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CHE312 Problem set #2 1. The gas CO2 is diffusing at steady state through a tube 0.20 m long having a diameter of 0.01 m and containing N2 at 298oK. The total pressure is constant at 101.32 kpa. The partial pressure of CO2 at one end is 456 mmHg and 76 mm Hg at the other end. The diffusivity DAB is 1.67×10-5 m2/s at 298oK. Calculate the flux of CO2 in cgs and SI units for equimolar counterdiffusion. Ans: 1.7110-7 mol/cm2s = 1.7110-6 kmol/m2s 2. Helium and nitrogen gas are contained in a conduit 5 mm in diameter and 0.1 m long at 298oK and a uniform constant pressure of 1.0 atm abs. The partial pressure of He at one end of the tube is 0.060 atm and 0.020 atm at the other end. The diffusivity of Helium in nitrogen at 298oK is 6.87×10-5 m2/s. Calculate the following for steady-state equimolar counterdiffusion. a) Flux of he in kmol/s∙m2. b) Flux of N2. c) Partial pressure of He at a point of 0.05 from either end. Ans: a) 1.12410-6 kmol/s∙m2. b) 1.12410-6 kmol/s∙m2 c) 0.04 atm 3. Assuming air to be composed exclusively of O2 and N2, with their partial pressure in the ratio 0.21:0.79, what are their mass fractions? Ans: Mass fraction of O2 = 0.233, Mass fraction of N2 = 0.767 4. A mixture of CO2 and N2 is in a container at 25oC, with each species having a partial pressure of 1 bar. Calculate the molar concentration in kmol/m3, the mass density in kg/m3, the mole fraction, and the mass fraction of each species. Ans: Let A = CO2, B = N2 CA = CB = 4.0310-2 kmol/m3 A = 1.775 kg/m3, B = 1.13 kg/m3 Mass fraction of CO2 = 0.611, Mass fraction of N2 = 0.389 5. A thin plastic membrane is used to separate helium from a gas stream. Under steady-state conditions the concentration of helium in the membrane is known to be 0.02 and 0.005 kmol/m3 at the inner and outer surfaces, respectively. If the membrane is 1 mm thick and the binary diffusion coefficient of helium with respect to the plastic is 10-9 m2/s, what is the diffusive flux? Ans: 1.510-8 kmol/s∙m2 6.1 The steady-state temperature distribution in a one-dimensional wall of thermal conductivity 50 W/mK and thickness 50 mm is observed to be T(oC) = a + bx2, where a = 200oC, b = 2000oC/m2, and x is in meters. (a) What is the heat generation rate q in the wall? (b) Determine the heat fluxes at the two wall faces. In what manner are these heat fluxes related to the heat generation rate? Ans: (a) 2105 W/m3 (b) q”|0 = 0; q”|L = 10,000 W/m2; heat generation rate = 2105 W/m3 7.1 One-dimensional, steady-state conduction whit uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of 5 W/mK. For these conditions, the temperature distribution has the form, T(x) = a + bx + cx2. The surface at x = 0 has a temperature of T(0) = T0 = 120oC and experiences convection with a fluid for which T = 20oC and h = 500 W/m2K. The surface at x = L is well insulated. (a) Applying an overall energy balance to the wall, calculate the internal energy generation rate, q . (b) Determine the coefficients a, b, and c by applying the boundary conditions to the prescribed temperature distribution. Use the results to calculate and plot the temperature distribution. (c) Consider conditions for which the convection coefficient is halved, but the internal energy generation rate remains unchanged. Determine the new values of a, b, and c, and use the results to plot the temperature distribution. Hint: recognize that T(0) is no longer 120oC. (d) Under conditions for which the internal energy generation rate is doubled, and the convection coefficient remains unchanged (h = 500 W/m2K), determine the new values a, b, and c, and plot the corresponding temperature distribution. Referring to the results of Parts (b), (c), and (d) as Cases 1, 2, and 3, respectively, compare the temperature distribution for the three cases and discuss the effects of h and q on the distributions. Ans: (a) 1106 W/m3 (b) a = 120oC, b = 104oC/m, and c = 105 oC/m2 (c) a = 220oC, b = 104oC/m, and c = 105 oC/m2 (d) a = 220oC, b = 2104oC/m, and c = 2105 oC/m2 8.1 The rear window of an automobile is defogged by passing warm air over its inner surface. (a) If the warm air is at T,i = 40oC and the corresponding convection coefficient is hi = 30 W/m2K, what are the inner and outer surface temperatures of 4-mm-thick window glass, if the outside ambient air temperature is T,o = 10oC and the associated convection coefficient is ho = 30 W/m2K? (b) In practice T,o and ho vary according to weather conditions and car speed. For values of ho = 2, 65, and 100 W/m2K, compute and plot the inner and outer surface temperatures as a function of T,o for 30oC T,o 0oC. Ans: Ts,i = 7.7oC, Ts,o = 4.9oC 9.1 In a manufacturing process, a transparent film is being bonded to a substrate as shown in the sketch. to cure the bond at a temperature To, a radiant source is used to provide a heat flux q o (W/m2), all of which is absorbed at the bonded surface. The back of the " substrate is maintained at T1 while the free surface of the film is exposed to air at T and a convection heat transfer coefficient h. (a) Show the thermal circuit representing the steady-state heat transfer situation. Be sure to label all elements, nodes, and heat rates. Leave in symbolic form. (b) Assume the following conditions: T = 20oC, h = 50 W/m2K, and T1 = 30oC. " Calculate the heat flux q o that is required to maintain the bonded surface at To = 60oC. Ans: (b) 2833 W/m2 10.1 The wind chill, which is experienced on a cold, windy day, is related to increased heat transfer from exposed human skin to the surrounding atmosphere. Consider a layer of fatty tissue that is 3 mm thick and whose interior surface is maintained at a temperature of 36 oC. On a calm day the convection heat transfer coefficient at the outer surface is 25 W/m 2K, but with 30 km/h winds it reaches 65 W/m2K. In both cases the ambient air temperature is 15oC. (a) What is the ratio of the heat loss per unit area from the skin for the calm day to that for the windy day? (b) What will be the skin outer surface temperature for the calm day? For the windy day? (c) What temperature would the air have to assume on the calm day to produce the same heat loss occurring with the air temperature at 15oC on the windy day? Ans: (a) 0.553 (b) Calm day Ts,2 = 22.1oC, windy day Ts,2 = 10.8oC (c) T = -54.6oC.