Question

Figure 18-26 shows three different arrangements of materials 1, 2, and 3 to form a wall. The thermal conductivities are ${\mathbf{k}}_{\mathbf{1}}\mathbf{>}{\mathbf{k}}_{\mathbf{2}}\mathbf{>}{\mathbf{k}}_{\mathbf{3}}$. The left side of the wall is${\mathbf{20}}^{\mathbf{°}}$ higher than the right side. Rank the arrangements according to (a) the (steady state) rate of energy conduction through the wall and (b) the temperature difference across material 1, greatest first.

Short answer

1. The ranking of the arrangements according to the rate of energy conduction through the wall is $\left(\mathrm{a}\right)=\left(\mathrm{b}\right)=\left(\mathrm{c}\right)$.
2. The ranking of the arrangements according to the temperature difference across material 1 is$\left(\mathrm{a}\right)=\left(\mathrm{b}\right)=\left(\mathrm{c}\right)$.

Step-by-step solution

The given data

1. The thermal conductivities are ${\mathrm{k}}_1>{\mathrm{k}}_2>{\mathrm{k}}_3$.
2. The left side of the wall is ${20}^{°}$higher than right side.

 Step 2: Understanding the concept of conduction rate

Fourier's law of heat conduction, which states that the temperature differential in a homogeneous solid body is directly proportional to the cross-sectional area and the rate of heat transfer, defines heat conduction. Using the formula for the conduction rate, we can find the ranking of arrangements according to the rate of energy conduction through the wall and according to the temperature difference across material 1.

Formula:

The conduction rate ${\mathrm{P}}_{\mathrm{cond}}$of the material, ${\mathrm{P}}_{\mathrm{cond}}=\frac{\mathrm{A}\left({\mathrm{T}}_{\mathrm{H}}-{\mathrm{T}}_{\mathrm{C}}\right)}{\sum \mathrm{L}/\mathrm{k}}$ …(i)

where, T_H is the temperature of hot material, T_C is the temperature of cold material, k is the thermal conductivity of the material, A is face area and L is the thickness.

(a) Calculation of the ranking according to their rate of energy conduction

Form Figure 18-26, we see that the values of A,$\sum \mathrm{L}$ and$\left({\mathrm{T}}_{\mathrm{H}}-{\mathrm{T}}_{\mathrm{C}}\right)$. are same in all three arrangements.

The thermal conductivity of the wall is

$\begin{array}{rcl}\sum \mathrm{k}& =& {\mathrm{k}}_1+{\mathrm{k}}_2+{\mathrm{k}}_3\\ & =& \mathrm{constant}\end{array}$

Thus, we get that the rate of conduction rates are:

${\mathrm{P}}_{\mathrm{cond},\mathrm{a}}={\mathrm{P}}_{\mathrm{cond},\mathrm{b}}={\mathrm{P}}_{\mathrm{cond},\mathrm{c}}={\mathrm{P}}_{\mathrm{cond},\mathrm{d}}$

Therefore, the rate${\mathrm{P}}_{\mathrm{cond}}$of energy conduction through the wall isthesame for all three arrangements.

Therefore, the ranking of the arrangements according to the rate of energy conduction through the wall is $\left(\mathrm{a}\right)=\left(\mathrm{b}\right)=\left(\mathrm{c}\right)$.

(b) Calculation of the ranking according to their temperature difference across material 1

The thermal conductivity of material 1 is k₁ and from part (a) we know that the rate of conduction is given using equation (i) as follows:

$\begin{array}{rcl}{\mathrm{P}}_{\mathrm{cond}}& =& \frac{\mathrm{A}\left({\mathrm{T}}_{\mathrm{H}}-{\mathrm{T}}_{\mathrm{C}}\right)}{\sum \mathrm{R}}\left(\because \mathrm{R}=\mathrm{L}/\mathrm{k}\right)\\ & =& \frac{\mathrm{A}\left(∆{\mathrm{T}}_1\right)}{{\mathrm{R}}_1}\end{array}$

Thus, we get the temperature across materialfrom the above equation as follows:

$∆{\mathrm{T}}_1=\frac{{\mathrm{R}}_1\left({\mathrm{P}}_{\mathrm{cond}}\right)}{\mathrm{A}}$

Since, all the right hand terms of the above equationare constant for material 1. Thus,$∆{\mathrm{T}}_1$is constant for any arrangement.

Therefore, we can say that the ranking of the arrangements according to the temperature difference across material 1 is $\left(\mathrm{a}\right)=\left(\mathrm{b}\right)=\left(\mathrm{c}\right)$.