Nomenclature

interfacial area per volume, ft²/ft³ or m²/m³

a^l_j

interfacial area for heat transfer on stage j, m²

a_flow, a_heat, a_mass

eddy diffusion parameters, Eqs. (15-61)

a_p

surface area/volume, m²/m³

a_p1, a_p2, a_p3, a_T1,

constants in Eq. (2-28) and Table 2-3

a_T2, a_T6

A

area, m²

A, B, C

constants in Antoine Eq. (2-34)

A, B, C, D, E

constants in Eq. (2-62a)

A, B, C, D

constants in matrix form of mass balances, Eqs. (6-13) and (12-58)

A_E, B_E, C_E, D_E

constants in matrix form of energy balances, Eq. (6-28)

A_active

active area of tray, ft² or m²

A_c

cross-sectional area of column, ft² or m²

A_d

downcomer area, ft² or m²

A_du

flow area under downcomer apron, Eq. (10-28), ft²

A_f

area for flow, m²

A_hole

area of holes in sieve plate, ft²

A_I

interfacial area between two phases, ft² or m²

A_mixer

cross-sectional area of mixer, m²

A_mt

area for mass transfer, m²

A_net

net area, Eq. (10-13), ft² or m²

A_total

total area in horizontal drum, ft² or m²

A_vap

area for vapor flow in horizontal drum, ft² or m²

b

equilibrium constant for linear equilibrium, y = mx + b

b_flow, b_heat, b_mass

eddy diffusion parameters, Eqs. (15-61) and (15-62)

B

bottoms flow rate, kmol/h or lbmol/h

C

number of components

C_BM

bare module cost (Chapter 11)

C_C

concentration of solute in continuous phase, kmol/m³ continuous phase

C*_C

concentration of solute in continuous phase in equilibrium with C_D, kmol/m³

C_D

concentration of solute in dispersed phase, kmol/m³ dispersed phase

C_fL

vapor load coefficient, Eq. (16-38) and Figure 16-7

C_A, C_B, C_m

molar concentrations, of A, B, and mixture, mol/m³

C_o

orifice coefficient, Eqs. (10-24) and (10-25)

C_p

heat capacity, Btu/(lb · °F), Btu/(lbmol · °F), J/(g · °C), J/(mol · °C), etc.

C_p

base purchase cost (Chapter 11)

C_p,W

water heat capacity

C_p,size

packing size factor, Table 10-5

C_s

capacity factor at flood, Eq. (10-48)

C_sb,f

capacity factor at flood, Eq. (10-8) and Figure 10-16

d_hydraulic

hydraulic diameter of drop, Eq. (13-58), m

d_i

impeller diameter, m

d_p, d_d

drop diameter, m

characteristic drop diameter, Eq. (16-97b), m

d_settler

diameter of horizontal settler, m

d_tube

tube diameter, m

df

damping factor

D

diffusivity, Fickan, m²/s or ft²/h

D°

infinite dilution Fickian diffusivity, m²/s

D

diffusivity, Maxwell-Stefan, m²/s or ft²/h

D°

infinite dilution Maxwell-Stefan diffusivity, m²/s or ft²/h

D

distillate flow rate, kmol/h or kg/h

D

diameter of sphere, m (Chapter 15)

D, Dia

diameter of column, ft or m

D′_col

column diameter, see Table 16-2, ft

D_eddy

eddy diffusivity, Eqs. (16-111a) and (16-111b), m²/s

D_total

total amount of distillate (Chapter 9), moles or kg

e

absolute entrainment, mol/h

E

extract flow rate (Chapters 13 and 14), kg/h

Ê

mass extract in batch extraction, kg

E^V_j

energy transfer rate on stage j from bulk liquid to bulk vapor, J/s, Eq. (16-106b)

E_k

value of energy function for trial k, Eq. (2-50)

E_ML, E_MV

Murphree liquid and vapor efficiencies, Eqs. (4-58) and (4-59)

E₀

activation energy, Kcal/mol

E_o

Overall efficiency, Eqs. (4-56) and (10-1)

E_pt

point efficiency, Eq. (10-5) or (16-76a)

Ê_t

holdup extract phase in tank plus settler, kg

f

friction factor

f_AB

friction coefficient between molecules A and B

f = V/F

fraction vaporized

f

fractional approach to flooding

f(x)

equilibrium function (Chapter 9)

f_k(V/F)

Rachford-Rice function for trial K, Eqs. (2-41) to (2-44)

F

packing factor, Tables 10-3 and 10-4

F

degrees of freedom, Eq. (2-4)

F

charge to still pot (Chapter 9), moles or kg

mass of feed in batch extraction, kg

F

feed flow rate, kmol/h, lbmol/h, kg/h, etc.

