4.1.1.1 Irreversible thermodynamics

The degree of irreversibility is measured in terms of the rate of energy dissipation,

$D=TdS/dt,$ (4.5)

(4.5)

where dS/dt is the entropy production of the system while it is held at the constant temperature T (i.e., T may be thought of as the temperature of a large heat reservoir, with which the system is in contact). In order to describe the nature of the dissipation process, the concept of free energy is introduced (cf. Prigogine, 1968; Callen, 1960).

The free energy, G, of a system is defined as the maximum work that can be drawn from the system under conditions where the exchange of work is the only interaction between the system and its surroundings. A system of this kind is said to be in thermodynamic equilibrium if its free energy is zero.

Consider now a system divided into two subsystems, a small one with extensive variables (i.e., variables proportional to the size of the system) U, S, V, etc., and a large one with intensive variables T_ref, P_ref, etc., which is initially in thermodynamic equilibrium. The terms small system and large system are meant to imply that the intensive variables of the large system (but not its extensive variables U_ref, S_ref, etc.) can be regarded as constant, regardless of the processes by which the entire system approaches equilibrium.

This implies that the intensive variables of the small system, which may not even be defined during the process, approach those of the large system when the combined system approaches equilibrium. The free energy, or maximum work, is found by considering a reversible process between the initial state and the equilibrium. It equals the difference between the initial internal energy, U_init=U+U_ref, and the final internal energy, U_eq, or it may be written (all in terms of initial state variables) as

$G=U-T_{ref}S+P_{ref}V,$ (4.6)

(4.6)

plus terms of the form Σμ_i,refn_i if chemical reactions are involved, and similar generalizations in case of electromagnetic interactions, etc.

If the entire system is closed, it develops spontaneously toward equilibrium through internal, irreversible processes, with a rate of free energy change

$\frac{\mathrm{d}G}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}(U_{init}-U_{eq}(t))=\left(\frac{\partial }{\partial S(t)}{U}_{eq}(t)\right)\frac{\mathrm{d}S(t)}{\mathrm{d}t},$

assuming that the entropy is the only variable. S(t) is the entropy at time t of the entire system, and U_eq(t) is the internal energy that would be possessed by a hypothetical equilibrium state defined by the actual state variables at time t, i.e., S(t), etc. For any of these equilibrium states, ∂U_eq(t)/∂S(t) equals T_ref according to (4.3), and by comparison with (4.5) it is seen that the rate of dissipation can be identified with the loss of free energy, as well as with the increase in entropy,

$D=-\mathrm{d}G/\mathrm{d}t=T_{ref}\mathrm{d}S(t)/\mathrm{d}t.$ (4.7)

(4.7)

For systems met in practice, there are often constraints preventing the system from reaching the absolute equilibrium state of zero free energy. For instance, the small system considered above may be separated from the large one by walls that keep the volume V constant. In such cases, the available free energy (i.e., the maximum amount of useful work that can be extracted) becomes the absolute amount of free energy, (4.6), minus the free energy of the relative equilibrium that the combined system can be made to approach in the presence of the constraint. If the extensive variables in the constrained equilibrium state are denoted U⁰, S⁰, V⁰, etc., then the available free energy becomes

$\mathrm{\Delta }G=(U-U^0)–T_{ref}(S-S^0)+P_{ref}(V-V^0),$ (4.8)

(4.8)

eventually with the additions involving chemical potentials, etc. In (4.6) or (4.8), G is called the Gibbs potential. If the small system is constrained by walls, so that the volume cannot be changed, the free energy reduces to the Helmholtz potential U−TS, and if the small system is constrained so that it is incapable of exchanging heat, the free energy reduces to the enthalpy (4.1).

The corresponding forms of (4.8) give the maximum work that can be obtained from a thermodynamic system with the given constraints.

A description of the course of an actual process as a function of time requires knowledge of equations of motion for the extensive variables, i.e., equations that relate the currents, such as

$\begin{array}{l}{J}_s=\mathrm{d}S/\mathrm{d}t(\mathrm{entropy}\mathrm{flow}\mathrm{rate})\mathrm{or}{J}_Q=\mathrm{d}Q/\mathrm{d}t(\mathrm{heat}\mathrm{flow}\mathrm{rate}),\\ J_m=\mathrm{d}m/\mathrm{d}\mathit{t}(\mathrm{mass}\mathrm{flow}\mathrm{rate})\mathrm{or}{J}_{\theta }=\mathrm{d}\theta /\mathrm{d}t(\mathrm{angular}\mathrm{velocity}),\\ J_q=\mathrm{d}q/\mathrm{d}t=I(\mathrm{charge}\mathrm{flow}\mathrm{rate}\mathrm{or}\mathrm{electrical}\mathrm{current}),\mathrm{etc}.\end{array}$ (4.9)

(4.9)

to the (generalized) forces of the system. As a first approximation, the relation between the currents and the forces may be taken as linear (Onsager, 1931),

${\mathit{J}}_i=\sum _j{L}_{ij}{\mathbf{F}}_j.$ (4.10)

(4.10)

The direction of each flow component is J_i/J_i. The arbitrariness in choosing the generalized forces is reduced by requiring, as did Onsager, that the dissipation be given by

$D=-\mathrm{d}G/\mathrm{d}t=\sum _i{\mathit{J}}_i·{\mathit{F}}_i.$ (4.11)

(4.11)

Examples of the linear relationships (4.10) are Ohm’s law, stating that the electric current J_q is proportional to the gradient of the electric potential (F_q α grad ϕ), and Fourier’s law (3.34) for heat conduction or diffusion, stating that the heat flow rate E^sens=J_Q is proportional to the gradient of the temperature.

