Mass Balance
Development
The Law of Conservation of Mass states that mass can neither be produced or
destroyed. This law provides the basis for one of the most important tools in
environmental engineering - the mass balance.
The mass balance provides a means for constructing a budget for a material (mass) which
is no more complicated than bookkeeping for a checking account:
$
t
 $ in  $ out  $ reacted
where: $reacted are the gains and losses to such things as interest (+) and check charged (-).
Let's apply this to the case of chemical mass:
M t  t  M t  Min , t t  t  Mout , t t  t  M reacted , t t  t
The analysis is performed over the time period t. Moving the initial mass to the other
side and dividing by t yields:
M t  t  M t M in , t  t  t M out , t  t  t M reacted , t  t  t



t
t
t
t
The left hand side of the equation is the rate of change in chemical mass, i.e.
M t  t  M t M

t
t
and as t  0,
M dm

t
dt

The terms on the right hand side are each a mass flux ( m , units of mass per time), i.e. the
rate at which mass enters, exits or reacts within the system. This can then be written:


dm 
 min  mout  m reaction
dt
This is the governing equation for mass balances throughout environmental engineering.
It remains to identify an approach for quantifying the terms in this equation.
1
Reactor Analog
Two reactor analogs, plug flow (PFR) and completely mixed flow (CMFR), are
commonly used in environmental engineering. We will begin our development of the
mass balance using the latter reactor analog.
[T] Completely-mixed flow reactor
One fundamental of the CMFR is that it is well mixed, i.e. material introduced to the
reactor is instantaneously mixed throughout the reactor and chemical concentrations are
the same everywhere in the reactor.
The Control Volume
The mass balance is written for a control volume, i.e. a specific region of space
which has boundaries across which the mass flux in and mass flux out can be determined.
[T] Control volume
The control volume for the CMFR is the entire reactor, taken to include the inlet and
outlet.
Terms in the Mass Balance
There are four terms in the mass balance:


dm 
 min  mout  m reaction
dt
the rate of mass accumulation, the mass flux in, the mass flux out and the rate of mass
reaction.
Rate of mass accumulation
As mentioned earlier, it is concentration, not mass, which is of great importance in
environmental systems. For this reason, we write the mass balance in terms of
concentration, remembering that:
m  VC
and substituting:
bg
dm d VC

dt
dt
2
For our purposes, and in most environmental applications, it can be assumed that the
volume of the reactor is constant. Thus:
dm
dC
 V
dt
dt
Note that the assumption of constant volume requires that inflow equal outflow.
Mass flux in
The mass flux in is most commonly quantified as the product of the volumetric flow rate
(Q) and the chemical concentration in the inflow (Cin):

min  Qin  Cin
Note the way that the units work out in this equation:
g m3 g


d d m3
Mass flux out
The mass flux out is calculated in a similar way:

mout  Qout  Cout
again, with units:
g m3 g


d d m3
But note that since the chemical concentration is the same anywhere in a CMFR,
including at the outlet, we can refer to Cout as simply C:

mout  Qout  C
Rate of mass reaction
This term quantifies gains or losses of a chemical as a result of biological or chemical
reactions. Chemical reaction rates are typically expressed in terms of concentration, not
mass, as it is concentration which often drives the reaction:
3

mrxn  V 
dC
dt due to rxn only
note the dimensional character of this term:
g
g
 m3  3
d
m d
Proceeding from our discussion of chemical kinetics, there are three types of reaction
kinetics which we should consider:


Conservative

Zero Order

First Order
m
rxn

m
rxn

m
rxn
V
dC
0
dt
V
dC
 k C0  k
dt
V
dC
 k  C1  k  C
dt
We consider examples of these in developing mass balances.
Putting it all Together
The original mass balance expression:


dm 
 min  mout  m reaction
dt
can thus be written as:
V
F
I
G
HJ
K
dC
dC
 Q  Cin  Q  C  V 
dt
dt
rxn only
Note that Qin = Qout = Q since volume is constant. Before trying our first application of
the mass balance, we need to address a key concept: steady state.
4
Steady state versus Non-steady state
It can be seen from the above equation that changes in flow, inlet concentration, or
reactivity result in changes in chemical concentration within the reactor. If these
conditions remain constant for a sufficient period, they come into balance and the
concentration and mass within the control volume remain constant. This is termed steady
state and since concentration is not changing:
dm
dC
 V
0
dt
dt
[T] Mall
[T] CMFR: discuss how the ‘QC’ and VkC’ terms allow the reactor to ‘catch up’.
In sufficient time has not passed since a change in flow, inlet concentration, or reactivity
then the system is considered non-steady state and:
dm
dC
 V
0
dt
dt
This has important implications in solving the mass balance equation.
CMFR: Steady State - Conservative
The steady state conservative case for a CMFR with a single inlet and outlet is quite
straightforward:
V
F
IJ
G
HK
dC
dC
 Q  Cin  Q  C  V 
dt
dt
rxn only
0  Q  Cin  Q  C
C  Cin
and has relatively little application. Of more interest is the case where two inlet streams
meet with a single outlet. We will examine this case with another example from the
Onondaga Lake system.
5
Example: Chloride in 9 Mile Creek
[T] Chloride and 9 Mile Creek
Upstream of the waste lagoons, 9 Mile Creek has a flow of 3.5 m3/s and a chloride
concentration of 13.8 mg/L. The lagoons discharge to the creek at a rate of 0.7 m3/s with
a chloride concentration of 50,000 mg/L. Use a mass balance approach to determine the
chloride concentration in the mixing basin.
V
V
F
IJ
G
HK
FdCI
 QC  VG J
Hdt K
dC
dC
 Q  Cin  Q  C  V 
dt
dt
dC
 Q up  C up  Q in  Cin
dt
rxn only
rxn only
We can make some assumptions here to aid in simplifying the solution. First, this is a
steady state problem because the control volume is small (has a short hydrualic residence
time) and thus we can assume that the system is in equilibrium with its flow and inlet
concentration. Second, because chloride is a conservative substance, the reaction term
goes to zero. Thus:
0 Q up  Cup  Qin  Cin  Q  C
C
Q up  C up  Qin  Cin
Q

