Heat and Mass Transfer 39 (2003) 195–199 DOI 10.1007/s00231-002-0333-8
Model of gas absorption in gas-liquid plug flow with first-order and zero-order chemical reaction T. Elperin, A. Fominykh
195 Abstract A model of mass transfer during gas absorption in gas-liquid plug flow accompanied by irreversible chemical reaction of the first order and zero order is suggested. The expressions for coefficients of mass transfer during chemical absorption from a single Taylor bubble are derived in the approximation of the thin concentration boundary layer in a liquid phase. Under the assumptions of a perfect liquid mixing in liquid plugs recurrent relations for the dissolved gas concentrations in the n-th liquid plug and mass fluxes from the n-th Taylor bubble are derived. The total mass fluxes in gas-liquid plug flow during chemical absorption are determined. In the limiting case of absorption without chemical reaction the derived formulas recover the expressions for mass transfer during physical absorption in gas-liquid plug flow. Theoretical results are compared with available experimental data.
r S us
uL u0 U1 x, y
zero-order reaction-rate constant surface area of a Taylor bubble liquid velocity at gas liquid interface in a falling liquid film between a Taylor bubble and a wall in coordinate system, fixed with a moving Taylor bubble liquid velocity in a liquid plug in coordinate system, fixed with a moving Taylor bubble average liquid velocity in the falling liquid film rise velocity of a Taylor bubble in a stagnant fluid coordinates
Opertaional symbols max(n1, n2) defined as equal to larger of n1 and n2 Nomenclature A cross section area of a tube c0 initial concentration of dissolved gas in liquid ci concentration of the dissolved gas at the tail of the i-th liquid plug ci0 concentration of the dissolved gas at the front of the i-th liquid plug Qci mass flux from the i-th Taylor bubble total mass flux from N Taylor bubbles QcRN total mass flux from N Taylor bubbles for Q0cRN physical absorption QL liquid flow rate k first-order reaction-rate constant length of a Taylor bubble, LG length of a liquid plug LL m(s) amount of gas absorbed per unit area after contact time s N number of Taylor bubbles
Received: 26 March 2001 Published online: 6 September 2002 Springer-Verlag 2002 T. Elperin (&), A. Fominykh The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
Greek symbols b coefficient of mass transfer for a single Taylor bubble for a case of physical absorption s contact time Subscripts c G L s
concentration gas liquid value at interface
1 Introduction Interest to a problem of mass transfer during physical and chemical absorption in gas-liquid plug flow is determined by high rates of mass transfer in plug flow attained due to complete destruction of the concentration boundary layer at the trailing edge of a Taylor bubble by a vortex in a liquid plug (see, e.g. [1–4]). Problem of mass transfer during physical absorption of a soluble gas from a rising Taylor bubble was solved by van Heaven and Beek [5]. Authors of work [5] took into consideration, that the surface elements, which are formed at the Taylor bubble top are stretched during their movement down the Taylor bubble. The stretching in the direction of flow is due to their acceleration and the stretching normal to the flow
