Notation

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Notation

Commonly used notations are indexed here. Notations specific to certain sections are defined at appropriate places in those sections.

A	area for flow or mass transfer
A₁, A₂	usually integration constants
a_gl	interfacial area for gas-liquid mass transfer, m^–1
a_ls	interfacial area for liquid-solid mass transfer, m^–1
B₁, B₂	usually integration constants
Bi_h	Biot number for heat transfer, hL_ref/k_solid
Bi_m	Biot number for mass tranfser, k_mL_ref/D
C	total molar concentration of a multicomponent mixture, mol/m³
$\bar{c}$ $\bar{c}$	average molecular speed of gas molecules in kinetic theory
c_A	dimensionless concentration of species indicated in the subscript (A here), C_A/C_ref
C_A	local concentration of species indicated in the subscript (A here), mol/m³
〈C_A〉	cross-sectionally averaged concentration, mol/m³
C_A,e	concentration of species indicated in the subscript at the exit
C_A,i	inlet concentration of species A for flow reactor, mol/m³
c_Ab	dimensionless cup-mixed average concentration
C_Ab	concentration of species A indicated in the bulk phase, mol/m³
C_Ab	cup-mixed (flow) average concentration of species for flow systems
C_AG	concentration of species indicated in the subscript in the bulk gas, mol/m³
C_Ai	concentration of species A at the interface in Chapter 20, mol/m³
C_Ai	initial concentration of species A for transient problems, mol/m³
C_AL	concentration of species indicated in the subscript in the bulk liquid, mol/m³
$C_{A L}^{}$ $C_{A L}^{}$	hypothetical concentration of A if in equilibrium with the bulk gas, mol/m³
C_As	concentration of species A at a solid surface, mol/m³
C_D	drag coefficient
C_G	total molar concentration in the gas phase, mol/m³
C_L	total molar concentration in the liquid phase, mol/m³
c_p	specific heat of a species, mass basis, at constant pressure conditions, J/kg K
C_p	specific heat of a species, mole basis, J/mol K
c_v	specific heat of a species, mass basis, at constant volume conditions J/kg K
d	diameter of the molecules treated as rigid sphere in Chapter 7
$\tilde{D}$ $\tilde{D}$	diffusivity matrix for multicomponent systems in Section 15.5, m²/s
Da	Damkohler number, ratio of mean residence time to reaction time
d_B	bubble diameter in a gas-liquid dispersion
D_E	axial dispersion coefficient, m²/s
$D_{E}^{}$ $D_{E}^{}$	dimensionless axial dispersion coefficient, D_E/〈v〉 L,
D_eA	effective diffusitiy of species A in a heterogeneous media
D_i	molecular diffusivity of species i, m²/s
D_K	Knudsen diffusion coefficient for small pores
d_p	particle or solid diameter
d_t	diameter of the tube or pipe
D_t	turbulent mass diffusivity, m²/s
E	enhancement factor for gas-liquid mass transfer with reaction
E	Electric field
E(t)	exit age distribution function in Chapters 13 and 14
e_x	unit vector in the (x)-direction
f	Fanning friction factor
F	Faraday constant = 96,485 C/mol
$F$ $F$	Drift flux correction factor
f_A	fugacity of species A in Chapter 13
g	acceleration due to gravity, m/s²
G	molar flow rate in mass exchanger; mol/s
$\dot{G}$ $\dot{G}$	superficial gas velocity, kg/m²s
G^E	excess free energy for non-ideal liquids in Section 7.3.2, J/mol
Gr	Grashoff number
$\hat{h}$ $\hat{h}$	enthalpy per unit mass, J/kg
h	heat transfer coefficient, W/m² K
Ha	Hatta number for gas–liquid reactions
H_A	Henry law solubility coefficient, usually atm m³/mol
h_G	heat transfer coefficient in the gas film
${\hat{h}}_{g l}$ ${\hat{h}}_{g l}$	heat released on condensation of a species, J/kg
h_L	heat transfer coefficient in the liquid film
${\hat{h}}_{l g}$ ${\hat{h}}_{l g}$	heat of vaporization, J/kg
