TAILIEUCHUNG - Heat Transfer Handbook part 125

Heat Transfer Handbook part 125. The Heat Transfer Handbook provides succinct hard data, formulas, and specifications for the critical aspects of heat transfer, offering a reliable, hands-on resource for solving day-to-day issues across a variety of applications. | HEAT TRANSFER TO MOVING MATERIALS UNDERGOING THERMAL PROCESSING 1237 Figure Steady-state temperature distributions in a continuously moving cylindrical rod for different values of L D 4Pe VD a 4Bi hD k . Figure shows the results for the temperature distribution in a circular rod y D 4 for one particular case. Note that when L D 7 for this case the condition of an infinitely long rod is attained and the temperature at the end of the rod approaches asymptotically that of the environment. Under those conditions it is easlly shown that eq. reduces to 0 x Pe - VPe2 4 Bi - exp i------------2---------x j Necessarily eqs. and are valid only when the value of Bi às suffieienlly small to ensure that the temperature variation across the rod is negligible compared to that along its length. Furthermore when Pe 0 the problem reduces to that of a stationary extended surface. The problem depicted in Fig. may be extended to include a more realistic boundary condition at x 0 and L taking into account the upstream and downstream thermal conditions. If - x œ 0o and - must remain finite as x œ the temperature distribution in the region exposed to the thermal environment is governed by the following three equations d2Oi dOi --1 - Pe dx 2 dx for x 0 1238 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING d 202 d 02 TT - Pe dx 2 dx Bi0 0 d203 d 03 T t Pe- dx 2 dx for 0 x L for x L for which Jaluria has provided the solutions as 01 Pe m2em2L m1emiL j m2emiL m2 em2L ePex 02 1 Pe m2em2L em1x m1em1L em2x m2em1 L m2 em2L 03 1 Pe m2 m1 ePeL m2em1L m2em2L 1 As in the more limited case three physical parameters emerge from the analysis. The Peclet number Pe represents a dimensionless speed of the workpiece and serves to compare the ability of the workpiece to advect energy through its motion to its abiiity to transfer energy along its length due to thermal conduction. The Biot number Bi may be thought of as a ratio of mtemal ihermai .

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