Considering the isothermal expansion process required in the Carnot cycle (Fig. 4.1), heat must be flowing to the system at a rate J_Q=dQ/dt, with J_Q=LF_Q according to (4.10) in its simplest form. Using (4.11), the energy dissipation takes the form

$D=T\mathrm{d}S/\mathrm{d}t=J_Q{F}_Q=L^{-1}{J_Q}^2.$

For a finite time Δt, the entropy increase becomes

$\mathrm{\Delta }S=(\mathrm{d}S/\mathrm{d}t)\mathrm{\Delta }t={(LT)}^{-1}{J_Q}^2\mathrm{\Delta }t={(LT\mathrm{\Delta }t)}^{-1}{(\mathrm{\Delta }Q)}^2,$

so that in order to transfer a finite amount of heat ΔQ, the product ΔS Δt must equal the quantity (LT)⁻¹ (ΔQ)². For the process to approach reversibility, as the ideal Carnot cycle should, ΔS must approach zero, which is seen to imply that Δt approaches infinity. This qualifies the statement made in the beginning of this subsection that, in order to go through a thermodynamic engine cycle in a finite time, one has to give up reversibility and accept a finite amount of energy dissipation and an efficiency that is smaller than the ideal (4.4).

4.1.1.2 Efficiency of an energy conversion device

A schematic picture of an energy conversion device is shown in Fig. 4.2, and it is sufficiently general to cover most types of converters in practical use (Angrist, 1976; Osterle, 1964). There is a mass flow into the device and another one out from it, as well as incoming and outgoing heat flow. The work output may be in the form of electric or rotating shaft power.

It may be assumed that the converter is in a steady state, implying that the incoming and outgoing mass flows are identical and that the entropy of the device itself is constant, i.e., that all entropy created is being carried away by the outgoing flows.

From the first law of thermodynamics, the power extracted, E, equals the net energy input,

$E=J_{Q,in}-J_{Q,out}+J_m(w_{in}-w_{out}).$ (4.12)

(4.12)

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, w_in, and of the outgoing mass flow, w_out, 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,

$w=w^{\mathit{pot}}+w^{kin}+h.$ (4.13)

(4.13)

The power output may be written

$E=–{\mathit{J}}_{\theta }·{\mathit{F}}_{\theta }-{\mathit{J}}_q·{\mathit{F}}_q,$ (4.14)

(4.14)

with the magnitude of currents given by (4.9) and the generalized forces given by

$\begin{array}{ll}{\mathbf{F}}_{\theta }={\int }^{}\mathit{r}\times d{\mathit{F}}_{mech}(\mathit{r})\hfill & (\mathrm{torque}),\hfill \\ {\mathbf{F}}_q=-\mathrm{grad}(\varphi )\hfill & (\mathrm{electric}\mathrm{field})\hfill \end{array}$ (4.15)

(4.15)

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

$\mathrm{d}S/dt={(T_{ref})}^{-1}{J}_{Q,out}-T^{-1}{J}_{Q,in}+J_m(s_{m,out}-s_{m,in}),$

where s_m,in is the specific entropy of the mass (fluid, gas, etc.) flowing into the device, and s_m,out is the specific entropy of the outgoing mass flow. J_Q,out may be eliminated by use of (4.12), and the rate of dissipation obtained from (4.7),

$\begin{array}{ll}D\hfill & =T_{ref}dS/dt=J_{Q,in}(1-T_{ref}/T)+J_m(w_{in}-w_{out}-T_{ref}(s_{m,in}-s_{m,out}))-E\hfill \\ \hfill & =\max (E)-E.\hfill \end{array}$ (4.16)

(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),

$D=J_{Q,in}{F}_{Q,in}+J_m{F}_m+{\mathit{J}}_{\theta }·{\mathit{F}}_{\theta }+{\mathit{J}}_q·{\mathit{F}}_q,$ (4.17)

(4.17)

by defining generalized forces

$\begin{array}{ll}{F}_{Q,in}\hfill & =1-T_{ref}/T,\hfill \\ F_m\hfill & =w_{in}-w_{out}-T_{ref}(s_{m,in}-s_{m,out})\hfill \end{array}$ (4.18)

(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),

$\eta =\frac{E}{J_{Q,in}+J_m{w}_{in}},$ (4.19)

(4.19)

where 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

${\eta }^{(2.law)}=\frac{E}{\max (E)}=-\frac{{\mathbf{J}}_{\theta }·{\mathbf{F}}_{\theta }+{\mathbf{J}}_q·{\mathbf{F}}_q}{J_{Q,in}{F}_{Q,in}+J_m{F}_m},$ (4.20)

(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 (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

$h^{(2.law)}=W/\max (W).$ (4.21)

(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.

