2001_-2003__Collected_Exams - 2001 2003 Collected Exams...

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2001 – 2003 Collected Exams 2001 Collected Exams MSM 855 Midterm Exam Friday, April 6, 2001 Room 1314 Engineering, 1:50 pm (Exam starts, 1:40 pm, ends 2:50 pm) 1. (a) Consider a three-dimensional diffusion problem expressed in cylindrical coordinates. Give an example of a physical situation for which one could utilize symmetry to simplify the solution of a diffusion problem. (15 points) (b) BRIEFLY discuss all assumptions or simplifications that are involved (if any) of your statement of the role of symmetry in this problem. (10 points) 2 (a) . Sketch (Co/2)[1 + erf(z)] as a function of z. Label the axes. Indicate the values of (Co/2) [1 + erf(z)] at z = 0, z and for -∞ z . (12 points) (b). Sketch (C 1 /2)[1 - erf(z)] as a function of z. Label the axes. Indicate the values of (C 1 /2) [1 - erf(z)] at z = 0, z and for -∞ z . (12 [points) 3. For the sum C(x,t) = {(Co/2)[1 + erf(z)] } + {(C 1 /2) [1 - erf(z)]}, where z = Dt 4 x , (i) sketch C(x,t) for time t = 0. Label the axes. (10 points) (ii) sketch C(x,t) times t 1 and t 2 , where t 2 > t 1 > 0. Label the axes. (15 pts) (iii) Evaluate C(x=0, t), the concentration at x = 0. Explain what C(x=0, t) means physically, including the way in which C(x=0,t) depends on time. (15 points) (iv) C(x,t) is the solution to what physical problem. Describe it briefly in words. Is it related to part (i) of this problem? (10 points) 4. Beginning with Fick's First Law and the continuity equation, derive Fick's Second Law. Express ALL equations in coordinate-invariant form. Show the two cases (i) D a function of C, and (ii) D is not a function of C. (30 points) 5. In the expression derived in class for a rectilinear, bounded medium ) / t exp( L ) x n ( sin A ) t , x ( C 1 n n τ - π = = where A n are the Fourier coefficients. (a) Give a physical explanation of the role of the coefficients A n . Be specific. (15 pts) (b) The solution involves the product of a function of space, x, and a function of time, t. Describe the assumptions (both physical and mathematical) that lead to this type of solution. (10 points) 1
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(c) For an arbitrary initial concentration f(x'), describe how one can employ the time constant τ to estimate the amount of time needed to "homogenize" the specimen. Relate this criterion to equation 1 above (both the sin term and the exp term). Finally, be specific about the criterion used to decide whether or not the specimen has been homogenized. (25 points) 6. For the initial distribution f(x') given by f(x') = 3A 1 x 3 + 5A 2 x for x' > 0 f(x') = 0 otherwise, (a) Set up the integral that can be used to solve this problem for times t > 0. Use the coordinate transformation Dt 4 x 2 2 = η Write the integrand and the integrator in terms of the transformed coordinate, η . Also, demonstrate the appropriate change of integration limits. (26 pts) (b) State the assumptions involved in the solution to this problem, including the nature of the host. Is D = D(C) an important consideration in this problem? Is your solution
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2001_-2003__Collected_Exams - 2001 2003 Collected Exams...

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