Unit Operations and water and wastewater treatment lecture 2

1. CEE 155
CEE 155 Unit Operations & Processes in
Unit Operations & Processes in
Water and Wastewater Treatment
Water and Wastewater Treatment
Lecture 3
Lecture 3
Mass Balances, Ideal Flow Models, &
Mass Balances, Ideal Flow Models, &
Mass Balances, Ideal Flow Models, &
Mass Balances, Ideal Flow Models, &
Comparison of Ideal Reactor Performance
Comparison of Ideal Reactor Performance
Dr. Minghua Li and Dr. Eric M.V. Hoek
Dr. Minghua Li and Dr. Eric M.V. Hoek
Civil & Environmental Engineering
Civil & Environmental Engineering

2. Outline for Today!s Lecture
Batch and continuous processes
Reactor types and ideal flow models
Mass balances and ideal reactor models
Ideal reactor performance equations
Ideal reactor performance equations
Comparing the performance of a CSTR & PFR

3. Batch and Continuous Processes
Batch Processes - small volumes, laboratory
Materials are added to empty tank, system is closed, a reaction
proceeds, the products are removed
Continuous Processes most W&WWT
Materials flow in and out of the system continuously
Materials flow in and out of the system continuously
Most water and wastewater treatment processes are operated at
steady-state, so that there is NO NET CHANGE in system or
surroundings with time

4. Ideal Reactors and Nonideal Reactors
Ideal Reactors:
Reactors defined for purposes of modeling. Ideal
assumptions, such as no dispersion or diffusion, are nearly
achievable under closely controlled laboratory conditions.
Definitions assume extreme fluid conditions, such as
complete mixing or no mixing of reactants.
complete mixing or no mixing of reactants.
Nonideal Reactors:
Mixing and/or residence time distribution in the reactors
does not meet ideal assumptions.

5. Ideal Reactors
Batch reactor (BR=CMBR):
No reactants or products flow into or out of the reactor. Complete mixing
occurs instantaneously and uniformly throughout the reactor, and the
reaction rate proceeds at the identical rate everywhere in the reactor.
Completely mixed flow reactor (CSTR = CMFR = MFR):
Reactants and products flow into and out of the reactor. Complete
Reactor Types and Flow Models
Reactants and products flow into and out of the reactor. Complete
mixing occurs instantaneously and uniformly throughout the reactor. The
reaction rate proceeds at the identical rate everywhere in the reactor, and
the concentrations throughout the reactor are the same as the effluent
concentration.
Plug flow reactor (PFR/PBR):
Fluid moves through reactor as a plug does not mix with fluid elements
in front or behind it. As a result, the reaction rate and concentration are
generally non-uniform throughout the reactor. The composition at any
travel time/distance down the reactor is identical to the composition in a
batch reactor after the same period of time has passed.

6. Reactor Types
(a) Batch Reactor
(b) CSTR
(c) CSTR!s in Series
(d) Rectangular PFR
Reactor Types and Flow Models
(d) Rectangular PFR
(e) Tubular PFR
(f) Serpentine PFR
(g) Downflow PBR
(h) Upflow PBR
(i) Fluidized PBR

7. General Mass Balance
Accumulation = Inflow Outflow + Reaction
VdC/dt = Q0C0 QC + kCn

8. The General Mole Balance Equation
To perform a mole balance on any system, the system
boundaries must be specified and the volume enclosed by
these boundaries defines the system volume.
Ri
Fi0 Fi
rate of flow of i
into the system
(moles/time)
rate of
degredation of i
by chemical
reaction within
the system
(moles/time)
rate of flow of i
out of the system
(moles/time)
rate of
accumulation of
i within the
system
(moles/time)
+ – =
In + Reaction – Out = Accumulation
Fi0 + Ri – Fi = dNi/dt
Qi0Ci0 - kCiV – QiCi = d(CiV)/dt
* Note: If Ri is a degradation reaction,
dt
dN
dV
r
F
F
dV
r
R
V
r
R i
V
i
i
i
V
i
i
i
i =
−
−
−
=
−
= ∫
∫ 0
0
0
hence,
;
or

