Mechanistic, Mass Balance Models in Surface Water Quality Engineering
1. The Basic Mass Balance
A common approach in surface water quality modeling is to develop a mass balance on a
completely-mixed flow reactor, i.e.
dM
 sources  sinks .
dt
Given that,
M  C V ,
the equation may be written as,
d  C V 
 sources  sinks
dt
and for a constant volume,
V
dC
 sources  sinks
dt
Here, the source is loading and the sinks include outflow and, potentially, reaction and settling.
[T] Chapra, Figure 3.1
2. Loading
Mass may enter a lake from a variety of sources, including point and tributary discharges, direct
runoff and from the atmosphere and sediments. In a modeling context, these inputs represent
the loading (W, e.g. g·d-1) to the system,
Loading  W
Loading is described mathematically in two ways, depending on the nature of the source. For
tributary and point sources,
W  Qi  Cin , i
where Qi is the flow (e.g. m3·d-1) and Cin,i the concentration of the material of interest (e.g., g·m3
) in tributary or point source i. For atmospheric and sediment inputs, an areal approach is
taken, tributary or point source
W JA
where J is the flux from the atmosphere or sediment (e.g. g·m-2·d-1) and A is the area of the
sediment or, for atmospheric inputs, the lake surface (e.g., m2).
3. Outflow
Mass is removed from the system in the outflow stream,
Outflow  Q  C
where Q (e.g. m3·d-1) is the sum of the inflows and C (e.g., g·m3) is the concentration of the
material of interest in the lake (note the impact of the assumption of complete mixing on the use
of C versus Cin here).
Reaction  k  V  C
or
Reaction  V  k  C
[T] Excel – behavior of first order decay reactions.
4. Settling
Losses due to settling may be formulated as a flux across the sediment-water interface,
Settling  J  As
with J and As (sediment area) having units as described above. Because J is not an intrinsic
property of the material of interest (i.e. A is a system characteristic), it is recognized that J may
be represented as,
J  v C
(note dimensionality here and that v is an intrinsic property)
[T] Chapra, Figure 3.2
and the settling relationship written as,
Settling  v  As  C
5. The Total Mass Balance
These representations of the source and sink terms may then be combined to yield the mass
balance for a well-mixed lake.
V
dC
 W  Q  C  V  k  C  v  As  C
dt
Note in this equation that C and t are the dependent and independent variables, respectively,
because we seek to predict concentration as a function of time. The loading term if the forcing
function, as it describes the manner in which external conditions influence or ‘force’ the system,
and V, Q, k, v, and As are parameters or coefficients particular to the system or material of
interest.
6. The Steady State Solution
If a system receives a constant loading for a sufficient time (and the sink terms remain
unchanged), it will reach a dynamic equilibrium termed a steady state.
V
dC
 W  Q  C  V  k  C  v  As  C
dt
(discuss how the system approaches steady state)
Here the rate of accumulation is zero,
V
dC
 0  W  Q  C  V  k  C  v  As  C
dt
and the mass balance can be solved for C at steady state,
Css 
W
Q  V  k  v  As
Recalling that,
C
1
W
a
the assimilation factor may be defined as,
a  Q  V  k  v  As
Thus the steady state solution has served to define the assimilation factor in terms of
measureable coefficient and parameters that reflect,
C  f (biology, chemistry, physics)
Finally, we can use W, and Css times the various components of the assimilation factor to look
at the relative importance of the source-sink terms as they determine the fate of the material of
interest.
[T] Chapra, Figure 3.3
7. Transfer Function
The transfer function,  (dimensionless), specifies the manner in which an input to the system
(Cin) is transformed or “transferred” to an output (C). Remembering that,
W  Q  Cin
and substituting this to the steady state solution,
Css 
W
Q  V  k  v  As
and dividing both sides by Cin,
Css
Q

