CSTR: the same as chemostat - continuous flow in and out ...wemt.snu.ac.kr/lecture 2012-2/ENV/Ch...

5.4 A Continuous-Flow Stirred –Tank Reactor with Effluent Recycle

• CSTR: the same as chemostat

- continuous flow in and out

- reach a steady state, VdC/dt =0

• CSTR with effluent recycle (Fig 5 4)• CSTR with effluent recycle (Fig 5.4)

- some of the effluent stream is recycled back at a flow rate Qr.

How does “effluent recycle” affect reactor performance?- It does not affect reactor performance at allIt does not affect reactor performance at all.

-Which (the entire system or the reactor itself) is used as the control volume makes no difference to the results.

- But, CSTR with effluent recycle can increase the total reactor volume by the volume of recycle line and increase the extent of mixing.


Mass Balance

QS0 + QrS = QiSi and QXa0 + QrXar = QiXaiQS0 + Q S = Q S and QXa0 + Q Xa = Q Xa

ri SQQSS +0

ii

QQQS = [5.12]

ri QQQ += [5.13]

for the steady-state case

VrSQSQ utiii +−=0 [5.14]


0

substitutions from Equations 5.12 and 5.13

VrSSQ ut+−= )(0 0 [5.15]

• Equation 5.15 is identical to Equation 3.15, the case for h t t ith t lchemostat without recycle.

( ) [ ]1530 0 .SSQVrut −+=• Simple recycle for a CSTR does not change substrate

removal compared with that obtained without recycle.

( )

• Organism concentrations within the reactor and in the reactor effluent are not affected by effluent recycle, since the same y y ,mass flow that leaves the reactor returns to the reactor.

5.5 A Plug-Flow Reactor

• PFR

- the substrate and active-organism concentrations vary over- the substrate and active-organism concentrations vary over the length of the reactor.

- Substrate

VSSQQSSV ΔΔΔ

Δ )( [5 16]

A ti i i

VrSSQQSt

V utΔ+Δ+−=Δ

Δ )( [5.16]

- Active microorganisms

VrXaXaQQXaXaV netΔ+Δ+−=Δ

Δ )( [5.17]

rut and rnet are the reaction rates for substrate (Eq. 3.6) and active organisms (Eq 3 8)

VrXaXaQQXat

V netΔ+Δ+Δ

Δ )( [5 ]

active organisms (Eq. 3.8).


- At steady-state case, for which influent flow rate (Q), substrate concentration (S) and active organism concentration (Xa) do not change with time, the left sides of Eq. 5.16 and 5.17 are zero.

VrSSQQS Δ+Δ+−= )(0 VrSSQQS utΔ+Δ+ )(0

VrXaXaQQXa netΔ+Δ+−= )(0

- Area of the control volume (A): A = ΔV/Δz

- Velocity of flow within the reactor (u): u = Q/A = Q/ (ΔV/Δz) = Q x Δz / ΔV


- Substrate at steady-state

SΔVSSQQS ΔΔ )(0utrz

Su =ΔΔ

[5.18]VrSSQQS utΔ+Δ+−= )(0

,u = Q x Δz / ΔV

- Active microorganisms at steady-state

rXau =Δ [5 19]

VrXaXaQQXa netΔ+Δ+−= )(0 ,

Assumption: 1) Δz approach zero

netrz

u =Δ

[5.19],u = Q x Δz / ΔV

- Assumption: 1) Δz approach zero

2) Monod reaction applies for substrate utilization (Eq. 3.6)

3) O i h h d d (E 3 8)3) Organisms net growth represents growth and decay (Eq. 3.8)

XaSqdSu −= ˆ [5 20] bXaXaSqYdXau −= ˆ [5.21]XaSK

qdz

u+ [5.20] bXaXa

SKqY

dzu

+[5.21]

5.5 A Plug-Flow Reactor- If we ignore organism decay (b=0) and combine Eq. 5.20 and Eq. 5.21,

then analytical solution is possible.

dzdSuY

dzdXau −= [5.22]

dSYdXaS

So

Xa

Xa ∫∫ −=0

[5.23]

)( 00 SSYXaXa −+= [5.24]

- Substituting Eq. 5.24 into Eq. 5.20 gives a differential eq with only two variables, s and z.

SdS

)]([ˆ 00 SSYXaSqdSu += [5 25]

XaSK

SqdzdSu

+−= ˆ [5.20]

)]([ SSYXaSK

qdz

u −++

−= [5.25]


- The ratio dz/u = dt (the time for an element of water to move a distance dZ )

S b tit ti dt f d / i E 5 25 i ld f diff ti l ti th t i th- Substituting dt for dz/u in Eq 5.25 yields for a differential equation that is the same as Eq. 5.10 for batch reactor.

