International Journal of Refrigeration 30 (2007) 274e281www.elsevier.com/locate/ijrefrig

Coupled heat and mass transfer during nonisothermalabsorption by falling droplet with internal circulation

T. Elperina,*, A. Fominykha, Z. Orenbakhb

aThe Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University

of the Negev, P.O. Box 653, Beer Sheva 84105, IsraelbNegev Academic College of Engineering, 71 Bazel St., P.O. Box 45, Beer Sheva 84100, Israel

Received 9 July 2005; received in revised form 7 May 2006; accepted 11 July 2006

Available online 11 September 2006

Abstract

We considered mass and heat transfer during nonisothermal absorption of a gas by a falling droplet with internal circulation.Gas phase is assumed to be free of inert admixtures and mass transfer is liquid phase controlled. Mass flux is directed froma gaseous phase to a droplet, and the interfacial shear stress causes a fluid flow inside the droplet. Droplet deformation underthe influence of interface shear stress is neglected. Absorbate accumulation and temperature increase in the bulk of liquid phaseare taken into account. The problem is solved in the approximations of a thin concentration and temperature boundary layers inthe liquid phase. The thermodynamic parameters of the system are assumed constant. The system of transient partial parabolicdifferential equations of convective diffusion and energy balance with time-dependent boundary conditions is solved by com-bining the similarity transformation method with Duhamel’s theorem, and the solution is obtained in a form of Volterra integralequation of the second kind which is solved numerically. Theoretical results are compared with available experimental data forwater vapor absorption by falling droplets of aqueous solution of LiBr.� 2006 Elsevier Ltd and IIR. All rights reserved.

Keywords: Absorption system; Absorption; Gas; Drops; Heat transfer; Mass transfer; Modelling

Transfert de chaleur et de masse lors de l’absorption a gouttelettestombantes, avec circulation interne

Mots cles : Systeme a absorption ; Absorption ; Gaz ; Gouttelettes ; Transfert de chaleur ; Transfert de masse ; Modelisation

*Corresponding author. Fax: þ972 8 6472813.

E-mail addresses: elperin@bgu.ac.il (T. Elperin), fominykh@

bgu.ac.il (A. Fominykh), zakhar@nace.ac.il (Z. Orenbakh).

0140-7007/$35.00 � 2006 Elsevier Ltd and IIR. All rights reserved.

doi:10.1016/j.ijrefrig.2006.07.006

1. Introduction

Spray absorption and spray absorber design and anal-ysis attracted considerable attention in recent years be-cause of higher contact surfaces per unit volume andhigher rates of mass transfer in spray absorbers in

275T. Elperin et al. / International Journal of Refrigeration 30 (2007) 274e281

Nomenclature

a thermal diffusivity of a liquid [m2 s�1]b coefficient in Eq. (7)cp specific heat [kJ kg�1 K�1]d coefficient in Eq. (7) [K�1]D molecular diffusion coefficient in a liquid phase

[m2 s�1]FoD ¼ Dt=R2 Fourier number (diffusion)FoT ¼ at=R2 Fourier number (temperature)K ¼ cp=ðdLÞ dimensionless numberk coefficient in Eq. (1)L heat of absorption [kJ kg�1]Le Lewis number (D/a)PeD ¼ kRU=D Peclet number for a moving droplet

(diffusion)PeT ¼ kRU=a Peclet number for a moving droplet

(temperature)Pr ¼ n=a Prandtl numberr radial coordinate [m]R droplet radius [m]ReG ¼ 2URrG=mG Reynolds number for a moving

dropletSc ¼ n=D Schmidt numberT temperature of a liquid [K]T arbitrary function [K]T00 equilibrium temperature at the initial concen-

tration [K]t time [s]U translational velocity of a droplet [m s�1]vy; vq velocity components [m s�1]V droplet volume [m3]X ¼ x�x0

x00�x0dimensionless weight fraction of an absorbate

x weight fraction of an absorbatex arbitrary functionx00 equilibrium weight fraction of an absorbate at

the initial temperature

y distance from the surface of a droplet [m]Y¼ y/R dimensionless distance from the surface of

a droplet

Greek symbolsdc thickness of a diffusion boundary layer [m]dT thickness of a temperature boundary layer [m]Dc ¼ dc=R dimensionless thickness of a diffusion

boundary layerDT ¼ dT=R dimensionless thickness of a temperature

boundary layerz variable, Eqs. (13) and (14)hc similarity variable, Eq. (9)hT similarity variable, Eq. (9)q angle [rad]Q ¼ ðT � T0Þ=ðT00 � T0Þ dimensionless temperaturel thermal conductivity of a liquid [kJ m�1

s�1 K�1]m viscosity of a fluid [kg m�1 s�1]n kinematic viscosity of a fluid [m2 s�1]r density of a solution at the bulk of a liquid

