# An Unified Approach to Limits on Power Generation and Power Consumption in Thermo-Electro-Chemical Systems

## Abstract

**:**

_{i}of reservoirs’ are the only necessary state coordinates describing purely thermal units, in chemical (electrochemical) engines, heat pumps or separators it is necessary to use both temperatures and chemical potentials μ

_{k}. Methods of mathematical programming and dynamic optimization are applied to determine limits on power yield or power consumption in various energy systems, such as thermal engines, heat pumps, solar dryers, electrolysers, fuel cells, etc. Methodological similarities when treating power limits in engines, separators, and heat pumps are shown. Numerical approaches to multistage systems are based on methods of dynamic programming (DP) or on Pontryagin’s maximum principle. The first method searches for properties of optimal work and is limited to systems with low dimensionality of state vector, whereas the second investigates properties of differential (canonical) equations derived from the process Hamiltonian. A relatively unknown symmetry in behaviour of power producers (engines) and power consumers is enunciated in this paper. An approximate evaluation shows that, at least ¼ of power dissipated in the natural transfer process must be added to a separator or a heat pump in order to assure a required process rate. Applications focus on drying systems which, by nature, require a large amount of thermal or solar energy. We search for minimum power consumed in one-stage and multi-stage operation of fluidized drying. This multi-stage system is supported by heat pumps. We outline the related dynamic programming procedure, and also point out a link between the present irreversible approach and the classical problem of minimum reversible work driving the system.

## 1. Introduction

_{1'}and T

_{2'}; Chapter 3 of ref. [13]) for the case of chemical systems. In the main text the ideas referring to endoreversible systems are generalized to those with internal dissipation.

## 2. Steady Thermal Systems with Internal Dissipation

_{1}and T

_{2}and internal irreversibility factor Φ, the propelling heat in terms of Carnot temperature is [1,13]:

_{2}and the Carnot structure holds for thermal efficiency η in terms of Carnot T', the power-maximizing efficiency follows as:

^{3}), omits to a considerable extent analytical difficulties of the Stefan-Boltzmann equation. Moreover, we can extend the present approach to dynamical systems, as outlined in the next section.

_{1}− A

_{2}= 0 (isomerisation or phase change of A

_{1}into A

_{2}), takes the following form:

## 3. Some Results for Dynamical Thermal Systems

_{1}= T(t) (approximated here by the sequence T

^{n}). The optimization procedure searches for an extremal curve rather than an extremum point.

**Figure 1.**A scheme of a multistage control with distinguished time interval, Θ

^{n}. When the method of the dynamic programming is applied, in the forward algorithm, elipse-shaped balance areas pertain to sequential subprocesses which grow by inclusion of following units in the way described by Bellman’s recurrence equation. When algorithms of Pontryagin’s type are applied, the optimal evolution is characterized by the difference canonical equations and extremum conditions of the Hamiltonian H

^{n}with respect to controls.

_{σ}which is a constant numerical value of H along an optimal path. The related optimal Carnot control has the structure:

## 4. Radiation Engines Decribed by the Stefan-Boltzmann Equations

_{0}.

^{n}. In the transformed problem, without t

^{n}, accuracy of DP solutions is high.

_{i}

^{n}accompany temperature T

^{n}as in chemical engines), DP algorithms become inefficient and inaccurate. We must then abandon DP approaches and turn to the Pontryagin type (Hamiltonian-based) approaches. For the Hamiltonian approaches, which apply Pontryagin’s canonical equations, problems of large dimensionality of state vectors are inessential, as outlined in the section below.

## 5. Hamiltonian-Based Approaches

^{n}.

^{n}is identified with any state variable growing monotonically. After defining the function:

^{n}, 1), we call this form the “standard form”.

_{0}is a generation rate for the generalized profit (power in the case of energy yield problems).

_{i}are adjoint (Pontryagin’s) variables.

