Formula Sheet: Mass Transfer

Diffusion and Fick's Laws

Fick's First Law (Steady-State Diffusion)

Molar flux (equimolar counterdiffusion): \[J_A = -D_{AB} \frac{dC_A}{dz}\]

• J_A = molar flux of component A (mol/m²·s or lbmol/ft²·h)
• D_AB = diffusivity of A in B (m²/s or ft²/h)
• C_A = molar concentration of A (mol/m³ or lbmol/ft³)
• z = distance in direction of diffusion (m or ft)

Mass flux form: \[j_A = -D_{AB} \frac{d\rho_A}{dz}\]

• j_A = mass flux of component A (kg/m²·s or lb/ft²·h)
• ρ_A = mass concentration of A (kg/m³ or lb/ft³)

For gases (ideal gas): \[J_A = -\frac{D_{AB}}{RT} \frac{dp_A}{dz}\]

• p_A = partial pressure of A (Pa or psia)
• R = universal gas constant (8.314 J/mol·K or 1545 ft·lbf/lbmol·°R)
• T = absolute temperature (K or °R)

Fick's Second Law (Unsteady-State Diffusion)

\[\frac{\partial C_A}{\partial t} = D_{AB} \frac{\partial^2 C_A}{\partial z^2}\]

• t = time (s or h)
• Applies when concentration changes with time
• Assumes constant diffusivity

Diffusion Through Stagnant Film

One-dimensional diffusion of A through stagnant B: \[N_A = \frac{D_{AB} P}{RT(z_2 - z_1)} \ln\left(\frac{P - p_{A2}}{P - p_{A1}}\right)\]

• N_A = molar flux of A (mol/m²·s or lbmol/ft²·h)
• P = total pressure (Pa or psia)
• p_A1, p_A2 = partial pressures of A at positions z₁ and z₂
• z₂ - z₁ = diffusion path length (m or ft)

Alternative form using log mean: \[N_A = \frac{D_{AB} P}{RT \Delta z} \frac{(p_{A1} - p_{A2})}{(p_{B})_{lm}}\] where the log mean of inert component B: \[(p_B)_{lm} = \frac{p_{B2} - p_{B1}}{\ln(p_{B2}/p_{B1})}\]

Equimolar Counterdiffusion

\[N_A = \frac{D_{AB}}{RT \Delta z}(p_{A1} - p_{A2})\]

• Applies when equal molar amounts of A and B diffuse in opposite directions
• Net molar flux is zero (N_A = -N_B)

Diffusivity Estimation

Gas Diffusivity

Fuller-Schettler-Giddings Equation: \[D_{AB} = \frac{0.00143 T^{1.75}}{P M_{AB}^{0.5}[(\Sigma v)_A^{1/3} + (\Sigma v)_B^{1/3}]^2}\]

• D_AB = diffusivity (cm²/s)
• T = temperature (K)
• P = pressure (atm)
• M_AB = (2/M_A + 2/M_B)^-1, reduced molecular weight
• Σv = sum of atomic diffusion volumes

Temperature and pressure correction: \[D_{AB,2} = D_{AB,1} \left(\frac{T_2}{T_1}\right)^{1.75} \left(\frac{P_1}{P_2}\right)\]

Liquid Diffusivity

Wilke-Chang Equation: \[D_{AB} = \frac{7.4 \times 10^{-8} (\phi M_B)^{0.5} T}{\mu_B V_A^{0.6}}\]

• D_AB = diffusivity (cm²/s)
• φ = association parameter for solvent B (2.6 for water, 1.9 for methanol, 1.5 for ethanol, 1.0 for unassociated solvents)
• M_B = molecular weight of solvent B (g/mol)
• T = temperature (K)
• μ_B = viscosity of solvent B (cP)
• V_A = molar volume of solute A at normal boiling point (cm³/mol)

Temperature correction for liquid diffusivity: \[\frac{D_2}{D_1} = \frac{T_2 \mu_1}{T_1 \mu_2}\]

