# Thermal Efficiency Calculator

Created by Luis Hoyos
Reviewed by Steven Wooding
Based on research by
Cengel YA, Boles MA, Kanoğlu M. Thermodynamics: an engineering approach. 9th ed. New York: McGraw-hill (2019)
Last updated: Jul 14, 2023

With this tool, you'll calculate the thermal efficiency of reversible (Carnot) or irreversible heat engines using the thermal efficiency equations presented in the article below. You can use the calculator as long as you know the amount of heat in the energy interactions or the temperature of the thermal reservoirs (for reversible heat engines).

Keep reading this article to learn how to use this calculator and the thermal efficiency formulas behind it.

## How to calculate the thermal efficiency with this calculator

This tool calculates the efficiency of heat engines of two types:

1. Irreversible heat engine:

• Select this option if you're dealing with a real process (a process containing irreversibilities).

• You only need to enter two of the three input variables (Qin, Qout, and Wnet, out) for the tool to provide the answer.

• You can also use this mode for reversible processes, as the formula is the same (look at the equation in the following section). Even so, for reversible processes, it'll probably be easier to use the second mode, as it only requires knowing the temperatures of the thermal reservoirs.

2. Reversible heat engine:

• If you need to calculate the thermal efficiency of a reversible heat engine such as the Carnot engine, this is the mode you need.

• You only need to input the temperatures of the thermal reservoirs (Th and Tc)

• Please keep in mind that this mode cannot be used for the actual efficiency of irreversible heat engines, as it only provides their theoretical maximum value.

## Thermal efficiency equation

In the most general sense, efficiency indicates the ability to avoid wasting resources when accomplishing an objective. Our efficiency calculator, for example, defines it as the ratio of the useful energy output to the required energy input to a system.

Heat engines receive heat from a high-temperature source and convert a portion into work while rejecting the remaining heat to a low-temperature thermal reservoir (this heat rejection is mandatory). In thermodynamics, the efficiency we calculate for heat engines is known as thermal efficiency, defined as the ratio of the net work output to the heat input:

ηth = Wnet,out / Qin

where:

• ηth — Thermal efficiency;
• Wnet,out — Net work output of the heat engine, in joules (J) or British thermal units (BTU); and
• Qin — Total heat input, also in J or BTU.

From an energy balance, note that the net work output (Wnet) equals the heat input portion that doesn't go to the cold thermal reservoir (Wnet = QinQout). Therefore, another way to calculate ηth is:

ηth = 1 − Qout / Qin

💡 The previous and following equations calculate the efficiency as a fraction. To know how to convert it to a percentage, visit our decimal-to-percent converter.

#### Other energy units

If you still don't know much energy the heat engine will handle, you can analyze everything using heat and work per unit mass:

ηth = wnet,out / qin
ηth = 1 − qout / qin

where:

• wnet,out — Net work output per unit mass, in joules per kilogram (J/kg) or BTU per pound (BTU/lb);
• qin — Heat input per unit mass, in J/kg or BTU/lb; and
• qout — Total heat output per unit mass, in J/kg or BTU/lb.

🙋 You can use any energy units you want for the equations above. The only requirement is the units be the same for both variables (i.e., the same for Qin & Qout or qin & qout).

#### Net work output

Apart from producing a work output, heat engines require some work input for their different components to work (see the diagram above). The net work output equals the work produced by the components of the heat engine minus the work they consume.

For example, in a steam power plant – the most common heat engine – the net work output equals the work delivered by the turbines minus the work required to pump the working fluid.

## Thermal efficiency formula for reversible cycles

The fundamental characteristic of reversible processes is that they don't contain irreversibilities. Fiction, unrestrained gas expansion, and heat transfer under finite/realistic temperature differences are examples of irreversibilities. The most common reversible process is the Carnot cycle we saw in our Carnot efficiency calculator, operated by a Carnot heat engine.

The Carnot engine is the most efficient heat engine we can build between two thermal energy reservoirs at two different temperatures, and its efficiency equals to:

ηth,rev = 1 − Tc / Th,

where:

• ηth,rev — Thermal efficiency of the reversible heat engine;
• Tc — Temperature of the cold reservoir, in kelvin (K) or Rankine degrees (°R); and
• Th — Temperature of the hot reservoir, in kelvin (K) or Rankine degrees (°R).

Unfortunately, real heat engines contain irreversibilities that lower their efficiency, making them less than ηth,rev. Even so, the Carnot heat engine provides the maximum value we could reach if we operated under the best conditions, telling us how near we are to the perfect process.

🙋 Be careful! As the formula above relies on an absolute temperature scale, we must use absolute temperature units (i.e., kelvin (K) or degrees Rankine (°R)) when calculating ηth,rev. We do the same for reversible devices in our coefficient of performance calculator. And our temperature conversion tool can convert between different units.

## FAQ

### How do I calculate the thermal efficiency of the Rankine cycle?

To obtain the Rankine cycle thermal efficiency:

1. Calculate the heat rejected in the condenser (qout). For the ideal Rankine cycle, it's the difference between the enthalpies at its input (h) and output (h):

qout = h₄ − h

2. Calculate the heat added to the boiler (qin). For the ideal Rankine cycle, it's the difference between the enthalpies at its output (h) and input (h):

qin = h₃ − h

3. Use the thermal efficiency formula:

ηth = 1 − qout / qin

You can also obtain ηth using the net work output of the cycle (wnet, out):

ηth = wnet,out/qin

### What is the Brayton cycle thermal efficiency formula?

The formula for the thermal efficiency of the Brayton cycle under the cold-air-standard assumptions is:

ηth = 1 − 1 / rp(k − 1)/k

where:

• ηth — Thermal efficiency;
• rp = P₂/PPressure ratio, the ratio of the compressor exit pressure (P) to the pressure at its inlet (P); and
• k = cp/cᵥSpecific heat ratio of the working fluid, which equals k = 1.4 for air.

### How much heat is received by a heat engine with a thermal efficiency of 45% that rejects 500 kJ/kg of heat?

909.1 kJ/kg. We can obtain the solution by using the thermal efficiency equation and solving for q:

• ηth = 1 − qc/q (thermal efficiency equation);
• qh = qc/(1 − ηth) (solving for qh); and
• qh = (500 kJ/kg)/(1 − 0.45) = 909.1 kJ/kg.

### How much power is produced by a heat engine with a thermal efficiency of 45% that receives 10⁹ kJ/h of heat?

125 megawatts (MW). We can obtain the solution by using the thermal efficiency equation and solving for net,out:

• ηth = net,out / in (thermal efficiency equation)
• net,out = ηthin (solving for net out)
• net,out = 0.45 × 10⁹ kJ/h = 120 MW
Luis Hoyos Type of heat engine
Reversible (Carnot heat engine)
Input two of the variables below
Hot reservoir temperature (Tₕ)
°R
Cold reservoir temperature (Tc)
°R
Thermal efficiency (ηₜₕ ᵣₑᵥ)
%
ηth, rev = 1 - Tc/Th
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