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Magazine Chapter 12: Problem 42
A fixed amount of moist air initially at 2 bar and a relative humidity of \(50 \%\) is compressed isothermally until condensation of water begins. Determine the pressure of the mixture at the onset of condensation, in bar. Repeat if the initial relative humidity is \(80 \%\).
Short Answer
Expert verified
For 50% RH: \[ P_{final} = 2 \text{ bar} + 0.5 \times P_{sat} \] For 80% RH: \[ P_{final} = 2 \text{ bar} + 0.8 \times P_{sat} \]
Step by step solution
01
Understand Relative Humidity
Relative humidity represents the amount of water vapor in the air compared to the maximum amount of water vapor the air can hold at that temperature. It is given as a percentage. In this problem, the initial relative humidity of the moist air is either 50% or 80%.
02
Formula for Pressure at Saturation
When air is compressed isothermally to the point of condensation, it reaches its dew point at which the relative humidity becomes 100%. Use the formula: \[RH = \frac{P_v}{P_{sat}} \times 100\text{%} \ \text{where } RH \text{ is the relative humidity, } P_v \text{ is the partial pressure of water vapor, and } P_{sat} \text{ is the saturation pressure at the given temperature.} \]
03
Formula Rearrangement
Rearrange the formula to solve for the partial pressure of water vapor at the initial state \[ P_v = RH \times \frac{P_{sat}}{100} \]
04
Determine Initial Conditions
For initial relative humidity of 50%: \[ P_{v1} = 0.5 \times P_{sat} \ \text{For initial relative humidity of 80%:} \ P_{v2} = 0.8 \times P_{sat} \]
05
Apply Isothermal Compression
As the air is compressed isothermally, the partial pressure of the water vapor increases until it reaches saturation pressure at the given temperature, i.e., when condensation begins.\[ P_v' = P_{sat} \]
06
Calculate Final Pressure
By compressing the air, the pressure increases while the volume decreases. Use Dalton鈥檚 law of partial pressures to find the initial pressure: \[ P_{initial} = P_{dry\text{ } air} + P_v \ \text{After compression, } P_{final} = P_{dry\text{ } air} + P_{sat} \]
07
Solve for Initial Case (50% RH)
For the initial condition of 50% relative humidity: \[ P_{initial} = P_{dry\text{ } air} + 0.5 \times P_{sat} \ P_{initial} = 2 \text{ bar} \ \text{So,} P_{dry\text{ } air} = 2 \text{ bar} - 0.5 \times P_{sat} \]
08
Calculate Final Pressure for 50% RH
The final pressure when condensation begins is: \[ P_{final} = P_{dry\text{ } air} + P_{sat} \ P_{final} = (2 \text{ bar} - 0.5 \times P_{sat}) + P_{sat} \ P_{final} = 2 \text{ bar} + 0.5 \times P_{sat} \]
09
Solve for Initial Case (80% RH)
For the initial condition of 80% relative humidity: \[ P_{initial} = P_{dry\text{ } air} + 0.8 \times P_{sat} \ P_{initial} = 2 \text{ bar} \ \text{So,} P_{dry\text{ } air} = 2 \text{ bar} - 0.8 \times P_{sat} \]
10
Calculate Final Pressure for 80% RH
The final pressure when condensation begins is: \[ P_{final} = P_{dry\text{ } air} + P_{sat} \ P_{final} = (2 \text{ bar} - 0.8 \times P_{sat}) + P_{sat} \ P_{final} = 2 \text{ bar} + 0.8 \times P_{sat} \]
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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 an important concept in thermodynamics. It measures the amount of water vapor present in the air compared to the maximum amount that could be there at the same temperature. Think of it as the ratio of actual water vapor pressure to the saturation vapor pressure, expressed as a percentage. For example, if the air is at 50% RH, it contains half the water vapor it could potentially hold at that temperature. For this problem, we need to find when the relative humidity changes due to isothermal compression. Initially, the air starts at two different RH values, 50% and 80%. By knowing these values and the total pressure, we can calculate the partial pressure of the water vapor.
isothermal compression
Isothermal compression means compressing a gas at a constant temperature. During this process, some properties of the gas change, while the temperature stays the same. For the initial conditions given in the exercise, the air has a fixed amount of moisture and starts at 2 bar pressure. When we compress it isothermally, we maintain the same temperature but decrease the volume, which increases the pressure. At some point, the total pressure equals the saturation pressure of the water vapor at that temperature, and condensation begins. This precise moment is our point of focus for the solution.
partial pressure
Partial pressure refers to the pressure that a single component of a gas mixture would exert, if it alone occupied the entire volume. Dalton's Law states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of each individual gas. In our exercise, we calculate the partial pressure of water vapor using the relative humidity and the saturation pressure at a given temperature. Initially, the water vapor's partial pressure is a fraction of the saturation pressure based on the given RH (either 50% or 80%) and the total initial pressure (2 bar). Calculating this helps us determine the mixed air's total pressure when condensation starts.
