General Mixtures
Our goal here is to determine some useful properties for gas mixtures. Achieving this goal will vastly broaden the total number of solvable systems with defined working fluids.
Important Concepts
A few concepts must be defined before the properties of a mixture can be calculated. First, the mass of a mixture and the total mass fraction, m_{tot} and c_i.
\displaystyle m_{tot}=\sum_{i=1}^k m_i
\displaystyle \boxed{c_i=\dfrac{m_i}{m_{tot}} \text{ and } \sum_{i=1}^kc_i=1}
Secondly, we have the total number of moles in a mixture and the mole fraction, n_{tot} and y_i.
\displaystyle n_{tot}=\sum_{i=1}^k n_i
\displaystyle \boxed{y_i=\dfrac{n_i}{n_{tot}}\text{ and }\sum_{i=1}^ky_i=1}
Finally, a relation that should be familiar, the relation of molar mass, mass, and moles.
n=\dfrac{m}{M}
Gas Constant
The gas constant of a substance is related to the universal gas constant and the molar mass of the substance.
R=\dfrac{\overline{R}}{M}
To calculate the gas constant of a mixture the following relation can be used:
\displaystyle\boxed{ R_{tot}=\sum_{i=1}^k c_iR_i}
\displaystyle \sum_{i=1}^k n_i=n_{tot}
\displaystyle \sum_{i=1}^k \dfrac{m_i}{M_i}=\dfrac{m_{tot}}{M_{tot}}
\displaystyle \sum_{i=1}^k \dfrac{m_iR_i}{\overline{R}}=\dfrac{m_{tot}R_{tot}}{\overline{R}}
\displaystyle \dfrac{1}{m_{tot}}\sum_{i=1}^k m_iR_i=R_{tot}
\displaystyle R_{tot}=\sum_{i=1}^kc_iR_i
Molecular Mass
\boxed{\displaystyle M_{tot}=\dfrac{m_{tot}}{n_{tot}}=\sum_{i=1}^ky_iM_i\text{ or }M_{tot}=\left[\sum_{i=1}^k\dfrac{c_i}{M_i}\right]^{-1}}
The mass and mole fractions are therefore related by the following:
\displaystyle c_j=y_j\dfrac{M_j}{M_{tot}}=\dfrac{y_jM_j}{\displaystyle\sum_{i=1}^ky_iM_i}
\displaystyle y_j=\dfrac{\dfrac{c_j}{M_j}}{\displaystyle\sum_{i=1}^k\dfrac{c_i}{M_i}}
Partial Pressures
According to Dalton’s Law, the pressure of a mixture is equal to the sum of partial pressures of the mixture components. The partial pressure is the pressure exerted by a component at the same temperature and volume as the mixture.
\displaystyle P=\sum P_i
Partial pressures are directly related to the mole fraction of the component.
\boxed{P_i=y_iP}
Internal Energy, Enthalpy, and Specific Heats
Each of the following properties have very similar formulas, and can all be thought of as weighted averages of the respective mixture components.
\displaystyle \boxed{u=\sum_{i=1}^kc_iu_i}
\displaystyle\boxed{ C_{v0,tot}=\sum_{i=1}^kc_iC_{v0,i}}
\displaystyle \boxed{h=\sum_{i=1}^kc_ih_i}
\displaystyle \boxed{C_{p0,tot}=\sum_{i=1}^kc_iC_{p0,i}}
Entropy
The formulas for entropy are similar to those previously mentioned.
\displaystyle \boxed{s_2-s_1=\ln\left(\dfrac{T_2}{T_1}\right)\sum_{i=1}^k c_iC_{p0,i}-\ln\left(\dfrac{P_2}{P_1}\right)\sum_{i=1}^k c_iR_i}
\displaystyle s_1=\sum_{i=1}^kc_is_{i,1}
\displaystyle s_2-s_1=\sum_{i=1}^kc_i(s_{i,2}-s_{i,1})
\displaystyle s_2-s_1=\sum_{i=1}^k c_i\left[C_{p0,i}\ln\left(\dfrac{T_2}{T_1}\right)-R_i\ln\left(\dfrac{P_2}{P_1}\right)\right]
\displaystyle s_2-s_1=\ln\left(\dfrac{T_2}{T_1}\right)\sum_{i=1}^k c_iC_{p0,i}-\ln\left(\dfrac{P_2}{P_1}\right)\sum_{i=1}^k c_iR_i
Air Mixtures
Because of the importance of air-water mixtures in the field of heating, ventilating, and air-conditioning (HVAC) these mixtures have been given special attention. The ideas discussed here are fundamental to any further HVAC study, as such some new terminology is required. Finally, a psychrometric chart is a useful visual tool that can be used to fully define the properties of air that will be discussed.
