Question
Assume there is a non-uniform temperature field T(x,y,z,t) in a fluid moving with velocity v(x,y,z,t). Assume there are no sources of heat and take the following parameters of the fluid to be constant: density p: specific heat c; conductivity k. Assume that generation of heat by dissipation of mechanical energy is negligible. Because the density is constant, thermal energy (heat) and mechanical energy are then seperatly conserved. Making these assumptions, begin with the statement "heat is conserved" and end with the differential equation for the temperature, using the following physical model: (1.) The heat per unit volume is pcT: (2.) Heat radiation is negligible, so transport of heat is carried out by two mechanisms: (A.) Conduction. heat flow due to conduction is proportioal to the temperature gradient and directed opposite to it, i.e. it is -k(del)T. (B.) Convection. Heat flow due to the fluid motion is pcTv.
Explanation / Answer
hey are you the sjcc guy taking langlois's class at leland right now? no guarantees that this is right, but for #1 we used the continuity equation on http://en.wikipedia.org/wiki/Continuity_equation - the general one. the symbols don't transfer over when i copy paste, but the section looks like: The general form for a continuity equation where --------see link-------- trident looking symbol is some quantity, f is a vector function describing the flux (flows) of , del dot is divergence, and s is a function describing the generation and removal of . (Generation and removal are also called "sources" and "sinks" respectively, and correspond to s > 0 ands < 0 respectively.) so the trident looking thing is heat, so you take the derivative of that, and you replace f with (convection + conduction) and you set s = 0 because that's like the continuity equation that he gave us in class. final answer looks something like: (del T/del t)cp+div(pcTv-k grad T)=0 the c and the p can go outside b/c they're constants. it looks kind of simple, so we're not sure if there's anything additional thing langlois expects us to do, but hey if that's the case then we all get it wrong together :) Source(s): wikipedia.com, this math major from flexcollegeprep, some really smart classmates in our class