CE5504 – Surface Water Quality Modeling
Lab 1 Handout: Numerical Methods - Applications of Excel and Visual Basic
Software
Reading:
Chapra, Surface Water Quality Modeling, Chapter 7 (intro, 7.1, 7.4)
Numerical Integration
The field of mathematical modeling was developed on the backs of those who could
derive solutions for the ordinary differential equations that constituted the models. These
analytical solutions have limitations, however, as described by Chapra (1997, Ch. 7):

non-idealized loading functions: analytical solutions require the idealized loading
functions (impulse, step, etc.) developed in Chapter 3. Actual loads behave in a
much more arbitrary fashion can only be accommodated numerically.

variable parameters: an analytical solution requires coefficient values remain
constant over the course of the simulation. This is certainly not the case, as the
specific growth rate for microbial populations, for example, may vary over the
course of the simulation due to changes in environmental forcing conditions (e.g.
light, temperature and nutrients).

multi-segment systems: the development of 1, 2, and 3-D models for requires
spatial segmentation that is inefficient to handle with analytical solutions (see
Chapra 1997, p. 92, Eq. 5.21).

non-linear kinetics: while linear (e.g. first order) kinetics are quite useful, there
are several water quality applications that require non-linear kinetics (e.g. the
Monod model). Analytical solutions cannot be obtained for these.
Thus we turn to numerical methods, developing numerical or computer-based solutions.
The basic idea here is the solutions are trivially simply, but exhaustively long were it not
for computing power.
The Euler Method
In the Euler method, we transform the differential equation (here the exponential growth
model) as follows:
dX
 X
dt
dX    X  dt
such that we can calculate a change in X and then add it to the original value of X to get
an updated value at the end of the time interval dt:
X new  X old  dX
This approach is implemented in a computer program in the form of a loop, calculating X
over an interval 0 to tmax:
For t  0 to tmax
dX    X  dt
X  X  dX
Next t
The selection of a value for dt is critical to the success of the process (Spain 1982, Figure
5.1). We find as dt0, the numerical solution approaches the analytical solution. The
trade-off becomes one of computation time – as we decrease the time step (dt) the
solution becomes more accurate, but the number of computations that must be performed
increases. Alternative numerical methods (Chapra 1997, Chapter 7: Heun’s Method,
Runge-Kutta Methods) can provide accurate results at significantly longer time steps,
thus reducing computation time.
A. Setting up the workbook.
1. Open a new workbook.
2. Label four worksheet, one for each of the Lab 1 problems.
3. Establish and label input-output regions.
4. Open the Visual Basic editor
Tools > Macro > Visual Basic editor
5. In the empty programming work space, type –
Sub Integration
and hit Return. Your program is intended to fit between these lines.
6. Returning to the worksheet, add a program control –
View > Toolbars > Forms
Select the ‘bulls-eye’ form and drag it onto the worksheet. Move away from it
and then return to activate. You can ‘right-click’ on the form to edit the text,
shape and color.
7. Right click on the form and select –
Assign macro …
and then select
Sheet1.Integration
and click on ‘OK’. This links the form and the program, running the software
every time you click on the form.
8. The worksheet is now ready to have the software added.
B. Writing the program.
Sub Integration()
Assign model inputs (C0, k, tmax, dt)
variablename = Cells(row,column)
Initialize state variable
variablename = variablename0
Begin outer loop
For i = 1 to tmax
Assign model output
Cells (row,column) = variablename
Begin inner loop
For j = 1 to 1/dt
Perform incremental calculation
dC = k * C * dt
Perform update
C = C + dC
Close loop inner loop
Next j
Close loop outer loop
Next i
EndSub
C. Applying the program
We will apply the software to examine four population growth models:
1. Exponential Model
dX
 max  X
dt
where: µmax = 1 d-1, X0 = 2 mg∙L-1 and tmax = 10 days. Calculate x = f(t), comparing
the analytical solution to the numerical solution for the Euler, Heun and 4th Order
Runge Kutta methods to evaluate the accuracy of the integrators for different time
steps (e.g. values of dt ranging from 1 day to 0.0001 days). Consider the impact of
this on run time and accuracy. Examine model sensitivity to values of X0 and µmax.
2. Logistic Model

dX
X 
 max  1 
 X
dt
 X max 
where: Xmax = 25000 mg∙L-1, tmax = 25 days, dt is as determined above for best
performance with an integrator of your choice. Examine model sensitivity to changes
in Xmax. Would you expect carrying capacity or resource availability to govern
population growth in most ecosystems and world populations.
3. Monod Model
 S 
dX
 max  
 X
dt
 Ks  S 
where: S0 = 500 mg∙L-1, Ks = 5 mg∙L-1, Y is 0.5 mgX∙mgS-1 and tmax = 10 days; all
other inputs as before. Note that this requires equations for both S and X:
 S 
dS
1
   max  
 X
dt
Y
 Ks  S 
The coupled differential equations are solved simultaneously, carrying the two
equations through the increment (dX = ) and update (X = X + …) steps together.
The biomass curve looks similar to that for the logistic model; how does it most
differ? Why? How do the half-saturation constant and yield coefficient influence the
maximum value of X? Why? Is this realistic? What does the half-saturation
constant influence? Why is this? There is one common shortfall of both the logisitic
and Monod models. What is this?
4. Batch Growth Model


dX 
X   S 
  max  1 

 kd   X



dt 
 X max   K s  S 

where: kd = 0.2 d-1, tmax = 25 days and all other inputs are as before. The equation for
S remains as before, unless recycle is added,

dS
1
X   S 
   max  1 

 X
dt
Y
 X max   K s  S 
without recycle


dS  1
X   S 
    max  1 

  kd   X

dt  Y
 X max   K s  S 

with recycle
What is the effect of recycle on model behavior? Can you explain this? Which
mechanism is most realistic? Compare model sensitivity to variation in the halfsaturation constant for both cases. Can you explain the resulting behavior?
5. Completely-Mixed Flow Reactor



dX Q
Q
X   S 
  X in   X   max  1 

 kd   X




dt V
V
 X max   K s  S 


 1


dS Q
Q
X   S 
  Sin   S     max  1 

 kd   X


 Y

dt V
V
 X max   K s  S 


where: V = 100 L, Q = 10 L∙d-1, and Sin = 200 mg∙L-1. Examine model sensitivity to
the half-saturation constant and initial and final substrate concentrations. Explain the
behavior in each case.