Cell model for gas absorption with first-order ... - Springer Link

Heat and Mass Transfer 35 (1999) 357±360 Ó Springer-Verlag 1999

Cell model for gas absorption with first-order irreversible chemical reaction and heat release in gas-liquid bubbly media T. Elperin, A. Fominykh

Abstract A model for mass and heat transfer during physical gas absorption in gas-liquid bubbly medium suggested in [1] is generalized for a case of chemical absorption accompanied by heat release. Diffusion and thermal interactions between bubbles are taken in to account in the approximation of a cellular model of a bubbly medium whereby a bubbly medium is viewed as a periodic structure consisting of identical spherical cells with periodic boundary conditions at a cell boundary. Distribution of concentration of the dissolved gas, temperature distribution in liquid and rates of mass and heat transfer during nonisothermal chemical absorption of a soluble pure gas from a bubble by liquid are determined. In the limiting case of chemical absorption without heat release the derived formulas recover the expressions for isothermal chemical absorption. In the limiting case of physical absorption with heat release the derived formulas recover the expressions for nonisothermal absorption obtained in [1]. List of symbols a thermal diffusivity, m2 sÿ1 b constant in formula (9), kg mÿ3 cp speci®c heat, kJ kgÿ1 Kÿ1 c concentration of a soluble gas, kg mÿ3 cs concentration at an interface, kg mÿ3 c0 equilibrium concentration at initial temperature, kg mÿ3 d constant in formula, (9), kg mÿ3 Kÿ1 D coef®cient of molecular diffusion, m2 sÿ1 Da Damkohler number, k`2 =D f …t; r† ˆ …T ÿ T0 †r=`, K g…t; r† ˆ cr=`; kg mÿ3 cp q K dimensionless number, dDH ÿ1 k reaction rate, s ` ˆ R ÿ r0 thickness of a liquid layer in a cell, m Le Lewis number, D=a qc mass ¯ux during absorption with heat release and chemical reaction, kg sÿ1 mÿ2

Received on 20 April 1998

T. Elperin A. Fominykh Pearistone Center for Aeronautical Engineering Department of Mechanical Engineering Ben-Gurion University of the Negev P.O. Box 653, Beer Sheva 84105, Israel Correspondence to: T. Elperin

qc1 qc2 qc3 qT qT1 r0 r R t T T0 Ts Greek g k q s u DHS DHR DH

mass ¯ux during absorption with chemical reaction and without heat release, kg sÿ1 mÿ2 mass ¯ux during absorption with chemical reaction and without heat release, kg sÿ1 mÿ2 mass ¯ux during absorption without heat release and without chemical reaction, kg sÿ1 mÿ2 heat ¯ux during absorption with heat release and chemical reaction, kJ sÿ1 mÿ2 heat ¯ux during absorption with heat release and without chemical reaction, kJ sÿ1 mÿ2 bubble radius, m coordinate, m cell radius, m time, c temperature of a liquid, K equilibrium temperature at initial concentration, K temperature at an interface, K symbols ˆ …r ÿ r0 †=` thermal conductivity, kJ mÿ1 sÿ1 Kÿ1 liquid density, kg mÿ3 Dt=`2 , variable void fraction heat of solution, kJ kgÿ1 heat of reaction, kJ kgÿ1 ˆ DHS ‡ DHR ; kJ kgÿ1

Subscripts c concentration p at constant pressure s value at the interface T temperature 0 value at inlet

1 Introduction Thermal effects are important in a number of gas-liquid processes involving chemical reaction in a liquid phase. Dissolution of gas into liquid is accompanied by heat release, DHS , at the gas-liquid interface. In addition, the heat of the exothermic chemical reaction, DHR , is released in a liquid phase. Some industrially important gas-liquid systems which exhibit signi®cant thermal effect during chemical absorption are sulfur-trioxide-dodecylbenzene, hydrogen chloride-ethylene glycol and ammonia-water. Penetration and ®lm theory for exothermic gas absorption

357

358

with irreversible chemical reaction of the ®rst order were developed in [2±9]. State-of-the art in penetration and ®lm models of exothermic chemical absorption is presented in [10]. Temperature elevation at the free gas-liquid surface for a case of gas absorption which is accompanied by a rapid ®rst-order chemical reaction in a turbulent liquid ®lm ¯ow was determined in [11]. The main goal of this study is to develop a model for combined mass and heat transfer during nonisothermal chemical gas absorption in a two-phase gas-liquid bubbly medium with a high gas content and/or large times of gas-liquid contact. Diffusional and thermal interactions between bubbles are taken into account in the approximation of a cellular model of a bubbly medium whereby a bubbly medium is viewed as a periodic structure consisting of identical spherical cells with periodic boundary conditions at a cell boundary. It is assumed that the temperature rise due to heat release is small enough that the ®rst-order reaction rate constant, the gas solubility and other physical parameters can be assumed constant. Following the approach developed in [4] heat released due to chemical reaction is accounted for as a heat ¯ux at the boundary.

