Nonlinear Heat Transfer Laws Research Articles

Thermal stability analysis of nuclear and fossil fuel power plants including the Dulong-Petit heat transfer law and economic features

The growing interest in leveling costs in electricity generation has led, on the one hand, to establish optimization criteria with economic parameters, and on the other, to study thermal stability for power plants models. In this work, a non-endoreversible heat engine model, as well as a non-linear heat transfer law (Dulong-Petit) are considered. Additionally, three performance regimes in the so-called profit function (aF) are analyzed: Maximum power output (P), maximum efficient power (Pη) and maximum generalized ecological function (E). By means of an approximate analytical expression for the efficiencies, we present local and global stability analysis for three different types of power plants (nuclear, combined-cycle and simple-cycle ones), including the most representative running costs (investment and fuel ones). We show that for less dissipative regimes (Pη and E), the disturbed internal temperatures return quickly to their respective steady states within the physical and economic intervals. This fact is best visualized by the direct Lyapunov method, i.e, there is no surface such that some steady state is globally asymptotically stable.

Optimization and Entropy Production: Application to Carnot-Like Refrigeration Machines.

Several optimization models of irreversible reverse cycle machines have been developed based on different optimization criteria in the literature, most of them using linear heat transfer laws at the source and sink. This raises the issue how close to actual operation conditions they are, since the heat transfer law on the phase-change processes is dependent on ΔT3. This paper addresses this issue by proposing a general model for study and optimization of thermal machines with two heat reservoirs applied to a Carnot-like refrigerator, with non-linear heat transfer laws and internal and external irreversibility. The optimization was performed using First and Second Law of Thermodynamics and the Lagrange multipliers method. Thus, several constraints were imposed to the system, also different objective functions were considered, allowing finding the optimum operating conditions, as well as the limited variation ranges of the system parameters. Results show that the nature of the heat transfer laws affects the optimum values of system parameters for obtaining maximum performances and also their magnitude. Sensitivity studies with respect to system several parameters are presented. The results contribute to the understanding of the system limits in operation under different constraints and allow choosing the most convenient variables in given circumstances.

Reconstruction of a nonlinear heat transfer law from uncomplete boundary data by means of infrared thermography

Heat exchange between a conducting plate and the environment is described here by means of an unknown nonlinear function F of the temperature u. In this paper we construct a method for recovering F by means of polynomial expansion, perturbation theory and the toolbox of thermal inverse problems. We test our method on two examples: In the first one, we heat the plate (initially at ) from one side, read the temperature on the same side and identify the heat exchange law on the opposite side (active thermography); in the second example we measure the temperature of one side of the plate (initially at ) and study the heat exchange while cooling (passive thermography).

An analytical study of the endoreversible Curzon–Ahlborn cycle for a non-linear heat transfer law

Abstract In the present article, an endoreversible Curzon–Ahlborn engine is studied by considering a non-linear heat transfer law, particularly the Dulong–Petit heat transfer law, using the `componendo and dividendo' rule as well as a simple differentiation to obtain the Curzon–Ahlborn efficiency as proposed by Agrawal in 2009. This rule is actually a change of variable that simplifies a two-variable problem to a one-variable problem. From elemental calculus, we obtain an analytical expression of efficiency and the power output. The efficiency is given only in terms of the temperatures of the reservoirs, such as both Carnot and Curzon–Ahlborn cycles. We make a comparison between efficiencies measured in real power plants and theoretical values from analytical expressions obtained in this article and others found in literature from several other authors. This comparison shows that the theoretical values of efficiency are close to real efficiency, and in some cases, they are exactly the same. Therefore, we can say that the Agrawal method is good in calculating thermal engine efficiencies approximately.

Identification of nonlinear heat transfer laws from boundary observations

We consider the problem of identifying a nonlinear heat transfer law at the boundary, or of the temperature-dependent heat transfer coefficient in a parabolic equation from boundary observations. As a practical example, this model applies to the heat transfer coefficient that describes the intensity of heat exchange between a hot wire and the cooling water in which it is placed. We reformulate the inverse problem as a variational one which aims to minimize a misfit functional and prove that it has a solution. We provide a gradient formula for the misfit functional and then use some iterative methods for solving the variational problem. Thorough investigations are made with respect to several initial guesses and amounts of noise in the input data. Numerical results show that the methods are robust, stable and accurate.

A Finite-Time Thermal Cycle Variational Optimization with a Stefan–Boltzmann Law for Three Different Criteria

This work shows the power of the variational approach for studying the efficiency of thermal engines in the context of the Finite Time Thermodynamics (FTT). Using an endoreversible Curzon–Ahlborn (CA) heat engine as a model for actual thermal engines, three different criteria for thermal efficiency were analyzed: maximum power output, ecological function, and maximum power density. By means of this procedure, the performance of the CA heat engine with a nonlinear heat transfer law (the Stefan–Boltzmann law) was studied to describe the heat exchanges between the working substance and its thermal reservoirs. The specific case of the Müser engine for all the criteria was analyzed. The results confirmed some previous findings using other procedures and additionally new results for the Müser engine performance were obtained.

