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Physical Water and Wastewater Treatment Processes 9-109

TA BLE 9.12 (continued) Approximate Values of Freundlich Adsortion Isotherm

Parameters for Various Organic Substances Sorbed onto Filtrasorb 300™ Activated Carbon

from Distilled Water at 22°C

Contaminant

(mg/g)(L/mg)

1/n

Fluoranthene, pH 5.3 664 0.61

Fluorene, pH 5.3 330 0.28

2,4-Dinitrophenol, pH 3 to 7 5.5 1

2,4-Dinitrophenol, pH 9.0 1.3 1.4

Guanine, pH 3.0 75 0.48

Guanine, pH 7 to 9 120 0.40

Heptachlor, pH 5.3 1220 0.95

Heptachlor epoxide, pH 5.3 1038 0.70

Hexachlorobenzene, pH 5.3 450 0.60

Hexachlorobutadiene, pH 5.3 258 0.45

Hexachlorocyclopentadiene, pH 5.3 370 0.17

Hexachloroethane, pH 5.3 96.5 0.38

Hydroquinone, pH 3.0 90 0.25

Isophorone, pH 5.5 32 0.39

Methylene chloride, pH 5.8 1.30 1.16

4,4’-Methylene-bis(2-chloroaniline), pH 7.5 190 0.64

Naphthalene, pH 5.6 132 0.42

a-Naphthol, pH 3 to 9 180 0.32

b-Naphthol, pH 3 to 9 200 0.26

a-Naphthylamine, pH 3.0 140 0.25

a-Naphthylamine, pH 7 to 9 160 0.34

b-Naphthylamine, pH 7.5 150 0.30

r-Nitroaniline, pH 3 to 9 140 0.27

Nitrobenzene, pH 7.5 68 0.43

4-Nitrobiphenyl, pH 7.0 370 0.27

2-Nitrophenol, pH 3.0 101 0.26

2-Nitrophenol, pH 5.5 99 0.34

2-Nitrophenol, pH 9.0 85 0.39

4-Nitrophenol, pH 3.0 80.2 0.17

4-Nitrophenol, pH 5.4 76.2 0.25

4-Nitrophenol, pH 9.0 71.2 0.28

N-Nitrosodiphenylamine, pH 3 to 9 220 0.37

N-Nitrosodi-n-propylamine, pH 3 to 9 24.4 0.26

p-Nonylphenol, pH 3.0 53 1.04

p-Nonylphenol, pH 7.0 250 0.37

p-Nonylphenol, pH 9.0 150 0.27

PCB 1221, pH 5.3 242 0.70

PCB 1232, pH 5.3 630 0.73

Pentachlorophenol, pH 3.0 260 0.39

Pentachlorophenol, pH 7.0 150 0.42

Pentachlorophenol, pH 9.0 100 0.41

Phenanthrene, pH 5.3 215 0.44

Phenol, pH 3 to 9 21 0.54

Phenylmercuric acetate, pH 3 to 7 270 0.44

Phenylmercuric acetate, pH 9.0 130 0.54

Styrene, pH 3 to 9 120 0.56

1,1,2,2-Tetrachloroethane, pH 5.3 10.6 0.37

Te trachloroethene, pH 5.3 50.8 0.56

1,2,3,4-Tetrahydronaphthalene, pH 7.4 74 0.81

Thymine, pH 3 to 9 27 0.51

To luene, pH 5.6 26.1 0.44

1,2,4-Trichlorobenzene, pH 5.3 157 0.31

1,1,1-Trichloroethane, pH 5.3 2.48 0.34

9-110 The Civil Engineering Handbook, Second Edition

The abscissal intercept, –a

, is the smallest possible EBCT that produces the required efﬂuent quality.

The ordinal intercept, a

, represents that portion of the bed that is not exhausted when breakthrough

occurs. It represents the length of the mass transfer zone. If the service time to column saturation, t

sat

is determined, the length of the MTZ is,

(9.255)

where H = the bed length (m or ft)

mtz

= the length of the mass transfer zone (m or ft)

sat

= the service time to bed saturation (sec)

It is desirable to minimize the MTZ so as to maximize the efﬁciency of carbon usage. The length of

the MTZ can be reduced by reducing the ﬂow rate, but this increases the required column size.