2 Cell model of absorption with chemical reaction and heat release Consider combined mass and heat transfer during absorption of a pure soluble gas from stationary spherical gas bubbles accompanied by irreversible ®rst order chemical reaction in the liquid phase and heat release. The analysis is performed for a case of large gas content in bubbly medium or for long duration of gas-liquid contact when thermal and concentration boundary layers in a liquid phase near spherical bubbles overlap. In this study we employ an approximation of the in®nite dilution of an absorbate. The thermodynamic parameters of a system are assumed constant, and only resistance to mass and heat transfer in the liquid phase is taken into account. Heat released during absorption is dissipated in a liquid phase, and mass and heat transfer are assumed not to affect the hydrodynamics in a liquid phase. The equilibrium condition at the gas-liquid interface is described by a linear dependence of concentration on temperature. Therefore combined mass and heat transfer during nonisothermal chemical absorption of gas in a bubbly gas-liquid medium is governed by equations of nonstationary diffusion and heat transfer written for the liquid phase:  2  oc o c 2 oc ‡ …1† ˆD ÿ kc; r0  r  R ot or2 r or  2  oT o T 2 oT ‡ …2† ˆa ; r0  r  R ot or 2 r or

oc oT ˆ 0; ˆ 0; at r ˆ R …5† or or Using the method suggested by Danckwerst [12] we derived expressions for the transient concentration distribution and the rate of absorption by transformation of the solutions obtained in [1] for diffusion without reaction: ÿ 2
1 X cn ‡ `2 =R2 sin…cn g† g…s; g†` 1 ÿ R` …1 ‡ gR=`† ÿ
ÿ2 ˆ 2 2 2 cs r0 1 ÿ `=R nˆ1 cn ` =R ÿ `=R ‡ cn ÿ  ÿ

1 ‡ c2n Daÿ1 exp ÿsDa 1 ‡ c2n Daÿ1 ÿ
 1 ‡ c2n Daÿ1 …6† 2

2

where s ˆ Dt=` , Da ˆ k` =D, g…t; r† ˆ cr=`, g ˆ r ÿ r0 =`, cn ; n ˆ 1; 2; . . . are the positive roots of a transcendental algebraic equation

c cot c ˆ `=R

…7†

The ®rst six roots, cn , of Eq. (7) for different values of `=R are given in Table 1. Temperature distribution in a liquid phase is described by Eq. (14) of the previous study [1]:

1 ÿ R` …1 ‡ gR=`† f …s; g†` ˆ 1 ÿ `=R …Ts ÿ T0 †r0 ÿ
1 X 2 c2n ‡ `2 =R2 sin…cn g† ÿ
ÿ 2 2 2 nˆ1 cn ` =R ÿ `=R ‡ cn ÿ
 exp ÿc2n Leÿ1 s

…8†

where f …t; r† ˆ …T ÿ T0 †r=`. The unknown values of concentration and temperature at the gas-liquid interface can be determined from two boundary conditions:

c ˆ dT ‡ b at r ˆ r0 oT oc ˆ DHD at r ˆ r0 k or or

…9† …10†

Table 1. The ®rst six roots cn of equation c cot c ˆ `=R `=R

1.0 0.995 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.90 0.85 0.8 Partial differential equation describing heat transfer re0.7 duces to Eq. (2) because heat released due to chemical 0.6 reaction is considered as a heat ¯ux at the boundary (see, 0.5 e.g., [4]). The system of equations (1)±(2) is solved with 0.4 the following initial and boundary conditions: 0.3 0.2 c ˆ 0; T ˆ T0 ; at t ˆ 0 …3† 0.1 c ˆ cs ; T ˆ Ts ; at r ˆ r0 …4† 0

c1

c2

c3

c4

c5

c6

0 0.1224 0.1730 0.2445 0.2991 0.3450 0.3854 0.4217 0.4551 0.4860 0.5150 0.5423 0.6609 0.7593 0.9208 1.0528 1.656 1.264 1.3525 1.4320 1.5044 1.5708

4.4934 4.4945 4.4956 4.4979 4.5001 4.5023 4.5045 4.5068 4.5090 4.5112 4.5134 4.5157 4.5268 4.5379 4.5601 4.5822 4.6042 4.6261 4.6479 4.6696 4.6911 4.7124

