Equivalent Circuits for Exergy Flow in Thermodynamic Systems

The flow of energy in thermodynamic systems has been of great interest to the scientific community over the past three centuries. The necessary flow of heat was the focus of Sadi Carnot's 1824 publication that gave birth to the field of thermodynamics [1]. Carnot realized that even a perfect heat engine must reject some amount of heat, because heat carries some quantity that work does not. Clausius would go on to coin this quantity “entropy” in 1865 [2], and Boltzmann later revealed that its origins lie in statistical mechanics [3,4].

The relationship between heat rejection, entropy, and temperature led to various researchers exploring the concept of “available energy” [5–11]. For a closed system, the energy provided by a heat source must leave the system as either heat rejection or work, so the heat rejection effectively “steals” energy from the work, which is the desired output of the cycle. Based on the minimum amount of heat rejection that must occur for a system that interacts with a specified environment, there is some maximum work output (available energy) that the system can therefore produce. Rant coined the term “exergy” for this concept in 1956 [12]. Exergy is now one of the fundamental concepts that appears in undergraduate thermodynamics textbooks [13], and exergy analyses are widely used to determine the upper limits of performance for various thermodynamic systems [14–27]. Broadly, the exergy of a thermodynamic system can be defined as the maximum amount of work that could be obtained from the system if it were brought to equilibrium with a reference environment [13].

In practice, thermodynamic systems are often very complex and can interact with other systems in various ways. For example, natural gas combined cycle power plants are among the most efficient and cost-effective methods of electricity generation, which utilizes a gas turbine (e.g., Brayton) topping cycle that rejects heat to power a steam turbine (e.g., Rankine) bottoming cycle [28,29]. In systems with multiple cycles, it is important to understand how a change in performance of the topping cycle affects the performance of the downstream bottoming cycle. These interactions are illustrated in Fig. 1, where Fig. 1(a) illustrates two thermodynamic cycles that interact with one another. Heat is rejected from the topping cycle at an intermediate temperature (⁠$Tr,1$⁠), which is used to power a bottoming cycle that rejects heat to a lower temperature (⁠$Tr,2$⁠).

In this example, the topping cycle is irreversible, so some extra amount of heat equal to $S˙genTr,1$ must be rejected. This reduces the amount of work that the topping cycle produces, but it increases the heat supplied to the bottoming cycle, thereby increasing the work produced by the bottoming cycle. Eliminating entropy generation in the topping cycle would decrease the heat rejection, and all of that energy would leave the cycle via work output instead. Engineering the system to generate less entropy is, of course, not arbitrary, but when done without changing the heat source or rejection temperatures, the work output increases. In this way, the energy flowing through the two coupled cycles is analogous to current flowing through branches of an electrical circuit, as depicted in Fig. 1(b). Some of the current, analogous to energy in the thermodynamic cycle, in circuit flows through the desired branch (labeled “work output”), while the remainder flows through the bottom branch (labeled “heat rejection”). In Fig. 1(b), entropy is analogous to a voltage source that drives some amount of current to flow through the bottom branch. Decreasing the entropy that must be removed from the system is equivalent to reducing the voltage of the source. This decreases the current through the heat rejection branch and commensurately increases the current through the work output branch. Even when the entropy generation is zero, there is still some flow through the heat rejection branch, because the heat source will always carry some entropy with it since $TS$ cannot be infinite.

In this analogy, entropy is not the only quantity that affects how energy flows through the system. In Fig. 1(a), both of the topping cycle's heat rejection terms are proportional to $Tr,1$⁠. In Fig. 1(b), this means that control over $Tr,1$ acts like a rheostat (i.e., a variable resistance). If the cycle could reject heat to 0 K, the resistance would be dialed to an infinite value in the circuit analogy, and all of the incoming heat would be converted to work. In a real system, there will be some minimum available temperature for heat rejection, $T0$⁠, that is greater than 0 K, and the actual rejection temperature will be even higher than the minimum available temperature (due to finitely sized heat exchangers). Thus, heat rejection cannot be entirely resisted in real systems, but it can be reduced. Improving the system (e.g., by increasing the conductance of heat exchangers) reduces the energy that leaves as heat rejection and correspondingly increases the work output. Thus, implementation of these improvements is like turning a knob that decreases entropy generation, or turning another knob that decreases the heat rejection temperature. This is analogous to turning down the knob on the voltage source and turning up the knob on the variable resistor in Fig. 1(b). While resistance networks have been used to quantify the rates of heat and mass transfer (i.e., kinetics) across a wide range of applications [30–35], a method of using resistance networks to quantify energy flow in a cycle as dictated by the need to remove entropy (i.e., thermodynamics) has yet to be developed.