F_D

diluent flow rate (Chapter 13), kg/h

F_lv, FP

, flow parameter

F_m

material factor for cost, Table 11-4

F_p

pressure factor for cost, Eq. (11-7) and Table 11-2

F_q

quantity factor for cost, Eq. (11-9) and Table 11-2

F_s,F_solv

flow rate solvent (Chapters 13 and 14), kg solvent/h

F_solid

solids flow rate in leaching (Chapter 14), kg insoluble solid/h

F_weir

weir modification factor, Eq. (10-26) and Figure 10-22

FR_A,dist

fractional recovery of A in distillate

FR_B,bot

fractional recovery of B in bottoms

gap

gap from downcomer apron to tray, Eq. (10-28), ft

g

acceleration due to gravity, 32.2 ft/s², 9.81 m/s²

g_c

conversion factor in English units, 32.2 ft · lbm/(lbf · s²)

G

flow rate carrier gas, kmol/h or kg/h

G′

gas flux, lb/s ft²

h

pressure drop in head of clear liquid, in. liquid

h

height of liquid on stage (Chapter 16), ft

h

height, m or ft

h

height of liquid in mixer, m

h

liquid enthalpy, kcal/kg, Btu/lbmol, etc.

h

step size in Euler’s method = Δt, Eqs. (8-29) and (8-30)

pure component enthalpy, kcal/kg, Btu/lbmol, etc.

h_f

enthalpy of liquid leaving feed stage, kcal/kg, Btu/lbmol, etc.

h_F

feed enthalpy (liquid, vapor, or two phase), kcal/kg, Btu/lbmol, etc.

h_{heat transfer}

heat transfer coefficient, kW/(s·m²), BTU/(h·ft²)

h_L

clear liquid height on stage, m or cm

h_o

hole diameter for sieve plates, in.

h_p

packing height, ft or m

h_total

height of flash drum, ft or m

h_w

height of weir, m, cm, ft, or in.

H, H_B

volatility form of Henry’s law constant, Eqs. (8-10), (8-11), and (12-1a)

H_sol

solubility form of Henry’s law constant, Eq. (12-4)

H

molar holdup of liquid on tray, Eqs. (8-27) and (8-28)

H

vapor enthalpy, kcal/kg, Btu/lbmol, etc.

H^V_i,j

partial molar enthalpy of component i in vapor on stage j, J/kmol, BTU/lbmol

height of tank, m

H_t,OD

overall height of a transfer unit for mass transfer driving force in concentration units, Eq. (16-83a analog), ft or m

H_G

height of gas-phase transfer unit, ft or m

H_L

height of liquid-phase transfer unit, ft or m

H_OG

height of overall gas-phase transfer unit, ft or m

H_OL

height of overall liquid-phase transfer unit, ft or m

HETP

height equivalent to a theoretical plate, ft or m

HTU

height of a transfer unit, ft or m

j_D, j_H

j-functions, Eqs. (15-63b) and following material

J_A,z

molar flux in direction with respect to molar average velocity of fluid, mole A/(s · m²)

k_B

Botzmann’s constant, J/k (see Appendix C)

k_conduction

thermal conductivity, J/(m · s · K)

individual mass transfer coefficients in liquid and vapor phases in mass units, see Table 16-1

k_c

mass transfer coefficient with concentration driving force, m/s, Eq. (15-24b), Table 16-1

k′_y

mass transfer coefficient in concentrated solutions, Eq. (15-32f)

k_x, k_y

individual mass transfer coefficient in molar units

k_x,c, k_xD

individual mass transfer coefficients in continuous and dispersed phases, kg/(s · m³), lbm/(s · ft³), kmol/(s · m³), or lbmol/(s · ft³)

k_LD, k_LC

individual mass transfer coefficients in continuous and dispersed phases with driving force in concentration units, m/s

k_L, k_V

individual liquid and vapor mass transfer coefficients in distillation, Eq. (16-108), m/s

k

mass transfer coefficient in Maxwell-Stefan analysis, D/Δz, m/s

K_d

y/x, distribution coefficient for dilute extraction

K, K_i

y_i/x_i, equilibrium vapor-liquid ratio

K_drum

parameter to calculate u_perm for flash drums, Eqs. (2-61) and (2-62a)

K_x, K_y

overall mass transfer coefficient in liquid or vapor, lbmol/(ft²·h), or kmol/(h·m²)

K_LD

overall mass transfer coefficient in extraction based on dispersed phase in concentration units, Eq. (16-80b analog), m/s

K_O–ED

overall mass transfer coefficient in extraction based on dispersed phase, Eq. (16-80a), kg/(s · m³) or kmol/(s · m³)

l_w

weir length, ft or m

L

length, ft or m

L

liquid flow rate, kmol/h or lbmol/h

liquid flow rate in bottom section of column, kmol/h or lbmol/h

mass liquid flow rate, lb/h (Chapter 15)

L′

liquid flux, lb/(s·ft²)

L_g

liquid flow rate in gal/min (Chapter 10)

m

linear equilibrium constant, y = mx + b

m

local slope of equilibrium curve, Eq. (15-30b)

m_CD

slope of equilibrium curve of continuous- versus dispersed-phase mass or mole fractions, Eq. (16-80c)

m_{CD,conc_units}

slope of equilibrium curve of continuous versus dispersed phase in concentration units, Eq. (16-80c analog)

