Formula Sheet: Steady-State Systems

Mass Balance for Steady-State Systems

General Mass Balance

Overall Mass Balance:

\[ \sum \dot{m}_{\text{in}} = \sum \dot{m}_{\text{out}} \]

• \(\dot{m}\) = mass flow rate (kg/s, lb_m/s)
• Applies when no accumulation occurs (steady-state)
• No generation or consumption of total mass

Volumetric Flow Rate Relationship:

\[ \dot{m} = \rho \dot{V} \]

• \(\rho\) = fluid density (kg/m³, lb_m/ft³)
• \(\dot{V}\) = volumetric flow rate (m³/s, ft³/s)

Mass Flow Rate from Velocity:

\[ \dot{m} = \rho A v \]

• A = cross-sectional area (m², ft²)
• v = velocity (m/s, ft/s)

Component Mass Balance

Component Balance (No Reaction):

\[ \sum (\dot{m} \cdot x_i)_{\text{in}} = \sum (\dot{m} \cdot x_i)_{\text{out}} \]

• \(x_i\) = mass fraction of component i (dimensionless)
• Sum of all mass fractions equals 1

Component Balance (With Reaction):

\[ \sum (\dot{m} \cdot x_i)_{\text{in}} + R_i = \sum (\dot{m} \cdot x_i)_{\text{out}} \]

• \(R_i\) = net generation rate of component i (kg/s, lb_m/s)
• \(R_i\) > 0 for net production
• \(R_i\) < 0="" for="" net="">

Molar Balance for Steady-State Systems

Overall Molar Balance (No Reaction):

\[ \sum \dot{n}_{\text{in}} = \sum \dot{n}_{\text{out}} \]

• \(\dot{n}\) = molar flow rate (kmol/s, lb-mol/s)
• Only valid when total moles are conserved (no reaction changing total moles)

Component Molar Balance:

\[ \sum (\dot{n} \cdot y_i)_{\text{in}} + G_i = \sum (\dot{n} \cdot y_i)_{\text{out}} \]

• \(y_i\) = mole fraction of component i (dimensionless)
• \(G_i\) = net molar generation rate of component i (kmol/s, lb-mol/s)

Conversion between Mass and Molar Flow Rates:

\[ \dot{n}_i = \frac{\dot{m}_i}{M_i} \]

• \(M_i\) = molecular weight of component i (kg/kmol, lb_m/lb-mol)

Average Molecular Weight:

\[ M_{\text{avg}} = \sum y_i M_i \]

• Used for gas mixtures

Energy Balance for Steady-State Systems

General Energy Balance

First Law of Thermodynamics for Open Systems:

\[ \sum \dot{m}_{\text{in}} \left( h + \frac{v^2}{2} + gz \right)_{\text{in}} + \dot{Q} + \dot{W}_s = \sum \dot{m}_{\text{out}} \left( h + \frac{v^2}{2} + gz \right)_{\text{out}} \]

• h = specific enthalpy (J/kg, Btu/lb_m)
• v = velocity (m/s, ft/s)
• g = gravitational acceleration (9.81 m/s², 32.2 ft/s²)
• z = elevation (m, ft)
• \(\dot{Q}\) = heat transfer rate into the system (W, Btu/hr)
• \(\dot{W}_s\) = shaft work rate into the system (W, Btu/hr)
• Kinetic energy term: \(\frac{v^2}{2}\) (J/kg, Btu/lb_m)
• Potential energy term: \(gz\) (J/kg, Btu/lb_m)

Simplified Energy Balance (Negligible KE and PE):

\[ \sum \dot{m}_{\text{in}} h_{\text{in}} + \dot{Q} + \dot{W}_s = \sum \dot{m}_{\text{out}} h_{\text{out}} \]

• Most common form for chemical process equipment
• Valid when velocity changes and elevation changes are small

Enthalpy Calculations

Specific Enthalpy Change:

\[ \Delta h = c_p \Delta T \]

• \(c_p\) = specific heat capacity at constant pressure (J/(kg·K), Btu/(lb_m·°F))
• \(\Delta T\) = temperature change (K or °C, °F or °R)
• Valid for ideal gases and incompressible liquids

Enthalpy of Mixtures:

\[ h_{\text{mix}} = \sum x_i h_i \]

• \(x_i\) = mass fraction of component i
• \(h_i\) = specific enthalpy of component i
• Assumes ideal mixing (no heat of mixing)

Enthalpy with Phase Change:

\[ h_{\text{vapor}} - h_{\text{liquid}} = \Delta h_{\text{vap}} \]

• \(\Delta h_{\text{vap}}\) = latent heat of vaporization (J/kg, Btu/lb_m)
• Also applies for sublimation, fusion, etc.

