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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. = (D ^{2} **x

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,

Pounds-foot per minute = **P _{m} **x

As by definition 1 h.p. = 33,000 pounds-foot per minute:

*Indicated* horsepower, **i.h.p.** = **(P _{m}
x D^{2} x Pi x L x N x f_{p})/(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. = (P _{m} x D^{2}
x L x N)/84023**

The distance travelled by the piston in the cylinder per minute gives the mean piston
speed** Vm**:

**V _{m} **=

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. = (P _{m} **x

Thus, in an engine having **n** cylinders:

**b.h.p. = (P _{m} **x

**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 **V _{m}** is

**(2).**
The mean effective pressure **P _{m}** is

**(3).**
The mean piston speed **V _{m}** is

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 **V****m**.

Inserting these three specified values into the formula above for an engine with **n**
cylinders:

**b.h.p. = (90 **x** D ^{2} **x

**= (D ^{2 }**x

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.

__Worked Example__:

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