Mass Balances
Fundamental Principle of
(Dynamic) Mass Balances
The rate at which something accumulates
in a region of interest (a “control volume”)
equals the net rate at which it enters by
physical movement plus the net rate at
which it is generated inside the control
volume by chemical reactions.
Processes for Transport Across the
Boundaries of an Aquatic System
 Advection:
Bulk flow, carrying the
substance of interest with it
 Molecular Diffusion: Random thermal
kinetic motion, leads to net transport
down a concentration gradient
 Dispersion: Random motion of small
packets of fluid, leading to same result as
molecular diffusion, but usually faster
The Mass Balance in Words
Rate of change of the amount of i stored
within the system (rate of accumulation)
= Net rate (in - out) at which i enters by
advection
+ Net rate (formation - destruction) at
which i is created by chemical reaction
The ‘Storage’ or ‘Accumulation’ Term
d Vci 
Rate of Accumulation 
dt
Special Cases
dci
Well-Mixed,
Rate of Accumulation  V
Fixed Volume:
dt
Steady State: Rate of Accumulation  0
The Advective Term
Net Advective Inflow 
Q c
in i ,in
inlets
-
Q
c
out i ,out
outlets
Special Case
Batch system: Net Advective Flow  0
The Reaction Term
Mass or moles of i formed
Reaction Rate ri 
 Volume   Time 
Net Reaction Term  rV
i CV
ri  fcn  ci , c j , ck ,..., T 
The Reaction Term
Special Cases
Non-reactive Substance (Conservative Tracer):
ri  0
nth-order Reaction Dependent Only on ci:
ri  k c
n
n i
The Overall Mass Balance
for Constant-Volume Systems
d  cV
i CV 
  Qin ci ,in -  Qout ci ,out 
dt
inlets
outlets

all
reactions
rV
i CV
Hydraulic Characteristics
of Reactors
Idealized Model Reactors

Limiting Case #1. Unidirectional advection with no
mixing: A Plug Flow Reactor (PFR). Often used to
model rivers, pipe flow, settling basins, disinfection
processes.
Q, cin
Q, cout
x
L
All parcels of fluid have identical residence time:
L LA V
 

vx vx A Q
Concentration
Anticipated Tracer Output for a
Pulse Input to a PFR
Output
Input
0
Time

Concentration
Anticipated Tracer Output from Step
Input to a PFR
Input
F t  
Output
cs  t 
cin ,t 0
0
Time

More Realistic Tracer Profiles after a
Pulse Input into a PFR-Like Reactor
Dimensionless
Conc., c /c o
Concentration
12
t / = 0.3
10
0.5
0.7
8
0.9
6
4
2
0
0.0
0.2
0.4
0.6
Dimensionless Distance, x /L
0.8
1.0
Idealized Model Reactors
Limiting Case #2. Advection with intense mixing: a
Completely Stirred Tank Reactor (CSTR, CMFR,
CFSTR, CMR). Often used to model lakes,
reservoirs, flocculation basins.
Q, Cin
V
Q, Cout
•
All parcels of fluid have identical chance of
exiting in any instant, so they have a wide
range of residence times; still =V/Q
CSTR Response to a Pulse Input
of Tracer, Steady Flow
Evaluate from t = 0+ to 
Q, Cin
Q, Cout
• Constant V, Q
• cin = 0 (at t>0+)
• ci = ci,out
• No reaction
d  cV
i 
 Qin ci ,in - Qout ci ,out  rV
i
dt
dci
V
 Qin ci ,in - Qout ci  ri V
dt
dci
V
 -Qci
dt
CSTR Response to Pulse Input
dci
V
 -Qci
dt
c t 
t
t
dci
Q
1
c 0 ci  - V 0 dt  -  0 dt
 
