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HW-CH6 - Isz Chaplin'fifoiffiuion concentration gradient...

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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 interstitial-type 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 Nonsteady-state diffusion Concentration gradient Pick‘s first and second laws Self-diffusion Concentration profile Interdiffusion (impurity Steady-state diffusion Diffusion diffusion) Vacancy diffusion Diffusion coefficient REFERENCES Carslaw, H. S. and J. C. laeger, Conduction of Heat Butterworth-Heinemann 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. Steady-State Diffusion 6.3 6.4 6.5 (a) Briefly explain the concept of a driving force. (6) What is the driving force for steady-state 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- mm-thick 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 low-pressure sides of the plate are 2.0 and 0.4 kg of hydrogen per cu- bic meter of palladium. and that steady-state 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/mz-s Him: Use Equation 5.12 to con- vert the concentrations from weight percent to kilograms of carbon per cubic meter of iron. Nonsteady-State 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 car-iron (ECG) and y-iron (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. . . _3-9 ?‘ 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“” kglrnz-s. At approximately what temperature would a specimen of y-iron 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 SOO-h heat treatment at 1000°C (1273 K), the concentration of Ni is 3.0 wt ”/0 at the 1.0-mm 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 2.0-n1rn 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 tx-Fe. 6.18 maintained constant. Assuming conditions 0 steady state. what is the diffusion flux [in (cm: S'I‘P)fcm2-s] 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 steady-state 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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