M

flow rate of mixed stream (Chapter 13), kg/h

M

multiplier times (L/D)_min (Chapter 7)

M

ratio HETP_practical/HETP_packing, Eq. (10-47a)

MW

molecular weight

average molecular weight

n

moles

n

number of drops

n_G

number of gas-phase transfer units

n_L

number of liquid-phase transfer units

n_O–ED, n_O–EC

number of overall extraction transfer units in dispersed and continuous phases, Eq. (16-81)

n_OG

number of overall gas-phase transfer units

n_OL

number of overall liquid-phase transfer units

n_org

moles organic in vapor in steam distillation

n_w

moles water in vapor in steam distillation

N

impeller revolutions per second

N

number of stages

N_A

molar flux of A, lbmol/(h·ft²) or kmol/(h·m²)

N_f, N_feed

feed stage

N^L_f,j

transfer to liquid from vapor on stage j, mol component i/s

N^V_f,j

transfer to vapor from liquid on stage j, mol component i/s

N_min

number of stages at total reflux

N_feed,min

estimated feed stage location at total reflux

N_Po

power number, Eq. (13-47)

N_tOD

number of overall extraction transfer units for mass transfer driving force in concentration units, Eq. (16-81a analog)

Nu

Nusselt number, Eq. (15-46g)

NTU

number of transfer units

O

total overflow rate in washing, kg/h

p

pitch of sieve plate holes, m

p, p_tot

pressure, atm, kPa, psi, bar, etc.

partial pressure

P

number of phases, Eq. (2-4)

P

power, W

Pe

dimensionless Peclet number in terms of molecular diffusivity, Eq. (15-46c)

Pe

dimensionless Peclet number in terms of eddy diffusivity, Eq. (16-111a)

Per_f

flow perimeter, Figure 13-33B, m

Pr

dimensionless Prandt number, Eq. (15-46f)

q

L_F/F = ( – L)/F, feed quality

q

volumetric flow rate/plate width, m²/s

Q

amount of energy transferred, Btu/h, kcal/h, J/s, etc.

Q_c

condenser heat load, Btu/h, kcal/h, J/s, etc.

Q_c, Q_C

volumetric flow rate continuous phase, m³/s

Q_d, Q_D

volumetric flow rate dispersed phase, m³/s

Q_flash

heat loss from flash drum, Btu/h, kcal/h, J/s, etc.

Q_L

volumetric flow rate of liquid, m³/s

Q_R

reboiler heat load, Btu/h, kcal/h, J/s, etc.

Q_z/A

heat flux in z direction, J/(m²·s)

r

radius of column, ft or m

R

gas constant, 1.9859 cal/(mol · K) or 8.314 m³Pa/(mol · K), Appendix C

R

raffinate flow rate (Chapters 13 and 14), kg/h

R_A

solute radius, m

mass raffinate in batch extraction, kg

Holdup raffinate phase in tank plus settler, kg

Re

dimensionless Reynolds number, Eq. (15-46b)

Re_settler

Reynold’s number for settler, Eq. (13-56a)

S

solvent flow rate kmol/h or lbmol/h

S

tray spacing, in., Eq. (10-47b)

S

moles second solvent in constant-level batch distillation

mass of solvent in batch extraction, kg

S

solvent flow rate, kg/h

S_L, S_V

flow rates of liquid and vapor sidestreams, kmol/h

Sc_L, Sc_v

Schmidt number for liquid or vapor = µ/(ρD)

Sh_c, Sh_x, Sh_y

dimensionless Sherwood numbers, Eq. (15-46a)

St_c, St_x, St_y

dimensionless Stanton numbers, Eq. (15-46d)

t

time, s, min, or h

t_batch

period for batch distillation, Eq. (9-29a)

t_down

down time in batch distillation

(t_f,95 – t₀)

residence time in extractor for 95.0% extraction, Eq. (16-104), s

t_L, t_V

average residence time per pass for liquid and vapor, s

liquid residence time, Eq. (16-111c), s

t_{residence,dispersed}

residence time of dispersed phase in settler, s

t_operating

operating time in batch distillation

t_res

residence time in downcomer, Eq. (10-30), s, or on plate

t_tray

tray thickness, in.

T

temperature, °C, °F, K, or °R

T_j^IL, T_j^IV

liquid and vapor temperatures on stage j at the interface, °C, °F, K, or °R

T_ref

reference temperature, °C, °F, K, or °R

u

vapor velocity, cm/s or ft/s

u_flood

flooding velocity, cm/s or ft/s, Eq. (10-8)

u_op

operating velocity, cm/s or ft/s, Eq. (10-11)

u_perm

permissible vapor velocity, cm/s or ft/s, Eq. (2-61)

u_t,hindered

hindered settling velocity, cm/s or ft/s, Eq. (13-54)

u_t, u_t,Stokes

Stokes’ law terminal velocity, cm/s or ft/s, Eq. (13-53)

U

underflow liquid rate (Chapter 14), kg/h

U

overall heat transfer coefficient, kW/(s·m²), BTU/(h·ft²)

