Thermodynamics II Unit 8 Review: Gas Mixtures & Air-Conditioning Processes

Key Concepts and Definitions

• Gas mixture consists of two or more gases that are mixed together at the molecular level
• Mole fraction represents the ratio of the number of moles of a particular component to the total number of moles in the mixture
• Partial pressure is the pressure that each individual gas in a mixture would exert if it alone occupied the volume of the mixture at the same temperature
• Psychrometrics is the study of the thermodynamic properties of moist air and the use of these properties to analyze conditions and processes involving moist air
  • Involves analyzing the relationships between dry-bulb temperature, wet-bulb temperature, dew-point temperature, relative humidity, humidity ratio, and enthalpy
• Dry-bulb temperature is the temperature of air measured by a thermometer freely exposed to the air but shielded from radiation and moisture
• Wet-bulb temperature represents the temperature that a parcel of air would have if it were cooled to saturation (100% relative humidity) by the evaporation of water into it
• Dew-point temperature is the temperature to which air must be cooled to become saturated with water vapor, assuming constant pressure and water vapor content
• Relative humidity $(\phi)$ is the ratio of the partial pressure of water vapor in the air to the saturation pressure of water vapor at the same temperature

Composition of Gas Mixtures

• Gas mixtures can be characterized by the mole fractions, mass fractions, or volume fractions of their constituent components
• Mole fraction $(\chi_i)$ of a component $i$ in a gas mixture is defined as $\chi_i = \frac{n_i}{n}$, where $n_i$ is the number of moles of component $i$ and $n$ is the total number of moles in the mixture
• Mass fraction $(y_i)$ of a component $i$ in a gas mixture is defined as $y_i = \frac{m_i}{m}$, where $m_i$ is the mass of component $i$ and $m$ is the total mass of the mixture
• Volume fraction $(v_i)$ of a component $i$ in a gas mixture is defined as $v_i = \frac{V_i}{V}$, where $V_i$ is the volume occupied by component $i$ and $V$ is the total volume of the mixture
• Mole fractions, mass fractions, and volume fractions are related to each other through the molecular weights of the components
  • $y_i = \frac{\chi_i M_i}{\sum_{j=1}^{n} \chi_j M_j}$, where $M_i$ is the molecular weight of component $i$
  • $v_i = \frac{\chi_i}{\sum_{j=1}^{n} \chi_j}$
• Apparent molecular weight of a gas mixture $(\bar{M})$ is the weighted average of the molecular weights of its components, given by $\bar{M} = \sum_{i=1}^{n} \chi_i M_i$

Dalton's Law and Partial Pressures

• Dalton's law of partial pressures states that the total pressure of a gas mixture is equal to the sum of the partial pressures of its constituent components
  • Mathematically, $P = \sum_{i=1}^{n} P_i$, where $P$ is the total pressure and $P_i$ is the partial pressure of component $i$
• Partial pressure $(P_i)$ of a component $i$ in a gas mixture is the pressure that the component would exert if it alone occupied the volume of the mixture at the same temperature
• Partial pressure can be calculated using the mole fraction of the component and the total pressure of the mixture: $P_i = \chi_i P$
• Ideal gas equation can be applied to gas mixtures by considering the partial pressures and volumes of the individual components
  • For a gas mixture, $PV = nRT$ becomes $PV = \sum_{i=1}^{n} n_i RT$, where $n_i$ is the number of moles of component $i$
• Amagat's law states that the total volume of a gas mixture is equal to the sum of the partial volumes of its components at the same temperature and pressure
  • Mathematically, $V = \sum_{i=1}^{n} V_i$, where $V$ is the total volume and $V_i$ is the partial volume of component $i$

