Mixtures

Need more than 2 Intensive variables.

Mixing Rules

Specific Volume:

v = \frac{V}{m} = (1 - Y) v_{l} + Y v_{v}

Internal Energy:

u = \frac{U}{m} = (1 - Y) u_{l} + Y u_{v}

Enthalpy:

h = \frac{H}{m} = (1 - Y) h_{l} + Y h_{v}

Entropy:

s = \frac{S}{m} = (1 - Y) s_{l} + Y s_{v}

Fractions

Mixture of $N$ components:

Independent of EOS

Mass of component $i$ : $m_{i}$
Amount of component $i$ : $n_{i}$

Totals:

Total mass: $m = \sum_{i} m_{i}$
Total amount: $n = \sum_{i} n_{i}$

Fractions:

Mass fraction: $Y_{i} = \frac{m_{i}}{m}$ $\sum_{i} Y_{i} = 1$
Mole fraction: $X_{i} = \frac{n_{i}}{n}$ $\sum_{i} X_{i} = 1$

Molecular Weights:

Molecular Weight:

Component $i$ : $W_{i} = \frac{m_{i}}{n_{i}}$
Mixture: $W = \frac{m}{n}$

Relations:

Molecular weight: $W = \sum_{i} X_{i} W_{i} = {[\sum_{i} \frac{Y_{i}}{W_{i}}]}^{- 1}$
Mass fractions: $Y_{i} = \frac{w_{i}}{w} X_{i}$
Mole fractions: $X_{i} = \frac{w}{w_{i}} Y_{i}$

Energies:

Extensive variables:

Simply add them up. $U = \sum_{i} U_{i}$ $H = \sum_{i} H_{i}$

Intensive variables:

Use mixing rules: $u = \sum_{i} Y_{i} u_{i}$ $h = \sum_{i} Y_{i} h_{i}$

Heat Capacities:

Updated definition: $C_{v} = \frac{\partial u}{\partial T} |_{v, m i x}$ $C_{p} = \frac{\partial h}{\partial T} |_{v, m i x}$
Mixing rules: $C_{v} = \sum_{i} Y_{i} C_{v, i}$ $C_{p} = \sum_{i} Y_{i} C_{p, i}$

Molar quantities:

Same math $\bar{u} = \sum_{i} X_{i} {\bar{u}}_{i}$ $\bar{h} = \sum_{i} X_{i} {\bar{h}}_{i}$ where $\bar{u}$ , $\bar{h}$ are per mole.

Dalton Model:

Mixture of Ideal Gases

Partial Pressures:

Pressure of component $i$ occupying volume $V$ at temperature $T$
For ideal gases, $P_{i} V = n_{i} R T$

Dalton's Law:

$P = \sum_{i} P_{i}$

In layman's terms: Mixture of ideal gases is an Ideal Gas

$P V = \sum_{i} P_{i} V = \sum_{i} n_{i} R T = n R T$

Key relations:

$P_{i} = X_{i} P$ $X_{i} = \frac{P_{i}}{P}$

Entropy of Mixing:

(Recall) Key Relations:

Gibbs Equation: $d u = T d s - P d v$
Entropy change: $d s = C_{p} \frac{d T}{T} - R \frac{d P}{P}$
Integrated form: $s_{i} = s_{i, r e f}^{0} + C_{p} \ln (\frac{T}{T_{r} e f}) - R \ln (\frac{P_{i}}{P_{0}})$

Mixing of Ideal Gases:

Initial state: each gas separated at $T$ and $P$
Final state: all mixed at $T$ and $P$
Per component: $Δ s_{i} = - R \ln X_{i}$
For mixture: $Δ S = \sum_{i} m_{i} Δ s_{i} = - n R \sum_{i} X_{i} \ln X_{i}$

Gas + Vapor Mixture:

Defintions:

Gas: substance that does not do any phase changes (exit in single phase)
- Ex: air
Vapor: gaseous form of a substance that also exists in liquid form
- Ex: water

(Common) Assumptions:

Gaseous phase = mixture of ideal gases
No dissolved gases in solid/liquid phase
Equilibrium between condensed/vapor phase independent of other gases

Saturated mixture:

If vapor partial pressure is saturation pressure $P_{v} = P_{s a t} (T)$
Cannot have more vapor!

Dew Point:

Temperature when air is saturated with water vapor

Relative humidity (RH):

Ratio of mole fraction of vapor over mole fraction when it is saturated $0 \leq ϕ \leq 100 %$

Humidity ratio:

Also known as specific humidity or absolute humidity, it is the mass of water per kilogram of dry air.

w = \frac{m_{v}}{m_{a} i r} = \frac{W_{v} * n_{v}}{W_{a i r} * n_{a i r}} = \frac{W_{v} * X_{v}}{W_{a i r} * X_{a i r}} = \frac{W_{v}}{W_{a i r}} \cdot \frac{P v * \frac{V}{R T}}{\frac{P V}{R T} - \frac{P v V}{R T}} = \frac{W_{v}}{W_{a i r}} \cdot \frac{P v}{P - P v} = \frac{W_{v}}{W_{a i r}} \cdot \frac{ϕ P_{s a t}}{P - ϕ P_{s a t}}

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Example: Dehumidifier

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Assumptions:

Steady State
Ideal Gases: air, water vapor

Incoming Mixture:

Partial Pressure: $P_{v, 1} = ϕ_{1} P_{s a t} (T_{1}) \approx 3.40 k P a$
Humidity ratio: $ω_{1} = \frac{w_{v}}{w_{a i r}} \frac{P_{v, 1}}{P_{a i r, 1}} \approx 0.0208$

Outgoing Mixture:

Partial Pressure: $P_{v, 2} = ϕ_{2} P_{s a t} (T_{2}) \approx 1.62 k P a$
Humidity ratio: $ω_{1} = \frac{w_{v}}{w_{a i r}} \frac{P_{v, 1}}{P_{a i r, 1}} \approx 0.0102$