## 5 Preliminary Engine Design

INTKOUUC.TION Power output: i/ic Beale number concept

Wn i.jam rhai.fi. observed several years ago (hat the power output of many Stirling engines conformed approximately to the simple equation:

where P = engine power (watts), p = mean cycle pressure (bar), /-cycle frequency or engine speed (hertz), V0 = displacement of power piston (cm3).

This can be rearranged as P/(pfVt>) — constant. The equation was found by Beale to be approximately true for all types and sizes of Stirling engines for which data were available including free-piston machines and those with crank mechanisms. In most instances the engines operated with heater temperatures of 650 V and cooler temperatures of 65 °C.

The combination P/(pfVu) is a dimensionless group that may be called the Beale number. Now it is self evident that the Beale number will lie a function of both heater and cooler temperatures. Recent work by the author leads to the conclusion that the relationship of Beale number to heater temperature may be of the form shown in Fig. 5.1 by the full line. Although the relationship is shown lor the sake of clarity as a single line it must of course be understood that the relationship shown is a gross approximation and particulai examples of engines that depart widely may be cited. Nevertheless, a surprisingly large number of engines will be found to lie within the bounds of the confidence limits drawn on either side of the proposed relationship. Well-designed, high-efficiency units with low cooler temperatures will be concentrated near the upper bound. Less well designed units of moderate efficiency with high cooler temperatures will be located at the lower extremity.

The applicability of a simple relationship such as the Beale number is clearly limited. Nevertheless, it provides a handy guideline for both 'back-of-envelope' style calculations for new projects and for newcomers to the field.

Many have an overly sanguine view of the capability of Stirling engines. One frequently hears proposals to convert small internal combustion engines to Stirling engines operating on low pressure air and using furnace heat or solar energy to produce several kilowatts of power. The Beale number is useful for dispelling such high hopes. For example, consider a small internal combustion engine having two cylinders with

Heater temperature (K)

l;to. 5.1. The Fictile numliei as a function o( heater temperature. A design guideline for estimating the power output IP) of Stirling engines in watts.

Heater temperature (K)

l;to. 5.1. The Fictile numliei as a function o( heater temperature. A design guideline for estimating the power output IP) of Stirling engines in watts.

diameter and piston strokes oi, say. 5 cm. The engine may be modified to equip the two cylinders with heaters, regenerators, coolers and displacers to operate as a Stirling engine. Assume the working fluid to be at a low pressure, say 2 bar (0.2 MN/m2), Then, the power output at a speed of 1200 revolutions per minute may be estimated using the Beale number concept as:

I nri

75 watts approximately.

Clearly sneh a low power will come as a disappointment to our enthusiastic newcomer. Furthermore, it is likely that the mechanical friction will be greater than the power generated and so the engine will never run or, if il does, will barely be self-sustaining.

* . — -----1 „r »u„ ..,..,1.. ...-,„«.;,»„,- ii,., miironi nf n Inrnr Stirlino agricultural or industrial waste for stationary power generation. Assume, arbitrarily, an output of 746 kW {1000 horsepower) per cylinder with a mean cycle pressure of 20 MN/ni" (200 bar), with helium working fluid and a frequency of 30 cycles per second, corresponding to a speed of 1800 revolutions per minute.

The size of engine cylinder required may then be computed from the Bcale number. There is some evidence that the Beak' number is conservative for large engines. Furthermore., in such a high capital cost application. a sophisticated design with adequate cooling might be expected. Therefore perhaps it would be reasonable to double the value of the constant in eqn (5.1) from 0.015 to 0.03. Finally, it is advantageous at this stage to recall, from the aspect of seal, bearing, and piston-ring wear, the attractions of an 'oversquare' engine, i.e. a large bore and short stroke. Here we will arbitrarily assume a stroke that is one half the piston diameter.

Therefore, the piston swept volume:

So the Beale number PI(pfV(1)~ 0.03 may be rearranged as:

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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