Newton

  Calculus Without Tears



Derivation of the differential equation model for 2-D heat transfer

Fourier's heat transfer law in 2-D
eq A010
It states that the heat flux qx in the x direction is proportional to negative of the partial derivative of temperature w.r.t. x, and the heat flux qy in the y direction is proportional to the negative of the partial derivative of temperature w.r.t. y.

Flux is a flow rate divided by an area, so that flux∙area equals a flow rate. A flow of what? Heat energy.

Since flux is proportional to the negative of the directional derivative, heat energy flows from a region with high temperature to a region with low temperature.

Given a thin plate, the control volume, as in the diagram, the heat flux through the surface A is given by the ľk times the directional derivative of T in x direction evaluated at the center of A, (x-dx/2,y), where we assume that the control volume is A1.jpg small enough that the temperature can be considered as constant on a side, and thus the rate heat is flowing into the small square through surface A is
eq A001


Note that a positive q represents heat flowing to the right and hence into the control volume through side A.

Similarly, the rate heat is flowing out of the square thru side B is
eq A002
The rate heat is flowing into the square thru side C is
eq A003
The rate heat flowing out of the square thru side D is
eq A004
Adding these four terms gives the rate that heat energy is flowing in/out from all sides
eq A005

Grouping terms and dividing by the volume gives the rate that heat energy is flowing into the square per unit volume
eq A005

Heat energy flowing in/out of the control volume changes its temperature. The relationship between heat energy and temperature is given by

eq A007

Thus the rate of change of temperature in the control volume is given by
eq A008
and letting dx -> 0, dy -> 0, we have the differential equation model for time dependent 2-D heat transfer

eq A009




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