Formula Sheet

\underline{Vapor-Liquid Equilibrium}:

P = P_{A}^{0}x + P_{B}^{0}(1 − x)

y^{∗} = \frac{P_{A}^{0}x}{P}

α = \frac{P_{A}^{0}}{P_{B}^{0}} = \frac{y^{∗}(1 − x)}{x(1 − y^{∗})}

\underline{Flash Unit}: Split Fraction:

ξ_{k} = \frac{α_{k/n}ξ_{n}}{1 + (α_{k/n} − 1)ξ_{n}}

Simplified bubble-point equation: \frac{P_{k}^{0}}{P} = \frac{α_{k/n}}{ᾱ}

Simplified dew-point equation: ∑_{i=1}^{n} \frac{y_{i}}{α_{i/n}} = \frac{P_{n}^{0}}{P}

Vapor Fraction and Split Fraction:

ξ_{n} = \frac{1}{\frac{1}{K_{n}ϕ} − \frac{1}{K_{n}} + 1}

\underline{Distillation}: Fenske Equation

N = \frac{ln [\frac{(ξ_{lk}(1 − ξ_{hk}))}{(ξ_{hk}(1 − ξ_{lk}))}]}{ln(α_{lk/hk})}

Split fractions:

ξ_{k} = \frac{α_{k}^{N}ξ_{hk}}{1 + (α_{k}^{N} − 1)ξ_{hk}}

N_{i} = \frac{12.3}{(α_{lk/hk} − 1)^{2/3}(1 − β_{i})^{1/6}}  i = lk, hk  R_{i} = \frac{1.38}{(α_{lk/hk} − 1)^{0.9}(1 − β_{i})^{0.1}}  i = lk, hk

N_{T} = 0.8max(N_{i}) + 0.2min(N_{i}) R = 0.8max(R_{i}) + 0.2min(R_{i})

\underline{Absorption}:

A_{E} = \frac{L_{0}}{V_{N+1}K_{n}}

N = \frac{ln \{\frac{l_{0}^{n} + (r − A_{E})v_{N+1}^{n}}{l_{0}^{n} − A_{E}(1 − r)v_{N+1}^{n}} \}}{ln \{A_{E}\}}

\begin{aligned}β_{N}^{k} & = & \frac{[1 − (A^{k})^{N+1}]}{1 − A^{k}} \\ β_{N−1}^{k} & = & \frac{[1 − (A^{k})^{N}]}{1 − A^{k}} \end{aligned}

\begin{aligned}v_{1}^{k} & = & \frac{v_{N+1}^{k}}{β_{N}^{k}} + \frac{β_{N−1}^{k}}{β_{N}^{k}} l_{0}^{k} \\ l_{N}^{k} & = & (1 − \frac{β_{N−1}^{k}}{β_{N}^{k}} ) l_{0}^{k} + (1 − \frac{1}{β_{N}^{k}} ) v_{N+1}^{k} \end{aligned}

\underline{Energy Balance}: Enthalpy of Vapor Stream: ΔH_{v} = ΔH_{f} + ΔH_{T} ΔH_{f} = ∑ y_{k}H_{f,k}(T_{0}) ΔH_{T} = ∑ y_{k}∫_{T_{0}}^{T} c_{p,k}^{0}dT Enthalpy of Liquid Stream: ΔH_{L,k}(T) = ΔH_{f,k}^{0} + ∫_{T_{0}}^{T} c_{p,k}^{0}(T)dT − ΔH_{vap}^{k} where

ΔH_{vap}^{k}(T) = ΔH_{vap}^{k}(T_{b}) [\frac{(T_{c}^{k} − T)}{(T_{c}^{k} − T_{b})}]^{0.38}