The question of how the (efficiency) of an ideal heat engine will change while simultaneously reducing the temperatures of the heater and refrigerator by the same value is a classical problem of thermodynamics. Many students and engineers mistakenly believe that the temperature difference remains unchanged, which means the machine's operating efficiency should not change. However, mathematical analysis of the Carnot formula shows the opposite: even if the temperature difference is maintained at 50 degrees, absolute values play a critical role. efficiency factor (efficiency) of an ideal heat engine while simultaneously reducing the temperatures of the heater and refrigerator by the same amount is a classical problem of thermodynamics. Many students and engineers mistakenly believe that the temperature difference remains unchanged, which means the machine's operating efficiency should not change. However, mathematical analysis of Carnot's formula shows the opposite: even if the temperature difference remains 50 degrees, absolute values play a critical role.
To understand the essence of the process, it is necessary to consider the nature itself Carnot cycle, which sets the maximum possible theoretical efficiency limit for any heat engines. A decrease in the absolute temperatures of both tanks by 50 Kelvin leads to a non-linear change in the final efficiency, and this shift always occurs in the direction of increasing efficiency, which may not seem obvious at first glance. In this article we will analyze the physical meaning of this phenomenon, carry out mathematical calculations and consider practical aspects important for understanding the operation of refrigeration units and engines.
It is important to immediately note that the scenario under consideration assumes ideal conditions, where the working fluid is an ideal gas, and the processes proceed reversibly. In real conditions, such as the operation of household refrigerators or automobile engines, additional factors of friction and heat loss come into force, but the fundamental pattern, which will be discussed below, remains the basic principle for the construction of any thermodynamic systems.
Theoretical foundations of the Carnot cycle
The basis for understanding the efficiency of heat engines is the formula derived by Sadi Carnot. It states that maximum efficiency depends solely on the temperatures of the heater ($T_1$) and refrigerator ($T_2$), expressed in the absolute Kelvin scale. The formula is as follows: $\eta = 1 - \frac{T_2}{T_1}$. It is the ratio of these quantities that determines what part of the heat can be converted into useful mechanical work.
When we talk about decreasing both temperatures by the same amount $\Delta T$ (in our case 50 K), we are actually shifting the operating range to lower absolute values. Since the denominator of the fraction ($T_1$) decreases and the numerator ($T_2$) also decreases, but was originally less than the denominator, mathematical logic dictates that the result of the division changes. This leads to the fact that the fraction $\frac{T_2 - \Delta T}{T_1 - \Delta T}$ becomes smaller than the original fraction $\frac{T_2}{T_1}$, and therefore one minus this fraction gives a larger value.
For clarity, imagine that we have two modes of engine operation. In the first mode, the temperatures are 500 K and 300 K. In the second mode, we reduce both temperatures by 50 K, getting 450 K and 250 K. The temperature difference in both cases is 200 K, however, the relative proportion of “cold” energy in the second case becomes smaller in relation to “hot”.
⚠️ Attention: All calculations in thermodynamics must be performed exclusively on the absolute temperature scale (Kelvins). Using degrees Celsius in Carnot's formula will lead to catastrophically incorrect results, since the zero of the Celsius scale does not correspond to the absence of thermal motion of molecules.
Thus, the theoretical basis confirms that shifting the temperature range down while maintaining the delta increases the theoretical limit of efficiency. This is a fundamental property that engineers try to take into account when designing systems operating under extreme conditions.
Mathematical proof of changes in efficiency
To strictly prove that efficiency will increase, we will conduct a comparative analysis of two states of the system. Let us denote the initial temperature of the heater as $T_1$, and the initial temperature of the refrigerator as $T_2$. The initial efficiency is $\eta_1 = 1 - \frac{T_2}{T_1}$. After the temperature decreases by $A = 50$ K, the new temperatures will be $T_1 - A$ and $T_2 - A$. The new efficiency will be equal to $\eta_2 = 1 - \frac{T_2 - A}{T_1 - A}$.
