Chapter 14.7, Problem 29P

The air in a room has a dry-bulb temperature of 26°C and a wet-bulb temperature of 21°C. Assuming a pressure of 100 kPa, determine (a) the specific humidity, (b) the relative humidity, and (c) the dew-point temperature.

To determine

The specific humidity.

Answer to Problem 29P

The specific humidity is $\text{0}\text{.0138}\text{kg}{\text{H}}_{\text{2}}\text{O}/\text{kg}\text{dry}\text{air}$.

Explanation of Solution

Express the specific humidity.

$\begin{array}{c}{\omega }_1=\frac{c_p\left(T_2-T_1\right)+{\omega }_2{h}_{fg2}}{h_{g1}-h_{f2}}\\ =\frac{c_p\left(T_2-T_1\right)+{\omega }_2{h}_{fg\text{@21°C}}}{h_{g\text{@26°C}}-h_{f\text{@21°C}}}\end{array}$ (I)

Here, specific heat at constant pressure of air is $c_p$, initial and final temperature is $T_1\text{and}{T}_2$ respectively, specific humidity at exit is ${\omega }_2$, initial specific enthalpy at saturated vapor is $h_{g1}$, final specific enthalpy at saturated liquid is $h_{f2}$ and final specific enthalpy at evaporation is $h_{fg2}$.

Express specific humidity at exit.

$\begin{array}{c}{\omega }_2=\frac{0.622P_{g2}}{P_2-P_{g2}}\\ =\frac{0.622P_{\text{sat@21°C}}}{P_2-P_{\text{sat@21°C}}}\end{array}$ (II)

Here, final pressure is $P_2$, final vapor pressure is $P_{g2}$ and saturation pressure at temperature of $\text{21°C}$ is $P_{\text{sat@21°C}}$.

Conclusion:

Refer Table A-4, “saturated water-temperature table”, and write the saturation pressure at temperature of $\text{21°C}$ using an interpolation method.

Write the formula of interpolation method of two variables.

$y_2=\frac{\left(x_2-x_1\right)\left(y_3-y_1\right)}{\left(x_3-x_1\right)}+y_1$ (III)

Here, the variables denote by x and y is temperature and saturation pressure respectively.

Show thesaturation pressure corresponding to temperature as in Table (1).

Temperature $T\left(\text{°C}\right)$	Saturation pressure $P_{\text{sat}}\left(\text{kPa}\right)$
20 $\left(x_1\right)$	2.3392 $\left(y_1\right)$
21 $\left(x_2\right)$	$\left(y_2=?\right)$
25 $\left(x_3\right)$	3.1698 $\left(y_3\right)$

Substitute $\text{20°C,21°C,}\text{and}\text{25°C}$ for $x_1,x_2\text{and}{x}_3$ respectively, and $\text{2}\text{.3392}\text{kPa}$ for $y_1$ and $\text{3}\text{.1698}\text{kPa}$ for $y_3$ in Equation (III).

$\begin{array}{c}{y}_2=\frac{\left(\text{21°C}-\text{20°C}\right)\left(\text{3}\text{.1698}\text{kPa}-\text{2}\text{.3392}\text{kPa}\right)}{\left(\text{25°C}-\text{20°C}\right)}+\text{2}\text{.3392}\text{kPa}\\ =\text{2}\text{.488}\text{kPa}\\ =P_{\text{sat@21°C}}\end{array}$

Thus, the saturation pressure at temperature of $\text{21°C}$ is,

$P_{\text{sat@21°C}}=\text{2}\text{.488}\text{kPa}$

Substitute $\text{2}\text{.488}\text{kPa}$ for $P_{\text{sat@21°C}}$ and $\text{100}\text{kPa}$ for $P_2$ in Equation (II).

