高等輸送二 — 質傳 Lecture 2 Diffusion in concentrated solutions 郭修伯 助理教授

Dilute vs. concentrated (solution) Any mass flux include both convection and diffusion (Maxwell, 1860). Diffusion causes convection. Dilute solution: convection caused by diffusion is small and can be neglected. Concentrated solution: both convection and diffusion have to be considered. Difference between Heat transfer and Mass transfer Heat conduction can occur without convection Diffusion and convection always occur together

Total mass transported Mass transported by convection Mass transported by diffusion Total mass transported = Convective reference velocity ??? Average solute velocity w.r.t fixed coordinate Local concentration

Convective reference velocity How to choose its value ? Our goal: choose va so that va is zero as frequently as possible! The mass transfer reduced to “diffusion” only. va can be: the molar average velocity good for ideal gases where the molar concentration is constant the mass average velocity good for constant-density liquid the volume average velocity good for constant-density liquid and for ideal gas What are those?

The temperature and pressure are such that the diffusion coefficient is 0.1 cm2/sec. Find the molar average velocity v*, the mass average velocity v, the volume average velocity v0 at the average concentration in the system. nitrogen hydrogen The volume in this system does not move, so v0 = 0. If the gases are ideal, the molar concentration is constant, so v* = 0. Diffusion in the thin-film For nitrogen at an average concentration of 0.5 c For hydrogen at an average concentration of 0.5 c Mass fraction of nitrogen Mass fraction of hydrogen

Solute accumulated in volume Az Fast evaporation by diffusion and convection l z A mass balance on a differential volume A z gives: Solute transported out at z + z Solute transported in at z Solute accumulated in volume Az = Dividing A z z  0

contribution of both diffusion and convection and it is constant. s.s choose volume average velocity v0 If the solvent vapor is stagnant Total molar concentration

B.C. Exponential concentration profile n1 = constant The diffusion flux is smallest at the bottom of the capillary and rises to a maximum value at the top of the capillary.

Concentrated solution Dilute solution or Linear concentration profile

Solute accumulated in volume Az Fast Diffusion into a Semiinfinite slab z l, >> Fast evaporation by diffusion and convection A mass balance on a differential volume A z gives: Solute transported out at z + z Solute transported in at z Solute accumulated in volume Az = Dividing A z z  0

choose volume average velocity v0 independent of z =1 Continuity equation

Solvent gas is insoluble

B.C.

How important is the convection term? z l The vapor pressure of benzene at 6ºC is about 37 mmHg The vapor pressure of benzene at 60ºC is about 395 mmHg Total flux Diffusion + convection Diffusion 6 ºC 2 % 60 ºC 40 %

General form of the mass balance equation x z y Input rate through ABCD C G Input rate through ADHE D Input rate through ABFE H z B Output rate through EFGH F y Output rate through BCGF A E x Output rate through CDHG input - output + generation = accumulation

Diffusion and the convection term General equation include the effects of chemical reaction, convection, and concentration-driven diffusion Combine with Fick’s law

Membrane example Fresh water Salt water membrane x y Find differential equations for calculating the drop in flux caused by the concentration polarization (i.e., by the salt accumulation near the membrane surface) s.s. 2D no reaction Responsible to concentration polarization Usually much smaller than the convection term

Spinning disc example Solute from dissolving disc flow The dissolution rate is diffusion-controlled. Calculate the rate at which the disc dissolves. s.s. no reaction Angularly symmetric Angularly symmetric Disc infinitely wide Disc infinitely wide

B.C. Levich, 1962

Reynolds number Schmidt number Independent of the disc radius: constant flux

Mass transfer versus Heat transfer Diffusion-induced convection; chemical reaction only for mass transfer Radiation only for heat transfer J. Crank, (1975) The Mathematics of Diffusion, 2nd ed. Oxford: Clarendon Press (include reactions) H.S. Carslaw, and J. C. Jaeger (1986) The Conduction of Heat in Solids, 2nd ed. Oxford: Clarendon Press (a more complete selection of boundary conditions)

Diffusion through a polymer film Diaphragm（隔板） cell Initial pressure zero Measure the ethylene concentration in the upper compartment as a function of time. Polymer film Initial pressure one atmosphere From mass balance and Fick’s law: Boundary conditions: l: film’s thickness H: Henry’s law coefficient

Mole balance on the top compartment: Crank, (1975) Mole balance on the top compartment: t = 0, p = 0

Large time From the intercept and the slope, we can obtain the equilibrium Henry’s law coefficient, H, and the diffusion coefficient, D, in a single experiment Pressure Time

A dissolving pill Estimate the time required to produce a steady flux of drug pill in the gut (腸). Assumption: the drug’s dissolution is controlled by diffusion into the stagnant contents of the gut.(i.e. , The dissolution is diffusion-controlled The surrounding s are stagnant ) The mass balance on a spherical shell: Diffusion control Boundary conditions:

Crank, (1975) Carslaw and Jaeger (1986) The flux in a dilute solution: The time to reach steady flux: Time ~ 80h is much longer than the experimental result (i.e., 10 min) Because: free convection driven by the density difference caused by the dissolution

Effective diffusion coefficients in a porous catalyst pellet A porous catalyst pellet containing a dilute gaseous solution. Determine the effective diffusion of solute by dropping this pellet into a small, well stirred bath of a solvent gas and measuring how fast the solute appears in this bath. A mass balance on a spherical pellet: Diffusion control Boundary conditions: Bath concentration

A mass balance on the solute in the bath of volume VB t = 0, C1 = 0 Carslaw and Jaeger (1986) Crank, (1975) Void fraction in the sphere