[kg m�3]t ¼ tUk=R dimensionless time

Subscripts0 value at the inletN value at infinityB value in the bulk of a dropletG gasL liquidr radial directions value at the interfaceq tangential direction

comparison with conventional falling film absorbers.Analysis and comparison of two different absorptionmodes (see, e.g. Refs. [1,2]) show that a size of spray ab-sorber can be two times smaller than the size of a conven-tional falling film absorber with the same efficiency. Thesimplicity of the spray absorber design and the low priceof the absorption chamber promise essential reduction ofspray absorber price in comparison with a conventionalfalling film absorber. Heat and mass transfer during gasabsorption in droplet flows is of interest in various fieldsof modern technology, e.g., refrigeration engineering, airconditioning, and chemical engineering. Examples ofindustrial operations where the effects of heat releaseduring absorption can be important include absorption

of ammonia into water, water vapor absorption by aque-ous solutions of LiBr, LiCl and potassium formate, ab-sorption of friendly refrigerants such as R32, R134aand R124 by dimethyl acetamide. Analysis of the cou-pled mass and heat transfer during droplet absorptioncan be potentially useful in design calculations of gaseliquid absorbers and for incorporating into system designcomputer codes. Accumulation of the dissolved gases ina falling droplet and temperature increase duringnonisothermal absorption, circulation of liquid insidea droplet caused by shear stress at the interface, yielda conjugate problem of unsteady convective diffusionand energy balance with two velocity components andtime-dependent boundary conditions.

276 T. Elperin et al. / International Journal of Refrigeration 30 (2007) 274e281

2. State-of-the-art in nonisothermal absorptionby liquid droplets

The simplest model of simultaneous heat and mass trans-fer during nonisothermal absorption of vapor by a stagnantliquid droplet was suggested by Nakoryakov and Grigoreva[3]. They assumed that (a) the state of equilibrium is estab-lished instantaneously over the droplet’s surface and is sub-sequently maintained through the entire process, (b) thedependence of concentration on temperature in equilibriumis linear and (c) heat release caused by absorption occurs atthe surface. The problem was solved in the approximation ofthe infinite dilution of absorbate in the absorbent. Theapproximation of thin temperature and diffusion boundarylayers allowed determining the solutions for concentrationand temperature distribution inside a liquid droplet in theform of error functions, and formulas for Sherwood andNusselt numbers were obtained in a simple analyticalform. Later Grigoreva [4] modified the approach of Nakor-yakov and Grigoreva [3] for a case of developed diffusionand temperature boundary layers. Solutions for concentra-tion and temperature distribution inside a droplet and formu-las for Sherwood and Nusselt numbers were found aseigenfunctions expansion. Coupled mass and heat transferduring vapor absorption by a moving liquid droplet wasinvestigated by Morioka et al. [5] and by Lu et al. [6].They took into account the effect of internal circulation onsimultaneous heat and mass transfer. In these studies it wasassumed that circulation inside the droplet is induced by therelative motion between the droplet and a surrounding fluid.Action of shear forces at the nozzle walls during atomizationon the induction of internal circulation in liquid droplets wasneglected. Since the kinematic viscosity of liquid is by twoto four orders of magnitude larger than thermal diffusivityand mass diffusivity, Morioka et al. [5] and Lu et al. [6]assumed that the flow field in a liquid droplet becomes steadymuch faster than the temperature and concentration fields.Therefore the steady velocity distribution is instantly estab-lished before the absorption starts. Morioka et al. [5] and Luet al. [6] assumed that the velocity, temperature and concen-tration fields are axially symmetric. Morioka et al. [5] useda steady velocity distribution inside a liquid droplet obtainedby Hamielec and Johnson [7] for intermediate Reynoldsnumbers. Lu et al. [6] assumed that the stationary velocityfield inside a liquid droplet is determined by HadamareRybczinski solution (for details see Ref. [8]). Analysis ofa steady-state velocity field inside a liquid droplet conductedby Hamielec and Johnson [7] shows that the internal flowvelocity is very small compared with a free-stream velocity.The maximum velocity inside a liquid droplet is less than0.2% of the free-stream velocity even at Re¼ 10. Moriokaet al. [5] noted that the flow at the center of a droplet is inthe opposite direction to the main stream and its speed isas high as the surface velocity. Lu et al. [6] found that theinfluence of internal circulation on the rate of mass transferdecreases with the increase of a thermal effect. When