$\begin{array}{c}{\omega }_2=\frac{0.622\left(\text{2}\text{.488}\text{kPa}\right)}{\text{100}\text{kPa}-\text{2}\text{.488}\text{kPa}}\\ =0.01587\text{kg}{\text{H}}_{\text{2}}\text{O}/\text{kg}\text{dry}\text{air}\end{array}$

Refer Table A-4, “saturated water-temperature table”, and write final specific enthalpy evaporation at temperature of $\text{21°C}$ using an interpolation method.

Show thefinal specific enthalpy evaporationcorresponding to temperature as in Table(2).

Temperature $T\left(\text{°C}\right)$	Final specific enthalpy evaporation $h_{fg2}\left(\text{kJ}/\text{kg}\right)$
20 $\left(x_1\right)$	2453.5 $\left(y_1\right)$
21 $\left(x_2\right)$	$\left(y_2=?\right)$
25 $\left(x_3\right)$	2441.7 $\left(y_3\right)$

Use excels and tabulates the values from Table (2) in Equation (III) to get,

$h_{fg2}=2451.2\text{kJ}/\text{kg}$

Refer Table A-4, “saturated water-temperature table”, and write initial specific enthalpy saturated vapor at temperature of $\text{26°C}$ using an interpolation method.

Show theinitial specific enthalpy saturated vapor corresponding to temperature as in Table (3).

Temperature $T\left(\text{°C}\right)$	Initial specific enthalpy saturated vapor $h_{g1}\left(\text{kJ}/\text{kg}\right)$
25$\left(x_1\right)$	2546.5$\left(y_1\right)$
26$\left(x_2\right)$	$\left(y_2=?\right)$
30$\left(x_3\right)$	2555.6$\left(y_3\right)$

Use excels and tabulates the values from Table (3) in Equation (III) to get,

$h_{g1}=2548.3\text{kJ}/\text{kg}$

Refer Table A-4, “saturated water-temperature table”, and write final specific enthalpy saturated liquid at temperature of $\text{21°C}$ using an interpolation method.

Show thefinal specific enthalpy evaporationcorresponding to temperature as in Table(4).

Temperature $T\left(\text{°C}\right)$	Final specific enthalpy saturated liquid $h_{f2}\left(\text{kJ}/\text{kg}\right)$
20 $\left(x_1\right)$	83.915 $\left(y_1\right)$
21 $\left(x_2\right)$	$\left(y_2=?\right)$
25 $\left(x_3\right)$	104.83 $\left(y_3\right)$

Use excels and tabulates the values from Table (4) in Equation (III) to get,

$h_{f2}=88.10\text{kJ}/\text{kg}$

Refer Table A-2 (a), “ideal gas specific heats of various common gases”, and write specific heat at constant pressure of dry air.

$c_p=1.005\text{kJ}/\text{kg°C}$

Substitute $1.005\text{kJ}/\text{kg°C}$ for $c_p$, $\text{21°C}$ for $T_2$ and $\text{26°C}$ for $T_1$, $0.01587$ for ${\omega }_2$, $2451.2\text{kJ}/\text{kg}$ for $h_{fg2}$, $2548.3\text{kJ}/\text{kg}$ for $h_{g1}$ and $88.10\text{kJ}/\text{kg}$ for $h_{f2}$ in Equation (I).

$\begin{array}{c}{\omega }_1=\frac{\left(1.005\text{kJ}/\text{kg°C}\right)\left(21-26\right)\text{°C}+\left(0.01587\right)\left(2451.2\text{kJ}/\text{kg}\right)}{\left(2548.3-88.10\right)\text{kJ}/\text{kg}}\\ =\text{0}\text{.0138}\text{kg}{\text{H}}_{\text{2}}\text{O}/\text{kg}\text{dry}\text{air}\end{array}$

Hence, the specific humidity is $\text{0}\text{.0138}\text{kg}{\text{H}}_{\text{2}}\text{O}/\text{kg}\text{dry}\text{air}$.

To determine

The relative humidity.

Answer to Problem 29P

The relative humidity is $\text{64}\text{.4\%}$.

Explanation of Solution

Express saturation pressure of water at temperature of $\text{30°C}$.

