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

Fourier's heat transfer law in 2-D

It states that the heat flux q_{x} in the x direction is proportional
to negative of the partial derivative of temperature w.r.t. x, and
the heat flux q_{y} 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 –k 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
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

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

The rate heat is flowing into the square thru side C is

The rate heat flowing out of the square thru side D is

Adding these four terms gives the rate that heat energy is flowing in/out from all sides

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

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

Thus the rate of change of temperature in the control volume is given by

and letting dx -> 0, dy -> 0, we have the differential equation model for time dependent 2-D heat transfer