WASTEWATER TREATMENT

D O N A L D W . S U I M D S T R O M

and

H E R B E R T E. KLEI

Department of Chemical Engineering The University of Connecticut

\

PRENTICE-HALL, INC., Englewood Cliffs, NJ. 07632

Library of Congress Cataloging in Publication Data Sundstrom, Donald William, 1931-

Wastcwater treatment.

Includes bibliographical references and index. 1. SewagePurification. I. Klei, Herbert E.,

1935- joint author. II. Title. TD745.S85 628'.3 78-13058 ISBN 0-13-945832-8

Editorial production supervision

and interior design by: JAMES M. CHEGE

Cover design by: EDSAL ENTERPRISES

Manufacturing buyer: GORDON OSBOURNE

1979 by Prentice-Hall, Inc., Englewood Cliffs, N.J. 07632 \

All rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing from the publisher.

Printed in the United States of America

1 0 9 8 7 6 5 4 3 2 1

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CONTENTS

P r e f a c e xv

S E C T I O N I : W A S T E W A T E R C H A R A C T E R I S T I C S A N D T R E A T M E N T P R O C E S S E S

1 . W a s t e w a t e r C h a r a c t e r i z a t i o n 3

1 - 1 . Water supply and consumption 3

1 - 2 . Water quality 5

1 - 3 . Wastewater components and analysis 10

Physical characteristics 11 Chemical characteristics 13 Biological characteristics 19

vii

viii Contents

2 . W a s t e w a t e r T r e a t m e n t Processes

2 - 1 . Classification of processes 28

Pre- and primary treatment 29 Secondary treatment 29 Tertiary treatment 31

2-2. Physical-chemical treatment 32

2-3. Sludge disposal 33

2-4. Industrial wastes 37

SECTION I I . B I O L O G I C A L P R O C E S S E S

3 . Aerat ion and M a s s T r a n s f e r

3 - 1 . Aeration methods 42

3-2. Limiting resistances for mass transfer 43

3-3. Gas-liquid interface 44

Two-film theory 44 Correlations for mass transfer coefficients Penetration and surface renewal theories

^ 3-4. Liquid mixing 56

Axial dispersion description and tracer analysis Mixing with mechanical agitation 63

3-5. Mass transfer near the biomass 66

Liquid-solid resistance 66 Diffusion within the biomass 67

4. Biological M e c h a n i s m s and K ine t ics

4 - 1 . Describing chemical reactions 75

4-2. Enzymes as biological catalysts 77

4-3. Mechanisms in biological reactions 78

4-4. Growth rates in batch reactors 82

4-5. Biological reaction kinetics 84

Michaelis-Menten model 84 Other kinetic models 88

Contents

4 - 6 . Kinetic constants from batch reactor studies

4 - 7 . Kinetic constants from flow reactor studies

4 - 8 . Temperature, pH and other effects 98

M o d e l s f o r B io log ica l R e a c t o r s

5 - 1 . Basis for models 107

5-2 . Batch reactor 108

5-3 . Well-mixed reactors without recycle 110

5-4 . Well-mixed reactors with recycle 113

5-5 . Stability with recycle solids 116

5 -6 . Plug-flow reactor 118

5 -7 . Flow reactor with axial mixing 722

5 -8 . Well-mixed reactors in series 124

5-9 . Comparison of reactors 128

5 -10 . Fixed-film reactors 129.

5 - 1 1 . Trickling-filter models 131

5 -12 . Rotating biological contactor 136

5-13 . River analysis 137

Etiological W a s t e T r e a t m e n t Processes

6 - 1 . Steady-state design equations 745

6 -2 . Biological design parameters 145 Kinetic constants 145 Sfudge age 146 Sludge volume index 147 Process loading factors (F/M ratio) 149 Sludge production 149 Oxygen requirements 152

6-3 . Activated sludge processes 154 Process descriptions 154 Nitrification-denitrification 162 Design procedure 164

6-4. Anaerobic digestion 168 Chemical parameters 169 Process parameters 170

X Contents

6-5. Fixed-film biological systems 772

Trickling filters 172 Rotating biological contactor 174

S E C T I O N I I I . P H Y S I C A L P R O C E S S E S -

7 . S e d i m e n t a t i o n , T h i c k e n i n g , a n d F l o t a t i o n

7 - 1 . Classifications of settling 188

7 - 2 . Settling of discrete, particles 188

7 -3 . Ideal settling basins 190

7-4 . Tube settlers 755

7 -5 . Flocculent suspensions 196

7 -6 . Gravity thickening 755

7 -7 . Continuous thickeners 200

7 -8 . Sedimentation practice 207

. 7 - 9 . Frotation 209

7 - 1 0 . Dissolved-air flotation process 209

7 - 1 1 . Design of air flotation units 27 7

7 -12 . Flotation practice 2 7 5

8 . F i l t r a t i o n and C e n t r i f u g a t i o n

8 - 1 . Types of filters 219

8-2 . Theory of granular filters 223

8-3 . Granular filter practice 227

8-4. Theory of vacuum filtration 229

8-5 . Vacuum filtration practice 233

8-6 . Centrifugation 234

8-7 . Theory of centrifugation 235

8-8 . Centrifuge practice 237

Contents

9 . A d s o r p t i o n 247

9 - 1 . Nature of adsorbent 241

9 - 2 . Nature of adsorbate 242

9 - 3 . Adsorption equilibria 243

9-4 . Equilibrium batch adsorption 249

9-5 . Rates of adsorption 253

9 - 6 . Continuous flow adsorbers 254

9 - 7 . Adsorption column design 256

9 - 8 . Adsorption practice 269

9 - 9 . Regeneration 270

1 0 . M e m b r a n e S e p a r a t i o n Processes

1 0 - 1 . Membrane processes 275

1 0 - 2 . Reverse osmosis 2 7 7

Membrane structure and rejection mechanism Osmotic pressure 279 Transport models and flux equations 285 Concentration polarization 286

