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Chapter 12: Problem 76

Moist air enters a device operating at steady state at \(1 \mathrm{~atm}\) with a dry-bulb temperature of \(55^{\circ} \mathrm{C}\) and a wet-bulb temperature of \(25^{\circ} \mathrm{C}\). Liquid water at \(20^{\circ} \mathrm{C}\) is sprayed into the air stream, bringing it to \(40^{\circ} \mathrm{C}, 1\) atm at the exit. Determine (a) the relative humidities at the inlet and exit. (b) the rate that liquid water is sprayed into the air stream, in \(\mathrm{kg}\) per \(\mathrm{kg}\) of dry air.

Short Answer

Expert verified
Inlet RH: Find from Step 3. Exit RH: Find from Step 3. Water spray rate: Find from Step 5.

Step by step solution

01

Determine inlet and exit saturation pressures

Using steam tables or a psychrometric chart, find the saturation pressures at the given temperatures. For inlet, at dry-bulb temperature of \(55^{\text{掳C}}\), and wet-bulb temperature of \(25^{\text{掳C}}\). For exit, at temperature of \(40^{\text{掳C}}\).
02

Calculate inlet partial pressure of water vapor

With the wet-bulb temperature and corresponding saturation pressure, use the psychrometric equation to find the partial pressure of water vapor at the inlet condition.
03

Calculate inlet and exit relative humidities

Use the formula for relative humidity: \( \text{RH} = \frac{p_{\text{v}}}{p_{\text{sat}}} \times 100\). Calculate the relative humidity at both inlet and exit using the partial pressures and saturation pressures found.
04

Determine the specific humidity at the inlet and exit

Use the specific humidity formula \( \text{SH} = \frac{0.622p_{\text{v}}}{p_{\text{atm}} - p_{\text{v}}} \) to find the specific humidity at the inlet and exit conditions.
05

Calculate the rate of liquid water sprayed

Find the difference in specific humidity between the inlet and exit conditions. The rate of liquid water sprayed in \(\text{kg/kg dry air}\) is equal to this difference.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relative Humidity
Relative Humidity (RH) is a measure of how much water vapor is present in the air compared to the maximum amount that the air can hold at that temperature. It is expressed as a percentage. To find the Relative Humidity, we use the formula \( \text{RH} = \frac{p_{\text{v}}}{p_{\text{sat}}} \times 100 \).
Here:
  • \( p_{\text{v}} \) is the partial pressure of the water vapor.
  • \( p_{\text{sat}} \) is the saturation pressure at that temperature.
For example, in the exercise, we first find the saturation pressures for both the inlet and exit conditions using steam tables or a psychrometric chart. Then, we use the RH formula to calculate the relative humidity at both the inlet and exit, ensuring an accurate analysis of moisture levels present in the air-water mixture.
Specific Humidity
Specific Humidity (SH) quantifies the mass of water vapor in a given mass of dry air. It's an important measure because it directly relates to the actual water vapor content without being affected by temperature. The formula for Specific Humidity is: \( \text{SH} = \frac{0.622p_{\text{v}}}{p_{\text{atm}} - p_{\text{v}}} \).
Here:
  • \( 0.622 \) is the ratio of the gas constants of water vapor to dry air.
  • \( p_{\text{v}} \) is the partial pressure of the water vapor.
  • \( p_{\text{atm}} \) is the atmospheric pressure, typically 1 atm.
In the exercise, you calculate the Specific Humidity at both the inlet and exit conditions using the above formula. This helps in determining the amount of liquid water spray added to the air stream.
Psychrometric Equation
The Psychrometric Equation assists in calculating various properties of moist air, crucial for understanding air-water mixtures in thermodynamics. One common form used is: \( p_{\text{v}} = p_{\text{sat}}(\text{T}_{\text{wb}}) - p_{\text{atm}}[\text{T}_{\text{db}} - \text{T}_{\text{wb}}] \text{A}_\text{wb} \), where:
  • \( \text{T}_{\text{db}} \) is the dry-bulb temperature.
  • \( \text{T}_{\text{wb}} \) is the wet-bulb temperature.
  • \( \text{A}_{\text{wb}} \) is a psychrometric constant, which depends on the properties of air and water vapor.
For the problem, you used this equation to determine the partial pressure of water vapor at the inlet, using the given temperatures and saturation pressures.
Saturation Pressure
Saturation Pressure (\(p_{\text{sat}}\)) is the pressure at which water vapor is in equilibrium with its liquid form at a given temperature. It's tightly linked to the temperature of the air-water mixture. In calculations:
  • Saturation Pressure increases with temperature.
  • It can be found using steam tables or specialized charts.
In the exercise, using steam tables or a psychrometric chart helps you find the saturation pressures at both the dry-bulb and wet-bulb temperatures at the inlet, and at the temperature at the exit. This forms the basis for subsequent calculations, such as relative and specific humidity, and understanding moisture content in the system.