U_a

superficial vapor velocity in active area of tray, m/s

v

superficial vapor velocity, ft/s

v_c,flood, v_d,flood

continuous- and dispersed-phase flooding velocities, m/s

v_o

vapor velocity through holes, Eq. (10-24), ft/s

v_o,bal

velocity where valve is balanced, Eq. (10-36)

v_A, v_B

component transfer velocities, m/s, Eqs. (15-15e) and (15-15f)

v_ref

reference or basis velocity, m/s, Eqs. (15-15c) and (15-15d)

v_y

vertical velocity, m/s

V

vapor flow rate, kmol/h or lbmol/h

V_i

molal volume, m³/mol, Eq. (13-1a)

V_A

molar volume solute at normal boiling point, m³/kmol

V_liq,tank

volume of liquid in tank, m³

V_max

maximum vapor flow rate

V_mixer

volume of liquid in mixing tank, m³

V_settler

volume settler, m³

V_tank

volume tank, m³

V_surge

surge volume in flash drum, Eq. (2-65), ft³

VP

vapor pressure, same units as p

w

plate width, m

W_L

liquid flow rate, kg/h or lb/h

W_L

liquid mass flux, lb/s ft² or lb/h ft² (Chapter 16)

W_V

vapor flow rate, kg/h or lb/h

x

weight or mole fraction in liquid

x

weight fraction in underflow in washing (Chapter 14)

x

[L/D – (L/D)_min]/(L/D + 1.0) in Eqs. (7-37)

x*

equilibrium mole fraction in liquid

x_i,k, x_i,k+1

trials for integration, Eq. (8-29)

x_I

interfacial mole fraction in liquid

x*_out

liquid mole fraction in equilibrium with inlet gas, Eq. (16-35b)

X

weight or mole ratio in liquid

X

kg solute/kg insoluble solid (Chapter 14)

y

weight or mole fraction in vapor

y

weight fraction in overflow in washing (Chapter 14)

y_A,ref, y_B,ref

fractions to calculate velocity of center of total flux, Eq. (15-17)

y_vol

volume fraction in vapor

y*

equilibrium mole fraction in vapor

y*_out

vapor mole fraction in equilibrium with inlet liquid in countercurrent system, Eq. (16-35a), or in equilibrium with outlet liquid in cocurrent contactor, Eq. (16-71)

y_lm

log mean difference, Eq. (15-32d)

y_I

interfacial mole fraction in vapor

mass fraction in vapor

Y

weight or mole ratio in vapor

Y

kg solute/kg solvent (Chapters 13 and 14)

z

weight or mole fraction in feed

z

axial distance in bed (Chapters 15 and 16)

z_l

distance from downcomer exit to weir, m

Z

(lb solution)/(lb oil-free solids) in underflow (Chapter 14)

K_A/K_B, relative volatility

α_thermal

thermal diffusity, m²/s

β

A_hole/A_active

γ

activity coefficient

δ

thickness of mass transfer film or thickness of falling film, m

δ_p

characteristic dimension of packing, in., Eq. (10-38)

δ_i

solubility parameter, Eq. (13-1a)

Δ

change in variable or difference operator

ΔE_v

latent energy of vaporization, Eq. (13-1a)

ΔH

steady-state height of dispersion band in settler, m

Δρ

|ρ_C – ρ_D|

ε

limit for convergence

ε_A, ε_B, ε_AB

Lennard-Jones interaction energies, Table 15-2 and Eq. (15-21d)

η

fraction of column available for vapor flow

η

parameter in series solution for liquid falling-film mass transfer, Eq. (15-55b)

θ

angle of downcomer, Figure 10-20B

λ

latent heat of vaporization, kcal/kg, Btu/lb, Btu/lbmol, etc.

µ

viscosity, cp or Pa·s = kg/(m·s)

µ_w

viscosity of water, cp

ρ_L

liquid density, g/cm³, lb/ft³, or kg/m³

ρ_V

vapor density, g/cm³, lb/ft³, or kg/m³

ρ_C, ρ_D

densities of continuous and dispersed phases, g/m³

ρ_m

mixture density, Eq. (13-48), g/m³

ρ_m

molar density, mol/m³

σ, γ

surface tension, dynes/cm or interfacial tension

ς

dimensionless distance, Eq. (15-14a)

χ

term defined in Eq. (13-45)

φ_c, φ_d

volumetric fraction of continuous and dispersed phases

φ_d,feed

volumetric fraction of dispersed phase in feed

ϕ

liquid-phase packing parameter, Eq. (16-38) and Figure 16-6

ϕ_B

solvent interaction parameter, Eq. (15-22b)

ϕ_dc

relative froth density in downcomer, Eq. (10-29)

ψ

ρ_water/ρ_L (Chapter 10)

ψ

e/(e + L), fractional entrainment, Figure 10-17

ψ

packing parameter for gas phase, Eq. (16-37) and Figure 16-5

Ω_D

collision integral, Table 15-2

Ω_V

collision integral for viscosity, Table 15-2

µ_C, µ_D

viscosity of continuous and dispersed phases, Pa · s

µ_H, µ_L

viscosity of heavy and light phases, Pa · s

µ_M

mixture viscosity, Eq. (13-49),

ω

revolutions per second

empirical constants in Eq. (17-49)