Heat Transfer

Heat Duty:

\[ \dot{Q} = \dot{m} c_p \Delta T \]

• Heat required to change temperature of a stream

Heat Exchanger Energy Balance:

\[ \dot{m}_h c_{p,h} (T_{h,\text{in}} - T_{h,\text{out}}) = \dot{m}_c c_{p,c} (T_{c,\text{out}} - T_{c,\text{in}}) \]

• Subscript h denotes hot stream
• Subscript c denotes cold stream
• Assumes no heat loss to surroundings

Log Mean Temperature Difference (LMTD):

\[ \Delta T_{\text{lm}} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)} \]

• \(\Delta T_1\) = temperature difference at one end
• \(\Delta T_2\) = temperature difference at other end
• For countercurrent flow: \(\Delta T_1 = T_{h,\text{in}} - T_{c,\text{out}}\), \(\Delta T_2 = T_{h,\text{out}} - T_{c,\text{in}}\)
• For cocurrent flow: \(\Delta T_1 = T_{h,\text{in}} - T_{c,\text{in}}\), \(\Delta T_2 = T_{h,\text{out}} - T_{c,\text{out}}\)

Heat Transfer Rate:

\[ \dot{Q} = U A \Delta T_{\text{lm}} \]

• U = overall heat transfer coefficient (W/(m²·K), Btu/(hr·ft²·°F))
• A = heat transfer area (m², ft²)

LMTD Correction Factor:

\[ \dot{Q} = U A F \Delta T_{\text{lm}} \]

• F = correction factor for non-ideal configurations (dimensionless)
• F = 1 for true countercurrent or cocurrent flow
• F < 1="" for="" crossflow="" and="" shell-and-tube="" with="" multiple="">

Work Terms

Shaft Work for Pumps and Compressors:

\[ \dot{W}_s = \dot{m} (h_{\text{out}} - h_{\text{in}}) \]

• For adiabatic operation (\(\dot{Q}\) = 0)
• Neglecting kinetic and potential energy changes

Isentropic Efficiency:

\[ \eta_{\text{isentropic}} = \frac{h_{\text{out,isentropic}} - h_{\text{in}}}{h_{\text{out,actual}} - h_{\text{in}}} \]

• For compressors and pumps (work input devices)

\[ \eta_{\text{isentropic}} = \frac{h_{\text{in}} - h_{\text{out,actual}}}{h_{\text{in}} - h_{\text{out,isentropic}}} \]

• For turbines and expanders (work output devices)

Flow Work:

\[ W_{\text{flow}} = Pv \]

• P = pressure (Pa, psi)
• v = specific volume (m³/kg, ft³/lb_m)
• Incorporated in enthalpy term: \(h = u + Pv\)

Momentum Balance for Steady-State Systems

General Momentum Balance

Momentum Balance Equation:

\[ \sum \vec{F} = \sum \dot{m}_{\text{out}} \vec{v}_{\text{out}} - \sum \dot{m}_{\text{in}} \vec{v}_{\text{in}} \]

• \(\vec{F}\) = net force acting on the system (N, lb_f)
• \(\vec{v}\) = velocity vector (m/s, ft/s)
• Vector equation (applies to each direction independently)

For Single Inlet and Outlet:

\[ \sum F = \dot{m} (v_{\text{out}} - v_{\text{in}}) \]

• Scalar form for one-dimensional flow

Forces in Momentum Balance

Pressure Forces:

\[ F_P = P \cdot A \]

• P = pressure (Pa, psi)
• A = cross-sectional area (m², ft²)
• Direction normal to the surface

Body Forces:

\[ F_g = mg = \rho V g \]

• m = mass (kg, lb_m)
• V = volume (m³, ft³)
• Weight of fluid in control volume

Mechanical Energy Balance (Bernoulli Equation)

Ideal Fluid Flow

Bernoulli Equation (Frictionless, Incompressible):

\[ \frac{P_1}{\rho} + \frac{v_1^2}{2} + gz_1 = \frac{P_2}{\rho} + \frac{v_2^2}{2} + gz_2 \]

• P = pressure (Pa, psi)
• \(\rho\) = density (kg/m³, lb_m/ft³)
• v = velocity (m/s, ft/s)
• g = gravitational acceleration (9.81 m/s², 32.2 ft/s²)
• z = elevation (m, ft)
• All terms have units of energy per unit mass (J/kg, ft·lb_f/lb_m)

Bernoulli Equation in Head Form:

\[ \frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2 \]

• All terms in units of length (m, ft)
• \(\frac{P}{\rho g}\) = pressure head
• \(\frac{v^2}{2g}\) = velocity head
• z = elevation head

Real Fluid Flow with Friction

Extended Bernoulli Equation with Friction:

\[ \frac{P_1}{\rho} + \frac{v_1^2}{2} + gz_1 + W_s = \frac{P_2}{\rho} + \frac{v_2^2}{2} + gz_2 + h_f \]

• \(W_s\) = shaft work per unit mass (J/kg, ft·lb_f/lb_m)
• \(h_f\) = friction loss per unit mass (J/kg, ft·lb_f/lb_m)
• \(W_s\) > 0 for pumps (work into fluid)
• \(W_s\) < 0="" for="" turbines="" (work="" from="">