In this work, we present Sankey and Venn diagrams to differentiate between the difference origins of exergy loss in thermodynamic cycles. In the case of a heat engine, these diagrams illustrate the difference between necessary heat rejection and unnecessary (or exergetic) heat rejection. We further show how to calculate effective temperatures of heat transfer that can be used to quantify the different exergy losses (exergy destruction and exergy rejection); this can be used to understand how changes in effective temperatures and entropy generation affect one or more thermodynamic subsystems (e.g., a change in heat rejection temperature will affect the work output of a topping cycle but will also affect the amount of exergy supplied to a bottoming cycle). From the same structure as the Sankey diagrams, we develop a method of representing thermodynamic systems using circuit components analogous to those in electrical circuits, namely, current sources, voltage sources, resistors, and capacitors. These networks can be used to analyze and visualize exergy flow in coupled thermodynamic systems. For example, a potential improvement to the system can be evaluated based on the direct effects it would have on a particular subsystem, as well as the indirect effects it would have on the downstream components. After describing this new method of analysis, it is applied to three separate case studies with real numbers to demonstrate its utility.

For the general thermodynamic system illustrated in Fig. 2(a), the total heat rejection, $((Q˙s/Ts)+S˙gen)Tr$⁠, can be categorized into three contributions. First, there is necessary heat rejection, $(Q˙s/Ts)T0$⁠, which occurs because the entropy introduced to the system from the heat source must be removed. Even in an ideal system, this heat must be rejected, leading to the efficiency of a heat engine always being less than unity (per Carnot's analysis [1]). Next, there is exergy destruction, $S˙genT0$⁠. Finally, there is the exergy rejected from the system, $((Q˙s/Ts)+S˙gen)(Tr−T0)$⁠, which arises because the entropy removed from the system is rejected at a temperature higher than the ambient. This breakdown is illustrated in the Venn diagram in Fig. 2(b), where the total heat rejection is the sum of the necessary heat rejection (leftmost circle) and the total exergy loss (the two overlapping circles on the right). The total exergy loss can then be categorized into the contribution from $Tr>T0$ and from $S˙gen$⁠; the overlap of these two contributions is the exergy loss that occurs due to the presence of both $Tr>T0$ and $S˙gen$⁠. Eliminating either one of these would eliminate this loss, though of course neither of these can be eliminated in practice. This reveals an important consequence of the second law, stated as: entropy generation, within a thermodynamic cycle, results in exergy loss due to both the exergy destruction and the extra heat rejection necessary to remove the generated entropy, which carries with it some exergy (because $Tr>T0$⁠) that is lost to the ambient.

Thus far, this work has only considered thermodynamic cycles with heat transfer and work at the boundary (i.e., closed systems). However, many real systems also involve mass flow at the boundaries (e.g., a hot air stream as the heat source, cooling water for heat rejection, etc.). To reconcile this, we employ the use of effective temperatures, a method that has been widely used to analyze thermodynamic cycles in the past [17,36–38]. Figure 3 illustrates a generic thermodynamic system with work, heat, and mass flow at the boundary. The accompanying first and second law equations are listed alongside the system diagram. These first and second law equations can be rewritten with the dead state subtracted from both the inlet and outlet terms, such that the dead state terms cancel; the reason for doing this is explained below.

Equation (5) reveals the reason for subtracting the dead state from both the inlet and outlet in Fig. 3. By doing so, the effective temperature of a fluid stream at the dead state is equal to the physical dead state temperature. Then, the total heat and enthalpy rejection, $[S˙gen+m˙r(sr−s0)]Tr$⁠, can be split into three terms: the exergy destruction, which takes its usual form of $S˙genT0$⁠, the exergy rejected from the system, $[S˙gen+m˙r(sr−s0)](Tr−T0)$⁠, and the rejection that would be necessary even for an internally and externally reversible system, $m˙r(sr−s0)T0$⁠.