**x**, t) and z = (

**z**, z

_{t}) the following equations:

**u**

^{n}. For example, if the optimal control lies within an interior of admissible control set:

_{i}are independent of θ

^{n}. In addition, the energy-like Hamiltonian (without z

_{t}term) is constant for the process whose rates are independent of time t

^{n}. Under convexity properties for rate functions and constraining sets the optimal controls are defined by the equations:

^{n}-dependent) costs and rates provides a bridge between constant-H algorithms [5,19,20] and more conventional ones such as those by Katz [21], Halkin [6], Canon et al. [7], Boltyanskii [8], and many others [9,18]. Since, as shown by Equation (24), control θ

^{n}can be included in the Hamiltonian definition, i.e., an effective Hamiltonian can be used:

^{n−1}. The related canonical set is that of Halkin [6,7]:

^{n}and θ

^{n}in the optimization algorithm is then lost since they both follow from the same stationarity condition for Hamiltonian H

^{n−1}in an optimal process. For example, in the weak maximum principle:

**Figure 3.**Decreasing temperatures of radiation relaxing in engine mode and increasing temperature of radiation utilized in heat pump mode in terms of time, for a constant value of Hamiltonian H = 1 × 10

^{−8}[J K

^{−1}m

^{−3}] [4].

## 6. Simple Chemical Systems

_{1}− A

_{2}=0 [2,14]. In an ‘endoreversible engine’ total entropy flux is continuous through the active zone. When a formula describing this continuity is combined with an isothermal energy balance we find:

_{1}equals to n, an invariant molar flux of reagents. Process efficiency ζ is defined as power yield per molar flux, n. This efficiency is identical with the chemical affinity of reaction in the chemically active part of the system. While ζ is not dimensionless, it describes correctly the system. In terms of Carnot variable, μ', which satisfies Equation (19) in the Appendix:

_{1}and its mole fraction x:

_{1eff}= x

_{1}− ${g}_{1}^{-1}$ n is decreased, whereas an effective concentration of the product in lower reservoir x

_{2eff}= x

_{2}+ ${g}_{2}^{-1}$ n is increased due to the finite mass flux. Therefore efficiency ζ decreases nonlinearly with n. When the effect of resistances is negligible or flux n is very small, reversible Carnot-like chemical efficiency, ζ

_{C}, is attained. The power function, described by the product ζ(n)n, exhibits a maximum for a finite value of the fuel flux, n.

_{1}/g

_{2}. The path optimality condition may be expressed in terms of the constancy of the following Hamiltonian:

_{1}close to the unity) optimal relaxation rate of the fuel resource is approximately constant. Yet, in an arbitrary situation optimal rates are state dependent so as to preserve constancy of H in Equation (37).

## 7. Power Limits in Fuel Cells

**Figure 4.**Entropy production and power yield in an electrochemical engine or fuel cell in terms of density of electric current.

^{0}is often a reference basis calculated from the Nernst equation. Yet, in more general cases, actual voltage without load must take into account losses of the idle run, which are the effect of flaws in electrode constructions and other imperfections. In [23] and many other works the operating voltage of a cell is evaluated as the departure from the idle run voltage E

_{0}:

_{act)}, ohmic polarization (V

_{ohm}), and concentration polarization (V

_{conc}). Power density is the product of voltage V and current density i. A large number of approaches for calculating polarization losses has been presented in literature, as reviewed in [22]. Experiments show power maxima in fuel cells [22,23]. Activation and concentration polarization occur at both anode and cathode locations. The resistive polarization represents ohmic losses throughout the cell. As the voltage losses increase with current, the initially increasing power begins finally to decrease for sufficiently large currents, so that maxima of power are observed [22,23]. The data include the losses of the idle run, attributed to flaws in electrode constructions and other imperfections.

^{TM}software for simulation purposes and organization of FC power experiments, as described in Wierzbicki’s M.Sc thesis [23]. A complete review of the theory and experiments is presented in the report [24].

