Nomenclature

Chapters 1 through 15

a

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

ap

surface area/volume, m2/m3

ap1, ap2, ap3, aT1, aT2, aT6

constants in Eq. (2-30) and Table 2-3

A,B,C

constants in Antoine Eq. (2-34)

A,B,C,D,E

constants in Eq. (2-60)

A,B,C,D

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

AE, BE, CE, DE

constants in matrix form of energy balances, Eq. (6-34)

Aactive

active area of tray, ft2 or m2

Ac

cross sectional area of column, ft2 or m2

Ad

downcomer area, ft2 or m2

Adu

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

Ahole

area of holes in column, ft2

AI

interfacial area between two phases, ft2 or m2

Anet

net area, Eq. (10-13), ft2 or m2

b

equilibrium constant for linear equilibrium, y = mx + b

B

bottoms flow rate, kg mole/hr or lb mole/hr

C

number of components

CBM

bare module cost, Chapter 11

CfL

vapor load coefficient, Eq. (15-38)

Co

orifice coefficient, Eq. (10-25)

Cp

heat capacity, Btu/lb° F or Btu/lbmole° F or cal/g° C or cal/g mole° C, etc.

Cp

base purchase cost, Chapter 11

Cp,size

packing size factor, Table 10-5

CpW

water heat capacity

Cs

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

Csb

capacity factor, Eq. (10-8)

d

dampening factor, Eq. (2-57)

D

dampening factor, Eq. (2-57)

D

diffusivity, cm2/s or ft2/hr

D, Dia

diameter of column, ft or m

D′col

column diameter, see Table 15-2, ft

Dtotal

total amount of distillate (Chapter 09), moles or kg distillate flow rate, kg mole/hr or lb mole/hr

e

absolute entrainment, moles/hr

E

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

Ê

mass extract, kg

Ek

value of energy function for trial k, Eq. (2-51)

EML, EMV

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

Eo

Overall efficiency, Eq. (4-56)

Ept

point efficiency, Eq. (10-5) or (15-76a)

Esettler

Holdup extract phase in settler, kg

f = V/F

fraction vaporized

f(x)

equilibrium function, Chapter 09

fk(V/F)

Rachford-Rice function for trial K, Eq. (2-42)

F

packing factor, Figure 10-25 and Tables 10-3 and 10-4

F

degrees of freedom, Eq. (2-2)

F

charge to still pot (Chapter 09), moles or kg

F

mass of feed in batch extraction, kg

F

feed flow rate, kgmole/hr or lbmol/hr or kg/hr etc.

FD

diluent flow rate (Chapter 13), kg/hr

Flv, FP

images, flow parameter

Fm

material factor for cost, Table 11-1

Fp

pressure factor for cost, Eqs. (11-5) and (11-6)

Fq

quality factor for cost, Eq. (11-7)

Fs,Fsolv

flow rate solvent (Chapter 13), kg/hr

Fsolid

solids flow rate in leaching, kg insoluble solid/hr

Fw

modification factor, Eq. (10-26) and Figure 10-20

gap

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

g

32.2 ft/s2, 9.81 m/s2

G

flow rate carrier gas, kgmole/hr or kg/hr

G′

gas flux, lb/s ft2

h

pressure drop in head of clear liquid, inches liquid

h

height of liquid on stage (Chapter 15), ft

h

height, m or ft

h

liquid enthalpy, kcal/kg, Btu/lb mole, etc.

h

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

images

pure component enthalpy

hf

enthalpy of liquid leaving feed stage

hF

feed enthalpy (liquid, vapor or two-phase)

ho

hole diameter, inches

hp

packing height, ft or m

htotal

height of flash drum, ft or m

H

Henry’s law constant, Eqs. (8-8) and (12-1)

H

molar holdup of liquid on tray, Eq. (8-27)

H

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

HG

height of gas phase transfer unit, ft or m

HL

height of liquid phase transfer unit, ft or m

HOG

height of overall gas phase transfer unit, ft or m

HOL

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

JA

flux with respect to molar average velocity of fluid

imagesimages

individual mass transfer coefficients in liquid and vapor phases, see Table 15-2

k′y

mass transfer coefficient in concentrated solutions, Eq. (15-43)

kx, ky

individual mass transfer coefficient in weight units

KD

y/x, distribution coefficient for dilute extraction

K, Ki

yi/xi, equilibrium vapor-liquid ratio

Kdrum

parameter to calculate uperm for flash drums, Eq. (2-59)

