Why does a gas engines torque and horsepower curve always meet at 5252 RPM?

Question

I'm watching a TV show called Tech Garage and they just made the claim that an engines torque and horsepower curves always meet at 5252 RPM. Assuming this is true, why does this happen? Is it something designed into engines, or is it just how the thermodynamics work out?


If it is a result of design, what about the engine is arranged to make it true?

Answer 1

It's just math, and is because horsepower is defined (in terms of torque) as 550 ft·lbs per second.

A single HP is 33,000 pounds moved 1 foot in 1 minute (as per James Watt, that's the average of what an actual horse can do). An RPM of an engine moving the same 1 lb would travel ~6.283ft (the circumference of a 1 foot radius circle).

33,000 / 6.283 = 5252

Answer 2

Just for fun, I did some math in google, to show that this is an artifact of the unit system being used to put numbers to torque and power.

The number 5252 can be calculated as:

1 horsepower / 1 lbf foot radian in turns/minute
5 252.11312

The exact number is 16,500/π (33,000/τ)

So, if the math were done in metric units instead (watts, and and newton-meters for torque), you would get:

1 W / 1 N m rad in turns/minute
9.54929659

This number happens to be 30/π (or 60/τ), due to the number of seconds in a minute. If you measured engine rotation speed in radians/second, the number would become 1. The same would apply to the non-metric system, if foot-pounds were used instead of horsepower to measure engine power.

Where the "curves meet" is entirely an artifact of placing both quantities (measured in horsepower and pound-feet) on the numerically same scales on the axis of a graph. If you graphed them against each other rather than against RPM, this would instead show up in that some point (corresponding to 5252 RPM) would show up at a point where the power in horsepower and the torque in pound-feet are the same.

Answer 3

Just adding to Marks great answer above:

Although horsepower is defined in terms of torque, horsepower is the more useful measure of an engines output when it comes to trying to figure out how fast your car will be, assuming you have an appropriate transmission. A transmission alters the torque output, but leaves the horsepower unchanged (neglecting friction losses, etc.). Therefore, the old saying "horsepower sells cars, torque wins races" is actually completely false in theory. A hypothetical engine that made 1000ft-lbs of torque, but only revved to 10 RPM would result in a painfully slow car.

For actual cars, though, people rarely ever discuss the actual horsepower or torque curves. They usually only talk about peak horsepower or peak torque. Cars with high torque usually produce it in the lower end, and this also raises the horsepower in the lower end. Because horsepower is related to RPM, peak horsepower typically comes at high RPMs. Therefore, a 'high torque' engine may have the same peak HP as a 'low torque' engine, but the high torque engine will have more horsepower in the lower end of the RPM range. Because of this, the car will be faster - but people who only consider peak values will say that this is due to the torque, when really it is due to a broad horsepower band. Remember - any engine can in theory produce an arbitrarily large amount of torque after being routed through a transmission.

For a simple hypothetical example, consider an engine A with a typical horsepower curve, then imagine an engine B with the same exact horsepower curve, but scaled x2 on the RPM axis. If engine A produced peak HP at 5,000 RPM, engine B would produce peak HP at 10,000 RPM. If engine A produced 90 HP at 2,000 RPM, engine B would produce 90 HP at 4,000 RPM.

Now, imagine routing engine B through a frictionless transmission with a 2:1 ratio. This will effectively slow down engine B, so that the features of the transmission-modified horsepower curve now occur at the same rotational speeds as engine A. Engine B is now producing the same exact torque and horsepower output as engine A after the transmission, despite the fact that engine B would clearly have less torque before the transmission. (Again, see Marks great explanation of the math)