How will the efficiency of a heat engine change if the temperature of the refrigerator is constant (17°C)?

Thermodynamics is the science that controls the operation of all heat engines, from steam turbines in power plants to internal combustion engines in cars. One of the key performance indicators of such systems is coefficient of performance (efficiency), which determines what part of the supplied heat is converted into useful work. But what will happen to the efficiency if the temperature is fixed refrigerator at 17°C (290 K) and the heater temperature is changed? This question is relevant for engineers designing power plants, students of technical universities and even for car enthusiasts who want to understand how climatic conditions affect engine operation.

In classical thermodynamics, the efficiency of an ideal heat engine operating at Carnot cycle, is described by the formula:

η = (T₁ – T₂) / T₁where T₁ is the temperature of the heater, T₂ is the temperature of the refrigerator. At a fixed T₂ = 290 K (17°C) efficiency depends only on T₁. But how exactly will the efficiency change if the heater temperature is increased or decreased? And how close are real engines to the theoretical maximum? Let's figure it out in order.

In the article you will find not only theoretical calculations, but also practical examples of calculations, comparison of ideal and real cycles, as well as tips for optimizing the operation of heat engines in conditions of constant refrigerator temperature. If you've ever wondered why an engine runs differently in winter than in summer, or how to increase the efficiency of an industrial turbine, this material is for you.

1. Basics of thermodynamics: what is the efficiency of a heat engine?

Coefficient of performance (Efficiency) is a dimensionless quantity that shows what fraction of the heat received from the heater is converted into mechanical work. In real conditions, efficiency is always less than 100% due to friction losses, heat exchange with the environment and non-ideal processes.

For an ideal Carnot cycle, efficiency reaches its maximum value and depends only on the temperatures of the heater (T₁) and refrigerator (T₂): η₍₍carnot₎₎ = 1 – (T₂ / T₁).

This formula shows that efficiency increases with increasing T₁ or decreasing T₂. However, in real conditions, the temperature of the refrigerator is often limited by external factors - for example, the temperature of the surrounding air or water in the cooling system.

In the case when T₂ = 290 K (17°C), the formula is simplified to: η = 1 – (290 / T₁).

This means that the higher the efficiency, the hotter the heater. But in practice it exists. limit on thermal stability of the materials from which engine parts are made.

• 🔥 Heater temperature (T₁): Determined by the type of fuel, the design of the combustion chamber and the material of the parts. In steam turbines it can reach 800–1000 K, in internal combustion engines - 2000–2500 K.
• ❄️ Refrigerator temperature (T₂): Usually close to the ambient temperature. In automobile engines this is the temperature of the radiator (~350 K), in industrial installations - the temperature of the water in the condenser (~300 K).
• ⚙️ Real efficiency: Always below ideal due to losses. For example, The efficiency of gasoline engines rarely exceeds 30–35%, and diesel engines - 40–45%.

Interesting fact: even a small change T₂ can significantly affect the efficiency. For example, if the temperature of the refrigerator drops from 17°C to 7°C (from 290 K to 290 K). 280 K), the efficiency of an ideal engine will increase by 3-5% at the same T₁. This explains why some power plants operate more efficiently in cold climates.

2. Calculation formula: how will the efficiency change at T₂ = 17°C?

Suppose we have a heat engine. with a fixed refrigerator temperature T₂ = 290 K (17°C). We want to find out how many times its efficiency will change when the heater temperature increases T₁ from the original value T₁₀ to the new one T₁₁.

Denote:

- Initial efficiency: η₀ = 1 – (290 / T₁₀)

- New efficiency: η₁ = 1 – (290 / T₁₁)

To find how many times the efficiency will increase, let's calculate the ratio: k = η₁ / η₀ = [1 – (290 / T₁₁)] / [1 – (290 / T₁₀)].

For example, if T₁₀ = 500 K, a T₁₁ = 1000 K, then:

- η₀ = 1 – (290 / 500) = 0.42 (42%)

- η₁ = 1 – (290 / 1000) = 0.71 (71%)

- k = 0.71 / 0.42 ≈ 1.69.