SdS )]([ˆ 00 SSYXaSK

SqdzdSu −+

+−= [5.25]

( )[ ] [ ]105ˆ 00 SSYXSqdS

+ ( )[ ] [ ]10.5SSYXSKdt a −+

+−=

- Integration of eq 5.25 gives eq 5.26

}ln1ln)(}ln{)1{(1 00

0

0000

00 XaSXaKYSYSXaKz−−−++=

which is almost identical to eq 5.11 for a batch reactor.

[5.26]

})(}{){(ˆ 00000 YSYSXaYYSXaqu ++

( ) } [ ]115l1ll11 00

00 ⎪⎬⎫⎪

⎨⎧

⎟⎟⎞

⎜⎜⎛

+⎟⎟⎞

⎜⎜⎛

+ a XSXKYSYSXKt ( ) } [ ]11.5lnlnln

ˆ0

00000

00 ⎪⎭

⎪⎬

⎪⎩

⎪⎨ −⎟⎟

⎠⎜⎜⎝ +

−−+⎟⎟⎠

⎜⎜⎝

++

= aa

aa

a

XYSYSX

YSYSXYYSXq

t


}ln1ln)(}ln{)1{(ˆ1 0

0

0

0000

00 XaYS

SXaYSXa

KYSYSXaYYSXa

Kqu

z−

+−−++

+=

[5 26]

- An expression for the effluent concentration (S) from the batch reactor

[5.26]

is obtained by letting z = L.

}l1l)(}l {)1{(1 00

00 XSXaKYSYSXKL }lnln)(}ln{){(ˆ

0000

0000 Xa

YSYSXaYSYSXa

YYSXaqu−

+−−++

+=

L/ V/Q θ (th h d li d t ti ti )

}ln1ln)(}ln{)1{(1 00

0

0000

00 XaXaSKYSYSXaK ee −−−++=θ

L/u = V/Q = θ (the hydraulic detention time)

})(}{){(ˆ 00000 YSYSXaYYSXaq ++

[5.27]

E 5 27 i id ti l t 5 11 ith θ l i t- Eq 5.27 is identical to eq 5.11 with θ replacing t.


- We thus see that a PFR works exactly like a batch reactor.

I ti h it i diffi lt t t PFR di t th- In practice, however, it is difficult to operate a PFR according to the assumptions involved.

1) A PFR h i i h i i i f h fl id l h fl1) A PFR has no mixing or short-circuiting of the fluid along the flow direction. This is impossible to achieve in a real continuous-flow reactor.

2) Wall effects slow the fluid near the wall boundaries relative to the2) Wall effects slow the fluid near the wall boundaries relative to the velocity near the middle.

3) Aeration or mixing to keep the biomass in suspension introduces a large amount of mixing in all directionsamount of mixing in all directions.


- Methods to achieve as much of a plug-flow character as possible include

1) i l t d 2) i t i i1)using a very long, narrow reactor and 2)using many reactors in series. But, some mixing and short-circuiting are inevitable.

-If achieving the reaction kinetics represented by eq 5,27 is of paramount importance, a batch reactor is prudent choice (e.g., choice of batch p , p ( g ,reactor instead of PFR might be better), although it presents its own problems:

For example, time is required to fill and empty a batch reactor, time that might otherwise be used for treatment. In order to minimize downtime (e.g., idle time), a batch reactor can be operated while it is filling.) p g

}ln1ln)(}ln{)1{(ˆ1 0

0

0

0000

00 XaYS

XaSYSXa

KYSYSXaYYSXa

Kq

ee −

+−−++

+=θ

YSYSXaYYSXaq ++

[5.27]

5.6 A Plug-Flow Reactor with Effluent Recycle

•If no organisms are introduced in PFR, then the system fails to doany treatment.

•using effluent recycle, a portion of microorganisms in the effluent i b ht b k t th i fl t tis brought back to the influent stream.

