R E A C T O R S&

REACTOR KINETICS

BY

R A CHRISTIAN, Ph. D. Assistant Professor, Civil Engineering Department

S V National Institute of Technology, Surat

5th October 2009

CONCEPT OF REACTORS

Reactors (Treatment Units)

The units or vessels that hold wastewater for treatment by chemical or biological processes are normally called as reactors and the units that are used for separation of solids from liquid by settling or flotation are termed as basins or tanks. However, in practice the terms basins, tanks, vessels or reactors are used interchangeably.

The reactors may be of any shape but mostly rectangular or circular reactors are used in wastewater treatment. The size (capacity or volume) of a reactor, more particularly in biological processes, normally depends on the treatment system selected, order of reaction rate assumed and the flow conditions (hydraulic regime) that will prevail in the reactor.

Types of Reactors Depending upon the flow and operating conditions and the method of mixing of the wastewater therein, the reactors have been classified as under :Continuous - Flow Stirred Tank Reactor (CFSTR)Plug - Flow Reactor (PFR)Completely Mixed Batch Reactor (CMBR) Arbitrary - Flow Reactor (AFR)Fluidized Bed Reactor (FBR)Packed Bed Reactor (PBR)Sequencing Batch Reactor (SBR)

As the selection of a reactor and its design for achieving the desired degree of treatment requires a clear understanding of each of the above classified reactors, they have been briefly described in this presentation.

Continuous - Flow Stirred Tank Reactor (CFSTR)

CFSTR is also called Completely Mixed Reactor. As the flow of wastewater is continuous in such type of reactors, the reactants entering the reactor and the products flowing out from the reactor is considered as continuous. It is also assumed that the contents are distributed throughout the tank as soon as the flow enters the reactor and their uniform concentrations are maintained in the reactor operating under steady state conditions. Fig. 1 given below shows the schematic of CFSTR.

Notations in figure represent,V = Reactor volumeQ0 = Wastewater flow rate into and out of reactorCo = Initial reactant concentration in influent Ce = Final Reactant concentration in effluent

= Reactant concentration in reactor

V Ж Ce

Q

Co

Q

Ce

Fig. 1 Schematic of CFSTR

The equations for HRT and effluent reactant concentrations are derived from the mass balance of reactant as given below:

Net rate of changein mass of reactantwithin the reactor

=

Rate of increasein mass of reactantdue to its

presence in the influent

+

Rate of decreasein mass of reactant

due to its removal in the effluent

-

Rate of decrease

in mass of reactantdue to

reaction of reactants in the reactor

Or mathematically,

o e enet r

dc dcV = QC -Q C - V

dt dt

o e e enet

For first order reaction kinetics,

dcV = QC -Q C - VKC

dt

Or mathematically,

QV

K1

1

C

C

o

e e

o CFSTR

C 1or =

C 1+(K×t )

e

o CFSTR

C 1or =

C 1+(K×V / Q)

0CFSTR

e

CQor V = - 1

K C

where : K = reaction rate constant, and t = reaction time (hydraulic retention time) to achieve desired reactant concentration V = volume of reactor Q = flow of wastewaterSimilarly, the equation derived to obtain the effluent concentration for second order reaction kinetics is given by

e

o CFSTR e

C 1

C 1 (K t C )

e

o CFSTR e

C 1or =

C 1+(K×(V /Q)×C )

0 eCFSTR 2

e

C -CQ or V =

K C

the hydraulic retention time for CFSTR is given by

oCFSTR

e

C1t = -1

K Cfor first order reaction

0 eCFSTR 2

e

C -C1and t =

K Cfor second order reaction

0CFSTR

e e

C1t = -1

K C C

where, K = second order rate constant, [(mg/L)xd]-1

Illustrative Example

A wastewater is being treated in a CFSTR following first order reaction kinetics with a reaction rate constant equal to 0.15 day-1. For a reactor volume of 50 m3, what should be the flow rate to achieve 96% treatment efficiency? For this flow rate, compute the reactor volume if the desired treatment efficiency is 98%?

Flow rate of wastewater required to achieve 96% efficiency in CFSTR of 50 m3 capacity is 0.313 m3/day. Now, for same operating conditions, when the desired treatment efficiency of 98% is to be achieved, the volume required will be 102.25 m3 .

Thus, the capacity of the reactor will be almost doubled (from 50 m3 to 102 m3) when treatment efficiency is increased from 96% to 98% for the given conditions of wastewater treatment.