9. After initial reactants added, no
reactants or products flow into or out of
the reactor.
Complete mixing occurs
instantaneously and uniformly
(a) Ideal Reactor
Completely Mixed Batch Reactor (CMBR/BR)
instantaneously and uniformly
throughout the reactor.
The reaction rate proceeds at the
identical rate everywhere in the reactor.
The probability of a particle of water
being in any one part of the tank at any
time is the same

10. Batch Reactors
Batch Reactors (BR)
Fi0 = 0 (Mass flow in)
Fi = 0 (Mass flow out)
Ri = -"ridV (Rate of reaction)
Ai = dNi/dt (Rate of accumulation)
V
dN
General Mass Balance
Ai = Fi0 Fi + Ri
Ai = 0 0 + Ri
i
i
i
i
i
i
i
i
V
i
i
r
dt
dC
V
r
V
dt
V
dC
V
V
C
N
V
r
dt
dN
dV
r
dt
dN
−
=
⇒
−
=
=
−
=
−
= ∫
1
1
olume
constant v
assuming
and
,
that
Noting
hence,
location;
ith
reaction w
of
rate
in the
variation
no
is
there
means
mixing
perfect
assuming
0

11. Batch Reactors
i
i
From Mass Balance
dC
r
dt
= −
Assume First Order Reaction
dC
Assuming 1st Order Kinetics
lnC-lnC0 = -kt
ln(C/C0) = -kt
C/Co = exp(-kt)
0 0
i
i i
C t
C
i
dC
r kC
dt
dC
kdt
C
= − = −
⇒ = −
∫ ∫
C= Concentration of reactant in BR, mole/L
C0 = Initial concentration of reactant, mole/L
N = amount of reactant in reactor, mole,
T = time, s
K = reaction rate constant

12. Reactants and products continuously
flow into and out of the reactor.
Complete mixing occurs
instantaneously and uniformly
throughout the reactor.
(b) Completely-Mixed Flow Reactor
(CMFR = MFR = CSTR = CMSTR)
throughout the reactor.
The concentration and reaction rate are
the same as in the effluent and identical
everywhere in the reactor.

13. Mixed Flow Reactor (MFR)/CSTR
Assumptions
Complete mixing
Steady process
Flow in = flow out
A + B
Products +
un-reacted
A and B
Differences between BR and CSTR
In the BR, all the reactants in the reactor have the same
residence time. Where in CSTR, the reactants are in the
reactor for a variety of the residence time.
BR is a non-steady state, the concentration of reactant changes
with time. In CSTR, concentration is the same throughout the
reactor and all the time (if the reactor has reached steady
state).
A and B

14. General Mass Balance
Ai = Fi0 Fi + Ri
d(CV)/dt = Q0Ci0 - QCi - "ridV
A + B
Mixed Flow Reactor (MFR)/CSTR
Fi0 = Q0Ci0
Fi = QCi
Ri = -"ridV
Ai = dNi/dt = 0 (Assume Steady State)
Products +
un-reacted
A and B

15. Mixed Flow Reactor (MFR)/CSTR
0
0
0
0
assuming perfect mixing means there
is no variation in the rate of reaction with location; hence,
Noting that , and assuming constant flow rate
V
i i i
i i i
i i
F F rdV
F F rV
F C Q
C
V C
= − −
− =
=
−
∫
Assuming 1st Order Kinetics
=V/Q = (Ci-Ci0)/-kCi
k = Ci0/Ci - 1
C/C0 = 1/(1+k )
0 0 0
0
0
i i i
i
C Q C Q
C
V
V
C
Q
r τ
− = ⇒
−
= =
mean residence time (a.k.a., hydraulic retention time)
i
i
r
V
Q
τ =

16. Mixed Flow Reactor (MFR)/CSTR
Example
If a reactor, has C0=150 mg/L and k=2.0 hr-1, make a plot of
the effluent concentration C as a function of hydraulic
residence time for CSTR (assuming 1st order reaction)..
0
0 C
C
C C
V
τ
−
= = ⇒ =
160
0
0
0
1
150
1 2.0
C
C
k
C
C C
V
Q r τ
τ
τ
−
= = ⇒ =
+
⇒ =
+
Hydraulic Residence Time τ, hr
0 2 4 6 8 10
Effluent
Concentration
C,
mg/L
0
20
40
60
80
100
120
140

17. Fluid moves through reactor as a plug
does not mix with fluid elements in
front or behind it.
The reaction rate and concentration
vary along the length of the reactor.
(c) Plug Flow & Packed Bed Reactors
(PFR/PBR)
The composition at any travel
time/distance down the reactor is
identical to the composition in a batch
reactor after the same period of time has
passed.