1 0 - 3 . Ultrafiltration 289

1 0 - 4 . Electrodialysis 292

S E C T I O N IV . C H E M I C A L P R O C E S S E S

1 1 . C h e m i c a l Equi l ibr ia in A q u e o u s S y s t e m s 301

1 1 - 1 . Chemical equilibrium 302

11 - 2 . Equilibrium concentration 303

11 - 3 . Electrolyte solutions 305

11 -4 . Equilibrium calculations 307

11 - 5 . Acids and bases 308

11 - 6 . Carbonate systems 310

11 - 7 . Dissolved carbonates 311

274

277

Contents

11 - 8 . Carbonate concentration diagrams 314

11 - 9 . Contact with C 0 2 gas 317

1 1 - 1 0 . Contact with solid calcium carbonate 318

1 1 - 1 1 . Carbonate equivalence points 319

1 1 - 1 2 . Alkalinity 320

1 1 - 1 3 . Neutralization 322

11 -14 . Solubility 323

1 1 - 1 5 . Phosphorus equilibria 325

11 -16 . Water softening by precipitation 327

1 1 - 1 7 . Gas stripping 331

C o a g u l a t i o n

1 2 - 1 . Properties of suspended solids 335

1 2 - 2 . Destabilization mechanisms 338

1 2 - 3 . Destabilization chemicals 341

12 -4 . Flocculation model 348

12 -5 . Applications to wastewater 350

Ion Exchange

1 3 - 1 . Materials 356

1 3 - 2 . Reactions 358

13 -3 . Capacity and selectivity 359

13-4 . Equilibria and kinetics 360

13 -5 . Fixed-bed design 361

13 -6 . Applications 363

O x i d a t i o n and D i s i n f e c t i o n

1 4 - 1 . Oxidation-reduction reactions

1 4 - 2 . Redox equilibria 370

1 4 - 3 . Iron and manganese removal

14 -4 . Cyanide conversion 374

368

373

Contents xi i i

14 -5 . Disinfection 375

14-6 . Kinetics of disinfection 376

14-7 . Chlorine 377

14 -8 . Chlorine practice 380

14-9 . Ozone 382

S E C T I O N V . S O L I D S T R E A T M E N T A N D S Y S T E M S A N A L Y S I S

15 . S l u d g e Disposal 387

15 - 1 . Sludge characteristics 387

15-2. Sludge treatment processes 389

15-3. Thermal processes 392

15-4 . Ultimate disposal 396

1 6 . S y s t e m s Analys is 398

16 - 1 . Optimization techniques 398

1V2. Cost analysis 400

1 6 - 3 . Process control 406

A p p e n d i c e s

A. Notation 421

B. Atomic weights 427

C. Physical properties of water 428

D. Solubility of oxygen in water 429

E. M P N index for bacterial concentrations 430

F. Conversion factors 431

G. Chemical kinetics 433

Index 435

5

MODELS FOR BIOLOGICAL REACTORS

Biological reactors involve a variety of geometries and hydraulic regimes. Batch or serrft-batch reactors are often used for laboratory studies, for anaerobic digestion, and for manufacture of pharmaceuticals. Flow reactors a ^ ^ ^ m r n ^ n i y ^ employed for aerobic treatment of municipal and industrial wastes.

To model a biological process, we need information on the stoichiometry and kinetics of the reactions, and on the hydraulic regime of the system. The stoichiometry of a reaction relates the quantities of reactants consumed, such as substrates, to the quantities of products formed, such as microorganisms. The hydraulic regime refers to the patterns of flow into and out of the pro-cess, and the mixing and distribution of fluids and solids within the reactor. The influent and effluent for the process are described in terms of the time variation of flow rates and concentrations of species. Any recirculation of biological solids must also be considered in the analysis.

In flow reactors, the two extremes in mixing are represented by well-stirred and plug-flow reactors. Intermediate degrees of mixing are often des-cribed by well-stirred reactors in series or by plug-flow reactors with axial dispersion. More complex mixing models can be devised but their use may not be justified because of limitations in knowledge of the system.

In this chapter, we will develop mathematical models for several types of

106 %

Sec. 5-1 Basis for Models 1 0 7

3GICAL REACTORS

geometries and hydraulic regimes, t used for laboratory studies, for e of pharmaceuticals. Flow reactors atment of municipal and industrial

ed information on the stoichiometry hydraulic regime of the system. The antities of reactants consumed, such ts formed, such as microorganisms. Tis of flow into and out of the pro-fluids and solids within the reactor, are described in terms of the time

ns of species. Any recirculation of n the analysis. n mixing are represented by well-ite degrees of mixing are often des->r by plug-flow reactors with axial 5 can be devised but their use may knowledge of the system, matical models for several types of

biological reactors. The overall model of the reactor is obtained by combining the equations for the hydraulic regime and the kinetics of the reactions.

5-1 BASIS FOR M O D E L S

A number of assumptions and approximations are used in deriving the models for this chapter.

1. The chemical kinetics of the substrate and biomass reactions are described by the Monod model including endogenous respiration. The Monod model is adequate for many steady-state processes but is often in error for rapidly changing processes.

2. The substrate is the growth-limiting substance and all other nutrients are present in excess.

/ ty The kinetic constants are independent of concentrations or the degree of conversion.

4. The yield coefficient (biomass formed/substrate consumed) is constant and independent of the age of the microorganisms. In practice, the yield coefficient depends upon the nature of the substrate and process conditions. .

5. The concentration off active biological solids is proportional to a readily measurable parameter such as mixedliquor volatile suspended solids. ~~

6. The rates of the biological reactions ate controlled either by chemical kinetics or by diffusional effects.

7. The contents of the reactor are isothermal. Since biological processes have a heat of reaction and feed conditions may change, temperature variations are possible. In most wastewater treatment reactors, short-term temperature changes are usually small.

8. Physical properties of the fluid are constant. If the average values are used, little error is introduced.

9. The transport of oxygen and substrate through the fluid is rapid relative to the rate of reaction so that concentration gradients in the bulk liquid are negligible.

10. If a thickener is used to concentrate biological solids for recycle, the reaction of substrate in the thickener is negligible.

Any or all of these assumptions can be eliminated or modified if necessary for a particular process. If the complexity of the model is increased appre-

108 Models for Biological Reactors Chap. 5 \

ciably, solution of the equations may be difficult. In many cases, knowledge of the biological process is not adequate to justify a more sophisticated analysis.

5-2 B A T C H R E A C T O R

For a reaction following the Monod kinetic model, the material balances on substrate and biomass in a batch reactor a re :

k0XS dS "dts Y(Km + S) dX _ kBXS , y dt - (Km + S) K ' X

(5-1)

(5-2)

In the previous chapter, these equations were integrated for several special cases and then applied to obtain the kinetic constants from experimental data. Once the kinetic constants are known, they can bejused to design other batch reactors operating with the same type of substrate and microorganisms.

Time FIGURE 5-1 Schematic diagram of a batch reactor.

If no simplifying assumptions can be made, Eqs. (5-1) and (5-2) are best solved by numerical procedures. For the common case where endogenous respiration can be neglected, Eqs. (5-1) and (5-2) can be integrated to give

In S = In [> + Y(S - S)|J] + ( ^ ^ J 5 * ) In [: x + Y(s - sy k0t(X + YS)

YK

In X = k0t + In X

(5-3)

Gr0 + YS) l n [ ( z 5 ) {YS + SX - x)~\ ( 5 _ 4 ) where S and X" are initial concentrations and S and X are concentrations at time t.

Sec. 5-2 Batch Reactor 109

Since Eqs. (5-3) and (5-4) are implicit in S and X, they must be solved by trial and error. With known kinetic constants, we can use these equations to design a batch reactor by calculating S and X at various intervals of time.

EXAMPLE 5-1

An organic waste is inoculated with heterogeneous microorganisms in an agitated batch reactor. After 15 min reaction time, a sample is found to contain 182mgCOD/ and 198 mg biomass/^. From previous studies, the kinetic constants for this wastejmdcyj^^ = 0.5 hr" 1 , kd =0 .01 hr*"1, Km = 75 mg COD/, Y = 0.6 mg biomass/mg COD. Estimate the COD and h biomass concentration after 90 min operation. / '

SOLUTION:

The magnitudes of Km and S are comparable and X is not large enough to { be taken as a constant. Therefore, the approximate solutions for the batch reactor material balances given in Chapte j^c^motge used. If the endogenous I respiration term is small compared with the growfrt term, we can apply Eqs. (5-3) and (5-4).

Substituting values of S and X a t / = 15 min, Eqs. (5-3) and (5-4) are

In (182) = In [ s + 0.6(5 - 182)~]

X -f 0.65 . [X* 0.6(5 - 182)1 _ 0.5(H)(* + 0.65) 0.6(75) L X J 0.6(75)

In (198) = 0.5(Jg) + In X*

in f Y l 9 8 v a 6 5 Yl L\* A0.65 0 - +X - 198/J

0.6(75) . T/198V 0.65 X + 0.65

The equations can be solved by trial and error for the initial concentrations of 5 and X

5 = 210 mg COD/^ X = 181 mg//

After 90 min residence time,

In 5 = In [210 + 0.6(210 - S ) j g j ] 181 0.6(210)

+

0.6(75) 0.5(fg)(181 4-0.6 x 210)

0.6(75)

. H81 -f 0.6(210 - 5)1 L 181 J

In * == 0 . 5 ^ ) -f in 181

0.6(75) 181 +0.6(210)

0.6(75) . [X( 0 . 6 x 2 1 0 \ LI8IVO.6 x 210 + 181 - XJ

110 Models for Biological Reactors Chap. 5 \

Solving by trial and error,

S = 40 mg COD// X = 282 mg//

To check on the assumption of negligible endogenous respiration, the magni-tudes of the growth rate and death rate terms at 90 min can be compared.