Kx, Ky

overall mass transfer coefficient in liquid or vapor, lbmoles/ft2 hr

lw

weir length, ft

L

liquid flow rate, kgmoles/hr or lbmoles/hr

images

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

L′

liquid flux, lb/(s)(ft2)

Lg

liquid flow rate in gal/min, Chapter 10

m

linear equilibrium constant, y = mx + b

m

local slope of equilibrium curve, Eq. (15-56)

m

ratio HETPpractical/HETPpacking Eq. (10-46)

M

flow rate of mixed stream (Chapter 14), kg/hr

M

multiplier times (L/D)min (Chapter 07)

MW

molecular weight

n

moles

nG

number of gas phase transfer units

nL

number of liquid phase transfer units

nOG

number of overall gas phase transfer units

nOL

number of overall liquid phase transfer units

norg

moles organic in vapor in steam distillation

nw

moles water in vapor in steam distillation

N

number of stages

NA

flux of A, lbmoles/(hr)(ft2) or kgmoles/(hr)(m2)

Nf,Nfeed

feed stage

Nmin

number of stages at total reflux

Nfeed,min

estimated feed stage location at total reflux

NTU

number of transfer units

O

total overflow rate in washing, kg/hr

p, ptot

pressure, atm, kPa, psi, bar etc.

images, pB

partial pressure

P

Number of phases

Q

LF/F = (L − L)/F, feed quality

Q

amount of energy transferred, Btu/hr, kcal/hr etc.

Qc

condenser heat load

Qflash

heat loss from flash drum

QR

reboiler heat load

r

radius of column, ft or m

R

gas constant

R

raffinate flow rate (Chapts. 13 and 14), kg/hr

images

mass raffinate, kg

images

Raffinate holdup in settler, kg

images

Raffinate holdup in tank, kg

S

solvent flow rate (Chapter 08) kgmoles/hr or lbmoles /hr

S

tray spacing, inches, Eq. (10-47)

S

moles second solvent in constant-level batch distillation

Ŝ

mass of solvent, kg

S

solvent flow rate (Chapter 14), kg/hr

ScL

Schmidt number for liquid = µ/ρD

Scv

Schmidt number for vapor = µ/ρD

t

time, s, min, or hr

tbatch

period for batch distillation, Eq. (9-27)

tdown

down time in batch distillation

toperating

operating time in batch distillation

tres

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

ttray

tray thickness, inches

T

temperature,° C,° F, K or° R

Tref

reference temperature

u

vapor velocity, cm/s or ft/s

uflood

flooding velocity, Eq. (10-8)

uop

operating velocity, Eq. (10-11)

uperm

permissible vapor velocity, Eq. (2-59)

U

underflow liquid rate, (Chapter 13), kg/hr

v

superficial vapor velocity, ft/s

vo

vapor velocity through holes, Eq. (10-29), ft/s

vo,bal

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

V

vapor flow rate, kgmoles/hr or lbmoles/hr

V

molal volume Eq. (13-6)

Vmax

maximum vapor flow rate

Vsettler

volume settler, m3

Vtank

volume tank, m3

Vsurge

surge volume in flash drum, Figure 2-11, ft3

VP

vapor pressure, same units as p

WL

liquid flow rate, kg/hr or lb/hr

WL

liquid mass flux, lb/s ft2 or lb/hr ft2, (Chapter 15)

WV

vapor flow rate, kg/hr or lb/hr

x

weight or mole fraction in liquid

x

[L/D − (L/D)min]/(L/D + 1) in Eqs. (7-42)

x*

equilibrium mole fraction in liquid

xi,k, xi,k+1

trials for integration, Eq. (8-29)

xI

interfacial mole fraction in liquid

x*out

liquid mole fraction in equilibrium with inlet gas, Eq. (15-35b)

X

weight or mole ratio in liquid

y

weight or mole fraction in vapor

y*

equilibrium mole fraction in vapor

y*out

vapor mole fraction in equilibrium with inlet liquid in counter current system, Eq. (15-35a) or in equilibrium with outlet liquid in cocurrent contactor, Eq. (15-71)

yI

interfacial mole fraction in vapor

images

mass fraction in vapor

Y

weight or mole ratio in vapor

z

weight or mole fraction in feed

z

axial distance in bed (Chapter 15)