18. Plug Flow Reactor (PFR)
Assumptions
No mixing between dVs
Steady process (Q0 = Q)
Complete mixing in dV
General Mass Balance
A = F F + R
A + B
Products +
un-reacted
A and B
Ai = Fi0 Fi + Ri
Fi0 = Q0Ci0
Fi = QCi
Ri = -"ridV
Ai = dNi/dt = 0
Δx
Fi0 (x) Fi (x + Δ x)
QCi (x) Q (C - dC)

19. Plug Flow Reactor (PFR)
Assuming steady state
0 = QCi - Q(Ci - dCi) - "ridV
0 = QCi - Q(Ci - dCi) - "ridV
0 = QdCi - "ridV
QdCi = "ridV
A + B
Products +
un-reacted
A and B
i i
dC r
dV Q
=
Δx
Fi0 (x) Fi (x + Δx)
V
∫
i
i
i
i
i
i
i
i
i
i
i
i
i
i
V
i
i
i
r
dC
Q
V
Q
dV
r
dC
dx
dC
A
Q
Q
C
F
A
r
dx
dF
A
r
dx
dF
x
x
A
r
x
x
F
x
F
x
Adx
dV
dV
r
F
F
−
=
=
⇒
=
−
=
=
−
=
=
−
→
Δ
Δ
=
Δ
+
−
Δ
=
−
−
= ∫
τ
0
0
0
0
0
0
rate
flow
constant
assuming
and
,
that
Noting
or
0
as
;
)
(
)
(
hence,
location;
given
a
at
ith time
reaction w
of
rate
in the
variation
no
is
there
means
within
mixing
perfect
assuming
;
0

20. Plug Flow Reactor (PFR)
Assuming 1st Order
Kinetics
A + B
Products +
un-reacted
A and B
Δx
Fi0 (x) Fi (x + Δ x)
QCi (x) Q (C - dC)
0 0
0
Assuming first order
ln
A
A
C V
C
r
dC
dV Q
r
dC kc
dV
C V
k k
C Q
Q Q
dC k
dV
C Q
τ
=
−
= =
= −
= −
= −
∫ ∫

21. Plug Flow Reactor (PFR)
Example
If a reactor, has C0=150 mg/L and k=2.0 hr-1, make a plot of
the effluent concentration C as a function of hydraulic
residence time for PFR (assuming 1st order reaction).
160
0
2.0
150
k
C C e
C e
τ
τ
−
−
=
⇒ = ×
Hydraulic Residence Time τ, hr
0 2 4 6 8 10
Effluent
Concentration
C,
mg/L
0
20
40
60
80
100
120
140

22. CSTR vs PFR
Effluent
Concentration
C,
mg/L
100
120
140
160
CSTR
PFR
Hydraulic Residence Time τ, hr
0 2 4 6 8 10
Effluent
Concentration
C,
mg/L
0
20
40
60
80

23. Batch and Continuous-Flow Reactors
Packed Bed (Plug Flow) Reactors (PBR)
Fi0 = Q0Ci0
Fi = QCi
Ri = -"ri’dV
Ai = dNi/dt = 0
A + B Products +
un-reacted
A and B
' '
V
= − − =
∫
Δx
Fi0 (x) Fi (x + Δx)
' '
0 0
'
0
0 ; mol Areacted g media-s
assuming perfect mixing within means there
is no variation in the rate of reaction with time at a given location; hence,
( ) ( ) ;
V
i i i i
i i i media m
F F r dW r
x
F W F W W r W W V
ρ ρ
= − − =
Δ
− + Δ = Δ = =
∫
2
'
0
0
;
or
Noting that , and assuming constant flow and density of media
edia r
i i
i i
i i
i media i
i s m
media i
a L L length
dF dF
r r
dW dW
F C Q
Q dC V dC
r
dV Q r
π
ρ
ρ τ
ρ
−
=
− = = −
=
= − ⇒ = =
−

