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Ind. Eng. Chem. Process Des. Dev. 1980, 19, 220-222
Composition Effect and Thermal Effect to Overall Mass Transfer Coefficient Mitsuyasu Hiranuma Tomakomai Technical College, 443, Nishikioka, Tomakomai, 059- 12, Japan
In the distillation process, the conventional overall mass transfer coefficient can be expressed by an explicit function of composition, when the chemical potential difference of each component is used as a driving force of mass transfer. It is shown that the calculated local efficiency depends on composition and it can have a maximum according to was the system. The relative contribution of the molar flux caused by thermal effect to the total molar flux in estimated for some binary systems.
Introduction It has been indicated by numerous experimental results (for example, Bakowski, 1963) that the performance of a distillation column is affected by the composition of feed materials and that a maximum may occur in the experimental curve of efficiency vs. composition (Ruckenstein and Smigelschi, 1967). Some papers (Zuiderweg and Harmens, 1958; Ruckenstein and Smigelschi, 1965) said that such a maximum can be explained by the Maragoni effect, and others (Sawistowski and Smith, 1959; Liang and Smith, 1962) maintained that it can be explained by the thermal effect. In the present paper, the following problems will be discussed from a different perspective based on the driving force of mass transfer. (1)Why does a maximum exist in the curve performance vs. composition in distillation? (2) What is the relative contribution of the thermal effect to the overall mass transfer? Chemical Potential Difference as Driving Force Consider a heat transfer or a mass transfer from the liquid to the vapor phase. In the case of heat transfer, the interface temperature will be able to be eliminated as the heat transfer rate of the both phases must be equal. 4 = (TL - T , ) / ( l / h ~ l= (TI- T G ) / ( ~ / ~=G )
(TL- T G ) / ( ~ /+~l L/ h ~ )(1)
On the other hand, the mass transfer rate may be expressed in the following equation in accordance with the conventional two-resistance theory.
n = (x - xJ/(l/kJ
= (Y, - S’)/(l/kG)
(2)
However, the unknown interfacial compositions (x,,y,) cannot be eliminated, since x , # y,. As long as the vapor-liquid equilibrium curve can be represented by y* = mx + b (3) the rate per unit area is rewritten as
n = K,G(Y* - y ) = K,L(X - x * ) ;
~ / K , G= l / k G + m/kL (4)
but the equilibrium curve is not linear in distillation, so that KoG will depend significantly on composition of the mixture. It would be thermodynamically more rational to use the chemical potential difference of each component as a driving force in deriving the equation, and consequently the rate of mass transfer n may be written as 0196-4305/80/1119-0220$01 .OO/O
-
( ~ ( x -) p(xi))/(1/kFL) = (p(yi) - P ( Y ) ) / ( ~ / ~ , J = (pU(x) - p b ) ) / ( 1 / k P L + l / k , J = ( d x ) - F(Y))/(~/K,,)
(5) The chemical potentials a t the interface will be satisfactorily able to be eliminated a t equilibrium, since p ( x J = p(yJ a t vapor-liquid equilibrium. New Rate Equation Consider the process in the rectification plate. The quantity of substance i in the vapor passing through the differential section of the tower is Gy, and the rate of mass transfer of i to the vapor is therefore G dy. This may be described as G dy = K,G(y* - Y ) U dz (6) where a dz is the interfacial surface area. Examine the following equation which is described in terms of the chemical potential instead of the mole fraction G d d y ) = K,(p(x) - p b ) ) dz ~ (7) where K , is a proportionality factor and p ( x ) = pL(y*) a t the vapor-liquid equilibrium. The assumption of vapor-phase ideality may be justified in the distillation process a t pressures up to 1 atm. For a perfect gas mixture, a chemical potential of its components is given by the following relation, in which the sign e indicates values of a pure gas at atmospheric pressure. p ( y ) = p e ( T ) + R T In P + R T In y (8) p(x)
= p(y*) = pe(T*)
+ RT* In P + RT* In y*
(9)
where the asterisk indicates equilibrium mole fraction, temperature, etc., with the liquid mole fraction x . As a first approximation, unless the heat effect was taken into account, (T* = T ) ,the substitution of eq 8 and 9 into eq 7 leads to G dy = K , y In ( y * / y ) a dz (10) This is a rate equation in which the quantity of y In (y*/y) is used as a driving force. The Property of K , The transformation mentioned above is not reasonable because of the two following reasons: the distillation process is not under isothermal and not necessarily equimolar counterdiffusion. However, the use of eq 10 might be expected to be still worthy of investigating the property of K , on composition. Comparing eq 6 with eq 10, the following relationship between K , and K,,G is obtained. (11) K,G/K, = y In ( Y * / Y ) / ( Y * - Y ) @ 1980 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 2, 1980 221
satisfies the requirement that the Murphree local efficiency for one component agrees with that for another component in a binary mixture. The use of Ji is reasonable to compare K, with KoG. If Ni is K,G(y* - y ) a dz, eq 14 gives
90 70
JI
Figure 1. Variation of local efficiency with composition: (a) methanol-water; (b) acetone-water; 0 , in the literature (Bakowski, 1963); - - -, eq 13; -- eq 1 2 with eq 20; - - - ,eq 12 with eq 17.