The carbon usage rate for conventional GAC columns can be estimated as follows:

(9.256)

where R

= the carbon usage rate (kg/s or lb/sec)

V = the empty bed volume (m

or ft

)

e = the bed porosity when ﬁlled with carbon (dimensionless)

= the bulk density of the dry carbon (kg/m

or lb/ft

)

In this case, a portion of the carbon removed from the adsorber (equal to the volume of the MTZ) is

not saturated with adsorbate and does not require regeneration. Pulsed column operation overcomes

this deﬁciency, and the “base carbon usage rate” for a pulsed column is calculated as follows:

(9.257)

where Q = the water ﬂow rate (m

/s or ft

/sec)

bcu

= the base carbon usage rate (kg/s or lb/sec).

TA BLE 9.12 (continued) Approximate Values of Freundlich Adsortion Isotherm

Parameters for Various Organic Substances Sorbed onto Filtrasorb 300™ Activated Carbon

from Distilled Water at 22°C

Contaminant

(mg/g)(L/mg)

1/n

1,1,2-Trichloroethane, pH 5.3 5.81 0.60

Tr ichloroethene, pH 5.3 28.0 0.62

Tr ichloroﬂuoromethane, pH 5.3 5.6 0.24

2,4,6-Trichlorophenol, pH 3.0 219.0 0.29

2,4,6-Trichlorophenol, pH 6.0 155.1 0.40

2,4,6-Trichlorophenol, pH 9.0 130.1 0.39

Uracil, pH 3 to 9 11 0.63

p-Xylene, pH 7.3 85 0.19

Note: Filtrasorb 300™ is a trademark of Calgon Corporation, Inc.,

Source: Dobbs, R.A. and Cohen, J.M. 1980. Carbon Adsorption Isotherms for Toxic Organics,

EPA-600/8–80–023, U.S. Environmental Protection Agency, Ofﬁce of Research and Develop-

ment, Municipal Environmental Research Laboratory, Cincinnati, OH.

ttH

mtz

sat b

sat

()

re1

bcu

()

re1

Physical Water and Wastewater Treatment Processes 9-111

The Logistic Model

A purely empirical procedure has been developed by Oulman (1980). His model is based on the obser-

vance that the breakthrough curve is sigmoidal and can be approximated by the logistic equation:

(9.258)

where a = an empirical constant (dimensionless)

b = an empirical coefﬁcient (per sec)

C = the efﬂuent contaminant concentration at time t (kg/m

or lb/ft

)

= the inﬂuent contaminant concentration (kg/m

or lb/ft

)

t = the service time (sec)

This is actually a rearrangement of the Bohart–Adams equation, and the parameters in the exponential

can be related to column properties and hydraulic loading as follows:

(9.259)

(9.260)

where H = the bed depth (m or ft)

= the Bohart–Adams adsorption rate coefﬁcient (per s)

= the adsorption capacity of a carbon bed at bed saturation (kg substance/kg C or lb

substance/lb C)

= the interstitial water velocity in a packed bed (m/s or ft/sec)

The logistic equation can be linearized by rearrangement, and the coefﬁcients a and b are the intercept

and slope of plot:

(9.261)

The only unknowns are the adsorption rate coefﬁcient, k

, and the bed capacity, q

, and these may be

calculated from the slope and intercept of the plot.

Clark, Symons, and Ireland (1986) derived a modiﬁed logistic formula by assuming that mass transfer

to the carbon was controlled by transport through the water ﬁlm, and that the solute concentration at

the grain surface was in equilibrium with the quantity of adsorbed material. Freundlich’s isotherm was

used to describe the equilibrium. The efﬂuent concentration vs. service time plot is,

(9.262)

(9.263)

(9.264)

abt

()

kqH

bkC

abt

nkv

()

9-112 The Civil Engineering Handbook, Second Edition

where k

= the liquid ﬁlm mass transfer coefﬁcient (m/s or ft/sec)

n = an empirical exponent in the Freundlich adsorption isotherm (dimensionless)

r = an empirical coefﬁcient (per sec)

= the velocity of the adsorption zone through an activated carbon bed (m/s or ft/sec)

Equation (9.262) can be linearized as follows for curve ﬁtting:

(9.265)

The logistic equation assumes that the breakthrough curve is symmetrical. When this is not the case,

the modiﬁed logistic equation may give a better ﬁt. The derivation of the modiﬁed logistic model shows

that the coefﬁcients A and r will vary with the liquid velocity in the bed, and they do (Clark, 1987).

