Chapter 6, Problem 20P

(a)

To determine

Find the outdoor-indoor temperature difference in degrees Fahrenheit.

Answer to Problem 20P

The outdoor-indoor temperature difference is $42°\text{F}$.

Explanation of Solution

Given data:

The inside room temperature $\left(T_{\text{indoor}}\right)$ is $68°\text{F}$.

The outside air temperature $\left(T_{\text{outdoor}}\right)$ is $110°\text{F}$.

Calculation:

The difference between the outdoor-indoor temperatures is,

$T_{\text{outdoor}}-T_{\text{indoor}}$ (1)

Substitute $68°\text{F}$ for $T_{\text{indoor}}$ and $110°\text{F}$ for $T_{\text{outdoor}}$ in equation (1).

$\begin{array}{c}{T}_{\text{outdoor}}-T_{\text{indoor}}=110°\text{F}-68°\text{F}\\ =42°\text{F}\end{array}$

Thus, the outdoor-indoor temperature difference is $42°\text{F}$.

Conclusion:

Hence, the outdoor-indoor temperature difference is $42°\text{F}$.

(b)

To determine

Find the outdoor-indoor temperature difference in degrees Rankine.

Answer to Problem 20P

The outdoor-indoor temperature difference is $42°\text{R}$.

Explanation of Solution

Given data:

The inside room temperature $\left(T_{\text{indoor}}\right)$ is $68°\text{F}$.

The outside air temperature $\left(T_{\text{outdoor}}\right)$ is $110°\text{F}$.

Formula used:

Formula to calculate the outdoor temperature in degrees Rankine is,

$T_{\text{outdoor}}\left(°\text{R}\right)=T_{\text{outdoor}}\left(°\text{F}\right)+459.67$ (2)

Here,

$T_{\text{outdoor}}\left(°\text{R}\right)$ is the outside temperature in degree Rankine.

$T_{\text{outdoor}}\left(°\text{F}\right)$ is the outside temperature in degree Fahrenheit.

Formula to calculate the indoor temperature in degrees Rankine is,

$T_{\text{indoor}}\left(°\text{R}\right)=T_{\text{indoor}}\left(°\text{F}\right)+459.67$ (3)

Here,

$T_{\text{indoor}}\left(°\text{R}\right)$ is the inside temperature in degree Rankine.

$T_{\text{indoor}}\left(°\text{F}\right)$ is the inside temperature in degree Fahrenheit.

Calculation:

Substitute $110$ for $T_{\text{outdoor}}$ in equation (2).

$\begin{array}{c}{T}_{\text{outdoor}}\left(°\text{R}\right)=110°+459.67\\ =569.67\end{array}$

The outside temperature is $T_{\text{outdoor}}=569.67°\text{R}$.

Substitute $68$ for $T_{\text{indoor}}$ in equation (3).

$\begin{array}{c}{T}_{\text{indoor}}\left(°\text{R}\right)=68+459.67\\ =527.67\end{array}$

The inside temperature is $T_{\text{indoor}}=527.67°\text{R}$.

Substitute $527.67°\text{R}$ for $T_{\text{indoor}}$ and $569.67°\text{R}$ for $T_{\text{outdoor}}$ in equation (1).

$\begin{array}{c}{T}_{\text{outdoor}}-T_{\text{indoor}}=569.67°\text{R}-527.67°\text{R}\\ =42°\text{R}\end{array}$

Thus, the outdoor-indoor temperature difference is $42°\text{R}$.

Conclusion:

Hence, the outdoor-indoor temperature difference is $42°\text{R}$.

(c)

To determine

Find the outdoor-indoor temperature difference in degrees Celsius.

Answer to Problem 20P

The outdoor-indoor temperature difference is $23.3°\text{C}$.

Explanation of Solution

Given data:

The inside room temperature $\left(T_{\text{indoor}}\right)$ is $68°\text{F}$.

The outside air temperature $\left(T_{\text{outdoor}}\right)$ is $110°\text{F}$.