Before establishing the circuit analogy, Sankey diagrams can be drawn to illustrate the flow of exergy and nonexergetic energy through a thermodynamic system. Then, once the flow diagrams have been established, the circuits can be drawn with branches that match the flow paths. Figure 4(a) illustrates a general thermodynamic system, which receives heat at a temperature $Ts$⁠, generates some entropy internally, rejects heat to a temperature $Tr$⁠, and produces some work output. Applying the first and second laws of thermodynamics yields expressions for the heat rejection and work output as functions of the heat input, entropy generation, source temperature, and rejection temperature. There is some minimum amount of heat rejection that must occur, (⁠$Q˙s/Ts$⁠)$T0$⁠, based on the entropy introduced to the system from the heat source (⁠$Q˙s/Ts$⁠), and based on the lowest possible temperature to which heat can be rejected (⁠$T0$⁠). More heat must be rejected to remove that extra entropy, because entropy is generated internally; this heat rejection is equal to $S˙genT0$ and is conventionally termed “exergy destruction.” But these two terms do not comprise all of the heat rejected from the system. The second law dictates that the actual heat rejection is $((Q˙s/Ts)+S˙gen)Tr$⁠, because the system rejects heat a temperature $Tr$⁠, which is greater than $T0$⁠. The difference between the actual and minimum heat rejection is the “extra” heat rejection, $((Q˙s/Ts)+S˙gen)(Tr−T0)$⁠. If the heat rejection from the system in Fig. 4(a) were used to power a reversible heat engine that itself rejects heat to $T0$ (i.e., an ideal bottoming cycle), it would produce an extra amount of power equal to $((Q˙s/Ts)+S˙gen)(Tr−T0)$ (i.e., the extra heat rejection from the system in Fig. 4(a)), and it would reject heat at a rate of $((Q˙s/Ts)+S˙gen)T0$ (i.e., the necessary heat rejection from the system in Fig. 4(a)). Thus, the extra heat rejection can, in theory, be entirely converted to work; this is the exergetic heat. The rest of the heat rejection is necessary and can never be converted to work (so long as $T0$ remains the lowest possible temperature for heat rejection); this is the nonexergetic heat. This delineation is illustrated in the flow diagram in Fig. 4(b).

The flow diagram in Fig. 4(b) is a simplified illustration; a more general version is provided in Fig. 5(b). In Fig. 5(a), a thermodynamic system that consists of two subsystems (a topping and bottoming cycle) is depicted. The topping cycle is irreversible, while the bottoming cycle is reversible and rejects heat to the lowest available temperature, $T0$⁠. At the top of the flow diagram in Fig. 5(b), before the heat even enters the system, some of the energy splits off and starts the nonexergetic energy branch. This is the minimum heat rejection term, which is completely independent of the thermodynamic system; whether the system is reversible or irreversible, and whether it rejects heat to a temperature of $T0$ or some temperature greater than $T0$⁠, the heat rejected to the environment must be at least $(Q˙s/Ts)T0$ (though in every realistic system it will be more).

When the heat reaches the system boundary, it still has a magnitude of $Q˙s$⁠. However, of this heat, $(Q˙s/Ts)T0$ of it can never be converted to work, while the remainder, $Q˙s(1−(T0/T0))$⁠, is exergetic (i.e., the portion that can, in theory, be entirely converted to work). Within the system, the exergy destruction splits off and becomes nonexergetic heat, some of the exergy is converted to work, and the rest remains as exergetic heat. The total rate of heat transferred to the bottoming cycle is $((Q˙s/Ts)+S˙gen)Tr$⁠, but only $((Q˙s/Ts)+S˙gen)(Tr−T0)$ of it can ever be converted to work, which we see is the case in the reversible bottoming cycle. It should be noted that by making the topping cycle reversible, $W˙1$ would increase by $S˙genTr$⁠, while $W˙2$ would decrease by $S˙gen(Tr−T0)$⁠, yielding a net increase of $S˙genT0$ (the exergy destruction term). While Fig. 5 only considers two subsystems, Supplemental Note 1 available in the Supplemental Materials on the ASME Digital Collection provides an illustration of the flow diagram for the general case of $N$ irreversible thermodynamic subsystems (an even more general case).