That is, the efficiency will increase by 1.69 times (by 69%) when the temperature doubles heater.

From the table it is clear that the higher the initial temperature of the heater, the lower the increase in efficiency at the same absolute magnification T₁. This is due to the nonlinear dependence of efficiency on temperature.

3. Practical examples: internal combustion engines, turbines and refrigerators

Theoretical calculations are useful, but how are they applied in practice? Let's consider several real systems where the temperature of the refrigerator is close to 17°C (or fixed at another level), and see how change T₁ affects efficiency.

1. Automotive internal combustion engines (ICE):

In gasoline and diesel engines, the temperature of the refrigerator is determined by the cooling system and is usually 350–370 K (77–97°C)However, in cold climates (for example, at –10°C) efficient T₂ can drop to 263 K, which increases efficiency. (77–97°C) T₂ = 290 K (17°C) And T₁ = 2200 K (typical combustion temperature in an internal combustion engine):

- η = 1 – (290 / 2200) ≈ 0.87 (87%) — theoretical maximum.

- Real efficiency of gasoline engines: 25–30% (due to losses due to friction, incomplete combustion, heat transfer).

2. Steam turbines of power plants:

Here T₂ depends on the condenser temperature (usually 300–320 KIf you artificially reduce T₂ to 290 K (for example, using cooling with Arctic water), the efficiency of the turbine will increase from: T₁ = 800 K will grow with:

- η = 1 – (320 / 800) = 60% → to 1 – (290 / 800) ≈ 63.75%.

The increase will be ~6%, which for a large power plant means millions of kilowatt-hours of additional electricity per year.

3. Heat pumps and refrigerators:

In these devices, the “refrigerator” is the cooled space (for example, the fridge compartment), and the “heater” is the environment. If the ambient temperature (T₁) will increase from 20°C (293 K) to 30°C (303 K) at T₂ = 277 K (4°C), efficiency (in the case of a refrigerator this is coefficient of performance ε = T₂ / (T₁ – T₂)) will drop from:

- ε = 277 / (293 – 277) ≈ 15.4 → up to 277 / (303 – 277) ≈ 11.6.

That is, the efficiency will decrease by 25%, which will lead to an increase in energy consumption.

• 🚗 ICE: Real efficiency is limited by the heat resistance of materials (steel, aluminum) and fuel detonation. Increase T₁ requires the use of heat-resistant ones. alloys or ceramics.
• ⚡ Turbines: Modern gas turbines operate at T₁ ≈ 1600 K, and cooling of the blades makes it possible to raise this threshold to 1900 K.
• ❄️ Refrigerators: Their efficiency drops in the summer, when T₁ (room temperature) rises. Optimal operation is at 20–22°C indoors.

4. Limitations and real losses: why does the efficiency never reach 100%?

Even in an ideal Carnot cycle, the efficiency cannot be equal to 100%, since this would require the temperature of the refrigerator to be equal to absolute zero (0 K), which is impossible. In real engines, the losses are even more significant:

1. Heat losses:

Part of the heat is transferred to the environment through the engine housing, cooling system or exhaust gases. For example, in an internal combustion engine up to 30% of the heat leaves with the exhaust, and another 10–15% - through the radiator.

2. Mechanical losses:

Friction in bearings, air resistance (for piston engines) and hydraulic losses in pumps reduce useful work. In modern engines this takes 5–10% of power.

3. Imperfection of processes:

In real cycles (Otto, Diesel, Rankine), compression and expansion do not occur adiabatically, but with heat exchange. This reduces the efficiency by 10–20% compared to the Carnot cycle.

4. Thermodynamic limitations of materials:

No material can withstand arbitrarily high temperatures. For example:

- Aluminum alloys melt at ~900 K.

- Heat-resistant steels work up to ~1200 K.

- Ceramics (for example, in turbines) can withstand up to ~1600 K, but is fragile.

⚠️ Attention: When designing heat engines, engineers always look for a balance between increasing T₁ (to increase efficiency) and design reliability. Exceeding permissible temperatures leads to deformation of parts, accelerated wear and accidents.