4.2.1.1 Thermoelectric generators

One direct conversion process makes use of the thermoelectric effect associated with heating the junction of two different conducting materials, e.g., metals or semiconductors. If a stable electric current, I, passes across the junction between two conductors A and B, in an arrangement of the type depicted in Fig. 4.4, then quantum electron theory requires that the Fermi energy level (which may be regarded as a chemical potential μ_i) is the same in the two materials (μ_A=μ_B). If the spectrum of electron quantum states is different in the two materials, the crossing of negatively charged electrons or positively charged “holes” (electron vacancies) will not preserve the statistical distribution of electrons around the Fermi level,

$f(E)={(\exp ((E-{\mu }_i)/kT)+1)}^{-1},$ (4.23)

(4.23)

with E being the electron energy and k being Boltzmann’s constant. The altered distribution may imply a shift toward a lower or a higher temperature, such that the maintenance of the current may require addition or removal of heat. Correspondingly, heating the junction will increase or decrease the electric current. The first case represents a thermoelectric generator, and the voltage across the external connections (Fig. 4.4) receives a term in addition to the ohmic term associated with the internal resistance R_int of the rods A and B,

$\mathrm{\Delta }\varphi =-I{R}_{int}+{\int }_{T_{ref}}^T\alpha dT\prime ,$

The coefficient α is called the Seebeck coefficient. It is the sum of the Seebeck coefficients for the two materials A and B, and it may be expressed in terms of the quantum statistical properties of the materials (Angrist, 1976). If α is assumed independent of temperature in the range from T_ref to T, then the generalized electrical force (4.15) may be written

$F_q=R_{\mathit{int}}{J}_q-\alpha T{F}_{Q,in},$ (4.24)

(4.24)

where J_q and F_q,in are given in (4.9) and (4.18).

Considering the thermoelectric generator (Fig. 4.4) as a particular example of the conversion device shown in Fig. 4.2, with no mass flows, the dissipation (4.11) may be written

$D=J_Q{F}_Q+J_q{F}_q.$

In the linear approximation (4.10), the flows are of the form

$\begin{array}{cc}{J}_Q& =L_{QQ}{F}_Q+L_{Qq}{F}_q,\\ J_q& =L_{qQ}{F}_Q+L_{qq}{F}_q,\end{array}$

with L_Qq=L_qQ because of microscopic reversibility (Onsager, 1931). Considering F_Q and J_q (Carnot factor and electric current) as the “controllable” variables, one may solve for F_q and J_Q, obtaining F_q in the form (4.24) with F_Q=F_Q,in and

$L_{qq}={(R_{int})}^{-1}{L}_{qQ}=L_{Qq}=\alpha T/R_{int}.$

The equation for J_Q takes the form

$J_Q=CT{F}_Q+\alpha T{J}_q,$ (4.25)

(4.25)

where the conductance C is given by

$C=(L_{QQ}{L}_{qq}-L_{Qq}{L}_{qQ})/(LqqT).$

Using (4.24) and (4.25), the dissipation may be written

$D=CT{F_Q}^2+R_{\mathit{int}}{J_q}^2,$ (4.26)

(4.26)

and the simple efficiency (4.19) may be written

$\eta =\frac{-J_q{F}_q}{J_Q}=\frac{-J_q(R_{\mathrm{int}}{J}_q-\alpha T{F}_Q)}{CT{F}_Q+\alpha T{J}_q}.$ (4.27)

(4.27)

If the reservoir temperatures T and T_ref are maintained at a constant value, F_Q can be regarded as fixed, and the maximum efficiency can be found by variation of J_q. The efficiency (4.27) has an extremum at

$J_q=\frac{C{F}_Q}{\alpha }\left({\left(1+\frac{{\alpha }^{2}T}{R_{\mathit{int}}C}\right)}^{1/2}-1\right),$ (4.28)

(4.28)

corresponding to a maximum value

$\max (\eta )=F_Q\frac{{(1+ZT)}^{1/2}-1}{{(1+ZT)}^{1/2}+1},$ (4.29)

(4.29)

with Z=α²(R_intC)⁻¹. Equation (4.29) is accurate only if the linear approximation (4.10) is valid. The maximum second law efficiency is obtained from (4.29), dividing by F_Q [cf. (4.20)].

The efficiencies are seen to increase with temperature, as well as with Z. Z is largest for certain materials (A and B in Fig. 4.4) of semiconductor structure and is small for metals as well as for insulators. Although R_int is small for metals and large for insulators, the same is true for the Seebeck coefficient α, which appears squared. C is larger for metals than for insulators. Together, these features combine to produce a peak in Z in the semiconductor region. Typical values of Z are about 2×10⁻³ (K)⁻¹ at T=300 K (Angrist, 1976). The two materials A and B may be taken as a p-type and an n-type semiconductor, which have Seebeck coefficients of opposite signs, so that their contributions add coherently for a configuration of the kind shown in Fig. 4.4.