Mass Transfer Coefficients

Film Theory

Gas phase mass transfer coefficient: \[k_y = \frac{D_{AB}}{\delta}\] Liquid phase mass transfer coefficient: \[k_x = \frac{D_{AB}}{\delta}\]

• k_y = gas phase mass transfer coefficient (mol/m²·s or lbmol/ft²·h)
• k_x = liquid phase mass transfer coefficient (mol/m²·s or lbmol/ft²·h)
• δ = film thickness (m or ft)

Flux Equations Using Mass Transfer Coefficients

Gas phase (mole fraction driving force): \[N_A = k_y (y_A - y_{Ai})\] Gas phase (partial pressure driving force): \[N_A = k_G (p_A - p_{Ai})\] Liquid phase (mole fraction driving force): \[N_A = k_x (x_{Ai} - x_A)\] Liquid phase (concentration driving force): \[N_A = k_L (C_{Ai} - C_A)\]

• y_A = mole fraction of A in bulk gas
• y_Ai = mole fraction of A at interface
• x_A = mole fraction of A in bulk liquid
• x_Ai = mole fraction of A at interface

Overall Mass Transfer Coefficients

Gas phase overall coefficient: \[\frac{1}{K_y} = \frac{1}{k_y} + \frac{m}{k_x}\] \[\frac{1}{K_G} = \frac{1}{k_G} + \frac{H}{k_L}\] Liquid phase overall coefficient: \[\frac{1}{K_x} = \frac{1}{mk_y} + \frac{1}{k_x}\] \[\frac{1}{K_L} = \frac{1}{Hk_G} + \frac{1}{k_L}\]

• K_y, K_G = overall gas phase mass transfer coefficients
• K_x, K_L = overall liquid phase mass transfer coefficients
• m = slope of equilibrium line (y* = mx)
• H = Henry's law constant (p = Hx)

Relationship Between Coefficients

\[k_G = k_y c_{total,gas}\] \[k_L = k_x c_{total,liquid}\] \[K_G = K_y c_{total,gas}\] \[K_L = K_x c_{total,liquid}\]

• c_total = total molar concentration (mol/m³ or lbmol/ft³)

Convective Mass Transfer Correlations

Dimensionless Numbers

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

• Sc = ratio of momentum diffusivity to mass diffusivity
• μ = dynamic viscosity (Pa·s or lb/ft·s)
• ρ = density (kg/m³ or lb/ft³)
• ν = kinematic viscosity (m²/s or ft²/s)

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

• Sh = ratio of convective to diffusive mass transfer
• k_c = mass transfer coefficient (m/s or ft/s)
• L = characteristic length (m or ft)
• Analogous to Nusselt number in heat transfer

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

• v = velocity (m/s or ft/s)

Flat Plate (Laminar Flow)

\[Sh_L = 0.664 Re_L^{0.5} Sc^{1/3}\]

• Valid for Re_L < 5="" ×="">
• Sc > 0.6
• L = length of plate

Flat Plate (Turbulent Flow)

\[Sh_L = 0.037 Re_L^{0.8} Sc^{1/3}\]

• Valid for 5 × 10⁵ <>_L <>
• 0.6 < sc=""><>

Flow in Tubes (Laminar)

Short tubes (developing flow): \[Sh_D = 1.86 \left(Re_D Sc \frac{D}{L}\right)^{1/3} \left(\frac{\mu_b}{\mu_w}\right)^{0.14}\]

• Valid for Re_D <>
• D = tube diameter
• L = tube length
• μ_b = viscosity at bulk conditions
• μ_w = viscosity at wall conditions

Long tubes (fully developed): \[Sh_D = 3.66\]

• For constant wall concentration
• Valid when L/D is large

Flow in Tubes (Turbulent)

Chilton-Colburn analogy: \[Sh = 0.023 Re^{0.8} Sc^{1/3}\]

• Valid for Re > 10,000
• 0.6 < sc=""><>
• L/D > 60

Sieder-Tate form: \[Sh_D = 0.027 Re_D^{0.8} Sc^{1/3} \left(\frac{\mu_b}{\mu_w}\right)^{0.14}\]