However, because the model ignores the effect of surface diffusion with the carbon grain pores, which

is probably the rate-limiting process, Eqs. (9.263) and (9.264) cannot be used to calculate A and r and

they should be determined empirically.

Packed Column Theory

A commonly used model is homogeneous surface diffusion model (HSDM) developed by Crittenden

and Weber. This model was developed for the adsorption of single components.

It is assumed that the rate of adsorption is controlled by the rate of diffusion of adsorbed molecules

along the walls of the pores inside the carbon grains. The water phase is assumed to be dilute everywhere,

so that bulk ﬂow in the pores and concentration effects on diffusivities can be ignored. The pore surface

area is supposed to be uniformly distributed throughout the carbon particles, and the mass balance for

adsorbate adsorbed to the carbon surface is written in spherical coordinates as follows:

(9.266)

where D

= the surface diffusivity (m

/s or ft

/sec)

q = the mass of adsorbate per unit mass of carbon (kg adsorbate/kg C or lb adsorbate/lb C)

r = the radial distance from the center of the carbon grain (m or ft)

t =elapsed time (sec)

The water phase at any level in the bed is assumed to be of uniform composition (except for the

boundary layer) and without channeling. Mass transfer from the interstitial water to the carbon grains

is represented as transport across a surface ﬁlm. The mass balance for the adsorbate dissolved in the

water ﬂowing through the bed is as follows (James M. Montgomery, Consulting Engineers, Inc., 1985),

(9.267)

where a

= the radius of an activated carbon grain (m or ft)

C = the bulk water phase adsorbate concentration (kg/m

or lb/ft

)

= the adsorbate concentration at the water–carbon interface (kg/m

or lb/ft

)

= the water phase dispersion coefﬁcient (m

/s or ft/sec)

= the liquid ﬁlm mass transfer coefﬁcient (m/s or ft/sec)

= the interstitial water velocity in a packed bed (m/s or ft/sec)

z = the distance from the inlet of a granular activated carbon bed (m or ft)

ln ln

Art

-1

∂

=+◊

∂

accumulation in water

dispersive flux advective flux

transfer to carbon

{

1234123

124443444

=--

()

Physical Water and Wastewater Treatment Processes 9-113

e = the ﬁlter bed porosity (dimensionless)

f = the sphericity, the ratio of the surface area of an equivalent volume sphere to the surface

area (not counting internal pores) of the carbon grain (dimensionless)

At the interface between the carbon grain and the interstitial water, there is an equilibrium between

the adsorbed material and the dissolved material, which can be represented by some suitable isotherm

such as the Freundlich:

(9.251)

The interfacial area per unit bed volume is 3(1 – e)/a

Equations (9.266) and (9.271) are connected by the requirement that solute disappearing from the

water appears as adsorbate in the carbon. The transfer of solute to the carbon can be represented on an

areal ﬂux basis,

(9.268)

where r

= the particle density (kg/m

or lbm/ft

), or a bed volume basis,

(9.269)

The value of q(r,z,t) is given by Eq. (9.269) with the condition that q(a

,z,t) be q

The relevant boundary conditions are as follows:

(9.270)

(9.271)

(9.272)

(9.273)

(9.274)

Numerical procedures for solving these equations are presented by Weber and Crittenden (1975) and

by Roy, Wang. and Adrian (1993).

A number of parameters must be evaluated before the model can be used:

•Freundlich isotherms constants

• Liquid ﬁlm mass transfer coefﬁcient

• Liquid ﬁlm diffusivity

•Surface diffusivity

These parameters can be estimated using various published correlations, which are listed below, or by

analyzing batch or continuous ﬂow adsorption tests.

qkC

kC C

∂

()

kCC

qr dr

()

∂

Czt C

()

=+ ◊ =

∂

at 0

∂

zH==0 at

∂

r==00 at

qrt t,

()

==00 at

Czt t,

()

==00 at

9-114 The Civil Engineering Handbook, Second Edition

The Freundlich isotherm parameters must be determined experimentally for the substance of interest

and the intended carbon. Typical values for a variety of substances are given in Table 9.12.