4.2.1.2 Thermionic generators

Thermionic converters consist of two conductor plates separated by vacuum or plasma. The plates are maintained at different temperatures. One, the emitter, is at a temperature T large enough to allow a substantial emission of electrons into the space between the plates due to the thermal statistical spread in electron energy (4.23). The electrons (e.g., of a metal emitter) move in a potential field characterized by a barrier at the surface of the plate. The shape of this barrier is usually such that the probability of an electron penetrating it is small until a critical temperature, after which it increases rapidly (“red-glowing” metals). The other plate is maintained at a lower temperature T_ref. In order not to have a buildup of space charge between the emitter and the collector, atoms of a substance like cesium may be introduced in this area. These atoms become ionized near the hot emitter (they give away electrons to make up for the electron deficit in the emitter material), and, for a given cesium pressure, the positive ions exactly neutralize the space charges of the traveling electrons. At the collector surface, recombination of cesium ions takes place.

The layout of the emitter design must allow the transfer of large quantities of heat to a small area in order to maximize the electron current responsible for creating the electric voltage difference across the emitter–collector system, which may be utilized through an external load circuit. This heat transfer can be accomplished by a so-called heat pipe—a fluid-containing pipe that allows the fluid to evaporate in one chamber when heat is added. The vapor then travels to the other end of the pipe, condenses, and gives off the latent heat of evaporation to the surroundings, after which it returns to the first chamber through capillary channels, under the influence of surface tension forces.

The description of the thermionic generator in terms of the model converter shown in Fig. 4.2 is very similar to that of the thermoelectric generator. With the two temperatures T and T_ref defined above, the generalized force F_Q is defined. The electrical output current, J_q, is equal to the emitter current, provided that back-emission from the collector at temperature T_ref can be neglected and provided that the positive-ion current in the intermediate space is negligible in comparison with the electron current. If the space charges are saturated, the ratio between ion and electron currents is simply the inverse of the square root of the mass ratio, and the positive-ion current will be a fraction of a percent of the electron current. According to quantum statistics, the emission current (and hence J_q) may be written

$J_Q=A{T}^2\exp (-e{\varphi }_e/(kT)),$ (4.30)

(4.30)

where ϕ_e is the electric potential of the emitter, eϕ_e is the potential barrier of the surface in energy units, and A is a constant (Angrist, 1976). Neglecting heat conduction losses in plates and the intermediate space, as well as light emission, the heat J_Q,in to be supplied to keep the emitter at the elevated temperature T equals the energy carried away by the electrons emitted,

$J_{Q,in}=J_q({\varphi }_e+\delta +2kT/e),$ (4.31)

(4.31)

where the three terms in brackets represent the surface barrier, the barrier effectively seen by an electron due to the space charge in the intermediate space, and the original average kinetic energy of the electrons at temperature T (divided by e), respectively.

Finally, neglecting internal resistance in plates and wires, the generalized electrical force equals the difference between the potential ϕ_e and the corresponding potential for the collector ϕ_c,

$-F_q={\varphi }_c-{\varphi }_e,$ (4.32)

(4.32)

with insertion of the above expressions (4.30) to (4.32). Alternatively, these expressions may be linearized in the form (4.10) and the efficiency calculated exactly as in the case of the thermoelectric device. It is clear, however, that a linear approximation to (4.30), for example, would be very poor.

4.2.2.1 Ericsson hot-air engine

The engines in the examples mentioned above were based on the Rankine or the Stirling cycle. It is also possible that the Ericsson cycle (which was actually invented for the purpose of solar energy conversion) will prove advantageous in some solar energy applications. It is based on a gas (usually air) as a working fluid and may have the layout shown in Fig. 4.5. In order to describe the cycle depicted in Fig. 4.3, the valves must be closed at definite times and the pistons must be detached from the rotating shaft (in contrast to the situation shown in Fig. 4.5), so that heat may be supplied at constant volume. In a different mode of operation, the valves open and close as soon as a pressure difference between the air on the two sides begins to develop. In this case, the heat is added at constant pressure, as in the Brayton cycle, and the piston movement is approximately at constant temperature, as in the Stirling cycle (this variant is not shown in Fig. 4.3).

The efficiency can easily be calculated for the latter version of the Ericsson cycle, for which the temperatures T_up and T_low in the compression and expansion piston cylinders are constant, in a steady situation. This implies that the air enters the heating chamber with temperature T_low and leaves it with temperature T_up. The heat exchanger equations (cf. section 4.2.3, but not assuming that T₃ is constant) take the form

$\begin{array}{c}-J_m^g{C}_p^f\mathrm{d}{T}^f(x)/\mathrm{d}x=h\prime (T^f(x)-T^g(x)),\\ J_m^{g}Cpg\mathrm{d}{T}^g(x)/\mathrm{d}x=h\prime (T^f(x)-T^g(x)),\end{array}$

where the superscript g stands for the gas performing the thermodynamic cycle, f stands for the fluid leading heat to the heating chamber for heat exchange, and x increases from zero at the entrance to the heating chamber to a maximum value at the exit. C_p is a constant-pressure heat capacity per unit mass, and J_m is a mass flow rate. Both these and h′, the heat exchange rate per unit length dx, are assumed constant, in which case the equations may be explicitly integrated to give