7.7253 7.7259 7.7265 7.7278 7.7291 7.7304 7.7317 7.7330 7.7343 7.7356 7.7369 7.7382 7.7447 7.7511 7.7641 7.7770 7.7899 7.8028 7.8156 7.8284 7.8412 7.8540

10.9041 10.9046 10.9050 10.9060 10.9069 10.9078 10.9087 10.9096 10.9105 10.9115 10.9124 10.9133 10.9179 10.9225 10.9316 10.9408 10.9499 10.9591 10.9682 10.9774 10.9865 10.9956

14.0662 14.0666 14.0669 14.0676 14.0683 14.0690 14.0697 14.0705 14.0712 14.0719 14.0726 14.0733 14.0769 14.0804 14.0875 14.0946 14.1017 14.1088 14.1159 14.1230 14.1302 14.1372

17.2208 17.2210 17.2213 17.2219 17.2225 17.2231 17.2237 17.2242 17.2248 17.2254 17.2260 17.2266 17.2295 17.2324 17.2382 17.2440 17.2498 17.2556 17.2614 17.2672 17.2730 17.2788





where DH ˆ DHS ‡ DHR . Equation (9) is a condition of F…s† 1 ÿ LeB…s† KA…s† equilibrium at the gas-liquid interface and Eq. (10) implies qc ˆ   LeF…s† that all heat released during absorption is dissipated in the qc1 B…s† 1 ÿ KA…s† liquid phase. Note that a coef®cient d in formula (9) is KA…s† negative. Equations (6)±(10) yield the following explicit 1 ÿ LeB…s† qT expressions for Ts and cs : ˆ K A…s† ÿ T0 Le c00 F…s† Ts ˆ ; c ˆ s F…s† K A…s† 1 ÿ Le 1 ÿ Le F…s† K A…s† cp q where Le ˆ Da, K ˆ dDH ; c00 ˆ dT0 ‡ b; T00 ˆ …ÿb†=d 1 r0 =` r0 X 2…c2n ‡ `2 =R2 † ÿ
A…s† ˆ ‡ 1 ÿ `=R ` nˆ1 `2 =R2 ÿ `=R ‡ c2n

T00

qT1

…11† and

ÿ
 exp ÿc2n Leÿ1 s ‡ 1; …12† ÿ
1 2 c2n ‡ `2 =R2 r0 =` r0 X ÿ
‡ F…s† ˆ 1 ÿ `=R ` nˆ1 `2 =R2 ÿ `=R ‡ c2n ÿ  ÿ

1 ‡ c2n Daÿ1 exp ÿsDa 1 ‡ c2n Daÿ1 ÿ
 ‡1 1 ‡ c2n Daÿ1

KA…s† 1 ÿ LeF…s†

qc2 F…s† ˆ qc3 B…s†

…21†

…22† …23†

Dependence of qc =qc1 ; qT =qT1 and qc2 =qc3 vs. s in gasliquid bubbly media for rb ˆ 1 mm and u ˆ 0:01 for different values of Le/K and Da is shown in Figs. 2±4.

3 Discussion and conclusions Results presented in Figs. 2±4 show that mass and heat ¯uxes during absorption accompanied by irreversible chemical reaction of the ®rst order increase with the in-

…13† Mass and heat ¯uxes from an unit cell can be determined from Eqs. (6)±(8) and Eqs. (11)±(13):

  oc DF…s†r0ÿ1 c00 qc ˆ ÿD ˆ or rˆr0 1 ÿ LeF…s† KA…s† ÿ
  kA…s†r0ÿ1 T00 ÿ T0 oT qT ˆ ÿk ˆ K A…s† or rˆr0 1 ÿ Le F…s†

…14† …15†

In a case without chemical reaction (Da ˆ 0) Eqs. (14)± (15) recover formulas for nonisothermal absorption without chemical reaction:

  oc DB…s†r0ÿ1 c00 qc1 ˆ ÿD ˆ or rˆr0 1 ÿ LeB…s†

…16†

Fig. 1. Schematic view of a unit cell in bubbly media

KA…s†

and

qT1

ÿ
  kA…s†r0ÿ1 T00 ÿ T0 oT ˆ ÿk ˆ or rˆr0 1 ÿ K A…s†

…17†

Le B…s†

where

B…s† ˆ

1 r0 =` r0 X 2…c2n ‡ `2 =R2 † ÿ
‡ 1 ÿ `=R ` nˆ1 `2 =R2 ÿ `=R ‡ c2n ÿ
 exp ÿc2n s ‡ 1