Head Form with Pump and Friction:

\[ \frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 + h_p = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2 + h_L \]

• \(h_p\) = pump head (m, ft)
• \(h_L\) = head loss due to friction (m, ft)

Friction Losses

Darcy-Weisbach Equation:

\[ h_f = f \frac{L}{D} \frac{v^2}{2} \]

• f = Darcy friction factor (dimensionless)
• L = pipe length (m, ft)
• D = pipe diameter (m, ft)
• v = average velocity (m/s, ft/s)
• Units of \(h_f\): J/kg or ft·lb_f/lb_m

Head Loss Form:

\[ h_L = f \frac{L}{D} \frac{v^2}{2g} \]

• Units of \(h_L\): m or ft

Minor Losses:

\[ h_{f,\text{minor}} = K \frac{v^2}{2} \]

• K = loss coefficient (dimensionless)
• Accounts for fittings, valves, expansions, contractions

Total Head Loss:

\[ h_L = h_{L,\text{major}} + \sum h_{L,\text{minor}} = f \frac{L}{D} \frac{v^2}{2g} + \sum K_i \frac{v_i^2}{2g} \]

Reactor Material Balance

Stoichiometry

General Reaction:

\[ aA + bB \rightarrow cC + dD \]

• a, b, c, d = stoichiometric coefficients

Extent of Reaction:

\[ \xi = \frac{n_i - n_{i,0}}{\nu_i} \]

• \(\xi\) = extent of reaction (mol, kmol)
• \(n_i\) = moles of species i
• \(n_{i,0}\) = initial moles of species i
• \(\nu_i\) = stoichiometric coefficient (negative for reactants, positive for products)

Conversion:

\[ X_A = \frac{n_{A,0} - n_A}{n_{A,0}} = \frac{\text{moles reacted}}{\text{moles fed}} \]

• \(X_A\) = conversion of limiting reactant A (dimensionless)
• Range: 0 ≤ \(X_A\) ≤ 1

Moles of Species i:

\[ n_i = n_{i,0} + \nu_i \xi \]

For Limiting Reactant A:

\[ n_A = n_{A,0}(1 - X_A) \]

For Other Species:

\[ n_i = n_{i,0} + \frac{\nu_i}{\nu_A} n_{A,0} X_A \]

• \(\nu_A\) = stoichiometric coefficient of A (taken as negative for reactant)

Continuous Stirred Tank Reactor (CSTR)

Component Material Balance:

\[ \dot{n}_{i,\text{in}} - \dot{n}_{i,\text{out}} + \nu_i r V = 0 \]

• r = reaction rate per unit volume (mol/(m³·s), lb-mol/(ft³·hr))
• V = reactor volume (m³, ft³)
• Steady-state assumption (no accumulation)

For Constant Density Systems:

\[ \dot{V}_0 C_{A,0} - \dot{V} C_A + \nu_A r V = 0 \]

• \(\dot{V}\) = volumetric flow rate (m³/s, ft³/s)
• C = concentration (mol/m³, lb-mol/ft³)

Space Time:

\[ \tau = \frac{V}{\dot{V}_0} \]

• \(\tau\) = space time (s, hr)
• \(\dot{V}_0\) = volumetric flow rate at inlet conditions

CSTR Design Equation (First-Order Reaction):

\[ \tau = \frac{C_{A,0} X_A}{r_A} \]

• \(r_A\) = reaction rate of A (mol/(m³·s))
• For first-order: \(r_A = k C_A = k C_{A,0}(1 - X_A)\)

Plug Flow Reactor (PFR)

Differential Material Balance:

\[ \dot{n}_{A,0} \frac{dX_A}{dV} = -r_A \]

• V = reactor volume up to position (m³, ft³)
• \(r_A\) = rate of consumption of A (positive value)

PFR Design Equation:

\[ V = \dot{n}_{A,0} \int_0^{X_A} \frac{dX_A}{-r_A} \]

For Constant Density:

\[ V = \dot{V}_0 \int_0^{X_A} \frac{dX_A}{-r_A/C_{A,0}} \]

First-Order Irreversible Reaction in PFR:

\[ \tau = \frac{V}{\dot{V}_0} = \frac{1}{k} \ln \left( \frac{1}{1 - X_A} \right) \]

• k = first-order rate constant (s^-1, hr^-1)

Reaction Rate Expressions

Elementary Reaction Rate:

\[ r = k C_A^a C_B^b \]

• k = rate constant
• a, b = reaction orders

Arrhenius Equation:

\[ k = A e^{-E_a/(RT)} \]

• A = pre-exponential factor (frequency factor)
• \(E_a\) = activation energy (J/mol, cal/mol)
• R = universal gas constant (8.314 J/(mol·K), 1.987 cal/(mol·K))
• T = absolute temperature (K, °R)

