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nrod2712
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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.
I know because heat is conserved that the same amount of heat going in an area must come out so I am going to have to use a flux integral, but i have no idea how to set it up. Any suggestions??
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.
I know because heat is conserved that the same amount of heat going in an area must come out so I am going to have to use a flux integral, but i have no idea how to set it up. Any suggestions??
eat is conserved, so the rate of change of thermal energy in a given volume must be equal to the sum of all the heat fluxes entering and leaving the volume. This can be expressed mathematically as a flux integral:d/dt (pcT) = ∫(-k∇T + pcTv).dA Rearranging this equation and using Divergence theorem, we get:∇.(-k∇T + pcTv) = d/dt(pcT) This is the differential equation for the temperature field T(x,y,z,t).