Flow Around Spheres

Ranz-Marshall correlation: \[Sh_d = 2.0 + 0.6 Re_d^{0.5} Sc^{1/3}\]

• Valid for 1 <>_d <>
• d = sphere diameter
• First term (2.0) represents pure molecular diffusion

Packed Beds

Wilson-Geankoplis correlation: \[j_D = \frac{Sh}{Re Sc^{1/3}} = 1.09 Re^{-2/3}\]

• Valid for 0.0016 < re=""><>
• 165 < sc=""><>

\[j_D = 0.250 Re^{-0.31}\]

• Valid for 55 < re=""><>

Interphase Mass Transfer (Two-Phase Systems)

Equilibrium Relationships

Henry's Law: \[p_A = H x_A\]

• p_A = partial pressure of A in gas phase (Pa or psia)
• H = Henry's law constant (Pa or psia)
• x_A = mole fraction of A in liquid phase
• Applies to dilute solutions

Alternative form: \[y_A = m x_A\]

• m = H/P = equilibrium constant
• y_A = mole fraction of A in gas phase

Raoult's Law: \[p_A = x_A P_A^{sat}\]

• P_A^sat = vapor pressure of pure A (Pa or psia)
• Applies to ideal solutions

Two-Resistance Theory

Overall flux with gas phase driving force: \[N_A = K_y a (y_A - y_A^*)\] \[N_A = K_G a (p_A - p_A^*)\] Overall flux with liquid phase driving force: \[N_A = K_x a (x_A^* - x_A)\] \[N_A = K_L a (C_A^* - C_A)\]

• a = interfacial area per unit volume (m²/m³ or ft²/ft³)
• y_A* = gas phase mole fraction in equilibrium with bulk liquid
• x_A* = liquid phase mole fraction in equilibrium with bulk gas

Absorption and Stripping

Operating Line for Absorption Column

Material balance: \[L(x_2 - x_1) = G(y_1 - y_2)\] Operating line equation: \[y = \frac{L}{G}x + \left(y_1 - \frac{L}{G}x_1\right)\]

• L = molar flow rate of liquid (mol/s or lbmol/h)
• G = molar flow rate of gas (mol/s or lbmol/h)
• x₁ = inlet liquid mole fraction (bottom)
• x₂ = outlet liquid mole fraction (top)
• y₁ = inlet gas mole fraction (bottom)
• y₂ = outlet gas mole fraction (top)

Minimum Liquid-to-Gas Ratio

\[\left(\frac{L}{G}\right)_{min} = \frac{y_1 - y_2}{x_2^* - x_1}\]

• x₂* = liquid mole fraction in equilibrium with y₁
• At minimum L/G, infinite stages are required

Height of Transfer Unit (HTU)

Gas phase HTU: \[H_{OG} = \frac{G}{K_G a S}\] \[H_G = \frac{G}{k_G a S}\] Liquid phase HTU: \[H_{OL} = \frac{L}{K_L a S}\] \[H_L = \frac{L}{k_L a S}\]

• H_OG = overall gas phase height of transfer unit (m or ft)
• H_G = gas phase height of transfer unit (m or ft)
• H_OL = overall liquid phase height of transfer unit (m or ft)
• H_L = liquid phase height of transfer unit (m or ft)
• S = column cross-sectional area (m² or ft²)

Number of Transfer Units (NTU)

Gas phase NTU (dilute systems): \[N_{OG} = \int_{y_2}^{y_1} \frac{dy}{y - y^*}\] For linear equilibrium (y* = mx + b) and constant L/G: \[N_{OG} = \frac{\ln\left[\left(\frac{y_1 - mx_1 - b}{y_2 - mx_1 - b}\right)\left(1 - \frac{mG}{L}\right) + \frac{mG}{L}\right]}{1 - \frac{mG}{L}}\] When mG/L = 1: \[N_{OG} = \frac{y_1 - y_2}{y - y^*}\] where (y - y*) is the average driving force. Liquid phase NTU: \[N_{OL} = \int_{x_1}^{x_2} \frac{dx}{x^* - x}\]