Greek

αAB

KA/KB, relative volatility

β

Ahole/Aactive

γ

activity coefficient

δp

characteristic dimension of packing, inch, Eq. (10-39a)

δi

solubility parameter, Eq. (13-6)

Δ

change in variable

ΔEv

latent energy of vaporization, Eq. (13-6)

ε

limit for convergence

η

fraction of column available for vapor flow

θ

angle of downcomer, Figure 10-18B

λ

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

µ

viscosity, cp

µw

viscosity of water, cp

ρL

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

ρV

vapor density

σ

surface tension, dynes/cm

φ

liquid phase packing parameter, Eq. (15-38)

φdc

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

ψ

ρwaterL, Chapter 10

ψ

e/(e + L), fractional entrainment, Chapter 10

ψ

packing parameter for gas phase, Eq. (15-37)

Chapter 16

a, aj

term in quadratic equations for well-mixed membrane systems, Eqs. (16-10b), (16-74a) and (16-74a)

images

constant in expression to calculate osmotic pressure, kPa/mole fraction, Eq. (16-15a)

a

constant in expression to calculate osmotic pressure, kPa/weight fraction, Eq. (16-15b)

ai

activities, Eq. (16-51)

A

membrane area available for mass transfer, cm2 or m2

b, bj

term in quadratic equations for well-mixed membrane systems, Eqs. (16-10c), (16-74b), and (16-74b)

c, cj

term in quadratic equations for well-mixed membrane systems, Eqs. (16-10d), (16-74c) and (16-74c)

c

concentration, g solute/L solution

cout

outlet concentration of solute, g/L

cp

permeate concentration of solute, g/L

cw

concentration of solute at wall, g/L

c

water concentration in permeate in Figure 16-17

CPL,p

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

CPV,p

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

dt

diameter of tube, cm

dtank

tank diameter, cm

D

diffusivity in solution, cm2/s

Dm

diffusivity in the membrane, cm2/s

Fp

volumetric flow rate of permeate, cm3/s

Fout

volumetric flow rate of exiting retentate, cm3/s

Fsolv

volumetric flow rate of solvent in RO, cm3/s

images

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

F

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

h

½ distance between parallel plates, cm

hin

enthalpy of inlet liquid stream in pervaporation, kJ/kg

hout

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

HA

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

Hp

enthalpy of vapor permeate stream in pervaporation, kJ/kg

k

mass transfer coefficient, typically cm/s, Eq. (16-33)

Ksolv

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

j

counter for stage location in staged models in Figure 16-20

J

volumetric flux, cm3/(s cm2) or m3/(m2 day), Eq. (16-1b)

J

mass flux, g/(s cm2)or g/(m2 day), Eq. (16-1c)

images

mole flux, gmole/(s cm2) or kg mole/(day m2), Eq. (16-1d)

KA

solute permeability, g/(m day wt frac)

Km,i

rate transfer term for multicomponent gas permeation, dimensionless, Eq. (16-11d)

L

tube length, cm

M

concentration polarization modulus in wt. fraction units, dimensionless, Eq. (16-17)

Mc

concentration polarization modulus in concentration units, dimensionless, Eq. (16-48)

MW

molecular weight, g/gmole or kg/kg mole

N

number of well-mixed stages in models in Figure 16-20

p

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

pA

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

pp

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

pr

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

PA

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

R

rejection coefficient in wt frac units, dimensionless, Eq. (16-24)

inherent rejection coefficient (M = 1), dimensionless, Eq. (16-24)

Rc

rejection coefficient in conc. units, dimensionless, Eq. (16-48)

R

tube radius, cm

Re

Reynolds number, dimensionless, Eq. (16-35b)

Sc

Schmidt number, dimensionless, Eq. (16-35c)

Sh

Sherwood number, dimensionless, Eq. (16-35a)

tms

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

T

temperature,°C

Tref

reference temperature,°C

ub

bulk velocity in tube, cm/s

vsolvent

partial molar volume of the solvent, cm3/gmole

x

wt frac of retentate in pervaporation. In binary system refers to more permeable species.

xg

wt frac at which solute gels in UF

xp

wt frac solute in liquid permeate in RO and UF

xr

wt frac solute in retentate in RO and UF

y

wt frac of permeate in pervaporation. In binary system refers to more permeable species.