c t 
t
ln
c  0

 t
c  t   c  0  exp   
 M
 t
  exp  - 
 V
 
CSTR Response to Pulse Input
1.0
0.9
0.8
 t
c  t   c  0  exp  - 
 
0.6
0.5
0.4
0.3
1/e
0.2
0.1
0.0
0

2
Time
3
4
5
6
0
c t 
t
ln
c  0

-1
-2
ln(cp /c o)
cp /c o
0.7
-3
-1
-4
1
-5
(b)
-6
-7
0

2
3
Time
4
5
6
CSTRs-in-Series: Response to a
Pulse Input of Tracer
0.035
0.030
20
E (t ), min-
1
0.025
0.020
0.015
N =1
0.010
2
4
0.005
0.000
0
20
40
60
80
100
t (min)
120
140
160
180
Representing Intermediate
Degrees of Mixing
 PFR
with Dispersion: Zero dispersion is
PFR; increasing dispersion increases
mixing; infinite dispersion is a CSTR
 CSTRs
in Series: Increasing N (keeping
V
and
Q
constant)
segregates
(conceptual) segments of reactor and
decreases mixing; as N increases, overall
mixing decreases, and reactor becomes
more PFR-like
Summary of Key Points
 Reactor
hydraulics can be characterized
by the range of residence times of entering
water ‘packets’
 Ideal,
limiting cases include PFRs (no
mixing) and CSTRs (infinite mixing)
 For
both CSTRs and PFRs, the average
residence time, , is V/Q. For PFRs all the
fluid spends time  in the reactor; for
CSTRs, different packets of fluid spend
different amounts of time in the reactor,
but the average is 
Summary of Key Points
 Conformity
to a limiting case can be
assessed by a pulse or step input test
combined with a mass balance analysis
 Intermediate
mixing can be modeled as
dispersion and/or CSTRs in series
Designing and Evaluating
Systems in which Chemical
Reactions are Occurring
Extent of Reaction in a
Batch Reactor
dc
dc
VV Q
cinin - Qc
Q cout
Vr
Vr
Qc
out 
dtdt
dc
r
dt
c t 
t
dc
c 0 r  0 dt  t
 
1st-Order Reaction in a Batch
Reactor
 In
a disinfection process, bacterial kill
follows the first-order reaction expression:
rX = -(1.38 min-1)cX. How long is required
for 99% disinfection?
ct 
c t 
c t 
dc
dc
1
dc
1 c t 
t 
 
- 
 - ln
r c 0 -k1c
k1 c 0 c
k1 c  0 
c 0
1
tln  0.01  3.3 min
-1
1.38 min
Extent of Reaction in a
CSTR at Steady State
Q
Q
Cin
Cout
d Vc 
dt
 Qcin - Qcout  Vr
V
Cout
cout
V
- cin  r  r
Q
cout - cin

r
1st-Order Reaction in a CSTR at
Steady State
 What
average residence time is required for
the same, 99% kill of bacteria, if the reactor
is a CSTR? If the flow rate is 1.5 m3/min,
how large must the reactor be?
cout - cin
1 - 100


 71.7 min
-1
-k1c
- 1.38 min  1
 m3 
 60 s 
3
V  Q  1.5
  71.7 min  
  6450 m
s 
 min 

Extent of Reaction in a
PFR at Steady State
Because a PFR is essentially a batch reactor
on a conveyor belt, the extent of reaction in a
PFR with a given detention time is identical to
the extent of reaction over an equivalent time
period in a batch reactor.
dc
r
dt
cout

cin

dc
  dt  
r 0
 A contaminant
decomposes according to the
rate
expression:
ri=-kci0.5,
with
k=750(mol/L)0.5/s. What residence time is
required in a PFR to reduce ci from 10-3 to
10-4 mol/L?

cout

cin
dc

r
cout

cin
dc
1
0.5
-kc
k
2
 7.5x10
-5
 mol/L 
s
0.5
cout

cin
dc
2 0.5 0.5
 - cout - cin 
0.5
c
k
 -4 mol 0.5  -3 mol 0.5 
10
 - 10
   577s
L 
L  


Summary of Key Points


The extent of reaction that occurs in a reactor
depends on both the intrinsic reaction rate and
the hydraulics. The net result can be obtained by
applying appropriate parameter values and
appropriate assumptions to a mass balance on
the reactant.
PFRs are identical to batch reactors moving
through space.
 The instantaneous dilution of reactants in a
CSTR causes those reactors to be less efficient
than PFRs or batch reactors