9.1: Appendix A- Solving Engineering Problems - A Problem-Solving Heuristic

Engineering problem solving is based on the study of models that describe real systems. In every case, the real system must be modeled by making simplifying assumptions before any mathematical or empirical analysis can be performed. Realistic and useful answers can only be obtained if the modeling assumptions "catch" the important features of the problem. The behavior of any model is constrained by the physical laws it incorporates and the modeling assumptions used in its development. Two different models for the same system may behave in entirely different ways. The engineers' job is to develop the "best" model for the problem at hand.

Because most mistakes are made in the process of developing the model it is essential that you learn to solve problems in a methodical fashion that documents your solution process including your modeling assumptions. Engineering calculations are part of the archival record of any engineering project and are frequently referred to years after the original work is completed. Many a junior engineer begins a new job by reviewing engineering calculations performed by others.

To help you develop your engineering problem solving skills, a multi-step process is proposed to help you (1) organize your thoughts, (2) document your solution, and (3) improve your ability to solve new problems. A summary of the steps is presented in Figure A-1. A sample problem showing the format can be found at the end of this appendix. As with any heuristic, this one does not guarantee a solution; however, its usefulness has been proven so frequently that we want you to use it in this course.

Figure A-1

SUMMARY OF PROBLEM SOLVING STEPS

KNOWN: In your own words, state briefly what is known. (Step #1) FIND: State concisely what you are trying to find. (Step #2) GIVEN: Translate the problem word statement into sketches and symbolic notation. All pertinent information given explicitly in the problem statement should be listed here. (Step #3) ANALYSIS: Develop a model and solve for desired information. Develop a strategy. (STRATEGY) (Step #4) Make modeling assumptions. (Clearly identified.) (Step #5) Develop and solve the model. (Step #6) Develop symbolic solutions. Calculate numerical values. Check the reasonableness of your answers. COMMENT: Discuss your results. (Step #7)

A more detailed discussion of each step is presented in the following sections. (Based on material in Fundamentals of Engineering Thermodynamics by M. J. Moran and H. N. Shapiro, J. Wiley & Sons, Inc., New York, 1988.)

KNOWN: In your own words, state briefly what is known. Read the problem statement and think about what it says. Do not just blindly copy the problem statement over again or list every detail of the problem. Construct a short sentence that summarizes the situation.

FIND: State concisely what you are trying to find. (If you don't know what you are looking for, how do you know when you've found it?) Do not just copy (a)...., (b)...., etc. from the problem and do not assume that you must find things in the order implied in the problem statement.

GIVEN: Translate the word statement of the problem into engineering sketches and symbolic notation. When completed, you should be able to throw away the original problem statement because you have recorded all of the pertinent information.

Draw and label a sketch of the physical system or device. (If you cannot visualize the problem, you probably can't solve it!) If you anticipate using a conservation or accounting principle, identify the boundaries (control surfaces) of the system you select for your analysis and identify the interactions between this system and the surroundings, e.g. forces, work, mass flow, etc.

Define symbols for the important variables and parameters of the problem. Record the numerical values given for the important variables and parameters.

Label the diagram with all relevant information from the problem statement. This is where you record all of the information explicitly given in the problem statement.

Be especially wary of making implicit assumptions as you prepare this section. Recognize the difference between information that is given explicitly in the problem and your interpretation of the information.

ANALYSIS: It is in this section that an appropriate mathematical model is developed and used to find the desired information. As you prepare this section, carefully annotate your solution with words that describe what you are doing. This commentary is invaluable in exposing your thought processes and if need be in recreating it at a later time.

• Develop a strategy. Every solution should include some initial statements that reveal your plan for solving the problem. As a starting point, clearly state what you believe to be the physical laws or concepts that will be important in solving this problem. What's the property to be counted? What’s the appropriate system? What's the appropriate time period? What constitutive relationships may be required?
  Your initial strategy may not be the best approach or the only approach. It may not even be correct approach, but as you proceed through the analysis process your plans may change. As they do just document them.
  To stress the importance of consciously thinking about the problem, every analysis section should start with a brief subsection labeled STRATEGY.
  
• Make modeling assumptions. Every problem solution requires that you make modeling assumptions. These assumptions are based on the information given in the problem statement, your interpretation of the given information, and your understanding of the underlying phenomena. Every model begins with universally accepted natural laws, and the assumptions provide the traceable link between the fundamental laws and problem-specific model you have developed. All assumptions should be clearly identified as they are applied. You should be able to give a logical reason for every modeling assumption you make. If you cannot, it probably is an incorrect assumption.
  Some problem solving formats call for a separate section listing all assumptions before you begin your analysis. There are two problems with this approach. First, experience shows that it is often difficult to know exactly what assumptions to make until you are building the model. Secondly, separating the assumptions from their application in the model tends to hide how they influence the modeling process. If a summary list is desired, it should be prepared after the analysis is completed.
  
• Develop symbolic solutions. Symbolic solutions are critical in engineering analysis and should always be developed and examined before you insert numerical values. The physics is in the symbolic solution, not the numerical answer. If the symbolic solution is incorrect, there’s no hope for the numbers. If possible, solve an equation for the unknown quantity and isolate it on one side of the equal sign. It is desirable to work with symbolic equations as long as possible before substituting in numbers for many reasons. Symbolic solutions are especially useful when you are looking for errors, for solving parametric problems where certain parameters change, and are much easier to modify as your model develops. Look for groups of terms or ways to rearrange you symbolic answer that simplify the equation and allow you to check for dimensional consistency. Groups of terms with physical meaning or logical intermediate values should be assigned a unique symbol. Numerical values for these intermediate answers can then be calculated and checked separately.
  
• Calculate numerical values. Examine your symbolic solution and see if it makes sense. Once you are satisfied with the symbolic solution, substitute in the numbers and calculate the numerical answer. It is good practice to identify the source, e.g. table, chart, or book, of all numerical data used in the solution, especially if it is not common knowledge. It is also good practice to calculate intermediate or partial numerical answers when you are faced with a very long computation or complicated equation. This prevents calculator errors from creeping into a problem and gives you an opportunity to check the answers against your physical intuition.
  
• Check the reasonableness of your answers. Once you have a numerical answer, consider the magnitude and sign of all values and decide whether they are reasonable. One way to do this is to compare your answer against the results of a simpler model or models that would be expected to bracket your answer. Try different units for the answer, say gallons per minute instead of liters per second, to match your experience.

As you prepare the analysis, do not waste time recopying the solution over again if you reach a dead end or make a mistake. Just cross out the error, clearly identify the mistake, and keep going. Textbook examples and professors' notes give the mistaken impression that problem solving is a linear process that follows a single path with no mistakes and no side trips. Everyone makes mistakes, takes unexpected side trips, and forgets to make an important assumption.

Successful problem solvers acknowledge these diversions and learn from them. You should never start a problem more than once; however, your solution may take several turns before you are satisfied with the answer. The record of your journey is important. Don't "clean up" the solution. Clean up your standard problem solving method because a sloppy solution is usually the result of sloppy thinking. Get in the habit of attacking every problem in the same way. Scrap paper is meant for doodles, not engineering calculations.

COMMENTS: Discuss your results briefly. Comment on what you learned, identify key aspects of the solution, and indicate how your model might be improved by changing assumptions. Consciously check the validity of your answer by considering simpler models. Don't wait for someone else (like your boss or instructor) to find an error in your work by performing a five-minute "back-of-the-envelope" calculation you could have performed before submitting your answer.