Dry and Wet Bulb Temperatures
- Dry Bulb Temperature (T_{db} or T): The commonly used measurement for air temperature, measured using a dry thermometer.
- Wet Bulb Temperature (T_{wb}): The temperature of the air measured by a thermometer covered with a wet cloth. It is the lowest possible temperature of the air that can be achieved solely by evaporating water. T_{wb} will always be lower than or equal to T_{db}.
Humidity
The pressure of air can be calculated using the partial pressures of its components. Of interest are the partial pressures of dry air, P_a, and of the water vapour in the air, P_v.
P=P_a+P_v
Specific Humidity/Absolute Humidity/Humidity Ratio (\omega)
Specific humidity describes the mass ratio of water vapour to the mass of air.
\omega=\dfrac{m_v}{m_a}
\boxed{\omega=0.622\dfrac{P_v}{P_a}}
\omega=\dfrac{0.622P_v}{P-P_v}
Relative humidity, \phi, measures the amount of moisture in air relative to the maximum amount of moisture that the air can hold. When air holds the maximum amount of moisture possible it is saturated. The saturation pressure, P_g, and saturation mass m_g are measurements of the water vapour in the air. The saturation pressure is defined as the saturation pressure for water at the dry bulb air temperature, P_g=\left. P_{sat}\right|_T.
\boxed{\phi=\dfrac{m_v}{m_g}=\dfrac{P_v}{P_g}}
\phi=\dfrac{\omega P}{P_g(0.622+\omega)}
Dew Point Temperature
The dew point temperature, T_{dp}, is the temperature at which moisture begins to condensate out of air kept at constant pressure. Dew point temperature is defined as the saturation temperature for water kept at the vapour pressure in the air, T_{dp}=\left. T_{sat}\right|_{P_v}.
Internal Energy and Enthalpy
\boxed{u=C_{v0,a}T+\omega u_v}
u_v=\left.u_g\right|_T
\boxed{h=C_{p0,a}T+\omega h_v}
h_v=\left.h_g\right|_T
Mass and Volume Flow Rates
\boxed{\dot{m}_{tot}=\dot{m}_a(1+\omega)}
\rho_{tot}=\dfrac{m_a+m_v}{V}=\dfrac{1}{T}\left(\dfrac{P_a}{R_a}+\dfrac{P_v}{R_v}\right)
\boxed{\dot{V}_{tot}=\dfrac{\dot{m}_{tot}}{\rho_{tot}}}
Adiabatic Saturation Temperature
The adiabatic saturation temperature is the temperature of air that has been brought to saturation through an adiabatic evaporation process. Liquid water is denoted with subscript f, whereas gaseous water is denoted with subscript g.
\boxed{\omega_1=\dfrac{C_{p0,a}(T_2-T_1)+\omega_2h_{fg,2}}{h_{g,1}-h_{f,2}}}
Conservation of mass:
\dot{m}_f=\dot{m}_a(\omega_2-\omega_1)
Conservation of energy:
\dot{Q}=0
\dot{W}=0
0=\dot{m}_a(h_{a,2}-h_{a,1})+\dot{m}_a\omega_2(h_{g,2}-h_{f,2})+\dot{m}_a\omega_1(h_{f,2}-h_{g,1})
0=\dot{m}_aC_{p0,a}(T_2-T_1)+\dot{m}_a\omega_2h_{fg,2}+\dot{m}_a\omega_1(h_{f,2}-h_{g,1})
0=C_{p0,a}(T_2-T_1)+\omega_2h_{fg,2}+\omega_1(h_{f,2}-h_{g,1})
\omega_1=\dfrac{C_{p0,a}(T_2-T_1)+\omega_2h_{fg,2}}{h_{g,1}-h_{f,2}}