For illustration: if a steam turbine is increased T₁ from 800 K to 1000 K, the theoretical efficiency will increase from 60% to 71%. However, this will require replacing steel blades with nickel alloys, which will increase the cost of the turbine by 20–30%.

5. How to optimize efficiency at a fixed T₂ = 17°C?

If the temperature of the refrigerator is fixed (for example, due to climatic conditions or design restrictions), you can increase the efficiency in the following ways:

1. Increasing the heater temperature (T₁):

- Using more high-calorie fuel (for example, switching from gasoline to diesel or gas).

- Using turbocharging or intercooler to increase the temperature and pressure of the intake air.

- Improving the thermal insulation of the combustion chamber (for example, ceramic coatings in ICE).

2. Reduction of losses:

- Reduction of mechanical friction due to high-quality oils and bearings.

- Optimization of the exhaust system to reduce heat losses from exhaust gases.

- Use of regenerative heat exchangers (for example, in gas turbines).

3. Changing the working fluid:

- In steam turbines, switching to superheated steam instead of saturated steam increases T₁ without changing the pressure.

- In refrigeration machines, replacing freon with ammonia or CO₂ can improve heat transfer.

4. Combined cycles:

- Combined-cycle plants (CCGTs) combine gas and steam turbines, using the waste heat of the first to operate the second. Their efficiency reaches 55–60%.

- Cogeneration (joint production of electricity and heat) allows you to use up to 80–90% fuel energy.

Increase the heater temperature (T₁) by improving the fuel or pressurization|Reduce heat losses with exhaust and cooling|Optimize mechanical components (bearings, lubrication)|Apply combined cycles (for example, CCGT)|Use heat recovery-->

Practical example: at modern thermal power plants, where the temperature of the refrigerator is fixed by the conditions of the cooling pond (~15–20°C), the transition from coal boilers to gas turbines with T₁ = 1500 K allows to increase efficiency from 35% to 50–55%.

6. Frequent errors when calculating efficiency

When working with efficiency formulas, it is easy to make mistakes, especially if you confuse absolute temperatures (in Kelvin) with degrees Celsius or do not take into account real losses. Let's consider typical misconceptions:

1. Using degrees Celsius instead of Kelvin:

Carnot's formula requires a temperature of kelvins. If we substitute T₂ = 17°C without translation, we will get the wrong result. Correct:

T₂ = 17 + 273.15 = 290.15 K ≈ 290 K.

2. Ignoring losses in real engines:

Many people believe that the efficiency of an internal combustion engine can be calculated using the Carnot formula, but this is incorrect. Real efficiency is always lower due to:

- Incomplete combustion of fuel.

- Heat losses through the cylinder walls.

- Mechanical friction losses.

3. Neglecting the dependence of efficiency on load:

Engine efficiency is maximum at 70–80% of the load. At idle it drops to almost zero, and when overloaded, fuel consumption increases and efficiency decreases again.

4. Omitting the influence of humidity and pressure:

In gas turbines and internal combustion engines, air humidity is reduced T₁ due to the evaporation of water, which reduces efficiency by 1–3%. Pressure also plays a role: in the mountains (low pressure), engines lose up to 10–15% power.

⚠️ Attention: When designing systems with a fixed T₂ (for example, air conditioners), it is important to take into account not only the temperature, but also the humidity refrigerator. Moisture condensation on the heat exchanger can reduce efficiency by 20–30%.

For accurate calculations, use specialized software, for example:

- CoolProp (for the thermodynamic properties of refrigerants).

- EngineSim (modeling of internal combustion engines).

- ThermoCalc (calculations for metallurgy and energy).

7. The future of heat engines: is it possible to overcome the limitations?

Modern technologies make it possible to gradually increase the efficiency of heat engines, despite fundamental limitations. Here are some promising areas:

1. New materials:

- Ceramic composites (for example, based on silicon carbide) can withstand temperatures up to 2000 K, which will make it possible to raise T₁ in gas turbines.

- Graphene coatings reduce friction in engines by 30–40%, reducing mechanical losses.