Formula used:

Formula to calculate the outdoor temperature in degrees Celsius is,

$T_{\text{outdoor}}\left(°\text{C}\right)=\frac{5}{9}\left(T_{\text{outdoor}}\left(°\text{F}\right)-32\right)$ (4)

Here,

$T_{\text{outdoor}}\left(°\text{C}\right)$ is the outside temperature in degree Celsius.

$T_{\text{outdoor}}\left(°\text{F}\right)$ is the outside temperature in degree Fahrenheit.

Formula to calculate the indoor temperature in degrees Celsius is,

$T_{\text{indoor}}\left(°\text{R}\right)=\frac{5}{9}\left(T_{\text{indoor}}\left(°\text{F}\right)-32\right)$ (5)

Here,

$T_{\text{indoor}}\left(°\text{C}\right)$ is the inside temperature in degree Celsius.

$T_{\text{indoor}}\left(°\text{F}\right)$ is the inside temperature in degree Fahrenheit.

Calculation:

Substitute $110$ for $T_{\text{outdoor}}$ in equation (4).

$\begin{array}{c}{T}_{\text{outdoor}}\left(°\text{C}\right)=\frac{5}{9}\left(110-32\right)\\ =43.3\end{array}$

The outside temperature is $T_{\text{outdoor}}=43.3°\text{C}$.

Substitute $68$ for $T_{\text{indoor}}$ in equation (5).

$\begin{array}{c}{T}_{\text{indoor}}\left(°\text{C}\right)=\frac{5}{9}\left(68-32\right)\\ =20\end{array}$

The inside temperature is $T_{\text{indoor}}=20°\text{C}$.

Substitute $20°\text{C}$ for $T_{\text{indoor}}$ and $43.3°\text{C}$ for $T_{\text{outdoor}}$ in equation (1).

$\begin{array}{c}{T}_{\text{outdoor}}-T_{\text{indoor}}=43.3°\text{C}-20°\text{C}\\ =23.3°\text{C}\end{array}$

Thus, the outdoor-indoor temperature difference is $23.3°\text{C}$.

Conclusion:

Hence, the outdoor-indoor temperature difference is $23.3°\text{C}$.

(d)

To determine

Find the outdoor-indoor temperature difference in Kelvin.

Answer to Problem 20P

The outdoor-indoor temperature difference is $23.3°\text{K}$.

Explanation of Solution

Given data:

The inside room temperature $\left(T_{\text{indoor}}\right)$ is $68°\text{F}$.

The outside air temperature $\left(T_{\text{outdoor}}\right)$ is $110°\text{F}$.

Formula used:

Formula to calculate the outdoor temperature in Kelvin is,

$T_{\text{outdoor}}\left(\text{K}\right)=T_{\text{outdoor}}\left(°\text{C}\right)+273$ (6)

Here,

$T_{\text{outdoor}}\left(°\text{C}\right)$ is the outside temperature in degree Celsius.

$T_{\text{outdoor}}\left(\text{K}\right)$ is the outside temperature in Kelvin.

Formula to calculate the indoor temperature in Kelvin is,

$T_{\text{indoor}}\left(\text{K}\right)=T_{\text{indoor}}\left(°\text{C}\right)+273$ (7)

Here,

$T_{\text{indoor}}\left(°\text{C}\right)$ is the inside temperature in degree Celsius.

$T_{\text{indoor}}\left(\text{K}\right)$ is the inside temperature in Kelvin.

Calculation:

Refer from part (c),

The inside room temperature $\left(T_{\text{indoor}}\right)$ is $20°\text{C}$.

The outside air temperature $\left(T_{\text{outdoor}}\right)$ is $43.3°\text{C}$.

Substitute $43.3$ for $T_{\text{outdoor}}$ in equation (6).

$\begin{array}{c}{T}_{\text{outdoor}}\left(\text{K}\right)=43.3+273\\ =316.3\end{array}$

The outside temperature is $T_{\text{outdoor}}=316.3\text{K}$.

Substitute $20$ for $T_{\text{indoor}}$ in equation (7).

