Thermal Efficiency Calculator

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.

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 (Q_in, Q_out, and W_{net, 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 (T_h and T_c)

   • 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.

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:

η_th = W_net,out / Q_in

where:

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

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

η_th = 1 − Q_out / Q_in

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:

η_th = w_net,out / q_in
η_th = 1 − q_out / q_in

where:

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

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.

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:

η_th,rev = 1 − T_c / T_h,

where:

• η_th,rev — Thermal efficiency of the reversible heat engine;
• T_c — Temperature of the cold reservoir, in kelvin (K) or Rankine degrees (°R); and
• T_h — 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 η_th,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.

To obtain the Rankine cycle thermal efficiency:

1. Calculate the heat rejected in the condenser (q_out). For the ideal Rankine cycle, it's the difference between the enthalpies at its input (h₄) and output (h₁):

   q_out = h₄ − h₁

2. Calculate the heat added to the boiler (q_in). For the ideal Rankine cycle, it's the difference between the enthalpies at its output (h₃) and input (h₂):

   q_in = h₃ − h₂

3. Use the thermal efficiency formula:

   η_th = 1 − q_out / q_in

You can also obtain η_th using the net work output of the cycle (w_{net, out}):

η_th = w_net,out/q_in

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

η_th = 1 − 1 / r_p^{(k − 1)/k}

where:

• η_th — Thermal efficiency;
• r_p = P₂/P₁ — Pressure ratio, the ratio of the compressor exit pressure (P₂) to the pressure at its inlet (P₁); and
• k = c_p/cᵥ — Specific heat ratio of the working fluid, which equals k = 1.4 for air.

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

• η_th = 1 − q_c/qₕ (thermal efficiency equation);
• q_h = q_c/(1 − η_th) (solving for q_h); and
• q_h = (500 kJ/kg)/(1 − 0.45) = 909.1 kJ/kg.

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

• η_th = Ẇ_net,out / Q̇_in (thermal efficiency equation)
• Ẇ_net,out = η_thQ̇_in (solving for Ẇ_{net out})
• Ẇ_net,out = 0.45 × 10⁹ kJ/h = 120 MW