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corresponds to their increase in circumference. van Heaven and Beek [5] assumed free fall of the liquid along the Taylor bubble surface. The bubble-liquid interface was assumed to be a surface of revolution obtained by the rotation of a curve y(x) around a x-axis. Applying the general theory of mass transfer across free surfaces by Beek and Kramers [6] to Taylor bubbles, van Heaven and Beek [5] derived equation for mass transfer coefficient for a Taylor bubble, multiplied on the bubble’s surface area. Combined mass and heat transfer during nonisothermal absorption from a single Taylor bubble was first investigated by Infante Ferreira [7]. Author of work [7] approximated a Taylor bubble by half a sphere nose and cylindrical body. He assumed that the lower part of a Taylor bubble is merely a region of falling film flow. It follows from investigation of Infante Ferreira [7] that the effect of the bubble nose on the temperature and concentration profile in a liquid film is very small. Gas phase controlled mass transfer from a short Taylor bubble was first investigated by Filla et al. [8]. Authors of work [8] took into consideration ingomogenity of concentration of a soluble gas in the gas phase. Theoretical analysis of concentration profiles within the bubble was based on two alternative assumptions: (a) that there was streamline motion in the toroidal vortex, and (b) that there was complete mixing in the bulk of a bubble except for the concentration boundary layer near the interface. Comparison of theoretical results with experimental measurements of gas phase controlled mass transfer coefficient shows that concentration boundary layer model gives good prediction of gas phase mass transfer coefficient for Pe > 100. Authors of papers [9–10] derived the recurrent formulas for the concentration and temperature in the n-th liquid plug and determined expressions for mass and heat fluxes from the n-th Taylor bubble and the total mass and heat fluxes from N Taylor bubbles in a linear cluster of Taylor bubbles under the assumption of a perfect liquid mixing in a liquid plug by a vortex and homogeneous distribution of temperature and concentration of the dissolved gas in each liquid plug. Case of inhomogeneous distribution of temperature and concentration of the dissolved gas in each liquid plug was analyzed in [11]. Coupled influence of heat release and finite absorbate concentration level on the rate of mass transfer during gas absorption from a single rising Taylor bubble was investigated theoretically in [12]. In all the above referenced papers mass transfer from a single Taylor bubble and in gas-liquid plug flow was investigated for a case of physical absorption. In spite of a large number of works devoted to mass transfer with chemical reaction in two-phase bubbly, droplet, film and foam flows (see works [13–19]) diffusional transport enhanced by chemical reaction in gasliquid plug flow was not investigated.
following approximations and assumptions: 1) infinite dilution of an absorbate; 2) thermodynamic parameters of a system are constant; 3) mass transfer does not affect the hydrodynamics in the liquid phase; 4) equilibrium condition at the gas-liquid interface; 5) diffusion boundary layer is thin; 6) all Taylor bubbles rise with a constant velocity in a vertical pipe and are separated by liquid plugs (Fig. 1); 7) the lengths of all Taylor bubbles and of all liquid plugs are equal, but the length of a Taylor bubble can differ from the length of a liquid plug; 8) rate of chemical reaction is slow and concentration of the dissolved gas is not depleted; 9) form of a Taylor bubble is cylindrical; 10) region between a Taylor bubble and a wall of a tube is a region of a falling liquid film.
