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Unformatted text preview: Isz . Chaplin'ﬁfoifﬁuion concentration gradient according to Fick’s ﬁrst law. The mathematics for nonsteady
state are described by Fick’s second law, a partial differential equation. The solution
for a constant surface composition boundary condition involves the Gaussian error
function. Factors That Inﬂuence Diffusion The magnitude of the diffusion coefﬁcient is indicative of the rate of atomic motion,
being strongly dependent on and increasing exponentially with increasing tempera
ture. ' Diffusion in Ionic and Polymeric Materials Diffusion in ionic materials occurs by a vacancy mechanism; localized charge neu
trality is maintained by the coupled diffusive motion of a charged vacancy and some
other charged entity. With regard to diffusion in polymers, small molecules of foreign substances dif
fuse between molecular chains by an interstitialtype mechanism from one amor
phous region to an adjacent one. Diffusion (or permeation) of gaseous species is "
often characterized in terms of the permeability coefﬁcient, which is the product
of the diffusion coefﬁcient and solubility in the polymer. Permeation ﬂow rates are ‘ expressed in terms of a modiﬁed form of Fick’s ﬁrst law. IMPORTANT TERMS AND CONCEPTS Activation energy Diffusion ﬂux Interstitial diffusion
Carburizing Driving force Nonsteadystate diffusion
Concentration gradient Pick‘s ﬁrst and second laws Selfdiffusion
Concentration proﬁle Interdiffusion (impurity Steadystate diffusion
Diffusion diffusion) Vacancy diffusion Diffusion coefﬁcient REFERENCES
Carslaw, H. S. and J. C. laeger, Conduction of Heat ButterworthHeinemann Ltd, Woburn, UK,
in Solids, 2nd edition, Oxford University Press, 2004. '
Oxford, 1986. Glicksman, M., Diﬂizsion in Solids, Wlley
Crank, 1., The Mathematics of Diffusion. 2nd edi Interscience, New York, 2000.
tion, Oxford University Press, Oxford, 1980. Shewmon, P. (3., Diffusion in Solids, 2nd edition,
Gale, W. F. and T. C. Totemeier, (Editors), The Minerals, Metals and Materials Society,
Smirhells Merci: Reference Book, 8th edition, Warrendale, PA. 1989. QUESTIONS AND PROBLEMS Additional problems and questions for this chapter may be found on both Student and
Instructor Companion Sites at ww.wifey.com/coﬂege/caﬂﬁter. introduction Diffusion Mechanisms l 6.1 Brieﬂy explain the difference between self 6.2 (1:) Compare interstitial and vacancy atomic
diffusion and interdiffusion. mechanisms for diffusion. (b) Cite two reasons why interstitial diffusion
is normally more rapid than vacancy dif
fusion. SteadyState Diffusion 6.3 6.4 6.5 (a) Brieﬂy explain the concept of a driving
force.
(6) What is the driving force for steadystate
diffusion?
The puriﬁcation of hydrogen gas by diffusion
through a palladium sheet was discussed in
Section 6.3. Compute the number of kilograms
of hydrogen that pass per hour through a 6
mmthick sheet of palladium having an area
of 0.25 m2 at 600°C. Assume a diffusion co
efﬁcient of 1.? x 10’8 mzis, that the concen
trations at the high and lowpressure sides of
the plate are 2.0 and 0.4 kg of hydrogen per cu
bic meter of palladium. and that steadystate
conditions have been attained. A sheet of ECG iron 2 mm thick was exposed
to a carburizing gas atmosphere on one side
and a decarburizing atmosphere on the other
side at 675°C. After having reached steady
state, the iron was quickly cooled to room
temperature. The carbon concentrations at the
two surfaces of the sheet were determined to
be 0.015 and 0.0068 wt%. Compute the diffu
sion coefﬁcient if the diffusion flux is 7.36 x
10—9 kg/mzs Him: Use Equation 5.12 to con
vert the concentrations from weight percent to
kilograms of carbon per cubic meter of iron. NonsteadyState Diffmion 6.6 6.7 WM 6.8 Determine the carburizing time necessary to
achieve a carbon concentration of 0.30 wt%
at a position 4 mm into an iron—carbon alloy
that initially contains 0.10 wt% C. The sur
face concentration is to be maintained at 0.90
wt% C, and the treatment is to be conducted
at 1100°C. Use the diffusion data for y—Fe in
Table 6.2. Nitrogen from a gaseous phase is to be diffused
into pure iron at 675°C. If the surface concen
tration is maintained at 0.2 wt% N. what will
be the concentration 2 mm from the surface af
ter 25 h? The diffusion coefﬁcient for nitrogen
in iron at 675°C is 1.9 x 10“1 mzis For a steel alloy it has been determined that
a carburizing heat treatment of 15 h dura Questions and Problems ' 183 tion will raise the carbon concentration to 0.35
wt% at a point 2.0 mm from the surface. Es
timate the time necessary to achieve the same
concentration at a 6.0mm position for an iden
tical steel and at the same carburizing temper
ature. Factors That inﬂuence Diffusion 6.9
6.10
as 6.11 6.12 6.13 6.14 6.15 Cite the values of the diffusion coefﬁcients
for the interdiffusion of carbon in both cariron
(ECG) and yiron (FCC) at 900°C. Which is
larger? Explain why this is the case. At what temperature will the diffusion coefﬁ
cient for the diffusion of zinc in copper have
a value of 2.6 x 10'“5 mzr’s‘? Use the diffusion
data in Table 6.2. The activation energy for the diffusion of cop
per in silver is 193,000 Jimol. Calculate the dif»
fusion coefﬁcient at 1200 K (922° C), given that
D at 1000 K (727°C) is 1.0 x 10*14 units. The diffusion coefﬁcients for carbon in nickel
are given at two temperatures: 371°C). .___ 9(ﬂ'1’€)__
600 5.5 x 10”
301. . . _39 ?‘ 10;” (:1) Determine the values of Do and (25.