# k0XS 0.5(282X40)

Growth rate - xf~+- = 75 + 40 = 49 mg/;(hr)

Death rate = kdX = 0.01(282) = 2.8 mg/;(hr)

The endogenous rate is. about 5J/0 of the growth rate of biomass at 90 min and a smaller percentage at lower times. Neglecting the endogenous term is thus a reasonable approximation.

5-3 W E L L - M I X E D R E A C T O R S W I T H O U T RECYCLE

In a well-mixed flow reactor, the composition is uniform throughout the reactor. Thus^the exit stream from this type of reactor has the same composi-tion as the fluid within the reactor. The mixing action must be sufficient to disperse the incoming feed rapidly throughout the reactor.

A schematic diagram of a well-mixed biological reactor of volume V is shown in Fig. 5-2. The stream arriving at the process has a flow rate Q, a

Q V Q S.X s.x

FIGURE 5-2 Schematic diagram of well-mixed reactor without recycle.

substrate concentration S, and a biomass concentration X. A material balance around the reactor states

Accumulation = Input Output + Formation by reaction

Sec. 5-3 Well-Mixed Reactors without Recycle 111

For the substrate and biomass, the material balances are

K ^ = e 5 -QS+ r,V

V^= QX -QX + rxV

(5-5)

(5-6)

If the rate of reaction follows the MondJcmeUcjnj3del,

r = k S X

Y(Km + S) r k0SX 1 y r

' - ^ T T ^ k d X

Substituting these rate expressions, the material balances become

~di ~ Q S - Y ( K m + S)

= QX - QX + xS*Vs - kdXV (s-io) +

1 1 2 Models for Biological Reactors Chap. 5^

In a particular application, the appropriate substrate and biomass bal-ances can be solved simultaneously for the unknown quantities. For example, if feed conditions and kinetic constants are known, the material balances can be solved to give the volume_of_the reactor required to reduce the effluent^u^slrate'cor7cen^ to a desired level.

The matelTalTalaiices are often written in terms of residence time, which is defined as 6 = V/Q. The residence time is the average time that a fluid element spends in tKe reactor. For a well-mixed reactor, there is a distribu-tloiToT residence times around the average value. The steady-state Eqs. (5-11) and (5-12) in terms of residence time are

S

X* X k0SXO Km + S kdxe = o

(5-15)

(5-16)

Ifjjojtaicroojganisms enter with the feed, there is a critical flow rate at which microorganisms are washed out of the reactor faster than they are generale

Sec. 5-4 Well-Mixed Reactors with Recycle 1 1 3

output substrate concentration. This response represents an inherent "self-control" by the reactor since changesTn feed concentration do not affect the outputjubstrate concejUratirjr^The^m for this self-control action is an increase in biological solids concentration that is sufficient to handle the higher loading of substrate. ^

c

/ M 7 ?

5-4 W E L L - M I X E D R E A C T O R S W I T H R E C Y C L E

Microorganisms formed in a biological process are frequently fed back to the entrance of the reactor. Since the reactions are autocatalytic in nature, the performance of the process can be modified by recycle of biological solids. In a properly operated biological process, the main purpose of recycle -is to increase the concentration of biomass in the reactor. The addition of a recycle stream also dilutes the concentration of entering substrate and decreases the residence time of fluid elements in the reactor.

A schematic diagram of a biological reactor with recycle of biomass is shown in Fig. 5-3. The effluent from the well-mixed reactor is settled in a

Reactor Q+ Qr Clarifier

S,X0 < V.S.X s.x Clarifier

K \ \

or.s,xr QW.X,

FIGURE 5-3 Schematic diagram of well-mixed reactor with recycle.

clarifier and a portion of the concentrated sludge is returned to the reactor with flow rate Q r and concentration Xr. If the reaction of substrate in the clarifier is negligible, the recycle stream will contain the same substrate con-centration as the effluent from the reactor.

The material balances a r o u n d j h e reactor include terms for the addition of substrate and biomass with the recycle stream:

Accumulation = Feed input + Recycle input Output

+ Formation by reaction

V = QS + QrS - (2 + Qr)S - Y * X + S )

V ^ = QX + QrXr - (QQ + Qr)X + S * V S ) - kdXV

(5-21)

(5-22)

1

Models for Biological Reactors Chap. 5 x

Defining a recycle ratio R = Qr/Q and a residence time based on fresh feed as 9 = V/Q9 the material balances become

. f - i ^ - ^ - i d f e s ) ( 5 - 2 3 ) ^ = + * - (1 + + - (5-24)

The residence time based on fresh feed is a constant for a specified flow rate of entering feed. The true residence time, given by Vj(Q + Qr\ changes with the recycle ratio of the process. If the recycle ratio is used as a control variable, the residence time based on total flow will be a variable instead of a constant.

For a steady-state reactor, the material balances are

S-

S-YMTS) = (5-25)

X +RX,- (1 + R)X + xS*ds - kdX9 = 0 (5-26)

If the biomass entering with fresh feed is negligible, the material balance on biological solids is

RXr - (1 + R)X + xS*0s - kdX9 = 0 ' (5-27)

If the reactor is also operating in the growth region where endogenous respira-tion is unimportant, the material balance on biomass is

RXr - (1 + R)X + X^ = 0 (5-28)

Biological reactors generally produce an excess of biological solids that must be removed from the system during steady-state operation. The quantity of excess biomass formed is equal to the net growth of biomass in the reactor.

Net biomass = *1 - kdXV (5-29)

This excess biomass is usually wasted from the separator that produces the concentrated solids for recycle. In a waste treatment plant, these solids are sent to some type of sludge disposal process.

Two operational parameters are widely used in the design and operation of biological treatment systems. The process loading factor is the mass of

1 Reactors Chap. 5

esidence time based on fresh feed

(5-23)

(5-24)

i constant for a specified flow rate iven by Vj(Q + Qr), changes with e ratio is used as a control variable, >e a variable instead of a constant. I balances are

I k0SX9 _ o ' ~ Y{Km + S)

- kdX9 = o

(5-25)

(5-26) cBSX9

negligible, the material balance, on

Sec. 5-4 Well-Mixed Reactors with Recycle 115

substrate consumed over a finite time period (usually a day) by the mass of microorganisms in the reactor:

~ k0SXV i u lY(Km + S)j

~ XV kaS

Y(Km + S) (5-30)

The quantity U is also referred to as substrate removal velocity or food to microorganism ratio.

The other common parameter is the solids residence time. The true mean solids residence time, # m , is the ratio of the amount of solids in the system to the sum of the rates of biomass synthesis and solids input with fresh feed. If the solids are assumed to be largely in the reactor and the reactor is taken to be well mixed, then the true mean solids residence time for the conventional activated sludge process of Fig. 5-3 becomes

9m = XV XQ -{- k0XSV

(5-31) (Km + S)

At steady-state conditions, the denominator of Eq. (5-31) equals the rate at which biomass is lost from the system by outflow and endogenous respiration,

'^L-kdx9 = o (5-27)

h region where endogenous respira-on biomass is

k0SX9 _ Km + S

^ = 0 (5-28)

an excess of biological solids that steady-state operation. The quantity ,et growth of biomass in the reactor.

.-kdXV (5-29)

9m = XV QwXr + QeX, + kdXV (5-32)

where: Qw flow rate of waste from recycle line Q

= overflow rate from clarifier

Xe = biomass concentration in overflow from clarifier

Generally the mean cell-residence time has not been widely adopted since the value of kd is often not known for the influent wastewater. Instead, the term sludge age has been adopted, which is defined as the ratio of biomass in the reactor to the net rate of biomass generation:

vx Km + S kpSXV _ k x y ~ k 0 S - k^Km + S) (5-33) Km + S

_ from the separator that produces waste treatment plant, these solids process. ly used in the design and operation ocess loading factor is the mass of

The processjoadjng factor is simply related to sludge age through the kinetic parameters Yand kd:

= YU - k d (5-34)

V 5 fyM

116 Models for Biological Reactors Chap. 5 >>

Equations (5-30) and (5-33) can be solved for S to give

5 __ YUKm _ Km + KJcA * - kQ - YU ~ k09e - kj9c - 1 P " j : > ;

If the kinetic constants are known, specifying any one of the quantities S, U or 9 e determines the other two.