00 rrr XQQXSQQS ++ ]12.5[iaai

aii

QXQQX

XandQ

SQQSS+

=+

=

]13.5[ri QQQ += eer SSandXX ==]13.5[QQQ + SSandXX


- The reaction occurring are same manner as with the simple PFR exceptthat , iii SandXQSandXQ 000 → SandXQSandXQ ,, →

- Ignore organism decay (b=0) , eq 5.28. And 5.29 can be obtained from Eq 5 24 and 5 26obtained from Eq. 5.24 and 5.26

)( 00 SSYXaXa −+= [ 5.24 ])(

[ 5 26 ]}ln1ln)(}ln{)1{(

ˆ1 0

0

0

0000

00 XaYS

SXaYSXa

KYSYSXaYYSXa

Kqu

z−

+−−++

+=

]285[)-( 00 ee SSYXX +=

[ 5.26 ]

]28.5[)(aa SSYXX +

]29.5[ln1ln)ln(1ˆ1

⎬⎫

⎨⎧

−⎟⎟⎞

⎜⎜⎛

−−+⎟⎟⎞

⎜⎜⎛

+= iai

ia

e

iieii

aiii XXSKYSYSXKV ][)(ˆ ⎭

⎬⎩⎨ ⎟

⎠⎜⎝ +⎟

⎠⎜⎝ + aiii

aaii

ai YSYSXYYSXqQ


]29.5[ln1ln)ln(11⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+⎟⎟⎠

⎞⎜⎜⎝

⎛+

+= i

ai

ia

e

iia

eiiaii

ai X

YSXS

YSXKYSYSX

YYSXK

qQV

- Recycle ratio R, ]30.5[QQRr

=

- Detention time θ, ]31.5[)1(iQRV

QV +==θ


Solve equation using spreadsheet

washout time• Graph with Se, θ , R

- Solve equation using spreadsheetto determine Se as a function of θ and RUpper graph; arithmetic scaleLower ; log scaleLower ; log scale

-The washout detention time is larger when R decreases

1- For the same condition, the for the CSTR is 0.2 days (close to the value for R=8 for plug-flow

minxθ

is 0.2 days (close to the value for R 8 for plug flowwith recycle).

2

-In theory, when R approaches infinity, the PFRwith recycle becomes identical to a CSTR.


washout timewashout time

1

22


• Relative benefits of a CSTR and a plug-flow reactor with recycle

- R=8 is very similar to a CSTR ‘s performance- CSTR improves reliability if contaminant removal of 80 to 90 %

were satisfactory1

were satisfactory.

- PFR with low recycle ratio is much more desirable if contaminant l f 99 9% i d

2removal of 99.9% were required.

- There is an optimal recycle ratio that provide low θ and high efficient contaminant removal.

• Effluent recycle with a CSTR does not change system performance• Effluent recycle with a CSTR does not change system performance,

but with a PFR, recycle is essential and has a great impact on performance.

5.7 Reactors with Recycle of Settled Cells

• Microorganism recycle from a settling tank : the most widely• Microorganism recycle from a settling tank : the most widely used suspended-growth reactor.

• Any method that increases the con of microorganisms in the reactor• Any method that increases the con. of microorganisms in the reactor increases the volumetric reaction rate and, in this manner, decreases the required reactor volumeq

• The primary advantage: a much smaller reactor volume is required.

The disadvantage: the cost of the settler and the recycling system.

Smaller reactor plus settler vs a larger reactor without a settler

5.7.1 CSTR with Settling and Cell Recycling

• Assumptions• Mass balance for microorganisms• Mass balance for substrates• Solids retention time (SRT)

CSTR

with Settling and Cell Recyclingwith Settling and Cell Recycling

Applicable Equations : S, Xa


Assumptions:Assumptions:• Biodegradation of the substrates takes place in the reactor only,

no biological reactions take place in the settling tank, and biomass in the ttl i i i ifi tsettler is insignificant.

• No active microorganisms are in the influent to the reactor (Xa0 = 0).

• The substrate is soluble so that it can not settle out in the settling tank.


Accumulation = In – Out + Generation - ConsumptionM b l f i i

]32.5[]-)-([)(-0 VbXVrYXQXQdtdX

V autwa

wea

ea ++=

Mass balance for microorganism:

dt

]335[)(00 VrSQSQSQdSV wwee ++=

Mass balance for substrates:

]33.5[)(- VrSQSQSQdt

V ut++=


systemin thebiomassactivebiomass active of rate production

systemin thebiomass active=xθ

At t d t t

]355[= aVXsystem in the biomass activeθ

At steady state,

]35.5[+ w

awe

ae XQXQbiomass active of rate removal

=xθ


])([)(00 VbXVrYXQXQ autwa

wea

e −−++−=

At steady-state , from eq 5. 32

])([)( QQ autaa

]36.5[-)-(b

XrY

VXXQXQ ut

wa

wea

e

=+

XVX aa

]35.5[+

= wweea

XQXQVX

xθ + aa XQXQ

Seeing the similarity between the left side of Eq.5.36 and the right side of eq. 5.35