35
.  138
.  0.7  50000
mg
 8344
35
.  0.7
L
This particular mass balance, i.e. one for a conservative substance at steady state, is often
referred to as a mixing basin calculation and is commonly employed wherever two
chemical streams join together.
CMFR: Steady State - 1st Order Decay
The steady state with first order decay for a CMFR is very common in both natural
(lakes) and engineered (treatment units) systems:
V
F
IJ
G
HK
dC
dC
 Q  Cin  Q  C  V 
dt
dt
here the reaction term is given by:
6
rxn only
dC
  k  C1   k  C
dt rxn only
and, at steady state, the mass balance is:
0  Q  Cin  Q  C  V  k  c
and solving for C:
C  Cin 
Q
Q V k
Example: Portage Lake Treatment Plant
The Portage Lake Treatment Plant receives an inflow of 8000 m3d-1 with a pollutant
concentration of 250 mgL-1. The first unit operation removes the pollutant according to
first order kinetics with a rate constant of 0.5 d-1. The volume of the tank is 2500 m3.
Calculate the effluent pollutant concentration.
C  C in 
Q
8000
mg
 250 
 216
Q  Vk
8000  2500  0.5
L
Another way of looking at this application is to consider the tank size required to effect a
particular reduction in pollutant concentration.
Calculate the tank size (m3) required to achieve a 75% reduction in the effluent
concentration:
For a 75% reduction, C  0.25 Cin
0.25 C in  C in 
0.25 
Q
Q  Vk
Q
Q  Vk
0.25 Q  0.25 V  k  Q
V
0.75 Q
0.75 8000

 48000 m3
k
0.5 0.25
7
Note that the effluent concentration can be reduced through reductions in Q and increases
in V and k. Why is this? How can we influence these parameters?
Hydraulic Retention Time
Hydraulic retention time (, units of time; also called detention time and residence time)
refers to the average period which a volume of water spends within the control volume.
 
V
Q
Since this is the time available for reactions to proceed,  influences the degree of
treatment.
Example: Portage Lake Treatment Plant
In the previous example, we increased treatment efficiency from ~15% removal to ~75%
removal by changing the tank volume from 2500 to 48000 m3. What was the
corresponding change in hydraulic retention time?
1 
V1
2500

 0.33 d
Q
8000
2 
V2
42000

 5.25 d
Q
8000
Reductions in flow or increases in volume will result in reduced effluent concentrations
and improved treatment. Since we have little control over flow, effluent quality is
typically managed through tank size. What are the economic implications of this?
Kinetics
The degree of treatment is also influenced by the reaction rate coefficient, k. The
coefficient can be influenced by a variety of environmental factors, most notably
temperature. The effect of temperature on the reaction rate coefficient is calculated using
a 'theta function':
k T  k 20  T20
8
The effect is one of increasing values of k with increasing temperature and the value of 
ranges from 1.00 (no temperature effect) to ~1.2 (very strong temperature effect).
Example: Portage Lake Treatment Plant
Some reactions, such as the conversion of ammonia to nitrate, are highly sensitive to
temperature. Calculate the % removal for the case above (V = 42,000 m3) for
temperatures of 4°C and 20°C if  = 1.08.
At 20°C, k = 0.5 d-1 and the removal is 75% … as calculated previously.
At 4°C:
k 4  k 20  T20  0.5108
. ( 420)  0.29
C  C in 
Q
8000
mg
 250 
 99
Q  Vk
8000  42000  0.29
L
This is 60% removal. Thus one can expect a strong seasonal signal in effluent quality.
[T] Nitrification and seasonality in discharge
Time Variable Solution
In the time variable solution, the left-hand side of the equation does not go to zero and we
must integrate the mass balance equation:
V
F
IJ
G
HK
dC
dC
 Q  Cin  Q  C  V 
dt
dt
rxn only
over the interval t = 0 to t = t. For a step change from one steady state condition (C ss1) to
a second (Css2), the solution is:
F
G
H
1 I
F
F
IJ
 k J t
G
G
H
H
K
C t  Css1  e
 Css2  1  e  K
1
   k t
Let's look at this equation in its parts and then as a whole.
[T] Time variable response
9
I
J
K
We can conclude our treatment of the CMFR with a discussion of the choice between
steady state and time variable solutions.
Response Time
Response time is how long a system takes to reach equilibrium with a new set of
inputs or kinetics. Because a system is never 100% at equilibrium (it takes an infinite
period), we typically say that a system is at steady state when it has covered 95% of the
distance between Css,1 and Css,2. This leads to:
t 95% 
 ln 0.05
3