Separation Processes

Flash Distillation

Overall Material Balance:

\[ F = V + L \]

• F = feed flow rate (mol/s, kmol/hr)
• V = vapor flow rate (mol/s, kmol/hr)
• L = liquid flow rate (mol/s, kmol/hr)

Component Material Balance:

\[ F z_i = V y_i + L x_i \]

• \(z_i\) = mole fraction of component i in feed
• \(y_i\) = mole fraction of component i in vapor
• \(x_i\) = mole fraction of component i in liquid

Vapor Fraction:

\[ \psi = \frac{V}{F} \]

• \(\psi\) = fraction of feed vaporized (dimensionless)

Equilibrium Relationship:

\[ y_i = K_i x_i \]

• \(K_i\) = equilibrium ratio (K-value) for component i
• For ideal solutions: \(K_i = P_i^{\text{sat}} / P\)

Rachford-Rice Equation:

\[ \sum_{i=1}^{n} \frac{z_i (K_i - 1)}{1 + \psi (K_i - 1)} = 0 \]

• Used to solve for vapor fraction \(\psi\) in flash calculations

Component Compositions:

\[ x_i = \frac{z_i}{1 + \psi (K_i - 1)} \] \[ y_i = \frac{K_i z_i}{1 + \psi (K_i - 1)} \]

Distillation Column

Overall Material Balance:

\[ F = D + B \]

• D = distillate flow rate (mol/s, kmol/hr)
• B = bottoms flow rate (mol/s, kmol/hr)

Component Balance:

\[ F z_F = D x_D + B x_B \]

• \(z_F\) = mole fraction of key component in feed
• \(x_D\) = mole fraction of key component in distillate
• \(x_B\) = mole fraction of key component in bottoms

Reflux Ratio:

\[ R = \frac{L}{D} \]

• R = reflux ratio (dimensionless)
• L = liquid reflux flow rate to column

Minimum Reflux Ratio (Underwood Equation for Binary):

\[ R_{\min} = \frac{1}{\alpha - 1} \left( \frac{x_D}{1 - x_D} - \frac{\alpha z_F}{1 - z_F} \right) \]

• \(\alpha\) = relative volatility = \(K_{\text{light}} / K_{\text{heavy}}\)

Rectifying Section Operating Line:

\[ y_{n+1} = \frac{R}{R+1} x_n + \frac{x_D}{R+1} \]

• Relates vapor composition leaving stage n+1 to liquid composition leaving stage n

Stripping Section Operating Line:

\[ y_m = \frac{L'}{V'} x_m - \frac{B x_B}{V'} \]

• L' = liquid flow rate in stripping section
• V' = vapor flow rate in stripping section

Fenske Equation (Minimum Stages at Total Reflux):

\[ N_{\min} = \frac{\ln \left[ \left( \frac{x_D}{1-x_D} \right) \left( \frac{1-x_B}{x_B} \right) \right]}{\ln \alpha} \]

• \(N_{\min}\) = minimum number of theoretical stages
• Applicable for binary systems with constant relative volatility

Absorption and Stripping

Operating Line for Absorption:

\[ Y_{n+1} = \frac{L}{G} X_n + \left( Y_1 - \frac{L}{G} X_0 \right) \]

• Y = mole ratio of solute in gas phase (mol solute/mol carrier gas)
• X = mole ratio of solute in liquid phase (mol solute/mol solvent)
• L = liquid flow rate (solvent basis)
• G = gas flow rate (carrier gas basis)

Minimum Liquid-to-Gas Ratio:

\[ \left( \frac{L}{G} \right)_{\min} = \frac{Y_1 - Y_2}{X_1^* - X_0} \]

• \(X_1^*\) = liquid composition in equilibrium with entering gas \(Y_1\)

Extraction

Single-Stage Extraction:

\[ F + S = E + R \]

• F = feed flow rate
• S = solvent flow rate
• E = extract flow rate
• R = raffinate flow rate

Component Balance:

\[ F x_F + S x_S = E y_E + R x_R \]

• x = mass fraction of solute in raffinate phase
• y = mass fraction of solute in extract phase

Distribution Coefficient:

\[ K_d = \frac{y_E}{x_R} \]

• \(K_d\) = distribution coefficient (dimensionless)
• Ratio of solute concentration in extract to raffinate at equilibrium

Fluid Flow in Pipes and Channels

Flow Regimes

Reynolds Number:

\[ Re = \frac{\rho v D}{\mu} = \frac{v D}{\nu} \]

• \(\rho\) = fluid density (kg/m³, lb_m/ft³)
• v = average velocity (m/s, ft/s)
• D = pipe diameter (m, ft)
• \(\mu\) = dynamic viscosity (Pa·s, lb_m/(ft·s))
• \(\nu\) = kinematic viscosity (m²/s, ft²/s)
• Re < 2100:="" laminar="">
• 2100 < re="">< 4000:="" transitional="">
• Re > 4000: turbulent flow