At steady-state conditions, the net rate of biomass generation is equal to the rate at which biomass flows out of the system. If biomass is removed by wasting from the recycle line and by losses in the clarifier overflow, the sludge age is given by

$'

=

Q.X!+Q.X. (5"36) Thus, sludgejige can be controlled by the rate of wasting of biomass. Since the sludge age omits the endogenous term from the denominator, sludge age will be greater than mean solids residence time. However, the ease of applying Eq. (5-36) to treatment plant operations encourages the use of sludge age. The sludge age has an important effect on the settling characteristics of the biomass and will be discussed further in the next chapter.

5 - 5 S T A B I L I T Y W I T H R E C Y C L E S O L I D S

In designing a biological reactor, the concentration of biomass in" the recycle stream, is needed. This' concentration depends upon the specific type of separator and its operating characteristics. Thus, the reactor and separator should be considered together in the design of a biological process with recycle. To illustrate the effect of process variables on reactor performance, two idealized models of the separator are often used. In one model, the separator is assumed to give a constant ratio of output to input solids concentration, i.e.,

f} = -ji = constant

This type of behavior might be approximated by a sedimentation vessel. In the other model, the separator is assumed to provide a recycle stream with a constant biomass concentration, i.e., XR constant. A relatively constant biomass concentration might be achieved with a centrifuge.

If the concentration ratio across the separator is assumed constant, biomass Eq. (5-28) becomes

^ ( / ? - l ) - l + ^ ^ _ = 0 or

KJl + R BR)

Sec. 5-5 Stability with Recycle Solids

The steady-state biomass concentration can then be calculated from Eq. (5-25):

Y _ ( S -S)(Km + S)Y k0Sd

(5-38)

At washout conditions, the biomass concentration drops to zero and no con-version of substrate occurs in the reactor. The critical fresh-feed residence time for washout is obtained by substituting S = S in Eq. (5-37):

0 w g ( * . + S)(l + R - PR) ( 5 . 3 9 )

According to this model of the separator, there is no conversion of substrate for fresh-feed residence times equal to or less than 6W.

If the concentration of biomass in the recycle stream is assumed to be a constant, the solution for substrate concentration involves a quadratic equa-tioji. For example, combining material balance equations (5-25) and (5-28) gives

S ^ - 6 A / y - J g (5.40)

where: a = 1 + R k09

b = k09 (s + + (Km - + R) c = -SKm(l+R)

Once S is calculate^!. JTcan be found from Eq. (5-38). * With constant recycle solids concentration, complete washout of the

reactor is not possible. The presence of cells in the entering stream ensures that cells will also exist in the reactor. As long as some biological solids are present in the reactor, some finite conversion of substrate will occur.

A comparison of reactor performance for the two separator models is shown in Fig. 5 - 4 2 . The values for the kinetic constants and recycle ratio are fairly typical for activated sludge reactors. For the constant-ratio model, a value of 4 was selected for the ratio of output to input concentrations across the separator. The effluent substrate and biomass concentrations from the reactor (curves A) were calculated from Eqs. (5-37) and (5-38). For the con-stant-concentration model, the recycle solids concentration was taken as 10,000 mg/C. In this case, Eqs. (5-38) and (5-40) were used to calculate the substrate and biomass concentrations for the reactor (curves B). As pre-dicted by Eq. (5-39) for the constant-ratio model (curves A), washout occurs at a fresh-feed residence time of 0.58 hours. Since washout is not possible at constant Xr, the conversion of substrate is finite at all residence times (curves

Models for Biological Reactors Chap. 5

2500 r

2000 -

- 1500

1000

500

A Biomass

/B

_ * 0 = 0.5 hr" 1 Km = 75 mg/

Y =0.6

_

\B \B Substrate COD

i -4 1 Substrate COD

i -4 0.5 1.0 1.5

Residence time (hr) 2.0

FIGURE 5-4 Comparison of reactor performance for two separator models.2 Curves A at constant concentration of 10.000 malt. Curves B at a constant concentration ratio of Xr/X - 4.

B). A separator giving a high stable concentration of recycle solids is desirable since reactor performance is improved during periods of high volumetric flow rates.

\ 5-6 P L U G - F L O W R E A C T O R

In a plug-flow reactor, the velocity is constant at any given cross-section and no mixing of fluid elements occurs longitudinally along the flow path. Thus, all elements of fluid have the same residence time in the reactor. Since com-position of the fluid varies from position to position along the reactor, the material balances must be made on a differential element of fluid.

A schematic diagram of a plug-flow reactor with recycle is shown in Fig. 5-5. For the differential element of volume dV, the steady-state material balance is

Input Output + Formation by reaction = 0

The terms for the substrate are

Input = (2 + Qr)S = + R)S Output = (2 + Qr)(S + dS)= Q\\ + R)(S + dS)

Formation by reaction == k0XS Y(Km + S)

dV

Sec. 5-6 Plug-Flow Reactor 119

q+ A . s,x S.. X;

~Ws~7,

1 X + dX

^*-dZr or.sxr

FIGURE 5-5 Schematic diagram of a plug-flow reactor.

The steady-state material balance on substrate is

k0XS G(l + R) dS + dV=0 Y(Km + sy

Similarly, the steady-state material balance on biomass is

(5^1)

+ R) dx - [Y^S ~KDX) DV = (5_42)

The material balances can be expressed as a function of axial position, Z, by the relation

dV == A dZ

where A is the cross-sectional area of the reactor

+R)dS+ Y{zf+S)A d z = c5-43)

e(l +R)dX- ( + s - k4x}AdZ = 0 (5-44)

The boundary conditions for these equations are obtained by material balances around the entrance where fresh feed mixes with recycle.

QS + QrS, = (2 + Qr) S, QX + QrX, = (2 + Qr) X,

where S, and X, are the concentrations of substrate and biomass in the total fluid entering the reactor. After dividing by Q and rearranging, the boundary conditions become

S - 5 + R S - at 7 - 0 l+R a t Z ~

Z t = Z l + R K a t Z = 0

(5-45)

(5-46)

120 Models for Biological Reactors Chap. 5

Equations (5-43) to (5-46) form a set of nonlinear differential equations that are difficult to solve analytically. For this general case, the equations are more readily solved by computer techniques. The equations can be integrated directly, however, in certain special cases.

For example, if the amount of biomass formed by the reaction is small relative to the amount entering the reactor, then the concentration of biomass is nearly constant along the length of the reactor. Denoting the average con-centration of biomass in the reactor as Xay the substrate material balance is

+R)dS + y^MA dZ = 0 (5-47)

Integrating with Xa as a constant gives

( S , - * ) + * In f =Y*!*-fR) (5-48)

The small change in biomass along the reactor is approximated by

(X - X,) = Y(S, - S) - Q0ff'+ZR)

Substituting for (S{ S) from Eq. (5-48) gives

x < * - X.) = ^ - } - KmY In - (5-49)

The validity of the assumption of nearly constant biomass concentration can now be checked by comparing the inlet concentration with the outlet concentration predicted by Eq. (5-49).

Plug-flow reactors are generally operated with recycle of microorganisms. In the absence of recycle, the fresh feed is the only source of biomass. Since the concentration of biomass in fresh feed is often very low, the reaction rate at the entrance of the reactor would also be low. Without longitudinal mixing in an ideal plug-flow reactor, there is no feedback mechanism for biomass. Thus, the reaction rate would remain low for a substantial length of the reactor. By recycle of biomass, the reaction rate is increased and the length of reactor for a given conversion is decreased.