The mass transfer coefﬁcient can be estimated using any one of several correlations:

Williamson, Bazaire, and Geankoplis (1963):

(9.275)

where D

= the molecular diffusivity of the adsorbate in the bulk water (m

/s or ft

sec)

= the diameter of a carbon grain (m or ft)

Re = the particle Reynolds number (dimensionless)

= rn

Sc = the Schmidt number (dimensionless)

= m/rD

m = the dynamic viscosity of water (N·s/m

or lbf·sec/ft

)

r = the water density (kg/m

or lbm/ft

)

Wakao and Funazkri (1978):

(9.276)

where Re = the modiﬁed particle Reynolds number (dimensionless), = ren

/m.

The liquid phase solute diffusivities can be estimated in several ways:

Stokes-Einstein (Einstein, 1956):

(9.277)

where k =Boltzmann’s constant (1.380 658•10

–23

J/K or approximately 0.5658•10

–23

ft·lbf/°R)

T = the absolute temperature (K or °R).

Polson (1950):

(9.278)

where D

= the diffusivity in cm

= the relative molecular weight (dimensionless).

Wilke-Chang (Wilke and Chang, 1955; Hayduk and Laudie, 1974):

(9.279)

where D

= the diffusivity of species i in water in cm

= the molecular weight of water in g

= 18 g

mbp

= the molar volume of the liquid phase of species i at its boiling point in cm

/g-mole

m = the viscosity of water in centipoise (cp)

Note that 1 cp = 0.001 N·s/m

(exactly).

◊ = ◊ ££

Sc Re Re

058 066

240 008 125

..;

=+ ◊ ££211 3 10

06 13 4

Re Sc Re;

6pm

=¥◊

274 10

513

mbp

=¥ ¥

74 10

Physical Water and Wastewater Treatment Processes 9-115

Perry and Chilton (1973) recommend the Le Bas volumes in place of n

mbp

if values for the latter cannot

be found. Table 9.13 gives the Le Bas volumes for common structural components of organic substances.

Surface diffusivities of volatile organic substances and polycyclic aromatic substances can be estimated

by the Suzuki–Kawazoe (1975) correlation:

(9.280)

where D

= the surface diffusivity in cm

T = the adsorption temperature in K

= the boiling point of the organic substance in K

It is expected that the coefﬁcient in front of the exponential will depend only on the carbon used and

not on the adsorbate. The correlation was developed using Takeda HGR 513™, which is derived from

coconut shell and has a mean grain size of 3.41 mm and a specﬁc surface area of 1225 m

/g.

A simpliﬁed approach has been developed by Hand, Crittenden and Thacker (1984) that eliminates

the need for computer simulations. They ﬁtted an empirical formula to the results of many computer

solutions. The equations and boundary conditions are ﬁrst nondimensionalized by introducing the

following dimensionless variables and groups:

TA BLE 9.13 Components of the Molar Volume of Liquids

Structural Component

Molar Volumes

(cm

/g-mol)

Molar Volumes

(cm

/g-mol)

Bromine, add 27.0 —

Carbon, add 14.8 16.5

Chlorine, end (R-Cl), add 21.6 19.5

Chlorine, inner (R-Cl-R), add 24.6 19.5

Fluorine, add 8.7 —

Hydrogen, add 3.7 1.98

Iodine, add 37.0 —

Nitrogen, double bonded, add 15.6 5.69

Nitrogen, triple bonded, add 16.2 5.69

Nitrogen, primary amines (RNH

), add 10.5 5.69

Nitrogen, secondary amines (R

NH), add 12.0 5.69

Nitrogen, tertiary amines (R

N), add 10.8 5.69

Oxygen, if not combined as follows, add 7.4 5.48

Oxygen, in methyl esters, add 9.1 5.48

Oxygen, in methyl ethers, add 9.9 5.48

Oxygen, in other esters or ethers, add 11.0 5.48

Oxygen, in acids, add 12.0 5.48

Oxygen, combined with N, P, or S, add 8.3 5.48

Phosphorus, add 27.0 —

Sulfur, add 25.6 17.0

3-Member rings, deduct 6.0 —

4-Member rings, deduct 8.5 —

5-Member rings, deduct 11.5 —

6-Member rings (benzene, cyclohexane), deduct 15.0 20.2

Anthracene ring, deduct 47.5 20.2

Naphthalene ring, deduct 30.0 20.2

Le Bas, 1915. The Molecular Volumes of Liquid Chemical Compounds, Longmans, London.

