Chapter 5.3, Problem 111AYU

117. Relative Humidity The relative humidity is the ratio (expressed as a percent) of the amount of water vapor in the air to the maximum amount that the air can hold at a specific temperature. The relative humidity, R, is found using the following formula:

$R = {10}^{\left(\frac{4221}{T + 459.4} - \frac{4231}{D + 459.4} + 2\right)}$

where T is the air temperature (in $°F$) and D is the dew point temperature (in $°F$).

(a) Determine the relative humidity if the air temperature is $50°$ Fahrenheit and the dew point temperature is $41°$ Fahrenheit.

(b) Determine the relative humidity if the air temperature is $68°$ Fahrenheit and the dew point temperature is $59°$ Fahrenheit.

(c) What is the relative humidity if the air temperature and the dew point temperature arc the same?

To determine

To find:

a. The relative humidity if the air temperature is $50°$ Fahrenheit and the dew point temperature is $41°$ Fahrenheit.

Answer to Problem 111AYU

a. $70.96$

Explanation of Solution

Given:

$R = {10}^{\left(\frac{4221}{T + 459.4}-\frac{4221}{D + 459.4} + 2\right)}$

a. Given the relative humidity if the air temperature is $50°$ Fahrenheit and the dew point temperature is $41°$ Fahrenheit.

$R = {10}^{\left(\frac{4221}{50 + 459.4}-\frac{4221}{41 + 459.4} + 2\right)}$

$= {10}^{\left(1.851\right)}$

$= 70.96$

To determine

To find:

b. The relative humidity if the air temperature is $68°$ Fahrenheit and the dew point temperature is $59°$ Fahrenheit.

Answer to Problem 111AYU

b. $R=72.61$

Explanation of Solution

Given:

$R = {10}^{\left(\frac{4221}{T + 459.4}-\frac{4221}{D + 459.4} + 2\right)}$

b. Given the relative humidity if the air temperature is $68°$ Fahrenheit and the dew point temperature is $59°$ Fahrenheit.

$R = {10}^{\left(\frac{4221}{68 + 459.4}-\frac{4221}{59 + 459.4} + 2\right)}$

$= {10}^{\left(1.861\right)}$

$= 72.61$

To determine

To find:

c. The relative humidity if the air temperature is and the dew point temperature is are the same.

Answer to Problem 111AYU

c. $R=100$

Explanation of Solution

Given:

$R = {10}^{\left(\frac{4221}{T + 459.4}-\frac{4221}{D + 459.4} + 2\right)}$

c. The relative humidity if the air temperature is and the dew point temperature is are the same.

Let $T\text{ =}$ air temperature $\text{=}$ dew point temperature.

$R = {10}^{\left(\frac{4221}{T + 459.4}-\frac{4221}{T + 459.4} + 2\right)}$

$= {10}^{\left(2\right)}$

$= 100$