Ind. Eng. Chem. Res. 1990,29, 101-105
101
Extraction of Ammonia from a Dilute Aqueous Solution by Emulsion Liquid Membranes. 2. Theory and Mass-Transfer Model? Chau J. Lee* and Chih C. Chant Department of Chemical Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC
A mass-transfer model built with an immobilized hollow spherical globule configuration and gross film theory is being proposed to interpret the laboratory data of ammonia extraction from a dilute aqueous solution by emulsion liquid membrane (ELM) technology. The model assumes acid-base reaction equilibrium to exist in both internal droplet and external aqueous phases. The effect of solute leakage due to membrane rupture (in terms of a time-varying leakage function) has been considered to account for the reverse transport of ammonia back into the external phase (Figure 3), which adversely affects the extraction efficiency. The theory and model computations can predict fairly well the general trends of extraction phenomena in the batch system operations described in part 1. 1. Introduction
Emulsion liquid membrane (ELM) technology has been extensively studied in recent years because of its potential applications in many industrial, environmental, and biomedical systems (Cahn and Li, 1974; Chan, 1987; Kopp, 1982). The mathematical models of these systems are important in the interpretation of laboratory data and in the design and scale-up of such systems. A number of mathematical models have been proposed in the literature to describe the behavior of separation processes using emulsion liquid membranes (Chan and Lee, 1984). In fact, all models are based on the "emulsion globule model", which may be catagorized into five different models owing to different assumptions on the geometric configurations (see Figure 1): (a) uniform flat-plate model, (b) hollow sphere model, (c) immobilized hollow sphere model, (d) immobilized spherical globule model, (e) immobilized hollow spherical globule model. Among them, models a and b assume that the internal solution droplets dispersed within the W/O emulsion globule are completely mobile and well-mixed. Thus, they may be all combined and treated as a uniform concentration phase (layer) of internal solution. Models c-e assume that the droplets of internal solution are immobile and unmixed, and the concentration of the component to be transported becomes a function of the radial positions. Figure 2 depicts the qualitative concentration profiles. Most of the mathematical models published earlier did not consider the effect of emulsion stability on overall mass transfer. However, our results of experimental studies, presented in part 1, reveal that the leakage of internal solution due to rupture of some emulsions could greatly retard the extraction of NH3 from the external solution. Therefore, the effect of leakage must be considered in the formulation of the model in order to enable the treatment of experimental data. In this paper, a mechanistic formulation based on the immobilized hollow spherical globule model (Figure l(e)), which also takes into consideration the acid-base reaction equilibrium as well as the leakage effect, is being proposed. The results of theoretical model computations are utilized to analyze and to describe the behavior of the experimental * T o whom correspondence should be addressed. Presented a t PACHEC '88, Oct 19-22, Acapulco, Mexico. t Present address: Department of Chemical Engineering, Feng-Chia University, Taichung, Taiwan, ROC. 0888-588519012629-0101$02.50/0
data of ammonia extraction in a batch system (as presented in part 1). 2. Theory The theory is built on an immobilized hollow spherical emulsion globule model which assumes acid-base reaction equilibrium to exist in both the internal encapsulated phase and the external continuous phase. We also consider that the continuous mass-transfer resistance is due to the turbulent boundary layer, the interfacial resistance is due to the surfactant layer, and the leakage of solute is due to the membrane rupture. 2.1. Reaction System and Chemical Equilibrium. The reaction system of the ammonia extraction by an emulsion globule is shown in Figure 1of part 1. Ammonia is a weak base, and it is known to exist as ammonium ion, NH4+,in acidic solution, but it exists as NH3 molecules in basic solution. It is noted that only NH, molecules can be soluble in the oil membrane and be transported across the membrane to be extracted from the external bulk solution. Thus, to enable the extraction of ammonia from the external bulk solution by the emulsion liquid membrane method, the pH of the external solution must be controlled to ensure the solution is under basic condition (i.e., pH > 12). Also, the acid existing in the internal aqueous droplets will react with the NH3 molecules transported across the oil membrane to form ammonium ion (NH4+)and be trapped. This is the mechanism that makes the NH3 extraction process irreversible if no leakage occurred due to membrane rupture. By the principle of chemical equilibrium, the following reaction may be occurring in the external solution,
NH4
+ OH-
-+
NH3 + HzO
(1)
and the total ammonia content (NH, molecules plus NH4+ ions) in the external solution, C,, may be expressed as
where
K , = [H+],[OH-], = 1.00 X 0 1990 American Chemical Society
(at 25 "C)
102 Ind. Eng. Chem. Res., Voi. 29, No. 1, 1990
Real Emulsion Drop
r
Model Emulsion Drop
( a ) Uniform Flat-Plate Model
( b ) Hollow S p h e r e Model ( c ) Immobilized Hollow Sphere Model ( d ) Immobilized Spherical Globule Model ( e ) Immobilized Hollow Spherical Globule Model
Figure 1. Configurations of various emulsion drop models.
