'*'n'a

0.9 ~ " m, Jt

0.8 l

0.7 x. , m

"~ 0.6

0.5

~ 0.4

~ 0.3

~ 0.2

o 0.1

" " 0 m O°

o.o o.1

Fig. 11.

it \

(

m

I I I I I

: m' be b

= - - d* e

/

-N \

t'i

!

|

0.3 0.4

Time (see.) Dimensionless bubble mass and bubble diameter vs bubble residence time, ~ = de/do,i and

$ m b u b ~ m b u b / m b u b , i .

thermore , there was agreement between accepted semi-empirical correlat ions and the model for average heat and mass t ransfer coefficients. Using this model , the impact of the liquid subcooling, the initial bubble diameter , the liquid flow field, and other parameters on bubble collapse rates can be determined. Ult i- mately this improved unders t and ing of bubble absorp t ion dynamics will lead to more effective and more compac t bubble absorbers .

Acknowledgements--The authors gratefully acknowledge the support of Mr Tetsuo Setoguchi and the Osaka Gas Co.

REFERENCES

1. G. Alefeld and R. Radermacher, Heat Conversion Systems. CRC Press, Boca Rato, FL (1994).

2. V.G. Levich, Motion of gaseous bubbles with high Rey- nolds number, Zh. Eksp. Teor. Fiz. 19, 18-24 (1949).

3. B.T. Chao, Motion of spherical gas bubbles in a viscous fluid at large Reynolds numbers, J. Fluid Mech. 25, 69- 79 (1962).

4. D. W. Moore, The boundary layer on a spherical gas bubble, J. Fluid Mech. 16, 161-176 (1963).

5. D. Bhaga and M. E. Weber, Bubbles in viscous liquids, shapes, wakes, and velocities, J. Fluid Mech. 105, 61-85 (1981).

6. R. M. Griffith, Mass transfer from drops and bubbles, Chem. Engng Sci. 12, 198-205 (1960).

7. J. H. Leonard and G. Houghton, Mass transfer and velocity of rise phenomena for single bubbles, Chem. Engng Sci. 18, 133-142 (1963).

8. P. H. Calderbank and A. C. Lochiel, Mass transfer coefficients, velocities and shapes of carbon dioxide bubbles in free rise through distilled water, Chem. Engng Sci. 19, 235-255 (1964).

9. A. I. Johnson, F. Besik and A. E. Hamielcc, Mass transfer from a single rising bubble, Can. J, Chem. Engng 47, 559-564 (1969).

I0. D. Azbel, Two-Phase Flows in Chemical Engineering, chapters 1, 2, 3, and 7. Cambridge University Press (1981).

11. T. K. Sherwood, R. L. Pigford and C. R. Wilke, Mass Transfer. McGraw-Hill, New York (1975).

12. C. W. Bowman, D. M. Ward, A. I. Johnson and O. Trass, Mass transfer from fluid and solid spheres at low Reynolds numbers, Can. J. Chem. Engng 9, 9-13 (1961).

t3. B. T. Chao, Transient heat and mass transfer to a trans- lating droplet, J. Heat Transfer 91, 273-281 (1969).

14. M. S. Plesset and S. A. Zwick, A non-steady heat diffusion problem with spherical symmetry, J. appl. Phys. 23, 95-99 (1952).

15. H. K. Forster and N. Zuber, Growth of a vapor bubble in a superheated liquid, J. appl. Phys. 25, 474-479 (1954).

16. M. S. Plesset and S. A. Zwick, The growth of vapor bubbles in superheated liquids, J. appl. Phys. 25, 493- 498 (1954).

17. L. W. Florscheutz and B. T. Chao, On the mechanics of vapor bubble collapse, J. Heat Transfer 87, 209-220 (1965).

18. G. G. Brucker and E. M. Sparrow, Direct contact con-


densation of steam bubbles in water at high pressure, Int. J. Heat Mass Transfer 20, 371-381 (1977).

19. S. S. Wang, J. H. Stuhmiller and R. E. Ferguson, Numerical simulation of steam bubbles in a vertical tube, in Fundamentals of Two-Phase Heat Transfer, Vol. ASME, HTD 35, pp. 79-86. New York (1985).