]37.5[-)-(1 bXrY

a

ut

x

=θ

ˆ)-(1 SqrY

If it takes Monod kinetics,

]38.5[--)(1 bSKSqYb

XrY

a

ut

x +==

θ


]38.5[-ˆ

-)-(1 bSKSqYb

XrY ut

+==

θ SKX ax +θ

Solving this equation explicitly for S,

]39.5[1)-(

1∧

−

+=

bqY

bKS

x

x

θ

θ

( ) [ ] θθθθ

θχ

χχ

χ =+−

+= where

bqYb

KS 24.31ˆ

1

Eq 5.39 is identical to eq 3.24, which was developed for the chemostat without settling and recycle.

So then, what is unique about the CSTR with settling and microorganism recycle ?

5.7.1 CSTR with Settling and Cell Recycling So then, what is unique about the CSTR with settling and microorganism recycle ?

]39.5[1

∧

+=

bKS xθ

( ) [ ]2431

1.

bq̂Yb

KS χ

θθθ+−

+=

1)-( −bqYxθ

( )1 bqY χχ θθ +

)()(:

θθ

timeretentionhydraulicthefromseparatedissystemtheinismsmicroorganoftimeretentiontheAnswer x

θθ =xθ

)(θtimeretentionhydraulicthefromseparatedis

- In eq 3.24 without microorganisms recycle,

- Thus one can have a large , in order to obtain high efficiency of substratel d h i h ll

xθ

xθ θBut in eq 5.39 with microorganisms recycle, is not necessarily equal to

removal, and at the same time have a small .

For example, = 4-10 d (small S, substrate),

x

xθ

θ

while = 4-10 h (small V, volume).Thus, while operating at the same treatment efficiency for substrate, the reactor size can be 1/24 of the volume of CSTR without settling and recycle.

θ

5.7.1 CSTR with Settling and Cell Recycling )(1 Y

)(Y

]37.5[-)-(

=1

bXrY

θ a

ut

x

]40.5[+1

)-(=

x

utxa θb

rYθX

From Mass balance for substrates (eq 5.33 & 5.34)

From eq 5.37,

00 SQSQSQ wwee

From Mass balance for substrates (eq 5.33 & 5.34)

VrSQSQSQ utwwee ++−= )(0 00

tansin

]41.5[---

ksettlingtheinoccursreactionnoceV

SQSQSQrut =

]425[)-()-(,,

000

0

SSSSQthenQQQSSS

gwewe +===

]42.5[)()(- θV

Qrut ==

)()( 0Y θSubstituting eq 5.42 into eq 5.40,

]43.5[1

)-(1

)-( 0

x

x

x

utxa b

SSYbrY

Xθθ

θθ

θ+

=+

=


]43.5[1

)-( 0

x

xa b

SSYXθθ

θ+

=x

θθ x : Solids concentration ratio

- Active biomass concentration in the reactor depends onθActive biomass concentration in the reactor depends on the ratio of solids retention time to the hydraulic detention time,

θ x

xθ

-For a CSTR without settling and recycle, = 1 and then eq 5.43 is equato eq 3.25.

θx

[ ]2531

0

.bSSYXa ⎟⎟⎞

⎜⎜⎛ −

=θ

- If = 24, and is same, then Xa with settling is 24 times higher θθ x

[ ]1 ba ⎟

⎠⎜⎝ + χθ

xθ

than it would be without biomass recycling


Mass rate of active biomass productionAt steady state, the mass rate of active biomass production must equal the rate at which the biomass leaves the system from the effluent stream and the waste stream.

)/(: TMproductionbiomassactiverabpCombining eq 5,44 and eq 5.35 yields eq. 5.45

]35.5[+

= wa

wea

ea

XQXQVX

xθ

]44.5[wa

wea

eabp XQXQr +=

]45.5[x

aVXθ

=


Table 5 2 summarizes a series of equations to design aTable 5.2. summarizes a series of equations to design a CSTR with settling and recycle.

- Assumptions for Table 5.2

i) Operating at steady statei) Operating at steady stateii) Treating a soluble substrateiii) No input of active biomassiii) No input of active biomass

- The equations in Table 5.2 can be used for a CSTR without a settler

θθ =x

qby letting .


