Chapter 6.11, Problem 138RP

To determine

The time required to cool the truck from $25°\text{C}$ to $5°\text{C}$.

Answer to Problem 138RP

The time required to cool the truck from $25°\text{C}$ to $5°\text{C}$ is $\underset{\_}{\text{3}\text{.95}\text{min}}$.

Explanation of Solution

Determine the mass of the air in the truck.

$\begin{array}{c}{m}_{\text{air}}={\rho }_{\text{air}}\times {\nu }_{\text{truck}}\\ ={\rho }_{\text{air}}\times {\left(l\times b\times h\right)}_{\text{truck}}\end{array}$ (I)

Here, the density of air is ${\rho }_{\text{air}}$, the volume of the truck is ${\nu }_{\text{truck}}$, the length of the truck is $l$, the breath of the truck is $b$, and the height of the truck is $h$.

Determine the amount of heat removed from the truck.

$\begin{array}{c}{Q}_{\text{cooling,air}}=\left(m_{\text{air}}{c}_p\Delta T\right)\\ =m_{\text{air}}{c}_p\left(T_1-T_2\right)\end{array}$ (II)

Here, the specific heat at the average temperature is $c_p$, the initial temperature of the truck is $T_1$, and the final temperature of the truck is $T_2$.

Determine the average temperature in the truck during precooling.

$T_{\text{avg}}=\frac{T_1+T_2}{2}$ (III)

Determine the average rate of heat gain by transmission.

$\begin{array}{c}{\dot{Q}}_{transmission,avg}=UA\Delta T\\ =UA\left(T_1-T_{\text{avg}}\right)\end{array}$ (IV)

Here, the rate of transmission heat gain occurs is $UA$.

Determine the time required to cool the truck.

$\begin{array}{l}{\dot{Q}}_{refrig}\times \Delta t=Q_{cooling,air}+{\dot{Q}}_{transmission}\times \Delta t\\ \Delta t=\frac{Q_{cooling,air}}{{\dot{Q}}_{refrig}-{\dot{Q}}_{transmission}}\end{array}$ (V)

Conclusion:

From the Table A-2, “Ideal-gas specific heats of various common gases” to obtain value of density of air and the specific heat at the average temperature of $15°\text{C}$ as $1.2\text{kg}/{\text{m}}^{\text{3}}$ and $1.0\text{kJ}/\text{kg}\cdot °\text{C}$.

Substitute $1.2\text{kg}/{\text{m}}^{\text{3}}$ for ${\rho }_{\text{air}}$, 12 m for $l$, 2.3 m for $b$, and 3.5 m for $h$ in Equation (I).

$\begin{array}{c}{m}_{\text{air}}=\left(1.2\text{kg}/{\text{m}}^{\text{3}}\right)\times {\left(12\text{m}\times 2.3\text{m}\text{}\times 3.5\text{m}\right)}_{\text{truck}}\\ =\left(1.2\text{kg}/{\text{m}}^{\text{3}}\right)\times \left(96.6{\text{m}}^{\text{3}}\right)\\ =115.92\text{kg}\\ \cong \text{116}\text{kg}\end{array}$

Substitute 116 kg for $m_{\text{air}}$, $1.0\text{kJ}/\text{kg}\cdot °\text{C}$ for $c_p$, $25°\text{C}$ for $T_1$, and $5°\text{C}$ for $T_2$ in Equation (II).

$\begin{array}{c}{Q}_{\text{cooling,air}}=\left(116\text{kg}\right)\times \left(1.0\text{kJ}/\text{kg}\cdot °\text{C}\right)\times \left(25-5\right)°\text{C}\\ =\left(116\text{kg}\right)\times \left(1.0\text{kJ}/\text{kg}\cdot °\text{C}\right)\times \left(20°\text{C}\right)\\ =2,320\text{kJ}\end{array}$

Substitute $25°\text{C}$ for $T_1$, and $5°\text{C}$ for $T_2$ in Equation (III).

$\begin{array}{c}{T}_{\text{avg}}=\frac{\left(25+5\right)°\text{C}}{2}\\ =15°\text{C}\end{array}$

Substitute $120\text{W}/\text{°C}$ for $UA$, $25°\text{C}$ for $T_1$, and $15°\text{C}$ for $T_{avg}$ in Equation (IV).

$\begin{array}{c}{\dot{Q}}_{transmission,avg}=\left(120\text{W}/\text{°C}\right)\times \left(25°\text{C}-15°\text{C}\right)\\ =\left(120\text{W}/\text{°C}\right)\times \left(10°\text{C}\right)\\ =1200\text{W}\\ =1200\text{W}\times \left(\frac{{10}^{-3}\text{kJ}/\text{s}}{1\text{W}}\right)\end{array}$

                  $=1.2\text{kJ}/\text{s}$

Substitute 2320 kJ for $Q_{cooling,air}$, $11\text{kJ}/\text{s}$ for ${\dot{Q}}_{refrig}$, $1.2\text{kJ}/\text{s}$ for ${\dot{Q}}_{transmission}$ in Equation (V).

$\begin{array}{c}\Delta t=\frac{2320\text{kJ}}{\left(11-1.2\right)\text{kJ}/\text{s}}\\ =\frac{2320\text{kJ}}{9.8\text{kJ}/\text{s}}\\ =236.73\text{s}\\ =236.73\text{s}\times \left(\frac{1\min }{60\sec }\right)\end{array}$

    $\begin{array}{l}=3.9455\text{min}\\ \cong 3.95\text{min}\end{array}$

Thus, the time required to cool the truck from $25°\text{C}$ to $5°\text{C}$ is $\underset{\_}{\text{3}\text{.95}\text{min}}$.