$\begin{array}{c}{T}_{\text{indoor}}\left(\text{K}\right)=20+273\\ =293\end{array}$

The inside temperature is $T_{\text{indoor}}=293\text{K}$.

Substitute $293\text{K}$ for $T_{\text{indoor}}$ and $316.3\text{K}$ for $T_{\text{outdoor}}$ in equation (1).

$\begin{array}{c}{T}_{\text{outdoor}}-T_{\text{indoor}}=316.3\text{K}-293\text{K}\\ =23.3\text{K}\end{array}$

Thus, the outdoor-indoor temperature difference is $23.3°\text{K}$.

Conclusion:

Hence, the outdoor-indoor temperature difference is $23.3°\text{K}$.

(e)

To determine

Check whether one degree temperature difference in Celsius equal to one temperature difference in kelvin, and one degree temperature in Fahrenheit equal to one degree temperature difference in Rankine.

Explanation of Solution

Case 1:

Refer to part (c),

The outdoor-indoor temperature difference is $23.3°\text{C}$.

Refer to part (d),

The outdoor-indoor temperature difference is $23.3°\text{K}$.

From part (c) and part (d), outdoor-indoor temperature differences are equal.

Note:

Let the temperature difference in Kelvin is,

$T_1\left(\text{K}\right)-T_2\left(\text{K}\right)$ (8)

Formula to calculate the temperature in degree Celsius is,

$T\left(\text{K}\right)=T\left(°\text{C}\right)+273$

Here,

$T\left(\text{K}\right)$ is the temperature in Kelvin.

$T\left(°\text{C}\right)$ is the temperature in degree Celsius.

Substitute $T_1\left(°\text{C}\right)+273$ for $T_1\left(°\text{R}\right)$ and $T_2\left(°\text{C}\right)+273$ for $T_2\left(°\text{R}\right)$ in equation (8).

$\begin{array}{c}{T}_1\left(\text{K}\right)-T_2\left(\text{K}\right)=\left(T_1\left(°\text{C}\right)+273\right)-\left(T_2\left(°\text{C}\right)+273\right)\\ =T_1\left(°\text{C}\right)+273-T_2\left(°\text{C}\right)-273\\ =T_1\left(°\text{C}\right)-T_2\left(°\text{C}\right)\end{array}$

Thus, one degree temperature difference in Celsius is equal to one temperature difference in kelvin.

Case 2:

Refer to part (a),

The outdoor-indoor temperature difference is $42°\text{F}$.

Refer to part (b),

The outdoor-indoor temperature difference is $42°\text{R}$.

From part (a) and part (b), outdoor-indoor temperature differences are equal.

Note:

Let the temperature difference in Rankine is,

$T_1\left(°\text{R}\right)-T_2\left(°\text{R}\right)$ (9)

Formula to calculate the temperature in Rankine is,

$T\left(°\text{R}\right)=T\left(°\text{F}\right)+459.67$

Here,

$T\left(°\text{R}\right)$ is the temperature in degree Rankine.

$T\left(°\text{F}\right)$ is the temperature in degree Fahrenheit.

Substitute $T_1\left(°\text{F}\right)+459.67$ for $T_1\left(°\text{R}\right)$ and $T_2\left(°\text{F}\right)+459.67$ for $T_2\left(°\text{R}\right)$ in equation (9).

$\begin{array}{c}{T}_1\left(°\text{R}\right)-T_2\left(°\text{R}\right)=\left(T_1\left(°\text{F}\right)+459.67\right)-\left(T_2\left(°\text{F}\right)+459.67\right)\\ =T_1\left(°\text{F}\right)+459.67-T_2\left(°\text{F}\right)-459.67\\ =T_1\left(°\text{F}\right)-T_2\left(°\text{F}\right)\end{array}$

Therefore, one degree temperature in Fahrenheit is equal to one degree temperature difference in Rankine.

Conclusion:

Hence, one degree temperature difference in Celsius equal to one temperature difference in kelvin, and one degree temperature in Fahrenheit equal to one degree temperature difference in Rankine has been explained.