Reactor Concepts
Reactor – any device in which an incoming constituent undergoes
chemical (or biochemical) transformation, phase transformation, or
phase separation
Constituent – soluble, colloidal, and particulate substances
Starting point for analysis of reactor performance – material balance
relationship (input, output, and reaction terms)
Net rate of mass transport
through the control
volume
Net rate of
mass input
across the
CV
boundaries


Net rate of
mass output
across the
CV
boundaries
Net rate of mass
transformation
within the
control volume
(Vc )

Net rate of
reaction
within the
CV (Vc )

Net rate of mass
change within
the control
volume (Vc )

Net rate of
accumulation
within the
CV (Vc )
In  Out  Generation or Removal  Accumulation
Ideal Reactors - CMBRs
“Batch”  No input, no output
“Completely mixed”  no spatial concentration, density or thermal
gradients
Useful for determining reaction rates
Masses of the target components varying with time only by their
reactions
dC
MBE  rVR  VR
dt
Residence time ( t )

t
0
dt  
Ct
C0
t
Ct
C0
dC
r
dC
r
Ideal Reactors - CMFRs
Instantaneous mixing immediately after fluid elements and constituents
entering the reactor  rapid dilution of influent reactants
CR = Cout
Advantages – effective and fast mixing of reactants, reduction of
loading shock (buffering)
MBE  QC in  QC out  rVR  VR
dCout
dt
Ideal Reactors - PFRs
Uniform fluid velocity across any cross section normal to the axial flow
Fluid elements and constituents intermixed completely throughout the
cross-section of the reactor
No mixing in the direction of flow axis
C
C
 v x
r
t
x
MBE  point form for dVR
At steady state, 0  vx
C
r
x

0
Residence time ( t ) t  L  C dC
C
o ut
vx
L
in
r
Cout dC
1
dx  
Cin
vx
r
Comparison of Reactor Performance
Comparison of residence time based on the same extent of reactions
(i.e., reaction rate order, n, and rate coefficient, k)  the required
volume of reactor to achieve a given (expected) transformation rate
at a given flow rate
Comparison of Reactor Performance
Except 0th order reaction, the residence time for CMFR > the residence
time for PFR
500
500
450
(A)
350
450
PFR
400
Residence time (hr)
Residence time (hr)
400
CMFR
300
250
200
150
CMFR
PFR
350
300
250
200
150
100
100
50
50
0
(B)
0
0
20
Fractional removal (%)
40
0
20
40
Fractional removal (%)
Except 0th order reaction, the reaction efficiency for CMFR < the
reaction efficiency for PFR
CMFR in Series
A number of CMFRs connected in series  increase of overall process
efficiency (reduction of reactor volume, residence time, increase of
reaction rate, etc.)
With the same flow rate (Q) and reactor volume (VR), the reactor
behavior and performance → closed to those of PFR as the number
of the CMFR increases.
CMFR in Series
If the reaction is the 1st order,
Cn
C
C C C
 1  2  3 n
C IN C IN C1 C2 Cn 1
 1   1   1   1   1 






1 kt  1 kt  1 kt  1 kt  1 kt 

1  C IN
t  
k  Cn

n


  1



1
n
Total residence time,

n  CIN
nt  
k  Cn



  1



1
n
1
If n , by L’Hospital’s theorem,
 C IN  n
1

  1


C
n  C IN  n 
0

  1  lim  n 
lim

 1 0
n  k  C
k
0
 n 
 n
n

1


1
n
 C IN   1


C n C
 Cn 

 IN  ln  IN
C
C


 lim 
 lim  n   n
1
1

k
0
0
k 
n
n
 n 


  1 [ln( Cin )]
k
Cout
CMFR with Solids Recycle
Increase of residence time for fluids and solids within the reactor for the
enhancement of reaction efficiency without increase of reactor volume
MBE,
QIN C IN  QR COUT  (QIN  QR )COUT  rVR  0
(steady state)
QIN C IN  QIN COUT  rVR  0
Residence time
t
VR
1
 (C IN  COUT )
QIN r
Steady-state per-pass retention time for a CMFR with flow recycle ( t P ,
detention time per pass through the CMFR)
tP 
VR
QIN  QR
CMFR with Solids Recycle Using a
Clarifier
MBE,
QIN C IN  QR C R  (QIN  QR )CMIX  rVR  0
CMIX 
(steady state)
QIN C IN  QR C R
QIN  QR
If the input solids negligible, CIN = 0,
C MIX 
QR C R
RC R