$\begin{array}{ll}{T}_{c,in}\hfill & =T_{c,out}-\frac{J_m^g{C}_p^g}{J_m^f{C}_p^f}(T_{up}-T_{low})\hfill \\ \hfill & =\frac{J_m^g{C}_p^g(1-H)}{J_m^g{C}_p^g+J_m^f{C}_p^f}{T}_{low}+\left(H+\frac{J_m^f{C}_p^f(1-H)}{J_m^g{C}_p^g+J_m^f{C}_p^f}\right)T_{c,out}.\hfill \end{array}$ (4.33)

(4.33)

Here T_c,out=T is the temperature provided by the solar collector or other heat source, and T_c,in is the temperature of the collector fluid when it leaves the heat exchanger of the heating chamber, to be re-circulated to the collector or to a heat storage connected to it. H is given by

$H=\exp (-h({(J_m^f{C}_p^f)}^{-1}+{(J_m^g{C}_p^g)}^{-1})),$

where h=∫h′ dx. Two equations analogous to (4.33) may be written for the heat exchange in the cooling chamber of Fig. 4.5, relating the reject temperature T_r,in and the temperature of the coolant at inlet, T_r,out=T_ref, to T_low and T_up. T_c,in may then be eliminated from (4.33) and T_r,in from the analogous equation, leaving two equations for determination of T_up and T_low as functions of known quantities, notably the temperature levels T and T_ref. The reason for not having to consider equations for the piston motion in order to determine all relevant temperatures is, of course, that the processes associated with the piston motion have been assumed to be isothermal.

The amounts of heat added, Q_add, and rejected, Q_rej, per cycle are

$Q_{add}=m{C}_p^g(T_{up}-T_{low})+n5T_{up}\log (V_{\mathit{max}}/V_{\mathit{min}}),$ (4.34a)

(4.34a)

$Q_{rej}=m{C}_p^g(T_{up}-T_{low})+n5T_{low}\log (V_{\mathit{max}}/V_{\mathit{min}})+Q_{rej}\prime ,$ (4.34b)

(4.34b)

where m is the mass of air involved in the cycle and n is the number of moles of air involved. $\mathcal{R}$ is the gas constant, and V_min and V_max are the minimum and maximum volumes occupied by the gas during the compression or expansion stages (for simplicity, the compression ratio V_max/V_min has been assumed to be the same for the two processes, although they take place in different cylinders). The ideal gas law has been assumed in calculating the relation between heat amount and work in (4.34), and Q′_rej represents heat losses not contributing to transfer of heat from working gas to coolant flow (piston friction, etc.). The efficiency is

$\eta =(Q_{add}-Q_{rej})/Q_{add},$

and the maximum efficiency that can be obtained with this version of the Ericsson engine is obtained for negligible Q′_rej and ideal heat exchangers providing T_up=T and T_low=T_ref,

$\max (\eta )=(1-T_{ref}/T)/\left(1+\frac{m{C}_p^g}{n5\log (V_{\mathit{max}}/V_{\mathit{min}})}(1-T_{ref}/T)\right).$ (4.35)

(4.35)

The ideal Carnot efficiency may even be approached, if the second term in the denominator can be made small. (However, to make the compression ratio very large implies an increase in the length of time required per cycle, such that the rate of power production may actually go down, as discussed in section 4.1.1). The power may be calculated by evaluating (4.34) per unit time instead of per cycle.

4.2.3.1 Heat pump operation

If heat of modest temperature is required, and a high-quality form of energy is available, a device is needed that can derive additional benefits from the high quality of the primary energy source. This can be achieved by using one of the thermodynamic cycles described in section 4.1.2, provided that a large reservoir of approximately constant temperature is available. The cycles (cf. Fig. 4.3) must be traversed “anti-clockwise,” such that high-quality energy (electricity, mechanical shaft power, fuel combustion at high temperature, etc.) is added, and heat energy is thereby delivered at a temperature T higher than the temperature T_ref of the reference reservoir from which it is drawn. Most commonly, the Rankine cycle, with maximum efficiencies bounded by (4.22), is used (e.g., in an arrangement of the type shown in Fig. 4.6). The fluid of the closed cycle, which should have a liquid and a gaseous phase in the temperature interval of interest, may be a fluorochloromethane compound (which needs to be recycled owing to climatic effects caused if it is released to the atmosphere). The external circuits may contain an inexpensive fluid (e.g., water), and they may be omitted if it is practical to circulate the primary working fluid directly to the load area or to the reference reservoir.

The heat pump contains a compressor, which performs step 7-5 in the Rankine cycle depicted in Fig. 4.3, and a nozzle, which performs step 2–9. The intermediate steps are performed in the two heat exchangers, giving the working fluid the temperatures T_up and T_low, respectively. The equations for determining these temperatures are of the form of equation (4.37). There are four such equations, which must be supplemented by equations for the compressor and nozzle performance, in order to allow a determination of all the unknown temperatures indicated in Fig. 4.6, for given T_ref, given load, and a certain energy expenditure to the compressor. Losses in the compressor are in the form of heat, which in some cases can be credited to the load area.