Similarly, the reaction and the total ammonia content in the internal solution may be expressed as NH, H+ s NH4+ (4)
+
Figure 2. Mcdeliig the emulsion globule-qualitative concentration profiles.
8. The distribution coefficients of the NH, molecules between the oil membrane and aqueous solution stay constant and are defined as cyi = [NH3],/[NH3li for internal solution a, = [NH,],/[NH,],
for external solution
(8)
9. Under constant stirring speed, the volumetric leakage rate of the emulsion globules (internal solution leaks into external bulk solution) may be expressed as
2.2. Mass-Transfer Model. The immobilized hollow spherical globule model, Figure l(e), is selected to analyze the mass-transfer characteristics of the system. In order to simplify the mathematics and the model development, the following assumptions are made: 1. An ideal batch system exists under complete mixing and constant temperature operation. 2. Physical and transport properties are constant during the process operation. 3. The internal and external aqueous solutions are completely immiscible with the oil membranes. 4. The diameter of the emulsion drops as well as the internal droplets is defined in terms of the Sauter mean diameter, d32;Le., N
nidi3 i=l d,, = N C nidi2
(7)
i=l
5. The droplets dispersed in the emulsion globules are immobile and are uniformly distributed. 6. No concentration gradient exists within the droplets. 7 . No interfacial mass-transfer resistance exists between the oil membrane and the internal aqueous phase.
10. The changes in the total volume of the internal solution, Vi,and of the external solution, V,, are neglected (it seems reasonable when the leakage is not serious). Knowing the transport steps of the NH, molecules from the external solution toward the internal solution containing acid and the above-mentioned assumptions, the following transport equations may be formulated based on the law of mass conservation. For the external bulk solution,
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 103 J*NH3
For emulsion globules,
=
J*NH4
@ ~ 7 j j i= 7'@,~7jji
(23)
@u7jjiNH4 = 7 a @ 0 ~ 7 j j N H 4
(24)
=
J*H = @a4(l+ a2)jjiH= 7'@,u4(1
= @u7jjioH = ~ ' @ . , ~ ~ 7 j j i O ~
J*OH
the initial conditions are t = 0, [NH,], = 0, Ci = 0
with initial conditions
and the boundary conditions are (1) r = 0, d[NH,],/ar
and boundary conditions
= 0, ym = yi = yiNH4 = 0
(25) (26) (27)
=0
r = Ri
(2)
where
_1 - -1 + -1+ -
1
Ko k~ k a e k m effective diffusivity within the emulsion globule, kL is the external phase mass-transfer coefficient, l / k is the interfacial resistance, lzm is the mass-transfer coefficient of the oil-phase membrane, KOis the overall mass-transfer coefficient, and J , is the molar leakage rate of component n. Figure 3 is a pictorial illustration of the relative relationship between the leakage and diffusion of ammonia in the extraction system. The leakage brings the ammonia molecules that were diffused into the internal solution back to the external solution. Also, acid leaking to the external solution will cause the conversion of NH3 molecules to NH4+ions, which is not permeable through the oil membranes. Both have an adverse effect on the extraction efficiency. To accommodate an easier solution of eq 11-17, and for a simpler and clearer presentation of the results of the model calculations, the transport equations are transformed into dimensionless form by defining a series of dimensionless parameters and variables which are documented in the Appendix. The final dimensionless transport equations become
D& is the
I(-
7
+ u2)jjiH
\
u1'4uS -
Equations 20-29 may be solved numerically by the Crank-Nicolson numerical method (Carnahan et al., 1969; Chan and Lee, 1987). The results of the computation with various parameters are used to describe theoretically the behavior of ammonia extraction by ELM in a batch system. 2.3. Parameter Evaluation. Effective Diffusivity, DefpThe diffusion of the NH3 molecule in the emulsion globules is represented by an effective diffusivity, De@The magnitude of such a diffusion coefficient in heterogeneous phases may be calculated with an equation recommended by Jefferson-Witzell-Sibbet (Crank, 1975, Chapter 12),i.e.,
where
p = 0.4034
