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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 threedimensional 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 coordinateinvariant 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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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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