_{1'}− ϕ

_{2'}. Total power production is the sum of thermal, substantial and electric components, i.e.,:

_{ik}I

_{k}in Equation (39)). While many fuel cell systems are nonlinear, i.e., possess current dependent resistances, the dependence is often weak, so the linear model can be a good approximation. Below we develop a simple theory of power limits for these systems. Based on experiments, these power limits are exemplified in Figure 5.

**Figure 5.**Experimental voltage-current density and power-current density characteristics of the SOFC for various temperatures. Continuous lines represent the Aspen Plus

^{TM}calculations testing the model versus the experiments. The lines were obtained in Wierzbicki’s M.Sc thesis supervised by S. Sieniutycz and J. Jewulski [8]. Points refer to experiments of Wierzbicki and Jewulski in Warsaw Institute of Energetics (Wierzbicki [8], and his ref. [9]).

_{FC}= ΔG/ΔH or η

_{FC}= −W/ΔH, which are commonly applied to many fuel cell systems, can easily achieve numerical values much higher than ¼ [power ratio of Equations (44) and (45)]. They are, however, first-law efficiencies defined in a different way than the power ratios P

_{mp}/P

_{F}satisfying Equations (44) and (45).

## 8. Power Consumption Limits in Thermochemical Systems

**Figure 7.**A scheme of one-stage drying operation with a heat pump 1 and a continuous co-current dryer D

^{1}with falling particles. The multistage idea means, of course, the repetition of this single-stage set in the next stages.

**Figure 8.**Changes of gas states in a multistage work-assisted drying operation on the gas enthalpy - gas humidity diagram, i-X. Primed states refer to temperatures of fluids circulating in the heat pumps which heat gases supplied to dryers 1, 2...n.

^{1}, is described by an expression [compare with Equation (8)]:

^{1}is the energy supply to the drying gas in the condenser of the heat pump, and u

^{1}= −q

^{1}/g>0 is a measure of this energy supply in the temperature units. We may include the effect of internal irreversibilities within the heat pump in which case the performance coefficient of the support heat pump contains imperfection factor Φ as the multiplicative factor of the bath temperature T

^{e}.

^{0}[also ${X}_{1}^{n}$ = X

^{n−1}= X

_{s}(T

^{n−1}), for n = 2,..N] we find the mechanical energy consumption at the stage:

^{1}and θ

^{1}(consider difference constraint describing ΔT

^{n}= u

^{n}θ

^{n}for n = 1):

^{1}= X

_{s}(T

^{1}) resulting from the outlet phases equilibrium in the first dryer is incorporated in the second work expression. In terms of the adiabatic temperature at the second stage:

^{2}= θ

^{1}+ θ

^{2}there are two free controls: θ

^{1}and T

^{1}. Thus we can accomplish the power minimization procedure.

^{1}and an optimal heat transfer area of the first heat pump a

^{1}present in the control variable θ

^{1}. The requirement of sufficiently low final moisture content in a solid defines the amount of the evaporated moisture per unit time. The optimization can be generalized to the N-stage cascade system.

_{ik}I

_{k}in Equation (57). While many thermal separation systems and fuel cell electrolyzers are nonlinear, i.e., possess current dependent resistances, the dependence is often weak, so a linear model can be a good approximation. Below, by applying Equation (57) we shall attempt to develop a simple evaluation of power limits for heat pumps and separation systems under the specified assumptions.

**I**= (I

_{s}, I

_{n}, I

_{e}), the enlarged vector of potentials $\tilde{\mathsf{\mu}}$ = (Τ, μ, ϕ), and the related resistance tensor R, Equation (57) can be written in a simple form:

**μ**

_{1}−

**μ**

_{2}are given constants. Therefore, in systems with constant resistances, we are confronted with a simple minimization problem for a quadratic power consumption function p. While the dimensionality of the potential vector will often be quite large in real systems, the structure of Equation (58) will be preserved whenever the power expression will be considered in the above matrix-vector notation.

