Mass Transfer Coefficients (MTC) and Correlations II

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1 Mass Transfer Mass Transfer Coefficients (MTC) and Correlations II 7.2-

2 Correlations of Mass Transfer Coefficients Mass transfer coefficients (MTCs) are not physical properties like the diffusion coefficient. They differ from case to case and even within a system, depending on their definition. With the help of experimental observations, correlations for mass transfer coefficients have been developed for standard cases (e.g. fluid flow through a packed bed of particles, gas bubbles rising in a tank, falling films, flow over surfaces and within tubes, ). Such MTC correlations are typically expressed with dimensionless numbers, frequently in the following form: kd D C dv X D Y or X Y Sh C Re Sc

3 Convective Sh Nu Conductive heat transfer heat transfer h l Equivalent in heat transfer h: convective heat transfer coefficient λ: thermal conductivity diffusivity of momentum thermal diffusivity Pr is the equivalent to Sc in heat transfer α: thermal diffusivity Dimensionless numbers Table 8.3- from Cussler, 3 rd ed

4 Selected mass transfer correlations for fluid-fluid interfaces a Table from Cussler

5 Selected mass transfer correlations for fluid-solid interfaces a 5 Table from Cussler 7.2-5

6 MTC Correlations Excellent for preliminary design of small pilot plants. For design of full scale equipment you must supplement them with data of the SPECIFIC chemical system. Fluid-Fluid interface MTC error at best 30% The MTC is expressed mostly as Sherwood-# kl D or a Stanton-# k The Sh is typically expressed by powers of Re and Sc corresponding to convection and diffusion, respectively. When the convection is not a typical forced one but one generated by density gradients, it is free convection and the Re is replaced by the Grashof-#. Fluid-Solid interface The error is about 0% and as low as % especially when dealing with laminar flow in a single tube. This high accuracy is attributed to the heat transfer origin of these correlations and the fact that SIMPER geometries are involved. Also heat transfer is an older subject than mass transfer. For example, laminar flow of one fluid in a tube is much better understood than turbulent flow of gas and liquid in a packed tower! Again the MTC is written in the Sh- or St-notation

7 Example: Dissolution rate of a spinning disk Remember from Generalized Mass Balances : A solvent flow approaches a spinning disk made out of a sparingly soluble solute. Calculate the diffusioncontrolled rate at which the disk slowly dissolves at steady state. The diffusion flux is: 2/ 3 / 2 c D j D z 0 z c(sat) z / 6 j j / 2 2 D d 0.62 d D / Re Sc d D / 3 c / 3 (sat) c (sat)

8 Now: A solid disc of benzoic acid (BA) 2.5 cm in diameter is spinning at 20 rpm and 25 C. How fast will it dissolve in a large volume of a) water and b) air? -5 2 D BA/W=0 cm /s 2 D BA/A=0.233cm /s Solubility of benzoic acid in water is g/cm 3. Equilibrium vapor pressure of benzoic acid in air is 0.3 mmhg at 25 C. The molecular weight of BA is 22 g/mol. Will the mass transfer be faster in air or in water?

9 From Table : /2 /3 k 0.62D D a) For water /3 /2 and N kc (sat) cm 20 / 60 2 / s 0.0cm / s s 0.0cm / s 0 cm / s k N cm / s 0.003g / cm b) For air /2 2 cm 20 / 60 2 / s 0.5cm / s ka s 0.5cm 2 / s 0.233cm 2 / s / cm / s g / cm s 0.47cm / s N 0.3mmHg mol g 0.47cm / s 760mmHg cm 298 mol g / cm s The flux in air is /3 of that in water even though the k in air is 500 times that in water

10 Mass Transfer across Interfaces (very important) Often we encounter the following scenario: Bulk interface Bulk Basic equation: N Kc where K is the overall MTC and c an APPROPRIATE concentration difference and most notably the equilibrium or asymptotic concentration. Case a: Heat transfer Hot benzene on cold water. The benzene cools while the water warms until they reach the same T. Equal T is the criterion for equilibrium. The amount of energy transferred is always equal to the T. No problem!