HTU	height of a transfer unit
j_A	mass diffusion flux vector of A (mass reference), kg/m²s
J_A	molar diffusion flux vector of A (mole reference), mol/m²s
J_Ax	component of molar diffusion vector in direction x
j_D	j-factor for mass transfer
j_h	j-factor heat mass transfer
k	thermal conductivity of a species, subscript l = liquid, g = gas, s = solid W/m K
k	rate constant for reaction, general
k₀	rate constant for a zero-order reaction, mol/m³ s
k₁	rate constant for a first-order reaction, 1/s
k₂	rate constant for a second-order reaction, m³/mol s
k_ω	mass transfer coefficient from an interface to bulk liquid (mass fraction driving force), mol/s m²(massfrac)
K, K_eq	equilibrium constant for a chemical reaction
K_A	adsorption equilibrium constant
k_B	Boltzmann constant = 1.38 × 10^–23J/K
k_G	mass transfer coefficient from a gas to the interface (partial pressure driving force), mol/Pa m² s
K_G	overall mass transfer coefficient from an bulk gas to bulk liquid (gas phase partial pressure driving force); mol/Pa m² s
K_L	overall mass transfer coefficient from a bulk gas to bulk liquid (liquid concentration driving force), m/s
k_m	mass transfer coefficient from a solid to fluid (concentration driving force), m/s
$k_{m}^{°}$ $k_{m}^{°}$	mass transfer coefficient under low mass flux conditions, m/s
k_s	rate constant for a heterogeneous first order reaction, m/s
k_sl	solid-liquid mass transfer coefficient, m/s
k_x	mass transfer coefficient from an interface to bulk liquid (mole fraction driving force), mol/s m²(molefrac)
K_x, K_y	overall mass transfer coefficient from gas to liquid based on mole fraction driving force
k_y	mass transfer coefficient from gas to interface (mole fraction driving force), mol/s m²(molefrac)
L	length of the plate or tube or catalyst slab, m
L	liquid molar flow rate in mass exchanger, mol/s
$\dot{L}$ $\dot{L}$	superficial liquid mass velocity, kg/m² s
L_C	equivalent length parameter for a solid catalyst, V_p/S_e
Le	Lewis number, ratio of thermal to mass diffusivity
m	partition or solubility coefficient of a solute between two phases, y/x
$\dot{m}$ $\dot{m}$	mass flow rate, kg/s
M	ratio of diffusion time to reaction time (Chapters 21 and 22)
$\bar{M}$ $\bar{M}$	average molecular weight of a mixture, kg/mol
$\bar{M}$ $\bar{M}$	average molecular weight of a mixture
$M$ $M$	total moles present in a control volume, g-mol
$\dot{M}$ $\dot{M}$	moles per sec entering the unit, i = inlet, e = exit
M_A	molecular weight of species indicated in the subscript, kg/mol
$M_{A}$ $M_{A}$	moles of A in the system or control volume
m_A,tot	total mass of A in an unit or control volume, kg
${\dot{M}}_{A, e}$ ${\dot{M}}_{A, e}$	moles leaving control volume per unit time
${\dot{M}}_{A, i}$ ${\dot{M}}_{A, i}$	moles entering a control volume per unit time
${\dot{m}}_{A e}$ ${\dot{m}}_{A e}$	mass flow rate of A exiting a unit, kg/s
${\dot{m}}_{A i}$ ${\dot{m}}_{A i}$	mass flow rate of A entering a unit, kg/s
${\dot{m}}_{A W, t o t}$ ${\dot{m}}_{A W, t o t}$	total mass of A transferred to walls from a unit or process, kg/s
n	normal vector outward from a control surface
N	number of mixed tanks into which the reactor is divided into in Chapter 13
n_A	mass flux vector of species A, stationary frame, kg-A/m²s
N A	combined mole flux vector of species A, stationary frame, mole of A/m²s
N_av	Avogadro number = 6.23 × 10²³ molecules/mol
N_Aw	molar flux at the tube wall, mol/m² s
n_Ax	component of mass flux vector of A in the x-direction, kg-A/m²s
N_Ax	component of mole flux vector of A in the x-direction
ns	total number of species in a multicomponent mixture