K¼�0.1, that corresponds to water vapor absorptionby aqueous solution of LiBr, the influence of internal circu-lation on the rate of mass transfer is negligible. The latterimplies that high exothermic absorption by a dropletfor Re � 1 can be analyzed using a stagnant drop model.Venegas et al. [2] simulated numerically the coupledmass and heat transfer during the absorption of ammonia va-por by a droplet of lithium nitrateeammonia solution, andconsidered the low-pressure and high-pressure absorbersof a double-stage absorption system. Low Reynoldsnumber (Re< 1) and intermediate Reynolds numberð10 � Re � 250Þ situations were investigated usingHadamardeRybczynski (see Ref. [8]) and Uribe-Ramırezand Korchinsky [9] hydrodynamic solutions, respectively.Results obtained by Venegas et al. [2] were compared withNewman model [10] for mass transfer. Simulations ofVenegas et al. [2] showed that Newman model is adequatefor predicting mass transfer during gas absorption by mov-ing droplets with diameters smaller than 100 mm, becausethe internal circulation for small droplets is insignificant.The comprehensive review of the experimental studies onmass transfer during nonisothermal absorption by liquiddroplets was performed by Venegas et al. [2] The state-of-the-art in dispersed-phase controlled mass transfer duringisothermal gas absorption by liquid droplets was summa-rized by Amokrane and Caussade [11] and by Uribe-Ram-ırez and Korchinsky [9]. In all the previous studies masstransfer during nonisothermal gas absorption by fallingliquid droplets with internal circulation was investigatedby solving numerically a system of partial differential equa-tions (see Refs. [1,5,6]). In the approach suggested in thisstudy the problem is reduced to numerical solution ofVolterra integral equation of the second kind.

3. Description of the model: general solution

Consider absorption of a single component gas by a mov-ing droplet accompanied by a heat release. At time t¼ 0a droplet begins to absorb gas from a host medium occupy-ing a large volume as compared with the droplet’s volume.Temperature field is initially homogeneous inside a liquiddroplet. Heat release at the interface during gas absorptionresults in inhomogeneous temperature distribution insidethe droplet. The problem is solved in the approximation ofan infinite dilution of absorbate in the absorbent. Gaseousphase is assumed to be free of inert admixtures. The thermo-dynamic parameters of a system are assumed constant, andonly the resistance to mass and heat transfer in the liquidphase is taken into account. Heat released or absorbedduring absorption is assumed to dissipate in a liquidphase, and it causes a change of the temperature in a droplet.The equilibrium condition at the gaseliquid interface isdescribed by a linear dependence of concentration on tem-perature. Heat and mass transfer problem is axially symmet-ric. Convective diffusion and heat transfer are determined by

277T. Elperin et al. / International Journal of Refrigeration 30 (2007) 274e281

fluid velocity at the droplet’s surface. The thicknesses of dif-fusion and temperature boundary layers are assumed smallcompared with the droplet’s size, and tangential molecularmass and heat transfer rate along the surface of a sphericaldroplet is assumed small compared with a molecular massand heat transfer rate in the normal direction. The bulk ofa droplet, beyond the diffusion and temperature boundarylayers, is completely mixed under the influence of the inter-nal circulation inside the droplet, and concentration and tem-perature are homogeneous in the bulk. Circulation insidea drop is caused by a shear stress at gaseliquid interface dur-ing droplet fall. The shape of a drop is assumed to be spher-ical. The assumptions that there is circulation insidea droplet and that the droplet has a spherical shape are validin the following range of falling in air water droplet radii,Reynolds numbers and velocities, respectively: 0:1 mm �R � 0:5 mm, 10 � ReG � 300 and 0:7 m s�1 � U �4:5 m s�1. Analysis of fluid flow around a moving droplet[12] shows that at different Reynolds numbers tangentialand radial fluid velocity components in the vicinity of thegaseliquid interface can be approximated by the followingequation (see Ref. [13, p. 392]):