Friction Factor Correlations

Laminar Flow (Re <>

\[ f = \frac{64}{Re} \]

• f = Darcy friction factor

Turbulent Flow - Colebrook Equation:

\[ \frac{1}{\sqrt{f}} = -2.0 \log_{10} \left( \frac{\varepsilon/D}{3.7} + \frac{2.51}{Re \sqrt{f}} \right) \]

• \(\varepsilon\) = pipe roughness (m, ft)
• \(\varepsilon/D\) = relative roughness
• Implicit equation requiring iteration

Turbulent Flow - Haaland Approximation:

\[ \frac{1}{\sqrt{f}} \approx -1.8 \log_{10} \left[ \left( \frac{\varepsilon/D}{3.7} \right)^{1.11} + \frac{6.9}{Re} \right] \]

• Explicit approximation to Colebrook equation

Turbulent Flow - Smooth Pipes (Blasius):

\[ f = \frac{0.316}{Re^{0.25}} \]

• Valid for 4000 < re=""><>⁵

Continuity Equation

Conservation of Mass in Pipe Flow:

\[ \rho_1 A_1 v_1 = \rho_2 A_2 v_2 \]

• For steady flow in a streamtube or pipe

Incompressible Flow:

\[ A_1 v_1 = A_2 v_2 = \dot{V} \]

• Constant volumetric flow rate

Non-Circular Conduits

Hydraulic Diameter:

\[ D_h = \frac{4A}{P} \]

• A = cross-sectional area (m², ft²)
• P = wetted perimeter (m, ft)
• Used in place of D for non-circular cross-sections

Pump and Compressor Calculations

Pump Head and Power

Pump Head:

\[ H = \frac{P_2 - P_1}{\rho g} + \frac{v_2^2 - v_1^2}{2g} + (z_2 - z_1) + h_L \]

• H = total head developed by pump (m, ft)
• Includes pressure, velocity, elevation, and friction loss terms

Brake Horsepower (Power Required):

\[ \text{BHP} = \frac{\dot{m} g H}{\eta} \]

• BHP = brake horsepower (W, hp)
• \(\eta\) = pump efficiency (dimensionless)

In US Customary Units:

\[ \text{BHP (hp)} = \frac{\dot{V} \text{ (gpm)} \times H \text{ (ft)} \times \rho \text{ (lb/gal)}}{3960 \times \eta} \]

• gpm = gallons per minute
• \(\rho\) in lb/gal (water ≈ 8.34 lb/gal)

Simplified for Water:

\[ \text{BHP (hp)} = \frac{\dot{V} \text{ (gpm)} \times H \text{ (ft)}}{3960 \times \eta} \]

Compressor Work

Isothermal Compression (Ideal Gas):

\[ W = n R T \ln \left( \frac{P_2}{P_1} \right) \]

• W = work per mole (J/mol)
• n = number of moles
• R = gas constant (8.314 J/(mol·K))
• T = temperature (K)

Adiabatic Compression (Ideal Gas, Isentropic):

\[ W = \frac{n R T_1}{\gamma - 1} \left[ \left( \frac{P_2}{P_1} \right)^{(\gamma-1)/\gamma} - 1 \right] \]

• \(\gamma\) = ratio of specific heats \(c_p/c_v\)
• \(T_1\) = inlet temperature (K)

Polytropic Compression:

\[ W = \frac{n R T_1}{n_p - 1} \left[ \left( \frac{P_2}{P_1} \right)^{(n_p-1)/n_p} - 1 \right] \]

• \(n_p\) = polytropic exponent
• For isothermal: \(n_p = 1\)
• For isentropic: \(n_p = \gamma\)

Discharge Temperature (Isentropic):

\[ T_2 = T_1 \left( \frac{P_2}{P_1} \right)^{(\gamma-1)/\gamma} \]

Heat Exchanger Design

Overall Heat Transfer Coefficient

Resistance Model:

\[ \frac{1}{U} = \frac{1}{h_i} + \frac{t_w}{k_w} + \frac{1}{h_o} \]

• U = overall heat transfer coefficient (W/(m²·K), Btu/(hr·ft²·°F))
• \(h_i\) = inside convective heat transfer coefficient
• \(h_o\) = outside convective heat transfer coefficient
• \(t_w\) = wall thickness (m, ft)
• \(k_w\) = wall thermal conductivity (W/(m·K), Btu/(hr·ft·°F))

For Fouled Surfaces:

\[ \frac{1}{U} = \frac{1}{h_i} + R_{f,i} + \frac{t_w}{k_w} + R_{f,o} + \frac{1}{h_o} \]

• \(R_f\) = fouling resistance (m²·K/W, hr·ft²·°F/Btu)

Cylindrical Geometry (Based on Outside Area):

\[ \frac{1}{U_o A_o} = \frac{1}{h_i A_i} + \frac{r_o \ln(r_o/r_i)}{k_w} + \frac{1}{h_o A_o} \]

