Modeling Tools for Environmental Engineers and Scientists Episode 6 ppt

The former type of reactors is referred to as completely mixed reactors and the latter, as plug flow reactors.. In contrast,plug flow reactors are characterized by concentration gradient

CHAPTER 5

Fundamentals of Engineered Environmental Systems

CHAPTER PREVIEW

Applications of the fundamentals of transport processes and reactions

in developing material balance equations for engineered tal systems are reviewed in this chapter Alternate reactor configura- tions involving homogeneous and heterogeneous systems with solid, liquid, and gas phases are identified Models to describe the perform- ances of selected reactor configurations under nonflow, flow, steady, and unsteady conditions are developed The objective here is to pro- vide the background for the modeling examples to be presented in Chapter 8.

environmen-5.1 INTRODUCTION

CHAPTER4 contained a review of environmental processes and reactions

In this chapter, their application to engineered systems is reviewed Anengineered environmental system is defined here as a unit process, operation,

or system that is designed, optimized, controlled, and operated to achievetransformation of materials to prevent, minimize, or remedy their undesiredimpacts on the environment

The application and analysis of environmental processes and reactions inengineered systems follow the well-established practice of reaction engineer-ing in the field of chemical engineering While both chemical and environ-mental systems deal with processes and reactions involving liquids, solids,and gases, some important differences between the two systems have to benoted Environmental systems are often more complex than chemical sys-tems, and therefore, several simplifying assumptions have to be made in

analyzing and modeling them The exact composition and nature of theinflows are well defined in chemical systems, whereas in environmental sys-tems, lumped surrogate measures are used (e.g., BOD, COD, coliform) Theflow rates are often constant, steady, or predictable in chemical systems,whereas, in environmental systems, they are not, as a rule.

Engineered environmental systems are built up of reactors A reactor is

defined here as any device in which materials can undergo chemical, ical, biological, or physical processes resulting in chemical transformations,phase changes, or separations The starting point in developing mathematicalmodels of such reactors and systems is the material balance (MB) Principles

biochem-of micro- and macro-transport theory and process/reaction kinetics (reviewed

in Chapter 4) can be applied to derive expressions for inflows, outflows, andtransformations to complete the MB equation The mathematical form of thefinal MB equation can be algebraic or differential, depending on the nature offlows, reactions, and the type of reactor

A complete analysis of reactors is beyond the scope of this book, and ers should refer to other specific texts on reactor engineering for furtherdetails Excellent examples of such texts include those by Webber (1972),Treybal (1980), Levenspiel (1972), and Weber and DiGiano (1996)

read-5.2 CLASSIFICATIONS OF REACTORS

Reactors can be classified into several different types for the purposes ofanalysis and modeling At the outset, they can be classified based on the type

of flow and extent of mixing through the reactor These factors determine the

amount of time spent by the material inside the reactor, which, in turn, mines the extent of reaction undergone by the material At one extreme con-

deter-dition, complete mixing of all elements within the reactor occurs; and at the other extreme, no mixing whatsoever occurs The former type of reactors is referred to as completely mixed reactors and the latter, as plug flow reactors Complete mixing here implies that concentration gradients do not exist

within the reactor, and the reaction rate is the same everywhere inside thereactor A corollary of this condition is that the concentration in the effluent

of a completely mixed reactor is equal to that inside the reactor In contrast,plug flow reactors are characterized by concentration gradients, therefore,they have spatially varying reaction rates within the reactor Thus, completelymixed reactors fall under lumped systems, and plug flow reactors fall underdistributed systems

Reactors with either complete mixing on one extreme or no mixing on the

other extreme are known as ideal reactors Reactors in which some diate degree of mixing between the two extremes occurs are called nonideal

interme-reactors While most reactors are analyzed and designed to be ideal, in tice, all reactors exhibit some degree of nonideality due to channeling, short-circuiting, stagnant regions, inlet/outlet effects, wall effects, etc The degree

prac-of nonideality can be quantified through residence time distribution (RTD)

studies Even the best-designed reactors often exhibit some degree of ideality that requires complex models; hence, they are often approximated bymodified ideal reactors For example, large, nonideal continuous mixed-flowreactors (CMFRs) can be approximated by smaller, ideal CMFRs operating inseries; large nonideal plug flow reactors (PFRs) may be approximated byideal PFRs with dispersive transport added on Thus, it is beneficial to fullyappreciate ideal reactors and develop models for them so that large, full-scalereactors could be realistically designed, operated, and evaluated

non-Ideal reactors can be further divided into homogeneous vs heterogeneous,depending on the number of phases involved; flow vs nonflow, depending onwhether or not the flow of material occurs during the reaction; or steady vs.unsteady, depending on the time-dependency of the parameters Illustrativeapplications of the fundamentals of environmental processes in homogeneousand heterogeneous reactors under flow, nonflow, steady, and unsteady condi-tions are presented in the following sections

