CHEN 7100 FA16 Final Exam

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1 CHEN 7100 FA16 Final Exam Show all work and state all assumptions for full credit. The exam is closed book, notes, and homework. Only the course reader and your mind should be open. No electronic devices (cell phones, calculators, etc.) are allowed. Time limit is 150 minutes, 200 points total. Put your name on each page you want graded. Write only on one side of the paper. Observe the margins outlined in the Course Reader. Explain your arguments carefully but succinctly. Name: 1

2 1. (40 points) Using language, arguments, and mathematics consistent with graduate level transport phenomena, address the following explain questions. You may need to draw pictures to adequately explain one or more concepts. a. (10 points) Explain the creeping flow assumption and when it is appropriate to invoke it in solving momentum transport problems. b. (10 points) Explain why the term ττ vv appears in both the Equation of Mechanical Energy and the Equation of Internal Energy with opposite sign. c. (10 points) Explain how the Continuum Approximation is necessary for the validity of the Principle of Stress Equilibrium. d. (10 points) Explain two scenarios where a species may diffuse in a direction opposing its concentration gradient. Hint: recall our discussion on driving forces for molecular motions. 2

3 2. (45 points) A fluid of constant viscosity is confined in a rectangular slot. The slot has vertical walls at x = ±B, y = ±W, and a bottom and top at z = 0 and z = H, with H >> W >> B. The walls are nonisothermal, with temperature distribution given by T = T + Ay where T is the average wall temperature. Note δz g = gz. Neglect end effects. a. (15 points) Simplify the equations of continuity and motion based on a steady state condition. Use the Boussinesq Approximation for density as shown below to derive the full differential equation governing the velocity profile. ρρ(tt) = ρρ ρρ ββ (TT TT ) b. (15 points) Develop the necessary arguments to form postulates about the system, starting with v = vz(x,y) δz. Simplify the differential equation from part (a) using your postulates. Think carefully about how terms weigh against others and how approximations can be made to discard terms. You must justify why you eliminate terms from the governing equation. You should end up with: 0 = μμ 2 vv zz xx 2 + ρρ ββ gg zz (TT TT ) c. (5 points) Write the boundary conditions needed to solve the differential equation. d. (5 points) Insert the temperature profile and solve for the velocity profile in terms of x and y (i.e. vz(x,y)). e. (5 points) Why do we need not consider the equation of energy in this problem? 3

4 3. (35 points) Newton s Law of Cooling is shown below with heat flux, q, heat transfer coefficient, h, temperature at the solid-fluid interface, T, and bulk temperature, T. q = h(t T ) A small segment of a solid wall with temperature gradient in x- direction is depicted in the figure below. solid dddd dddd δx fluid a) (5 points) Qualitatively plot the temperature profile in the solid and fluid when the interfacial heat flux is approximated by Netwon s Law of Cooling. Identify T, the temperature of the wall at the solid-fluid interface, and T in your plot. b) (5 points) Qualitatively plot the temperature profile in the fluid when no approximations are used. Identify T and T in your plot. c) (10 points) Explain the differences in the plots from parts (a) and (b) as they relate to the approximations inherent in the Newton s Law of Cooling. d) (15 points) In the cooling fin example from class, we had the choice of several boundary conditions at the solid-fluid interfaces as shown below. T 1) T = T 2) q = h(t-t ) 3) dt/dx = 0 Explain the physical meaning of each boundary condition. Then, compare boundary conditions (1) and (3) by clearly describing the key differences. 4

5 4. (35 points) A pool of trichloroethylene, or TCE, (ρ = 1.46 g/cm 3 ) resides at the bottom of the Fishkill River. We can postulate that the rate-controlling step in the dissolution of the TCE into the Fishkill River is the diffusion of TCE from the pool surface through a stagnant liquid film of thickness δ (in the y direction). The molar solubility of TCE in water is CA0 (letting TCE be component A and water be component B). Assume DDAB is constant and that A is only slightly soluble in in B so that convective flux may be neglected. a. (5 points) Setup the differential equation that can be solved to give CA as a function of y if the concentration at y = δ is CA = CA. Then, find the expression for the rate of dissolution of TCE in water per unit area. Draw the concentration profile of A in the stagnant film. b. (5 points) Repeat part (a) for CA = 0 at y = δ. (this means the concentration of A in the river is negligible, not necessarily 0) c. (10 points) Develop an expression for the rate of dissolution of A per unit area if Fishkill River contains a remediation compound, C, which at the plane y = mδ, reacts instantaneously and irreversibly with A in the following the reaction scheme A + C D. The stream consists primarily of C and B, with mole fraction of C given as xc. The expression must be put into a final form that does not include m as a parameter. (Hint: recognize that both species A and C are diffusing toward the plane at y = mδ, with DDAB and DDCB) Draw the concentration profile of A and C in the stagnant film. d. (15 points) Instead of an instantaneous, irreversible reaction between A and C, assume the reaction is first order in species A. Derive the differential equations needed to solve for CA(y) and CC(y), including boundary conditions. Define mδ as the plane where CA = 0 and δ as the plane where CC = CC. Prepare qualitative plots of CA and CC vs. y for (1) large ϕa with small ϕc, (2) large ϕa and ϕc, (3) small ϕa with large ϕc, where ϕ is the Thiele Modulus (ratio of reaction rate to diffusion rate). Be sure to identify mδ and δ in the plots! 5

6 5. (45 points) A spherical catalyst pellet has a radius R and constant thermal conductivity k. The chemical reaction within the porous pellet generates heat at a rate of Sc cal/cm 3 -s (i.e. coupled heat/mass transport modeled emperically). Heat is lost at the outer surface of the pellet to a gas stream at constant temperature Tg by convective heat transfer approximated by Newton s Law of Cooling. Find the steady-state temperature profile, assuming Sc is constant throughout the pellet following the guidelines below. a. (5 points) Set up the differential equation by simplifying the appropriate form of the energy equation. b. (10 points) Identify boundary conditions and solve for constants to get the full temperature profile. c. (15 points) Draw the temperature profile in the pellet. d. (15 points) If Sc is a function of reactant concentration within the pellet, draw and/or explain how the temperature profile will change with a corresponding change in Thiele Modulus. You should use at least two different extremes for Thiele Modulus in your explanation. 6

Chemical Reaction Engineering Prof. Jayant Modak Department of Chemical Engineering Indian Institute of Science, Bangalore

Chemical Reaction Engineering Prof. Jayant Modak Department of Chemical Engineering Indian Institute of Science, Bangalore Lecture No. # 26 Problem solving : Heterogeneous reactions Friends, in last few