• \(r_i\) = inside radius
• \(r_o\) = outside radius
• \(A_i\) = inside surface area
• \(A_o\) = outside surface area

Effectiveness-NTU Method

Heat Capacity Rate:

\[ C = \dot{m} c_p \]

• C = heat capacity rate (W/K, Btu/(hr·°F))

Heat Capacity Rate Ratio:

\[ C_r = \frac{C_{\min}}{C_{\max}} \]

• \(C_{\min}\) = smaller of the two heat capacity rates
• \(C_{\max}\) = larger of the two heat capacity rates

Number of Transfer Units (NTU):

\[ \text{NTU} = \frac{UA}{C_{\min}} \]

• Dimensionless parameter representing heat exchanger size

Effectiveness:

\[ \varepsilon = \frac{\dot{Q}}{\dot{Q}_{\max}} = \frac{C_h (T_{h,\text{in}} - T_{h,\text{out}})}{C_{\min} (T_{h,\text{in}} - T_{c,\text{in}})} \]

• \(\varepsilon\) = heat exchanger effectiveness (dimensionless)
• \(\dot{Q}_{\max}\) = maximum possible heat transfer

Actual Heat Transfer:

\[ \dot{Q} = \varepsilon C_{\min} (T_{h,\text{in}} - T_{c,\text{in}}) \]

Effectiveness for Counterflow (General):

\[ \varepsilon = \frac{1 - \exp[-\text{NTU}(1 - C_r)]}{1 - C_r \exp[-\text{NTU}(1 - C_r)]} \]

• Valid for \(C_r\) <>

Effectiveness for Counterflow (\(C_r\) = 1):

\[ \varepsilon = \frac{\text{NTU}}{1 + \text{NTU}} \]

Effectiveness for Parallel Flow:

\[ \varepsilon = \frac{1 - \exp[-\text{NTU}(1 + C_r)]}{1 + C_r} \]

Thermodynamic Properties and Relationships

Ideal Gas Law

Equation of State:

\[ PV = nRT \]

• P = absolute pressure (Pa, psia)
• V = volume (m³, ft³)
• n = number of moles (mol, lb-mol)
• R = universal gas constant
• R = 8.314 J/(mol·K) = 8.314 kPa·m³/(kmol·K)
• R = 1.987 Btu/(lb-mol·°R) = 10.73 psia·ft³/(lb-mol·°R)
• T = absolute temperature (K, °R)

Mass Basis:

\[ PV = \frac{m}{M} RT \]

• m = mass (kg, lb_m)
• M = molecular weight (kg/kmol, lb_m/lb-mol)

Specific Volume Form:

\[ Pv = \frac{RT}{M} \]

• v = specific volume (m³/kg, ft³/lb_m)

Density Form:

\[ \rho = \frac{PM}{RT} \]

Real Gas Equations

Compressibility Factor:

\[ Z = \frac{Pv}{RT/M} \]

• Z = compressibility factor (dimensionless)
• Z = 1 for ideal gas
• Z ≠ 1 for real gas

Van der Waals Equation:

\[ \left( P + \frac{a}{v^2} \right) (v - b) = RT \]

• a = attraction parameter
• b = volume exclusion parameter
• Constants specific to each gas

Thermodynamic Properties

Internal Energy (Ideal Gas):

\[ \Delta u = c_v \Delta T \]

• u = specific internal energy (J/kg, Btu/lb_m)
• \(c_v\) = specific heat at constant volume

Enthalpy Definition:

\[ h = u + Pv \]

Relationship for Ideal Gases:

\[ c_p - c_v = R/M \]

Ratio of Specific Heats:

\[ \gamma = \frac{c_p}{c_v} \]

• Monatomic gases: \(\gamma\) ≈ 1.67
• Diatomic gases: \(\gamma\) ≈ 1.4
• Polyatomic gases: \(\gamma\) ≈ 1.3

Entropy Change (Ideal Gas):

\[ \Delta s = c_p \ln \left( \frac{T_2}{T_1} \right) - R \ln \left( \frac{P_2}{P_1} \right) \]

• s = specific entropy (J/(kg·K), Btu/(lb_m·°R))

Isentropic Process (\(\Delta s\) = 0):

\[ \frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{(\gamma-1)/\gamma} \] \[ \frac{T_2}{T_1} = \left( \frac{V_1}{V_2} \right)^{\gamma-1} \] \[ \frac{P_2}{P_1} = \left( \frac{V_1}{V_2} \right)^{\gamma} \]

Psychrometrics and Humidity

Humidity Definitions

Absolute Humidity (Humidity Ratio):

\[ H = \frac{m_{\text{water}}}{m_{\text{dry air}}} \]

• H = absolute humidity (kg H₂O/kg dry air, lb_m H₂O/lb_m dry air)
• Also called specific humidity or moisture content