Completely

Mixed Batch Reactors

(CMBR)

Sequencing Batch Reactors (SBR)

Completely Mixed Fed Batch Reactors (CMFBR)

Completely Mixed Flow Reactors (CMFR)

Plug Flow Reactors (PFR)

With recycle

Without recycle

5.3 MODELING OF HOMOGENEOUS REACTORS

In the following sections, development of the MB equation for variousconfigurations of homogeneous reactors is summarized The goal of this sec-tion is not to provide a formal treatment of reactor engineering, but instead toillustrate the different forms of MB equations, mathematical formulations,and the solution procedures that are involved in the modeling of commonengineered environmental reactors

Table 5.2 Classification of Heterogeneous Reactors

5.3.1 COMPLETELY MIXED BATCH REACTORS

In completely mixed batch reactors (CMBRs), the reactor is first chargedwith the reactants, and the products are discharged after completion of thereactions During the reaction, inflow and outflow are zero, and the volume,

V (L3), remains constant, but the concentration of the material undergoing the

reaction changes with time, starting at an initial value of C0 The MB tion for a CMBR during the reaction is as follows:

equa-d(V

dt

C)

where r is the rate of removal of the material by reactions (ML–3T–1), and k

is the first-order reaction rate constant (T–1) The solution to the MB equation

where C is the concentration of the material at any time, t, during the reaction.

5.3.2 SEQUENCING BATCH REACTOR

In sequencing batch reactors (SBRs), a sequence of processes can takeplace in the same reactor in a cyclic manner, typically starting with a fillphase Reaction can occur during the fill phase of the cycle as the volumeincreases and can continue at constant volume after completion of the fillphase On completion of the reaction, another process can take place, or the

contents can be decanted to complete the cycle The volume, V t , at any time,

t, during the fill phase = V0+ Qt , where V0is the volume remaining in the

reactor at the beginning of the fill phase (i.e., t = 0), and Q is the volumetric

fill rate (L3T–1) The MB equation during the fill phase, with reaction, forexample, is as follows:

[(V0 Qt)C] = –kC(V0 Qt)  QCin (5.5)The solution to the above MB equation is:

Figure 5.1 Concentration profile in an SBR during the fill phase.

t can provide more insight into the dynamics of the process An Excel®model

of the process is presented in Figure 5.1 A complete model for a biologicalSBR with Michaelis-Menten type reaction kinetics is detailed in Chapter 8,where the profiles of COD, dissolved oxygen, and biomass are developedemploying three coupled differential equations

5.3.3 COMPLETELY MIXED FLOW REACTORS

Completely mixed flow reactors (CMFRs) are completely mixed with continuous inflow and outflow CMFRs are, by far, the most common environ-mental reactors and are often operated under steady state conditions,

i.e., d( )/dt = 0 Under such conditions, the inflow should equal the outflow, while the active reactor volume, V, remains constant A key characteristic of CMFRs is

that the effluent concentration is the same as that inside the reactor CMFRs can

be characterized by their detention time,τ, or the hydraulic residence time (HRT),which is given by τ = HRT = V/Q The material balance equation is as follows:

In some instances, multiple CMFRs are used in series, as shown in Figure 5.2,

to represent a single nonideal reactor, or to improve overall performance, or

to minimize total reactor volume

For n such identical CMFRs shown in Figure 5.2, the overall concentrationratio is related to the individual ratio of each reactor by the following series:

C

C

ou in

×C

C

2 1

,n

where C p is the effluent concentration of the pth reactor ( p = 1 to n).

Substituting from the result found above for a single CMFR into the aboveseries gives the following:

C

C

ou in

Worked Example 5.1

A wastewater treatment system for a rural community consists of twocompletely mixed lagoons in series, the first one of HRT = 10 days, and thesecond one of HRT = 5 days It is desired to check whether this system canmeet a newly introduced regulation of 99.9% reduction of fecal coliform by

a first-order die off The rate constant, k, for the die-off reaction has been found to be a function of HRT described by k = 0.2τ – 0.3 (adapted fromWeber and DiGiano, 1996)

Solution

Because Equation (5.11) assumes identical rate constants in all the tors, it cannot be applied here However, Equations (5.9) and (5.10) can beapplied to yield the following:

reac-C

C

2, i o n

C

C

2,o 1

ut

1 + (1

1.7)(10)

1 + (0

1.7)(5)

C

C

2 1

C

C

3,o 2

ut

0

0

.0

01

02

13

C

C

2 1



and replacing k in terms of the given function, results in a quadratic equation:

[0.2τ – 0.3]τ = 11.35

or, 0.2τ2

– 0.3τ = 11.35giving a detention time of τ = 8.3 days in the third lagoon to meet the new regulation

In plug flow reactors (PFRs), elements of the material flow in a uniformmanner, so that each plug of fluid moves through the reactor without inter-mixing with any other plug As such, PFRs are also referred to as tubularreactors The concentration within the reactor is, therefore, a function of thedistance along the reactor Hence, an integral form of the MB has to be used

as shown in Figure 5.3 (see also Section 2.3 in Chapter 2)

For the element of length, dx, and area of cross-section, A, and velocity of flow, u = Q/A, the MB equation is:

d[(A

d

d t

x)C]

 = r(Adx)  QC – QC  d

d

C x

Element for MB

Element for MB dV=A dx C Q C

Q C+ (dC/dx) dx dx

L

V, C(x)

C L = C0e –(k /u)L = C0e –kτ (5.16)where τ = L/u is the hydraulic detention time, HRT.

5.3.5 REACTORS WITH RECYCLE

Reactors with some form of recycling often are advantageous over otherreactor configurations, providing dilution of the feed and performance im-provement Recycling in CMFRs or PFRs is used more commonly in contin-uous flow heterogeneous reactors Liquid recycling in CMFRs and PFRs,shown in Figure 5.4, can be modeled as follows by applying MB across theboundaries indicated:

Figure 5.4 CMFR and PFR with recycle.

a) CMFR with Recycle b) PFR with Recycle

Q

C out

V C

V, C(x) RQ; C

RQ; C out

Boundary for MB

Element for MB

C in

L

RQ; Cout

Q Cout Cin

Q C0 Q

Cin

d

d t

x)C]

 = rAdx  (Q  RQ)C – (Q  RQ)C  d

d

C x

dx (5.19)which at steady state reduces to:

0 = –kAC – (Q  RQ) d

d

C x

whose solution can be found as follows:

Cout Cin d

or,

Cout= Cine –[Ak /Q(1R)]L = Cine –[k/u(1R)]L (5.23)

A value for concentration Cinat x = 0 can be found by applying an MB at the

mixing point at the inlet:

Cin= QC

Q

0(



1

RQ R

C

)out

Hence, the final solution is as follows:

It can be noted that when R = 0, the above equation becomes identical to the

one for the PFR without recycle, Equation (5.16)

Worked Example 5.2

A first-order removal process is to be evaluated in the following reactorconfigurations: a CMFR, two CMFRs in series, three CMFRs in series, and aPFR Compare the reactors on the basis of hydraulic retention time forremoval efficiencies of 75, 80, 85, 90, and 95%

HRT =1

k(1

–)

The overall HRT for n CMFRs in series can be found by rearranging Equation

(5.11) to get:

HRT = n1

k(1 –

5.4 MODELING OF HETEROGENEOUS REACTORS

The analysis and modeling of heterogeneous systems is often more plex than homogeneous systems Furthermore, reactions in natural environ-mental systems are also typically heterogeneous Hence, they are presented inthis chapter, in somewhat more detail However, because it is impossible todetail all the different reactor configurations, only a representative number ofexamples are presented

and therefore, the overall rate of gain or loss of material in the fluid phase will

be controlled by transport alone or reaction alone or by both The mass fer process may be limited by external resistance due to boundary layers at the fluid-solid interface or by internal pore resistances in the case of micro-

trans-porous solids

Examples of environmental liquid-solid reactors include adsorption,biofilms, catalytic transformations, and immobilized enzymatic reactions.Some of these reactors involve physical processes (e.g., carbon adsorption),while others involve chemical [e.g., UV-light-catalyzed reduction of Cr(VI)

to Cr(III) by titanium dioxide] or biological reactions (e.g., removal of ics by biofilms) In this section, the development of two models of liquid-solid reactors is presented—one with biological reaction and one with achemical reaction under nonideal conditions

organ-5.4.1.1 Slurry Reactor

Reactors used in activated sludge treatment, powdered activated carbontreatment (PACT), metal precipitation, and water softening can be catego-rized as slurry reactors Here, biological flocs or precipitated solids representthe solid phase and act as catalysts to promote the reaction These reactors areoften modeled as CMFRs and are operated under steady state conditions Inthis example, the activated sludge process is modeled, where the rate at whichthe dissolved substrate is consumed in the reactor is described using theMonod’s expression A typical CMFR-based activated sludge system isshown in Figure 5.5

Figure 5.5 Schematic of CMFR-based activated sludge process

Influent

Q

C in

Effluent Q-Q w

Solids wastage Qw

Cw

Influent

Q

Cin