# Mathematical modeling using linear first order equations

Two tanks each hold 3 liters of salt water and are connected by two pipes (see figure below) the salt water in each tank is kept well stirred. Pure water flows into tank A at a rate of 5 liters per minute and the salt mixture exits tank B at the same rate.

Salt water flows from tank A to tank B at the rate of 9 liters per minute and flows from tank B to tank A at the rate of 4 liters per minute. If tank A initially contains 1 Kg of salt and tank B contains no salt, find the quantities of salt in tank A and tank B respectively, at any time t ≥ 0.

See attached file for full problem description with symbols and diagram.

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#### Solution Preview

Please see the attached file.

Let CA and CB be the concentration of salt in tank A and B (kg of salt/L)

Initial Volume of tank VA = VB = 3 (liters)

Perform material balance for each tank:

Rate of change of material = Rate of mass in - Rate of mass out

Tank A:

Tank B:

Equipped with the two differential ...

#### Solution Summary

The solution provides very detailed theory for developing governing equations of continuous flow stirred tank reactors. It then provides explanations and calculations for the final answer in a 3-page Word document.

Optimization and Linear Programming/ Heuristics

Question 1:

In this case you are learning to use LP to determine the best set of Order Quantities so that you obtain the Optimum (max) amount of profit for one product line at EBBD.

As you write your report to Wilco to present your results, consider other applications for Optimization and LP.

Explore the various ways that LP can be used in Logistics to obtain optimal results, not only at EBBD, but at any company.

Linear programming. (n.d.). Absolute Astronomy. Retrieved fromhttp://www.absoluteastronomy.com/topics/Linear_programming

Linear programming: Introduction (n.d.). Purplemath. Retrieved fromhttp://www.purplemath.com/modules/linprog.htm

Question 2:

Consider how you would make the quarterly ordering decision without using optimization techniques. This is called heuristic decision making - meaning it is an approximation.

Discuss where heuristic decision making might be better than using Optimization. What are the benefits v costs? In other words, when is it not cost effective to spend the additional time and money to obtain the Optimum solution?