2. Alternative cycles:

- A cycle Erickson or Stirling with heat recovery can exceed the efficiency of the Carnot cycle under certain conditions.

- Detonation engines (for example, a rotary detonation engine) promise efficiency up to 50–60% due to more complete fuel combustion.

3. Hybrid systems:

- Combining heat engines with electric machines (as in hybrid cars) allows excess heat to be used to generate electricity.

- Thermoelectric generators convert waste heat from exhaust gases into electricity, adding 2–5% to overall efficiency.

4. Artificial intelligence in control:

- Systems predictive control (for example, in cars Formula 1) optimize fuel supply and ignition in real time, increasing efficiency by 3–7%.

- Digital twins Thermal plants allow you to simulate and minimize losses without physical experiments.

However, even with these innovations the second law of thermodynamics remains insurmountable: no heat engine can have 100% efficiency. The maximum that humanity strives for is 70–80% for the most advanced energy plants.

Why can't the efficiency be 100%?

According to the second law of thermodynamics, it is impossible to create a perpetual motion machine of the second of a kind that would completely convert heat into work. Some energy will always be transferred to the refrigerator, since the entropy of an isolated system cannot decrease. Even in an ideal Carnot cycle, efficiency tends to 100% only at T₂ → 0 K, which is unattainable.

FAQ: Answers to frequently asked questions

Is it possible to achieve 100% efficiency at T₂ = 17°C if T₁ is increased endlessly?

No. Even with endless growth T₁ Efficiency will tend to the limiting value η = 1 – (290 / ∞) = 1 (100%), but it is impossible to achieve it due to:

• Limitations on the strength of materials.
• Increase in heat losses at high temperatures.
• Fundamental thermodynamic limitations (second law of thermodynamics).

In practice, the maximum efficiency of modern power plants does not exceed 60–65% (for example, in combined-cycle plants).

How does ambient temperature affect the efficiency of a car engine in winter?

In winter, the temperature of the refrigerator (radiator, surrounding air) falls, which increases theoretical efficiency. However, in practice, the effect is often neutralized:

• Increasing viscosity of oil and fuel (more mechanical losses).
• Non-optimal operation of the ignition system on a cold engine.
• Additional loads (interior heating, battery).

As a result, real efficiency can either increase by 1–3%or fall due to increased losses.

Why is absolute temperature (kelvins) used in the Carnot formula and not Celsius?

Carnot formula derived from thermodynamic relations, where temperature is included as a ratio T₁ / T₂. The absolute scale (Kelvin) begins at 0 K (absolute zero), which corresponds to the complete absence of thermal movement. The Celsius scale is shifted by 273.15, and its use will lead to incorrect calculations. For example:

• If T₂ = 0°C = 273.15 K, then η = 1 – (273.15 / T₁).
• If we substituted 0°C directly, we would get η = 1 – (0 / T₁) = 1 (100%), which is absurd.

Which heat engines have the highest efficiency today?

Efficiency records among real heat engines:

• Combined cycle plants (CCGT): up to 63% (for example, Siemens HL-class).
• Diesel engines of large ships: up to 50% (for example, Wärtsilä RT-flex96C).
• Aerospace-grade gas turbines: up to 45–50% (for example, GE9X for Boeing 777X).
• Thermoelectric generators: efficiency ~5–10%, but they use "waste" energy.

For comparison: the average efficiency of gasoline car engines is 20–30%, and of old steam engines - only 5–15%.

Can it be used these calculations apply to refrigerators and air conditioners?

Yes, but adjusted for the fact that refrigerators and air conditioners operate according to reverse Carnot cycle. Their efficiency is assessed not by efficiency, but by refrigeration coefficient (ε): ε = T₂ / (T₁ – T₂),

where T₂ is the temperature of the fridge compartment, and T₁ is the ambient (heater) temperature.

At T₁ = 290 K (17°C) i T₂ = 273 K (0°C):

ε = 273 / (290 – 273) ≈ 12.4.

This means that for 1 Joule of consumed electricity, the refrigerator transfers 12.4 Joules heat from the chamber to the environment. Real values of ε for household refrigerators: 2–4.