## 9. Final Remarks

_{CAN}= 0.293) and the experimental thermal efficiency (η

_{exp}= 0.350). By simple economical considerations De Vos also explained why the actual efficiency of the engine is larger than its CAN efficiency (De Vos [14]).

## Nomenclature

a | temperature exponent in an exchange equation [-] |

a_{0} = 4σ/c | radiation constant related to the Stefan-Boltzmann constant [J m ^{−3}K^{−4}] |

a_{v} | exchange area per unit volume [m ^{−1}] |

E^{0}, E_{0} | Nernst and idle run voltages [V] |

$\dot{G}$ | resource flux [g s ^{−1}], [mols^{−1}] |

g_{1},g | partial and overall conductance [J s ^{−1}K^{−a}] |

f_{0}, f_{i} | profit rate and process rates [-] |

H, $\tilde{H}$ | standard and enlarged Hamiltonian functions |

H_{TU} | height of transfer unit [m] |

h | numerical value of Hamiltonian [J m ^{3}K^{−1}] |

h, h_{v} | specific and volumetric enthalpies [J g ^{−1}, J m^{−3}] |

$\tilde{I}$ | current vector, Equation (39) |

i | electric current density [A m ^{−2}] |

n | flux of fuel reagents [g s ^{−1}, mols^{−1}] |

p = $\dot{W}$ | power output [J s ^{−1}] |

${p}_{m}^{0}$ | molar constant of photons density [mol m ^{−2}K^{−3}s^{−1}] |

q | heat flux between a stream and power generator [J s ^{−1}] |

Q | total heat flux involving transferred entropies [J s ^{−1}] |

$\tilde{R}$ | resistance tensor, Equation (39) |

S, S_{σ} | entropy and entropy produced [J K ^{−1}] |

s, s_{v} | specific and volumetric entropy [J K ^{−1}g^{−1}, J K^{−1}m^{−3}] |

T | vector transformation function |

T | variable temperature of resource [K] |

T_{1}, T_{2} | bulk temperatures of reservoirs 1 and 2 [K] |

T_{1’}, T_{2’} | temperatures of circulating fluid (Figure 2) K |

T^{e} | temperature of the environment [K] |

${T}^{\prime}$ | Carnot temperature control [K] |

$\dot{T}$ = u | rate of control of T in non-dimensional time [K] |

t | time, holdup time [s] |

u and υ | rate controls, dΤ/dτ and dT/dt [K, K s ^{−1}] |

V | voltage, maximum work function, respectively [V, J mol ^{−1}] |

v | velocity of resource stream [m s ^{−1}] |

W | work produced, positive in engine mode [J] |

w | specific work at flow or power per unit molar flux [J mol ^{−1}] |

X,Y | |

x | mass fraction [-] |

z | adjoint variable [-] |

## Greek Symbols

α_{1}, α‘ | partial and overall heat coefficients referred to respective cross-sections [J m ^{−2}s^{−1}K^{−1}] |

β | coefficient of radiation transfer related to molar constant of photons density
${p}_{m}^{0}$ and Stefan-Boltzmann constant;
$\beta =\sigma {a}_{v}{c}_{h}^{-1}{({p}_{m}^{0})}^{-1}$[s ^{−1}] |

ε | total energy flux, conservative along a conductor [J s ^{−1}] |

η = p/q_{1} | first-law thermal efficiency [-] |

χ = ρc_{v}(α'a_{v})^{−1} | time constant assuring the identity of ratio t/χ with number of transfer units [s] |