yp

mole fraction solute in gas permeate for gas permeation

yr

mole fraction solute in gas retentate for gas permeation

yr,w

mole fraction solute in gas retentate at membrane wall

yt,A

mole fraction solute A in gas that transfers through the membrane

Greek letters

α

selectivity, dimensionless, Gas Permeation: Eq. (16-4b), RO: Eq. (16-20), pervaporation: Eq. (16-53a)

Δx

difference in wt frac 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

images, with flows in molar units, dimensionless

θ′

cut = Fp/Fin in flows in mass units, dimensionless

µ

viscosity, centipoise or g/(cm s)

ν = µ/ρ

kinematic viscosity, cm2/s

ρsolv

mass solvent density, kg/m3

images

molar solvent density, kg mol/m3

λp

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

Ω

stirrer speed in radians/s

Chapter 17

a

constant in Langmuir isotherm, same units as q/c, Eq. (17-6c)

a

argument for error function, dimensionless, Eq. (17-70), Table 17-6

ap

surface area of the particles per volume, m−1

Ac

cross sectional area of column, m2

Aw

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

b

constant in Langmuir isotherm, (concentration)−1, Eq. (17-6c)

cA

concentration of species A, kg/m3, kg moles/ m3, g/liter, etc.

ci

concentration of species i, kg/m3,kg moles/ m3, g/liter, etc., or

ci

concentration of ion i in solution, typically equivalents/m3

images

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

images

average concentration of solute in pore, same units as ci

cpore

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

ci,surface

fluid concentration at surface of particles, εp = 0, same units as ci

cRi

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

cRT

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

cT

total concentration of ions in solution, typically equivalents/m3

Ci

constant relating solute velocity to interstitial velocity, dimensionless, Eq. (17-15e)

CP,f

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

CP,p

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

CP,s

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

CP,w

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

dp

particle diameter, cm or m

D

desorbent rate in SMB, same units as F

D/F

desorbent to feed ratio in SMB, dimensionless

Dcol

column diameter, m or cm

D

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

Deffective

effective diffusivity, m2/s or cm2/s, Eq. (17-4)

DK

Knudsen diffusivity, m2/s or cm2/s, Eq. (17-51)

Dmolecular

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

Ds

surface diffusivity, m2/s or cm2/s, Eq. (17-53)

erf

error function, Eq. (17-70) and Table 17-7

ED

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

EDT

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

Eeff

effective axial dispersion coefficient, same units ED, Eq. (17-68)

F

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

hp

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

hw

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

HETP

height of equilibrium plate, cm/plate, Eq. (17-78b)

kf

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

km,c

lumped parameter mass transfer coefficient with concentration driving force, m/s or cm/s, Eqs. (17-56a) and (17-57a)

km,q

lumped parameter mass transfer coefficient with amount adsorbed driving force, m/s or cm/s, Eqs. (17-56b) and (17-57b)

KAB

mass action equilibrium constant for monovalent-monovalent ion exchange, dimensionless, Eq. (17-40a)

KA,c

adsorption equilibrium constant in terms of concentration, units are (concentration)−1

K′i,c

linearized adsorption equilibrium constant in terms of concentration, units are units of q/c, Eq. (17-6b)

KAo

pre-exponential factor in Arrhenius Eq, (17-7a), same units as KA

KA,p

adsorption equilibrium constant in terms of partial pressure, units are (pressure)−1

K′A,p

linearized adsorption equilibrium constant in terms of partial pressure, units are units of qA/pA, Eq. (17-5b)

Kd

size exclusion parameter, dimensionless

KDB

mass action equilibrium constant for divalent-monovalent ion exchange, same units as cT/cRT, Eq. (17-41)

KDE

Donnan exclusion factor, dimensionless, Following Eq. (17-44)

L

length of packing in column, m or cm

LMTZ

length of mass transfer zone, Figure 17-23, m or cm

M

molecular weight of solute, g/g mole or kg/kg mole

Mi

multipliers in Eqs. (17-29), dimensionless

N

equivalent number of plates in chromatography, Eq. (17-78)