11 Case b: Bromine extraction A benzene solution of bromine is placed on top of a water solution containing the SAME concentration of bromine. ater the initially equal concentrations have CHANGED and the Br concentration in C 6 H 6 is higher than in H 2 O. Why? Bromine is more soluble in C 6 H 6 than in H 2 O. Now the c here should be: c = c (in benzene) - Hc (in water) Otherwise the initial c is zero and we still have flux. The partition coefficient H is: Bromine concentration: concentration of Br in benzene H= concentration of Br in water at equilibrium 7.2-

12 Case c: Bromine vaporization Initially Br evaporates from water into air. Initially the Br concentration in water is higher than that in air; at the end it is lower. This might be a problem of units: Concentrations in the liquid are expressed in mol/ and those in air by the partial pressure?? Mass transfer should be described in terms of the more fundamental chemical potentials. If this was done, the concentration difference would disappear

13 7.4 The Overall MTC The flux in the gas is: N k (p p ) p 0 i (4) Because the interfacial region is thin, it is at steady state. Thus, the flux will be equal to that in the liquid. N k (c c ) (5) i 0 where k P and k are the gas and liquid MTC s! So, k p p k c c P 0 i i

14 Always we must remove the dependency on the interfacial concentration or partial pressures, as these are difficult to determine. Usually there is equilibrium at the interface: c i p i H c i k P p k P k c H k 0 0 H is Henry s constant or the partition coefficient in the simplest case (6) So the flux N from equation (5) should be derived as: N (p Hc ) 0 0 / k H / k p (7) K P k p H k is the overall gas-side MTC

15 Analogy with electric circuits: Current N p k 0 P Hc H k 0 Voltage difference 2 resistances in series Many times the engineer s job is to determine which is the rate limiting resistance: in the gas or the liquid??

16 Now we can write the flux equation in two ways: A) B) N K (p p *) p 0 N K (c * c ) 0 where where K K p / k H / k / k / k H p and c * 0 p / H K is called the overall liquid-side mass transfer coefficient and c * is the hypothetical liquid concentration in equilibrium with the bulk gas concentration. p p * Hc 0 K P is the overall gas-side mass transfer coefficient and p * is the hypothetical gas-phase concentration that would be in equilibrium with the bulk liquid concentration. and

17 Example : Oxygen Mass Transfer Estimate the overall liquid-side MTC for O 2 transfer from water into air assuming that each MTC is k = D/0.0 cm, Henry s law constant is H = atm, D air = 0.23 cm 2 /s and D water = cm 2 /s. Goal: Calculate k and k P and substitute in the appropriate equation. k D 0.0cm cm / s cm / s 0.0cm Finding k P and H is more difficult for unit conversion k p k D G G RT (0.0cm)RT cm / s 3 (0.0cm)(82cm atm / (mol K))(298K) mol / (cm s atm)

18 From the way H is given (unit consistency) H H` c atm mol / 8 cm atm cm 3 / mol Insert these values into the equation for K K / k / k H p / (2. 0 cm / s) / (9.4 0 mol / cm s atm cm atm / mol) cm / s The mass transfer is dominated by the liquid-side resistance!!

19 Example : Perfume Extraction Jasmone (C H 6 O) is a valuable aroma from jasmine flowers that is used in soaps and cosmetics. We are recovering this from its water solution (jasmine flowers in water) with benzene drops the k B =3.0 x 0-4 cm/s while k W = 2.4x0-3 cm/s However, jasmone is 70 times more soluble in C 6 H 6 than in H 2 O. What is the overall MTC? Assuming steady state N = k (c - c ) = k (c - c ) (8) W 0 W iw B ib 0B The interfacial concentration is in equilibrium, so: c ib H ciw (9)

20 Eliminate the interfacial concentration using equations 8 and 9: k c k c k Hc k c W 0 W W iw B iw B 0B k c k c ( k k H) c W 0 W B 0B W B iw Replace this in the N for benzene c iw c k c k c ib H k k H W 0 W B 0B W B (0) k c k c W 0W B 0B N k H c B - 0B k k H W B Hk c Hk c k c Hk c kk k k H k k H W B W B W 0 W B 0B W 0B B 0B B W k (Hc c ) B 0 W 0B / k H / k B W (Hc c ) 0 W 0B

21 The overall MTC, K is K' 4 3 / (3.0 0 cm / s) 70 / (2.4 0 cm / s) cm / s Again the mass transfer in water controls the process because jasmone is more soluble in C 6 H

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