μ | chemical potential [J mol ^{−1}] |

${\mu}^{\prime}$ | Carnot chemical potential [J mol ^{−1}] |

Φ | factor of internal irreversibility [-] |

ϕ | electric potential [V] |

σ | Stefan-Boltzmann constant for radiation [J m ^{−2} s^{−1}K^{−4}] |

σ_{s} | entropy production of the system [J K ^{−1}s^{−1}] |

ξ | intensity index [-] |

ζ | chemical efficiency [-] |

τ | dimensionless time or number of transfer units [-] |

## Subscripts

C | Carnot point |

g | gas |

m | molar flow |

s | entropy, solid |

v | per unit volume |

1,2 | bulks of first and second fluid |

1',2' | circulating fluid |

0 | idle run voltage |

## Superscripts

e | environment |

i | initial state |

f | initial state |

0 | ideal (equilibrium) voltage |

‘ | Carnot state |

## Abbreviations

CNCA | Chambadal-Novikov-Curzon-Ahlborn engine |

DP | Dynamic Programming |

HJB | Hamilton-Jacobi-Bellman equation |

## Acknowledgments

## Appendix: Entropy Source and Power Yield in Terms of Carnot Variables

_{1}and q

_{2}identified with the so-called senstive heat (the heat attributed to thermal agitations in the continuum medium). In this paper we shall also define and use other flux related to heat q, the so-called total heat flux which contains the product of temperature T and the sum of partial entropies of the species multiplied by the involved mass fluxes:

_{2}from Equation (A1) with the help of the engine’s energy balance:

_{1}in the system with a complete conversion. A transformed form of this equation is the power formula:

_{s}in the system.

_{2}and n

_{2}with the help of Equations (A5) and (A6) yields:

_{2}:

_{2}satisfies the Carnot formula. When an input energy flux is given, Equation (A15) is suitable to determine power production or consumption in various steady and unsteady systems. Whenever internal dissipation effects prevail in the power generation zone, an effective environment temperature ΦT

_{2}should appear in Equation (A15) in place of T

_{2}(Section 2).

_{1}= T(t) which describes the change of temperature of the resource fluid in time, whereas a control curve may be represented by efficiency η(t) or Carnot temperature T'(t). The latter quantity, defined by Equation (A14), is particularly suitable in describing driving forces and resource relaxations in dynamical energy systems. Whenever T'(t) differs from T(t) the resource is relaxing to the environment with a finite rate associated with the efficiency deviation from the Carnot efficiency. Only when T'(t) = T(t) the efficiency is Carnot, but this corresponds with an infinitely slow relaxation rate of the resource to the thermodynamic equilibrium. In real processes (those with finite relaxation rates) thermal efficiencies are always lower than Carnot, corresponding with Carnot temperatures T'(t) lower than the resource temperatures T(t). In chemical systems the role similar to T’ is played by the Carnot chemical potential, described below.

_{1}− A

_{2}= 0, undergoes in the system [1,13].

_{1}− A

_{2}= 0 (isomerisation or phase change of A

_{1}into A

_{2}), takes the following simple form:

_{1}, satisfying in the considered case an expression ${Q}_{1}\equiv {\epsilon}_{1}-{\mu}_{1}{n}_{1}$, we finally obtain:

_{1}can also be expressed as the sum Q

_{1}= q

_{1}+ T

_{1}s

_{1}n

_{1}. The resulting Equations (A19) and (A20) are formally equivalent to an expression obtained for the process of purely dissipative exchange of energy and matter between two bodies with temperatures T

_{1}and T' and chemical potentials μ

_{1}and μ'.

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Sieniutycz, S.
An Unified Approach to Limits on Power Generation and Power Consumption in Thermo-Electro-Chemical Systems. *Entropy* **2013**, *15*, 650-677.
https://doi.org/10.3390/e15020650

**AMA Style**

Sieniutycz S.
An Unified Approach to Limits on Power Generation and Power Consumption in Thermo-Electro-Chemical Systems. *Entropy*. 2013; 15(2):650-677.
https://doi.org/10.3390/e15020650

**Chicago/Turabian Style**

Sieniutycz, Stanisław.
2013. "An Unified Approach to Limits on Power Generation and Power Consumption in Thermo-Electro-Chemical Systems" *Entropy* 15, no. 2: 650-677.
https://doi.org/10.3390/e15020650