$\begin{array}{c}{\varphi }_1=\frac{{\omega }_1{P}_1}{\left(0.622+{\omega }_1\right)P_{g1}}\\ =\frac{{\omega }_1{P}_1}{\left(0.622+{\omega }_1\right)P_{\text{sat@26°C}}}\end{array}$ (IV)

Here, initial pressure is $P_1$ and saturation pressure at temperature of $\text{26°C}$ is $P_{\text{sat@26°C}}$.

Conclusion:

Refer Table A-4, “saturated water-temperature table”, and write the saturation pressure at temperature of $\text{26°C}$ using an interpolation method.

Show thesaturation pressure corresponding to temperature as in Table (5).

Temperature $T\left(\text{°C}\right)$	Saturation pressure $P_{\text{sat}}\left(\text{kPa}\right)$
25 $\left(x_1\right)$	3.1698 $\left(y_1\right)$
26 $\left(x_2\right)$	$\left(y_2=?\right)$
30 $\left(x_3\right)$	4.2469 $\left(y_3\right)$

Use excels and tabulates the values from Table (5) in Equation (III) to get,

$P_{\text{sat@26°C}}=3.3638\text{kPa}$

Substitute $0.01377$ for ${\omega }_1$, $\text{100}\text{kPa}$ for $P_1$ and $3.3638\text{kPa}$ for $P_{\text{sat@26°C}}$ in Equation (IV).

$\begin{array}{c}{\varphi }_1=\frac{\left(0.01377\right)\left(\text{100}\text{kPa}\right)}{\left(0.622+0.01377\right)\left(3.3638\text{kPa}\right)}\\ =0.644\\ =64.4\%\end{array}$

Hence, the relative humidity is $\text{64}\text{.4\%}$.

To determine

The dew point temperature.

Answer to Problem 29P

The dew point temperature is $\text{18}\text{.8°C}$.

Explanation of Solution

Express initial partial pressure of water vapor.

$\begin{array}{c}{P}_{\nu 1}={\varphi }_1{P}_{g1}\\ ={\varphi }_1{P}_{\text{sat@26°C}}\end{array}$ (V)

Express the dew point temperature

$T_{\text{dp}}=T_{\text{sat@}{P}_{\nu }}$ (VI)

Conclusion:

Substitute $0.644$ for ${\varphi }_1$ and $\text{3}\text{.3638}\text{kPa}$ for $P_{\text{sat@26°C}}$ in Equation (V).

$\begin{array}{c}{P}_{\nu 1}=\left(0.644\right)\left(\text{3}\text{.3638}\text{kPa}\right)\\ =2.166\text{kPa}\end{array}$

Substitute $2.166\text{kPa}$ for $P_{\nu 1}$ in Equation (VI).

$T_{\text{dp}}=T_{\text{sat@}2.166\text{kPa}}$ (VII)

Here, saturation pressure at pressure of $2.166\text{kPa}$ is $T_{\text{sat@}2.166\text{kPa}}$.

Refer Table A-4, “saturated water-temperature table”, and write temperature at saturation pressure of $2.166\text{kPa}$ using an interpolation method.

Show thetemperature corresponding to saturation pressure as in Table (6).

Saturation pressure $P_{\text{sat}}\left(\text{kPa}\right)$	Temperature $T\left(\text{°C}\right)$
1.7057 $\left(x_1\right)$	15 $\left(y_1\right)$
2.166 $\left(x_2\right)$	$\left(y_2=?\right)$
2.3392 $\left(x_3\right)$	20 $\left(y_3\right)$

Use excels and tabulates the values from Table (6) in Equation (III) to get,

$T_{\text{sat@}2.166\text{kPa}}=18.8\text{°C}$

Substitute $18.8\text{°C}$ for $T_{\text{sat@}2.166\text{kPa}}$ in Equation (VII).

$T_{\text{dp}}=18.8\text{°C}$

Hence, the dew point temperature is $\text{18}\text{.8°C}$.