Mathematics is an incredibly efficient language. With just a single, short sentence, it can say what takes several volumes in English. One such “sentence” is the equation for Carnot efficiency (η):

η = 1-T_{c}/T_{h}

where T_{c} and T_{h} represent cold and hot temperatures. With just five letters (admittedly, two of them are subscripts, and one is Greek), one number, and two mathematical symbols, this equation unlocks one of the greatest secrets of thermodynamics. The equation is a quantitative expression of the second law of thermodynamics; I hinted at it in a recent column (see, May 2020 issue, “Double Up on Cogeneration”). But what does it mean, and how does it apply to energy-efficient design and energy management?

The Carnot cycle, first proposed by French military engineer Sadi Carnot in 1824, was the first successful theoretical model of an “ideal” heat engine. Our equation expresses the maximum efficiency of this ideal heat engine. For every unit of heat flow that enters the heat engine, η is the maximum number of units of mechanical power that can be produced.

A heat engine is any device that converts heat to mechanical power. The ones we encounter most often are internal combustion engines, steam turbines and gas turbines. In a recent column (“Take a Closer Look at Cascaded Efficiency,” March 2020), I compared the “cascade efficiencies” of electric- and gasoline-engine cars. Heat engines dominate both cascades. Steam turbines and gas turbines produce the electric power to charge electric cars, and internal combustion engines power gasoline-fueled vehicles. Because the same limiting efficiency equation applies to all heat engines, there isn’t as big a difference as might be expected between the overall efficiencies of these different types of cars. The situation changes somewhat when alternative methods are used to generate the electricity — but that is a different subject.

The beauty of the equation is its remarkable simplicity. η depends on just two things — T_{h}, which is the inlet (hot) temperature and T_{c}, which is the exhaust (cold) temperature. Both are expressed as absolute temperatures, in either the Kelvin or Rankine scale, and assumed constant. The equation applies equally to heat engines that use a gas, liquid or mixed working fluid. It is also striking that pressure does not appear explicitly in the equation — although, as we shall see, pressure does vary along with temperature in practical heat engines.

The equation assumes all of the conditions are “ideal” — no friction, no heat loss, no fluctuations in conditions. We never see “ideality” in industrial applications. Rather, η represents the theoretical maximum efficiency; it defines an upper limit. Real efficiencies are always lower. Much of the research and development work in energy efficiency focuses on minimizing non-idealities in equipment designs, thus bringing design efficiencies closer to η. The resulting improvements include, for example, reducing frictional pressure drops in gas and steam turbines.

The equation also tells us that the Carnot efficiency increases as the ratio T_{c}/T_{h} decreases, which means high inlet temperatures and low exhaust temperatures are desirable. This fact has driven a multi-year trend in gas turbine designs towards higher inlet temperatures, which increase T_{h}, and also towards larger pressure ratios, which lead to lower exhaust temperatures, T_{c}. Similar considerations also apply in the design of chemical plants that use steam turbines. For example, there is a focus on minimizing heat losses from steam piping by better insulation, and decreasing pressure drop — such as, by reducing the number of bends, fittings and valves. This ensures the highest possible temperature T_{h} (subject to design limits) entering the steam turbines, and greatest pressure differential across the steam turbine, which maximizes power generation and lowers the exhaust temperature T_{c}. Where condensing steam turbines are used, a strong incentive exists to optimize the design of the condenser and the vacuum system to ensure the lowest possible exhaust pressure, which leads to a low exhaust temperature, T_{c}. Turning to operation and maintenance, fixing leaks in vacuum systems, together with cleaning fouled condensers, can sometimes improve steam turbine efficiencies by 5% or more.

All of this flows from a simple equation with just five letters, one number, and two mathematical symbols — and we have only scratched the surface.