24. Batch and Flow Reactor Summary
General Mole Balance Eq!n:
Kinetic Rate Law for ri is:
A function of the properties of reacting materials
Intensive (affected by system temperature, pressure, etc.)
0 0
V
i
i i i
dN
F F rdV
dt
− − =
∫
An algebraic equation, not a differential equation
Mole Balances for Four Common Reactors
Reactor Mole Balance Comment
BR dNi/dt = -riV no spatial variation, non-steady
CSTR V = (Fi0 Fi)/-ri no spatial variation, steady-state
PFR dFi/dV = -ri steady-state, rate varies spatially
PBR dFi/dW = -ri
’ steady-state, rate varies spatially

25. Conversion in a Chemical Reactor
Conversion (= #removal$)
The number of moles of A that have reacted per mole of A fed to the system:
.
;
1
1
fed
of
moles
reacted
of
moles
0
0
0
0
const
V
C
C
N
N
N
N
N
A
A
X
A
A
A
A
A
A
A
A =
−
≡
−
=
−
=
=
1 log removal, X = 1 C/C0 = 0.9 ; 2 log removal, X = 1 C/C0 = 0.99
3 log removal, X = 1 C/C0 = 0.999 ; 4 log removal, X = 1 C/C0 = 0.9999
XA = Conversion or removal
NA0 = Moles of A fed
NA = Moles of A left (remained)
CA0 = Concentration of A Fed
CA = Concentration of left (remained)

26. Conversion in a Chemical Reactor
Total Number of Moles, N, and Molar Flow Rate, F
NA = VCA FA = QCA
Conversion X = “removal” (constant V & Q)
Batch Reactors: X = 1 – NA/NA0 X = 1 – CA/CA0
Flow Reactors: X = 1 – F /F X = 1 – C /C
Flow Reactors: X = 1 – FA/FA0 X = 1 – CA/CA0
Hydraulic Retention Time, τ = V/Q (note: Q# τ = V)
The “hydraulic retention time”, τ, is the average amount of a
chemical species spends in a reactor or the “mean residence time”

27. Reactor Performance in terms of Conversion
Batch Systems
The longer a reactant is in the reactor, the more is converted to
product until equilibrium is reached or the reaction is exhausted
In batch systems the conversion, X, is a function of the time the
reactants spend in the reactor
NA0 = initial number of moles of A
NA0 = initial number of moles of A
X·NA0 = no. of moles of A that have reacted at time t
NA = NA0 X·NA0 = no. moles A left in the reactor at time t
dCA/dt=-rA
0
0 0
0 ( )
Assuming consta
(1 )
1 1
n V
( )
t ,
kt
kt
A
A
t
A A
k
first order reaction
C e
C
X e for first order reaction
C C
C C e
−
−
−
−
= − = = −
=

28. Reactor Performance in terms of Conversion
Flow Systems
The conversion increases with the time reactants spend in the
reactor, but for flow systems this time increases with reactor size;
consequently, the conversion X is a function of reactor size or
volume V whether in CSTR, PFR, and PBR systems
FA0 = molar flow rate of A fed to a system at steady-state
FA0 = molar flow rate of A fed to a system at steady-state
X·FA0 = molar rate at which A is reacting in the system
FA = FA0 X FA0 = molar flow rate of A exiting the system
FA = (1 X)FA0
or, since FA0 = Q0·CA0 and FA = Q0·CA (assuming constant Q)
CA = CA0 X CA0 = molar flow rate of A exiting the system
CA = (1 X)CA0
1 X = CA/CA0 (recall our original definition of X = 1 CA/CA0)

29. Reactor Performance in terms of Conversion
CSTR or #Backmix Reactor$ (I will explain this later)
Recall the CSTR performance equation: FA0 FA = rAV
Substitute for the exiting molar flow rate of A, FA in terms of the
conversion X and the entering flow rate FA0
FA0 FA = X·FA0
X·F = r V
X·FA0 = rAV
We can re-arrange to determine the CSTR volume necessary to
achieve a specified conversion X
V = X FA0/( rA)exit
( rA)exit = rate of reaction based on the concentration of A at the exit
because the exit composition from the reactor is identical to the
composition inside the reactor

