Carnot process

The Carnot process played an outstanding role for the development of thermodynamics. Usually the Carnot process is exemplified for an ideal gas, in the p-V diagram. We now show that the Carnot is a rectangular process. 

In textbooks, often the Carnot cycle is explained based on a thermal reservoir at the higher temperature, at the working gas, and a thermal reservoir at a lower 

Example 9.1. We try as an exercise the Carnot process in the T - g diagram, i.e., in Fig. 9.5 we plot the amount of heat over the temperature. The machine transfers now in the first step (a) isothermally heat energy from the reservoir (R2). Thus, we must vertically in the diagram lift off. In the second step without transfer of heat the temperature is lowered. The heat content of the gas thus remains constant. That means we are forced to go horizontal to the left. Next, we must go down until the initial temperature is reached. 

Otherwise, we could not come back to the starting state in an isothermal process. However, this means that in step (d) the same amount of heat is given to the reservoir (7 i) as withdrawn from reservoir (/ 2). The individual processes are thus compelling, but the total process does not balance out itself energetically in this representation. If the same energy is taken from the reservoir as the other reservoir is transferred, then no more energy remains left over for work. 

Example 9.2. We draw now the Carnot process for an ideal gas in an energy diagram. We draw the steps in the T5—pV diagram. First, observe that pV = n/ 7 for an ideal gas. Thus, we may use a parametric plot with x = nRT and y = 75. At constant mol number, in the isothermal steps, the temperature is constant and we have a vertical straight line. In the isentropic steps, the entropy 5 is constant, and we have a straight line passing the origin. 

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Substitution gives the expression for the efficiency of a Carnot process ... 

This is illustrated in Fig. 2 which compares the theoretical efficiencies as a function of temperature of a fuel cell supplied with pure hydrogen and pure oxygen and a Carnot process as the most efficient heat cycle operating between a heat reservoir at T and one at 300 K. Only at temperatures higher than 1000 °C is... 

Example 2.1. In the first step of the Carnot process, thermal energy will flow into a gas. This inflow is accompanied with an expansion of the gas, in order to run the process isothermal. In the last step of the Carnot process, the gas is compressed adiabatically. Strictly, we mean that the compression occurs at constant entropy, dS = 0. Here an increase of the temperature occurs, even when the entropy remains... 

The isothermal process is a process in that dT = 0. In the case of an ideal gas, the expansion may achieved in contact with a thermal reservoir to run an isothermal process. This process occurs as a basic step in the Carnot process, and it is a reversible process. 

We made here again use of the law of Schwarz. We can represent the process now in refined representation either in F—h diagram or in tx-n diagram, cf. Fig. 9.2. As the process of the paddle wheel can be represented with two different energy conjugated variables, also, for example, the Carnot process can be represented in... 

We discuss now the question how we can achieve that the machine can take a certain amount of energy out of the reservoir. We discuss both the paddle wheel and the Carnot process. Subsequently we want to generalize the treatment. 

Relationship (9.1) is applied to the machine in the Carnot process. During the isothermal expansion step we must deliver that quantity entropy, in the course of expansion. If we would not do so, then we eould not hold the temperature in the machine. The machine requires thus the subsequent delivery of the entropy. The entropy becomes in any case larger after Maxwell s equation, if we increase the volume. 

An important prerequisite is that the machine can exchange the energy forms under consideration. Otherwise, such a process is not possible. To illustrate the above statement we want to perform the Carnot process with an incompressible body. The incompressible body has the function of state... 

The pathway differences receive further attention in Figure 5.5. Shown are the probability distributions constructed for the temperature states. The results reflect, unsurprisingly, that the temperature distribution is considerably skewed for the Carnot process. A system so programmed would pose less uncertainty in query-and-measurement exercises. The bias is marked because T is at its minimum or maximum value for sizable portions of the cycle. By contrast, the diamond path reflects a greater range and more even dispersion of temperature states. The information values in Figure 5.5 apply... 

The quantity of heat Q is from a heat source with a constant temperature Tu and Q2 = Q W, and is delivered to a heat sink with a constant temperature T2. (See Fig. 4-5.) This imaginary, ideal process is called a Carnot process. It is an example of a reversible process because it can also proceed in the opposite direction and both the system and its surroundings can be returned to their original state. Thus using a Carnot process, we can determine the zero point of the absolute temperature scale in accordance with Eq. 4-15. 

Let us look at the entropy change in the surroundings of a Carnot process. The heat quantity Qi enters the process at temperature T. Since Ti is constant,... 

Fig. 4-7. Carnot process or reversible cycle. Starting at point A, the cycle consists of an isentropic compression, an isothermal expansion, an isentropic expansion, and an isothermal compression. All steps are reversible. The open arrows symbolize the heat flow in a real technical process where there are temperature drops in the heat transfer to and from the process medium. The entropy flows from the heat source to the heat sink respectively, correspond to the hatched lines in that case.

Carnot process producing all the refrigeration at the condensation point. Since with air the heat absorbed for precooling is approximately equal to the heat of condensation, a great reduction in power consumption can be obtained by precooling even if this process has a poor efficiency. 

The second law of thermodynamics stems from the studies in the 1800 s of heat engines, and in the this period s theories on the motive power of heat. Therefore, the second law is introduced, along with the concept of entropy, through the Carnot cycle for a heat engine operating with an ideal gas. The energy considerations used in the Carnot process are universal and thus they lead to general conditions of equilibrimn for thermodynamic systems. 

The Carnot cycle is a reversible cyclic process during which a thermodynamic system cycles between two heat reservoirs of different temperatures. A Carnot process with an ideal gas will be described here as an illustration, but as will be shown later, there is no restriction on the nature of the system. 

In a reversible Carnot process with an ideal gas, the sum of the supplied quantities of heat Q divided by the thermodynamic temperature T by which the heat is supplied is equal to zero... 

In accordance with (4.9), the thermal efficiency t of a reversible Carnot cycle (4.9) solely depends on the ratio between the thermodynamic temperature in in the two heat reservoirs, in this case T and T3. This property of the Carnot process has made possible the definition of an absolute temperature scale, which is independent of the thermometer substances used (Lord Kelvin, 1854). 

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