An indication of the departures from the Carnot limit of the coefficients of performance, ε^{heat pump}, encountered in practice, is given in Fig. 4.7, as a function of the temperature difference T_up−T_low at the heat pump and for selected values of T_up. In the interval of temperature differences covered, the ε^{heat pump} is about 50% of the Carnot limit (4.22b), but it falls more and more below the Carnot value as the temperature difference decreases, although the absolute value of the coefficient of performance increases.

Several possibilities exist for the choice of the reference reservoir. Systems in use for space heating or space cooling (achieved by reversing the flow in the compressor and expansion-nozzle circuit) have utilized river water, lake water, and seawater, and air, as well as soil as reservoirs. The temperatures of such reservoirs are not entirely constant, and it must therefore be acceptable that the performance of heat pump systems will vary with time. The variations are damped if water or soil reservoirs at sufficient depth are used, because weather-related temperature variations disappear as one goes just a few meters down into the soil. Alternative types of reservoirs for use with heat pumps include city waste sites, livestock manure, ventilation air from households or from livestock barns (where the rate of air exchange has to be particularly high), and heat storage tanks connected to solar collectors.

With solar heating systems, the heat pump may be connected between the heat store and the load area (whenever the storage temperature is too low for direct circulation), or it may be connected between the heat store and the collector, so that the fluid let into the solar collector is cooled in order to improve the collector performance. Of course, a heat pump operating on its own reservoir (soil, air, etc.) may also provide the auxiliary heat for a solar heating system of capacity below the demand.

The high-quality energy input to the compressor of a heat pump may also come from a renewable resource, e.g., by wind or solar energy conversion, either directly or via a utility grid carrying electricity generated by renewable energy resources. As for insulation materials, concern has been expressed over the use of CFC (Chloro-Fluoro-Carbon) gases in the processing or as a working fluid, and substitutes believed to have less negative impacts have been developed.

4.2.3.2 Heat exchange

A situation like the one depicted in Fig. 4.8 is often encountered in energy supply systems. A fluid is passing through a reservoir of temperature T₃, thereby changing the fluid temperature from T₁ to T₂. In order to determine T₂ in terms of T₁ and T₃, in a situation where the change in T₃ is much smaller than the change from T₁ to T₂, the incremental temperature change in the fluid traveling a short distance dx through the pipe system is related to the amount of heat transferred to the reservoir, assumed to depend linearly on the temperature difference,

$J_m{C}_p^{fluid}\mathrm{d}{T}^{fluid}/\mathrm{d}x=h\prime (T_3-T^{fluid}).$

Integrating from T₁ at the inlet (x=x₁) gives

$T^{fluid}(x)-T_3=(T_1-T_3)\exp (-h\prime (x-x_1)/(J_m{C}_p^{fluid})),$ (4.36)

(4.36)

where h′ is the heat transfer per unit time from a unit length of the pipe for a temperature difference of one unit. The heat transfer coefficient for the entire heat exchanger is

$h={\int }_{x_1}^{x_2}h\prime \mathrm{d}x,$

which is sometimes written h=U_hA_h, with U_h being the transfer coefficient per unit area of pipe wall and A_h being the effective area of the heat exchanger. For x=x₂, (4.36) becomes (upon re-ordering)

$(T_1-T_2)=(T_1-T_3)(1-\exp (-h/(J_m{C}_p^{fluid}))).$ (4.37)

(4.37)

4.2.4.1 Conversion of ocean thermal energy

Downward gradients of temperature exist in most oceans, and they are particularly stable (i.e., without variations with time) in the tropical oceans. The utilization of such temperature gradients for electricity generation (e.g., by use of a Rankine cycle) has been considered several times since the first suggestions by d’Arsonval (1881).

The temperature differences available over the first 500–1000 m of water depth are only about 25°C. For a closed Rankine cycle, with a working fluid (e.g., ammonia) that evaporates and condenses at convenient temperatures, placed near the ocean surface, colder water will have to be pumped through a pipe from the depth to a heat exchanger for condensation of the working fluid. Further, a warm water heat exchanger is required for evaporating the working fluid. If the heat exchange surface is such that, say, 5°C is “lost” at each heat exchanger, the temperature difference available to the thermodynamic cycle is only 15°C, corresponding to a limiting Carnot efficiency of roughly 0.05. For an actual engine, the efficiency is still lower, and from the power generated should subtract the power needed to pump hot and cold water through the heat exchangers and to pump cold water from its original depth to the converter level. It is expected that overall efficiencies around 0.02 may be achieved (cf., for example, McGowan, 1976).

In order to save energy to pump the hot water through the heat exchanger, it has been suggested that the converters be placed in strong currents, such as the Gulf Stream (Heronemus, 1975). Of course, the power extraction from ocean thermal gradients could have adverse environmental effects. This danger may be increased if ocean currents are incorporated into the scheme, because of the possible sensitivity of the currents to small perturbations, and because of the dependence of climatic zones on the course of currents like the Gulf Stream and the Kuro Shio. (Similar concerns are discussed in connection with global warming caused by increased anthropogenic injection of greenhouse gases into the atmosphere.)