30. Reactor Performance in terms of Conversion
Plug Flow Reactor (PFR)
Recall the PFR performance equation: dFA/dV = rA
Substitute for the entering (fed) molar flow rate of A, FA in
terms of the conversion X and the entering flow rate FA0
FA = FA0 X·FA0
F ·dX/dV = r
FA0·dX/dV = rA
We now separate the variables and integrate with the limit V = 0
when X = 0 to obtain the PFR volume required to achieve a
specified conversion X
V = FA0 !0
X
dX/( rA)
To carry out the integration we need to know how the reaction rate ( rA)
varies with the concentration (i.e., conversion) of the reacting species

31. Comparing Performance of CSTR & PFR
Reciprocal Rate of Reaction (same for any reactor)
The rate of disappearance of A, rA, is a function of the
concentrations of the various species present or the conversion X
Many reactions in water and wastewater treatment are first order
(or appear as first order) with a rate of disappearance given by:
rA = kCA = k(1 X)CA0
0
200
400
600
800
1,000
0.0 0.2 0.4 0.6 0.8 1.0
Conversion, X
1/(-
r
A
),
L-s/mol
rA = kCA = k(1 X)CA0
A plot of 1/( rA) versus X yields a curve like the one below
Irreversible
1/( rA) & as X 1
Reversible
1/( rA) & as X Xequil
X r A , mol/L-s 1/( r A ), L-s/mol
0.00 0.005300 189
0.10 0.005200 192
0.20 0.005000 200
0.30 0.004500 222
0.40 0.004000 250
0.50 0.003300 303
0.60 0.002500 400
0.70 0.001800 556
0.80 0.001250 800
0.85 0.001000 1,000
characteristic first
order behavior

32. Comparing Performance of CSTR & PFR
Sizing a CSTR
Analytically calculate the volume required to achieve 80%
conversion of A for Q0 = 6 L/s and CA0 = 0.1445 mol/L
V = X·FA0/( rA)exit; 1/( rA)exit = 800 L-s/mol
V = 0.8 × 6 × 0.1445 × 800 = 555 L
Graphically determine the volume of CSTR required to achieve
0
200
400
600
800
1,000
0.0 0.2 0.4 0.6 0.8 1.0
Conversion, X
1/(-
r
A
),
L-s/mol
Graphically determine the volume of CSTR required to achieve
80% conversion of A
V/FA0 = 1/( rA)exit·X
X r A , mol/L-s 1/( r A ), L-s/mol
0.00 0.005300 189
0.10 0.005200 192
0.20 0.005000 200
0.30 0.004500 222
0.40 0.004000 250
0.50 0.003300 303
0.60 0.002500 400
0.70 0.001800 556
0.80 0.001250 800
0.85 0.001000 1,000
X = 0.8
1/( rA) = 0.8

33. Comparing Performance of CSTR & PFR
Sizing a PFR
Analytically calculate the volume required to achieve 80%
conversion of A for Q0 = 6 L/s and CA0 = 0.1445 mol/L
V = FA0·#0
X
dX/( rA) = FA0·$ ΔX/( rA)x
Let ΔX = 0.1 and use the trapezoid rule
V = (6×0.1445)×(0.1)[(189+192)/2 +'+ (800+556)/2] = 225 L
0
200
400
600
800
1,000
0.0 0.2 0.4 0.6 0.8 1.0
Conversion, X
1/(-
r
A
),
L-s/mol
V = (6×0.1445)×(0.1)[(189+192)/2 +'+ (800+556)/2] = 225 L
The volume is based on
the area under the curve
X r A , mol/L-s 1/( r A ), L-s/mol
0.00 0.005300 189
0.10 0.005200 192
0.20 0.005000 200
0.30 0.004500 222
0.40 0.004000 250
0.50 0.003300 303
0.60 0.002500 400
0.70 0.001800 556
0.80 0.001250 800
0.85 0.001000 1,000

34. Comparing Performance of CSTR & PFR
Comparing CSTR and PFR Sizes
600
800
1,000
),
L-s/mol
V/FA0 - CSTR
0
200
400
600
0.0 0.2 0.4 0.6 0.8 1.0
Conversion, X
1/(-
r
A
),
L-s/mol
V/FA0 - PFR