NPe

Peclet number, dimensionless, Eq. (17-62)

pA

partial pressure of species A, mm Hg, kPa, or other pressure units

ph

high pressure, mm Hg, kPa, or other pressure units

pL

low pressure, mm Hg, kPa, or other pressure units

PeL

Peclet number based on length, dimensionless, Eq. (17-78a)

qA

amount of species A adsorbed, kg/kg adsorbent, gmoles/kg adsorbent, or kg/liter

qA,max

maximum amount of species A that can adsorb, kg/kg adsorbent, gmoles/kg adsorbent, or kg/liter

images

amount adsorbed in equilibrium with feed concentration, same units as qA average amount of species i adsorbed, kg/kg adsorbent, gmoles/kg adsorbent, or kg/liter

images

amount adsorbed that would be in equilibrium with fluid of concentration ci, same units as qA

Q

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

rp

pore radius, m or cm

R

resolution, dimensionless, Eq. (17-82)

R

gas constant (e.g., images)

Re

Reynolds number, dimensionless, Eq. (17-60)

Sc

Schmidt number, dimensionless, Eq. (17-60)

Sh

Sherwood number, dimensionless, Eq. (17-60)

t

time, s, minutes or hours

tbr

breakthrough time, s, minutes or hours

tcenter

time center of pattern exits column, s, min or hours, Eq. (17-85b)

telution

elution time, s, minutes or hours

tF, tfeed

feed time, s, minutes or hours

tMTZ

time of mass transfer zone, Figure 17-23, s, minutes or hours

tR

retention time, s, minutes or hours

tsw

switching time in SMB, s, minutes or hours

T

temperature,°C or K

Tamb

ambient temperature,°C or K

Ts

solid temperature,°C or K

uion,i

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

us

average solute velocity, m/s or cm/s

images

average of solute velocities for A and B, cm/s, Eq. (17-83)

us,ion,i

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

ush

shock wave velocity, m/s or cm/s

ush,ion,i

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

uth

thermal wave velocity, m/s or cm/s

utotal_ion

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

vA,product

interstitial velocity of A Product if it was in the column, m/s or cm/s = (A Product)/(εe Ac)

vB,product

interstitial velocity of B Product if it was in the column, m/s or cm/s = (B Product)/(εe Ac)

vD

interstitial velocity of desorbent if it was in the column, m/s or cm/s = D/(εe Ac)

vFeed

interstitial velocity of feed if it was in the column, m/s or cm/s = F/(εe Ac)

vinter

interstitial velocity, m/s or cm/s, Eq. (17-2b)

vsuper

superficial velocity, m/s or cm/s, Eq. (17-2a)

Vavailable

volume available to molecule, m3, Eq. (17-1c)

Vcolumn

column volume, m3

Vfeed

volume feed gas, m3

Vfluid

volume available to fluid, m3, Eq. (17-1a)

Vpurge

volume purge gas, m3

wA, wB

width of chromatographic peak, s, min or hours

W

weight of the column per length, kg/m

x

deviation from the location of the peak maximum, dimensionless Eq. (17-79)

xl

deviation from peak maximum in length units, Eq. (17-80b)

xt

deviation from peak maximum in time units, Eq. (17-80a)

x

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

xi

= ci/cT equivalent fraction of ion in solution, dimensionless

Xbreakthrough (z,t)

general solution for column breakthrough for linear isotherms, same units as c, Eq. (17-72)

y

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

yi

= cRi/cRT equivalent fraction of ion on resin, dimensionless

Z

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

Greek letters

βstrong

ratio velocities of strong and weak solutes, Eq. (17-27), dimensionless

Δc

change in solute concentration, same units as c

ΔHads

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

ΔpA

change in partial pressure, kPa, atm, etc.

Δq

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

Δt

change in time, s, min or hour

ΔTf

change in fluid temperature,°C or K

Δz

increment of column length, m

γ

volumetric purge to feed ratio in PSA, dimensionless, Eq. (17-26)

εe

external porosity, dimensionless

εp

internal or pore porosity, dimensionless

εT

total porosity, dimensionless, Eq. (17-1b)

ρb

bulk density of adsorbent, kg/m3, Eq. (17-3b)

ρf

fluid density, kg/m3

images

molar density of fluid, kg mole/m3

ρp

particle density, kg/m3, Eq. (17-3a)

ρs

structural density of solid, kg/m3

σ

standard deviation of Gaussian chromatographic peak, Eq. (17-79)

σl

standard deviation in length units, m or cm, Eq. (17-80b)

σt

standard deviation in time units, min or s, Eq. (17-80a)

τ

tortuosity, dimensionless, Eq. (17-4)

ζ

Greek letter zeta used as dummy variable in Eq. (17-70)

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