## Info

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Cold reservoir

Fig. 4.1 A schematics of a heat engine or heat pump. The heat pump is a heat engine running in reverse.

The reservoirs supply or receive heat without alteration of their temperatures. Heat flows in the reservoirs are reversible whether or not the engine is.

The circle with the arrows in Fig. 4.1 is a shorthand representation of the heat engine. It is intended to signify that the working substance (a fluid such as an ideal gas or water) moves through many thermodynamic states in a never-ending cyclic process. The detailed structure of the heat engine can vary greatly, but the simplest version contains the following four steps:

1. one in which heat absorbed isothermally from the high-temperature reservoir

2. the next in which work is produced adiabatically

3. followed by isothermal rejection of heat to the low-temperature reservoir

4. the last in which work is done on the working substance to return it to the state at the start of step 1

The heat engine can operate in either of two ways: i) as a single device that sequentially moves through the four processes described above, or ii) with a fluid flowing through four distinct devices, each assigned to one of the four steps.

### 4.1.1 Single device - sequential states - ideal gas

The sequential type is illustrated in Fig. 4.2 by a single piston-cylinder that performs each of the four steps. (see Ref. 1, pp. 36 - 40 for a explanation of this cycle)

Fig. 4.2 A heat engine consisting of a single piston/cylinder performing four operations.

Fig. 4.2 A heat engine consisting of a single piston/cylinder performing four operations.

Proceeding clockwise in the diagram follows the device in time. If the gas in the cylinder were ideal and if the four steps were conducted reversibly, the cycle would appear in the process diagrams shown in Fig. 4.3. The cycle consists of two isotherms (like the one in Fig. 3.2) and a pair of isentropes (see Fig. 3.4). Work is exchanged with the surroundings in each step. The net work produced by the cycle is:

W = W1.2 + W2-3 - W3-4 -W4-1. Since each of the component work terms is the integral of pdV, W is the area inside the p-V graph in Fig. 4-3. Because each step is assumed to be reversible, the work produced by an ideal gas in the piston/cylinder is:

The work terms originate from Eqs (3.5) and (3.16). The net heat of the cycle is:

Q = Th(S2 - S1) - Tl(S3 - S4) = (Th - Tl)(S2 - S3) (4.2)

Here TH is the temperature of the hot reservoir that delivers heat during the 1 - 2 step and TL is the temperature of the cold reservoir that receives heat from the engine during the 3 - 4 step.

The same cyclical process is represented in T- S coordinates on the right in Fig. 4-3. The net heat exchanged with the surroundings is Q = Q1-2 - Q3-4. Because each of the component heat exchange terms are integrals of TdS, the shaded area inside the rectangle is Q.

In describing this power cycle, the convention for the signs of Q and W are different from that used for the 1st law (Sect. 1.8), in which heat added to the system is positive and work done by the system is positive. For analyzing power cycles, heat and work terms are positive in the direction in which they act (indicated by arrows). In this way, a welter of minus signs is avoided. Thus Q1-2 is written as QH because heat is delivered to the gas from a high-temperature reservoir, and for a similar reason, Q3-4 is denoted as QL, heat rejected to a low-temperature reservoir.

Fig. 4.3 Process diagrams for the power cycle of Fig. 4.2

### 4.1.2 Four devices - circulating fluid - water

The flow-cycle heat engine shown in Fig. 4.4 contains the four processes in separate units that act on the circulating working fluid (see also Ref. 1, Chap. 4). This heat engine is an idealized steam power plant, either nuclear or fossil. The boiler (or steam generator in a nuclear plant) receives heat from the primary source. The condenser rejects heat to a sink such as a river, lake, or cooling tower. The turbine produces shaft (i.e., rotating) work, which is then converted to electric power with high efficiency. However, some of the turbine power is consumed internally by the pump needed to keep the fluid circulating in the loop. The net work of the cycle is W = W2-3 - W4-1.

Fig. 4.4 A heat engine based on a fluid circulating through four heat and work devices (step 1-2 converts saturated liquid to saturated vapor)

Process diagrams analogous to those in Fig. 4.3 for the sequential piston/cylinder power cycle are shown in Fig. 4.5 for water/steam as the working fluid. The areas of the 4-sided process figures represent the net work W on the left and the net heat Q on the right.

Fig. 4.5 Process diagrams for the power cycle of Fig 4.4 on the EOS of water

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