(wt%)-(wt%)(x,t) /Cs-C(x,t) could examine the ratio: If the ratio is one, there is (wt%) -(wt%) CS-Co S no error, and the farther the ratio is from one, the more extensive the error. It is obvious the ratio will be one for C(x,t) = Co. Show that the ratio is otherwise always less than one for an interstitial impurity A diffusing in a host B, again assuming the lattice parameter and vacancy content do not change with composition. (HINT: You should be able to show that the derivative of the ratio with respect to (wt%) (x,t) is negative, meaning the ratio always decreases from the value at C(x,t) = C₁ as C(x,t) increases.) (NOTE: The case explored here is for Cs > Co, although a similar conclusion can be reached for Co > Cs, except here the ratio would monotonically increase from one as C(x,t) decreases to Cs.) c) From part (b) we may conclude that the most extensive error occurs when C(x,t) = Cs for an interstitial impurity A diffusing in a host B. Produce a graph of (wt%)s vs. (wt%), for which the extreme value of the ratio will be 0.8. Shade in the portion of the graph (above or below your curve), for which it is guaranteed our error ratio will always fall between 0.8 and 1.0 for any composition (wt%), <(wt%)(x,t)<(wt%)s · a) Show that for a substitutional impurity A diffusing in a host B, replacing concentrations (C) with at% results in the exact same ratio as C₁-C(x,t) CS-Co assuming the lattice parameter and vacancy content of the material do not change with composition. In other words, assume that there is always the same number of atoms (A and B combined) per unit volume. b) As a measure of the inaccuracy in assuming (wt%)s-(wt%)(x,t) _ Cs-C(x,t) (wt%)-(wt%) we , Cs-Co
(wt%)-(wt%)(x,t) /Cs-C(x,t) could examine the ratio: If the ratio is one, there is (wt%) -(wt%) CS-Co S no error, and the farther the ratio is from one, the more extensive the error. It is obvious the ratio will be one for C(x,t) = Co. Show that the ratio is otherwise always less than one for an interstitial impurity A diffusing in a host B, again assuming the lattice parameter and vacancy content do not change with composition. (HINT: You should be able to show that the derivative of the ratio with respect to (wt%) (x,t) is negative, meaning the ratio always decreases from the value at C(x,t) = C₁ as C(x,t) increases.) (NOTE: The case explored here is for Cs > Co, although a similar conclusion can be reached for Co > Cs, except here the ratio would monotonically increase from one as C(x,t) decreases to Cs.) c) From part (b) we may conclude that the most extensive error occurs when C(x,t) = Cs for an interstitial impurity A diffusing in a host B. Produce a graph of (wt%)s vs. (wt%), for which the extreme value of the ratio will be 0.8. Shade in the portion of the graph (above or below your curve), for which it is guaranteed our error ratio will always fall between 0.8 and 1.0 for any composition (wt%), <(wt%)(x,t)<(wt%)s · a) Show that for a substitutional impurity A diffusing in a host B, replacing concentrations (C) with at% results in the exact same ratio as C₁-C(x,t) CS-Co assuming the lattice parameter and vacancy content of the material do not change with composition. In other words, assume that there is always the same number of atoms (A and B combined) per unit volume. b) As a measure of the inaccuracy in assuming (wt%)s-(wt%)(x,t) _ Cs-C(x,t) (wt%)-(wt%) we , Cs-Co
Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
8th Edition
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
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Chapter5: Analysis Of Convection Heat Transfer
Section: Chapter Questions
Problem 5.2P: 5.2 Evaluate the Prandtl number from the following data: , .
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Transcribed Image Text:(wt%)-(wt%)(x,t) /Cs-C(x,t)
could examine the ratio:
If the ratio is one, there is
(wt%) -(wt%)
CS-Co
S
no error, and the farther the ratio is from one, the more extensive the error. It is obvious
the ratio will be one for C(x,t) = Co. Show that the ratio is otherwise always less than
one for an interstitial impurity A diffusing in a host B, again assuming the lattice
parameter and vacancy content do not change with composition. (HINT: You should be
able to show that the derivative of the ratio with respect to (wt%) (x,t) is negative,
meaning the ratio always decreases from the value at C(x,t) = C₁ as C(x,t) increases.)
(NOTE: The case explored here is for Cs > Co, although a similar conclusion can be
reached for Co > Cs, except here the ratio would monotonically increase from one as
C(x,t) decreases to Cs.)
c) From part (b) we may conclude that the most extensive error occurs when C(x,t) = Cs
for an interstitial impurity A diffusing in a host B. Produce a graph of (wt%)s vs.
(wt%), for which the extreme value of the ratio will be 0.8. Shade in the portion of the
graph (above or below your curve), for which it is guaranteed our error ratio will always
fall between 0.8 and 1.0 for any composition (wt%), <(wt%)(x,t)<(wt%)s ·
a) Show that for a substitutional impurity A diffusing in a host B, replacing concentrations
(C) with at% results in the exact same ratio as
C₁-C(x,t)
CS-Co
assuming the lattice
parameter and vacancy content of the material do not change with composition. In other
words, assume that there is always the same number of atoms (A and B combined) per
unit volume.
b) As a measure of the inaccuracy in assuming
(wt%)s-(wt%)(x,t) _ Cs-C(x,t)
(wt%)-(wt%)
we
,
Cs-Co
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