1
1
k
k


Note that for a conservative substance, we can say that steady state is achieved after three
retention times:
t 95% 
3
 3 
1

and that for a reactive substance it will reach steady state more quickly.
Selection of SS or Time-Variable Approach
We set two criteria for the use of a steady state approach. First, the time to steady
state must be meaningful in terms of the application of a steady state solution to the
problem at hand. We can evaluate this by calculating t95%. Second, inputs or kinetics
must have been constant long enough for the system to reach steady state. Again, a
knowledge of t95% and the time variability of the inputs and kinetics help us to evaluate
this.
Treatment Facilities
Evaluation of treatment performance is conducted on a monthly or weekly (and
occasionally, daily) basis. As can be seen in Table 4-3, the hydraulic retention times for
most unit operations in water and wastewater treatment are short and the response times
are on the order of hours to days. This meets the criteria for a steady state solution.
Inputs to treatment plants vary seasonally, but are reasonably constant over the response
time of the unit operations. An exception to this is diurnal flows which are
10
accommodated with equalization basins. This also meets the criteria for a steady state
solution.
Natural Systems
Evaluation of water quality conditions in lakes is typically considered on an annual basis.
While this would suggest that a steady state solution is valuable, the hydraulic retention
time of lakes varies widely and thus so does the utility of a steady state approach:
Onondaga Lake (0.25 yr), Lake Ontario (8 yr), Lake Michigan (136 yr), Lake Superior
(179 yr). Here, that application is lake and chemical (due to kinetics) specific.
Seasonal variation in inputs and kinetics must also be considered. This is especially
important with respect to spring runoff and biological features and depends on the
specific case.
Batch Reactor
A reactor which has no inlet our outlet is termed a batch reactor. Mathematically:
V
F
IJ
G
HK
dC
dC
 Q  Cin  Q  C  V 
dt
dt
V
F
I
G
HJ
K
dC F
dC I
 GJ
dt H
dt K
dC
dC
 V
dt
dt
rxn only
rxn only
and for a first order decay:
dC
  k C
dt
and integrating:
C t  C0  e  k  t
Consider the shape of the C = ƒ(t) curve.
Plug Flow Reactor
11
rxn only
The CMFR reactor is used to simulate conditions in tanks and tank-like systems, e.g.
lakes. The plug flow (PF) reactor is most suitable to model biochemical transformations
in pipes and pipe-like systems, e.g. rivers.
In a PFR, complete mixing occurs in a radial direction, but there is no mixing in the axial
direction. Thus each 'plug' of fluid is considered to be a separate entity ( mini-batch
reactor) as it flows down the pipe.
The mass balance for a PFR with first order decay is as follows:
V
F
IJ
G
HK
dC
dC
 Q  Cin  Q  C  V 
dt
dt
V
rxn only
dC
  V k C
dt
C t  C0  e  k  t
The influent and effluent concentrations are related by the hydraulic retention time of the
pipe, i.e. where t = :
Cout  Cin  e  k 
and  is related to the length of the pipe (L) and the fluid velocity ():

L

where fluid velocity is determined by the flow (Q) and the cross-sectional area of the pipe
(A):

Q
A
and thus:

LA
Q
Note that the apparent time dependence here is really a position dependence which is
governed by the velocity and downstream distance in the pipe:
12
t
L

where  is the fluid velocity and L is the downstream distance. Thus:
C L  C0  e
 k
L

Comparison of CMFR and PFR
Efficiency
Consider the case of PF and CMF reactors having the same volume (V), inflow
(Q), influent concentration (Cin), and reaction kinetics (k). The PFR is more efficient in
removing the chemical and the effluent from the PFR would have a lower concentration
than that for the CMFR. This occurs because the influent to the CMFR is immediately
mixed with the more dilute reactor contents, minimizing the value of the reaction term:
V  k C
in the CMFR (Figure 4-8). Thus a smaller (and more economical) PF reactor would be
required to meet the same effluent standards.
Influent spikes
Many pollution control systems encounter fluctuations or spikes in their influent
concentrations. These spikes would simply travel down a PFR, but the complete mixing
in a CMFR tends to dampen the impact of spikes on the effluent concentration (Figure 49).
Selection of a Reactor
The selection of a PFR or CMFR in engineered systems is guided by efficient
utilization of tank volume (favoring PFRs) and sensitivity to inlet spikes (favoring
CMFRs). In natural systems, the choice is based on which best describes the lake or
river.
13