QIN  QR 1  R
where, R = QR/QIN (recycle ratio)
Residence time
t
VR  VC
QIN
Mean solid residence time (SRT,
tS 
tS
)
CMIX VR
QW CW  QOUT COUT
If rate of solids build-up were small, CR  CMIX  close to steady state
Nonideal Reactors
In reality, no assumptions such as “completely (and instantaneously)
mixed” and “plug flow” are valid.  CMBR, CMFR, and PFR do not
exist.
Existence of dispersion condition  behaviors of reactors somewhere
between the PFR and the CMFR
Due to short-circuiting, recycle, stagnant zones
Nonideal Reactors
Flow and mixing characteristics deviated from ideal conditions –
determined by experiments  residence time distribution (RTD)
analysis using tracers
An RTD is obtained by applying stimulus-response analysis. 
introduction of reactor responses (readily detectable tracers) to
reactors as either pulse or step inputs
Pulse (delta) input – instantaneous injection of a fixed mass of tracer to
the influent of a reactor
Step input – continuous injection of a tracer at a constant concentration
to the influent of a reactor
Tracers – environmentally acceptable, non-reactive, and readily
measureable at low concentrations (e.g., Cl-, Br-, dye, etc.)
C Curve
When a pulse or delta input is used, the effluent tracer concentration profile
(stimulus-response relationship) is the C curve.
C curve  t (time) vs.
Cout
C
where C 
MT
VR
C curve shows how fluid elements are distributed in time as they pass through
the reactor.
Some elements exit in a time shorter than HRT, t , while others greater than t .

Mass balance check!  M T  Q  Cout (t )dt
0
E Curve
E curve – the exit age or residence time distribution curve
Area under the C curve,
Ac 
1
C


0
Cout (t )dt
E curve is defined as
E (t ) 
Cout (t ) / C
Ac
Total area under the E curve,


0
E (t )dt 
1
C


0
Cout (t )dt
Ac
1
C
 
1
C




0
0
Cout (t )dt
1
Cout (t )dt
 E curve shows a time-normalized or factional age distribution.
t
Fraction of fluid elements with age < t   E (t )dt
0
E Curve
Mean hydraulic residence time, t 
VR
Q
Mean constituent (dissolved solutes or suspended solids) residence time, =
mean value of the residence time of a constituent, which is given by the first


moment of the centroid of E curve,
tm



0

0
tE(t )dt
E (t )dt


0
tE(t )dt
1

  tE(t )dt
0
Variance of the distribution, which is given by the second moment of the
centroid of E curve,





0
0
0
0
0
 t2   (t  tm ) 2 E(t )dt   t 2 E (t )dt   2ttm E (t )dt   tm 2 E (t )dt   t 2 E (t )dt  tm 2
Skewness of the distribution, which is given by the third moment of the centroid
of E curve,
1 
3
( s)  1.5  (t  t m ) 3 E (t )dt
t
0
Class quiz
Question: Characterize the RTD for the filters.
Dimensionless E Curve
Using dimensionless time, 0  t / t , area under the dimensionless E
curve, if t  tm


0
E ( 0 )d ( 0 ) 
1
C


0
Cout ( 0 )d ( 0 )
Ac
1
C
 
1
C




0
0
Cout ( 0 )d ( 0 )
Cout ( )d ( )
0
0
Variance of the distribution,


 2   ( 0  t m / t ) 2 E ( 0 )d ( 0 )   ( 0  1) 2 E ( 0 )d ( 0 )
0
0
0



0
0
0
  ( 0 ) 2 E ( 0 )d ( 0 )   2 0 E ( 0 )d ( 0 )   E ( 0 )d ( 0 )