Cin Q  V  k  v  As
where,

Q
Q  V  k  v  As
Thus, if  << 1 then sink processes are acting to significantly reduce levels of the material of
interest before it leaves the lake. Conversely, if   1, there is little transfer and the outflow
concentration approaches the inflow concentration. Note in this equation that increases in V, k,
v and As all act to reduce the outflow concentration. Flow, which appears in both the numerator
and denominator of the equation acts to both increase (delivery) and decrease (flushing)
concentration.
8. Residence Time
Residence time (, e.g. yr) is defined as the mean amount of time that a molecule or particle
remains or ‘resides’ in a system. The residence time of water in a lake is given by,
w 
V
Q
and the residence time of a material of interest by,
c 
V
Q  V  k  v  As
[T] Excel - residence time example (how do Q, k, V and A act to influence the response time for
remedial actions).
9. Time Variable Solution
To this point we have focused on steady state, an average water quality that will result if
loadings and factors mediating assimilation are held constant for a sufficiently long period. Our
consideration of the time-variable solution will proceed in two steps. First we will look at what
happens when a pollutant is removed from a lake and then we will consider the response to
time-variable loads.
a. Temporal Aspects of Pollutant Reduction
Beginning with the mass balance model developed previously (with W now = f(t)),
V
dC
 W (t )  Q  C  V  k  C  v  As  C
dt
dividing both sides by V,
A
dC
Q
 W (t )   C  k  C  v  s  C
dt
V
V
noting that A/V = H,
dC
Q
v
 W (t )   C  k  C   C
dt
V
H
and collecting terms,
dC
W (t )
  C 
dt
V
Where  is termed the eigenvalue and is equivalent to the sum of all the processes that purge
the material of interest from the lake,

Q
V
k
V
H
This equation consists of two parts, a general solution (cg) on the left there W(t)=0 and a
particular solution (cp) on the right for specific forms of W(t),
c  cg  c p
The general solution may be developed by integrating the left side of the equation yielding,
c  c0  e   t
Note that the units of  (d-1) and its component terms suggest that they may be conceptualized
as first order losses yielding a characteristic shape to the lake response.
[T] Excel – temporal response following loading termination for various values of 
Although the parameter  clearly dictates the lake response to elimination of loads, it is not
particularly useful in communicating that phenomenon as (1) it carries the counterintuitive
property that as  increases the response time decreases and (2) in the general solution, the
concentration of the material of interest never reaches zero (mathematically).
As an alternative, we use the general solution to derive an alternative parameter group,
response time, that represents the time that it will take for the lake to accomplish a certain
percentage of its recovery.
Consider, for example, calculation of t50, the time required to achieve 50% completion of the
response. Here c is reduced to 50% of its initial value and,
0.5  c0  c0  e   t
cancelling c0, taking ln of both sides and re-arranging yields,
t50 
0.693

In generalized form,
1
t 

 ln
100
100  
[T] Excel – response times
A comparison of the response time for conservative and non-conservative materials leads to a
valuable rule of thumb for lake response. Beginning with the 95% response time,
t95 
3

and remembering that,

Q
V
k
V
H
and that

V
Q
t95 for a conservative substance becomes,
t95 
3


3
Q
V

3
1
 3 

This result tells us that it would take 3 for a lake to complete 95% of its recovery if the only loss
term were flushing (why not just ?). The response time becomes shorter as additional loss
terms such as reaction and settling become important.
b. Time Variable Loading
The right hand or particular solution side of the total solution will now be considered,
dC
W (t )
  C 
dt
V
c  cg  c p
Various forms of W(t), the particular solution, may be considered for idealized loads. This
approach is useful in predicting future conditions and in better understanding the lake response
to loadings. Five particular solutions will be considered.
(1) Impulse Loading (spill)
Here, a mass (m) is discharged to a lake with volume (V) and purging functions (). The initial
concentration for the lake is given by,
c0 
m
V
and since it is a pulse load (with no further inputs) the system response is given by,
c  c0  e   t
[T] Excel - impulse loading and system response
(2) Step Loading (new continuous source)
The step loading is an ‘on-off’ change in input such as the corresponding to a new point source,
with a solution,
C
W
 (1  e   t )
 V
Here, the shape of the response is described by its height,
C
W
Q  V  k  v  As
or, substituting ,
C
W
 V
its temporal nature by .
[T] Excel – step loading and system response
(3) Linear Loading
This case is one of two that we will consider which describes a trend toward increasing (or
decreasing) loading. Here a linear or ramp increase is described by,
W (t )  Wbase  linear  t
Thus W(t) begins at some initial loading, W base (g·d-1), and increases in a linear fashion with a
slope linear (g·d-2 or grams per day per day). For the case of W base = 0,
c
Wbase

 1  exp   t   linear
 (  t  1  e   t )
 V
 2 V
[T] Excel – linear loading and system response
(4) Exponential Loading
This second case describing a trend utilizes an exponential increase (or decrease). The loading
is given by,
W (t )  Wbase  e
exp t
As for the linear model, W(t) begins at some initial loading, W base (g·d-1), but this time increases
exponentially according to the loading growth (or decay) rate constant  exp (d-1). The time
variable solution is given by,
c

Wbase
  t
 e exp  e  t
V  (  exp )

[T] Excel – exponential loading and system response
[T] Excel – combined solution for step loading