3 Chemical reaction of the first order Gas absorption with a first-order chemical reaction from the i-th Taylor bubble is described by the equation of convective diffusion in a system of coordinates fixed with a moving Taylor bubble: us
@c @2c ¼ D 2 kc @x @y
ð1Þ
with boundary conditions x ¼ 0 : c ¼ ci1 ;
y ¼ 0 : c ¼ cs ;
y ! 1 : c ¼ ci1 expðkx=us Þ
ð2Þ ð3Þ
Solution of (1)–(3) for kLG/us 1 yields the formula for the average mass flux (for details see [20–21]): Qci ¼ cs fQL ð1 þ Ks =4Þ ci1 fQL ð1 Ks =2Þ ð4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where f = bS/QL, b ¼ 2 Dus =ðpLGÞ, Ks = kLG/us. Concentrations of the dissolved gas at the tail and at the front of a liquid plug are related as follows: ci ¼ ci0 expðKL Þ
2 Description of a model In this study a model developed for the analysis of mass transfer during physical gas absorption in gas-liquid slug flow [9] is generalized for a case of gas absorption accompanied by a slow irreversible chemical reactions of the first and zero order. A model is developed using the Fig. 1. Schematic view of gas-liquid plug flow
ð5Þ
where kL = kLL/uL. Relation between the concentrations of the dissolved gas at the front of the i-th liquid plug and at the tail of (i–1)-th liquid plug reads: ci0 ¼
Qci þ ci1 expðK0 Þ QL
ð6Þ
where K0 = kLG/u0, QL ¼ U1 A. From Eq. (4)–(6) we obtain the following relation between ci and ci–1: ci ¼ ci1 ½expðK0 Þ expðKL Þ fð1 Ks =2Þ expðKL Þ þ cs fð1 þ Ks =4Þ expðKL Þ
197
ð7Þ
From (7) we obtain a recurrent equation for dissolved gas concentration at the tail of the i-th liquid plug: ci ¼ c1 ð1 li Þ
ð8Þ
Fig. 2. Dependence of dimensionless rate of absorption from dimensionless rate of first-order chemical reaction
where c1 is equal to concentration of dissolved gas in the i-th liquid plug, when i fi ¥: where the step function H(c) is defined as H(c) = 1 when c ‡ 0 and H(c) = 0 when c £ 0. Solution of Eq. (12) with cs fð1 þ Ks =4Þ expðKL Þ c1 ¼ ð9Þ boundary conditions (2) and 1l rx and l = exp (–(KL + K0))[1– f(1 – Ks/2)exp(K0)]. From y ! 1 c ¼ max ci1 ; 0 ð13Þ us Eq. (4) and (8) we obtain recurrent relation for mass flux for the arbitrary i-th Taylor bubble: yields the formula for the average mass flux (for details see [20] and [23]): i1 fð1 Ks =2Þ expðKL Þð1 l Þ Qci ¼ cs fQL ð1 þ Ks =4Þ 1 2rLG 1l Qci ¼ fQL ðcs ci1 Þ þ fQL ð14Þ 3us ð10Þ rLG ci1 From (10) we determine equation for a dimensionless total Equation (14) is valid if cs us < f cs . Concentrations of the dissolved gas at the tail and at the front of a liquid plug mass flux from N Taylor bubbles: are related as follows: QcRN Nfð1 þ Ks =4Þ½1 expððKL þ K0 ÞÞ ¼ rLL N Q0cRN ð1 lÞ 1 ð1 fÞ ci ¼ max ci0 ;0 ð15Þ uL f 2 ð1 þ Ks =4Þð1 Ks =2Þ expðKL Þð1 lN Þ ð11Þ Relation between the concentrations of the dissolved gas at þ ð1 lÞ2 1 ð1 fÞN the front of the i-th liquid plug and at the tail of (i–1)th liquid plug reads: where N is a number of Taylor bubbles flowing through a cross section of a liquid plug during the residence time of a c ¼ Qci þ c rLG ð16Þ N i0 i1 P QL u0 liquid plug in a channel, Q ¼ Q ; Q0 ¼ c Q ½1 cRN
N
i¼1
ci
cRN
s
L
ð1 fÞ (see [9]). Expressions (8), (10) and (11) are derived for c0 = 0. Dependence of QcRN =Q0cRN from Ks for different number of Taylor bubbles for f = 0.01, LG = LL is shown at Fig. 2. It is taken into consideration, that dimensionless rates of reaction KS, KL and K0 are related as follows: KL = KsLLus/(LGuL), K0 = Ksus/u0. Relations between us, u0 and uL are known (see, e.g. [22]).