Thus, the quantity $[∫r(δQ˙/T)+∑r(m˙(s−s0))](Tr−T0)$⁠, which is the extra heat leaving the system in Fig. 4, is equal to $∫rδQ˙(1−(T0/T))+∑r[m˙(h−h0−T0(s−s0))]$⁠, the exergy of the heat and mass leaving the system, enabling the framework presented in this work.

To use a resistance network to describe exergy in a thermodynamic system, the electrical circuit analogs must be identified. The most immediately intuitive is the current analog, which is energy flow. Electrical current moves through different paths of a circuit based on the resistances of each path. Likewise, energy flows into a thermodynamic system from the heat source, and it leaves through work output and heat rejection. The amount of energy that leaves via heat rejection depends somewhat on the entropy that must be removed from the system and the heat rejection temperature; one of these quantities will appear in the driving force (i.e., voltage), while the other will appear in the resistance. Even though temperatures are the driving force in thermal resistance networks, it is actually the entropy that is the driving force in the exergy circuits described in this work. Just as a voltage source can be used to place a constraint on the voltage dropped across a resistor, an entropy source can be used to enforce the entropy that must be removed from the system via heat rejection. There will be an entropy source associated with the heat provided to the system and another associated with the internal entropy generation.

Based on the Ohms' law (⁠$V=IR$⁠) analogy, the resistance within an exergy circuit should have units of inverse temperature. If energy flow is analogous to current, and the entropy that must be removed from the system is analogous to a voltage drop, then a resistance can be used to find the energy that must leave the system via heat rejection. From the second law of thermodynamics, the heat rejection is equal to the product of total entropy that must be removed and heat rejection temperature (⁠$Q˙r=(Q˙s/Ts+S˙gen)Tr$⁠). This indicates that the inverse of heat rejection temperature can be expressed as a resistance in the exergy circuit. Conceptually, a given amount of heat carries more entropy when it is at a lower temperature. Thus, if the rejection temperature goes down (i.e., the inverse of rejection temperature goes up), less heat rejection is necessary to remove the required amount of entropy from the cycle, and work output goes up. This means that the inverse rejection temperature resists energy leaving the system via heat rejection and motivates the conversion of heat into work. An alternative conception is that the temperature of heat rejection is a conductance (the inverse of resistance), and a high heat rejection temperature conducts more energy to leave the system as heat rejection, rather than the conversion into work output (Table 1).

Figure 6 illustrates the thermodynamic circuits and how they compare to the exergy flow diagrams. In Fig. 6(a), the flow diagram for the heat source is depicted next to the equivalent circuit. An energy source element is analogous to a current source, which introduces a flow of energy at a rate of $Q˙s$ to the circuit. Because this heat carries with it some entropy, an entropy source is placed within the circuit, which is analogous to a voltage source. Finally, the resistance associated with the minimum available temperature for heat rejection is placed. The entropy drop across the resistance causes heat to flow to the nonexergetic branch at a rate of $(Q˙s/Ts)T0$⁠, leaving $Q˙s(1−(T0/Ts))$ to flow down the exergetic branch.

Meanwhile, the equivalent circuit for a thermodynamic system is depicted in Fig. 6(b). The exergetic heat from the heat source is the energy (analogous to current) introduced to the circuit. There is an entropy source associated with the entropy generation term, which constrains the entropy drop across the $T0−1$ resistance, thereby forcing energy to flow to the nonexergetic branch at a rate of $S˙genT0$⁠. Another entropy source is placed above the $(Tr−T0)−1$ resistor, where $S˙in$ is the rate of entropy entering the system. For example, this would be equal to $Q˙s/Ts$ for a system receiving heat at a rate of $Q˙s$ and a temperature of $Ts$⁠. This means that the total entropy drop across the bottom resistor is constrained to $S˙in+S˙gen$⁠, which forces exergetic heat to leave the system at a rate of $(S˙in+S˙gen)(Tr−T0)$⁠, which is equal to $((Q˙s/Ts)+S˙gen)(Tr−T0)$⁠. The remaining power, $(Q˙s/Ts)(1−(Tr/Ts))−S˙genTr$⁠, leaves the system as work.