Column Height

\[Z = H_{OG} \times N_{OG}\] \[Z = H_{OL} \times N_{OL}\]

• Z = packed height of column (m or ft)

Relationship Between HTU Values

\[H_{OG} = H_G + \frac{mG}{L} H_L\] \[H_{OL} = H_L + \frac{L}{mG} H_G\]

Absorption Factor

\[A = \frac{L}{mG}\]

• A = absorption factor (dimensionless)
• A > 1.0 indicates favorable absorption
• A < 1.0="" indicates="" stripping="">

For stripping: \[S = \frac{mG}{L} = \frac{1}{A}\]

• S = stripping factor

Kremser Equation (Isothermal Absorption)

Number of theoretical stages for absorption: \[N = \frac{\ln\left[\left(\frac{y_1 - mx_1}{y_2 - mx_1}\right)\left(1 - \frac{1}{A}\right) + \frac{1}{A}\right]}{\ln A}\] When A = 1: \[N = \frac{y_1 - mx_1}{y_2 - mx_1} - 1\] For stripping: \[N = \frac{\ln\left[\left(\frac{x_2 - y_1/m}{x_1 - y_1/m}\right)\left(1 - S\right) + S\right]}{\ln S}\]

• N = number of theoretical stages
• Assumes constant L, G, m, and temperature

Distillation

Vapor-Liquid Equilibrium

Relative volatility: \[\alpha_{AB} = \frac{y_A/x_A}{y_B/x_B} = \frac{K_A}{K_B}\]

• α_AB = relative volatility of A to B
• K_A = y_A/x_A = equilibrium ratio for A
• K_B = y_B/x_B = equilibrium ratio for B

For binary system (A + B = 1): \[y_A = \frac{\alpha x_A}{1 + (\alpha - 1)x_A}\]

• Assumes constant relative volatility

McCabe-Thiele Method

Rectifying section operating line: \[y_{n+1} = \frac{R}{R+1} x_n + \frac{x_D}{R+1}\]

• R = reflux ratio (L_D/D)
• x_D = distillate composition (mole fraction)
• L_D = liquid flow in rectifying section
• D = distillate flow rate

Slope of rectifying line: \[slope = \frac{R}{R+1} = \frac{L_D}{V}\] y-intercept of rectifying line: \[y_{int} = \frac{x_D}{R+1}\] Stripping section operating line: \[y_m = \frac{L_S}{V_S} x_m - \frac{W x_W}{V_S}\]

• L_S = liquid flow in stripping section
• V_S = vapor flow in stripping section
• W = bottoms flow rate
• x_W = bottoms composition (mole fraction)

Alternative form: \[y_m = \frac{L_S}{L_S - W} x_m - \frac{W x_W}{L_S - W}\]

Minimum Reflux Ratio

At feed stage (for saturated liquid feed): \[R_{min} = \frac{x_D - y_f^*}{y_f^* - x_f}\]

• x_f = feed composition (liquid mole fraction)
• y_f* = vapor composition in equilibrium with x_f

Underwood equation for minimum reflux: \[R_{min} + 1 = \frac{\alpha_A x_{D,A}}{\alpha_A - \theta} + \frac{\alpha_B x_{D,B}}{\alpha_B - \theta}\] where θ is found from: \[\frac{\alpha_A x_{F,A}}{\alpha_A - \theta} + \frac{\alpha_B x_{F,B}}{\alpha_B - \theta} = 1 - q\]

• Valid for multicomponent systems
• q = thermal condition of feed

Feed Line (q-Line)

Feed line equation: \[y = \frac{q}{q-1} x - \frac{x_f}{q-1}\] Slope of feed line: \[slope = \frac{q}{q-1}\] Thermal condition parameter: \[q = \frac{H_V - H_F}{H_V - H_L}\]

• H_V = enthalpy of saturated vapor
• H_F = enthalpy of feed
• H_L = enthalpy of saturated liquid

Values of q:

• q = 1: saturated liquid feed (slope = ∞, vertical line)
• q = 0: saturated vapor feed (slope = 0, horizontal line)
• q > 1: subcooled liquid feed
• 0 < q="">< 1:="" partially="" vaporized="">
• q < 0:="" superheated="" vapor="">