As in the case of a well-mixed reactor, washout cannot occur if there is a fixed concentration of biomass in the stream entering the reactor. Washout is possible, however, if the fresh feed contains no biomass and if the separator produces a constant ratio of output to input solids concentration (j5). In this case, then, the biomass concentration in the recycle approaches zero as the

Sec. 5-6 Plug-Flow Reactor 121

biomass concentration leaving the reactor goes to zero. The critical fresh-feed residence time at washout for constant /? is given by 3

e =

(i+rxv + k j l n i R ( 5 . 5 0 )

At steady state, there is no conversion of substrate for fresh-feed residence times equal to or less than 6W. With no recycle (R = 0), 6W is infinite and the reactor washes out for all residence times.

EXAMPLE 5-2

Fresh feed enters an activated sludge plant at a flow rate of 0.088 jn 3/sec (2 mgd) and with a substrate concentration of 300 mg/C. Sludge is recycled to the reactor from a separator at a flow rate of 0.013 m 3/sec (0.3 mgd) and a biomass concentration of 6000 mg/(. If the conversion of substrate is 95 %, determine the residence time and volume for:

(a) a well-mixed reactor (b) a plug-flow reactor

The kinetic constants are: kQ = 0.4 h r " 1 ; Km = 75 mg/; Y = 0.6; kd ~ 0

SOLUTION:

(a) For a well-mixed reactor, the substrate balance is

\ The biomass balance (A' 0 = 0) is

RXr - (1 -f- R)X + | = 0 (5-28)

R = 0.3/2 = 0.15; Xr = 6000 mg/; S = 300 mg/t; S == 0.05(300) = 15 mg/^. Substituting in Eqs. (5-25) and (5-28),

300 - 15 - 0 A ^ x d - o 3 0 0 1 5

0.6(75 -f 15) ~ 0

0.15(6000) - (1 + 0.15)* + 7 5 ( 1 ^ g = 0

X = 930 mg/e Q = 2.76 hr

V = 0QO =

2.76 X 0.088 X 3600 = 874 m 3 (2.3 x 10 5 gal)

Solving,

122 Models for Biological Reactors Chap. 5

(b) For a plug-flow reactor, the substrate balance is

G(l +R)dS+ y{K*+

S) DV= (5_43) The biomass balance is

ed + R)dX- KmX+SsdV = 0 (5-44)

At V = 0,

X / = = X 1 ( 5 _ 4 6 )

Substituting in the above equations:

o 4

(0.088 x 3600X1 + 0.15) dS + Q 6(75 -j . 5) ^F = 0

(0.088 x 3600X1 + 0.15) dX - r/K = 0 ^ -

3^

+ U 5 5 ( 1 5 ) = = 2 6 3 m ^

Since the amount o{ biomass formed in the reactor is fairly large relative to the biomass entering, the equations were solved numerically. Small increments in Kwere assumed and 5 and A"were determined. The required reactor volume was found at S = 15 mg/.

X = 930 mg/{ V = 288 m3 (76,000 gal) 9 = 0.91 hr

Equations (5-48) and (5-49) will give a reasonable approximation of the results.

The plug-flow reactor is about \ the size of the well-mixed reactor.

5 -7 F L O W R E A C T O R W I T H A X I A L M I X I N G

Plug flow represents an idealized flow pattern in which all fluid elements have identical residence times. In actual reactors, some degree of mixing occurs in the axial direction of the reactor. For example, activated sludge reactors are often designed as long tanks with aeration of the liquid along the lengthof

Chap. 5 Sec. 5-7 Flow Reactor with Axial Mixing 123

(5-43)

(5-44)

(5-45)

(5-46)

5 dV = 0

git

tit-

or is fairly large relative to merically. Small increments "he required reactor volume

ible approximation of the

ie well-mixed reactor.

ING

which all fluid elements have degree of mixing occurs in

activated sludge reactors are he liquid along the length-of

the tank. The turbulence created by this aeration process causes fluid mixing in the flow direction.

The plug-flow model can be modified to account for axial mixing by adding a dispersion term. In a dispersion-flow model, the rate of axial mixing is assumed to be proportional to the concentration gradient of the diffusing component in the reactor. For the differential element of Fig. 5-5, the dif-fusion terms for substrate are :

Input by diffusion = DXA dS dZ

Output b y diffusion = ~[d,A j | + ^(pxA ^ dZ^

where Dg is the eddy diffusivity or axial dispersion coefficient for axial mixing with units of length 2/time. If the eddy diffusivity is assumed constant, the net rate of longitudinal dispersion is given by

Net diffusion = -DXA ^ dZ

By adding this term to the plug-flow material balance, Eq. (5-43) becomes

Q%l+R)dS-D2A^2dZ + k0XS dZ*~ ' Y(Km + S) A dZ = 0 (5-51)

In a similar manner, when an axial dispersion term is added to Eq. (5-44), the material balance for biomass becomes

e ( l + R) dX - DZA ^ d Z - ( ^ p ^ - kdx) AdZ = 0 (5-52)

Several different boundary conditions have been suggested for this type of second-order equation. The boundary conditions proposed by Danckwerts 4

are used frequently. At the entrance where Z = 0,

(2 + Qr)St = (2 + Qr)S,.o -

(2 + Qr)x, = (2 + Qr)x,-o - D,(4g) A

(5-53)

At the exit where Z = L,

(5-54)

124 Models for Biological Reactors Chap. 5 ^

If Eq. (5-51) and (5-52) are multiplied by the length of the reactor L 9 they can be rearranged to give

dS_-e k X S = 0 (5-55) \uL)d(ZILy d(Z(L) Y(Km + S) V K D > }

0(l 4- R) where: u = ^ v . mean fluid velocity

6 = = mean residence time u

. *

The combination uL/D2 is a dimensionless group known as the Peclet number, which can be used to characterize the degree of mixing in the vessel. For a plug-flow reactor with no axial mixing, the eddy diffusivity is zero so that the^eclet number is infinite. The diffusion terms can then be dropped from the material balances. Thus, a plug-flow reactor is approximated by the dis-persion model with a very large Peclet number. In a well-mixed reactor, the eddy diffusivity is infinite and the Peclet number becomes zero. Therefore, the dispersion model with a very low Peclet number approximates the con-ditions in a well-mixed reactor.

Because of the complexity of the nonlinear differential equations, an analytical solution is not available even for the case of constant biomass concentration in th^ reactor. The equations are best solved by numerical methods implemented on a digital computer.

5-8 W E L L - M I X E D R E A C T O R S I N S E R I E S

A sequence of well-mixed reactors can give higher conversion than a single well-mixed reactor with the same total volume. Also, the model for well-mixed reactors in series is useful in simulating the performance of certain designs of activated sludge reactors.

A schematic diagram of N well-mixed reactors in series is shown in Fig. 5-6. The concentrated biomass from the separator is recycled to the first reactor. For the /?th reactor in the sequence, the material balances on sub-strate and biomass are given by

vn ^ = e(i + - e(i + R)sa - Y{fnX+s) (5"57)

= eo + ~ eo + w + g g f f i ^ - k^vn * (5-58)

Reactors Chap. 5

y the length of the reactor L,

- Q kaXS 0 (5-55)

= (5-56)

locity

roup known as the Peclet number, ee of mixing in the vessel. For a e eddy diffusivity is zero so that *ms can then be dropped from the tor is approximated by the dis-iber. In a well-mixed reactor, the lumber becomes zero. Therefore, et number approximates the con-

llinear differential equations, an for the case of constant biomass >ns are best solved by numerical er.

