Runge-Kutta method in Excel VBA

Université d’Ottawa Faculté de génie Département de génie chimique et biologique

University of Ottawa Faculty of Engineering

Department of

Chemical and Biological Engineering

Fall 2019 1

CHG1371 PROJECT 2 DESCRIPTION

1. Background Consider the following system of three continuous stirred tanks, connected in series and filled with water:

To study the flow and mixing pattern in this system, salt was dissolved in the first tank to obtain a desired initial

salt concentration. Fresh water was then allowed to flow through the system, resulting in the dissolved salt to

flow through all three tanks and then eventually leaving the system. Each tank outlet stream is monitored with

an on-line thermal conductivity detector to provide instantaneous salt concentration measurements, 𝐶𝑆𝑖. By

plotting this data over time (Figure 1), we can determine how long it takes for all the salt to leave the system,

and the maximum salt concentrations held by each tank throughout the run.

Figure 1. Salt concentration measurements from the experimental system

0

50

100

150

200

250

300

350

400

450

0 20 40 60 80 100

Sa lt

C o

n ce

n tr

at io

n (

g/ L)

Time (s)

Cs1

Cs2

Cs3

500 L

𝐶𝑆1 (g/L)

250 L

𝐶𝑆2 (g/L)

1000 L

𝐶𝑆3 (g/L)

50 (L/s)

𝐶𝑆 = 0

50 (L/s)

𝐶𝑆1

50 (L/s)

𝐶𝑆2

50 (L/s)

𝐶𝑆3

CHG1371: Numerical Methods and Engineering Computation in Chemical Engineering

Winter 2020 2

It is also possible to numerically generate these salt concentration plots using differential equations. Assuming

that pure water is fed to the first tank and that each tank is perfectly mixed (i.e. the salt concentration in a tank

is uniform and equal to the concentration in the outlet stream from that tank), the following expressions for the

change in salt concentration over time can be determined for each tank:

𝑑𝐶𝑆1

𝑑𝑡 = −

�̇�

𝑉1 𝐶𝑆1 (Eq 1)

𝑑𝐶𝑆2

𝑑𝑡 =

�̇�

𝑉2 (𝐶𝑆1 − 𝐶𝑆2) (Eq 2)

𝑑𝐶𝑆3

𝑑𝑡 =

�̇�

𝑉3 (𝐶𝑆2 − 𝐶𝑆3) (Eq 3)

Here, 𝑉𝑖 and 𝐶𝑆𝑖 are the volume and salt concentration of each tank, and �̇� is the flowrate of water. The initial

salt concentration in tanks 1, 2 and 3 are 450 g/L, 0 g/L and 0 g/L, respectively. The tank volumes and water

flowrate are provided in the Block Flow Diagram.

2. Task For this project, you will determine the salt concentrations for each tank over time, by solving the system of

ordinary differential equations using a 2nd order or higher Runge-Kutta method. You will then compare and

discuss the differences between the numerical data with the experimental findings in Figure 1. The experimental

plot data is available in the provided Excel sheet. Hint – what is the solubility of salt in water?

The numerical solution must be solved in Microsoft Excel using VBA. The solution should be flexible and robust

enough that different parameters (e.g. different �̇�, 𝑉𝑖 and initial salt concentrations) could be easily changed

and a new solution would be automatically calculated.

3. Evaluation

A report will accompany the solution, a soft copy in .pdf format and must be submitted electronically prior to

the due date of Friday April 3rd, 2020 at 16:00. It is expected that the report should be about 6 pages, 1.5 spaced

and including figures, broken down as follows:

• 1 Page for Introduction/Background

• 2 Pages for VBA code design and Excel sheet set-up

• 2 Pages for Results/Discussion/Validation

• 1 Page for Improvements & Extensions/Conclusions

The VBA code should be included as text in an Appendix (not included in the number of pages). This project

should be done in groups of 2, and each member must share equal responsibility for the code and the report.

The deadline for joining a group on BrightSpace is March 20th, 2020 at 16:00. If you do not join a group by this

time, you will be randomly assigned. More information on the expectations of the report and the marking

scheme can be found in the Project Guidelines & Expectations document.