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Proportionality & Design Optimization

How understanding proportionality relationships shapes design – without the equations.



Understanding proportionality is key to optimizing the design of nearly everything. Proportionality describes how one quantity relates to another. This brief article explores an easy way to optimize your design process, whether you're installing insulation, building an efficient vehicle, or shaping the future of sustainable energy.


Physical equations govern the performance of almost everything. Sometimes, though, these equations can be complicated and difficult to remember. A shorthand way to keep the most important aspects of these equations in mind and allow them to optimize the design of whatever you're building or doing is to remember the proportionality between variables. First we'll define some terms, then explain how understanding the relationship between variables can be useful.


Direct Proportionality


Take the simple equation:

In this equation "y" is said to be "directly proportional" (or just "proportional") to "x", because as "x" increases, "y" also increases. Since as "x" increases, "y" increases by 2 times the amount, the proportionality constant is 2. The symbol we use to express proportionality is . So in the equation above, we can write: "y 2x".



Inverse Proportionality

In this equation "y" is said to be "inversely proportional" to "x", because as "x" increases, "y" decreases. In an inversely proportional relationship, one variable changes proportional to the reciprocal of the other. We can also say that "y" varies inversely with "x".


Types of Relationships


Design considerations often involve various mathematical relationships beyond simple proportionality. Variables can be related in linear, quadratic, square, cube, exponential, logarithmic, or other ways. We will focus on a few of the simple relationships.


Inverse-Square Law


A common relationship between variables is given by the inverse-square law, which says that the "y" is proportional to the inverse of "x" squared.

This is a handy relationship to know for many applications. For example, say a sound engineer is trying to design an audio system for a theatre, and she needs to figure out where to put speakers to ensure a similar auditory experience for show attendees from all seats. The equation that governs how loud a speaker of a given power (P ) sounds (intensity, I ) at a certain distance from it (radius, r ) is:

Rather than remembering this equation and constantly calculating, the sound engineer can simply remember that the sound intensity that a theater attendee hears is proportional to the inverse square of the attendee's distance from it.

Now she can quickly sketch out rough locations for speakers with the knowledge that if a speaker is twice as far away from a seat, the person in that seat will hear the sound at a quarter the intensity.



Square-Cube Law


Another common relationship between variables is given by the square-cube law. Many things in the built and natural world are governed and affected by surface area to volume ratios. The surface area of a cube is given by: SA = 6 L^2, where L is the length of a side of the cube. The volume of a cube is: V = L^3. So, the surface area is proportional to the square of the side length, whereas the volume is proportional to the cube of the side length. This relationship is called the square-cube law, and tells us how things scale in size, strength, weight, etc.


Example: Insulation Design


When determining insulation requirements for a home, building, or anything else, understanding the proportionality of heat transfer through a material is key. When it comes to insulating something, the goal is to minimize heat transfer (Q). Fourier's Law states:

But instead of memorizing that equation, we can more easily remember what heat transfer is proportional to:




This tells us that heat transfer is directly proportional to the surface area of the material and the temperature difference across the material, and inversely proportional to the material's R-value. Since some of these variables can't be changed in real-world situations, as in insulating an attic, this expression of proportionality determines what we can and should change. We can't change the surface area of the attic floor, so to minimize heat transfer out of the house in the winter, we need to increase the total R-value of the insulation.


Example: Wind Turbine Design


The electrical power generated by a horizontal axis wind turbine is given by:


P = 0.5 Cp ρ π R^2 V^3

where

  • P is the power generated

  • Cp is the Betz coefficient (efficiency)

  • ρ (rho) is the air density

  • R is the radius of the wind-swept circular area (the length of the blades)

  • V is the wind velocity


Given this equation, we can use the fact that the power generated is proportional to the square of the blade length and the cube of the wind velocity to optimize the design of wind turbines. This tells us that to maximize power output, the turbine should be big, and more importantly, located in a place with high wind speeds. Wind speed is greater higher off the ground, so one method of increasing power output is to make the turbine taller, or put it on a mountain.


Example: Maximizing Driving Efficiency


Even if you don't design things for a living, you can still benefit from understanding proportionality relations in everyday life. If you want to maximize your fuel efficiency while driving your car in terms of miles per gallon of gasoline (mpg), knowing a simple relation is helpful. Besides the wasteful engine itself (only 25-30% efficient), the biggest killer of your fuel efficiency is air resistance (drag). Your car needs to increase its power output and fuel consumption to match the force of drag on it. Drag is proportional to the cube of your car's speed. Cubic functions increase very rapidly. Knowing how drag is proportional to speed makes it obvious how to increase your fuel efficiency on the highway: drive slower. Slowing down from 75 mph to 65 mph will increase your fuel efficiency. Slowing down from 65 to 55 mph will also increase your fuel efficiency, albeit not by as much as the higher speeds due to the cubic proportionality relationship. This relationship holds true for electric vehicles as well. If you want to maximize your range on your EV, drive slower.


Due to the 1973 oil crisis, the US government was looking for a way to cut oil consumption. Using the cubic proportionality relationship between drag and speed, they enacted the National Maximum Speed Limit of 55 mph in order to save fuel for national energy security.


Conclusion:


Understanding proportionality relationships is a great short-hand way of knowing how to optimize something. It tells you how one thing changes when you change another. Whether you're designing a system for sustainable energy or you're just driving to work, knowing how one variable affects another is an important tool. It can help you save on heating and cooling costs at home, as well as provide the design framework for the world's future sustainable energy systems.


Questions for you:
  • What is one thing you do in your daily life or work governed by the laws of physics? (I know, everything is, but pick one)

  • What are those equations, and what is a simple proportionality relationship you can memorize?

  • How can that proportionality relationship help optimize your design or actions?

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