D = T dS/dt = Jq Fq = L-1 Jq2. For a finite time At, the entropy increase becomes AS = (dS/dt) At = (LT)-1 JQ2 At = (LTAt)-1 (AQ)2,
The magnitude of the currents is given by (4.9), and their conventional signs may be inferred from Fig. 4.2. The specific energy content of the incoming mass flow, win, and of the outgoing mass flow, w0ut, are the sums of potential energy, kinetic energy and enthalpy. The significance of the enthalpy to represent the thermodynamic energy of a stationary flow is established by Bernoulli's theorem (Pippard, 1966). It states that for a stationary flow, if heat conduction can be neglected, the enthalpy is constant along a streamline. For the uniform mass flows assumed for the device in Fig. 4.2, the specific enthalpy, h, thus becomes a property of the flow, in analogy with the kinetic energy of motion and, for example, the geopotential energy, w = wvo0 + wkm + h. The power output may be written E = - Jo- Fo - Jq -F^
with the magnitude of currents given by (4.9) and the generalised forces given by
(torque), (electric field)
corresponding to a mechanical torque and an electric potential gradient. The rate of entropy creation, i.e. the rate of entropy increase in the surroundings of the conversion device (as mentioned, the entropy inside the device is constant in the steady-state model), is dS/di — (Tref) JQ,out T JQ,in + Jm (sm,out Sm,in)/
where sm in is the specific entropy of the mass (fluid, gas, etc.) flowing into the device, and smout is the specific entropy of the outgoing mass flow. Jqo0u1 may be eliminated by use of (4.12), and the rate of dissipation obtained from (4.7),
jQ,in (1-Tref/T) + Jm Win - ™out - Tref (Sm,in - Smfiut)) - E — maX(E) -E. (4.16)
The maximum possible work (obtained for dS/dt = 0) is seen to consist of a Carnot term (closed cycle, i.e. no external flows) plus a term proportional to the mass flow. The dissipation (4.16) is brought in the Onsager form (4.11),
by defining generalised forces
Fm — Win — Wout — Tref (sm,in — Sm,out) (4.18)
in addition to those of (4.15).
The efficiency with which the heat and mass flow into the device is converted to power is, in analogy to (4.4), E
J Q, in + JmWin where the expression (4.16) may be inserted for E. This efficiency is sometimes referred to as the "first law efficiency", because it only deals with the amounts of energy input and output in the desired form and not with the "quality" of the energy input related to that of the energy output.
In order to include reference to the energy quality, in the sense of the second law of thermodynamics, account must be taken of the changes in entropy taking place in connection with the heat and mass flows through the conversion device. This is accomplished by the "second law efficiency", which for power-generating devices is defined by n ( 2. law) = E = - Je ' Fg + Jq ' Fq , (4.20)
where the second expression is valid specifically for the device considered in Fig. 4.2, while the first expression is of general applicability, when max(E) is taken as the maximum rate of work extraction permitted by the second law of thermodynamics. It should be noted that max(E) depends not only on the system and the controlled energy inputs, but also on the state of the surroundings.
Conversion devices for which the desired energy form is not work may be treated in a way analogous to the example in Fig. 4.2. In the form (4.17), no distinction is made between input and output of the different energy forms. Taking, for example, electrical power as input (sign change), output may be obtained in the form of heat or in the form of a mass stream. The efficiency expressions (4.19) and (4.20) must be altered, placing the actual input terms in the denominator and the actual output terms in the numerator. If the desired output energy form is denoted W, the second law efficiency can be written in the general form n<2. iuw) = w/ max (W). (4.21)
For conversion processes based on other principles than those considered in the thermodynamic description of phenomena, alternative efficiencies could be defined by (4.21), with max(W) calculated under consideration of the non-thermodynamic types of constraints. In such cases, the name "second law efficiency" would have to be modified.
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