vq ¼�kUsin q; vy ¼2kUy

Rcos q; ð1Þ

where coefficient k is equal to 0.04 in the range of externalflow Reynolds numbers ðRe ¼ 2URrG=mGÞ from 10 to 300(see Ref. [13, p. 386]). Eq. (1) is valid for y� R. The depen-dence of terminal fall velocity of liquid droplets on theirdiameter is presented in Ref. [13, pp. 415e421]). Followingthe approach suggested in our previous studies [14e16] wearrive at the nonstationary convective diffusion equation fora liquid phase which accounts for convection in radial andtangential directions:

vx

vtþUk

R

��sin q

vx

vqþ 2ycos q

vx

vy

�¼ D

v2x

vy2; ð2Þ

where y ¼ R� r; y� R (see Fig. 1). Taking into accountthat Le� 1 we arrive at the differential energy balanceequation (for details see Ref. [15]):

vT

vtþUk

R

��sin q

vT

vqþ 2ycos q

vT

vy

�¼ a

v2T

vy2: ð3Þ

The initial and boundary conditions to Eqs. (1) and (2)read:

x ¼ x0; T ¼ T0 at t ¼ 0; ð4ÞT ¼ Ts; x ¼ xs at y¼ 0; ð5ÞxB ¼ xBðtÞ; T ¼ TBðtÞ at y/N: ð6Þ

The unknown values of concentration and temperature atthe gaseliquid interface are determined from the followingequations:

x ¼ dT þ b at y¼ 0; ð7Þ

l

ðp

0

vT

vy

����y¼0

sin q dq¼ LrD

ðp

0

vx

vy

����y¼0

sin q dq: ð8Þ

Eq. (7) describes a condition of equilibrium at the inter-face. The relationship between the surface concentration andsurface temperature is approximated by the linear expres-sion (see Ref. [17]), and Eq. (8) implies that all heat releasedor absorbed during absorption is dissipated in the liquidphase. Note that for most gases coefficient d in Eq. (7) isnegative, and heat is released during gas absorption. Theequations of convective diffusion and energy, Eqs. (2) and(3), respectively, describe the change in concentration of ab-sorbed gas and temperature near the dropletegas interface.The form of these equations suggests that their solutionmay be obtained using a similarity transformation (seeRef. [18]).

4. Method of solution

Since the boundary conditions (6) to Eqs. (2) and (3) arefunctions of time, the solution can be found by combiningthe similarity transformation method with Duhamel’s theo-rem. Eqs. (2) and (3) are solved by first letting the concentra-tion of solute and temperature in the bulk of dispersed phasein Eq. (6) be constant. Introduce self-similar variables (seeRef. [19]):

hc ¼y

dcðt;qÞ¼ Y

Dc

; hT ¼y

dTðt;qÞ¼ Y

DT

; ð9Þ

Variables hc and hT allow to transform a system of par-tial differential equations (2) and (3) into a system of ordi-nary differential equations and to obtain solutions forconcentration and temperature, x and T, with constant

R

y

0

θ

Fig. 1. Schematic view of a droplet.

278 T. Elperin et al. / International Journal of Refrigeration 30 (2007) 274e281

boundary conditions at y/N in the following form (seeRef. [18]):

x ¼ dTB þ b� xB

1�ffiffiffiffiffiLep

=KerfcðhcÞ þ xB;

T ¼ ðxB � bÞ=d� TB

1�K=ffiffiffiffiffiLep erfcðhTÞ þ TB; ð10Þ

where

D2c ¼

4

PeDsin4 q

8>><>>:cos q� 1

3cos3 q�

26641� tg2q

2expð2PeDFoDÞ

1þ tg2q

2expð2PeDFoDÞ

� 1

3

0B@1� tg2q

2expð2PeDFoDÞ

1þ tg2q

2expð2PeDFoDÞ

1CA

337759>>=>>;; ð11Þ

D2T ¼

4

PeTsin4 q

8>><>>:cos q� 1

3cos3 q�

26641� tg2q

2expð2PeTFoTÞ

1þ tg2q

2expð2PeTFoTÞ

� 1

3

0B@1� tg2q

2expð2PeTFoTÞ

1þ tg2q

2expð2PeTFoTÞ

1CA

337759>>=>>;; ð12Þ

erfcðzÞ ¼ 1� 2ffiffiffiffipp

ðz

0

e�t2

dt:

Applying Duhamel’s theorem to Eq. (10) we obtain solu-tions of Eqs. (2) and (3) with time-dependent boundary con-ditions at the infinity in the following form:

xðy;q; tÞ ¼ v

vt

ðt

0

�dTBðzÞ þ b� xBðzÞ

1�ffiffiffiffiffiLep

=Kerfc

�Y

Dcðq; t� zÞ

�

þ xBðzÞ�

dz; ð13Þ

Tðy;q; tÞ ¼ v

vt

ðt

0

�ðxBðzÞ � bÞ=d

1�K=ffiffiffiffiffiLep erfc

�Y

DTðq; t� zÞ

�

þ TBðzÞ�

dz: ð14Þ

The variables xB and TB are the unknown functions oftime which can be determined from a material and energybalance over the droplet:

�VrL

dxB

dt¼ 2pR2rLD

ðp

0

vx

vy

����y¼0

sin q dq; ð15Þ

�VcprL

dTB

dt¼ 2pR2l

ðp

0

vT

vy

����y¼0

sin q dq: ð16Þ

It must be noted that Eqs. (15) and (16) are approximate,and they ignore mass and energy accumulation within theconcentration and temperature boundary layers insidea droplet. Eq. (15) was used by Uribe-Ramirez and Korchin-sky [9] for the analysis of mass transfer during extraction inliquideliquid dispersed system. Substituting expressions forconcentration and temperature (Eqs. (13) and (14)) into Eqs.(15) and (16) and using Eq. (8) we obtain after integration:

XBðtÞ ¼3ð1�KÞffiffiffiffi

pp

PeDK�1�

ffiffiffiffiffiLep

=Kðt

0

½XBðzÞ �K=ðK� 1Þ�

�ðp

0

sin q dq dz

Dcðq;t� zÞ; ð17Þ

QBðtÞ ¼3ð1�KÞffiffiffiffi

pp

PeDK�1�

ffiffiffiffiffiLep

=Kðt

0

½QBðzÞ � 1=ð1�KÞ�

�ðp

0

sin q dq dz

Dcðq;t� zÞ; ð18Þ

where

XBðtÞ ¼xBðtÞ � x0

x00 � x0

;

QBðtÞ ¼TBðtÞ � T0

T00 � T0

;

t¼ PeDFoD ¼ PeTFoT ¼ tUk=R;

x00 ¼ dT0 þ b; T00 ¼ ðx0 � bÞ=d:

Eqs. (17) and (18) are Volterra integral equations of thesecond kind. Eqs. (17) and (18) can be written in the follow-ing general form:

f ðtÞ ¼ðt

0

f ðzÞFðt;zÞdzþ gðtÞ: ð19Þ

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800

X B(τ

)=(x

B(τ)

-x0)

/(dT 0

+b-x

0)

τ=tUk/R

1-CO2

2-H2S 3-Cl2 4-HCl 5-SO2

6-NH3

1 2 3

4

5

6

PeD=105

Fig. 2. Dependence of concentration in the bulk of a droplet on

time for absorption of different gases by water.

279T. Elperin et al. / International Journal of Refrigeration 30 (2007) 274e281

After discretization on a uniform mesh with an incrementh and using trapezoidal integration rule Eq. (19) can be re-duced to a system of linear algebraic equations with a trian-gular matrix:

f ð0Þ ¼ gð0Þ; ð1� 0:5hFi;iÞfi

¼ h

0:5Fi;0 þ

Xi�1

j¼1

Fijfj

!þ gi; ð20Þ

where i ¼ 1;.;N, fi ¼ f ðihÞ, gi ¼ gðihÞ, Fmn ¼ Fðmh; nhÞ.In order to solve the system of equations, Eq. (20), it is re-quired to calculate N2/2 values of the kernel Fij. SinceFðt; zÞ ¼ Fðt� zÞ or Fij ¼ F ði� jÞhÞð , then Fiþ1;j ¼F ðiþ 1� jÞh ¼ F ði� ðj � 1ÞÞh ¼ Fi;j�1

��. Consequently

at each time step i it is necessary to calculate only one valueof kernel Fi0 because all the other values were alreadydetermined: Fi;1 ¼ Fi�1;0, Fi;2 ¼ Fi�1;1. Therefore whencondition Fðt; zÞ ¼ Fðt� zÞ is valid it is necessary tocalculate only N values of the kernel. Numerical calculationswere performed with N¼ 400.