It should be noted that the entropy (i.e., voltage) at the bottom right node of the circuit in Fig. 6(a) must be zero; however, the entropy at the top node of the circuit in Fig. 6(b) is $S˙gen$⁠. To attach these two circuits together and build a circuit for the overall system, an additional entropy source must be placed in between to create a virtual ground. The same applies when connecting two thermodynamic subsystem circuits together (i.e., connecting two of the circuit depicted in Fig. 6(b) in series). It should be noted that sometimes the heat leaving one subsystem is not equal to the heat entering a second. This could be because some extra heat is sourced to the bottoming cycle, or because a portion of the heat rejected from the topping cycle is lost to the environment and not transferred to the bottoming cycle. The former case appears in one of the examples depicted below. The circuit that is used to describe this situation is described in Supplemental Note 2 available in the Supplemental Materials on the ASME Digital Collection.

Now that the rules for constructing a thermodynamic resistance network have been described, several example problems are discussed to illustrate its implementation. The first is a combined cycle gas turbine depicted in Fig. 7. This example is adapted from Ref. [13], with some of the numbers slightly altered. Air flows through a Brayton cycle and rejects heat in a heat exchanger to water in a Rankine cycle. The Rankine cycle rejects heat to cooling water that enters the heat exchanger at 274 K, a temperature lower than the ambient air temperature (e.g., the cooling water comes from a lake that is partially frozen, even though the air temperature has warmed up). All pumps, compressors, and turbines are reversible. The python package coolprop is used to evaluate the enthalpy and entropy at each state; these values are given in Table S1 in Supplemental Note 3 available in the Supplemental Materials.

Equations (1)–(4) can be used to find the effective heat flows and temperatures for the source and rejection. This gives a topping cycle rejection temperature of $Tr,1$ = 508.7 K. For the bottoming cycle, mass flow is again the only means of entropy removal, and the effective temperature equates to $Tr,2$ = 291.6 K. The source temperature for the heat entering the topping cycle is assumed to be $Ts,1$ = 1400 K, while the effective source for the bottoming cycle is $Ts,2$ = 508.7 K (the same as the rejection temperature for the topping cycle). Detailed calculations for these effective temperatures are given in Supplemental Note 3 available in the Supplemental Materials.

The exergy flow diagram is depicted in Fig. 7(b), while the equivalent thermodynamic circuit is depicted in Fig. 7(c). From the entropy of the different states (Supplemental Table S1 available in the Supplemental Materials), the entropy introduced from the heat source is 66.9 kW K⁻¹, the entropy generated in the topping cycle is 33.1 kW K⁻¹, the entropy transferred to the bottoming cycle is the sum of the entropy entering and generated within the topping cycle, 100.0 kW K⁻¹, and the entropy generated in the bottoming cycle is 12.5 kW K⁻¹. The thermodynamic circuit mirrors the exergy flow diagram, where the energy flow through each branch of the circuit corresponds to the different branches of the flow diagram. The circuit also reveals the effects of different changes on the thermodynamic system. If the rejection temperature of the topping cycle was decreased, the 4.26 $×$ 10⁻³ K⁻¹ resistance would increase, which restricts the flow of exergetic heat out of the topping cycle. This would have two effects: it would increase the flow of energy through the branch that represents work output in the topping cycle, but it would equally reduce the energy that flows through the exergetic heat rejection branch into the bottoming cycle. Similar observations can be made regarding the entropy of the heat source (which depends on the heat source temperature) and the internal entropy generation.