[Quoted in Perry, R.H. and Chilton, C.H. 1973. Chemical Engineer’s Handbook, 5th ed.,

McGraw-Hill, Inc., New York.]

Bennet, C.O. and Myers, J.E. 1974. Momentum, Heat, and Mass Transfer, 2nd ed.,

McGraw-Hill, Inc., New York.

=¥ ◊

11 10

532

9-116 The Civil Engineering Handbook, Second Edition

(9.281)

(9.282)

(9.283)

(9.284)

(9.285)

(9.286)

(9.287)

(9.288)

(9.289)

(9.290)

where Bi = the Biot number (dimensionless)

= the adsorbate’s inﬂuent concentration (kg/m

or lb/ft

)

= the solute distribution parameter (dimensionless)

= the diffusivity modulus, the ratio of the Stanton to Biot numbers (dimensionless)

H = the depth of the carbon bed (m or ft)

= the mass of adsorbate per unit mass of activated carbon at equilibrium with the inﬂuent

concentration C

(kg adsorbate/kg C or lb adsorbate/lb C)

St = the modiﬁed Stanton number (dimensionless)

T = the throughput, the ratio of the adsorbate fed to the amount of adsorbate that may be

adsorbed at equilibrium (dimensionless)

The resulting transport equations and boundary conditions are as follows (Hand, Crittenden, and

Thacker, 1984):

(9.291)

Bi =

()

1 e

()

sgh

St =

()

t 1

◊ =- - ◊◊ -

()

∂

Physical Water and Wastewater Treatment Processes 9-117

(9.292)

(9.293)

(9.294)

(9.295)

(9.296)

(9.297)

The nondimensional Freundlich isotherm is

(9.298)

If the solute distribution parameter, D

, is greater than about 50, the Biot number, Bi, is greater than

about 0.5, and the Freundlich exponent, 1/n, is less than 1, the numerical solution is controlled by Bi,

1/n,

and the Stanton number, St. Hand, Crittenden, and Thacker (1984) summarized the results of

numerous computer simulations as parameters in the following empirical formula:

(9.299)

Equation (9.299) gives the processing time in terms of the throughput parameter, T, required to achieve

a speciﬁed efﬂuent concentration, C. The breakthrough ratio, C/C

, must lie between about 0.02 to 0.99.

Values for the parameters are given in Table 9.14. There are several apparent discontinuities in the

parameter values, which make interpolation hazardous. However, a three-dimensional plot of the coef-

ﬁcients reveals a smoother pattern, and it should be used as the basis of interpolation.

Equation (9.299) is valid only if the bed contacting time is deep enough to develop a mass transfer

zone. Minimum bed contacting times depend on the minimum Stanton number, which can be estimated

from empirical correlation:

(9.300)

Values of the correlation parameters are given in Table 9.15. The minimum bed contacting time is

given by:

(9.301)

∂

◊◊

∂

= ◊ -

()

∂

r =

Cz T= ≥

()

=001,

CT D z T

()

<£ <

qr zT010100££ ££ =

()

=,,

TA A

110.