N_t	total molar flux, N_A + N_B in a binary
NTU	number of transfers of unit parameter
p	the concentration gradient in the p-substitution method
P	power input for agitated vessels, W
P	total pressure of a gas mixture, Pa
p, P	fluid pressure, Pa
P_c	critical pressure of a species, Pa
Pe	Peclet number, L_ref v_ref/D_A = ReSc
Pe*	dispersion Peclet number, $1 / D_{E}^{}$ $1 / D_{E}^{}$
p_vap	vapor pressure of a species, Pa
q	concentration ratio parameter in Chapters 20 and 21
Q	volumetric flow rate in a pipe, m³/s
r	radial coordinate in cylindrical and spherical system
R	radius of cylinder or catalyst particle
R*	gas constant defined as R_G/M_w
r_A	local rate mass of production of A by reaction per unit volume, mass units, kg/m³s
R_A	local rate of mole production of A by reaction per unit volume, mole units, mol/m³s
R_A	rate of production of a species A by reaction
〈R_A〉	cross-sectional average of the rate of production of a species
$R_{A}^{v}$ $R_{A}^{v}$	volumetric rate of absorption of A, mol/m³s
Re	Reynolds number, L_ref v_ref ρ/μ
R_g	gas constant, 8.314 Pa m³/mol K
s	shape parameter for Laplacian for transient diffusion, 0 = slab, 1 = long cylinder, 2 = sphere
S	surface area of the control volume in Chapter 5
Sc	Schmidt number, ν/D_A
S_e	external surface area of a catalyst particle
Sh	Sherwood number, k_mL_ref/D_A
S_i	internal surface area of a catalyst particle per unit mass
St	Stanton number, k_m/v_ref = Sh/Re Sc
t	time variable
$\bar{t}$ $\bar{t}$	mean residence time V/Q in Chapters 13 and 14
T_∞	temperature of the approaching fluid
T	local temperature in the medium subscript c = coolant, G = gas, L = liquid
T_a	temperature of the surroundings
T_c	critical temperature of a species
t_E	exposure time for a gas–liquid interface
T_i	temperature of a gas–liquid interface
T_w	temperature of a wall or tube
U	overall heat transfer coefficient from hot fluid to cold fluid, W/m²K
$\hat{u}$ $\hat{u}$	internal energy per unit mass, J/kg
$\hat{U}$ $\hat{U}$	internal energy per unit mole, J/mol
$\hat{v}$ $\hat{v}$	specific volume, m³/kg = 1/ρ
v	velocity vector; also mass fraction–averaged velocity in a multicomponent mixture, m/s
v′	fluctuating velocity vector in turbulent flow
$\hat{v}$ $\hat{v}$	time-averaged velocity vector in turbulent flow
v*	mole fraction–averaged velocity in a multicomponent mixture, m/s
〈v〉	average velocity in the flow direction
v_θ	velocity component in the tangential (θ) direction
V	volume of a reactor or a macroscopic control volume
$\hat{V}$ $\hat{V}$	molar volume, m³/mol
v_A	velocity of species A in a multicomponent mixture, stationary frame, m/s
V_b	molecular volume at boiling point of solvent
v_e	velocity component in the fluid outside the boundary layer, m/s
V_f	friction velocity defined as $\sqrt{τ_{f} / ρ}$ $\sqrt{τ_{f} / ρ}$ used in turbulent flow, m/s
v_p	volume of a catalyst particle
v_x	x-component of the velocity; v_y, v_z defined similarly
v_z	axial (z-) component of velocity in cylindrical coordinates
x	distance variable in the x-direction, y and z defined similarly
x_i	mole fraction of species indicated by the subscript (usually in the liquid phase)
y	distance variable in the y-direction
y⁺	dimensionless length used in turbulent analysis near a wall
y_B(l.m)	log-mean mole fraction of the non-diffusing component
y_i	mole fraction of species indicated by the subscript (usually in the gas phase)
z	axial distance variable in cylindrical coordinates
Z	frequency of molecular collisions in Section 7.4
z*	dimensionless axial distance variable in cylindrical coordinates, z/R
z_i	number of charges on an ionic species

