Chapter 19, Problem 9SP

What temperature gradient must exist in an aluminum rod for it to transmit 8.0 cal per second per cm2 of cross section down the rod? kT for aluminum is 210 W/K•m.

To determine

The temperature gradient of an aluminum rod to transmit $8.0{\text{cal per second per cm}}^2$ of cross section down the rod, if the value of $k_T$ for aluminum is $210\text{ W/K}\cdot \text{m}$.

Answer to Problem 9SP

Solution:

$16°\text{C/cm}$

Explanation of Solution

Given data:

The transition from the aluminum rodper ${\text{cm}}^2$ of cross section is $8\text{ cal/s}$.

Cross section area of the rod is $1{\text{cm}}^2$.

Thermal conductivity of the rod is $210\text{ W/K}\cdot \text{m}$.

Formula used:

Write the expression forquantity of heat transmitted from the cross section of thematerial.

$\frac{\Delta Q}{\Delta t}=k_{T}A\frac{\Delta T}{L}$

Here, $\frac{\Delta Q}{\Delta t}$ is the transmission of heat from the material’s cross-section, $k_T$ is the thermal conductivity of the material, $A$ is the area of cross section through which transmission occurs, and $\frac{\Delta T}{L}$ is the temperature gradient.

Explanation:

Recall the expression for transmission of heat through the cross section of thealuminum rod.

$\frac{\Delta Q}{\Delta t}=k_{T}A\frac{\Delta T}{L}$

Substitute $8\text{ cal/s}$ for $\frac{\Delta Q}{\Delta t}$, $1{\text{ cm}}^2$ for $A$, and $210\text{ W/K}\cdot \text{m}$ for $k_T$

$\begin{array}{c}8\text{ cal/s}=\left(210\text{ W/K}\cdot \text{m}\right)\left(1{\text{ cm}}^2\right)\frac{\Delta T}{L}\\ \frac{\Delta T}{L}=\frac{\left(8\text{ cal/s}\right)\left(\frac{4.19\text{ W}}{1\text{}\text{ cal/s}}\right)}{\left(210\text{ W/K}\cdot \text{m}\right)\left(\frac{1\text{ W/}\left(°\text{C}\cdot \text{m}\right)}{1\text{ W/K}\cdot \text{m}}\right)\left(1{\text{ cm}}^2\right)\left(\frac{1{\text{m}}^2}{{10}^4{\text{ cm}}^2}\right)}\\ =\left(1596.19°\text{C/m}\right)\left(\frac{1\text{ m}}{100\text{ cm}}\right)\\ \approx 16°\text{C/cm}\end{array}$

Conclusion:

Thetemperature gradient of the aluminum rod is $16°\text{C/cm}$.