In Fig. 7(a), the temperature of the air leaving the air-to-water heat exchanger is equal to the temperature of the air entering the compressor. This means that the heat rejection from the topping cycle is equal to the heat source to the bottoming cycle. However, the air could leave the air-to-water heat exchanger at a temperature higher than 300 K, which would mean the heat rejection from the Brayton cycle is greater than the heat source to the Rankine cycle (based on the way the system boundaries were drawn when determining the heat rejection and source terms). This situation is explored in the example problem in Supplemental Note 2 available in the Supplemental Materials. Additionally, because the Rankine cycle rejects heat to a lower temperature than that of the air entering the Brayton cycle, the temperature of the air leaving the air-to-water heat exchanger could actually be lower than 300 K too. This provides another, more peculiar mismatch between the Brayton rejection and Rankine source terms, in which the Rankine heat source is greater in magnitude than the Brayton rejection term. This is why the temperature of the inlet cooling water in this example is 274 K, and this unique case is analyzed in Supplemental Note 2 available in the Supplemental Materials.

Notably, the system in Fig. 7 interacts with two different thermal environments: air at 300 K and cooling water at 274 K. This reveals the arbitrary nature of the dead state temperature when more than one thermal environment is present. In this case, we chose the temperature of the cooling water as the dead state temperature, but the ambient air would have been an equally valid choice for the dead state. Luckily, the exergetic resistance network accounts for this difference. If the higher $T0$ is chosen (ambient air temperature), then $Tr,2$ can take on a value lower than $T0$⁠, which would give a negative exergy rejection (analogous to current flowing in the opposite direction) in the bottom middle branch of the resistance network. In this case, negative exergy rejection is the heat rejection that a particular cycle is able to avoid (and equivalently the extra work it is able to produce) because it is rejecting heat to a lower temperature than $T0$⁠. Thus, when dealing with more than one thermal environment, $T0$ becomes arbitrary, but the results from the exergetic circuit are unambiguous and provide insight into the performance relative to the selected $T0$⁠.

A portion of the heat source must be rejected, which is equal to 18.3 MW (the first branch that breaks to the left in Figs. 7(b) and 7(c)). The remaining heat, 75.3 MW, is entirely exergetic and is equal to $E˙x,s$⁠. The exergy destruction consists of the remaining branches in Figs. 7(b) and 7(c) that divert to the left (9.1 MW and 3.4 MW). The exergy rejected to the ambient is found at the very bottom of the middle branch (2.0 MW). Using these values in Eq. (7), an exergetic efficiency of 81% results. Meanwhile, dividing the work output (60.8 MW) by the total energy input (93.6 MW) yields the conventional energetic efficiency, which is 65%. For reference, this is only slightly higher than the reported best-in-class natural gas combined cycle efficiency of 64% [39], revealing the high exergetic efficiency of real combined cycle gas turbines.

These exergetic equivalent circuits can be used not only to assess the flow of exergy through heat engine systems but also through systems that take work as the input and produce some desired effect (cooling, dehumidification, etc.). To illustrate this, Fig. 8 depicts a desalination system, in which reverse osmosis (RO) [40] is used as the topping cycle, and the brine from the RO system is fed as the input to the air gap diffusion distillation (AGDD) [34,41] bottoming cycle, which further concentrates the brine. A real AGDD system would likely use multiple stages to concentrate the brine more than is depicted in Fig. 8, but a single stage is used in this example for simplicity. The pump and turbine are assumed to be reversible (also for simplicity). The properties at each state and effective temperature calculations are given in Supplemental Note 4 available in the Supplemental Materials on the ASME Digital Collection.

To analyze the system, the source and rejection terms must be defined. The inlet state (brackish water entering the pump) and the work input combine to form the source terms of the topping RO cycle; the work carries no entropy with it, and the inlet state carries no entropy above that of the dead state (⁠$s1−s0$ = 0). Then, the rejection terms for the brackish RO system are the heat transfer at the RO control volume (6.2 MW), the rate of enthalpy leaving the RO system with the permeate (fresh) water, and the rate of enthalpy leaving the RO system with the brine exiting the turbine. The rejection terms therefore also total to 6.2 MW, which is intuitive, because the system is not designed to produce power. Even though the 6.2 MW of heat are rejected from the RO to the environment, we treat that heat as first passing through the bottoming cycle before reaching the environment. While we use this convention for the sake of simplicity, it is not necessary, as detailed in Supplemental Note 2 available in the Supplemental Materials. Despite implementing this nuance so that all of the topping cycle rejected energy is sourced to the bottoming cycle, the topping rejection terms still do not match the bottoming cycle source terms. This is because of the supplemental heat source of 3.9 MW to the bottoming AGDD cycle. We account for this mismatch by adding a supplemental source term to both the Sankey diagram and circuit (red dashed circle in Fig. 8(b) and 3.9 MW current source in Fig. 8(c)). The general procedure of accounting for rejection/source mismatch between a topping and bottoming cycle is detailed in Supplemental Note 2 available in the Supplemental Materials.