…18†

In a case without heat release (K ! 1) Eq. (14) recovers the formula for an isothermal absorption accompanied by the ®rst order irreversible chemical reaction:

qc2 ˆ cs DF…s†r0ÿ1

…19†

If Da ˆ 0 and K ! 1, Eq. (14) implies that

qc3 ˆ

cs DB…s†r0ÿ1

Using Eqs. (14)±(20) we obtain that

…20†

Fig. 2. Dependence of the dimensionless mass ¯ux qc =qc1 vs. s, Eq. (21). Solid line ± Le/K ˆ 1; dash line ± Le/K ˆ 2:1 ± Da = 0.02; 2 ± Da = 0.05; 3 ± Da = 0.1; 4 ± Da = 0.2; 5 ± Da = 0.5; 6 ± Da = 2.0

359

360

Fig. 3. Dependence of the dimensionless heat ¯ux qT =qT1 vs. s, Eq. (22). Solid line ± Le/K ˆ 1; dash line ± Le/K ˆ 1:5; 1 ± Da = 0.02; 2 ± Da = 0.05; 3 ± Da = 0.1; 4 ± Da = 0.2; 5 ± Da = 0.5; 6 ± Da = 2.0

Fig. 4. Dependence of the dimensionless mass ¯ux qc2 =qc3 vs. s, Eq. (23). 1 ± Da = 0.02; 2 ± Da = 0.05; 3 ± Da = 0.1; 4 ± Da = 0.2; 5 ± Da = 0.5; 6 ± Da = 2.0

crease of Damkohler number. Analysis of Eqs. (14) and 3. Danckwerts PV (1967) Gas absorption accompanied ®rst(21) and results presented in Fig. 2 show that mass transfer order reaction: concentration of product, temperature-rise and depletion of reactant. Chem Eng Sci 22: 472±473 during nonisothermal chemical absorption decreases with 4. Clegg GT; Mann R (1969) A penetration model for gas an increase of the rate of heat release during dissolution absorption with ®rst order chemical reaction accompanied and chemical reaction. Concentration and temperature by large heat effects. Chem Eng Sci 24: 321±329 distributions attain stationary values at s  30. In this 5. Shah YT (1972) Gas-liquid interface temperature rise in study combined mass and heat transfer during nonisothe case of temperature-dependent physical, transport and thermal chemical absorption in a bubbly gas-liquid mereaction properties. Chem Eng Sci 27: 1469±1474 dium at large gas contents and/or long duration of gas6. Cook AE; Moore E (1972) Gas absorption with a ®rst order chemical reaction and large heat effect. Chem Eng Sci 27: liquid contact is analyzed in the approximation of an 605±613 in®nite dilution of the absorbate. Diffusion and thermal 7. Tamir A; Danckwerts PV; Virkar PD (1975) Penetration interactions between gas bubbles are taken into account in model for absorption with chemical reaction in the presence the approximation of a cellular model of a bubbly medium of heat generation, bulk ¯ow and effects of the gaseous enwhereby a bubbly medium is viewed as a periodic strucvironment. Chem Eng Sci 30: 1243±1250 ture consisting of identical spherical cells with periodic 8. Asai S; Potter E; Hikita H (1985) Nonisothermal gas abboundary conditions at a cell boundary. Rates of mass and sorption with chemical reaction. AIChE J 31: 1304±1312 heat transfer during nonisothermal absorption for bubbly 9. Chatterjee SG; Altwicker ER (1987) Film and penetration theories for a ®rst-order reaction in exothermic gas absorpmedium are determined. In the limiting case of chemical tion. Canadian J Chem Eng 65: 454±461 absorption without heat release the derived formulas 10. Vas Bhat RD; van Swaaij WPM; Benes NE; Kuipers JAM recover the expressions for the isothermal chemical (1997) Non-isothermal gas absorption with reversible chemabsorption in bubbly medium. ical reaction. Chem Eng Sci 52: 4079±4094

11. Sandall O (1975) Heat effects in gas absorption accompanied by a rapid ®rst-order reaction in a turbulent liquid ®lm. 1. Elperin T; Fominykh A (1996) Cell model of nonisothermal Canadian J Chem Eng 53: 702±705 absorption in gas-liquid bubbly media. Heat and Mass 12. Danckwerts PV (1951) Absorption by simultaneous difTransfer 31 (5): 307±313 fusion and chemical reaction into particles of various 2. Danckwerts PV (1952) Temperature effects accompanying the shapes and into falling droplets. Trans Faraday Soc 47: absorption of gases by liquids. Appl Sci Research 3: 358±360 1014±1026

References