Analysis: Models of air resistance

As you’ll learn in your design exploration of an electric car, air resistance is negligible at low speeds, like our tractor test. But at typical highway speeds, it can be the biggest component of drag. We need to be able to estimate it. We’re into the realms of fluid mechanics. There’s much science behind it. But you learn the basics if we quote the final result and you use it. Consider our rotating test. Each of the spheres experiences an airflow with a speed v. We can’t get the speed precisely because the airflow on one sphere might be affected by the wake of the other. But it will be approximately the speed of the sphere as it rotates with the top disk.

The standard equation for predicting the force caused by this airflow is FA equals 1/2 A rho CD v squared. I think it’s beautiful. FA is the drag measured in newtons. Rho is air density measured in kilograms per cubic meter. A is the projected area of the object measured in square meters. CD is the drag coefficient, which has no dimensions. And v is the relative airspeed in meters per second. The drag coefficient depends upon the shape of the object. For a car, it might be, for example, 0.32. For a circular disk, it’s roughly 1.3. And for a sphere, it’s typically 0.4. We can try the equation out on our test rig. We can find air density easy enough.

It’s about 1.2 kilograms per cubic meter. We’ll take v as the speed of the object. And we calculate this from v equals v subscript d times the ratio R2 over R1, where vd is the speed with which the weights drop. R2 is the radius of the centre of the test object to the axis of rotation. And R1 is the radius where the thread wraps around the pencil. Calculations like this are a part of dynamics called kinematics. You can take it from us or search the web for an explanation. Projected area depends on the test object. For a sphere and a disk, it is pi r squared where r is the radius of the sphere or disk.

We can estimate the drag force FA from the force in each thread by taking moments about the axis of rotation. To get the tension in the thread from the weight, we must remember the friction factor k, which we find using the method we use for the rolling resistance test. Including the weight on each thread W and the factor k, we get sum of the moments about the pencil axis equals 0, which becomes moment due to threads wrapped around the pencil, minus the moment due to the drag of the object under test equals 0. In our case, that is 2 times W times R1 divided by k, minus 2 times FA times R2 equals 0.

Or, FA equals W, R1 over R2 divided by k. The weight we used is the additional weight we added to bring the speed of the rig with the test object up to the speed of the rig with the bare disk. This way, we account for the drag of the rig itself. Putting all this into our beautiful equation gives us this equation, which simplifies to this equation. We won’t go into detail. But it shows you the general idea. We’ll be using a simpler version in the design exploration. Again, we won’t go into detail. But our data gives us the following values for drag coefficient. For a sphere, 0.4. For the disk, 1.25. This is almost perfect agreement with published data.

There’s a bit of luck in this. But it’s impressive, don’t you think.

With this overview, you’re almost ready to do your design exploration now. But you’ll need to know about work engine power.