We need to compare two fractions: $\frac{T_2}{T_1}$ and $\frac{T_2 - A}{T_1 - A}$. A well-known mathematical property of a proper fraction (where the numerator is less than the denominator, which is true for refrigerator and heater temperatures) states that if the same positive number is subtracted from the numerator and denominator of a proper fraction, the fraction will decrease. Since $T_2 < T_1$, then the fraction $\frac{T_2}{T_1}$ is correct.
Why does the fraction decrease?
Mathematically, this can be proven by reduction to a common denominator. The difference $(\frac{T_2}{T_1}) - (\frac{T_2 - A}{T_1 - A})$ is equal to $\frac{T_2(T_1 - A) - T_1(T_2 - A)}{T_1(T_1 - A)}$. Expanding the parentheses in the numerator, we get $T_1 T_2 - T_2 A - T_1 T_2 + T_1 A = A(T_1 - T_2)$. Since $T_1 > T_2$ and $A > 0$, then the numerator is positive, which means the original fraction is greater than the new one.
Consequently, if the fraction $\frac{T_2 - A}{T_1 - A}$ is less than $\frac{T_2}{T_1}$, then the expression $1 - \frac{T_2 - A}{T_1 - A}$ will be greater than $1 - \frac{T_2}{T_1}$. This means that $\eta_2 > \eta_1$. Efficiency will inevitably increase.
Let's consider a numerical example for securing the material. Let $T_1 = 400$ K, $T_2 = 300$ K.
Initial efficiency: $\eta_1 = 1 - 300/400 = 1 - 0.75 = 0.25$ (or 25%).
After reduction by 50 K: $T_1' = 350$ K, $T_2' = 250$ K.
New efficiency: $\eta_2 = 1 - 250/350 \approx 1 - 0.714 = 0.286$ (or 28.6%).
We see a clear increase in efficiency by 3.6 percent point.
Physical interpretation of the process
Why does this happen? The physical meaning lies in the “quality” of energy. At lower absolute temperatures, that same 50 degree difference (or any other delta) represents a larger relative portion of the system's total thermal reserve. We seem to be “compressing” the range, making the cold part relatively even colder compared to the hot part than it was in the high-temperature regime.
There is a concept in thermodynamics entropy. When heat is transferred from the heater to the working fluid, the entropy of the system changes. By lowering temperatures, we reduce the amount of "unavailable" energy that must inevitably be given to the refrigerator. A smaller proportion of heat goes into the refrigerator relative to that received from the heater, which increases useful work.
However, it is worth considering that in real physical systems, cooling the heater often means a decrease in the intensity of chemical reactions (in an internal combustion engine) or a change in the properties of the working fluid. Therefore, although the theoretical efficiency of the Carnot cycle increases, the actual engine power may decrease due to a decrease in gas pressure or liquid viscosity.
Comparative analysis of temperature conditions
To better understand the scale of the changes, consider a table showing how the efficiency changes at different starting temperatures if we reduce them by 50 K. This will help to see the nonlinearity process.
| Mode | $T_1$ (start), K | $T_2$ (start), K | Efficiency start, % | Efficiency after -50K, % | Growth, % |
|---|---|---|---|---|---|
| Low temp. | 350 | 300 | 14.3 | 20.0 | +5.7 |
| Average temp. | 500 | 400 | 20.0 | 23.5 | +3.5 |
| High temp. | 800 | 600 | 25.0 | 27.3 | +2.3 |
| Extreme | 1000 | 800 | 20.0 | 21.6 | +1.6 |
The table shows that the highest percentage increase in efficiency is observed in the region low temperatures. When the absolute values of $T_1$ and $T_2$ are small, subtracting 50 K is a significant fraction of the total, which greatly changes the relationship. There is also an effect in the region of high temperatures, but it is less pronounced in percentage terms.
This analysis is important for cryogenic technology and systems operating with liquefied gaseswhere temperatures are initially low. There, each decrease in the temperature of the refrigerator gives a colossal gain in the efficiency of the cycle.
Practical application in refrigeration machines
In the context of the work of household and industrial refrigerators the considered principle works “the other way around”, since a refrigerator is reverse cycle heat engine. Here we are not interested in the engine efficiency, but in the cooling coefficient. However, the physics of the process of changing condensation and evaporation temperatures remains similar.
