# Thermal Efficiency

## Thermal Efficiency

The “thermal efficiency”  of any engine is defined as the amount of useful energy output divided by the amount of energy input .  It is not a fixed quantum but varies according to the engine’s load and conditions of operation.  Three types of efficiency are described here:  Cylinder or Indicated Efficiency; Drawbar Thermal Efficiency; and Boiler Efficiency.

In the case of steam locomotives, the term thermal efficiency may refer to “cylinder efficiency” or as “drawbar efficiency”.  Their definitions are importantly different:

• Cylinder efficiency is defined as the the amount of energy delivered by the cylinder to the piston divided by the amount of energy delivered to the cylinder in the form of steam delivered to the steamchest;
• Drawbar thermal efficiency is defined as the the amount of energy delivered at the locomotive’s drawbar (the hook at the back of its tender) divided by the amount of energy available in the fuel placed into its firebox.

Both the cylinder and drawbar efficiencies vary with speed and power output, maximum cylinder efficiency being achieved at much higher speed than maximum drawbar efficiency.  This is because, as speed rises, a locomotive’s rolling resistance also rises and the tractive force avalable at its drawbar falls until, at a certain speed, the drawbar force becomes zero and thus the drawbar efficiency also becomes zero.

### Cylinder efficiency

Cylinder efficiency is governed by the shape of the Indicator Diagram and in particular by the losses that are evidenced by it – most especially expansion losses, condensation losses and leakage losses.

It should be noted however, that even without these losses, cylinder efficiency is limited by Carnot’s equation which states that the maximum theoretical efficiency of any heat engine is governed by the temperature difference between its heat source and its heat sink [see Thermodynamics page for further information.]

Note: Isentropic efficiency is another (but very different) measure of cylinder efficiency.  Instead of describing the ratio of work output to work output, it describes the ratio of work output with maximum possible work output based on steam conditions – see Thermodynamics definitions.

### Drawbar efficiency

Drawbar efficiency can be seen as the sum of the efficiencies of a locomotive’s various components.  Wardale provides examples of these in his book “The Red Devil and Other Tales from the Age of Steam” where (in Table 78, page 457) he quotes figures for standard and (proposed) modified Chinese Class QJ locomotives, and where (on page 501) he suggests what might be achieved from the further development to the level of “Third Generation Steam” traction.

The figures from these pages are combined in a single table below, however it is recommended that the qualifying texts from both Table 78 (page 457) and page 501 of Wardale’s book be read in association with them.

 Item Standard QJ Modified QJ Third Generation Steam Boiler combustion efficiency 78% 87% 95% Boiler absorption efficiency 78.2% 80% 90% Auxiliary efficiency factor 93.1% 94% 96% Cylinder efficiency 16.4% 19.05% 22% Transmission efficiency 89% 93% 94% Drawbar efficiency 94% 95% 96% Overall drawbar thermal efficiency = product of all the above 7.8% 11.0% 16.3%*

Maximum drawbar thermal efficiency is usually reached at modest speed and power outputs such that increasing rolling resistance and increasing fuel carry-over (in the case of coal firing) are offset by increasing cylinder efficiency.

* Note: Wardale’s estimate for TGS drawbar efficiency differs significantly from the figure of 25% that he quotes as being Porta’s estimate for condensing third generation steam locomotives – see Second Generation Steam page of this website.  However Wardale makes it clear (on page 501 of his book) that his figure applies to non-condensing locomotives and that “higher efficiency could only be obtained by expanding the steam to sub-atmospheric pressure and low temperature by means of condensing to counter the negative effect on the cycle efficiency of the restricted inlet steam temperature as done in stationary steam plant”.

### Boiler Efficiency

Boiler efficiency can be defined as the amount of energy delivered from the boiler in the form of steam divided by the amount of energy delivered to the firebox in the form of fuel/chemical energy.  Boiler efficiency depends on the design on boiler and firebox, the type of fuel, and the draughting system.  In the case of coal-fired boilers, boiler efficiency declines linearly with rate of fuel feed, as discussed on the Grate Limit and Boiler Efficiency page.