Question

Consider a medium in which the finite difference formulation of a general interior node is given in its simplest form as $$ \frac{T_{m-1}-2 T_{m}+T_{m+1}}{\Delta x^{2}}+\frac{\dot{e}_{m}}{k}=0 $$ (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

#Answer# The heat transfer in this medium is steady.

Step-by-step solution

Question (a)

Is heat transfer in this medium steady or transient? The given finite difference formulation is: $$ \frac{T_{m-1}-2 T_{m}+T_{m+1}}{\Delta x^{2}}+\frac{\dot{e}_{m}}{k}=0 $$ To determine whether the heat transfer is steady or transient, we should look for any time dependency within the equation. In this formulation, there is no term containing any time variable, and the temperatures are not time-dependent. Therefore, we can conclude that the heat transfer in this medium is steady.

Question (b)

Is heat transfer one-, two-, or three-dimensional? When we consider the heat transfer dimensions, we should inspect whether there are terms corresponding to each direction (x, y, and z). The given equation is only given in terms of x-direction (\(\Delta x\)). There are no y or z terms. Hence, we can deduce that the heat transfer in this medium is one-dimensional.

Question (c)

Is there heat generation in the medium? In this finite difference formulation, the term \(\frac{\dot{e}_{m}}{k}\) represents the heat generation rate per unit volume (\(\dot{e}_m\)) in the medium, divided by the thermal conductivity (k). If there were no heat generation in the medium, this term would be absent or equal to zero. However, since the term is present in the equation, we can infer that there is heat generation in the medium.

Question (d)

Is the nodal spacing constant or variable? To determine if nodal spacing is constant or variable, we should examine the denominator of the first term in the equation, which is \(\Delta x^2\). In this equation, the nodal spacing is only represented by \(\Delta x\), and no other variable is present in the denominator. Since there is no dependency on other variables or indexes, we can conclude that the nodal spacing is constant.

Question (e)

Is the thermal conductivity of the medium constant or variable? The thermal conductivity of the medium is represented by the symbol "k" in the equation. If the thermal conductivity were variable, it would typically be denoted with a subscript (e.g., \(k_m\)) or as a function of temperature or position (e.g., \(k(T)\) or \(k(x)\)). However, in this equation, the thermal conductivity is denoted simply as "k" without any further specifications. Thus, we can conclude that the thermal conductivity of the medium is constant.