In a heat engine, reducing entropy generation increases power output, but in this case it does not, because the system does not produce power. But reducing the entropy generation (i.e., voltage sources) in Fig. 8(c) clearly reduces the energy flowing through the exergetic and nonexergetic energy branches of the circuit. Normally, the extra energy flow would be redirected to the power output branch, but in this case, because power output is zero, it instead means that the energy flow from the source must decrease. This is realized by a reduction in the net power consumption at the pump.

Next, the effective temperatures can be compared. Energy is rejected from the brackish RO cycle with an effective temperature of $Tr,1$ = 555.3 K, sourced to the AGDD brine concentrating cycle with an effective temperature of $Ts,2$ = 452.9 K, and ultimately rejected from the brine concentrator with an effective temperature of $Tr,2$ = 430.0 K. Because exergy is being added to the fluid in both the RO and AGDD systems, the rejection temperature for the RO system should be greater than $T0$ (300 K), which is observed to be true. This occurs because the effective temperature of a fluid stream is indicative of its exergy content.

To calculate the exergetic efficiency of the system in Fig. 8, we can again use the Sankey and circuit diagrams in Figs. 8(b) and 8(c), respectively. However, in Fig. 8, there is no work output. In this case, the exergy of the mass leaving the system is not lost to the environment, but is instead the desired product. Thus, for this system, $E˙x,r$ = 0, the total exergy destruction is 3.7 MW, and the total exergy sourced to the system is 6.7 MW. Using these values to evaluate Eq. (7), an exergetic efficiency of 45% results. We could include the thermomechanical exergy of the brine as “wasted” exergy, since the salinity/purity (i.e., chemical exergy) is the desired output, not the temperature. We could also separate the brine and permeate streams in the Sankey and circuit diagrams, since the permeate is usually the desired product, while the brine is considered a waste by-product. For simplicity, we did not make these distinctions in Fig. 8, but, as a reference, the exergetic efficiency drops to 23% when considering the permeate chemical exergy as the desired output and designating the brine stream as exergy lost.

To this point, all of the thermodynamic systems represented as equivalent circuits have been steady-state. However, a transient system can be represented with an equivalent circuit by adding capacitors. This is depicted by the example in Fig. 9, where a block with a high thermal conductivity (i.e., spatially uniform temperature) is heated over time. The boundary condition at the top is a variable heat flux condition, while the boundary condition at the bottom is convection. The variable heat flux at the top is such that the rate of entropy transferred to the block from the heat source, $S˙in$⁠, is constant. The heat flux at the top is also such that when the temperature of the block reaches $Tf$⁠, the top heat flux is equal to the bottom convective heat flux, and steady-state is achieved.

Equation (14) is equal to the energy that flows into the right capacitor branch, which represents the exergetic heat absorbed by the block, and the charge within the capacitor is the heat stored within the block that could be theoretically converted to work (i.e., the exergy stored by the block). This means that the energy that flows into the other capacitor branch is the nonexergetic heat, and the charge stored within that capacitor is the heat stored in the block that must eventually be rejected if the block is brought back to ambient temperature. Because the resistances and capacitances are functions of $T$⁠, they vary with time. However, because $S˙1$ and $S˙2$ are equal, the two resistances are in parallel and form an overall resistance that does not vary with time, $R=(R1−1+R2−1)−1=T0−1$⁠. Likewise, the overall capacitance is constant, $C=C1+C2=(mcp/UA)Tf$⁠, and the product of resistance and conductance forms the time constant $RC$⁠.