Plug - Flow Reactor (PFR)

In a plug flow reactor, the content of wastewater follows the principle of 'first - in - first - out'. So, the particles pass through the tank in the same order or sequence in which they enter the tank and longitudinal mixing is assumed to be almost negligible.

The concentration of a reactant varies with time and along the length of the reactor. Fig. 2 given below shows the schematic of a plug flow reactor.

NOTE: It is not economical to increase the volume of reactor by two times just to achieve 2% more treatment efficiency.

V

Fig.2 Schematic for PFR

Q

C0

Q

Ce

The equations for HRT and concentration of reactant in effluent are derived from mass-balance of a reactant at steady state conditions as under:

Change in the concentration of reactant

due to reaction of reactant in time, dt

=

Change in the concentration of reactant due to change in position

of fluid element in time, dt

dc dx

i.e. (-ve sign implies a decrease in reactant concentration)dt v

where, v = velocity of flow through reactor dx = differential change in distance along the length of reactor

Integrating the left hand side of the equation between concentration limits C0 to Ce and integrating right hand side of the equation for lengths zero to L and substituting the value of v/Q for L/v; we get the equations to determine HRT and volume of reactor as given below:

For first order reaction kinetics,

oPFR

e

C1t = ln

K C

o

e

CQV = ln

K C

Similarly for second order reaction kinetics,

PFRe 0

1 1 1t = - for second order reaction

K C C

0PFR

0 e

C1t = -1

K C C

e 0

Q 1 1and V = - for second order reaction

K C C

where, K = second order rate constant, [(mg/L)xd]-1

Illustrative Example

If the rate of reaction in the system is of second order, compare the required volume of continuous flow stirred tank reactor and the volume of plug flow reactor to achieve 94 % reduction in the reactant concentration in the system.

VCFSTR = 16 x VPFR

NOTE: In general, the volume required for CFSTR will always be more than that of PFR.

Completely Mixed Batch Reactor (CMBR)

Completely Mixed Batch Reactor is a closed system where no flow is added or allowed to go out during designed reaction time (detention period). The reactants are added to the reactor when it is empty and the contents are withdrawn after the reaction period is over. In CMBR, it is assumed that the reaction kinetics are of first order and at a given instant of time, the reactant concentration is uniform throughout the reactor. Fig. 3 shows the schematic of CMBR.

V, Ж C1

Q0

C0

Q0

C1

Fig. 3 Schematic of CMBR

The mass-balance for a reactant in CMBR can be expressed as :

Rate of change in the mass of reactant within the

reactor =

Rate of the reaction of reactant within the reactor

For first - order reaction kinetics, mathematically,

net reaction

dC dCV V V(KC) (4.31)

dt dt

we get the following equation to determine HRT and volume of the reactor,

e

oCMBR C

Cln

K

1t

Where, Co = initial reactant concentration Ce = desired or final reactant concentration K = reaction rate constant

e

0

CQln

K Cand, V = (as t = V/Q)

NOTE: Application of CMBR is limited in biological wastewater treatment. However, its use is more in bench scale laboratory studies and in digestion of sludge.

Arbitrary Flow Reactor (AFR)

A PFR designed with dispersion of flow is called an Arbitrary Flow Reactor. In practice, some intermediate amount of intermixing will always occur. The equation developed by Whener and Wilhem for such intermediate mixing occurring in AFR is as given below:

12dC 4aeo

a aCe 2 22d 2d(1 a) e (1 a) e

2

D Dtwhere, a 1 4 Ktd and d

vL L

here, K = reaction rate constant, (time-1) t = hydraulic retention time d = dispersion number or diffusivity constant (dimensionless) = 0 for PFR = α (infinity) for completely mixed system D = axial dispersion coefficient (area/time) v = fluid velocity (length/time) L = characteristic length of travel path of particle

Reactors in Series :  In the design of wastewater flow treatment system, sometimes, either the same or combination of different types of reactors are required to be used in series. The reactors provided in series may or may not be of equal size and may be operating on different types of processes. Fig. 4 given below shows two continuous flow stirred tank reactors in series.

V, C1

Q0

C0

Q0

C1

V, C2

Q0

C2

Fig. 4 Schematic of two CFSTRs in series

Assuming first - order reaction kinetics and 'n' number of equal sized CFSTR, we get the equation for detention time and thereby the total volume of reactors as follows:

1

C

C

K

1t

n

1

e

oiesCFSTRinser detention time in one reactor

1

C

C

K

ntn

n

1

e

oCFSTR to determine detention time in ‘n’ reactors

1

C

C

K

n

Q

Vn or

n

1

e

o to determine volume of ‘n’ reactors

Illustrative Example

Calculate and compare the volume of the reactor(s) required to achieve 90% reduction of a reactant in a flow of 1000 m3/d for the following conditions:

i) Single CFSTR is used

ii) Four CFSTRs are used in series, and

iii) Single PFR is used.

