#### Mass Transfer

3. A hemispherical droplet of liquid water, lying on a flat surface, evaporates by molecular diffusion through still air surrounding the droplet. The droplet initially has a radius R. As the liquid water slowly evaporates, the droplet shrinks slowly with time, but the flux of the water vapor is a nominal steady state. The temperature of the droplet and the surrounding still air are kept constant. The air contains water vapor of fixed concentration at an infinitely long distance from the droplet's surface. • Draw a picture of the physical process, • Select a coordinate system that will best describe this diffusion process, • List at least five reasonable assumptions for the mass-transfer aspects of the water- evaporation process, • Simplify the general differential equation for mass-transfer in terms of the flux NA. • Specify the simplified differential form of Fick's flux equation for water vapor (species A) • Identify the proper boundary conditions, • Write down the differential equation for CA-

P6.21 100 gmol/min of a gas stream containing 30 mol % ethane (C₂H6) and 70 mol% methane (CH4) is fed to a distillation column, where it is separated into an overhead product containing 90 mol % methane and a bottoms product containing 98 mol % ethane. Calculate the overhead and bottoms flow rates and the fractional recoveries of methane and ethane in their corresponding product streams.

P7.9 100 gmoles of a mixture of A and B (ZAF=0.6) is fed to an equilibrium stage. The two phases I and II leaving the stage are sampled and it is found that XAI=0.833 and XBII = 0.75. Write down equations showing the phase equilibrium relationship between the mole fractions of A and B in the two phases. Calculate the total moles of each of the phases leaving the stage as well as the separation factor αAB.

P7.13 A mixture of 35 mol % ethanol/65 mol % water is equilibrated at 1 atm and 86°C. Using Fig. 7.7, determine yE, XE, and the fraction of the mixture I

7.3.2.1 Vapor-Liquid Phase Equilibrium Diagrams VLE differs from SLE in two ways. First, VLE is sensitive to both temperature and pressure, whereas SLE is not affected much by pressure. Second, with VLE, all the components are usually distributed in both vapor and liquid phases, whereas with SLE, it is common that the solid phase is pure (single-component). To show the VLE relationships graphically, we usually either (1) hold the pressure P constant and plot the vapor (y) and liquid (x) mole fractions as a function of temperature T, or (2) hold T constant and plot y, and x, as functions of P. Both T-x-y and P-x-y diagrams are useful, but we will focus on T-x-y diagrams because it is more common to design separation units to operate at constant P. Figure 7.7 is a T-x-y diagram for ethanol (E) and water (W), with P= 1 atm. (For the same data in table form, see Table B.13.) The normal boiling point temperature T, 100°C for pure water is shown on the left-hand side (where ye 0.0 and yw 1.0) and 7,78.3°C for pure ethanol is shown on the right-hand side (where yg=1.0). -

P7.29 Styrene is the building block for polystyrene, a polymer that enjoys ubiquitous use in products such as coffee cups and packaging "peanuts." Styrene is made by dehydrogenation of ethylbenzene. Since reactor conversion is much less than 100%, a separation between styrene and ethylbenzene is needed. Using Antoine equations to calculate saturation pressures along with Raoult's law, calculate (a) the boiling point temperature of pure styrene at 2 bar, (b) the boiling point temperature for pure ethylbenzene at 2 bar, and (c) the dew point and bubble point temperatures for a 50:50 (mol:mol) mix of styrene-ethylbenzene system at 2 bar.

P7.38 A natural gas stream (100 kgmol/h) contains 98 mol% methane (CH4) and 2 mol % hydrogen sulfide (H₂S). H₂S is a highly toxic gas. The concentration of H₂S in the natural gas stream must be reduced to 0.01 mol% H₂S before the natural gas can be used as a fuel or chemical feedstock. A process engineer recommends absorption in a single equilibrium stage with water as the solvent. The absorber operates at 35 atm and 10°C. Determine the required flow rate of water to the absorber.

P7.37 The antibiotic streptomycin is extracted from a fer- mentation broth using an organic solvent, where KD = 40 (g streptomycin/kg extract/g streptomycin/kg raffi- nate). If the broth contains 1 wt% streptomycin and is fed at 150 kg/min to the solvent extraction unit, what is the required solvent flow rate to recover 80% of the streptomycin?

A plate of iron is exposed to a carburizing (carbon-rich) atmosphere on one side and a decarburizing (carbon-deficient) atmosphere on the other side at 700C°. If a condition of steady state is achieved, calculate the diffusion flux of carbon through the plate if the concentrations of carbon at positions of 5 and 10 mm (5x10³ and 10² m) beneath the carburizing surface are 1.2 and 0.8 kg/m³, respectively. Assume a diffusion coefficient of 3 x 10¹ m²/s at this temperature.

P7.42 Trace organic contaminants in waste water readily adsorb onto a solid called activated carbon, which is not very different from charcoal briquets. Equilibrium data for adsorption of phenol onto activated carbon is shown. One liter of wastewater containing 1 g phenol is contacted with 5 g activated carbon at 20°C. At equilibrium, what fraction of the phenol is adsorbed?