A

kg/h anhydrous crystals

A

cumulative area/volume of crystals, m²/m³

A

crystal surface area, m²

A_{heat exchanger}

area of heat exchanger, m²

A_T

total area/volume of crystals, m²/m³

B(L, t)

birth function, Eq. (17-15)

B^o

rate of formation of nuclei, #/(s·m³), Eqs. (17-18a) and (17-18b)

c

concentration, kg/m³

c*

equilibrium concentration of solute, kg/m³

c_I

interfacial concentration of solute, kg/m³

C

concentration, kg anhydrous crystals/kg water

C*

equilibrium concentration, kg anhydrous crystals/kg water

Cry

flow rate crystals, kg/h

D(L, t)

death function, Eq. (17-15)

F

feed rate, kg/h

F_W

liquid water flow rate, kg/h

G

linear growth rate, m/s

h

liquid enthalpy, kJ/kg

h_crystal

crystal enthalpy, kJ/kg

h_magma

enthalpy of magma, kJ/kg

h_M

enthalpy of mixing point, kJ/kg

H_V

vapor enthalpy, kJ/kg

i

order of nucleation

j

empirical exponent, Eq. (17-18a)

k

empirical exponent, Eq. (17-18a)

k_A

area shape factor

k_f

film mass transfer coefficient, m/s

k_N

empirical rate constant for nucleation, Eq. (17-18a)

k_r

rate constant

k_V

volume shape factor

K_salt

empirical constant, Eq. (17-52)

K_G

overall mass transfer coefficient, m/s

K_S

solubility product, Eq. (17-50a)

L

liquid flow rate, kg/h

L

characteristic dimension of crystal, m

L

cumulative length per volume, m/m³

L_p

size product crystals, m

L_s

size seed crystals, m

L_T

total length of particles/volume, m/m³

m

mass of a crystal, kg

m

valence of cation, Eq. (17-50a)

M

mixed stream amount or flow rate, kg or kg/h

M

cumulative mass of crystals/volume, kg/m³

M_T

magma density, weight of crystals in solution per volume, kg/m³

MW

molecular weight

n

population density, #/(m·m³), Eq. (17-12)

n

empirical exponent, Eq. (17-18a)

n

overall growth rate order

n

valence of anion, Eq. (17-50a)

n

moles water per mole hydrate

n_{anhyd→hydrate}

moles anhydrous compound per mole hydrate = 1.0

n^o

population density of nuclei, #/(m·m³)

n_r

order for growth

N

cumulative number crystals/volume

N_S

stirrer speed, rpm

N_T

total number of particles/volume, #/m³

P

product flow rate, kg/s

Q

volumetric flow rate, m³/s

Q_energy

heat or cooling load, kJ/s

R

ideal gas constant (see Appendix C)

Sc

kg/h crystals (hydrate form)

t

time

T

temperature, °C or K

T*

temperature at equilibrium, °C or K

T_cold

coolant temperature, °C or K

T_M

magma temperature, °C or K

T_melt

melting point temperature, K

U

overall heat transfer coefficient

V

volume magma, m³

V

vapor evaporation rate, kg/h

V_sample

volume of sample of magma sieved, m³

Vol

volume of crystals/volume, m³/m³

Vol_T

total volume of crystals/volume, m³/m³

W_i

weight of crystals in sieve, m³

W_p

flow rate of product crystals, kg/h

batch crystallization, kg product/kg initial water

W_S

flow rate of seed crystals, kg/h

batch crystallization, kg seed crystals/kg initial water

x

mass fraction solute in solid or in solution

x*

mole fraction solute in solution at saturation, Eq. (17-2)

x_Cry,salt

mass fraction salt in crystals (hydrates also contain water)

x_L,salt

mass fraction salt in liquid

x_M,salt

mass fraction salt at mixing point M

y_salt

mass fraction salt in vapor, typically 0

empirical constant, Eq. (17-52)

γ

activity coefficient

δ

film thickness, m

Δc

concentration supersaturation, c – c*, kg/m³

ΔL_growth

L_p-L_S, m

ΔL_i

size range sieve fraction i, m

ΔW_i

weight of crystals collected on screen i, kg

λ

molar heat of fusion, kJ/kmol

ρ_c

crystal solid density, kg/m³

ρ_F

feed density, kg/m³

ρ_out

fluid density out, kg/m³

τ

V/Q_out, drawdown or retention time, s or h

term in quadratic equation for well-mixed membrane system, Eq. (18-10b)

a_m

constant in expression to calculate osmotic pressure, kPa/mole fraction, Eq. (18-14b)

a′

constant in expression to calculate osmotic pressure, kPa/weight fraction, Eq. (18-14c)