Relative Humidity:

\[ \phi = \frac{P_v}{P_{\text{sat}}} \times 100\% \]

• \(\phi\) = relative humidity (%)
• \(P_v\) = partial pressure of water vapor
• \(P_{\text{sat}}\) = saturation pressure of water at the dry-bulb temperature

Humidity Ratio from Partial Pressures:

\[ H = 0.622 \frac{P_v}{P - P_v} \]

• P = total pressure (atmospheric)
• 0.622 = ratio of molecular weights (18/29)

Saturation Humidity:

\[ H_{\text{sat}} = 0.622 \frac{P_{\text{sat}}}{P - P_{\text{sat}}} \]

Psychrometric Calculations

Dew Point Temperature:

• Temperature at which water vapor begins to condense
• Corresponds to saturation pressure equal to actual vapor partial pressure
• \(P_{\text{sat}}(T_{\text{dew}}) = P_v\)

Wet-Bulb Temperature:

• Temperature indicated by thermometer covered with water-saturated wick
• Requires psychrometric chart or iterative calculation

Enthalpy of Moist Air:

\[ h = c_{p,\text{air}} T + H (h_{fg,0} + c_{p,\text{vapor}} T) \]

• h = enthalpy per unit mass of dry air (kJ/kg dry air)
• \(c_{p,\text{air}}\) ≈ 1.006 kJ/(kg·K)
• \(h_{fg,0}\) = latent heat of vaporization at 0°C ≈ 2501 kJ/kg
• \(c_{p,\text{vapor}}\) ≈ 1.86 kJ/(kg·K)
• T = dry-bulb temperature (°C)

Simplified Enthalpy:

\[ h \approx 1.006 T + H (2501 + 1.86 T) \]

Dimensionless Numbers

Fluid Flow

Reynolds Number:

\[ Re = \frac{\rho v L}{\mu} = \frac{v L}{\nu} \]

• Ratio of inertial forces to viscous forces
• L = characteristic length (pipe diameter, particle diameter, etc.)

Froude Number:

\[ Fr = \frac{v}{\sqrt{gL}} \]

• Ratio of inertial forces to gravitational forces
• Important in open channel flow

Euler Number:

\[ Eu = \frac{\Delta P}{\rho v^2} \]

• Ratio of pressure forces to inertial forces

Heat Transfer

Prandtl Number:

\[ Pr = \frac{c_p \mu}{k} = \frac{\nu}{\alpha} \]

• Ratio of momentum diffusivity to thermal diffusivity
• k = thermal conductivity
• \(\alpha\) = thermal diffusivity

Nusselt Number:

\[ Nu = \frac{hL}{k} \]

• Ratio of convective to conductive heat transfer
• h = convective heat transfer coefficient

Grashof Number:

\[ Gr = \frac{g \beta \Delta T L^3}{\nu^2} \]

• Ratio of buoyancy to viscous forces
• \(\beta\) = coefficient of thermal expansion (K^-1)
• \(\Delta T\) = temperature difference
• Important in natural convection

Rayleigh Number:

\[ Ra = Gr \cdot Pr = \frac{g \beta \Delta T L^3}{\nu \alpha} \]

Mass Transfer

Schmidt Number:

\[ Sc = \frac{\mu}{\rho D_{AB}} = \frac{\nu}{D_{AB}} \]

• Ratio of momentum diffusivity to mass diffusivity
• \(D_{AB}\) = mass diffusion coefficient (m²/s)

Sherwood Number:

\[ Sh = \frac{k_c L}{D_{AB}} \]

• Ratio of convective to diffusive mass transfer
• \(k_c\) = mass transfer coefficient (m/s)

Lewis Number:

\[ Le = \frac{Sc}{Pr} = \frac{\alpha}{D_{AB}} \]

• Ratio of thermal diffusivity to mass diffusivity

Process Control

Feedback Control

PID Controller Output:

\[ u(t) = K_c \left[ e(t) + \frac{1}{\tau_I} \int_0^t e(t') dt' + \tau_D \frac{de(t)}{dt} \right] + u_s \]

• u(t) = controller output
• \(K_c\) = controller gain
• e(t) = error = setpoint - measured value
• \(\tau_I\) = integral time constant
• \(\tau_D\) = derivative time constant
• \(u_s\) = steady-state output

Proportional-Only Control:

\[ u(t) = K_c e(t) + u_s \]

PI Control:

\[ u(t) = K_c \left[ e(t) + \frac{1}{\tau_I} \int_0^t e(t') dt' \right] + u_s \]

First-Order System Response

Transfer Function:

\[ G(s) = \frac{K}{\tau s + 1} \]

• K = process gain
• \(\tau\) = time constant
• s = Laplace variable

Step Response:

\[ y(t) = K M [1 - e^{-t/\tau}] \]

• M = magnitude of step input
• 63.2% of final value reached at \(t = \tau\)

Material and Energy Balance Strategies

Degrees of Freedom Analysis

Degrees of Freedom:

\[ NDF = NV - NE \]