Psychrometric Properties of Air

• Psychrometric properties describe the thermodynamic state of moist air, which is a mixture of dry air and water vapor
• Humidity ratio $(W)$ is the ratio of the mass of water vapor to the mass of dry air in a given volume of moist air
  • Mathematically, $W = \frac{m_v}{m_a}$, where $m_v$ is the mass of water vapor and $m_a$ is the mass of dry air
• Specific humidity $(q)$ is the ratio of the mass of water vapor to the total mass of moist air
  • Mathematically, $q = \frac{m_v}{m_a + m_v} = \frac{W}{1 + W}$
• Enthalpy of moist air $(h)$ is the sum of the enthalpies of dry air and water vapor per unit mass of dry air
  • Mathematically, $h = c_{p,a}t + W(h_{fg} + c_{p,v}t)$, where $c_{p,a}$ and $c_{p,v}$ are the specific heats of dry air and water vapor, respectively, $t$ is the dry-bulb temperature, and $h_{fg}$ is the enthalpy of vaporization of water
• Specific volume of moist air $(v)$ is the volume of moist air per unit mass of dry air
  • Mathematically, $v = \frac{RT}{P}(1 + 1.6078W)$, where $R$ is the gas constant for dry air, $T$ is the absolute temperature, and $P$ is the total pressure
• Degree of saturation $(\mu)$ is the ratio of the humidity ratio to the humidity ratio of saturated air at the same temperature and pressure
  • Mathematically, $\mu = \frac{W}{W_s}$, where $W_s$ is the humidity ratio of saturated air

Psychrometric Charts and Their Use

• Psychrometric chart is a graphical representation of the thermodynamic properties of moist air at a constant pressure (usually at sea level, 101.325 kPa)
• Horizontal axis represents the dry-bulb temperature, while the vertical axis represents the humidity ratio or specific humidity
• Lines of constant relative humidity, wet-bulb temperature, specific volume, and enthalpy are plotted on the chart
• Psychrometric charts are used to analyze various air-conditioning processes, such as heating, cooling, humidification, and dehumidification
• Mixing of two air streams can be represented on the psychrometric chart by drawing a straight line connecting the two points representing the initial states of the air streams
  • The final state of the mixture lies on this line, and its position depends on the mass flow rates of the two streams
• Sensible heating or cooling processes, which involve a change in temperature without a change in humidity ratio, are represented by horizontal lines on the psychrometric chart
• Humidification or dehumidification processes, which involve a change in humidity ratio without a change in enthalpy, are represented by vertical lines on the psychrometric chart

Humidity and Moisture Content Calculations

• Humidity calculations involve determining the amount of water vapor present in the air and its relationship with other psychrometric properties
• Absolute humidity $(\rho_v)$ is the mass of water vapor per unit volume of moist air
  • Mathematically, $\rho_v = \frac{m_v}{V} = \frac{P_v}{R_vT}$, where $P_v$ is the partial pressure of water vapor, $R_v$ is the gas constant for water vapor, and $T$ is the absolute temperature
• Relative humidity can be calculated using the partial pressure of water vapor and the saturation pressure of water vapor at the same temperature
  • Mathematically, $\phi = \frac{P_v}{P_{v,s}} \times 100\%$, where $P_{v,s}$ is the saturation pressure of water vapor
• Dew-point temperature can be calculated using the partial pressure of water vapor and the saturation pressure-temperature relationship for water vapor
  • Mathematically, $t_d = \frac{C_1 \ln(\frac{P_v}{C_2})}{\ln(\frac{P_v}{C_2}) - C_3}$, where $C_1$, $C_2$, and $C_3$ are constants specific to the temperature range
• Moisture content $(X)$ is the ratio of the mass of water vapor to the mass of dry air in a given volume of moist air (same as humidity ratio)
  • Mathematically, $X = \frac{m_v}{m_a} = 0.622 \frac{P_v}{P - P_v}$, where $P$ is the total pressure
• Specific enthalpy of moist air can be calculated using the humidity ratio and the enthalpies of dry air and water vapor
  • Mathematically, $h = c_{p,a}t + W(h_{fg} + c_{p,v}t)$, where $c_{p,a}$ and $c_{p,v}$ are the specific heats of dry air and water vapor, respectively, $t$ is the dry-bulb temperature, and $h_{fg}$ is the enthalpy of vaporization of water