Total Reflux

Fenske equation (minimum number of stages): \[N_{min} = \frac{\ln\left[\frac{x_D}{1-x_D} \cdot \frac{1-x_W}{x_W}\right]}{\ln \alpha}\]

• N_min = minimum number of theoretical stages at total reflux
• Does not include reboiler
• Assumes constant relative volatility

Actual Number of Stages

Gilliland correlation: \[Y = 1 - \exp\left[\frac{(1 + 54.4X)(X - 1)}{11 + 117.2X}\right]\] where: \[X = \frac{R - R_{min}}{R + 1}\] \[Y = \frac{N - N_{min}}{N + 1}\]

• N = actual number of theoretical stages
• R = actual reflux ratio

Column Efficiency

Overall column efficiency: \[E_O = \frac{N}{N_{actual}}\]

• N = number of theoretical stages
• N_actual = actual number of trays

Murphree tray efficiency (vapor basis): \[E_{MV} = \frac{y_{n+1} - y_n}{y_{n+1}^* - y_n}\]

• y_n+1 = actual vapor composition leaving stage n
• y_n = actual vapor composition entering stage n
• y_n+1* = equilibrium vapor composition with liquid leaving stage n

Murphree tray efficiency (liquid basis): \[E_{ML} = \frac{x_n - x_{n-1}}{x_n^* - x_{n-1}}\]

Material Balance

Overall material balance: \[F = D + W\] Component material balance: \[F x_F = D x_D + W x_W\]

• F = feed flow rate (mol/s or lbmol/h)
• D = distillate flow rate (mol/s or lbmol/h)
• W = bottoms flow rate (mol/s or lbmol/h)

Extraction (Liquid-Liquid)

Distribution Coefficient

\[K_D = \frac{C_{A,extract}}{C_{A,raffinate}}\]

• K_D = distribution coefficient
• C_A,extract = concentration of A in extract phase
• C_A,raffinate = concentration of A in raffinate phase

Single-Stage Extraction

Fraction extracted: \[E = \frac{V K_D}{V K_D + F}\]

• E = extraction efficiency (fraction of solute extracted)
• V = volume of extract phase (solvent)
• F = volume of raffinate phase (feed)

Multistage Crosscurrent Extraction

Fraction remaining after n stages: \[x_n = x_0 \left(\frac{F}{V K_D + F}\right)^n\]

• x_n = concentration remaining after n stages
• x₀ = initial concentration
• n = number of stages

Multistage Countercurrent Extraction

Operating line: \[y_{n+1} = \frac{F}{S} x_n + \left(y_1 - \frac{F}{S} x_0\right)\]

• F = raffinate flow rate
• S = solvent (extract) flow rate
• y = solute concentration in extract phase
• x = solute concentration in raffinate phase

Minimum solvent-to-feed ratio: \[\left(\frac{S}{F}\right)_{min} = \frac{x_0 - x_N}{y_1^* - y_0}\]

• y₁* = extract composition in equilibrium with x₀
• Requires infinite stages

Adsorption

Adsorption Isotherms

Langmuir isotherm: \[q = \frac{q_m b C}{1 + b C}\]

• q = amount adsorbed per unit mass of adsorbent (mol/kg or lb/lb)
• q_m = monolayer capacity
• b = Langmuir constant (related to adsorption energy)
• C = equilibrium concentration in fluid phase
• Assumes monolayer coverage and uniform surface

Freundlich isotherm: \[q = K_F C^{1/n}\]

• K_F = Freundlich capacity factor
• n = Freundlich intensity parameter (typically n > 1)
• Empirical equation

Linear isotherm: \[q = K C\]

• K = partition coefficient
• Valid at low concentrations

BET isotherm (multilayer adsorption): \[\frac{p}{V(p_0 - p)} = \frac{1}{V_m C_{BET}} + \frac{(C_{BET} - 1)p}{V_m C_{BET} p_0}\]

