Appendix D ‐ Can “ANYTHING” Happen in an Open System?
Granville Sewell · 2005
Appendix D-Can "ANYTHING" Happen in an Open System?The partial differential equations that govern heat conduction and diffusion, derived in Section 2.0 and studied extensively in this book, have some relevance to an interesting philosophical question.Consider the diffusion (conduction) of heat in a solid, R, with (absolute) temperature distribution U ( z , y, z , t).The first law of thermodynamics (conservation of energy) requires that where Q (Q = cpU) is the heat energy density and J is the heat flux vector.The second law requires that the flux be in a direction in which the temperature is decreasing, that is, JTVU 5 0 (D.2)In fact, in an isotropic solid, J is in the direction of greatest decrease of temperature, that is, J = -K V U (cf. equation 2.0.2).Note that D.2 simply says that heat flows from hot to cold regions-because the laws of probability favor a more uniform distribution of heat energy."Thermal entropy" is a quantity that is used to measure randomness in the distribution of heat.The rate of change of thermal entropy, S, is given by the usual definition (see Problem 10 of Chapter 2) as Using D.3 and the first law D .l , we get 316The Numerical Solution of Ordinary and Partial Differential Equations, 2nd Edition.