A

membrane area available for mass transfer, cm² or m²

b

term in quadratic equations for well-mixed membrane systems, Eq. (18-10c)

c

term in quadratic equations for well-mixed membrane systems, Eq. (18-10d)

c

concentration, g solute/L solution

c_out

outlet concentration of solute, g/L

c_p

permeate concentration of solute, g/L

c_w

concentration of solute at wall, g/L

c′

water concentration in permeate in Figure 18-16

C_PL,p

liquid heat capacity of permeate, kJ/(kg·°C)

C_PV,p

vapor heat capacity of permeate, kJ/(kg·°C)

d_t

diameter of tube, cm

d_tank

tank diameter, cm

D

diffusivity in solution, cm²/s

D_A,M

diffusivity of solute A in the membrane, cm²/s

F_p

volumetric flow rate of permeate, cm³/s

F_out

volumetric flow rate of exiting retentate, cm³/s

F_solv

volumetric flow rate of solvent in RO, cm³/s

F_m

molar flow rate, mol/s, mol/min, etc.

F′

mass flow rate, g/s, g/min, kg/min, etc.

h

1/2 distance between parallel plates, cm

h_in

enthalpy of inlet liquid stream in pervaporation, kJ/kg

h_out

enthalpy of outlet liquid retentate stream in pervaporation, kJ/kg

H_A

solubility parameter, cc(STP)/[cm³·(cm Hg)]

H_p

enthalpy of vapor permeate stream in pervaporation, kJ/kg

k

mass transfer coefficient, typically cm/s, Eq. (18-33)

K′_solv

permeability of the solvent through membrane, L/(atm·m²·day) or similar units

j

counter for stage location in staged models in Figure 18-19

J

volumetric flux, cm³/(s·cm²) or m³/(m²·day), Eq. (18-1b)

J′

mass flux, g/(s·cm²) or g/(m²·day), Eq. (18-1c)

J_m

mole flux, mol/(s·cm²) or kmol/(day·m²), Eq. (18-1d)

K′_A

solute permeability, g/(m·s·weight fraction), Eq. (18-15)

K′_solv

solvent permeability, g/(m·s·Pa), Eq. (18-13b)

K_M,i

rate transfer term for multicomponent gas permeation, dimensionless, Eq. (18-11d)

L

tube length, cm

M

concentration polarization modulus in weight fraction units, dimensionless, Eqs. (18-17) and (18-48)

M_c

concentration polarization modulus in concentration units, dimensionless, Eq. (18-48)

MW

molecular weight, g/mol or kg/kmol

N

number of well-mixed stages in models in Figure 18-19

p

pressure, Pa, kPa, atm, mm Hg, etc.

p_A

partial pressure of species A, Pa, atm, mm Hg, etc.

p_p

total pressure on the permeate (low pressure) side, Pa, kPa, atm, mm Hg, etc.

p_r

total pressure on the retentate (high-pressure) side, Pa, kPa, atm, mm Hg, etc.

P_A

permeability of species A in the membrane, [cc(STP)·cm/[cm²·s·cm Hg]

R

rejection coefficient in weight fraction units, dimensionless, Eq. (18-24a)

R^o

inherent rejection coefficient (M = 1.0), dimensionless, Eq. (18-48a)

R^o_c

inherent rejection coefficient in conc. units, dimensionless, Eq. (18-48a)

R

tube radius, cm

Re

Reynolds number, dimensionless, Eq. (18-35b)

Sc

Schmidt number, dimensionless, Eq. (18-35c)

Sh

Sherwood number, dimensionless, Eqs. (18-35a) and (18-35d)

t_ms

thickness of membrane skin doing separation, µm, mm, cm, or m

T

temperature, °C

T_ref

reference temperature, °C

u_b

bulk velocity in tube, cm/s

V_solvent

partial molar volume of the solvent, cm³/gmol

x

weight fraction of retentate in pervaporation; in binary system refers to more permeable species

x_g

weight fraction at which solute gels in UF

x_p

weight fraction of solute in liquid permeate in RO and UF

x_r

weight fraction of solute in retentate in RO and UF

y

weight fraction of permeate in pervaporation; in binary system refers to more permeable species

y_p

mole fraction of solute in gas permeate for gas permeation

y_r

mole fraction of solute in gas retentate for gas permeation

y_r,out

mole fraction of solute in retentate product for gas permeation

y_r,w

mole fraction of solute in gas retentate at membrane wall

selectivity, dimensionless, gas permeation: Eq. (18-4b), RO: Eq. (18-19), pervap: Eq. (18-60)

β

pervaporation separation factor, dimensionless, Eqs. (18-50a) and (18-50b)

γ

activity coefficient

δ

film thickness, m

Δx

difference in weight fraction of solute across the membrane

Δπ

difference in the osmotic pressure across the membrane, Pa, atm, mm Hg, etc.

π

osmotic pressure, Pa, kPa, atm, mm Hg, etc.