Substituting Qci from (14) to (16) and taking into consideration (15), we obtain relation between ci and ci–1: 2 ð17Þ ci ¼ ci1 ð1 fÞ þ cs f 1 þ Rs R0 RL 3 G G L where Rs ¼ crL ; R0 ¼ crL ; RL ¼ crL From (17) we s us s u0 s uL obtain a recurrent equation for dissolved gas concentration at the tail of the i-th liquid plug:
3.1 ci ¼ c1 ð1 ð1 fÞi Þ þ c0 ð1 fÞi Zero-order reaction Gas absorption with a zero-order chemical reaction from where expression for ci when i fi ¥ is: the i-th Taylor bubble is described by the equation of convective diffusion in a system of coordinates fixed with a 2Rs 1 f ðR0 þ RL Þ c1 ¼ cs 1 þ moving Taylor bubble: 3 us
@c @2c ¼ D 2 rHðcÞ @x @y
ð12Þ
ð18Þ
ð19Þ
From Eq. (14) and (18) we obtain recurrent relation for mass flux for the arbitrary i-th Taylor bubble:
Qci ¼ Qc1 þ fQL cs ð1 c0 =cs þ Rs Qc1 Þ ð1 fÞi1
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27 cm and the film surface velocity was about 0.32 m/s. In order to realize the first-order irreversible chemical reacð20Þ tion and to avoid a change in OH-ion concentration, conditions: where mass flux from the i-th Taylor bubble, when i fi ¥ experiments were evaluated at the following 4 1) Small partial pressure of CO (2Æ10 Pa) resulting in low 2 is determined as follows: value of concentration of saturation of gas in a water Qc1 ¼ QL cs ðR0 þ RL Þ ð21Þ solution; 2) Small contact time; 3) Low temperature (between 293 and 303 K). In all experiments the initial From (20) we determine equation for absorption rate ratio: concentration of CO in water solutions was equal to zero. 2 QcRN NðR0 þ RL Þ ð2=3ÞRs f 1 ðR0 þ RL Þ From Eq. (4) we obtain the following expression for the þ ¼1þ amount of gas absorbed per unit area after contact time s: ð1 c0 =cs Þ Q0cRN ð1 c0 =cs Þð1 ð1 fÞN Þ rffiffiffiffi D ks ð22Þ mðsÞ pffiffiffi ¼ 2cs 1þ ð25Þ p 4 s where Q0cRN is determined as follows (see [9]): Experimental values pffiffiffiffi of the first-order reaction velocity Q0cRN ¼ cs QL ð1 c0 =cs Þð1 ð1 fÞN Þ ð23Þ constant k and cs D for pure CO2 absorption in Na-buffer solutions at different temperatures are presented in Dependence of QcRN =Q0cRN from dimensionless rate of Table 1. Theoretical results (Eq. (25)) were compared with zero-order chemical reaction Rs for different number of the experimental data for CO2 absorption in Na-buffer Taylor bubbles for f = 0.1, LG = LL c0/cs = 0.2 is shown at aqueous solutions at 293, 298 and 303 K (see Fig. 4). InFig. 3. spection of Fig. 4 shows that theoretical predictions are in fairly good agreement with the experimental results [24].
4 Comparison with experimental results Nysing and Kramers [24] measured the rate of absorption of pure CO2 gas in Na-buffer solutions in a wetted-wall column. The physical and chemical conditions have been chosen in such a way that the simple theory of diffusion accompanied by an irreversible chemical reaction may be applied. In these experiments a liquid film was introduced on the internal wetted wall of a cylindrical channel with internal diameter 28.6 mm through an annular slit with about the same width (0.4 mm) as the thickness of a falling film. In experiments the amount of gas absorbed per unit area after contact time s was measured. The contact time s is determined as h s¼ us
ð24Þ
5 Conclusions Model of gas absorption in gas-liquid plug flow, accompanied by the first-order and zero-order chemical reaction is developed. Recurrent relations for dissolved gas concentrations in the arbitrary liquid plug, mass flux from arbitrary Taylor bubble and total mass flux from arbitrary number of Taylor bubbles are derived. In the case of absorption without chemical reaction the derived formulas recover the expressions for physical absorption obtained Table 1.
[Temperature] K
[k] (s–1)
[293]
[0.56] [0.8] [1.4]
where h is the height of the film and us is the film surface [298] velocity. The height of the falling film varied between 4 and [303]
Fig. 3. Dependence of dimensionless rate of absorption from dimensionless rate of zero-order chemical reaction
pffiffiffiffi –5 cs D 10 kgÆm–2 s–1/2 [0.79] [0.73] [0.7]
Fig. 4. Dependence of m(s) vs. s for CO2 absorption in Na-buffer solutions. Curves 1, 2 and 3-theory, Eq. (25) for T = 293, 298 and 303 K
previously in [9]. Theoretical results for mass transfer from a single Taylor bubble, accompanied by the firstorder irreversible chemical reaction show fairly good agreement with the experimental results for CO2 absorption by laminar falling liquid films of Na-buffer aqueous solutions.
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