• V = volume of gas adsorbed at pressure p
• V_m = volume for monolayer coverage
• p₀ = saturation pressure
• C_BET = BET constant (related to heat of adsorption)

Breakthrough Curve Analysis

Mass transfer zone (MTZ) length: \[L_{MTZ} = v_0 (t_b - t_s)\]

• L_MTZ = length of mass transfer zone
• v₀ = superficial velocity
• t_b = breakthrough time
• t_s = stoichiometric (equilibrium) time

Column capacity: \[q_{total} = \int_0^{t_{sat}} C_0 Q (1 - \frac{C}{C_0}) dt\]

• q_total = total amount adsorbed
• C₀ = inlet concentration
• C = outlet concentration
• Q = volumetric flow rate
• t_sat = saturation time

Crystallization

Supersaturation

Absolute supersaturation: \[\Delta C = C - C^*\] Relative supersaturation: \[S = \frac{C - C^*}{C^*}\] Supersaturation ratio: \[\sigma = \frac{C}{C^*}\]

• C = actual concentration
• C* = saturation (equilibrium) concentration

Crystal Growth Rate

Overall growth rate: \[G = \frac{dL}{dt}\]

• G = linear growth rate (m/s or μm/h)
• L = characteristic crystal size

Mass transfer controlled growth: \[G = k_d (C - C^*)\]

• k_d = mass transfer coefficient

Nucleation Rate

Primary homogeneous nucleation: \[B = k_n \exp\left(-\frac{\Delta G_{crit}}{kT}\right)\] Secondary nucleation: \[B = k_n M_T^j N^b (C - C^*)^i\]

• B = nucleation rate (number/m³·s)
• M_T = magma density (mass of crystals per volume of slurry)
• N = agitation rate
• j, b, i = empirical exponents

Population Balance

For steady-state MSMPR (Mixed Suspension Mixed Product Removal) crystallizer: \[n = n_0 \exp\left(-\frac{L}{G\tau}\right)\]

• n = population density at size L (number/m⁴)
• n₀ = population density of nuclei (L = 0)
• τ = residence time

Relationship between nucleation and growth: \[n_0 = \frac{B^0}{G}\]

• B⁰ = nucleation rate at zero size

Crystal Size Distribution

Dominant crystal size (MSMPR): \[L_{dominant} = 3G\tau\] Mass-average size: \[\overline{L}_m = 4G\tau\]

Yield

Theoretical yield (cooling crystallization): \[Y = F(x_F - x_M)\]

• Y = mass of crystals produced
• F = feed mass
• x_F = feed concentration (mass fraction)
• x_M = mother liquor concentration at final temperature

Humidification and Drying

Psychrometric Properties

Humidity (absolute humidity): \[H = \frac{mass_{water}}{mass_{dry air}} = \frac{M_{H_2O}}{M_{air}} \frac{p_{H_2O}}{P - p_{H_2O}}\] \[H = 0.622 \frac{p_{H_2O}}{P - p_{H_2O}}\]

• H = absolute humidity (kg H₂O/kg dry air or lb H₂O/lb dry air)
• p_H₂O = partial pressure of water vapor (Pa or psia)
• P = total pressure (Pa or psia)
• M_H₂O = 18 g/mol
• M_air = 29 g/mol

Saturation humidity: \[H_s = 0.622 \frac{p_{H_2O}^{sat}}{P - p_{H_2O}^{sat}}\]

• H_s = saturation humidity at given temperature
• p_H₂O^sat = saturation vapor pressure at temperature T

Relative humidity: \[RH = \frac{p_{H_2O}}{p_{H_2O}^{sat}} \times 100\%\] \[RH = \frac{H(P - p_{H_2O}^{sat})}{H_s(P - p_{H_2O})} \times 100\%\] Percentage humidity: \[H_{pct} = \frac{H}{H_s} \times 100\%\] Humid volume: \[v_H = \left(\frac{1}{M_{air}} + \frac{H}{M_{H_2O}}\right)\frac{RT}{P}\] \[v_H = \left(0.0345 + 0.0568 H\right)\frac{T}{P}\]

