Elements of calculation style

Style helps readers to understand, and calculations to succeed.

The elements of style in calculations are the choices in composition that strengthen collaboration between writers and readers by helping them meet one another's needs. The overriding need of both writers and readers is to not have to keep track of too much new, unfamiliar material at one time (1).

Elements of style include separate sections for assumptions, data, calculations and summaries. Each section can be prepared and read with a minimum of in-depth thought, yet each section moves the solution forward and serves as a resource for use with the later sections.

Elements of style at the formula level, which are even more helpful, include conventional symbols, reminders of variable definitions, reference names and pages, reminders of values of dependent variables, and equations that are visible, as shown in Figure 1. These elements let writers and readers understand and check formulas with a minimum of cognitive strain.

Elements of style in calculations are elaborated below and illustrated with a simple calculations on pages 52-53.

1. Use calculation software

Use software that dusplays the working formul las. The equations are the working parts of the calculation. A calculation is easier to use if its functions can be readily inferred by looking at the features visible to the user. Calculation software packages, such as Mathcad and Calculation-- Center, display a calculation's working formulas and results together. and print them for convenient writing and reading (3, 4). Spreadsheets typically don't display formulas, so only simple operations like totaling the numbers in a column can be readily inferred by looking at a spreadstreet. (Spreadsheets can be made to display formulas and results together through the use of named ranges (5) and user-defined functions, as shown in Figure 2). Programming langua&Es, such as Visual Basic and FORTRAN, display Ind print the working formulas and results separate from each other, and often use coded syntax (when referencing data objects, for example), making calcul ions in these languages diffi.ylt to read and document.

Use software that calculates units as well as numbers. Calculation software like Mathcad and CalculationCenter makes it possible to inside dimensional units in values and formulas, have the software perform unit conversions automatically, bypassing an otherwise major source of errors.

2. Set up for easy viewing

make calculations read from top to bottom. Use the approach that experts use when solving easy peoblems-work forward from the known data to determine the unknown values that are needed (6). Enter data before calcutation, and summarize inputs before summarizing results. In the same manner, provide reference materials before referring to them. Place the table of contents before the actual contents, list references before they are used, and list variable names before they are mentioned.

Use font, font size, and font style changes to help readers. Font changes can help readers distinguish equations from text. Font size changes can improve readability of equations. Both changes are performed automatically by the calculation software used to prepare Figure 1 and the sample calculation. A font style change to bold face for the headings can help readers rapidly scan through a calculation, as shown in the sample calculation.

Use graphic lines mostly to convey information. When graphic lines are used sparingly, the lines that are used stand out better (7). Horizontal lines can be used as blanks for user-supplied data. Vertical lines can be used as revision bars. Blank space can serve the same function that graphics lines are often used for in forms, providing separation between unrelated items of information and helping readers read horizontally across rows of information in tables.

Include equipment number and page number at the right on each page. Sets of calculations arranged by major equipment number including letter prefix can be leafed through easily to locate calculations that are of interest to the reader.

Make formulas readable without comments. Help readers be able to review formulas independent of the explanatory comments. Place formulas on separate lines from comments. Center formulas on the page, or indent them. Provide punctuation and text to allow the resulting material to be read straight through more easily than the formulas alone could be read.

3. Provide supporting information

Write clear sentences. Start by writing what you would say aloud. Then remove excessive words. Rearrange phrases to improve clarity or eliminate ambiguity. Add words wherever this will help readers understand without having to concentrate as hard and without having to reread (8). Reread the work yourself later and edit it again, repeating these steps.

List the contents. Simple, descriptive headings provide enough useful help to readers to avoid the need for paragraphs of explanatory text.

State the objective. Readers expect to find the most important information at the start, and if not there, then at the end. They spend more time reading the information at the start. When the key theme is identified up front, readers understand the subsequent material better as they proceed through it, and they proceed through it more quickly.

Sketch the system. Sketches with text help people understand problems more thoroughly and help people move further toward solutions (9). The more useful diagrams show spatial relationships, show key data at a glance, and place information near the associated objects so that symbolic labels are not needed (10). When solving a problem that requires the use of formulas to interpret physical information, experts tend to insert an intermediate step redescribing the problem qualitatively (11). Sketches can capture some of an expert's understanding of the problem by emphasizing key considerations while leaving out secondary details. Unfortunately, people who have less trouble proceeding with a problem tend to draw fewer sketches. As a result, they miss out on opportunities to help novices develop the skill of going beyond the literal features clearly evident in problem statements to infer additional relationships that are important for constructing effective solutions, which is the skill that novices are usually most lacking (6).