20. G. K. Raina and R. K. Wanchoo, Direct contact heat transfer with phase change : bubble growth and collapse, Can. J. Chem. F, ngng 64, 393-398 (1986).

21. C. Kleinstreurer, T. Y. Wang and H. Chiang, Interface heat and mass transfer of single and multiple drops in a hot gas stream, in 1989 National Heat Transfer Conf. : Multiphase Flow, Heat and Mass Transfer, Vol. 109, pp. 93-102 (1989).

22. I. Morioka, M. Kiyota, A. Ousaka and T. Kobayashim, Analysis of steam absorption by a subcooled droplet of aqueous solution of LiBr, J.S.M.E. Int. J. 35, 458-464 (1992).

23. N. I. Grigor'ew. and V. E. Nakoryakov, Exact solution of combined heat and mass transfer problem during absorption, Inzh. Fiz. Zh. 33, 893-896 (1977).

24. G. Grossman, Simultaneous heat and mass transfer in film absorption under laminar flow, Int. J. Heat Mass Transfer 26, 357-371 (1983).

25. G. C. Vliet and J. W. Andberg, A simplified model for absorption of vapors into liquid films flowing over cooled horizontal tubes, ASHRAE Trans. 93, Part 2, 1- 15 (1987).

26. J. S. Seewald and H. Perez-Blanco, A simple model for calculating the l~erformance of a lithium-bromide/water coil absorber, ,4SHRAE Trans. 100, 318-328 (1994).

27. V. Patnaik and H. Perez-Blanco, An empirical method- ology for the design of vertical tube absorbers, ASHRAE Trans. 100, 185-196 (1996).

28. D. D. Wittke and B. T. Chao, Collapse of vapor bubbles with translatory motion, J. Heat Transfer 89, 17-24 (1967).

29. R. Clift, J. R. Grace and M. E. Weber, Bubbles Drops and Particles, Chaps 1-12. Academic Press, New York (1978).

30. D.D. Wittke, CoUapse of vapor bubbles with translatory motion--a theoretical and experimental investigation. Ph.D. Thesis, University of Illinois, Champaign- Urbana, IL (1966).

31. T. L. Merrill, Bubble heat and mass transfer dynamics in binary solutions. Ph.D. Thesis, The Pennsylvania State University, University Park, PA (1994).

32. T. L. Merrill, Passive heat transfer enhancement techniques applied to compact bubble absorber design, J. Enhanced Heat Transfer 2, 199-208 (1995).

33. L. E. Scriven, On the dynamics of phase growth, Chem. Engng. Sci. 10, 1-13 (1959).

34. C.J. Geankoplis, Transport Process and Unit Operations, 2nd edn, pp. 440-451. AUyn and Bacon, Boston, MA (1983).

35. N. Zuber, Bubble growth in superheated in liquid, Int. J. Heat Mass Transfer 2, 83-90 (1961).

36. L. W. Florschuetz, On the mechanics of vapor bubble collapse--a theoretical and experimental investigation. Ph.D. Thesis, University of Illinois, Champaign- Urbana, IL (1963).

37. R. B. Bird, W. E. Stewart and E. Lightfoot, Transport Phenomena, Chaps. 16-22. Wiley, New York (1960).

38. D. A. Anderson, J. C. Tannehill and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Chaps 1-10. Hemisphere, New York (1984).

39. C. Hirsch, NumericalComputation oflnternalandExter- nalFlows, Vol. 1, Chaps 1-12. Wiley, New York (1988).

40. S. V. Patankar and D. B. Spalding, Heat and Mass Transfer in Boundary Layers, Chaps 1-5. Morgan- Grampian, London (1967).

41. R. S. Hirsch and D. H. Rudy, The role of diagonal dominance and cell Reynolds number in implicit methods for fluid mechanics problems. J. Comput. Phys. 16, 304-310 (1974).

42. D. B. Spalding, GENMIX: A General Computer Pro- gram for Two-dimensional Parabolic Phenomena, Chaps 1-1 I. Pergamon Press, New York (1977).

Combined heat and mass transfer during bubble absorption in binary solutions

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