1.4 Problem-Solving Methods
To help develop your problem-solving abilities, an explicit strategy, which is a modification of the strategy developed at McMaster University (Woods et al., 1975), is used throughout this book. The seven stages of this strategy are:
- I want to, and I can
- Define the problem
- Explore or think about it
- Do it
Step 0 is a motivation and confidence step. It is a reminder that you got this far in chemical engineering because you can solve problems. The more different problems you solve, the better a problem solver you will become. Remind yourself that you want to learn how to solve chemical engineering problems, and you can do it.
In step 1 you want to define the problem. Make sure that you clearly understand all the words. Draw the system and label its parts. List all the known variables and constraints. Describe what you are asked to do. If you cannot define the problem clearly, you will probably be unable to solve it.
In step 2 you explore and think about the problem. What are you really being asked to do? What basic principles should be applied? Can you find a simple limiting solution that gives you bounds to the actual solution? Is the problem over- or underspecified? Let your mind play with the problem and chew on it, and then go back to step 1 to make sure that you are still looking at the problem in the same way. If not, revise the problem statement and continue. Experienced problem solvers always include an explore step even if they don't explicitly state it.
In step 3 the problem solver plans how to subdivide the problem and decides what parts to attack first. The appropriate theory and principles must be selected and mathematical methods chosen. The problem solver assembles required resources such as data, paper, and calculator. While doing this, new subproblems may arise; you may find there are not enough data to solve the problem. Recycle through the problem-solving sequence to solve these subproblems.
Step 4, do it, is often the first step that inexperienced problem solvers try. In this step the mathematical manipulations are done, the numbers are plugged in, and an answer is generated. If your plan was incomplete, you may be unable to carry out this step. In that case, return to step 2 or step 3, the explore or plan steps, and recycle through the process.
In step 5, check your answer. Is it the right order of magnitude? For instance, commercial distillation columns are neither 12 centimeters nor 12 kilometers high. Does the answer seem reasonable? Have you avoided blunders such as plugging in the wrong number or incorrectly punching the calculator? Is there an alternative solution method that can serve as an independent check on the answer? If you find errors or inconsistencies, recycle to the appropriate step and solve the problem again.
The last step, generalize, is important but is usually neglected. In this step you try to learn as much as possible from the problem. What have you learned about the physical situation? Did including a particular phenomenon have an important effect, or could you have ignored it? Generalizing allows you to learn and become a better problem solver.
At first these steps will not "feel" right. You will want to get on with it and start calculating instead of carefully defining the problem and working your way through the procedure. Stick with a systematic approach. It works much better on difficult problems than a "start calculating, maybe something will work" method. The more you use this or any other strategy, the more familiar and less artificial it will become.
In this book, example problems are solved using this strategy. To avoid repeating myself, I will not list step 0, but it is always there. The other six steps will usually be explicitly listed and developed. On the simpler examples some of the steps may be very short, but they are always present.
I strongly encourage you to use this strategy and write down each step as you do homework problems. In the long run this method will improve your problem-solving ability.
A problem-solving strategy is useful, but what do you do when you get stuck? In this case heuristics or rules of thumb are useful. A heuristic is a method that is often helpful but is not guaranteed to help. A large number of problem-solving heuristics have been developed. I have listed ten (Wankat and Oreovicz, 1993) that are often helpful to students.
- Try solving simplified, limiting cases.
- Relate the problem to one you know how to solve. This heuristic encapsulates one of the major reasons for doing homework.
- Generalize the problem.
- Try putting in specific numbers. Heuristics 3 and 4 are the opposite of each other. Sometimes it is easier to see a solution path without all the details, and sometimes the details help.
- Solve for ratios. Often problems can be solved for ratios, but there is not enough information to solve for individual values.
- Do the solvable parts of the problem. This approach may provide information that allows you to solve previously unsolvable parts.
- Look for information that you haven't used.
- Try to guess and check. If you have a strong hunch, this may lead to an answer, but you must check your guess.
- Take a break. Don't quit, but do something else for a while. Coming back to the problem may help you see a solution path.
- Ask someone for a little help. Then complete the problem on your own.
Ten heuristics is probably too many to use on a regular basis. Select four or five that fit you, and make them a regular part of your problem-solving method. If you want to read more about problem solving and heuristics, I recommend How to Model It: Problem Solving for the Computer Age (Starfield et al., 1994) and Strategies for Creative Problem Solving (Fogler and LeBlanc, 1995).