St Bi

min

=+◊BB

o 1

min

()

St a

9-118 The Civil Engineering Handbook, Second Edition

TA BLE 9.14 Parameter Values for the Hand–Crittenden–Thacker

Breakthrough Formula

Equation (9.299) Parameters

1/n Bi A

0.05 0.5 –5.447 6.599 0.02657 0.01938 20.45

0.05 2.0 –5.468 6.592 0.004989 0.004988 0.5023

0.05 4.0 –5.531 6.585 0.02358 0.009019 0.2731

0.05 6.0 –5.607 6.582 0.02209 0.01313 0.2145

0.05 8.0 –5.606 6.505 0.02087 0.01708 0.1895

0.05 10.0 –5.664 6.457 0.01816 0.01994 0.1493

0.05 14.0 –0.6628 1.411 0.06071 0.02023 0.1439

0.05 25.0 –0.6628 1.351 0.03107 0.02035 0.1300

0.05 ≥100.0 0.6659 0.7113 2.987 0.01678 0.3610

0.10 0.5 –1.920 3.055 0.005549 0.02428 15.31

0.10 2.0 –2.279 3.394 0.04684 0.004751 0.3847

0.10 4.0 –2.337 3.378 0.04399 0.00865 0.2434

0.10 6.0 –2.407 3.374 0.04132 0.01255 0.1966

0.10 8.0 –2.478 3.371 0.03899 0.01628 0.1764

0.10 10.0 –2.566 3.371 0.03500 0.01939 0.1508

0.10 16.0 –2.567 3.306 0.02097 0.01948 0.1368

0.10 30.0 –2.569 3.242 0.009595 0.01961 0.1218

0.10 ≥100.0 –2.568 3.191 0.001555 0.01968 0.1101

0.20 0.5 –1.441 2.569 0.06092 0.002333 0.3711

0.20 2.0 –1.474 2.558 0.05848 0.005026 0.2413

0.20 4.0 –1.507 2.519 0.05552 0.008797 0.1875

0.20 6.0 –1.035 1.983 0.06928 0.01230 0.1679

0.20 8.0 –0.1692 1.078 0.01449 0.01550 0.1681

0.20 10.0 –1.403 2.188 0.05219 0.01842 0.1336

0.20 13.0 –1.369 2.119 0.03949 0.01845 0.1276

0.20 25.0 –1.514 2.209 0.01794 0.01851 0.1152

0.20 ≥100.0 0.6803 0.649 2.570 0.01495 0.3698

0.30 0.5 –1.7587 2.847 0.04953 0.003022 0.1568

0.30 2.0 –1.6579 2.689 0.04841 0.005612 0.1409

0.30 4.0 –0.5657 1.538 0.08445 0.008808 0.1391

0.30 6.0 –0.1971 1.119 0.1179 0.01153 0.1359

0.30 8.0 –0.1971 1.069 0.1198 0.1392 0.1327

0.30 10.0 –0.1734 1.000 0.1203 0.01594 0.1340

0.30 15.0 –0.1734 0.9194 0.07177 0.01416 0.08627

0.30 35.0 0.6665 0.4846 1.719 0.01344 0.2595

0.30 ≥100.0 0.6962 0.5170 2.055 0.01296 0.3032

0.40 0.5 –0.5343 1.604 0.09406 0.004141 0.1378

0.40 2.0 –0.1663 1.191 0.1223 0.006261 0.1348

0.40 4.0 –0.1663 1.132 0.1155 0.008634 0.1268

0.40 6.0 –0.1663 1.090 0.1123 0.01046 0.1243

0.40 9.0 0.4919 0.4918 0.4874 0.01137 0.1477

0.40 12.0 0.5641 0.4192 0.6398 0.01154 0.1490

0.40 15.0 0.6407 0.4325 1.048 0.01162 0.2127

0.40 25.0 0.6724 0.3970 1.153 0.01128 0.2169

0.40 ≥100.0 0.7414 0.4481 1.930 0.01020 0.3064

0.50 0.5 –0.04080 1.100 0.1590 0.005467 0.1391

0.50 4.0 –0.04080 0.9828 0.1116 0.008072 0.1114

0.50 10.0 0.09460 0.7549 0.09207 0.009877 0.09076

0.50 14.0 0.02300 0.8021 0.05754 0.009662 0.08453

0.50 25.0 0.02300 0.7937 0.03932 0.009326 0.08275

0.50 ≥100.0 0.5292 0.2918 0.08243 0.008317 0.07546

0.60 0.5 0.3525 0.6921 0.2631 0.005482 0.1218

0.60 2.0 0.5220 0.5042 0.3273 0.005612 0.1287

0.60 6.0 0.6763 0.3346 0.4823 0.005898 0.1389

0.60 14.0 0.7695 0.2595 0.7741 0.005600 0.1655