• Hydraulic Detention Time θθ =x

]20.3[= 0QV

θ

• Solids Retention Time, SRT

]35.5[+

= wwa

eea

ax QXQX

VXθ

]38.5[-ˆ

-)-(1 bSqYbrY ut == ].[SKX ax +θ


- : SRT at which microorganism washout results and the limit thereto:minxθ

- The minimum value of the mean cell residence time and its limiting value are identical to the case without settling ( eq 3 26 & 3 27)are identical to the case without settling ( eq 3.26 & 3.27)

00

min SK + ]26.3[→-)-(

0^

0

min SSKbbqYS

SKx

+=θ

]27.3[∞→1][ 0

limin S=θ ]27.3[

-ˆ][ lim S

bqYxθ


• Reactor of Effluent substrate concentration

]39.5[&]24.3[1-)-(

1∧

bqY

bKS x

θ

θ+=

• Reactor Minimum Substrate Concentration

1)( bqYxθ

b

• Reactor active Microorganism Concentration

]28.3[∞→

-ˆlim x

bqY

bKS θ=

• Reactor active Microorganism Concentration

]40.5[+1

)-(=

x

utxa θb

rYθX

]43.5[+1

)-(=

0

x

xa θb

SSYθθ

X

xθb

x

5.7.1 CSTR with Settling and Cell Recycling • Reactor active Microorganism Concentration

]40.5[+1

)-(= ut

xa θbrY

θX

]43.5[+1

)-(=

0x

a θbSSY

θθ

X

+1 xxa θb

+1 xθbθ

- Because inert biomass and total volatile solids are particles, like active biomass, the concentrations Xi and Xv take the same form as for a chemostat,biomass, the concentrations Xi and Xv take the same form as for a chemostat,but are multiplied by the solids-concentration factor ( eq 5.46 & 5.47 ).

• Reactor Inert Microorganism Concentration

]46.5[])-1(+[= 0 θbfXXθθ

X daix

i

R t V l til d d lid t ti• Reactor Volatile suspended solids concentration

X v = X i + X a

]47.5[]+1

))-1(+1)(-(+[=

00

x

xdi

xv θb

θbfSSYX

θθ

X


• Active Biological Sludge Production Rate

]45.5[= aabp θ

VXr

xθ

T t l Bi l i l S lid P d ti R t

- The total biological-solids production rate is analogous to eq 5.45

• Total Biological Solids Production Rate

]485[vVX ]48.5[x

vtbpr θ

=


- At a constant SRT, the effluent conc. (S) remains the same regardless of the influent conc. (S0). Only affects S because all other xθ

]395[&]243[1 bKS xθ+=

parameters in the equations are coefficients.

]39.5[&]24.3[1-)-(

∧

bqYKS

xθ=

Why ?

1) “ Self Control ” : As the influent conc. increases, so does the conc. of active organisms in the reactor (eq 3.25 & 5.43). The increased biomass is sufficient to consume the additionalThe increased biomass is sufficient to consume the additional substrate that is added to the reactor.

]435[)-( 0

x SSYθX ]43.5[

+1)(

=x

xa θbθ

X


2) The organisms’ growth rate and SRT are equal to the inverse of each other.

]223[=system in the biomass active

= 1-μθ ]22.3[biomass activeofrateproduction

μθ x

∧

S ]9.3[- bSKSqY+

=μ

Constant SRT ( ) Constant specific growth rate ( )C t t b t t ( S )

μxθConstant substrate ( S )

5.10 Engineering Design of Reactors

[ ] ]60.5[limminx

dx SF θθ = ( Christensen and McCarty, 1975)

: a design θxdxθ

- Conventional activated sludge treatment plants :medium sized treatment systems that are expected to operate reliablymedium sized treatment systems that are expected to operate reliably with fairly constant supervision by reasonably skilled operator.

- High Rate: Highly skilled operator or the removal efficiency and high reliabilityHigh Rate: Highly skilled operator or the removal efficiency and high reliability is not as critical.

- Low Rate : Extended aeration: Operator attention is quite limited : Operators are present for a very short period time.: “shopping center”, or “apartment complex”

5.10 Engineering Design of ReactorsFactors to consider when selecting SF :- High SF increases the degree of reliability of operation,

b i hi h ibut gives higher construction cost.

]35.5[+

= wweea

x QXQXVX

θ

- Low SF requires more continuous supervision and operators with increased skill.+ aa QXQX

5.10 Engineering Design of Reactors

- Higher SS ( X ) makes the reactor volume smaller and thus less expensive for a given . However, high SS may require larger settling tanks, b i d l d f SS t th ttli t

dxθ

because increased loads of SS to the settling reactor.

]35.5[+

= wweea

x QXQXVX

θ+ aa QXQX

CSTR: the same as chemostat - continuous flow in and out ...wemt.snu.ac.kr/lecture 2012-2/ENV/Ch...

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