T h e local efficiency on a gas basis is defined by the Murphree efficiency, E. It is indicated in the literature of Gerster (1949) that the Murphree efficiency can be represented in a term of KoG E = (y - y ) / ( y * - y) = 1 - exp(-KOGuz/G) (12) where z is the depth of liquid over the slots or on the plate. The substitution of eq 11 into eq 1 2 gives E = 1 - exp(--(K,uz/G)y In ( y * / y ) / ( y * -. y ) ) (13) Equation 12 cannot represent the composition effect of the mixture on local efficiency with the fixed value of KoG. Since eq 13 is an (explicit function of composition, the composition effect of the mixture on local efficiency must be expressed by eq 13 even if the factor K, is constant in all compositions. The closest approach to “local efficiency” can be achieved on a small plate where liquid is thoroughly mixed so that the Composition of liquid leaving the plate is that of the liquid on the plate. Variation of the efficiency on composition was calculated from eq 13 with y = y1 for some available systems and compared with the experimental data a t total reflux in the literature. Typical results are shown in Figure 1. A characteristic decline of the efficiency could be represented by eq 13 in the lower region of the concentration of the low-boiling component even if the factor K, is constant in all compositions. The fixed value of K , a z l G was determined for each system so that the efficiency calculated by eq 13 has the same order as the experimental data a t x: = 0.5 because the purpose is to verify the fact that K, is less dependent on composition than KoG is. Nonequimolar Colunterdiffusion Equation 13 is valid only in the particular case for equimolar counterd iffusion, as indicated by the following paragraphs. In a binary mixture, the Murphree local efficiency for one coinponent is equal to that for another component. However, eq 13 does not satisfy this requirement. When a molar flux relative to the stationary coordinate, Ni,is K,G(y* - y ) a dz N , = K,,G(y**- YJIZ dz = K,G((1 - y1*) (1 - ~ 1 ) ) dz a = -KoG(yl* - ~ l ) dz a = -N1 On the other hand, when Ni is given by the right side of eq 10, N , is not equal to -N2. For nonequimo1a.r counterdiffusion, the molar flux to the molar-average velocity Jiis given as (Bird, et al., 1960)
Ji = yjNi - yiN,
(14)
In a binary mixture, the diffusion fluxes Jiand J j are of equal magnitude and oppositely directed. The use of Ji
= Y2(K0G(Y1* - YJa dz) - Y~(KOG(Y~* - y2)a dz) = K o ~ b i -* yi)a dz = N1 (15)
On the other hand, when NLis K p In ( y * / y ) adz from eq 10, eq 14 gives J 1 = K,[YAY In ~ * / Y ) ) I - ylCV In (Y*/Y))& dz f N1 (16) Equation 16 gives the definition of a proportionality factor K, when the quantity in square brackets represents a driving force of the mass transfer process. Comparing eq 16 with eq 15, the following relationship between K, and KoG is obtained K 0 ~ / K ,=
E y,(y In LJ=
1
(y*/y)/(y*- y)),
(17)
Thermal Effect to Overall Mass Transfer Coefficient Equation 17 is not exact because of nonisothermal process. Then, the next transformation is given in this section. The substitution of eq 8 and 9 into eq 7 with the following transformation leads to Gdy, = K,(y In ( Y * / Y ) + ( y / T ) X ( & . L ~+ / RIn p*.AT))a dz - G(y/T)(dpe/R + In p d T ) (18) where AT = T* - T, Ahe = k e ( r * ) - p e ( T ) , p* = Py*, and p = Py. When N Lis defined to be the right side of eq 18, a similar procedure gives J1
= yAK,Cy In ( Y * / Y ) + ( y / T W P / R + In p*.AT))a dz - GCy/7‘)(dpe/R In p-dT)), -
+
yl(K,b In (y*/y)
+ (y/Z‘)(AP/R + In p*.AT))a dz G(y/T)(dpe/R + In p.dT)), (19)
Similarly, by using eq 19 and 15, the relationship between K,, and KoG is obtained as follows 2
K,G/K, = [ C y I ( y In ( y * / y ) / ( s * - Y ) + tj=1
( y / T ) ( ( W / A T ) / R + In p * ) ( l T / A ~ ) ) , l / [ 1+ 2
C y,((y/T)((dP/dT)/R + In p)(dT/dy)),l
I,J=1
where A y = y* - y, and the next relationship was dy/dz = K,G.lya/G Unless the thermal effect is taken into account, eq reduced to eq 17. Estimation of Thermal Effect The relative contribution of the thermal effect to the total molar flux in KoG can be estimated by using both eq 20 and 17. The ratio K,G/K, at total reflux was calculated for several binary systems and some of the results were shown in Figure 2. The value of KoGIK, nearly equals unity for such systems as benzene-carbon tetrachloride in which the boiling points and the molar latent heats of vaporization of the mixtures are nearly constant. It is proper that the significant thermal effect was observed in the system with a large difference between T and T*. The values of y*, dT/d.y, and ATlAy were calculated from the published vapor-liquid equilibrium data by using the Wilson equation as y = x a t total reflux (see Figure
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Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 2, 1980
process also was numerically calculated as follows: (1) Equation 12 is rewritten as E = 1- exp(-(Km/Kp)Kpaz/ G). (2) The values of (K&/K,,) are calculated from eq 20 or eq 17. They are also obtained from Figure 2. (3) The fixed value of K , a z / G was given for each system as
-
0.
