aero and automotive engines and engineers
The RAC HP (horsepower) Rating - Was there any technical basis? by Richard Hodgson
This article first appeared in a slightly different form in about 1989 in "Sphinx", the magazine of the Armstrong Siddeley Owners' Club. More recently, it appeared in yet slightly different form in the November/December 2000 edition of "The Flying Lady", the journal of the North American Rolls-Royce Owners' Club (RROC). Please use your Web Browser's "Back" button if you want to return to this article after visiting any hyperlink which are shown in bold dark blue. Please click here to go to the list of articles on this site's home page.
For many years - in fact into the 1960's, the RAC (Royal Automobile Club) horsepower rating of car engines was given. The figure increasingly had little connection with an engine's actual output, so ratings such as 16/65 were seen, referring to an RAC rating of 16hp and an output of 65bhp. The RAC rating was very important, as for many years the annual car taxation was based on its RAC rating.
The myth has grown that the RAC formula was arbitrary and just happened to calculate the actual output of some early petrol engines. It is said that this is obvious as the engine's stroke (and thus volume) is apparently not included in the RAC formula. In fact, the formula was based on sound engineering principles and account was taken of stroke. The limiting factor was that three specific Assumptions, which never varied, were made about engine and the conditions under which it operated. As the formula was taken into account for taxation purposes for many years, these assumptions were not changed and soon quietly forgotten about. The explanations below are not given in a strict mathematical fashion, but are correct (save for the exact of value of Pi. values dependent on its value and any typographical errors).
The RAC formula states:
h.p. = (D2 x n)/2.5
where D = the diameter of the cylinder in inches [1" = 25.4mm], and n = the number of cylinders
How does the RAC formula arise?
The formula is in fact derived from general principles.
The indicated brake horsepower per cylinder can be calculated if the mean effective pressure in the cylinder is known. This pressure is the mean of the varying pressures acting on the piston. Being the mean value, it is that notional unvarying pressure which acts on the piston during the entire power stroke.
Pm is the mean effective pressure in lbs (pounds) per square inch (lb/sq.in) [one pound ~ 0.454 Kg]; L the stroke in feet [one foot = 12 inches]; D the diameter of the cylinder in inches. The pressure acts on the piston of area (D/2) x (D/2) x Pi (Pi ~ 3.142) through a distance of L feet. 1 hp = 33,000 pounds-foot per minute (the old-fashioned but useful definition of one British horsepower). In a four-stroke engine, the frequency, fp, of power stroke in a cylinder per crankshaft revolution is ½ (0.5), as the crankshaft rotates twice to have one power stroke per cylinder. In one minute a single cylinder rotating at N r.p.m. will have had N/2 power strokes. The number of pounds-foot per minute of such a single cylinder engine is:
Pounds-foot per minute = Pm x (D/2) x (D/2) x Pi x L x N x fp
As by definition 1 h.p. = 33,000 pounds-foot per minute:
Indicated horsepower, i.h.p. = (Pm x D2 x Pi x L x N x fp)/(33,000 x 4)
Due to friction and other losses, the power developed at the crankshaft is less than the i.h.p. The "useful" power developed is termed the brake horse power (b.h.p.) as this power can be measured by a test brake. The difference between the i.h.p. and the b.h.p. is termed the mechanical efficiency (symbol is the Greek letter 'eta')
Mechanical Efficiency = b.h.p. / i.h.p
Returning to the i.h.p. formula given above, on calculating its 'number' part:
i.h.p. = (Pm x D2 x L x N)/84023
The distance travelled by the piston in the cylinder per minute gives the mean piston speed Vm:
Vm = 2 x L x N
In other words, the mean piston speed is twice the stroke length multiplied by the r.p.m, as the piston goes up and down each revolution. Replacing L and N gives:
b.h.p. = (Pm x D2 x Vm)/168046
Thus, in an engine having n cylinders:
b.h.p. = (Pm x D2 x Vm x n)/168046
The Three Specific RAC Assumptions
The RAC formula, seeking to simplify the above formula, made three assumptions which do not change, whatever the engine under consideration:
(1). Mechanical efficiency at Vm is 75%
(2). The mean effective pressure Pm is 90lb/sq.in
(3). The mean piston speed Vm is 1,000 ft/min, i.e. 2 x L x N = 1000 ft/min
But from a brief consideration of Assumption 3, it will be seen that the stroke of the engine has in fact been taken into account. 1000 ft/min was the then considered maximum mean piston speed that was consistent with acceptable wear and friction. Thus the formula "expected" a short stroke engine to operate at a higher r.p.m than a long stroke engine and vice-versa, the product of stroke length and r.p.m. being constant - in other words, the mean piston speed Vm.
Inserting these three specified values into the formula above for an engine with n cylinders:
b.h.p. = (90 x D2 x 1000 x 0.75 x n)/168046
= (D2 x n)/2.5 (accurate to about 0.4%)
Even by the mid to late 1930's, the values of the three "fixed" Assumptions were being exceeded, especially mean piston speed which was often taken as high as about 2750 ft/min for acceptable wear and friction. Mean effective pressure increased as had mechanical efficiency. In "The Elements of Motor Vehicle Design", 2nd edition, Oxford University Press, 1935, Donkin gave a figure of 80 to 90% for the mechanical efficiency of then modern engines. Thus an output for a better (unblown) engine of 3 to 4 times the RAC rating was not uncommon in the mid to late 30's. Note that some Armstrong Siddeley's of the pre-war period had mean piston speeds in excess of 2750 ft/min when going at their maximum speed.
The almost general use of high compression engines in later post-war years resulted in yet further increases in mean effective pressures. Note that blowing an engine (if done properly!) also increases mean effective pressure.
What is the RAC hp of a 6 cylinder engine with a bore of 2.625" (the bore of the ASM 17hp engine)?
Answer: RAC hp = (2.625 x 2.625 x 6)/2.5 = 16.5375 hp ~ 16.5 hp.
Thus the 17hp engines were not quite all that they were cracked up to be and were rounded up to 17hp.
Whilst the value of mean effective pressure is one of the key element to the above formulae, considerations of its measurement, indicated mean effective pressure, brake mean effective pressure, relationship to compression ratio, blowing - if any, fuel used and the like, are beyond the scope of this present article.
© 1989, 2000, 2001 (in parts) Richard Hodgson