The Basic Technique

The two alternative ways of setting up the nodal equations are the weighted residual approach and the physical approach.

Using the weighted residual approach, the 2-D heat conduction equation can be approximately satisfied by:

(14.2)

where the weight

within the control volume.

outside the control volume.

Thus, we get, for each i = 1, ....n

(14.3)

Integrating equation (14.3) by parts, we get:

where the Gauss divergence theorem has been used to convert the volume integral to a surface integral.

(14.4)

The meaning of Eqn. (14.4) is that the net heat generation rate in the control volume is equal to the net sum of the rate of heat energy going out of the control volume where is the boundary of the control volume

Equation (14.4) can be taken as an energy balance equation for the control volume. This balance equation can also be obtained physically, considering the balance of heat flux in Figure 14.2.

Figure 14.2: Balance of Heat Flux in a Control Volume.

For a typical node P with neighbors E,N,W,S (standing for east, north, west and south etc.) and corresponding control volume boundaries in those directions denoted by e,n,w,s etc., the heat balance for the control volume can be written as follows (for unit depth in z-direction):

when is the heat flux (per unit area) on the east face, is the heat flux on the west face etc., and the faces are taken to be one unit deep perpendicular to the plane of the figure. Thus, is the total heat flux through the east face. The fluxes are taken to be positive in the directions indicated by the arrows.

Physically, the above equation is equivalent to saying :

Net rate of heat energy leaving the control volume through the boundary = Rate of heat generation within the control volume (CV) at steady state

Thus,

(14.5)

Which is the same statement as equation (14.4). In the implementation of the FVM procedure, the heat fluxes are expressed in terms of the nodal temperatures (T_E, etc. at the CV centers) using piecewise interpolation around the control volume for the field variable (temperature in this case). Thus, assuming temperature to have linear variation between points E and P, the heat flux can be evaluated as follows:

(14.6)

while deriving (14.6) it has been assumed that the cell size is constant in x-direction (equal to ).

Similarly, is given by

(14.7)

Using similar expression for and also, the nodal equation for point P becomes:

	(14.8)

This equation can be rewritten in the familiar form used in finite difference as:

(14.9)

where

During numerical implementation, the subscripts E, W, etc. will be changed to numerical indices of i, j and solved in the same way (using point-by-point or line-by-line procedure etc.) as mentioned in previous lectures on finite differences.