When t/N, the values of concentration and tempera-ture in a liquid droplet are determined by the followingformulas derived in [14]:

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500

Β(τ

)=(T

B(τ)-T

0)/[(

x 0-b

)/dT 0

)]

τ =tUk/R

1 2 3

4

5

6

PeD= 105

1-CO22-H2S3-Cl24-HCl5-SO2

6-NH3

Fig. 3. Dependence of droplet temperature on time for absorption

of different gases by water.

xBðt/NÞ ¼ dT0 þ b� x0K�1

1�K�1;

TBðt/NÞ ¼ ðx0 � bÞ=d� T0K

ð1�KÞ ; ð21Þ

Note that dTðt/NÞ þ b ¼ xðt/NÞ. In Figs. 2 and 3 thetime dependence of concentration of the dissolved gas andtemperature in the bulk of a droplet are plotted using numer-ical solutions of Eqs. (17) and (18). The time dependence ofconcentration in the bulk of a droplet for absorption of gaseswith different thermal effects by water is shown in Fig. 2.Inspection of Fig. 2. shows that the higher the thermal effectof the absorption (see Table 1), the lower is the value of thedimensionless concentration in the bulk of a droplet. Thistrend can be explained as follows. Heat released during ab-sorption causes an increase of the droplet temperature andof the equilibrium temperature at the gaseliquid interface.The increase of the equilibrium temperature leads to a de-crease of the equilibrium concentration (see Eq. (7)). There-fore the dimensionless concentration of the dissolved gas inliquid droplets ðXBðt/NÞÞ at t/N is lower for gases withlarger thermal effect of absorption. Data in Table 1 are adop-ted from Refs. [20e22]. The results of numerical solution ofEq. (18) for absorption of different gases by falling waterdroplet show that the higher the thermal effect during absorp-tion, the higher is the value of the dimensionless temperaturein the bulk of a droplet. Time dependencies of the dimension-less concentrations of the dissolved gas and temperaturein the bulk of water droplet for absorption of ammonia fordifferent values of Peclet numbers PeD are shown in Figs. 4and 5. In Figs. 6 and 7 we compared the results of numericalcalculations of concentration and temperature in the bulkof a droplet using Eqs. (17) and (18) with experimentaldata obtained by Paniev [23] and Burdukov et al. [24] for ab-sorption of water vapor by falling droplets of aqueous solu-tion of LiBr with internal circulation. The experimentswere carried out in absorber made in the form of a hollow ver-tical, cylindrical column having the internal diameter of800 mm and a test section 1.1 m long. The control sectionswere located at distances 75, 200, 350, 550, 750 and1100 mm from the nozzle, and in each section theprobes to control the temperature and concentration of thesolution were set. Initial parameters of the solution wereT0¼ 298.35 K, x0¼ 0.466. Experimental results presented

Table 1

Values of xs, L, Le, K and d for absorption of different gases in water at normal temperature and pressure (NTP)

Gas xs L [kJ/kg] Le K d [K�1]

CO2 1.69� 10�3 418 1.37� 10�2 �120 �8.24� 10�5

H2S 3.85� 10�3 627 10�2 �42 �1.57� 10�4

Cl2 7.29� 10�3 334 8.7� 10�3 �33 �3.72� 10�4

SO2 0.1015 1254 1.05� 10�2 �0.58 �5.75� 10�3

NH3 0.3468 2090 1.13� 10�2 �0.336 �5.97� 10�3

HCl 0.4189 450 1.88� 10�2 �1.433 �1.55� 10�3

280 T. Elperin et al. / International Journal of Refrigeration 30 (2007) 274e281

in Figs. 6 and 7 correspond to gas absorption by droplets withR¼ 0.419 mm moving with the velocity equal to 7.1 m s�1.Peclet number in these experiments was PeD¼ 5.96� 104,and coefficient k was equal to 0.04. Numerical results, shownin Figs 6e7 were obtained for PeD¼ 5.96� 104 andk¼ 0.04. Due to a small time of contact in experiments ofPaniev [23] and Burdukov et al. [23] the droplets in theseexperiments did not attain saturation. The discrepancybetween the experimental and theoretical results is causedby large scatter of experimental results. The possible secondreason for the discrepancy is that theoretically determinedvalue of coefficient k (see Ref. [13, p. 386]) can differfrom a real value. Unfortunately the relation between themaximum interfacial velocity and the velocity of a dropletwas not measured in the experiments of Paniev [23] andBurdukov et al. [24]. In spite of a high scatter of experimental