ERIES

re higher conversion than a single olume. Also, the model for well-lating the performance of certain

reactors in series is shown in Fig. separator is recycled to the first

ice, the material balances on sub-

126 Models for Biological Reactors Chap. 5 *

Defining the residence time on the basis of fresh-feed flow rate, the material balances become

-dT--ffr{S-1 S">-Y(Km + Sn) ( 5 " 5 9 )

3 V^ii-l ^ n / "T "P j FT KdAn dX,_ (5-60)

where:

Sec. 5-8 WeII-Mixed Reactors in Series 127

In the limiting case where the number of reactors in series becomes infi-nite, the conversion in each reactor is infinitesimal and the performance characteristics of the reactor system are identical to those for a plug-flow reactor. Thus, a finite number of well-mixed reactors in series can be used to simulate intermediate degrees of mixing between a single well-stirred reactor and a plug-flow reactor.

EXAMPLE 5-3

Determine the residence time and reactor volume if the feed of Example 5-2 is treated in 3 equal-sized well-mixed reactors in series.

SOLUTION

Equations (5-61) and (5-62) with X0 = 0 and kd 0 can be applied to each reactor in series.

Reactor 1:

/c c* koS\X\0\ n

^ i , }

Y(Km + STxi + r) - 0 rxr v \ _i kpS\X\6\

A

Reactor 2:

/ R * C \ koS2X202 _ n Ki>\ ^ Y(.Km+S1){\+R)-0

(Y koS2X202 _ n x {X, x2) + { K m - S i ) { x + R ) - o Reactor 3:

/c c\ kpS-jXiB-j N

(v V ,* _ I _ kySiXiOi n (Xz - +

{ K m + SMI + R) ~

Input concentrations to first reactor:

S a = 1J = 300 +0-1*13) = 2 6 3 m g J (

For equal volumes, 0X = 02 = # 3 . Since S3 is specified as 15 mg/, the unknowns are Su Xu S2t X2, X2f and 0. The six simultaneous equations can be solved algebraically but the expressions become very complicated. Instead, the equations were solved by trial and error. Values for 0 were

128 Models for Biological Reactors Chap. 5 v

assumed until .S3 = 15 mg/. The results were:

0 , = 02 = 02 = 0.42 hr 0 t o u i = 3 x 0.42 - 1.26 hr

Vx = V2 = K3 = 132 m3 (35,000 gal) Ktoui = 396 m3 (105,000 gal)

As expected, the 3 well-stirred reactors in series are intermediate between plug-flow and well-stirred.

5-9 C O M P A R I S O N O F R E A C T O R S

The well-mixed and plug-flow- models represent the two extremes in flow behavior for a biological reactor. The well-mixed model assumes that the influent stream is dispersed instantaneously and uniformly throughout the contents of the vessel. The plug-flow model assumes that no mixing occurs along the flow path and that all fluid elements have identical residence times.

For a well-mixed reactor, the substrate concentration in the vessel is the same as the substrate concentration in the effluent. Thus, the fresh feed is immediately dispersed into an environment of lower concentration. In a plug-flow reactor, the substrate concentration decreases continuously along the length of the\vessel. If other conditions are the same, a higher substrate concentration gives a higher rate of reaction. As a result, a plug-flow reactor generally produces a higher conversion of substrate in a given volume than a well-mixed reactor.

When the biomass concentration entering the reactor with fresh feed or recycle is very low, the plug-flow reactor may require a larger volume than a well-mixed reactor. If little biomass is present in the influent to a plug-flow reactor, the rate of reaction is very low in the entrance region but gradually increases as biomass is generated by the reaction. If no biomass enters a plug-flow reactor, no biological reaction can occur and the reactor washes out. On the other hand, the influent to a well-stirred reactor is mixed with vessel fluid containing biomass so that the reaction can be sustained even in the absence of biomass in the feed stream.

The well-mixed reactor is generally more stable than a plug-flow reactor in response to toxic and shock loadings. For example, if a concentrated pulse of a toxic substance enters a plug-flow reactor, the concentration remains high as it moves along the reactor and into the clarifier. Because of the high con-centration, the toxic substance may destroy an appreciable quantity of the biomass in the system and cause a long-term upset in reactor performance.

Sec. 5-10 Fixed-Film Reactors 1 2 9

With a well-mixed reactor, the pulse of toxic material is dispersed rapidly throughout the vessel and its concentration level is reduced. The metabolic processes of the microorganisms may be only slightly affected by the toxic substance at the lower concentration level. In general, a well-mixed reactor gives a more uniform effluent under varying feed conditions.

Another advantage of a well-mixed reactor is the uniformity of oxygen consumption rates by the microorganisms. Since concentrations and reaction rates are constant throughout the vessel, the oxygen uptake rates are also the same everywhere in the reactor. With a plug-flow reactor, the oxygen demand is usually greatest in the inlet region where the substrate concentration is high. At the exit where the substrate concentration is low, the oxygen con-sumption rate is also generally low. Thus, the aerators for a plug-flow reactor should be designed to provide more oxygen transfer in the inlet region.

The well-mixed and plug-flow reactors are idealized models that are difficult to achieve in large-scale biological reactors. In actual mixed reactors, short-circuiting of fluid and stagnant zones may occur. Some portion of the fluid may flow from the inlet to the outlet with a short residence time because of incomplete mixing with the bulk of the reactor fluid. Also, some regions of the reactor may be relatively stagnant so that fluid elements entering these zones have long residence times.

In axial-flow reactors, aeration of the fluid causes longitudinal mixing and a distribution of residence times. Thus, long biological reactors with aeration are often better simulated by an axial dispersion model or a tanks in series model. If aeration is fairly uniform along the length of the reactor, the axialMispersion model may approximate the hydraulic regime. If discrete aerators give several local zones of intense mixing along the reactor, the model for well-mixed tanks in series may be preferable. Tracer techniques are useful in establishing an appropriate hydraulic model for a biological reactor.

5-10 F I X E D - F I L M R E A C T O R S

Fixed-film reactors consist of a solid surface with an attached layer of biomass. Substrate and oxygen diffuse into the layer and react to form pro-ducts and additional biomass. The most common types of fixed-film reactors are the trickling filter and the rotating biological contactor.

A trickling filter is a bed packed with rocks or plastic structures. Waste-water is distributed over the top of the packing and allowed to trickle through the bed. Recycle of a portion of the effluent liquid is often practiced. A rotating biological contactor consists of many plastic discs attached to a central drive shaft. The discs are parallel to each other with intermediate spacing to permit movement of fluid between them. The discs are partially

130 Models for Biological Reactors Chap. 5 \

submerged in the liquid so that they are exposed alternately to substrate in the wastewater and oxygen in the air during rotation.

Microorganisms grow on the surface of the packing or discs and meta-bolize the organic substrate from the wastewater. As the microorganisms multiply, the thickness of the biological layer increases. At some thickness, the diffusing oxygen is consumed by reaction before it penetrates through the entire layer. As a result, bacteria near the solid surface are in an anaerobic environment (Fig. 5-7). In addition, most of the substrate is consumed before reaching the packing so that anaerobic bacteria adjacent to the surface enter the endogenous phase of growth. These bacteria have less ability to adhere to the solid support and are washed off by shear forces. A new layer of microorganisms then starts to grow at this location in the bed.

Solid packing

Biological film

Liquid layer

Organics

Oxygen Air

Products

Anaerobic Aerobic Flow

or

FIGURE 5-7 Schematic diagram of phases in a trickling filter.

Most of the biological conversion occurs in the aerobic portion of the film, the depth of which depends upon the rate of reaction, the diffusivity of oxygen and the liquid flow rate. Oxygen is supplied to the film by diffusion through the liquid layer. The oxygen in the liquid either enters with the wastewater or is transferred from the air. Concentration gradients of both oxygen and substrate exist in the liquid and biomass layers. Under steady-state conditions, the* mass fluxes of substrate and oxygen across the liquid biomass interface equal the corresponding rates of substrate and oxygen consumption by reaction in the biological layer.