Open thermodynamic cycles have also been suggested for conversion of ocean thermal energy (Claude, 1930; Beck, 1975; Zener and Fetkovich, 1975), for example, based on creating a rising mixture of water and steam bubbles or “foam”; the mixture is separated at a height above sea level, so that the water can be used to drive a turbine rotor.

If viable systems could be developed for conversion of ocean thermal energy, then there would be a number of other applications of such conversion devices in connection with other heat sources of a temperature little higher than that of the surroundings, especially when such heat sources can be regarded as “free.” Examples are the reject or “waste” heat flows from other conversion devices operating at higher initial temperature differences, including fuel-based power plants.

4.3.1.1 Free-stream-flow turbines

Consider, for example, a free stream flow passing horizontally through a converter. In this case, the potential energy (4.39) does not change and may be left out. The pressure may vary near the converting device, but far behind and far ahead of the device, the pressure is the same if the stream flow is free. Thus,

$w=w^{kin}=1/2(u_x^2+u_y^2+u_z^2)=1/2\mathit{u}·\mathit{u},$

and

$w_{in}-w_{out}=½({\mathit{u}}_{in}-{\mathit{u}}_{out})·({\mathit{u}}_{in}+{\mathit{u}}_{out}).$ (4.46)

(4.46)

This expression, and hence the efficiency, would be maximum if u_out could be made zero. However, the conservation of the mass flow J_m requires that u_in and u_out satisfy an additional relationship. For a pure, homogeneous streamline flow along the x-axis, the rate of mass flow is

$J_m=\rho A_{in}{u}_{x,in}=\rho A_{out}{u}_{x,out},$ (4.47)

(4.47)

in terms of areas A_in and A_out enclosing the same streamlines, before and after the passage through the conversion device. In a more general situation, assuming rotational symmetry around the x-axis, the device may have induced a radial as well as a circular flow component. This situation is illustrated in Fig. 4.10. It is further discussed below, and the only case treated here is the simple one in which the radial and tangential components of the velocity field, u_r and u_t, which may be induced by the conversion device, can be neglected.

The axial force (“thrust”) acting on the converter equals the momentum change,

$F_x=J_m(u_{x,in}-u_{x,out}).$ (4.48)

(4.48)

If the flow velocity in the converter is denoted u, an effective area of conversion, A, may be defined by

$J_m=\rho A{u}_x,$ (4.49)

(4.49)

according to the continuity equation (4.47). Dividing (4.48) by ρA, one obtains the specific energy transfer from the mass flow to the converter, within the conversion area A. This should equal the expression (4.46) for the change in specific energy, specialized to the case of homogeneous flows u_in and u_out along the x-axis,

$u_x(u_{x,in}-u_{x,out})=1/2(u_{x,in}+u_{x,out})(u_{x,in}-u_{x,out})$

or

$u_x=1/2(u_{x,in}+u_{x,out}).$ (4.50)

(4.50)

The physical principle behind this equality is simply energy conservation, and the assumptions so far have been the absence of heat exchange, so that the energy change (4.12) becomes proportional to the kinetic energy difference (4.46), and the absence of induced rotation, so that only x-components of the velocity need to be considered. On both sides of the converter, Bernoulli’s equation is valid, stating that the specific energy is constant along a streamline (Bernoulli’s equation for an irrotational flow is given in section 2.5.3, with ϕ=constant defining the streamlines). Far from the converter, the pressures are equal but the velocities are different, while the velocity just in front of or behind the converter may be taken as u_x, implying a pressure drop across the converter,

$\mathrm{\Delta }P=1/2\rho (u_{x,in}+u_{x,out})(u_{x,in}-u_{x,out}).$ (4.51)

(4.51)

The area enclosing a given streamline field increases continuously across the converter at the same time as the fluid velocity continuously decreases. The pressure, on the other hand, rises above the ambient pressure in front of the converter, then discontinuously drops to a value below the ambient one, and finally increases toward the ambient pressure again, behind (“in the wake of”) the converter.

It is customary (see, for example, Wilson and Lissaman, 1974) to define an axial interference factor, a, by

$u_x=u_{x,in}(1-a),$ (4.52)

(4.52)

in which case (4.50) implies that u_x,out=u_x,in (1−2a). With this, the power output of the conversion device can be written

$E=J_m(w_{in}-w_{out})=\rho A{(u_{x,in})}^{3}2a{(1-a)}^2,$ (4.53)

(4.53)

and the efficiency can be written

$\eta =E/(J_m{w}_{in})=4a(1-a).$ (4.54)

(4.54)

It is seen that the maximum value of η is unity, obtained for a=1/2, corresponding to u_x,out=0. The continuity equation (4.47) then implies an infinite area A_out, and it will clearly be difficult to defend the assumption of no induced radial motion.

In fact, for a free stream device of this type, the efficiency (4.54) is of little relevance since the input flux may not be independent of the details of the device. The input area A_in, from which streamlines would connect with a fixed converter area A, could conceivably be changed by altering the construction of the converter. It is therefore more appropriate to ask for the maximum power output for fixed A, as well as fixed input velocity u_x,in, this being equivalent to maximizing the power coefficient defined by

$C_p=E/(1/2\rho A{(u_{x,in})}^3)=4a{(1-a)}^2.$ (4.55)

(4.55)

The maximum value is obtained for a=1/3, yielding C_p=16/27 and u_x,out=u_x,in/3. The areas are A_in=(1−a)A=2/3 A and A_out=(1−a)A/(1−2a)=2 A, so in this case it may be a reasonable approximation to neglect the radial velocity component in the far wake.