Assume the reaction rate constant, K = 0.5 day--1.

Assuming first order reaction kinetics for all three given conditions,

when a single CFSTR is used, VCFSTR = 18000.0 m3

when four CFSTR are used in series, Total volume = 6224.0 m3 (say)

when a single PFR is used, V = 4600.0 m3

COMMENT: Of the above 3 conditions, least volume of reactor is required for single PFR and less total volume is required when four CFSTRs are provided in series instead of one CFSTR.

More the number of CFSTR in series, smaller will be the total reactor volume for the system and the system approaches PFR with increase in number of reactors in series.

For CFSTR, KCFSTR = 0.45 day –1

For PFR, K = 0.115 day –1

CFSTR

PFR

K 0.45 3.91 4.0

K 0.115

COMMENTS: The values of reaction rate constants show that the reduction in reactant concentrations is about 4 times faster in CFSTR than in PFR, when similar conditions of flow, HRT and effluent concentration of reactant are maintained in both the reactor systems.

Illustrative Example

A reactor system reduces the influent reactant concentration from 200 mg/L to 20 mg/L with a detention time of 20 days. Assuming that the reaction rate is of first order, determine the value of K for a) CFSTR and b) PFR. Give your comments on the results.

Fluidized Bed Reactors (FBR)

A reactor in which the filled packing material expands and gets fluidized when the wastewater to be treated moves upward in the reactor is called a FBR. Normally, air is also introduced along with the influent flow from the inlet. Fig.5 shows the schematic of FBR

Fluidized bed of packing material

Effluent

Influent

Fig. 5 Schematic of a FBR

Gas(es)

Such reactors are becoming popular to treat wastewaters biologically either in aerobic or anaerobic conditions. They are also used for sludge treatment and removal of dissolved gases.

Packed Bed Reactors (PBR)

A reactor in which the filled inert packing material for the growth of biomass is kept packed (or fixed) is called a PBR. The flow of wastewater through the reactor may be upward or downward as shown in Fig. 6. The packing material commonly used is slag, rock or ceramic. However, the use of plastic as packing material, with various configuration and large specific area, is now more common.

a) Downflow PBR

Bed of packing material

Effluent

Influent

Gas(es) Gas(es)

Effluent

Influent

b) Upflow PBR

Fig. 6 Schematic for PBR

NOTE: When a reactor is completely filled with packing media with respect to flow, it is known as Anaerobic Reactor (or Filter).

Sequencing Batch Reactors (SBR) This is a fill and draw type of reactor working on the principle of an activated sludge process where reactions for aeration and waste conversion and clarification of effluent occur in the same reactor but in sequencing steps.

Operational steps:

The reactor is first filled with the wastewater up to the desired volume and the flow is stopped.

The content of wastewater is then aerated and mixed for the designed time period.

Aeration is then stopped and clarification or sedimentation of biomass is carried out to separate the sludge.

The clarified effluent is then withdrawn (or decanted) from the reactor.

Finally the deposited sludge is removed from the bottom of reactor.

Fig. 7 shows the operating steps of SBR used for activated sludge process system.

Aeration of W/W

Influent

Q, S0

Step – 1

Filling up the reactor

Step – 2

Reaction takes place for time t

Step – 3

Settling of sludge (clarification)

Step – 4

Removal of clarified effluent

Effluent

Q, Se

Step – 5

Removal of sludge

Qw

Fig. 7 Operating steps of SBR

SBR system does not require recycling of the activated sludge to maintain MLSS in the reactor. The sludge wasting depends on performance requirements.

TIP : To operate the SBR system on a continuous basis, two or more reactors are provided in parallel so that the second or next reactor is filled when the first or preceding reactor is completing its last step.

SUMMARY

Depending on flow conditions and mixing of wastewater there in various types of reactors are employed for wastewater treatment.

Various types of reactors and reactions occurring in reactors used for the treatment of domestic wastewater.

In practice, a reactor with large length to small width ratio is assumed as PFR.

When operating under similar conditions of flow and reaction order to achieve the same degree of treatment, plug flow reactor requires less volume than complete mix reactor.

Use of PBR, FBR and SBR for treating domestic wastewater is gaining popularity.

Normally for biological processes, reactions are heterogeneous in nature and of first order type.

T H A N K Y O U