Analysis of the Impact of Nonlinear Heat Transfer Laws on Temperature Distribution in Irradiated Biological Tissues: Mathematical Models and Optimal Controls

In this paper, we consider nonlinear control problems governed by some generalized transient bioheat transfer-type models with the nonlinear Robin boundary conditions. The control estimates the blood perfusion rate, the heat transfer parameter, the distributed energy source terms, and the heat flux due to the evaporation, which affect the effects of thermal physical properties on the transient temperature of biological tissues. The result can be very beneficial for thermal diagnostics in medical practices, for example, for laser surgery, photo and thermotherapy for regional hyperthermia often used in treatment of cancer. First, the mathematical models are introduced and the existence, uniqueness, and regularity of a solution of the state equation are proved as well as the stability and maximum principle under extra assumptions. Afterwards, the optimal control problem is formulated in order to control the online temperature given by radiometric measurement. We prove that an optimal solution exists and obtain necessary optimality conditions. Some strategy for numerical realization based on the adjoint variables are provided.

Natural linearization for the identification of nonlinear heat transfer laws

Natural linearization for the identification of nonlinear heat transfer laws

Natural linearization for the identification of nonlinear heat transfer laws

A fast algorithm for numerical identification of nonlinear heat transfer laws based on a natural linearization of the corresponding inverse problem is introduced. Its theoretical background is discussed, and the numerical tests show that it is suitable for problems with perturbed data. The proposed approach can be also used for other parameter identification problems, where one wants to recover an unknown nonlinear parameter β(u) from distributed noisy observations of the state u.

On Some Nonendoreversible Engine Models with Nonlinear Heat Transfer Laws

In this work, we analyze a nonendoreversible thermal engine model with a nonlinear heat transfer law between the heat reservoirs and the working fluid under two optimization criteria: the maximum power regime and the so-called ecological criterion. We find that this nonendoreversible model has a similar behaviour to that shown by De Vos (Am. J. Phys. 53, 570 (1985)) for endoreversible models with two thermal conductances with only one superior conductance and with only one inferior conductance, respectively. The model is compared with two sets of real power plants, the first one containing power plants of old design (before 1960's) and the second one being formed by modern nuclear power plants. Our results suggest that the first group was designed under conditions which are reminiscent of a maximum power regime and the second one under an ecological-like criterion. We also study some general properties of nonendoreversible thermal engine models.

Unified working regime of irreversible Carnot-like heat engines with nonlinear heat transfer laws

We present the results of efficiency and power output for irreversible Carnot-like heat engines with nonlinear inverse, Dulong–Petit and Stefan–Boltzmann heat transfer laws when optimized with a recent criterion. A unified working regime is found intermediate between those predicted by the maximum efficiency and maximum power for all realistic values of the parameters accounting for the irreversibilities: finite rate heat transfer between the working fluid and the external heat sources, internal dissipation in the working fluid, and heat leak between reservoirs.

A variational optimization of a finite-time thermal cycle with a nonlinear heat transfer law

In this paper we use a variational approach to study an endoreversible Curzon–Ahlborn–Novikov (CAN) heat engine under both maximum power and maximum ecological function conditions. By means of this procedure we analyze the performance of a CANheat engine with a nonlinear heat transfer law (the Dulong–Petit law) to describe the heat exchanges between the working substance and its thermal reservoirs. Our results are consistent with previous ones obtained by means of other procedures. In addition, we obtain expressions for the temperatures of the isothermal branches of the working fluid under maximum power conditions. Finally, we present an expression for a kind of nonendoreversible Carnot efficiency.

Stability estimates for the identification of nonlinear heat transfer laws

Consider the heat equation with a nonlinear function in the boundary condition which depends only on the solution u of the initial-boundary-value problem. The unknown function belongs to a set of admissible uniformly Lipschitz continuous functions. For this problem stability estimates of the form with different norms are derived. Finally an interesting example is discussed.

Identification of nonlinear heat transfer laws by optimal control

(1994). Identification of nonlinear heat transfer laws by optimal control. Numerical Functional Analysis and Optimization: Vol. 15, No. 3-4, pp. 417-434.

Endoreversible thermal cycle with a nonlinear heat transfer law

We calculate the efficiency of an endoreversible Carnot-type cycle in the maximum power regime by using a nonlinear heat transfer law (the so-called Dulong and Petit’s law of cooling). The results obtained from this model compare well (around 99% in some cases) with observed efficiencies for several power plants. The considered law of cooling includes conductive- convective and radiative contributions to the heat exchange between the working fluid and its surroundings. Our calculations improve considerably those obtained by means of a linear heat transfer law for the same power sources. We also analyze a nuclear power plant using an ecological optimization criterion for finite-time heat engines.