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1)A membrane of thickness L separates two solutions, Both solutions and the membrane have initially a zero concentration of a permeable solute

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1)A membrane of thickness L separates two solutions, Both solutions and the membrane have initially a zero concentration of a permeable solute. At time t = 0 and thereafter one side is maintained at a concentration of CA0 of this solute. Solve the concentration profile in the membrane as a function of time and position.

2)Use the thermal-diffusivity value estimated from the above problem to find out whether any part of a fruit of diameter 8 cm hanging in a tree will freeze if the ambient temperature suddenly drops to −6 ?C, from an initial temperature of 15 ?C. The heat transfer coefficient to the fruit surface is estimated as 10 W/m2 · K. This is, of course, a function of the air velocity.

3)A porous cylinder, 2.5 cm in diameter and 80 cm long, is saturated with alcohol and maintained in a stirred tank. The alcohol concentration at the surface of the cylinder is maintained at 1%. The concentration at the center is measured by careful sampling and is found to drop from 30% to 8% in 10 h. Find the center concentration after 15 and 20 h.

4)Consider the problem of transient diffusion in a composite slab with two different thermal conductivities. Thus region 1 extending from 0 to κ has a thermal conductivity k1, while the region 2 from κ to 1 has a conductivity of κ2. The slab is initially at a dimensionless temperature of 1, and both ends are exposed to a zero temperature. Set up the problem and solve by separation of variables.

5)A slab has a thermal diffusivity of 5×10−6 m2/s and is fairly thick. The initial temperature is 300 K and the surface temperature is raised to 600 K at time zero. Find the temperature 0.5 m below the surface after one hour has elapsed. Also find the depth to which the heat front has penetrated.

6)A hot dog at 5 ?C is to be cooked by dipping it in boiling water at 100 ?C. Model the hot dog as a long cylinder with a diameter of 20 mm. Find the cooking time, which is defined as the time, for the center temperature to reach 80 ?C. The heat transfer coefficient from the water to the surface is 90 W/m2 · K. The following data can be used: k = 0.5 W/m · K, ρ = 880 kg/m3, and cp = 3350 J/kg · K.

7)A finite cylinder is 2 cm in diameter and 3 cm long and at a temperature of 200 ?C, and is cooled in air at 30 ?C. The convective heat transfer coefficient is estimated as 10 W/m2 · K. Calculate and plot the center temperature. Use the following physical property values: k = 50 W/m · K, ρ = 2000 kg/m3, and cp = 1000 J/kg · K.

8)A concrete wall 20 cm thick is initially at a temperature of 20 ?C, and is exposed to steam at pressure 1 atm on both sides. Find the time for the system to reach a nearly steady state. Find the rates of condensation of steam at various values of time. The thermal diffusivity constant for concrete is needed. Use a value of 7.5 × 10−7 m2/s.

9)Analyze the transient problem with the Dirichlet condition for a long cylinder and for a sphere. Derive expressions for the eigenfunctions, eigenconditions, and eigenvalues. Find the series coefficients for a constant initial temperature profile.

10)A membrane separator is 3 mm in diameter, and the membrane permeability was estimated as 2 × 10−6 m/s. The solute being transported has a diffusivity of 2 × 10−9 m2/s in the liquid. Estimate the overall permeability. What fraction of the resistance can be attributed to mass transfer? The above membrane separator showed 70% solute recovery under certain operating conditions. A new membrane with 10 times the permeability is being considered as a replacement. Estimate the solute recovery in this new case.

11)Extend the analysis of gas absorption with reaction to the case of absorption of two gases with a common liquid-phase reactant. A detailed study of this topic has been published in an award-winning paper by Ramachandran and Sharma (1971). Simulate numerically some illustrative examples from this paper.

12)The effective diffusion coefficient of H2 in a mixture of H2 and CO in a porous catalyst was found to be 0.036 cm2/s at a temperature of 373 K and 2 atm total pressure. The catalyst has a monodispersed pore structure with pore void fraction of 0.3. What is the mean pore size of this material? Are the conditions such that this parameter can be calculated with a reasonable confidence?

13)Consider a second-order reaction in a catalyst in the form of a slab. Show that the differential equation can be expressed as d2cA dξ 2 = φ2c2 A (10.91) How is the φ parameter defined for this case? Use the p-substitution method and derive an implicit integral representation to the solution of this problem. Calculate the concentration profiles and the effectiveness factor for φ equal to 1 and 3.

14)Consider the same problem with now a Robin condition at the surface: dcA dξ 1 = Bi[1 − (cA)1] Derive an expression for the effectiveness factor as a function of Bi in addition to the φ parameter. Do the analysis for all three geometries.

15)Two bulbs are connected by a straight tube of diameter 0.001 m and length 0.15 m. Initially one bulb contains nitrogen and the bulb at the other end contains hydrogen. The system is maintained at a temperature of 298 K and a total pressure of 1 atm. The volume of each bulb is 8 × 10−6 m3. Calculate and plot the mole-fraction profile of nitrogen in the first bulb as a function of time. Verify the validity of the quasi-steady-state approximation by calculating the diffusion time and the process time constants.

16)A liquid is contained in a tapered conical flask with a taper angle of 30?. The radius in the flask for the liquid level at the bottom is 7 cm and the vapor height above this is 10 cm. Find an expression for the rate of evaporation and the mole-fraction profile in the vapor space and compare your answer with the case of a straight cylinder. Assume that the liquid is benzene under the conditions stated in Problem 3. The bulk gas is at zero concentration.