(a) Calculate the constant molar flux of methanol within the Stefan tube at steady state using a numerical technique. (b) Plot the mole fraction of methanol from the liquid methanol surface to the flowing air stream. (c) Compare the result of part (a) with the result calculated from Equation (10-7). (d) Verify several points on the numerical solution for the mole fraction profile with calculations from the analytical profile of Equation (10-6). (e) Repeat parts (a) through (d) for a temperature of 298.15K, where the vapor pressure of methanol is 16.0kPa. (f) Assume that the temperature within the Stefan tube varies linearly from 328.5K at the methanol surface to 295K in the air stream at the tube surface. Complete parts (a) and (b) for this condition. 10.1.3 Problem Statement Liquid A is evaporating into a gas mixture of A and B from a liquid layer of pure A near the bottom of a cylindrical Stefan tube, as shown in Figure 10-1. The rate of evaporation is relatively slow, so it is a good assumption that the level of the liquid surface is constant. A gas mixture is passing over the upper surface of the Stefan tube. Thus the partial pressure of A,pA2, and the mole fraction of A at point 2,xA2, are both known at z2. The surface of the liquid A contains no dissolved B since B is insoluble in liquid A; therefore, liquid A exerts its vapor pressure, pA1, at location z1. The mole fraction of A at the liquid surface is given by xA1=PPA0 where PA0 is the vapor pressure of component A and P is the total pressure. The simplest assumptions for this system would be that the temperature and pressure are constant and that the gases A and B are ideal. Thus, gas A is diffusing from the surface into the bulk stream above the surface of the Stefan tube though gas B which is stationary within the tube. This is the case of single component diffusion through a stagnant gas film. 383 Figure 10-1 Gas Phase Diffusion of A through Stagnant B Mass Balance on Component A within Diffusion Path Consider a differential element between points z1 and z2 with a differential length of z. Since there is no reaction in this case, a steady-state mass balance in the positive z direction yields dzdNA=0 where NA is the flux of A relative to stationary coordinates in kgmol/m2s. Fick's Law for Binary Diffusion Also for stationary coordinates, the general expression of Fick's law for the flux of A can be written as NA=DABCdzdx+CCA(NA+NB)flux=diffusion+bulkflow(convection) where C is the total concentration in kgmol/m3,DAB is the molecular diffusivity of A in B in m2/s,CA is the concentration of A in kgmol/m3, and NB is the flux of B in kg-mol/m m2s. For this problem, the total gas concentration C is constant, and component B is stagnant. Thus NB is zero. The mole fraction of A in the gas mixture, xA, can be used to replace CA/C. The modified expression for Fick's law can be written as NA=DABdzdCA+xANA (a) Calculate the constant molar flux of methanol within the Stefan tube at steady state using a numerical technique. (b) Plot the mole fraction of methanol from the liquid methanol surface to the flowing air stream. (c) Compare the result of part (a) with the result calculated from Equation (10-7). (d) Verify several points on the numerical solution for the mole fraction profile with calculations from the analytical profile of Equation (10-6). (e) Repeat parts (a) through (d) for a temperature of 298.15K, where the vapor pressure of methanol is 16.0kPa. (f) Assume that the temperature within the Stefan tube varies linearly from 328.5K at the methanol surface to 295K in the air stream at the tube surface. Complete parts (a) and (b) for this condition. 10.1.3 Problem Statement Liquid A is evaporating into a gas mixture of A and B from a liquid layer of pure A near the bottom of a cylindrical Stefan tube, as shown in Figure 10-1. The rate of evaporation is relatively slow, so it is a good assumption that the level of the liquid surface is constant. A gas mixture is passing over the upper surface of the Stefan tube. Thus the partial pressure of A,pA2, and the mole fraction of A at point 2,xA2, are both known at z2. The surface of the liquid A contains no dissolved B since B is insoluble in liquid A; therefore, liquid A exerts its vapor pressure, pA1, at location z1. The mole fraction of A at the liquid surface is given by xA1=PPA0 where PA0 is the vapor pressure of component A and P is the total pressure. The simplest assumptions for this system would be that the temperature and pressure are constant and that the gases A and B are ideal. Thus, gas A is diffusing from the surface into the bulk stream above the surface of the Stefan tube though gas B which is stationary within the tube. This is the case of single component diffusion through a stagnant gas film. 383 Figure 10-1 Gas Phase Diffusion of A through Stagnant B Mass Balance on Component A within Diffusion Path Consider a differential element between points z1 and z2 with a differential length of z. Since there is no reaction in this case, a steady-state mass balance in the positive z direction yields dzdNA=0 where NA is the flux of A relative to stationary coordinates in kgmol/m2s. Fick's Law for Binary Diffusion Also for stationary coordinates, the general expression of Fick's law for the flux of A can be written as NA=DABCdzdx+CCA(NA+NB)flux=diffusion+bulkflow(convection) where C is the total concentration in kgmol/m3,DAB is the molecular diffusivity of A in B in m2/s,CA is the concentration of A in kgmol/m3, and NB is the flux of B in kg-mol/m m2s. For this problem, the total gas concentration C is constant, and component B is stagnant. Thus NB is zero. The mole fraction of A in the gas mixture, xA, can be used to replace CA/C. The modified expression for Fick's law can be written as NA=DABdzdCA+xANA