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Most popular questions from this chapter

Figure P12.82 shows a compressor followed by an aftercooler. Atmospheric air at \(14.7 \mathrm{lbf} / \mathrm{in}^{2}, 900^{\circ} \mathrm{F}\), and \(75 \%\) relative humidity enters the compressor with a volumetric flow rate of \(100 \mathrm{ft}^{3} / \mathrm{min}\). The compressor power input is \(15 \mathrm{hp}\). The moist air exiting the compressor at \(100 \mathrm{lbf} / \mathrm{in}^{2}, 400^{\circ} \mathrm{F}\) flows through the aftercooler, where it is cooled at constant pressure, exiting saturated at \(100^{\circ} \mathrm{F}\). Condensate also exits the aftercooler at \(100^{\circ} \mathrm{F}\). For steady-state operation and negligible kinetic and potential energy effects, determine (a) the rate of heat transfer from the compressor to its surroundings, in Btu/min. (b) the mass flow rate of the condensate, in \(\mathrm{lb} / \mathrm{min}\). (c) the rate of heat transfer from the moist air to the refrigerant circulating in the cooling coil, in tons of refrigeration.

Dry air enters a device operating at steady state at \(27^{\circ} \mathrm{C}\), 2 bar with a volumetric flow rate of \(300 \mathrm{~m}^{3} / \mathrm{min}\). Liquid water is injected and a moist air stream exits at \(15^{\circ} \mathrm{C}, 2\) bar, and \(91 \%\) relative humidity. Determine (a) the mass flow rate at the exit, in \(\mathrm{kg} / \mathrm{min}\). (b) the temperature, in \({ }^{\circ} \mathrm{C}\), of the liquid water injected into the air stream. Ignore heat transfer between the device and its surroundings and neglect kinetic and potential energy effects.

A stream of air (stream 1 ) at \(60^{\circ} \mathrm{F}, 1 \mathrm{~atm}, 30 \%\) relative humidity is mixed adiabatically with a stream of air (stream 2) at \(90^{\circ} \mathrm{F}, 1 \mathrm{~atm}, 80 \%\) relative humidity. A single stream (stream 3 ) exits the mixing chamber at temperature \(T_{3}\) and \(1 \mathrm{~atm}\). Assume steady state and ignore kinetic and potential energy effects Letting \(r\) denote the ratio of dry air mass flow rates \(\dot{m}_{\mathrm{a} 1} / \dot{m}_{\mathrm{a} 2}\) (a) determine \(T_{3}\), in \({ }^{\circ} \mathrm{F}\), for \(r=2\). (b) plot \(T_{3}\), in \({ }^{\circ} \mathrm{F}\), versus \(r\) ranging from 0 to 10 .

Moist air at \(95^{\circ} \mathrm{F}, 1\) atm, and a relative humidity of \(30 \%\) enters a steam-spray humidification device operating at steady state with a volumetric flow rate of \(5700 \mathrm{ft}^{3} / \mathrm{min}\). Saturated water vapor at \(230^{\circ} \mathrm{F}\) is sprayed into the moist air, which then exits the device at a relative humidity of \(50 \%\). Heat transfer between the device and its surroundings can be ignored, as can kinetic and potential energy effects. Determine (a) the temperature of the exiting moist air stream, in \({ }^{\circ} \mathrm{F}\). (b) the rate at which steam is injected, in lb/min.

An insulated tank having a total volume of \(0.6 \mathrm{~m}^{3}\) is divided into two compartments. Initially one compartment contains \(0.4 \mathrm{~m}^{3}\) of hydrogen \(\left(\mathrm{H}_{2}\right)\) at \(127^{\circ} \mathrm{C}, 2\) bar and the other contains nitrogen \(\left(\mathrm{N}_{2}\right)\) at \(27^{\circ} \mathrm{C}, 4\) bar. The gases are allowed to mix until an equilibrium state is attained. Assuming the ideal gas model with constant specific heats, determine (a) the final temperature, in \({ }^{\circ} \mathrm{C}\). (b) the final pressure, in bar. (c) the amount of entropy produced, in \(\mathrm{kJ} / \mathrm{K}\).
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