Chapter 14, Problem 58GP

(a)

To determine

The rate of rise of leaf’s temperature.

Answer to Problem 58GP

The rate of rise in leaf’s temperature is $2.3°\text{C}/\text{s}$ .

Explanation of Solution

Given:

The area of the leaf is, $A=40{\text{cm}}^2=40{\text{cm}}^2\times \frac{{10}^{-4}{\text{m}}^2}{1{\text{cm}}^2}=4\times {10}^{-3}{\text{m}}^2$ .

The mass of the leaf is, $m=4.5\times {10}^{-4}\text{kg}$ .

The emissivity of the leaf is, $\epsilon =0.85$ .

The specific heat of the leaf is, $s=0.80\text{kcal}/\text{kg}\cdot \text{K}=0.80\text{kcal}/\text{kg}\cdot °\text{C}\times \frac{4186\text{J}}{1\text{kcal}}=334.88\text{J}/\text{kg}\cdot °\text{C}$ .

Formula Used:

The expression to calculate the rate of rise in leaf’s temperature is,

  $\begin{array}{c}{Q}_e=Q\\ ms\Delta T=1000\epsilon tA\\ \frac{\Delta T}{t}=\frac{1000\epsilon A}{ms}\end{array}$

Here,

Q is the heat lost by radiation

  $Q_e$ heat for evaporation

Calculation:

Substitute all the values in the above expression.

  $\begin{array}{c}\frac{\Delta T}{t}=\frac{1000\left(0.85\right)\left(4\times {10}^{-3}{\text{m}}^2\right)}{\left(4.5\times {10}^{-4}\text{kg}\right)\left(334.88\text{J}/\text{kg}\cdot °\text{C}\right)}\\ =2.3°\text{C}/\text{s}\end{array}$

Conclusion:

Thus, the rate of rise in leaf’s temperature is $2.3°\text{C}/\text{s}$ .

(b)

To determine

The temperature of the leaf.

Answer to Problem 58GP

The temperature of the leaf is $357\text{K}$ .

Explanation of Solution

Given:

The temperature of surrounding is, $T=20°\text{C}$ .

Formula Used:

The expression to calculate the temperature of the leaf is,

  $T_l=\sqrt{\frac{1000\cos \theta }{2\sigma }+T^4}$

Calculation:

Substitute all the values in the above expression.

  $\begin{array}{c}{T}_l=\sqrt{\frac{1000\left(\cos 0°\right)}{2\left(5.67\times {10}^{-8}\text{W}/{\text{m}}^2\cdot \text{K}\right)}+{\left(20°\text{C}\right)}^4}\\ =357\text{K}\end{array}$

Conclusion:

Thus, the temperature of the leaf is $357\text{K}$ .

(c)

To determine

The other method of the heat transfer through leaf.

Answer to Problem 58GP

The heat transfer from leaf occurs through convection as well as conduction.

Explanation of Solution

Introduction:

The heat can be transferred by conduction, convection and radiation.

The water evaporates from the surface of the leafs. The cool air takes heat of the leaf by convection. The heat transfer in the different positions of leaf occurs due to conduction.

Conclusion:

Thus, the heat transfer from leaf occurs through convection as well as conduction.