معادلات هامیلتون-ژاکوبی-بلمن و برنامه ریزی پویا برای به حداکثر رساندن آرامش قدرت تابش

We treat simulation and power optimization of nonlinear, steady and dynamical generators of mechanical energy, in particular radiation engines. In dynamical cases, associated with downgrading of resources in time, real work is a cumulative effect obtained from a nonlinear fluid, set of engines, and an infinite bath. Dynamical state equations describe resources upgrading or downgrading in terms of temperature, work output and process controls. Recent formulae for converter’s efficiency and generated power serve to derive Hamilton–Jacobi equations for the trajectory optimization. The relaxation curve of typical nonlinear system is non-exponential. Power extremization algorithms in the form of Hamilton–Jacobi–Bellman equations (HJB equations) lead to work limits and generalized availabilities. Optimal performance functions depend on end states and the problem Hamiltonian, h. As an example of limiting work from radiation, a generalized exergy flux of radiation fluid is estimated in terms of finite rates quantified by Hamiltonian h. In many systems governing HJB equations cannot be solved analytically. Then the use of discrete counterparts of these equations and numerical methods is recommended. Algorithms of discrete dynamic programming (DP) are particularly effective as they lead directly to work limits and generalized availabilities. Convergence of these algorithms to solutions of HJB equations is discussed. A Lagrange multiplier λ helps to solve numerical algorithms of dynamic programming by eliminating the duration constraint. In analytical discrete schemes, the Legendre transformation is a significant tool leading to the original work function.

An important class of research on energy limits involves nonlinear systems driven by fluids that are restricted in their amount or flow, i.e. play role of resources. A resource is a valuable substance used in a limited amount in a practical process. Value of the resource can be quantified thermodynamically by specifying its exergy, a maximum work that can be delivered when the resource relaxes to the equilibrium. Reversible relaxation of the resource is associated with the classical exergy. When some dissipative phenomena are allowed generalized exergies are found. They include the resource availability and a minimum work lost during its production. In the classical exergy only the first property is essential. To calculate an exergy, knowledge of a work integral is required. For thermal problems its integrand is the product of thermal efficiency and the differential of exchanged energy. Various dissipation models lead to diverse thermal efficiencies that deviate from the Carnot efficiency. In fact, generalized exergies quantify somehow these deviations. Formally, an exergy follows from the principal function of a variational problem for extremum work under suitable boundary conditions. Other components are optimal trajectory and optimal control. In thermal systems the trajectory is characterized by temperature of the resource, T(t), whereas a suitable control is Carnot temperature T′(t) defined in our previous work [1] and [2]. Whenever T′(t) differs from T(t) the resource relaxes to the environment with a finite rate and the system’s efficiency deviates from that of Carnot. Only in the case 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 with the environmental fluid. The structure of this paper is as follows. Section 2 discusses various aspects of steady and dynamical optimization of power yield. Quantitative analysis of processes with resource’s downgrading (in the first reservoir) and issues regarding generalization of the classical exergy for finite rates are presented in Section 3. Sections 4, 5 and 6 display various Hamilton–Jacobi–Bellman (HJB) and Hamilton–Jacobi equations for extremum power production (consumption). Extensions, highlighting systems with complex kinetics (e.g. radiation) and internal dissipation are treated in Section 7. Analytical formulae for generalized exergies of some nonlinear systems are discussed in Section 8. Next, in view of severe difficulties in getting analytical solutions for systems with nonlinear kinetics discretized (difference) equations and numerical approaches are considered. Section 9 displays difference equations obtained from discretization of the continuous model of power production from the black radiation and presents the dynamic programming equation (DP equation) of the problem. Section 10 discusses convergence conditions of discrete DP schemes to solutions of continuous HJB equations. Section 11 elucidates the solving method by discrete approximations and introduces a Lagrange multiplier as a time adjoint. Section 12 shows the significance of the Legendre transform in recovering original work functions. Section 13 describes numerical procedures using dynamic programming, whereas Section 14 discusses dimensionality reduction in numerical DP algorithms. Section 15 presents most essential conclusions. The size limitation of the present paper does not allow for inclusion of all suitable derivations to make this paper self-contained, thus the reader may need to turn to some previous works [1], [2], [4] and [5].

In this research we considered energy limits in dynamical energy systems driven by nonlinear fluids that are restricted in their amount or flow, and, as such, play role of resources. We discussed main aspects of analytical HJB theory for continuous systems and various examples of HJB equations in nonlinear power generation systems. Applications of HJB theory, subject to appropriate boundary conditions (the process or its inversion end at the equilibrium with the environment), lead to various finite-rate generalizations of the standard availability (exergy). Processes associated with generalized availabilities are characterized by presence of imperfect phenomena as, e.g., heat conduction or non-ideal compression and expansion. In modes departing from the equilibrium the generalized exergy is larger than in their inversions approaching the equilibrium. Bounds for mechanical energy yield or consumption, provided by generalized exergies, are stronger than those defined by the classical exergy (enhanced bounds). Analytical solutions are obtained for systems with linear kinetics, and their extensions are discussed for those with nonlinear kinetics and internal dissipation. For radiation fluids analytical difficulties appear, associated with the use of Stefan–Boltzmann equation in its exact form. These difficulties are avoided in the pseudo-Newtonian models [with state dependent exchange coefficients α(T3)] and by use of numerical DP algorithms. Specific results show complex, non-exponential form of the radiation relaxation during the power production process. We have also considered numerical approaches to power generation problems, which apply the dynamic programming method. Convergence of computational DP algorithms to solutions of corresponding HJB equations was shown. Lagrangian multipliers associated with duration constraint were used to reduce dimensionality of some power production problems. Legendre transform has been applied to recover original work functions. Other important application of the considered approach involves separation systems, chemical energy systems, and, especially, fuel cells. Fig. 5 depicts work limits for real and reversible heat pumps, separators and energy generators. Systems with work consumption are described by function R (T ,τ ), systems with work production – by function V(T,τ)V(T,τ). lbws is a line of lower bound for work supply, ubwp – a line of upper bound for work production. “Endoreversible limits” correspond with curves for Φ=1Φ=1; weaker reversible limits are represented by the straight line Rrev=VrevRrev=Vrev. Dashed lines mark regions of possible improvements when imperfect thermal machines are replaced by those with better performance coefficients, terminating at endoreversible limits with Carnot energy generators. Full-size image (32 K) Fig. 5. Influence of internal irreversibilities Φ on limiting finite-rate work generated in engines and consumed in heat pumps (T0=Te)(T0=Te). Example for continuous heat-pump system with Φ=0.5Φ=0.5 and engine system with Φ=1.5Φ=1.5. Figure options For a fixed change of system state reversible upper bound Vrev achieved in production modes equals to reversible lower bound Rrev achieved in consumption modes. For irreversible bounds the equality does not hold, and a lower bound of R is larger than upper bound of V. Note a similarity of this plot to charts characterizing generalized exergies [4]. This similarity is a suitable starting point to investigate energy generation in electrochemical systems.