The design of fixed-film reactors from fundamental theory is difficult because of the multiple phases involved and the flow patterns in the units. Differential equations can be written to describe the processes occurring in

Sec. 5-11 TrickUng-Filter Models 131

the liquid and biomass regions. If the velocity profiles, reaction kinetics, diffusivitics, and boundary conditions are known, these differential equations can be solved for substrate conversion.' Since the equations are difficult to solve and conditions are often not known accurately, assumptions are usually made to provide working models.

5-11 T R I C K L I N G - F I L T E R M O D E L S

Many theoretical and empirical models have been proposed to describe the performance of trickling filters 3 - 6 . A simplified reaction rate model can be developed by considering each cross-section as a pseudo-homogeneous reactor with no gradient in substrate across the b e d 7 . T h e reaction rate within the film is assumed to be reaction-rate controlled and to proceed at a rate determined by the substrate concentration in the liquid at that cross-section. For the differential element, dZ, in Fig. 5-8, a material balance on substrate gives for g= Q(l + R),

or

QS - Q(S + dS) + rsVb = 0

-QdS + rsVb =0 (5-64)

Q

\

RQ

(1 + R)CF\ S,

S + dS

z=o

dZ

z = za

Q

FIGURE 5-8 Schematic diagram of trickling filter for development of models.

132 Models for Biological Reactors Chap. 5

If the reaction follows Monod kinetics, the rate is given by

- k0XbS s

- Y(Km + S)

The volume of active biomass where reaction occurs is

Vb = aSA dZ (5-65) where: a = area of packing per unit volume of reactor

5 = thickness of active film

A = cross-sectional area.

Substituting in the material balance, Eq. (5-64) becomes

QdS+ Y^a+% d Z = 0

Sec. 5-11 TrickUng-Filter Models 133

At substrate concentrations below about 300 mg!C C O D , the thickness of the active film increases almost linearly with substrate concentration 8,

S =SS

where 5 is a proportionality constant. In this case, the material balance equation becomes

Q dS + kf(f2lSs) dZ = 0 (5-70) The integrated form of Eq. (5-70) is

" fc -MsK)-^

134 Models for Biological Reactors Chap. 5

Many empirical models have been developed for the design of trickling filters. As a particular example, Eckenfelder 9 correlated data by the following expression

where: Qa = liquid flow rate per unit area (Q/A) Z = bed depth

K and n = constants for the specific packing

The form of this empirical model is the same as the mass transfer model given by Eq.-(5-75).

When effluent liquid is recycled, substitution of S t from Eq. (5-45) gives

The constants K and n can be determined by operating a trickling filter at several different liquid loadings and measuring substrate concentrations at several depths. With no recycle, Eq. (5-76) suggests that a plot of\nSJSt vs. Z should give a straight line for each liquid loading. Since the absolute values of the slopes of these lines are K/Q", a. plot of In [slope] versus In Qa should also yield a straight line with slope n and intercept AT as illustrated in Example

Values of K and n for a variety of packings are tabulated in Chapter 6. With known values for these constants, Eq. (5-76) and (5-77) can be used to estimate the performance of a trickling filter.

EXAMPLE 5-4

A trickling filter with a depth of 8 ft is operating with a hydraulic loading of 0.18 gpm/ft2 (0.12 /m2-sec), a recycle ratio of 0.7 and an influent substrate concentration of 150 mg/. Based on the following laboratory data, determine the effluent substrate concentration from the trickling filter.

The laboratory unit has 10-ft depth with sampling taps at 2.5-ft intervals. Substrate concentrations were measured for hydraulic loadings of 0.1,0.2, and 0.3 gpm/ft2 (0.068, 0.136, 0.204 /m2-sec) with no recycle.

(5-76)

(5-77)

5-4. In [slope] = In K n In Qa

Fraction substrate remaining (Se/St)

Depth (ft) >, - 0.1 >, - 0.2 0.3

2.5 5 7.5

10

0.61 0.37 0.23 0.14

0.70 0.49 0.34 0.24

0.74 0.55 0.41 0.30

Sec. 5-11 TrickUng-Filter Models 1

SOLUTION:

Eckenfelder's trickling-filter model will be used:

Plot log (SjSt) vs. Z for each Qa as shown in Fig. 5-9. The slope of each line, -K/Qz, is equal to A(ln SJS,)/AZ.

1.0

0.5

C O

0.2

0.1 10 2 4 6 . 5

Z(f t ) FIGURE 5-9 Fraction of substrate remaining vs. 6*pth in trickling filter.

Plot log.(AT/22) vs. log Qa as shown in Fig. 5-10. The slope is n and intercept (at Qa = 1) is log K.

n = 0.45 K = 0.07

For the trickling filter with recycle,

S + RSt ^ 150 4- 0.75, A |

1 + 1 + 0 . 7 . Se KZ

1 n 5,(1.7) 0.07(8) 150 + 0.75, (0.18) 0- 4 5

.S, = 30 mg/Y

136 Models for Biological Reactors Chap. 5 \

0.3 -

0.051 i | | 0.1 0.2 0.5 1.0

FIGURE 5-10 Evaluation of constants n and K for Example 5-4.

5-12 R O T A T I N G B I O L O G I C A L C O N T A C T O R

The fixed-film model for reaction-rate control can be readily adapted to rotating biological discs. Assuming tha t the liquid phase in contact with the discs is well-mixed and that no mass transfer limitations exist, a material balance on the substrate is

QSt - QS + r,Vb = 0 (5-78)

The volume of active biomass on the discs is

Vb=aV5 (5-79) where: a = biofilm surface area per unit volume of liquid

V = volume of liquid.

For a reaction obeying Monod kinetics, the material balance becomes

QS, - QS, - Y ^ * f s f V 5 = 0 (5-80)

Substituting the parameter P = k05Xbf Y in Eq. (5-80) gives

| - , 'Pad ^ Km + S.

where: 0 V/Q, the liquid phase residence time.

.Sec. 5-13 River Analysis

If diffusion of substrate from the liquid to the liquid-biofilm interface limits the rate, the material balance is given by

By introducing liquid residence time and letting Sf at the interface approach zero, Eq. (5-82) becomes

For N stages in series with the same liquid residence time for each stage, the substrate removal is

If additional substrate removal occurs by suspended solids in the liquid, another consumption term should be included in the material balance. In most cases, however, removal by attached biomass is much greater than by suspended biomass.

Since LaMotta 9 has found that external diffusion is rate-limiting for liquid velocities below about 0.8 m/sec, rates in many rotating biological contactors may be controlled by mass transfer.

5 -13 R IVER A N A L Y S I S

The effluent from a wastewater treatment plant is usually discharged into a body of water so that remaining impurities will be diluted below harmful concentrations. The remaining carbon and nutrient compounds in the effluent will be consumed by the microorganisms in the river and will exert an oxygen demand upon the river. The amount of self-purification that can occur in the river will depend upon the river flow rate, its oxygen content and its reaera-tion capacity. If the BOD of a wastewater is above the reaeration capacity of the river, the oxygen level will drop as the water flows downstream from the wastewater discharge site. Eventually the oxygen content of the stream may be reduced far enough to kill fish and other marine life. When the reaera-tion capacity again exceeds the microorganism demand rate, the oxygen level in the stream will rise and approach its initial level. This "oxygen sag curve" in the river is shown in Fig. 5-11.

