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Develop a formula to show how the mass fraction of water vapor is connected to the humidity ratio.

Short Answer

Expert verified
The mass fraction of water vapor is \( x_{wv} = \frac{W}{W + 1} \), where \( W \) is the humidity ratio.

Step by step solution

01

Understand the Terminology

The mass fraction of water vapor refers to the proportion of the mass of the water vapor relative to the total mass of the moist air. The humidity ratio is defined as the mass of water vapor per mass of dry air. Our goal is to create a formula that connects these two concepts.
02

Define Humidity Ratio

The humidity ratio (W) can be expressed as:\[ W = \frac{m_{wv}}{m_{da}} \]where \(m_{wv}\) is the mass of water vapor and \(m_{da}\) is the mass of dry air.
03

Express Mass Fraction of Water Vapor

The mass fraction of water vapor (\(x_{wv}\)) is defined as:\[ x_{wv} = \frac{m_{wv}}{m_{wv} + m_{da}} \]This represents the ratio of the mass of water vapor to the total mass of the air mixture (water vapor + dry air).
04

Substitute for Mass of Water Vapor

Using the expression for the humidity ratio, we can substitute \(m_{wv} = W \, m_{da}\) into the mass fraction formula:\[ x_{wv} = \frac{W \, m_{da}}{W \, m_{da} + m_{da}} \]
05

Simplify the Expression

Factor out \(m_{da}\) from the numerator and the denominator:\[ x_{wv} = \frac{W}{W + 1} \]Here, \(x_{wv}\) is expressed solely in terms of the humidity ratio \(W\).

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Key Concepts

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

Mass Fraction of Water Vapor
The mass fraction of water vapor is a way to express how much water vapor is present in moist air compared to the entire mixture. It is a simple ratio calculated as the mass of water vapor divided by the total mass of the air mixture, which includes both the water vapor and the dry air. Understanding this concept is crucial in environments where moisture content plays an important role, such as in HVAC systems or climate studies. When calculating the mass fraction of water vapor, you use the formula:\[ x_{wv} = \frac{m_{wv}}{m_{wv} + m_{da}} \]where:
  • \(m_{wv}\) is the mass of the water vapor.
  • \(m_{da}\) is the mass of the dry air.
By determining the mass fraction, you get a better understanding of how predominant water vapor is in the air mixture, which can affect humidity-sensitive processes.
Moist Air Composition
Moist air is a mixture of dry air and water vapor, two components with distinct properties. In daily life, the composition of moist air determines weather phenomena, indoor comfort, and many industrial processes.To understand the composition, consider:
  • Dry Air: Composed primarily of nitrogen and oxygen, it is defined as air with no water vapor content.
  • Water Vapor: The gaseous phase of water, which depending on conditions, significantly impacts air's heat and energy properties.
Knowing the proportions of these components helps in analyzing air-conditioning needs, predicting weather, and monitoring environmental conditions. For computational purposes, the humidity ratio \(W\), is employed:\[ W = \frac{m_{wv}}{m_{da}} \]This represents the mass of water vapor per unit mass of dry air, essential for assessing air's total moisture content.
Thermodynamic Properties
The term "thermodynamic properties" refers to the characteristics of a system that define its ability to store and transfer energy. For moist air, these include properties like temperature, pressure, enthalpy, and specific volume. Water vapor's presence in moist air significantly alters these thermodynamic properties:
  • Temperature: Affects the air mixture's capacity to hold water vapor. Warmer air holds more water vapor.
  • Pressure: Vapor pressure contributes to the total pressure exerted by the moist air.
  • Enthalpy: Represents the energy content, influenced by the vapor, which can absorb/release heat during phase changes.
Understanding these properties is essential in applications like climate control systems, where precise management of air conditions is necessary. When designing systems or studying environmental impacts, practitioners rely on these properties to predict how air will behave under different conditions.
Psychrometrics
Psychrometrics is the study of moist air properties and how they interrelate. This field is crucial for engineering disciplines concerning HVAC and meteorology. Key psychrometric concepts include:
  • Humidity Ratio: It is critical for defining how "humid" air is, helping refine designs for air quality and comfort.
  • Relative Humidity: Expresses how close the air is to saturation. It's given as a percentage.
  • Dry Bulb and Wet Bulb Temperatures: These temperatures help compute air moisture content and comfort levels.
Psychrometric charts are often used in this field, plotting these properties to simplify the assessment of air's condition and predict its behavior in various scenarios. Understanding psychrometrics helps ensure proper environmental controls in buildings and manage processes affected by air moisture content.

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Most popular questions from this chapter

A flow of air at \(10^{\circ} \mathrm{C}, \phi=90 \%\) is brought into a house, where it is conditioned to \(25^{\circ} \mathrm{C}, 60 \%\) relative humidity. This is done with a combined heater-evaporator in which any liquid water is at \(10^{\circ} \mathrm{C}\). Find any flow of liquid and the necessary heat transfer, both per kilogram of dry air flowing. Find the dew point for the final mixture.

For flows with moist air where the water content is changed either by evaporation or by condensation, what happens to the exergy? Is the water vapor in air flowing over a lake in equilibrium with the liquid water?

Several applications of dehumidification do not rely on water condensation by cooling. A desiccant with a greater affinity to water can absorb water directly from the air accompanied by a heat release. The desiccant is then regenerated by heating, driving the water out. Make a list of several materials such as liquids, gels, and solids and show examples of their use.

A flow of moist air at \(100 \mathrm{kPa}, 35^{\circ} \mathrm{C}, 40 \%\) relative humidity is cooled by adiabatic evaporation of liquid \(20^{\circ} \mathrm{C}\) water to reach a saturated state. Find the amount of water added per \(\mathrm{kg}\) dry air and the exit temperature.

A water-filled reactor of \(50 \mathrm{ft}^{3}\) is at \(2000 \mathrm{lbf} / \mathrm{in} .^{2}\) \(550 \mathrm{~F}\) and is located inside an insulated containment room of \(5000 \mathrm{ft}^{3}\) that has air at \(1 \mathrm{~atm}\) and \(77 \mathrm{~F}\). Due to a failure, the reactor ruptures and the water fills the containment room. Find the final pressure.

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