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Metal Machining Episode 13

Tham khảo tài liệu 'metal machining episode 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Selected problems with no convection 353 A2.2 Selected problems with no convection When Ux uy u_ 0 and q 0 too equation A2.4 simplifies further to 1 dT d2T d 2T d 2Ĩ l_ _ _ k dt dx2 dy2 dz2 A2.5 where the diffusivity k equals K pC. In this section some solutions of equation A2.5 are presented that give physical insight into conditions relevant to machining. A2.2.1 The semi-infinite solid z 0 temperature due to an instantaneous quantity of heat Hper unit area into it over the plane z 0 at t 0 ambient temperature To It may be checked by substitution that H 1 T T - T0 e 4kt pc Pkt A2.6 is a solution of equation A2.5 . It has the property that at t 0 it is zero for all z 0 and is infinite at z 0. For t 0 dT dz 0 at z 0 and J pC T - T0 dz H 0 A2.7 Equation A2.6 thus describes the temperature rise caused by releasing a quantity of heat H per unit area at z 0 instantaneously at t 0 and thereafter preventing flow of heat across insulating the surface z 0. Figure A2.1 b shows for different times the dimensionless temperaturepC T- T0 H for a material with k 10 mm2 s typical of metals. The increasing extent of the heated region with time is clearly seen. At every time the temperature distribution has the property that 84.3 of the associated heat is contained within the region z Ỷ4kí 1. This result is obtained by integrating equation A2.6 from z 0 to V4ki. Values of the error function erf p 2 p 2 erf p J e u du V p 0 A2.8 that results are tabulated in Carslaw and Jaeger 1959 . Physically one can visualize the temperature front as travelling a distance V4kt in time t. This is used in considering temperature distributions due to moving heat sources Section A2.3.2 . A2.2.2 The semi-infinite solid z 0 temperature due to supply of heat at a constant rate q per unit area over the plane z 0 for t 0 ambient temperature To Heat dH qdt is released at z 0 in the time interval t to t dt . The temperature rise that this causes at z at a later time t is from equation A2.6 354 Appendix 2 qdt