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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.
5.4 A Continuous-Flow Stirred –Tank Reactor with Effluent Recycle
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]
5.4 A Continuous-Flow Stirred –Tank Reactor with Effluent Recycle
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).
5.5 A Plug-Flow Reactor
- 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
5.5 A Plug-Flow Reactor
- 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]
5.5 A Plug-Flow Reactor
- 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
5.5 A Plug-Flow Reactor
}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.
5.5 A Plug-Flow Reactor
- 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.
5.5 A Plug-Flow Reactor
- 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
5.6 A Plug-Flow Reactor with Effluent Recycle
- 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
5.6 A Plug-Flow Reactor with Effluent Recycle
]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
5.6 A Plug-Flow Reactor with Effluent Recycle
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.
5.6 A Plug-Flow Reactor with Effluent Recycle
washout timewashout time
1
22
5.6 A Plug-Flow Reactor with Effluent Recycle
• 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
5.7.1 CSTR with Settling and Cell Recycling
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.
5.7.1 CSTR with Settling and Cell Recycling
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++=
5.7.1 CSTR with Settling and Cell Recycling
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
5.7.1 CSTR with Settling and Cell Recycling
])([)(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
5.7.1 CSTR with Settling and Cell Recycling
]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+
=+
=
5.7.1 CSTR with Settling and Cell Recycling
]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
5.7.1 CSTR with Settling and Cell 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
=
5.7.1 CSTR with Settling and Cell Recycling
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 .
5.7.1 CSTR with Settling and Cell Recycling
• 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
5.7.1 CSTR with Settling and Cell Recycling
- : 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
5.7.1 CSTR with Settling and Cell Recycling
• 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
5.7.1 CSTR with Settling and Cell Recycling
• 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
=
5.7.1 CSTR with Settling and Cell Recycling
- 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
5.7.1 CSTR with Settling and Cell Recycling
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