State the approach, noting the key methods used. Name the key method or methods used, and describe how they were located and chosen. Avoid describing the details of the calculation in text form, since the actual formulas will be displayed later when they are used and will be nearly self-explanatory, while text descriptions of the formulas would take extra effort and extra skill to write and would be less clear and less helpful.

Name, list and enclose references. Identify convenient, clear references for each formula used and data value entered. Name each reference using a short descriptive name such as the lead author's name. Give titles and page ranges. Provide readers copies of references, so that they can find out things for themselves right away, while they are most interested.

Use conventional symbols. Match the conventional notation in the area of interest for ready recognition. Use the same main symbol for all variables of a given type, and use subscripts to differentiate the family members from one another. Greek letters and subscripts can be typed directly into a calculation when calculation software like Mathcad or CalculationCenter is used.

List complete symbols, including subscripts, and provide complete descriptions and standard units. Descriptions that include subscripts can eliminate guesswork. Standard units provide added descriptions of the symbols.

Promote alternative methods. Describe alternative approaches and possible outcomes. Present the alternatives as positive possibilities, so they will be considered more likely and will therefore more effectively counterbalance the base case that is being presented positively. Considering alternatives reduces overconfidence. This promotes progress on problems (12), improves decision-making, and may improve self-checking by writers and error-- checking by readers.

4. Include text comments and equation-style comments with the workin; formulas

Provide comments that supplement formulas but do not describe them. The working formulas do the actual calculation. Writers and readers need to be helped to review the formulas carefully, and need to not be lulled into a false sense of security by comments that seem to tell a complete story, and as a result, encourage them to skip over the equations (13).

Repeat the description of each symbol each time it is used in a formula. Provide the description and repeat the symbol, including any subscript.

List a source for each formula. Identify a convenient source that states the formula clearly. List the source's short, descriptive name from the reference list, and identify the page or pages where the formula is defined. Include the formula number from the source, where helpful.

Repeat the value of each symbol used in standard units each time it is used. Help people learn the relative magnitudes of terms and check the values of input data and intermediate results at every opportunity they have to do so.

Provide conversion factors each time they are used. Conversion factors often help reassure readers of the reasonableness of calculations and occasionally help writers find mistakes. When calculation software is used, it takes little effort to call up predefined conversion factors.

Check function definitions by calculating known values. Check function definitions for temperature-dependent properties, for instance, by evaluating the equations at temperatures where the property values are known.

Show a formula's comments together with the formula on the same page. Add a page break before an assembled formula block if needed to keep the block together on a single page. Self-contained formula definitions and evaluations that can be seen together at a glance are easier to review.

5. Provide assumptions, inputs and calculations

Note assumptions. Assumptions can include notes on how the mathematical models that are used oversimplify the behavior that they describe. They can also include notes on how the experimental approach underlying a method differs from the particulars of the process that is being analyzed. The assumptions that can be the most difficult to recognize are the underlying beliefs shared by previous workers, the writer, and the readers when they all are from the same era and have similar backgrounds. Explaining assumptions early, especially assumptions about factors that cannot be changed by the writer or the readers, produces more realistic assessments about the reliability of results. Reducing overconfidence improves checking, which improves accuracy.

Enter any assumed data values. Assumptions can also include reasonable guesses of data values that are not known for certain. Provide an equation block for each assumed data value, as shown in the sample calculation. Repeat the description and the symbol, including any subscripts. Display the value in standard units for reasonability checking.

Enter the hardware data. Provide an equation block for each hardware data value. Repeat the description and the symbol, including any subscripts. Repeat the source's name from the references and note the applicable page or pages. Enter the data in the dimensional units that were used in the source. Provide any conversion factors used, in units that are as familiar as possible. Display the data in standard units for reasonability checking.

Enter the property data. Provide an equation block for each property data value. Repeat the description and the symbol including any subscripts, the source of the data, the data in the units used in the source, any conversion factors, and the data in standard units. If a property is a function of variables such as temperature, pressure or composition, enter the property as a function that can be evaluated later based on the values of the variables at that point in the calculation.

Enter the operating data. Provide an equation block for each operating data value. Provide the description and the symbol including any subscripts, the source of the data, the data in the units used in the source, any conversion factors, and the data in standard units.

Enter the calculation formulas. Provide a formula block, like that in Figure 1, for each calculation formula. Provide text descriptions of the dependent variables and the independent variables, and provide a reference source for the formula. Then provide the values of the independent variables in standard units, any conversion factors, the formula, and the results.

6. Provide summary information

Repeat key assumptions. An assumption may be crucial, and well worth highlighting again by including it in the summary information at the end of a calculation.

Summarize the key input parameter values. Summarizing key input parameters near the end of calculations highlights them for the writer as well as for the reader. Also, sometimes it is convenient to set up a file containing a single case and then change input parameters and save separate files for new cases. In such situations, it is critical to point out the values of the changed parameters to distinguish the cases from one another.