0.0

0.05

0.1

0.15

0.2

0.25

0.3

0 100 200 300 400 500

X B(τ

)=(x

B(τ)

-x0)

/(dT 0

+b-x

0)

τ =tUk/R

1234

5

6 1-PeD=1.000

2-PeD=5.000

3-PeD=10.000

4-PeD=50.000

5-PeD=100.000

6-PeD=500.000

Fig. 4. Dependence of ammonia concentration in the bulk of a drop-

let of aqueous ammonia solution on time for different Peclet

(PeD¼ RkU/D) numbers.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500

B(τ)

=(T B

(τ)-T

0)/[(

x 0-b

)/dT 0

]

τ = tUk/R

1 234 5

61-PeD=1.000

2-PeD=5.000

3-PeD=10.000

4-PeD=50.000

5-PeD=100.000

6-PeD=500.000

Fig. 5. Dependence of droplet temperature on time for ammonia

absorption by aqueous ammonia solution for different Peclet

(PeD¼ RkU/D) numbers.

results, theoretical curves and experimental data show fairlygood agreement. To the best of our knowledge the studies ofPaniev [23] and Burdukov et al. [24] are the only publishedexperimental investigations where temperature increase ina falling droplet caused by heat release during absorptionwas measured. Certainly, the suggested model can be vali-dated and improved using new experimental results includ-ing simultaneous measurements of the rate of absorptionand fluid velocity at the droplet interface.

5. Summary and conclusions

The system of transient partial parabolic differentialequations of convective diffusion and energy balance withtime-dependent boundary conditions was solved by combin-ing the similarity transformation method with Duhamel’s

0.0

0.02

0.04

0.06

0.08

0.1

0 50 100 150 200

theory Eq. (17)experiment Burdukov et al. [24]

X B(τ

)=(x

B(τ)

-x0)

/(dT 0

+b-x

0)

τ= tUk/R

Fig. 6. Dependence of an absorbate concentration in the bulk of

a droplet on time for water vapor absorption by aqueous solution

of LiBr.

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200

theory Eq. (18)experiment Burdukov et al. [24]

B(τ)

=(T B

(τ)-T

0)/[(

x 0-b

)/dT 0

]

τ=tUk/R

Fig. 7. Dependence of temperature in the bulk of a droplet from

time for water vapor absorption by aqueous solution of LiBr.

281T. Elperin et al. / International Journal of Refrigeration 30 (2007) 274e281

theorem. It is shown that the dependence of radius-averaged concentration and temperature vs. time in a fallingdroplet is determined by Volterra integral equation of thesecond kind. The performed analysis allowed us to elucidatethe mechanism of the effect of heat release during absorptionon the rate of mass transfer. Heat released during absorptioncauses the increase of a droplet temperature and the equilib-rium temperature increase at the gaseliquid interface. Inaccordance with Eq. (7) the increase of an interfacial tem-perature causes a decrease of the equilibrium absorbate con-centration, reduces a driving force of mass transfer and massflux during absorption. Concentration and temperature in thebulk of a falling droplet at t/N attain their asymptoticvalues corresponding to the new values of temperature andconcentration in the bulk of a droplet. The obtained theoret-ical results were compared with the experimental data ofPaniev [23] and Burdukov et al. [24] for water vapor absorp-tion by falling aqueous solution of LiBr droplet with internalcirculation. The discrepancy between experimental andtheoretical results can be explained by a large scatter inthe experiments. Because of a small time of contact in theexperiments droplets did not attain saturation when theypassed through a contactor. The derived solution can beused in the design calculations of gaseliquid contactorsand incorporated into the system design computer codes.The simple form of the obtained solutions allows us to usethem to analyze the dependence of the rate of mass transferon different parameters, e.g., upon a radius of a droplet, dif-fusion coefficient, etc. The obtained results allow construct-ing a database that can be used for training artificial neuralnetworks (ANN) applied for gaseliquid absorption spraycolumns modeling (see e.g. Refs. [25,26]).

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