(b) What is the magnitude of D at 850° C? Carbon is allowed to diffuse through a steel
plate 10 mm thick. The concentrations of car
bon at the two faces are 0.85 and 0.40 kg Cicm3
Fe, which are maintained constant. If the pre»
exponential and activation energy are 6.2 x
10;, mzis and 80.000 Jlmol, respectively, com
pute the temperature at which the diffusion
ﬂux is 6.3 x 10“” kglrnzs. At approximately what temperature would a
specimen of yiron have to be carburized for
4 h to produce the same diffusion result as at
1000°C for 12 h? A copper—nickel diffusion couple similar to
that shown in Figure 6.1:: is fashioned. After a
SOOh heat treatment at 1000°C (1273 K), the
concentration of Ni is 3.0 wt ”/0 at the 1.0mm
position within the copper. At what temper
ature should the diffusion couple be heated
to produce this same concentration (i.e., 3.0
wt% Ni) at a 2.0n1rn position after 500 h? The 134 ' Chapteré f Diffusion preexponential and activation energy for the diffusion of Ni in Cu are 2.7 x 10"4 mza’s and _ 236.000 meol, respectively. 6.16 The outer surface of a steel gear is to be hard ened by increasing its carbon content; the car
bon is to be supplied from an external carbon
rich atmosphere that is maintained at an
elevated temperature. A diffusion heat treat
ment at 600°C (873 K) for 100 min increases
the carbon concentration to 0.75 wt% at a po
sition 0.5 mm below the surface. Estimate the
diffusion time required at 900°C (1173 K) to
achieve this same concentration also at a 0.5
mm position. Assume that the surface carbon
content is the same for both heat treatments,
which is maintained constant. Use the diffu
sion data in Table 6.2 for C diffusion in txFe. 6.18 maintained constant. Assuming conditions 0
steady state. what is the diffusion ﬂux [in (cm:
S'I‘P)fcm2s] at 298 K? The permeability coefﬁcient for a type of smal
gas molecule in a polymer is dependent on ab
solute temperature according to the following
equation: PM = PHD exp(— 3—?) where Pm and Q, are constants for a giver
gas—polymer pair. Consider the diffusion 01
water through a polystyrene sheet 30 mn
thick. The water vapor pressures at the twc
faces are 20 kPa and 1 kPa, which are main tained constant. Compute the diffusion ﬂux [in
(cm3 STP)fcm2~s] at 350 K? For this diffusion Diffusion in Polymeric Material: system 5 3 2 6.1'1‘r Consider the diffusion of oxygen through a PM“ : 9'0 x 10 (cm STP)(cm);crn s~Pa I low density polyethylene (LDPE) sheet 15 (2;: =42.3 kJ [mol
mm thick. The pressures of oxygen at the two . . .
faces are 2000 kPa and 150 kPa. which are 3:10, assumeacondrtion of steadystate diffu DESIGN PROBLEMS Furthermore, the diffusion coefﬁcients for the
diffusion of these gases in the metal are func
tions of the absolute temperature as follows: Steamy— State Diffusion
[Factors That inﬂuence Diffusion) 6.D1 A gas mixture is found to contain two di» atomic A and B species (A; and B2) for which
the partial pressures of both are 0.1013 MPa
(1 atm). This mixture is to be enriched in the
partial pressure of the A species by passing
both gases through a thin sheet of some metal
at an elevated temperature. The resulting en
riched mixture is to have a partial pressure
of 0.051 MPa (0.5 attn) for gas A and 0.0203
MPa (0.2 atm) for gas B. The concentrations
ofA and B (CA and C3. in molfm3) are func~
tions of gas partial pressures (pg2 and p52, in
MPa) and absolute temperature according to
the following expressions: cA : 1.5 x 103Mexp(———20'0:¥m01)
(6.17a) CE = 2.0 x103./p32exp(—§%Tﬂ)
(6.1?b) RT
(6.18s) 21.0 kJ {11101) RT
(6.18b) Is it possible to purify the A gas in this man
ner? If so, specify a temperature at which the
process may be carried out, and also the thick
ness of metal sheet that would be required. If
this procedure is not possible, then state the
reason(s) why. Da(m2!s) = 5.0 x IO'Texp(—M) DB(m2/s) = 3.0 x 10'6exp(— Noristeady~$tafe Diffusion (Factors That influence Diffusion] 6.D2 The wear resistance of a steel gear is to be im— proved by hardening its surface, as described
in Design Example 6.1. However, in this case the initial carbon content of the steel is 0.15
wt%, and a carbon content of 0.75 M96 is
to beestablished at a position 0.65 mm below
the surface. Furthermore, the surface concen
tration is to be maintained constant. but may Design Problems ' 185 be varied between 1.2 and 1.4 m% C. Spec
ify an appropriate heat treatment in terms
of surface carbon concentration and time.
and for a temperature between 1000°C and 1200" C. ...
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 Fall '08
 GRONSKY

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