Basic Air-Conditioning Processes

• Air-conditioning processes involve changing the thermodynamic properties of moist air to achieve desired indoor conditions
• Sensible heating or cooling is the process of adding or removing heat from the air without changing its humidity ratio
  • This process is represented by a horizontal line on the psychrometric chart
  • The sensible heat transfer rate is given by $\dot{Q}_s = \dot{m}_a c_{p,a} (t_2 - t_1)$, where $\dot{m}_a$ is the mass flow rate of dry air, $c_{p,a}$ is the specific heat of dry air, and $t_1$ and $t_2$ are the initial and final dry-bulb temperatures
• Humidification is the process of adding moisture to the air without changing its dry-bulb temperature
  • This process is represented by a vertical line on the psychrometric chart
  • The moisture addition rate is given by $\dot{m}_v = \dot{m}_a (W_2 - W_1)$, where $W_1$ and $W_2$ are the initial and final humidity ratios
• Dehumidification is the process of removing moisture from the air without changing its enthalpy
  • This process is represented by a line of constant enthalpy on the psychrometric chart
  • The moisture removal rate is given by $\dot{m}_v = \dot{m}_a (W_1 - W_2)$, where $W_1$ and $W_2$ are the initial and final humidity ratios
• Adiabatic mixing of two air streams involves the conservation of mass and energy, resulting in a final state that lies on a straight line connecting the initial states of the two streams on the psychrometric chart
  • The mass balance equation is $\dot{m}_{a,1} + \dot{m}_{a,2} = \dot{m}_{a,3}$, where $\dot{m}_{a,1}$, $\dot{m}_{a,2}$, and $\dot{m}_{a,3}$ are the mass flow rates of the two inlet streams and the outlet stream, respectively
  • The energy balance equation is $\dot{m}_{a,1}h_1 + \dot{m}_{a,2}h_2 = \dot{m}_{a,3}h_3$, where $h_1$, $h_2$, and $h_3$ are the specific enthalpies of the two inlet streams and the outlet stream, respectively

Real-World Applications and Examples

• HVAC (Heating, Ventilation, and Air Conditioning) systems in buildings
  • Maintain comfortable indoor conditions by controlling temperature, humidity, and air quality
  • Example: An office building with a central air-conditioning system that cools, dehumidifies, and filters the air before distributing it to individual rooms
• Automotive air conditioning
  • Provides comfortable conditions for passengers by cooling and dehumidifying the air inside the vehicle
  • Example: A car's air-conditioning system that uses a compressor, condenser, expansion valve, and evaporator to remove heat and moisture from the cabin air
• Industrial drying processes
  • Remove moisture from materials such as food, pharmaceuticals, and chemicals to improve their quality, stability, and shelf life
  • Example: A conveyor belt dryer that uses hot air to remove moisture from freshly harvested grains before storage
• Greenhouse climate control
  • Regulate temperature, humidity, and ventilation to create optimal growing conditions for plants
  • Example: A greenhouse that uses evaporative cooling pads and fans to maintain a cool and humid environment for tropical plants
• Indoor swimming pools
  • Control humidity levels to prevent condensation, corrosion, and mold growth
  • Example: A dehumidification system that removes excess moisture from the air in an indoor pool facility to maintain a comfortable environment for swimmers and prevent damage to the building structure
• Clean rooms in manufacturing and healthcare facilities
  • Maintain strict control over temperature, humidity, and particle concentration to ensure product quality and patient safety
  • Example: A pharmaceutical clean room that uses HEPA filters, laminar flow ventilation, and precise temperature and humidity control to prevent contamination during drug manufacturing
• Data centers and server rooms
  • Remove heat generated by electronic equipment and maintain stable temperature and humidity levels to prevent equipment failure and data loss
  • Example: A data center cooling system that uses chilled water, air handlers, and humidity control to maintain a constant environment for servers and other IT equipment