Question

Consider a medium in which the finite difference formulation of a general interior node is given in its simplest form as $$ T_{\text {node }}=\left(T_{\text {beft }}+T_{\text {top }}+T_{\text {right }}+T_{\text {bottom }}\right) / 4 $$ (a) Is heat transfer in this medium steady or transient? (b) Is heat transfer one-, two-, or three-dimensional? (c) Is there heat generation in the medium? (d) Is the nodal spacing constant or variable? (e) Is the thermal conductivity of the medium constant or variable?

Short answer

Based on the given finite difference formulation, determine the characteristics of the medium in which this formulation is applicable, including whether the heat transfer is steady or transient, one-, two- or three-dimensional, if there is heat generation, if the nodal spacing is constant or variable, and if the thermal conductivity is constant or variable.

Step-by-step solution

Question (a) - Steady or Transient

The given formula for the temperature of a node is independent of time, which means that the temperatures of nodes do not change with time. Therefore, the heat transfer in this medium is steady.

Question (b) - Dimensionality

The formula includes temperature terms from four neighboring nodes: left, top, right, and bottom. As a result, we can determine that the heat transfer is happening in two axes (x and y). Hence, the heat transfer is two-dimensional.

Question (c) - Heat Generation

Since the given finite difference formulation only includes the average temperature from the neighboring nodes, without any additional terms, there is no heat generation in the medium.

Question (d) - Nodal Spacing

The given formula doesn't provide any specific information about nodal spacing. However, the formulation has a simple, averaged equation, which suggests that the nodal spacing is constant. A more complicated equation would be required if nodal spacing were variable.

Question (e) - Thermal Conductivity

As with nodal spacing, the given formula doesn't provide any specific information about the variation in thermal conductivity. Since the given formulation is simple and only consists of averaged temperature from the neighboring nodes, it is reasonable to assume that the thermal conductivity of the medium is constant. A more complex equation or additional terms would be required if the thermal conductivity were variable.