Problem Solving Procedure

It usually appears that there are almost as many ways to solve problems as there are problems to be solved; but on closer examination it is possible to identify patterns and techniques for successful problem solving. It is important to be systematic, because such an approach will work even when "instinct" fails. An organized approach is also useful since it helps keep your thoughts in order.

Individual strengths and weaknesses need to be factored into your problem solving strategy. Take the ideas presented here and use them as the basis for your own approach.

General Problem Solving Approach

Almost every engineering problem-solving technique boils down to five broad steps

1. Describe the problem
2. Identify pertinent known and unknown facts
3. Identify the scientific principles needed for solution
4. Manipulate the numbers
5. Evaluate the result

Describing the problem requires you to understand what is given and what is needed. Often it is useful to develop a drawing or graph to help you at this stage.

It is critical to identify the facts of the problem. Some will be known, others will be missing. Sometimes the hardest part of a problem is locating missing information. Be sure that you stick to relevant facts. Don't waste time with facts that don't bear on your problem.

By identify the principles, the technique is telling you to look for the tools you have to solve the problem -- physical laws, relating equations, etc. Once you've decided what rules apply, you may need to go back and find more information.

Once you've done the first three steps, you've done the hard part. Now you just add numerical values and do the math. Don't forget to check your solution afterward. It is important that you be able to judge when an answer "makes sense".

Balance Problems

The five steps can be expanded to provide a detailed solution guide tailored to the solution of balance problems. Notice that the approach has the same basic outline as that above.

1. Read the problem thoroughly. Understand what is required for the answer.
2. Make a sketch or flowsheet of the problem.
3. Write down the known and label unknown stream variables.
4. Choose a calculation basis.
5. Check the specification of the problem (degrees of freedom). Can it be solved as is, or is more information needed?
6. Determine what additional data, if any, are needed. Find them.
   • Be sure to indicate the source and applicability of anything you bring from outside.
7. Write the required equations.
   • Material Balances -- NC +1 can be written, NC are independent.
   • Energy Balance
   • Specifications
     1. Assigned values of stream variables.
     2. Fractional Recoveries.
     3. Composition Relationships (x₁=K*x₂).
     4. Flow Ratios.
   • Physical Properties
   • Constraints
8. Keep track of units. They can help tell if an equation is complete. If the problem units are mixed, you may want convert all quantities to a common set of units, but probably should wait until you're sure which numbers you'll need.
9. Solve the equations for the unknowns.
   • Use a solution strategy. Solve the equations in a planned order. Often, this allows sequential rather than simultaneous solution.
10. Scale the answer.
11. Check the solution. Does it make sense?

A complete material balance problem, ready for solution, will consist of:

• the system with input and output streams
• variables which describe the flow rates and compositions of all streams
• a set of material balance equations
• a basis
• a set of specifications on the solution.

Dilute sulfuric acid has to be added to dry charged batteries at service stations in order to activate the battery. You are asked to prepare a new batch of acid as follows: A tank of old weak battery acid (H₂SO₄) solution contains 12.43 mass percent H₂SO₄ (the remainder is water). If 200 kg of 77.7% acid are added to the tank, and the final solution is 18.63% H₂SO₄, how many kilograms of battery acid have been made?

First, think in general terms about what is going on. Two streams are being mixed to create a third. Only mixing is taking place, so there are no reactions to worry about. The desired answer is the amount (in mass units) of product (the 19% solution) made.

Now, we're ready for a sketch, and we'll assign variable names to the streams. I'll let P (kg) be the amount of product, F (kg) the amount of old solution, N (kg) the amount of new solution.

On to choosing a basis -- which stream should I use?

• Possibilities:
  • the product stream -- since it is what I'm looking for
  • stream N -- since it's given
• Considerations --
  • If I use the product stream, I'll have to scale my answer so that it corresponds to the given flow of stream N
  • If I use N, the value of P I calculate will already be in scale.