The river may be analyzed as a plug-flow reactor with first order kinetics since the waste concentrations are usually less than the Km o? the waste substrate. 1 1 , 1 2 . If an oxygen balance is made on a differential distance of

QSi-QS-kLa{S,-Sf)V = Q (5-82)

(5-84)

1 3 8 Models for Biological Reactors Chap. 5

Distance downstream from point of discharge, Z

FIGURE 5-11 Oxygen sag curve in rivers.

the river, we arrive at

kBSX d dt YK. + P,-R,-B, (5-85)

where:

h u

Pr

K B,

S

mass transfer coefficient for surface reaeration and pre-dicted by Problem 3-3.

depth of the river

river velocity

rate of photosynthesis

rate of algae respiration

rate of river bottom respiration

: waste concentration in river

Making a balance on the waste in the same differential section, we have

dS dS k0SX 8 , S ^_ = _ ^ _ _ (5-86)

Chap. 5 Problems

For steady state conditions, Equation (5-86) becomes

S/S0 = e-k(*/u) (5-87)

where: S = BOD at some point z downstream

S0 = BOD at point of discharge after mixing with the river

k X k = ^ = modified rate constant

Therefore, introducing Eq. (5-87) into (5-85), neglecting Pry Rr and B and solving for steady state, the oxygen concentration downstream becomes

(C* - C) = (C* - CQ)e-k^/hu + k k ^ h k h [ e - k l / u - e~k^lhu\ (5-88)

Equation (5-88) is the Streeter-Phelps equation used in river analysis. The minimum or critical oxygen concentration, Cc> occurs at some distance, z c , from the discharge point where the rate of oxygen utilization equals the rate of reaeration.

(C* - C c ) == M**-*^" (5-89)

Differentiating Eq. (5-88) and setting it equal to zero, we have for the critical distance,

, - - ^ - & [ ' - < c , - % g ' - w i

1 4 0 Models for Biological Reactors Chap. 5

Calculate the following quantities: (a) Effluent sludge concentration (b) Reactor volume (c) Residence time in hr (d) Total sludge in reactor (e) Loading in kg COD/day/kg solids

5-2. A well-mixed activated sludge reactor using recycle is operated with a flow rate of 1 mgd (0.044 m3/sec) and a substrate feed concentration of 500 mg COD//. The recycle ratio is 0.5 and the recycle sludge concentration is 6000 mg//. The kinetic constants are the same as in Problem 5-1. If the residence time (based on fresh feed) is 3 hr, calculate the following quantities: (a) Conversion of substrate (b) Effluent sludge concentration (c) Sludge age in days (d) Sludge wastage per day (e) Loading in kg COD/day/kg solids

5-3. Fresh feed enters a plug-flow activated sludge reactor at a flow rate of 2 mgd (0.088 m3/sec) and with a substrate concentration of 300 mg//. Sludge is recycled from a thickener to the entrance of the reactor with a biomass con-centration of 6000 mg//. The volume of the plug-flow reactor is 40,000 gal (151 m 3 ) . Determine the recycle ratio needed to give a substrate conversion of 95%. The kinetic constants for the reaction are:

A-o = 0.3 hr" 1 Y = 0.5 mg/mg COD tfm = 75mg// kd~0

\ 5-4. A waste stream enters an 800,000 gal (3030 m 3 ) well-mixed activated sludge

reactor at a flow rate of 5 mgd (0.22 m3/sec) and a waste concentration of 200 mg//. Recycle sludge from a separator also enters at a flow rate of 1.5 mgd (0.066 m3/sec) and a biomass concentration of 8000 mg//. To increase the capacity of the plant, a second well-mixed reactor of 400,000 gal (1515 m3) is placed in series with the first. Should this reactor be placed before or after the original reactor ? What increase in waste flow rate can be handled if the conver-sion in the series reactor is the same as in the original single reactor? The kinetic constants for the reaction are the same as in Problem 5-3.

5-5. A waste stream containing 500 mg// of substrate is fed to a well-mixed activated sludge reactor operating to produce the maximum net growth rate of biomass (mass of biomass formed per unit time and volume). Determine the maximum cellular growth rate based on the following kinetic constants.

k0=0.3hx-1 Y= 0.6 Km = 75 mg// kd = 0.05 h r 1

5-6. A wastewater containing 300 mg// is treated in two trickling filters in series to give an effluent concentration of 20 mg//. Clarified liquid from each filter is

Chap. 5 References eactors Chap. 5 1

141

recycle is operated with a flow rate concentration of 500 mg COD/ / ,

e concentration is 6000 mg/ / . The n 5-1. If the residence time (based g quantities:

dge reactor at a flow rate of 2 mgd icentration of 300 mg// . Sludge is of the reactor with a biomass con-the plug-flow reactor is 40,000 gal ed to give a substrate conversion of n are:

0.5 mg/mg COD 0

30 m 3) well-mixed activated sludge sec) and a waste concentration of also enters at a flow rate of 1.5 mgd ion of 8000 mg//. To increase the reactor of 400,000 gal (1515 m 3 ) is

reactor be placed before or after the ow rate can be handled if the conver-in the original single reactor? The

same as in Problem 5-3. substrate is fed to a well-mixed

uce the maximum net growth rate of it time and volume). Determine the

the following kinetic constants.

y = o.6 kd = 0.05 h r 1

sated in two trickling filters in series to >//. Clarified liquid from each filter is

recycled to that filter at a recycle ratio of 0.5. Each bed is packed to a depth of 6 ft with rocks of about 2.5 in. diameter. Find the area of the trickling filters needed to handle a flow rate of 2 mgd (0.088 m3/sec).

The constants for the Eckenfelder trickling filter model are:

K = 0.645, n = 3.80, a = 27.6 ft 2/ft 3, Qa in mgd/acre

5-7. The wastewater from a city is discharged into a river flowing at 10 m/min, with a mean depth of 2.5 meters, and with a river bed drop of 1 meter/1000 meters.

The river is completely mixed in the vertical direction and has a waste concentration after the wastewater injection of 150 mg BOD// and a dissolved oxygen concentration of 8.0 ppm. The constants for the removal of the waste are,

k 0 - 0.5 h r 1 , Km = 100 mg BOD//, X = constant = 20 m g ^ L S S

Y - 0 6 m g M L S S r* o -) Y ~ *

6 me BOH C = 9 2 PPm mg BOD

If the only oxygen source is reaeration and the only sink is the biological oxidation of carbonaceous matter, estimate the minimum dissolved oxygen concentration along the river. How far along the river after the wastewater injection will the minimum oxygen concentration occur?

R E F E R E N C E S

1. LAWRENCE, A. W., and P. L. MCCARTY, "Unified Basis for Biological Treat-ment Design and Operation," /. San. Eng. Div., ASCE, 96 (SA3), 757 (1970).

2. RAMANATHAN, M., and A. F. GAUDY, "Steady State Model for Activated Sludge with Constant Recycle Concentration," Biotech, and Bioeng., 13, 125 (1971).

3. FAN, L. T., et al, "Effect of Mixing on the Washout and Steady State Perfor-mance of Continuous Cultures," Biotech, and Bioeng., 12, 1019 (1970).

4. DANCKWERTS, P. V., "Continuous Flow Systems,".Chem. Eng. Sci., 2, 1 (1953). 5. CLARK, J. W., W. VIESSMAN, and M. J. HAMMER. Water Supply and Pollution Control, 3rd. ed., Ch. 11. International Textbook, 1977.

6. ROBERTS, J., "Towards a Better Understanding of High Rate Biological Film Flow Reactor Theory," Water Res., 7, 1561 (1973).

7. KORNEGAY, B. H., and J . F. ANDREWS, "Application of the Continuous Culture Theory to the Trickling Filter Process," Proc. 24th Industrial Waste Conf., Purdue, 1398 (1969).

142 Models for Biological Reactors Chap. 5 \

8. HARRIS, N . P . , and G. S. HANSFORD, " A Study of Substrate Removal in a Microbial Film Reactor," Water Res., 10, 935 (1976).

9. ECKENFELDER, W. W., Industrial Water Pollution Control, Ch. 13, McGraw-Hill, 1966.

10. LAMOTTA, E. J . , "External Mass Transfer in a Biological Film Reactor," Biotech, and Bioeng., 18, 1359 (1976).

11. METCALF and EDDY, Wastewater Engineering, Ch. 15, McGraw-Hill, 1972.

12. THOMANN, R. V., Systems Analysis and Water Quality Management, Ch. 5, McGraw-Hill, 1972.

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