%-XI
Figure 2. The relative contribution of thermal effect to the overall mass transfer coefficient at total reflux in distillation: (a) carbon tetrachloride-benzene; (b) acetone-water; (c) carbon tetra-
chloride-heptane; (d) acetone-chloroform; (e) ethanol-benzene; eq 20; - - -, eq 17.
X
Figure 3. A diagram for the values of y*, dT/dy, and ATlAy. 3). The values of dpe/dT and Ape/AT were theoretically derived from the well-known thermodynamical relations and were calculated from dpe(T)/dT = -Se(T) = -(Se(298K) A p e / A T = -Se(T)
+ C,
In (T/298)) (22)
+ C,(1 - T* In ( T * / T ) / ( T *- T ) ) (23)
The numerical values of Se and C, were found from a textbook of thermodynamics. In Figure 2, the solid line presents the results from eq 20 and the dashed line represents the one from eq 17. The difference between the solid line and the dashed line represents the proportion of the molar flux caused by thermal effect to the total molar flux. Murphree Efficiency Estimated from Eq 20 The fact that the curve of performance vs. mole fraction can have an extreme value even with the fixed value of K,, can be explained by using both eq 12 and eq 20 or Figure 2. The local efficiency a t total reflux in a distillation
Two of the results were shown in Figure 1for comparison. The solid line in Figure 1 represents the result with a thermal effect and the dashed line represents the one without a thermal effect. Conclusion Equation 20 or 17 depends on composition and has an extreme value or not according to the system. Then, the curve performance of a distillation column depends on composition and it can have a maximum according to the system. The prediction of the important local efficiency in the region of very low composition could be estimated from the values of the efficiencies in the region of the middle compositions. The relative contribution of the molar flux caused by a thermal effect to the total molar flux in the conventional overall mass transfer coefficient was estimated by using eq 17 and 20 for some binary systems. Nomenclature a = interfacial surface area, cm2/cm3 C = specific heat, cal/(g-mol K) $= Murphree local efficiency h = heat transfer coefficient, cal/(s cm2) J = molar flux relative to the molar-average velocity, g-mol/s k = mass transfer film coefficient, g-mol/(s cm2) KoG = overall mass transfer coefficient, g-mol/(s cm2) K = constant in eq 7, 16, and 19, g-mol/(s cm3) d=molar flux relative to a stationary coordinate, g-mol/s P = total vapor pressure, atm/atm q = heat transfer rate, cal/(s cm2) S = entropy of vapor, cal/(g-mol K) T = temperature of the vapor, K T* = temperature of the liquid with mole fraction x , K x = mole fraction of a component in the liquid on the plate y = mole fraction of a component in the vapor entering the plate 9 = mole fraction of a component in the vapor leaving the plate y* = mole fraction which is in equilibrium with x z = depth of the liquid over the slots on the plate or height of the column, cm Gresk Letters = chemical potential of a component in the liquid, cal/
p(x)
g-mol p ( y ) = chemical potential of a component in the vapor, cal/
g-mol Subscripts G = vapor phase L = liquid phase p = for chemical potential i = for interfacial i, j , 1, 2 = for component
Literature Cited Bakowski, B., Brit. Chem. Eng., 8, 384 (1963). Bird, R. B., Stewart, W. E., Lightfoot, E. N., "Transport Phenomena", p 499, Wiley, New York, 1960. Gerster, J. A,. Colburn. A. P.. Bonnet, W. E., Carrnody, T. W.. Chem. Eng. Prog., 45, 716 (1949). Liang, S.Y., Smith, W., Chem. Eng. Sci., 17, 11 (1962). Ruckenstein, E., Smigelschi, 0.. Chem. Eng. Sci., 20, 66 (1965). Ruckenstein, E., Srnigelschi, 0.. Can. J . Chem. Eng., 45, 334 (1967). Sawistowski, H., Smith, W., Ind. Eng. Chem., 51, 915 (1959). Zuiderweg, F. I., Harmens, A., Chem. Sci., 12, 89 (1958).
Received for review December 19, 1978 Accepted October 22, 1979