  ( 0 ) 2 E ( 0 )d ( 0 )  1
0
where   0 
2
 t2
t m2
1
F Curve
When the tracer is introduced as a step-function stimulus (continuous
input), the effluent tracer concentration profile (stimulus-response
relationship) is the F curve.
F curve  t (time) vs. F (t )
where F (t ) 
Cout
Cin
F(t)  the fraction of tracer substances having an exit age young than t.
F Curve
Relationship btw F and E
t
F (t )   E (t )dt
0
or
dF (t )
 E (t )
dt
Mean residence time

1
0
0
tm   tE(t )dt  tdF (t )
Variance
1
   tdF (t )  tm2
2
t
0
RTD Analysis
RTD analysis  to compare the performance of real reactors to those
of ideal reactors
The RTD patterns of ideal reactors (theoretical) must be analyzed by
mathematical description of the responses of ideal reactors (CMFR,
PFR, and CMRF in series) to either pulse or step inputs.
RTD Analysis
RTDs for ideal CMFR
E curve  Mass balance relationship
dC
QCin  QC out  VR out where Cin  0 for pulse input
dt
The initial condition,
MT
 C  Cout
VR
t 0
 This is because the tracer
is instantaneously mixed throughout the reactor at time = 0.
Integrating the mass balance eqn.,

Cout
C
t1
1
dCout   dt
0 t
Cout
C
t
 ln out 
C t
t

Cout
e t
C
RTD Analysis
E curve
E (t ) 

C /C
C
C curve
e
 out t   out 

Area under the C curve
C t
t
t
e
dt

Area under the E curve,


0
E (t )dt  

0

0
t
t
e
dt  1
t
Dimensionless form by assuming
E ( 0 )  e
t
t
t  tm
0
Mean constituent residence time ( t m ) must be the same as hydraulic retention
time ( t )

tm   t
0

t
t
e
V
dt  t ( R )
Q
t
RTD Analysis
F curve  Mass balance relationship
dC
QC in  QC out  VR out
dt
Integrating the mass balance eqn.,

Cout
0
t1
1
dCout   dt
0 t
Cin  Cout
C  Cout t
 ln in

Cin
t
t

C
F (t )  out  1  e t
Cin
The shaded area above the F curve  Mean constituent residence time
Mean constituent residence time ( t ) must be the same as hydraulic retention
time ( t m )
1
tm   tdF (t )  t (
0
VR
)
Q
RTD Analysis
RTDs for ideal PFR
Pulse input
The response is a spike of infinite height
and zero width (a peak with no defined
area).
Mean constituent residence time must be
the same as hydraulic retention time
( tm  t ).
Step input
The response is a instantaneous step
increase in the exit concentration from
zero to a feed concentration at time t.
Mean constituent residence time must be
the same as hydraulic retention time
( t  t ).
m
RTD Analysis
RTDs for nonideal PFDR
Starting point for the characterization of real reactors – PFR with
dispersion (= plug flow dispersion reactor, PFDR)  advectiondispersion-reaction (ADR) model
For pulse input,
One-dimensional (x-axis) form of ADR
equation
C
 2C
C
 vx
 Dd 2  r 
x
x
t
Dimensionless form of ADR equation
C
C
( 2 C )
  0  Nd
0


( 0 ) 2
vt t
x
,  0  x  (t is fluid residence time),
L
L t
Dd
and N d (dispersio n number) 
vx L
0
where,  
If Nd  0, then the reactor  PFR
If Nd  , then the reactor  CMFR
RTD Analysis
Boundary conditions
C  0 at x  
for t  0
C  0 at x  
for t  0
For small amounts of dispersion, the analytical solution to the
dimensionless ADR equation
 (1   0 ) 2 
Cout
1

exp 

C (4N d )0.5
4Nd 

Variance,  20  2 Nd
This is valid only if the amount of dispersion is small and the C curve is
reasonably close to symmetrical.
RTD Analysis
For large amounts of dispersion,
RTD Analysis
For step input,
Boundary conditions
C  0 for x  0
at t  0
C  Cin for x  0
at t  0
C  0 at x  
for t  0
C  Cin at x  
for t  0
Analytical solution to the dimensionless ADR equation
Cout
Cin

t 

1



 
 0.5
t
 F (t )  0.51  erf 0.5 N d

t
0.5


( ) 

t  


1 
Variance (for closed vessel),  20  2 N d  2 N d 2 1  exp( 
)

Nd 