• v_H = humid volume (m³/kg dry air or ft³/lb dry air)
• T = absolute temperature (K or °R)
• Second equation: T in K, P in atm, v_H in m³/kg

Humid heat: \[c_s = c_{p,air} + H c_{p,H_2O}\] \[c_s = 1.005 + 1.88 H\]

• c_s = humid heat (kJ/kg dry air·K or Btu/lb dry air·°F)
• c_p,air = 1.005 kJ/kg·K (0.24 Btu/lb·°F)
• c_p,H₂O = 1.88 kJ/kg·K (0.45 Btu/lb·°F)

Adiabatic Saturation and Wet Bulb Temperature

Adiabatic saturation equation: \[H_s - H = \frac{c_s (T - T_{as})}{\lambda_{as}}\]

• T_as = adiabatic saturation temperature
• λ_as = latent heat of vaporization at T_as

For air-water system (Lewis number ≈ 1): \[T_{wb} \approx T_{as}\]

• T_wb = wet bulb temperature
• Wet bulb and adiabatic saturation temperatures are approximately equal for air-water

Dew point temperature:

• Temperature at which p_H₂O^sat = p_H₂O
• Air becomes saturated if cooled to dew point

Enthalpy of Humid Air

\[H_{humid} = c_s T + H \lambda_0\] \[H_{humid} = 1.005 T + H(2501 + 1.88 T)\]

• H_humid = enthalpy of humid air (kJ/kg dry air or Btu/lb dry air)
• λ₀ = latent heat at reference temperature (0°C or 32°F)
• Second equation: T in °C, H_humid in kJ/kg dry air

Drying Rate

Constant rate period: \[R_c = k_y (H_s - H_\infty)\]

• R_c = constant drying rate (kg/m²·s or lb/ft²·h)
• k_y = mass transfer coefficient
• H_s = saturation humidity at surface temperature
• H_∞ = bulk air humidity

Falling rate period (linear): \[R = R_c \frac{X - X_e}{X_c - X_e}\]

• X = free moisture content (kg H₂O/kg dry solid)
• X_c = critical moisture content
• X_e = equilibrium moisture content

Drying time (constant rate period): \[t_c = \frac{L_s (X_0 - X_c)}{R_c}\]

• L_s = mass of dry solid per unit area (kg/m² or lb/ft²)
• X₀ = initial moisture content

Drying time (falling rate period): \[t_f = \frac{L_s (X_c - X_e)}{R_c} \ln\left(\frac{X_c - X_e}{X_f - X_e}\right)\]

• X_f = final moisture content
• Valid for linear falling rate period

Membrane Separation

Permeability and Flux

Gas permeation flux: \[J_i = \frac{P_i}{l}(p_{i,feed} - p_{i,permeate})\]

• J_i = flux of component i (mol/m²·s or cm³(STP)/cm²·s)
• P_i = permeability of component i (mol·m/m²·s·Pa or Barrer)
• l = membrane thickness (m or cm)
• p_i = partial pressure (Pa or atm)

Permeance: \[Q_i = \frac{P_i}{l}\]

• Q_i = permeance (mol/m²·s·Pa or GPU)
• 1 GPU = 10⁻⁶ cm³(STP)/cm²·s·cmHg

Selectivity (ideal separation factor): \[\alpha_{ij} = \frac{P_i}{P_j}\]

• α_ij = ideal selectivity for i over j

Reverse Osmosis

Water flux: \[J_w = A(\Delta P - \Delta \pi)\]

• J_w = water flux (m³/m²·s or gal/ft²·day)
• A = water permeability coefficient (m³/m²·s·Pa or gal/ft²·day·psi)
• ΔP = transmembrane pressure difference (Pa or psi)
• Δπ = osmotic pressure difference (Pa or psi)

Salt flux: \[J_s = B(\Delta C)\]

• J_s = salt flux (kg/m²·s)
• B = salt permeability coefficient (m/s)
• ΔC = concentration difference (kg/m³)

Osmotic pressure (van't Hoff equation): \[\pi = i M R T\]