• All compositions are given in percent, so I immediately think of a basis of 1 or 100
• If I do that, I'll have to multiply my answer by 200 or by 2
• It probably would be just as easy to do the multiplication on the front end and use a basis of 200

Is the problem adequately specified? I can do this two ways: either by determining the degrees of freedom, or by just looking at the equations. The latter is easy for this problem:

• Unknowns: 2 (F,P)
• Equations: 2 component system, so can write 3 material balances (2 will be independent)

If I go the degrees of freedom route, I need to count a little more carefully:

• Variables (V): 6 (F, x_F, N, x_N, P, x_P)
• Equations (E): 2 independent balances (probably 1 total and one on the acid)
• Specifications: 4 (3 given compositions and the given value of N chosen as my basis)

Do I need more information? I've got compositions on all three streams. I don't need to worry about densities or anything like that. It looks like no other data are needed.

I've already noted that I can write two independent balances. Which combination would make the most sense?

• Total and Acid
• Total and Water
• Acid and Water

The answer I'm looking for, P shows up in all three equations. If it didn't, I'd probably want to make sure that I used an equation that included my answer directly.

Compositions are all given as mass fractions of acid. In order to do the water balance, I'd have to use (1-x) terms. This isn't hard, but it is slightly more work, so I'll stay away from the water balance for now.

I like to see "zeros" in problems, as a couple of zeros will usually lead to a sequential (instead of simultaneous) solution. Unfortunately, I don't have any in this problem.

OK, then, we'll use a total material balance and an acid balance.

Writing the total material balance:

• Accum = In - Out + Prod - Cons
• Steady state answer will work, so Accum = 0
• No reactions, so Prod = Cons = 0
• 0 = N + F - P = 200 + F - P

• Same general equation and assumptions
• 0 = In - Out
• 0 = (N*x_N + F*x_F) - P*x_P
• 0 = 200(0.777) + F(0.1243) - P(0.1863)
• 0 = 155.4 + 0.1243F - 0.1863P

Thus we need to solve two equations in two unknowns.

Sidebar: You won't always get answers that work nicely. Roundoff error is a significant culprit. Try not to round off numbers until you get a final answer.

We save some time on the next step because we chose the basis to eliminate scaling, and so we don't have to do it here.

Does the solution make sense? Well, P is greater than F -- if it weren't, we'd have a problem.

While we're here, let's see what would have happened with a different basis: What if I'd chosen N=100 kg?

My equations would have become:

If we'd decided to go with a basis of P=100 kg, it would change several steps -- notably, we cannot use the value of N given in the problem statement. We've "specified" P, so if we keep the given value of N, the degrees of freedom will be:

With the basis, P=100, our equations become:

See how the answers vary just a bit -- that's roundoff and scaling for you!

Common Mistakes

A news item in ASEE Prism (October 1996, p. 10) listed the areas where students commonly have difficulties in problem solving. Paraphrased, these are:

1. Inaccuracy in reading
   • not understanding what the problem statement means
   • ignoring unfamiliar concepts and terms (instead of looking them up)
   • losing track of facts or ideas
   • starting without reading the whole problem
2. Inaccuracy in thinking
   • sloppy or inaccurate work
   • carelessness
   • inconsistent interpretation (letting meaning vary from problem to problem, or within a problem)
   • failure to check
   • working too fast
3. Weakness in analysis
   • not breaking large problems down into smaller ones
   • not learning from earlier problems
   • not looking up unfamiliar concepts and terms
   • not making a sketch
4. Lack of perseverance
   • lack of confidence; giving up too quickly
   • guessing, not really trying
   • trying to plug in to formulas or using previous problems without thinking about the problem in question
   • jumping to an answer before finishing the required work
   • trying the "gut" approach, and quitting when it fails

References:

1. -----, "The Problems with Problem Solving", ASEE Prism, October 1996, p. 10.
2. R. M. Felder & R. W. Rousseau, Elementary Principles of Chemical Processes (2nd Ed.), John Wiley, 1986. pp. 105-107.
3. Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2005 3rd Edition, 2005, p. 101-104.
4. D. M. Himmelblau, Basic Principles and Calculations in Chemical Engineering (3rd Ed.), Prentice-Hall, 1974. pp. 40-41.
5. G. V. Reklaitis, Introduction to Material and Energy Balances, John Wiley, 1983, pp. 42

R.M. Price
Original: 6/13/94
Modified: 10/3/94, 6/26/96, 5/20/97; 1/4/2005

Copyright 1997, 2005 by R.M. Price -- All Rights Reserved