Greek Letters

α	thermal diffusivity of a fluid or solid, m²/s
γ_A	activity coefficient of species A in Chapter 7
∇	gradient operator
∇²	Laplacian operator
▵	ratio of boundary layer thickness, heat/mass to momentum
Δ	difference operator, out-in
Δ	ratio of mass and momentum transfer boundary layer thickness in Chapter 11
δ	thickness of momentum boundary layer in general
ΔΠ	osmotic pressure difference in Section 28.2.1, Pa
δ_f	film thickness for mass transfer; abbreviated as δ in Chapter 6
ΔH	heat of reaction, J/mol
ΔH_c	heat of condensation, J/mol
ΔH_v	heat of vaporization, J/mol
δ_m	thickness of mass transfer boundary layer
δ_t	thickness of thermal boundary layer
∊	gas or liquid holdup in a reactor
∊	a parameter in the Lennard-Jones model in Chapter 7
∊	power input to a vessel per unit mass
∊_p	porosity of a solid or a catalyst particle
ζ	dimensionless axial distance, z*/P e
η	effectiveness factor of a porous catalyst in Chapter 18
η	similarity variable defined by Equation 11.13 in Chapter 11
η	similarity variable for convective heat transfer in Chapter 10
η	dimensionless axial position in tubular reactor, z/L
θ	angular direction in polar coordinates
θ	latitude direction in spherical coordinates
θ	dimensionless temperature in heat transfer examples
κ	ratio of radius values, R_c/R_o, in the Krogh model in Section 23.2
κ	Boltzmann constant, also denoted as k_B
λ	mean free path defined by Equation 7.4
λ	location of reaction plane for an instantaneous reaction, Chapters 20 and 21
μ	coefficient of viscosity, Pa s
μ_A	chemical potential of species A in Section 7.3, J/mol
μ_w	chemical potential of water in Chapter 28
ν	coefficient of kinematic viscosity, μ/ρ, m²/s
ν	stoichiometric coefficient of products formed from A
ν_e	excess stoichiometric coefficient of products. ν – 1
ν_t	turbulent kinematic viscosity, μ/ρ, m²/s
$ν_{T}^{+}$ $ν_{T}^{+}$	dimensionless total (molecular + turbulent) kinematic viscosity
ξ	dimensionless radial position, r/R or x/L
ρ	density of the medium or the fluid, kg/m³
ρ_A	density of A in a multicomponent mixture, kg/m³
$ρ_{A}^{0}$ $ρ_{A}^{0}$	pure component density of A, kg/m³
σ	surface tension, N/m
σ²	variance of the tracer response in Chapter 13
(σ*)²	dimensionless variance of the tracer response, $σ^{2} / {\bar{t}}^{2}$ $σ^{2} / {\bar{t}}^{2}$
τ	dimensionless time in Chapters 2 and 8, t/t_ref
τ_w	wall shear stress
φ	longitude in the spherical coordinate system
φ	Thiele parameter for a first-order reaction
φ₀	Thiele parameter for a zero-order reaction defined by Equation 18.14
ψ	stream function defined by Equations 11.10 or 11.11
ω_A	mass fraction of species indicated by the subscript, kg-A/kg-total

Common Subscripts

b	bulk conditions
e	exit values
g, G	gas phase properties
i	inlet values
i	interface conditions
l, L	liquid phase properties
s	conditions at a surface of a solid or catalyst

Acronyms

AEM	anion exchange membrane in Chapter 30
BPM	bipolar membrane in Chapter 30
CEM	cation exchange membrane in Chapter 30
CFD	computational fluid dynamics
DEq	differential equation
ED	electrodialysis in Chapter 30
LHS	left-hand side of an equation
NRTL	non-random two-liquid model
PDE	partial differential equation
RHS	right-hand side of an equation
UNIFAC	universal functional-group activity coefficients

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Notation

Greek Letters

Common Subscripts

Acronyms

Table of Contents for
Notation