Engineering & Technology · Sketches, Models & Technical Communication
Physical, mathematical and computational models
We can model the teaching tank in several ways. A cardboard mock-up lets someone check whether the scale marks can be seen from a chosen position. An equation estimates stored water after a time interval. A small program repeats that equation for many times. These models describe aspects of the same proposed system, but their evidence is different.
Use units and simple arithmetic from Measurement, Units & Engineering Evidence. This chapter introduces its own coordinates, projections and model vocabulary; multiplication and division are enough for the worked cases. The geometry, system, data and results are original teaching constructions. They are not observed experiments, build specifications or operating instructions. No conformance to a drafting, piping or NASA standard is claimed.
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Choose among physical, mathematical and computational representations by the question they can answer.
A physical model preserves selected features
A physical model is a tangible representation. A full-size mock-up of a tank panel can preserve the intended label size and viewing distance. A smaller model might preserve overall shape while changing material and thickness. Before using either, name what is preserved. A cardboard panel can help with visibility and reach, yet its behaviour when wet or loaded tells us little about a different final material.
Scale alone does not make every behaviour scale proportionally. If all lengths of a solid are halved, corresponding areas become one quarter and volumes one eighth. A half-size tank model therefore has one eighth the volume only when all three dimensions scale equally and the shapes are similar. The different scaling of length, area and volume is one reason a miniature is not automatically a faithful strength or flow model.
Build a mathematical model from a simple accounting rule
A mathematical model describes chosen relationships with quantities and rules. Let V be stored volume in litres, t elapsed time in seconds and q a constant incoming rate in litres per second. For this first model, the outlet is closed, there is no leakage or overflow, and the tank begins with 1 L. Every second adds q litres. Taking q = 0.20 L/s gives V(t) = 1 L + (0.20 L/s)t.
The units explain the operation: litres per second multiplied by seconds gives litres, which can be added to the initial litre. At t = 10 s, the increase is 2 L, so the total is 3 L. This equation predicts volume, not the height of the water surface. Converting volume into height would require information about the tank’s internal shape.
A computational model executes specified rules
A program can evaluate that formula at 0, 5, 10, 15 and 20 seconds, producing 1, 2, 3, 4 and 5 L. In the figure, the line represents the formula and circles mark the selected evaluation times. The gaps between circles reflect our chosen reporting times; they do not imply that physical water arrives only every five seconds.
A simulation may instead update a changing state step by step, using more involved flow rules. It still needs assumptions, inputs, units and checks. More decimal places or a smooth animation add no independent evidence that those rules suit the tank. Computational methods can answer many cases quickly, while a physical comparison is needed to investigate whether the chosen model represents the intended situation well.
Select a model for three questions
Question one: can a seated reader see the 5 L mark? A full-size physical mock-up with representative viewing conditions is useful. The filling equation contains no information about eyesight, viewing angles or label contrast.
Question two: what volume does the constant-rate model predict after 12 s? Calculate V = 1 + 0.20 × 12 = 3.4 L. A computer is optional because the arithmetic is small.
Question three: what does the same rule predict at 101 reporting times from 0 to 20 s? A computational implementation is convenient. Check its 0 s, 10 s and 20 s values against hand calculations before relying on the larger output. The program’s agreement with the equation checks implementation, while agreement with a real tank would address a different question.
Try the reasoning
A geometrically similar half-size tank model has what fraction of the original volume?
Show the answer and reasoning
One eighth: (1/2) × (1/2) × (1/2) = 1/8. This statement concerns similar internal geometry, not automatically all physical behaviour.
What volume does the given formula predict after 7.5 s?
Show the answer and reasoning
V = 1 + 0.20 × 7.5 = 2.5 L, within the model’s stated conditions.
Do matching hand and computer answers establish that the physical inflow is constant?
Show the answer and reasoning
No. They support correct implementation of the chosen rule. Checking the constant-rate assumption requires suitable observations of the represented system.
What to carry forward
A model is useful for the features and conditions it represents. Choose it by the question, then check both its rules and their connection to the situation.
References
Teaching resources and research supporting this lesson’s explanations and examples.