Summarize the results. Conclude by returning to the big picture and recalling key intermediate results and final results for the benefit of casual readers and careful readers alike.

7. Get calculations checked

Seek checking, whether by experts or by interested colleagues. Unless people get very detailed feedback on their performance, they tend to be overconfident in their own abilities. They do not perform nearly enough selfchecks on material they believe is correct (14). As a result, errors of omission are almost never identified and corrected by the person who made them. With checking, a fresh viewpoint enters the situation, and errors of omission can be corrected.

Calculations will get easier

The calculation approach shown here easily scales up to handle tougher problems. An example of a more difficult calculation is available at www.cepmagazine.org (2).

This approach produces accurate results, is easy to read, and is easy to reuse. It is particularly helpful when experimental data on a process are unavailable, data cannot be obtained cheaply and quickly, and readily available calculation methods do not cover the process in question.

This list of uses barely hints at the broader roles that could quickly develop for approaches like this. Affordable calculation software already provides the capability to embed subprograms and the capability to define graphical symbols that have smart interconnections. Soon, such software could provide the capability to embed subprograms in smart symbols. Libraries of thermodynamic property calculation routines and unit operations could emerge easily from environments of friendly competition and sharing, in academic settings and in industry. It could ultimately be possible to simply connect components from reliable sources and produce accurate and reliable process simulations and other calculations.

Even wider impacts are imaginable. Calculation approaches developed for process applications could easily be adapted to other uses in science and in education.

Much can get easier when collaboration is improved by style in calculations.

[Reference]

Literature Cited

[Reference]

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2. Anthony, J., "Chloroform Plan." available via http://www.cepmagazine.org (2001).

3. Phillips, J. E., and J. D. Decicco, "Choose the Right Mathematical Software," Chem. Eng. Prog., 95 (7), pp. 69-74 (July 1999).

4. Sandier, S. L, "Spreadsheets for Thermodynamics Instruction: Another Point of View," Chem. Eng. Edu., 31 (11), pp. 18-20 (Winter 1997).

5. Lira, Carl T., "Advanced Spreadsheet Features for Chemical Engineering Calculations." submitted to Chem. Eng. Edu., http://www.egr.msu.edu/-lira/spreadsheets.pdf (2000).

6. Chi, M. T. H., et aL, "Expertise in Problem Solving," in Sternberg, R. J., "Advances in the Psychology of Human Intelligence," Vol. 1, Lawrence Erlbaum Associates, Publishers, Hillsdale, NJ, pp. 7-75; see pp. 18, 19, 35, and 71 (1982).

7. Tufte, E. R., "The Visual Display of Quantitative Information," Graphics Press, Cheshire, CT, p. 96 (1983).

8. Cook, C. K., "Line by Line: How to Improve Your Own Writing," Houghton Mifflin. Boston (1985).

9. Mayer, R. E., "Models for Understanding," Review of Educational Research, 59 (1), pp. 43-64 (1989).

10. Larkin, J. It, and H. A. Simon, "Why a Diagram is (Sometimes) Worth Ten Thousand Words," Cognitive Science, 11 (1), pp. 65-99 (1987).

11. Larkin, J. H., "Processing Information for Effective Problem Solving," Engineering Education, 70 (3), pp. 285-288 (December 1979). 12. Platt, J. R., "Strong Inference," Science, 146 (3642), pp. 347-353 (1964).

13. Kernighan, D. W., and P. J. Plauger, "The Elements of Programming Style," 2nd ed.. McGraw-Hill, New York, pp. 141-152 (1978). 14. Allwood, C. M., "Error Detection Processes in Statistical Problem Solv

ing," Cognitive Science, 8 (4), pp. 413-437; see pp. 419 and 431 (1984).

[Author Affiliation]

JAMES ANTHONY, LOCKWOOD GREENE

[Author Affiliation]

JAMES ANTHONY is a process engineer with Lockwood Greene, St. Louis, MO (Phone: (314) 919-3208; Fax: (314) 919-3208; Fax: (314) 919-3201; E-mail: janthony@lg.com). He has process design experience with chemical, pharmaceutical, and beverage applications, which have included the manufacture of iodine products, abrasives, inorganic salts, alkyds, polyesters, polyurethanes, synthetic pharmaceuticals, soy sauce, and tea. He also has aerospace design experience developing jet engine air inlets, piston-propeller systems, sensors, adhesive-bonded structures and molded plastic parts. He has a BS in chemical engineering from the Univ. of Missouri - Rolla and an MS in mechanical engineering from Washington Univ. He is a registered professional engineer.