Equations.org

Introduction to Engineering Calculation

• Base units

Dimension	SI	cgs	English
Length	m	cm	in, ft, mi
Mass	kg	g	lbm
Time	s	s	s
Temperature	K	K	F
Current	A	A
Light intensity	cd	cd

• Derived units

Volume	liter	L	1000 cm3
Force	Newton	N	1 kg m/s2
	dyne		1 g cm/s2
Energy/Work	Joule	J	1 N m = 1 kg m2/s2
	erg		1 dyne cm = 1 g cm2/s2
	calorie	cal	4.184 J
	Btu		1 Btu = 1055.05585 J
Power	Watt	W	1 J/s
	Horsepower	hp	1 hp = 745.7 W
Pressure	Pascal	Pa	1 N/m2 = 1 J/m3
	bar		105 Pa
	atmosphere	atm	1 atm = 1.01325 bar
	torr	torr	1/760 atm

• basic statistics

sample mean	Sample variance	Standard deviation
\(\bar{X}=∑1nXi \)	\(sX2=\frac{1}{N-1}∑1n(Xi-\bar{X})2\)	\(sx=\sqrt{sX2} \)

Processes and process variables

• Density

\[ρ \frac{m}{V}\frac{\dot{m}}{\dot{V}} \]

• Pressure

\[ P = P₀ + ρ g h \]

\[ P_gauge = P_abs - P_atm \]

• Temperature Scales
  • Kelvin: absolute scale, 0 → ∞
  • Celsius: \(T(^ˆC) = T(K) - 273.15)\)
  • Fahrenheit: \(T(^ˆF) = 1.8 T(^ˆC) + 32 )\)
  • Rankine: absolute scale, \(T(^ˆR) = T(^ˆF)+459.67\)
• Chemical composition

Material balances

• general balance

\begin{framed} output = input + generation - consumption - accumulation \end{framed}

• Reaction progress

\begin{equation*} n_j = n_j0 + ν_j ξ \end{equation*}

• conversion

\[X_j = \frac{n_j0-n_j}{n_j0} = -\frac{ν_jξ}{n_j0} \]

• Multiple reactions

\[ n_j = n_j0 + ∑_i ν_ij ξ_i \]

• yield \(=n_j/n_j^\text{max}\)
• selectivity (often) defined as amount of desired product over amount of undesired.

Properties of single-phase systems

• Ideal solution

\[ v \text{ (l/mol)} = ∑_i x_i v_i \]

\[ \frac{1}{\bar{ρ}} = ∑_iⁿ \frac{ω_i}{ρ_i} \]

• Ideal gases

\[ P V = n R T \text{ or } P v = R T \text{ or } v = \frac{RT}{P} \]

R

8.314472 J / (K mol)

0.082057 atm l / (K mol)

1.3806504e-23 J / K

• Ideal gas mixture

\[ V(N,T,P) = V₁(N₁,T,P) + V₂(N₂,T,P) \]

\[ \frac{P₁}{P} = \frac{N₁ RT/V}{N RT/V} = y₁\]

• van der Waals model

\[ P_\text{vdW} = \frac{RT}{v-b} - \frac{a}{v²} \]

\[b = v_c/3\quad\quad a = \frac{9}{8}R T_c v_c\]

• reduced variables

\[ T_r = T/T_c\quad P_r = P/P_c\quad v_r=v/v_c\]

• Soave-Redlich-Kwong (SRK) model

\[P_\text{SRK} = \frac{RT}{v-b} - \frac{α(T) a}{v(v+b)} \]

\begin{eqnarray*} a & = & 0.42747 \frac{(R T_c)²}{P_c}
b & = & 0.08664 \frac{R T_c}{P_c} \ m & = & 0.48508 + 1.55171 ω - 0.1561 ω²\ α & = & \[1+m (1-\sqrt{T_r})\]^2 \end{eqnarray*}

• Pitzer “acentric” factor

\[ω = -log \left ( \frac{P_sat}{P_c} \right ) \Big|_{T_{r}=0.7} -1 \]

• Virial expansion

\[ P= \frac{RT}{v} \left ( 1 + \frac{B₂(T)}{v} + \frac{B₃(T)}{v²} + \cdots \right ) \]

• compressibility

\[ Z = \frac{P(v,T) v}{RT} \]

• Law of corresponding states \[ Z_c = 0.27 \]

Two-phase systems

• Clapeyron equation

\[ \frac{d P^*}{dT} = \frac{Δ H_\text{latent}}{T(v_b-v_a)} \]

• Clausius-Clapeyron equation:

\[ ln \frac{P₂}{P₁} ≈ -\frac{Δ H_\text{vap}}{R}\left ( \frac{1}{T₂} - \frac{1}{T₁} \right ) \]

• Antoine equation

\[ log₁₀P^* = A - \frac{B}{T+C} \]

• Gibbs phase rule

\[ DOF = c - Π - r + 2\]

• Raoult’s Law

\[ x_A P^*_A(T) = P_A = y_A P \]

\[ P_\text{bubble} = ∑ x_i P_i^* \]

\[ P_\text{dew} = \left ( ∑_i\frac{y_i}{P_i^*} \right )^-1 \]

• Relative humidity

\[ RH(T) = P_\ce{H2O}/P^*_\ce{H2O}(T) \]

• Henry’s Law

\[ x_A H_A(T) = P_A = y_A P \]

• Colligative properties

\[Δ T_b ≈ \frac{R T_b²}{Δ H^*_vap}x \]

\[Δ T_m ≈ \frac{R T_m²}{Δ H^*_m}x \]

Energy balances

• Energy types \[ E_K = \frac{1}{2} m v²\quad\quad \dot{E}_K = \frac{1}{2}\dot{m} u² \]

  \[ E_V = m g h \quad\quad \dot{E}_V = \dot{m} g z \]

  \[ U = U(T,P,x_i)\quad\quad H=U+PV\]

• Closed, constant volume system

  \[ Δ U + Δ E_K + Δ E_V - q - w = 0 \]

• Open system at steady-state

\[ Δ\dot{H} + Δ\dot{E}_K + Δ{E}_P = \dot{q} + \dot{W}_s \]

• Bernoulli equation:

\[ \frac{1}{2} Δ u² + gΔ z + \frac{1}{ρ}Δ P = 0\]

Energy balances on non-reactive systems

• heat capacity

\[ C_v(T) = \left ( \frac{∂\hat{U}}{∂ T} \right )_v \]

\[ C_p(T) = \left ( \frac{∂\hat{H}}{∂ T} \right )_p \]

• For liquids and solids, \(C_p ≈ C_v\)
• For ideal gas, \(C_p = C_v + R\)

Energy balances on reactive systems

• Reaction energy

\[ Δ H^ˆ_r = ∑_j ν_j Δ \hat{H}_f,j^ˆ \]

• “Heat of reaction” method

\[ Δ \dot{H} = ξ\Delta\hat{H}^ˆ_r + ∑_out\dot{n}_out\hat{H}_out-∑_in\dot{n}_in\hat{H}_in \]

\[ Δ \dot{H} = ∑_iξ_iΔ\hat{H}^ˆ_r + ∑_out\dot{n}_out\hat{H}_out-∑_in\dot{n}_in\hat{H}_in \]

• “Heat of formation” method

\[ Δ \dot{H} = ∑_out\dot{n}_out\hat{H}_out-∑_in\dot{n}_in\hat{H}_in \]

Transient processes

• General balance around any system or element of a system

\[ \dot{F}_out(t) = \dot{F}_in(t) + r(t) - \frac{dF}{dt} \]