θ

cut = F_p/F_in with flows in molar units, dimensionless

θ′

cut = F′_p/F′_in in flows in mass units, dimensionless

µ

viscosity, centipoise or g/(cm·s)

ν = µ/ρ

kinematic viscosity, cm²/s

ρ_solv

mass solvent density, kg/m³

ρ_m,solv

molar solvent density, kmol/m³

λ_p

mass latent heat of vaporization of the permeate in pervaporation determined at the reference temperature, kJ/kg

ω

stirrer speed in radians/s

constant in Langmuir isotherm, same units as q/c, Eq. (19-6c)

a

argument for error function, dimensionless, Eq. (19-71), Table 19-7

a_p

surface area of the particles per volume, m^–1

A_c

cross-sectional area of column, m²

A_w

wall surface area per volume of column for heat transfer, m^–1

b

constant in Langmuir isotherm, (concentration)^–1, Eq. (19-6c)

c_A

concentration of species A, kg/m³, kmol/m³, g/L, etc.

c_i

concentration of species i, kg/m³, kmol/m³, g/L, etc., or

c_i

concentration of ion i in solution, typically equivalents/m³

c_i*

concentration of species i that would be in equilibrium with , same units as c_i

average concentration of solute in pore, same units as c_i

c_pore

fluid concentration at surface of adsorbent pores, same units as c_i

c_i,surface

fluid concentration at surface of particles, ε_p = 0, same units as c_i

c_Ri

concentration of ion i on the resin, typically equivalents/m³

c_RT

total concentration of ions on the resin, typically equivalents/m³

c_T

total concentration of ions in solution, typically equivalents/m³

Constant_i

constant relating solute velocity to interstitial velocity, dimensionless, Eq. (19-15e)

C_P,f

heat capacity of the fluid, cal/(g·°C), cal/(mol·°C), J/(g·K), etc.

C_P,p

heat capacity of particle including pore fluid, same units C_P,f

C_P,s

heat capacity of the solid, same units as C_P,f

C_P,w

heat capacity of the wall, same units as C_P,f

d_p

particle diameter, cm or m

D

desorbent rate in SMB, same units as F

D/F

desorbent to feed ratio in SMB, dimensionless

D_col

column diameter, m or cm

D

diffusivity including both molecular and Knudsen diffusivities, m²/s or cm²/s

D_effective

effective diffusivity, m²/s or cm²/s, Eqs. (19-4) and (19-50)

D_K

Knudsen diffusivity, m²/s or cm²/s, Eq. (19-51)

D_molecular

molecular diffusivity in free solution, m²/s or cm²/s

D_s

surface diffusivity, m²/s or cm²/s, Eq. (19-53)

erf

error function, Eq. (19-71) and Table 19-7

E_D

axial dispersion coefficient due to both eddy and molecular effects, m²/s or cm²/s

E_DT

thermal axial dispersion coefficient, m²/s or cm²/s

E_eff

effective axial dispersion coefficient, same units E_D, Eq. (19-68)

F

volumetric feed rate (e.g., m³/h, cm³/min, liter/h)

h_p

particle heat transfer coefficient, J/(K s m²) or similar units

h_w

wall heat transfer coefficient, J/(K s m²) or similar units

HETP

height of equilibrium plate, cm/plate, Eq. (19-78c)

k_f

film mass transfer coefficient, m/s or cm/s

k_m,c

lumped-parameter mass transfer coefficient with concentration driving force, m/s or cm/s, Eqs. (19-56a) and (19-57a)

k_m,q

lumped-parameter mass transfer coefficient with amount adsorbed driving force, m/s or cm/s, Eqs. (19-56b) and (19-57b)

K_AB

mass action equilibrium constant for monovalent-monovalent ion exchange, dimensionless, Eq. (19-40a)

K_A,c

adsorption equilibrium constant in terms of concentration, (concentration)^–1

K′_i,c

linearized adsorption equilibrium constant in terms of concentration, same units as q/c, Eq. (19-6b)

K_Ao

pre-exponential factor in Arrhenius Eq. (19-7a), same units as K_A

K_A,p

adsorption equilibrium constant in terms of partial pressure, (pressure)^–1

K′_A,p

linearized adsorption equilibrium constant in terms of partial pressure, same units as q_A/p_A, Eq. (19-5b)

K_d

size exclusion parameter, dimensionless

K_DB

mass action equilibrium constant for divalent-monovalent ion exchange, same units as c_T/c_RT, Eq. (19-41)

K_DE

Donnan exclusion factor, dimensionless, following Eq. (19-44)

L

length of packing in column, m or cm

L_MTZ

length of mass transfer zone, Figure 19-23, m or cm

M

molecular weight of solute, g/mol or kg/kmol

M_i

multipliers in Eqs. (19-29a) to (19-29d), dimensionless

N_i

equivalent number of plates in chromatography for solute i, Eq. (19-78a)

N_Pe

Peclet number based on particle diameter, dimensionless, Eq. (19-62)

p_A

partial pressure of species A, mm Hg, kPa, etc.

p_h

high pressure, mm Hg, kPa, etc.

p_L

low pressure, mm Hg, kPa, etc.