The maximum found above for C_p is only a true upper limit with the assumptions made. By discarding the assumption of irrotational flow, it becomes possible for the converter to induce a velocity field, for which rot(u) is no longer zero. It has been shown that if the additional field is in the form of a vortex ring around the converter region, so that it does not contribute to the far wake, then it is possible to exceed the upper limit power coefficient 16/27 found above (cf. section 4.3.3).

4.3.1.2 Methods of describing the conversion of wind flows

Conversion of wind energy into linear motion of a body has been utilized extensively, particularly for transportation across water surfaces. A large sailing vessels of the type used in the 19th century would have converted wind energy at peak rates of a quarter of a megawatt or more.

The force on a sail or a wing (i.e., profiles of negligible or finite thickness) may be broken down into a component in the direction of the undisturbed wind (drag) and a component perpendicular to the undisturbed wind direction (lift). In referring to an undisturbed wind direction it is assumed that a uniform wind field is modified in a region around the sail or the wing, but that beyond a certain distance such modifications can be disregarded.

In order to determine the force components, Euler’s equations (cf. section 2.5.2) may be used. If viscous and external forces are neglected, and the flow is assumed to be irrotational (so that Bernoulli’s equation is valid) and steady (so that the time-derivative of the velocity potential vanishes, cf. section 2.5.3), then the force on a segment of the airfoil (sail or wing) may be written

$\frac{d\mathit{F}}{dz}=\underset{C}{\overset{}{\oint }}P\mathit{n}ds=-1/2\rho {\oint }_C(\mathbf{v}·\mathbf{v})\mathit{n}ds.$

Here dz is the segment length (cf. Fig. 4.11), C is a closed contour containing the airfoil profile, n is a unit vector normal to the contour [in the (x, y)-plane], and ds is the path-length increment, directed along the tangent to the contour, still in the (x, y)-plane. Taking advantage of the fact that the wind velocity v approaches a homogeneous field W (assumed to be along the x-axis) far from the airfoil, the contour integral may be reduced and evaluated (e.g., along a circular path),

$d\mathit{F}/dz=\rho W\mathrm{\Gamma }{\mathit{e}}_y,$ (4.56)

(4.56)

$\mathrm{\Gamma }={\oint }_C\mathbf{v}·d\mathit{s}\approx \pi cW\sin \alpha .$ (4.57)

(4.57)

Here e_y is a unit vector along the y-axis, c is the airfoil chord length, and α is the angle between the airfoil and W. In the evaluation of the circulation Γ, it has been assumed that the airfoil is thin and without curvature. In this case, c and α are well defined, but in general the circulation depends on the details of the profile, although an expression similar to the right-hand side of (4.57) is still valid as a first approximation, for some average chord length and angle of attack. Equation (4.56) is known as the theorem of Kutta (1902) and Joukowski (1906).

The expressions (4.56) and (4.57) are valid in a co-ordinate system fixed relative to the airfoil (Fig. 4.11), and if the airfoil is moving with a velocity U, the velocities v and W are to be interpreted as relative ones, so that

$\mathit{W}={\mathit{u}}_{in}-\mathit{U},$ (4.58)

(4.58)

if the undisturbed wind velocity is u_in.

The assumption that viscous forces may be neglected is responsible for obtaining in (4.56) only a lift force, the drag force being zero. Primitive sailing vessels, as well as primitive windmills, have been primarily aimed at utilizing the drag force. It is possible, however, with suitably constructed airfoils, to make the lift force one or two orders of magnitude larger than the drag force and thereby effectively approach the limit where the viscous forces and hence the drag can be neglected. This usually requires careful “setting” of the airfoil, i.e., careful choice of the angle of attack, α, and, in order to study operation at arbitrary conditions, the drag component should be retained.

It is customary to describe the drag and lift forces on an airfoil of given shape as a function of α, in terms of two dimensionless constants, C_D(α) and C_L(α), defined by

$d{F}_x/dz=1/2\rho C_D{W}^{2}c,$ (4.59a)

(4.59a)

$d{F}_y/dz=1/2\rho C_L{W}^{2}c.$ (4.59b)

(4.59b)

The constants C_D and C_L are not quite independent of the size of the system, which is not unexpected since the viscous forces (friction) in air contribute most to turbulent motion on smaller scales (cf. the discussion in section 2.5.2). Introducing the Reynolds number,

$\mathrm{Re}=Wc/\eta ,$

where η is the kinematic viscosity of air defined in section 2.5.2 as a measure of the ratio between “inertial” and “viscous” forces acting between airfoil and air, the α-dependence of C_D and C_L for fixed Re, as well as the Re-dependence for the value of α that gives the highest lift-to-drag ratio, L/D=C_L/C_D, may appear as shown in Figs. 4.12 and 4.13. The contours of these “high lift” profiles are indicated in Fig. 4.12.