If we reduce the ambient temperature (which acts as a heater for the refrigerator cycle) and the temperature inside the chamber (refrigerator) at the same time, then the compressor will need to do less work to pump the same amount of heat. The difference in refrigerant pressure at the inlet and outlet of the compressor will decrease.
- 🌡️ Reducing the condensation temperature (radiator at the rear) always has a beneficial effect on energy consumption.
- ❄️ Reducing the evaporation temperature (inside the chamber) at a fixed external temperature, on the contrary, reduces efficiency.
- ⚙️ Simultaneous reduction of both parameters (for example, in winter in an unheated room) can lead to incorrect operation of thermostats.
It is important to understand that modern refrigerators are equipped with complex electronics. If you decide to experimentally test this law by reducing the temperature in the room and inside the refrigerator, you may encounter the fact that the system will stop turning on or, conversely, will work without stopping.
⚠️ Attention: Experiments with artificial cooling of the refrigerator (for example, installing it on the balcony in winter) can lead to freezing of the oil in the compressor and failure of the equipment. The design of most household models is designed to operate at ambient temperatures not lower than +10...+15°C.
Limitations and real operating conditions
Despite the beauty of Carnot's theory, in the real world we are faced with limitations of materials and properties of substances. Reducing the heater temperature in an internal combustion engine below a certain limit will cause the fuel to simply stop igniting or burning efficiently.
In addition, there are heat loss through cylinder walls, piston friction and aerodynamic drag. These factors do not directly depend on the Carnot temperature cycle, but they “eat up” the gain obtained from changing the temperature regime. In some cases, a decrease in temperature can even reduce the overall efficiency of a real machine due to increased friction losses (thickening of the lubricant).
☑️ Factors that reduce the actual efficiency
It is also worth mentioning the absolute zero limit. We cannot reduce the temperature indefinitely. As we approach 0 Kelvin ($−273.15$ °C), the properties of substances change dramatically: gases turn into liquids and solids, viscosity disappears (superfluidity), and electrical conductivity changes. The Carnot formula ceases to be applicable in its usual form when the working fluid changes its state of aggregation.
Final conclusions and recommendations
To summarize, we can confidently say: reducing the temperatures of the heater and refrigerator by 50 K leads to increasing the efficiency of an ideal heat engine. This is an indisputable fact, confirmed mathematically and physically. The ratio $\frac{T_2}{T_1}$ decreases, which leads to an increase in useful work.
However, when designing real systems, engineers must find a balance. Temperatures that are too low may require expensive materials, special lubricants and complex starting systems. The theoretical gain of 3-5% can be offset by the cost of maintenance and the complexity of the design.
It is important for students and researchers to remember this principle: in thermodynamics, not only temperature differences are important, but also their absolute values. Shifting the range to lower temperatures (while maintaining the delta) is beneficial for the efficiency of the cycle.
We hope that this analysis has helped you gain a deeper understanding of thermodynamic processes. Remember that physics is a science where intuition sometimes fails, but strict mathematical calculations always give the correct answer.
Why does efficiency increase if the temperature difference remains the same?
Efficiency depends on relationships temperatures, and not just on their difference. Decreasing both temperatures by the same amount reduces the denominator of the fraction ($T_1$) to a greater relative extent than the numerator ($T_2$), since $T_1 > T_2$. This leads to a decrease in the value of the entire fraction $\frac{T_2}{T_1}$, and, consequently, to an increase in $1 - \frac{T_2}{T_1}$.
Can the efficiency become greater than 100%?
No, this is impossible according to the second law thermodynamics. Even if we cool the refrigerator to absolute zero (which is unattainable), the efficiency will become equal to 1 (or 100%). Exceeding this value would mean creating a perpetual motion machine, which is contrary to the fundamental laws of physics.
How does this apply to a conventional car engine?
In a car in winter (at low air temperatures), the theoretical efficiency of the cycle could be higher, but in practice a cold engine has huge losses due to friction and self-heating. Therefore, the actual fuel consumption in winter is higher, despite the colder “refrigerator” (atmosphere).