Pe_L,i

Peclet number based on length for solute i, dimensionless, Eq. (19-78b)

q_A

amount of species A adsorbed, kg/kg adsorbent, mol/kg adsorbent, or kg/L

q_A,max

maximum amount of species A that can adsorb, kg/kg adsorbent, mol/kg adsorbent, or kg/L

q_F

amount adsorbed in equilibrium with feed concentration, same units as q_A

average amount of species i adsorbed, kg/kg adsorbent, mol/kg adsorbent, or kg/L

q_i*

amount adsorbed that would be in equilibrium with fluid of concentration c_i, same units as q_A

Q

volumetric flow rate, m³/s, L/min, etc.

r_p

pore radius, m or cm

R

resolution, dimensionless, Eq. (19-82)

R

gas constant (e.g., , see Appendix C)

Re

Reynolds number, dimensionless, Eq. (19-60)

Sc

Schmidt number, dimensionless, Eq. (19-60)

Sh

Sherwood number, dimensionless, Eq. (19-60)

t

time, s, min, or h

t_br

breakthrough time, s, min, or h

t_center

time center of pattern exits column, s, min, or h, Eq. (19-85b)

t_elution

elution time, s, min, or h

t_F, t_feed

feed time, s, min, or h

t_MTZ

time of mass transfer zone, Figure 19-23, s, min, or h

t_R

retention time, s, min, or h

t_sw

switching time in SMB, s, min, or h

T

temperature, °C or K

T_amb

ambient temperature, °C or K

T_s

solid temperature, °C or K

u_ion,i

velocity of ion i, m/s or cm/s

u_s

average solute velocity, m/s or cm/s

average of solute velocities for A and B, cm/s, Eq. (19-83)

u_s,ion,i

diffuse wave velocity of ion i, m/s or cm/s

u_sh

shock wave velocity, m/s or cm/s

u_sh,ion,i

shock wave velocity of ion i, m/s or cm/s

u_th

thermal wave velocity, m/s or cm/s

u_{total_ion}

velocity of total ion wave, m/s or cm/s

v_A,product

interstitial velocity of A product if it was in the column, m/s = (m³/s A product)/(ε_e A_c)

v_B,product

interstitial velocity of B product if it was in the column, m/s = (m³/s B product)/(ε_e A_c)

v_D

interstitial velocity of desorbent if it was in the column, m/s = (m³/s D product)/(ε_e A_c)

v_Feed

interstitial velocity of feed if it was in the column, m/s = (m³/s feed)/(ε_e A_c)

v_inter

interstitial velocity, m/s or cm/s, Eq. (19-2b)

v_super

superficial velocity, m/s or cm/s, Eq. (19-2a)

V_available

volume available to molecule, m³, Eq. (19-1c)

V_column

column volume, m³

V_feed

volume feed gas, m³

V_fluid

volume available to fluid, m³, Eq. (19-1a)

V_purge

volume purge gas, m³

w_A, w_B

width of chromatographic peak, s, min, or h

W

weight of the column per length, kg/m

x

deviation from the location of the peak maximum, dimensionless, Eq. (19-79)

x_l

deviation from peak maximum in length units, Eq. (19-80b)

x_t

deviation from peak maximum in time units, Eq. (19-80a)

x

weight or mole fraction solute in liquid, kg solute/kg liquid or

 

kmol solute/kmol liquid, dimensionless

x_i

= c_i/c_T equivalent fraction of ion in solution, dimensionless

X_breakthrough (z,t)

general solution for column breakthrough for linear isotherms, same units as c, Eq. (19-72)

y

weight or mole fraction solute in gas, kg solute/kg gas, or kmol solute/kmol gas, dimensionless

y_i

= c_Ri/c_RT equivalent fraction of ion on resin, dimensionless

z

axial distance in column, m or cm (Measured from closed end for PSA pressure change calculations)

ratio velocities of strong and weak solutes, Eq. (19-27), dimensionless

Δc

change in solute concentration, same units as c

ΔH_ads

heat of adsorption, J/kg, cal/mol, etc.

Δp_A

change in partial pressure, kPa, atm, etc.

Δq

change in amount adsorbed, kmol/kg adsorbent, kg/kg adsorbent, kmol/m³, or kg/m³

Δt

change in time, s, min, or h

ΔT_f

change in fluid temperature, °C or K

Δz

increment of column length, m

γ

volumetric purge to feed ratio in PSA, dimensionless, Eq. (19-26)

ε_e

external porosity, dimensionless

ε_p

internal or pore porosity, dimensionless

ε_T

total porosity, dimensionless, Eq. (19-1b)

ρ_b

bulk density of adsorbent, kg/m³, Eq. (19-3b)

ρ_f

fluid density, kg/m³

ρ_m,f

molar density of fluid, kmol/m³

ρ_p

particle density, kg/m³, Eq. (19-3a)

ρ_s

structural density of solid, kg/m³

σ

standard deviation of Gaussian chromatographic peak, Eq. (19-79)

σ_l

standard deviation in length units, m or cm, Eq. (19-80b)

σ_t

standard deviation in time units, min or s, Eq. (19-80a)

τ

tortuosity, dimensionless, Eq. (19-4)

ζ

Greek letter zeta used as dummy variable in Eq. (19-71)