• π = osmotic pressure (Pa or psi)
• i = van't Hoff factor (number of ions per molecule)
• M = molar concentration (mol/L or mol/m³)

Ultrafiltration and Microfiltration

Flux through membrane: \[J = \frac{\Delta P}{\mu (R_m + R_f)}\]

• J = permeate flux (m/s or gal/ft²·day)
• μ = viscosity (Pa·s or cP)
• R_m = membrane resistance (1/m)
• R_f = fouling resistance (1/m)

Concentration polarization: \[\frac{C_w}{C_b} = \exp\left(\frac{J}{k}\right)\]

• C_w = concentration at membrane wall
• C_b = bulk concentration
• k = mass transfer coefficient (m/s)

Leaching (Solid-Liquid Extraction)

Single-Stage Leaching

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

• F = feed solids flow rate
• S = solvent flow rate
• E = extract (overflow) flow rate
• R = raffinate (underflow) flow rate
• x, y = mass fractions of solute

Multistage Countercurrent Leaching

Operating line (similar to extraction): \[y_{n+1} = \frac{R}{E} x_n + \left(y_1 - \frac{R}{E} x_0\right)\] Number of ideal stages:

• Use graphical methods (equilibrium stages on x-y diagram)
• Similar approach to absorption/extraction

Ion Exchange

Exchange Capacity

Exchange capacity: \[Q = \frac{mol_{exchanged}}{kg_{resin}}\]

• Q = exchange capacity (eq/kg or meq/g)
• Typically expressed as milliequivalents per gram

Breakthrough

Stoichiometric capacity: \[V_s = \frac{Q \cdot W}{C_0}\]

• V_s = stoichiometric volume (L or gal)
• W = mass of resin (kg or lb)
• C₀ = influent concentration (eq/L or meq/mL)

Usable capacity: \[V_b = V_s - V_{MTZ}/2\]

• V_b = volume at breakthrough
• V_MTZ = volume of mass transfer zone

Evaporation

Single Effect Evaporation

Material balance: \[F = L + V\] \[F x_F = L x_L\]

• F = feed rate
• L = liquid (concentrate) rate
• V = vapor rate
• x = solute mass fraction

Energy balance: \[F h_F + Q = L h_L + V H_V\]

• h_F = specific enthalpy of feed
• h_L = specific enthalpy of liquid product
• H_V = specific enthalpy of vapor
• Q = heat input

Economy (steam economy): \[Economy = \frac{V}{S}\]

• V = mass of water evaporated
• S = mass of steam used

Multiple Effect Evaporation

Overall economy: \[Economy_{overall} = \frac{\sum V_i}{S_1}\]

• V_i = vapor from effect i
• S₁ = steam to first effect
• Economy increases approximately linearly with number of effects

Boiling point rise (BPR): \[BPR = T_{solution} - T_{pure water}\]

• Due to concentration and hydrostatic head
• Dühring's rule: plots used to estimate BPR

Analogies Between Heat and Mass Transfer

Reynolds Analogy

\[\frac{k_c}{v} = \frac{h}{\rho c_p v}\] \[Sh = Nu\]

• Valid for Pr = Sc = 1
• k_c = mass transfer coefficient (m/s)
• v = velocity (m/s)
• h = heat transfer coefficient (W/m²·K)

Chilton-Colburn Analogy

\[j_D = j_H = \frac{f}{2}\] \[j_D = \frac{Sh}{Re \cdot Sc^{1/3}} = St_D \cdot Sc^{2/3}\] \[j_H = \frac{Nu}{Re \cdot Pr^{1/3}} = St \cdot Pr^{2/3}\]

• j_D = Chilton-Colburn j-factor for mass transfer
• j_H = Chilton-Colburn j-factor for heat transfer
• f = Fanning friction factor
• St_D = Stanton number for mass transfer = k_c/v
• St = Stanton number for heat transfer = h/(ρc_pv)
• Valid for 0.6 < sc,="" pr=""><>

Lewis Number

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

• Le = Lewis number
• α = thermal diffusivity (m²/s)
• For air-water, Le ≈ 1