At any instant, the sum of the energy (i.e., current) that flows into both capacitor branches should be equal to $Q˙s−Q˙r$⁠. Additionally, the energy that does not flow through the capacitors is the heat that leaves the system. Within the circuit, this energy flows into a resistance network similar to the one in Fig. 6(b), and the circuit is split into two branches for exergetic and nonexergetic heat. Notably, the capacitor branches mirror the resistor branches, with one branch corresponding to exergetic energy and the other corresponding to nonexergetic energy.

Figure 9(c) shows the entropy rate (which is analogous to voltage) at the node between $C1$ and $R1$⁠, which is the same as the entropy rate between $C2$ and $R2$⁠. The entropy exponentially approaches the constant rate of entropy transfer into the system, $S˙in$⁠. Figure 9(d) shows the rate at which energy is stored in the block (analogous to current), with the nonexergetic heat depicted in blue and the exergy storage depicted in brown. Figure 9(e) illustrates the stored energy (analogous to charge) within the block. The stored nonexergetic thermal energy is shown in blue; this is the heat that must eventually be rejected to the ambient. The exergy is shown in blue, and the total thermal energy stored within the block is shown in black.

This RC circuit analogy also holds when $RC$ is not constant, though the mathematics become slightly more complicated. For this reason, the simple case of fixed entropy flux was depicted in Fig. 9, and a fixed heat flux case, which has a time varying $RC$⁠, is depicted in Supplemental Note 5 available in the Supplemental Materials on the ASME Digital Collection.

Thermodynamic systems behave in many ways analogous to electrical circuits. Energy flows into the system from a heat source, entropy serves as the forces that requires energy to leave via heat rejection, and the entropy per unit energy (i.e., inverse temperature [42]) of heat rejection resists the flow of heat rejection. The energy that does not leave as heat rejection instead can be converted to work. When work output is the desired effect, the driving force for heat rejection should be small (achieved with a high temperature heat source), and the heat rejection resistance should be large (achieved with a low temperature heat sink).

While this qualitative analogy is useful, it so happens that a thermodynamic system can also be quantitatively described using a thermodynamic circuit with components analogous to those in an electrical circuit. This lends insight into the changes that occur within a system that contains many subsystems that interact with one another. Programs that solve for the current and voltages in electrical circuits can therefore be used to analyze the performance of large, complex thermodynamic systems with many components, allowing upstream components to be altered and the effects on downstream components to be immediately identified. The circuits can be drawn to contain many different components, allowing insight into how energy flows out of a particular system as work or heat. Alternatively, the circuits can be greatly simplified, combining the energy and entropy sources, as well as resistances, to arrive at the simplest expression for system performance.

Finally, this analogy provides a way of conceptualizing exergy flow within a thermodynamic system. Exergy is the energy that does not necessarily need to leave the system as heat rejection. Based on the available environment or stream of fluid to which heat can be rejected, there is some necessary minimum amount of energy that must leave the system as heat rejection. All of the remaining energy could (in the reversible limit) be converted to work. In the circuit analogy, there is a single resistance through which energy must necessarily flow; this resistance is governed only by the available reservoir or fluid stream to which heat can be rejected. The driving force of energy through this resistance is the entropy from the heat source at the beginning of the circuit. No matter how much the rest of the system is idealized, there will always be some energy flow through this resistance. The remaining energy flow is exergy, which is either converted to work, destroyed due to entropy generation, or lost as the extra, unnecessary heat rejection.

J.D.K. acknowledges financial support from the IBUILD Fellowship. This research was performed under an appointment to the Building Technologies Office (BTO) IBUILD-Graduate Research Fellowship administered by the Oak Ridge Institute for Science and Education (ORISE) and managed by Oak Ridge National Laboratory (ORNL) for the U.S. Department of Energy (DOE). ORISE is managed by Oak Ridge Associated Universities (ORAU). All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of DOE, EERE, BTO, ORISE, ORAU, or ORNL.

J.D.K. and S.K.Y. would like to thank Dr. Aravindh Rajan and Dr. Mike Adams for their insightful conversations regarding the concepts discussed in this work.

• IBUILD Fellowship.

The author has no competing interests to declare.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.