• NDF = number of degrees of freedom
• NV = number of unknown variables
• NE = number of independent equations
• NDF = 0: system is exactly specified (solvable)
• NDF > 0: system is underspecified (need more information)
• NDF < 0:="" system="" is="" overspecified="" (redundant="" or="" conflicting="">

Multiple Unit Systems

Overall Balance:

• Draw boundary around entire system
• Write balances for overall inputs and outputs

Individual Unit Balances:

• Draw boundary around each unit
• Write balances for each unit separately

Recycle Streams:

• Fresh feed + Recycle = Total feed to process
• Product + Purge = Output from separator
• Overall conversion based on fresh feed
• Single-pass conversion based on total feed to reactor

Bypass and Mixing

Bypass Fraction:

\[ f = \frac{\dot{m}_{\text{bypass}}}{\dot{m}_{\text{total in}}} \]

Mixed Stream Property:

\[ \dot{m}_{\text{mix}} x_{\text{mix}} = \dot{m}_1 x_1 + \dot{m}_2 x_2 \]

• Applies to temperature, composition, enthalpy (if no heat of mixing)

Valve and Piping Pressure Drop

Valve Sizing

Valve Flow Coefficient (Liquid):

\[ \dot{V} = C_v \sqrt{\frac{\Delta P}{SG}} \]

• \(C_v\) = valve flow coefficient (gpm at 1 psi drop for water)
• \(\dot{V}\) = volumetric flow rate (gpm)
• \(\Delta P\) = pressure drop across valve (psi)
• SG = specific gravity (dimensionless, relative to water)

Gas Flow Through Valve:

\[ \dot{m} = C_v \sqrt{\Delta P \times P_1 \times \rho} \]

• \(\dot{m}\) = mass flow rate
• \(P_1\) = upstream pressure
• More complex correlations exist for compressible flow

Orifice Flow

Orifice Flow Rate:

\[ \dot{m} = C_d A_0 \sqrt{2 \rho \Delta P} \]

• \(C_d\) = discharge coefficient (dimensionless, typically 0.6-0.7)
• \(A_0\) = orifice area (m², ft²)
• \(\Delta P\) = pressure drop across orifice

Volumetric Flow Rate:

\[ \dot{V} = C_d A_0 \sqrt{\frac{2 \Delta P}{\rho}} \]

Settling and Particle Dynamics

Terminal Velocity

Force Balance on Settling Particle:

\[ F_g = F_b + F_d \]

• \(F_g\) = gravitational force
• \(F_b\) = buoyant force
• \(F_d\) = drag force

Terminal Velocity (Stokes' Law, Laminar Flow):

\[ v_t = \frac{g d_p^2 (\rho_p - \rho_f)}{18 \mu} \]

• \(v_t\) = terminal velocity (m/s, ft/s)
• \(d_p\) = particle diameter (m, ft)
• \(\rho_p\) = particle density
• \(\rho_f\) = fluid density
• \(\mu\) = fluid dynamic viscosity
• Valid for \(Re_p\) <>

Particle Reynolds Number:

\[ Re_p = \frac{\rho_f v_t d_p}{\mu} \]

Drag Coefficient:

\[ C_D = \frac{24}{Re_p} \]

• For Stokes' flow (\(Re_p\) <>

General Drag Force:

\[ F_d = C_D \frac{\pi d_p^2}{4} \frac{\rho_f v^2}{2} \]

Throttling and Expansion Devices

Throttling Process

Isenthalpic Process:

\[ h_1 = h_2 \]

• Enthalpy remains constant across throttle valve
• Pressure decreases
• Temperature may increase, decrease, or remain constant depending on fluid

Joule-Thomson Coefficient:

\[ \mu_{JT} = \left( \frac{\partial T}{\partial P} \right)_h \]

• \(\mu_{JT}\) = Joule-Thomson coefficient
• \(\mu_{JT}\) > 0: cooling upon expansion (most gases at room temperature)
• \(\mu_{JT}\) < 0:="" heating="" upon="" expansion="" (hydrogen,="" helium="" at="" room="">
• \(\mu_{JT}\) = 0: ideal gas

Mixing and Blending

Component Mixing

Mass Balance on Component:

\[ \sum \dot{m}_i x_{i,j} = \dot{m}_{\text{out}} x_{\text{out},j} \]

• Subscript j denotes component
• Sum over all inlet streams i

Weighted Average Property:

\[ P_{\text{avg}} = \frac{\sum \dot{m}_i P_i}{\sum \dot{m}_i} \]

• For extensive properties that are additive

Heat of Mixing

Actual Enthalpy with Heat of Mixing:

\[ h_{\text{actual}} = \sum x_i h_i + \Delta h_{\text{mix}} \]

• \(\Delta h_{\text{mix}}\) = heat of mixing (J/kg, Btu/lb_m)
• Can be positive (endothermic) or negative (exothermic)