Identification of heat transfer dynamics for non-modal analysis of thermoacoustic stability

A systematic approach for non-modal stability analysis of thermoacoustic systems with a localized heat source is proposed. The response of the heat source to flow perturbations is obtained from unsteady computational fluid dynamics combined with correlation-based linear system identification. A model for the complete thermoacoustic system is formulated with a Galerkin expansion technique, where the heat source is included as an acoustically compact element. The eigenvalues of the resulting system are obtained from discretization of the solution operator, the maximum growth factor is estimated from the pseudospectra using Kreiss’ theorem.The approach is illustrated with a simple Rijke tube configuration. Results obtained with a simple “baseline” model for the heat source dynamics based on King’s law - widely used in hot wire anemometry - are compared against the more advanced treatment developed here. Analysis of pseudospectra diagrams shows that the choice of the heat source model does influence the sensitivity of eigenvalues to perturbations and hence the non-normal behavior. The maximum growth factor for the system with the heat source model based on King’s law is more sensitive to changes in the heat source location than the CFD-based heat source model.     

Boundary integral solution of the two-dimensional heat equation

Here we consider initial boundary value problems for the heat equation by using the heat potential representation for the solution. Depending on the choice of the representation we are led to a solution of the various boundary integral equations. We discuss the solvability of these equations in anisotropic Sobolev spaces. It turns out that the double-layer heat potential D and its spatial adjoint D′ have smoothing properties similar to the single-layer heat operator. This yields compactness of the operators D and D′. In addition, for any constant c ≠ 0, cI + D′ and cI + D′ are isomorphisms. Based on the coercivity of the single-layer heat operator and the above compactness we establish the coerciveness of the hypersingular heat operator. Moreover, we show an equivalence between the weak solution and the various boundary integral solutions. As a further application we describe a coupling procedure for an exterior initial boundary value problem for the non-homogeneous heat equation.     

The stress-strain state of shells under the action of concentrated heat sources

We construct a fundamental solution of a static problem of heat conduction for thin isotropic shells of positive curvature under the action of concentrated heat sources. We study the problems involved in simplifying the heat equations. Four figures. Two tables. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 70–80.     

Obtaining upper bounds of heat kernels from lower bounds

We show that a near‐diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on‐diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off‐diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near‐diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. © 2007 Wiley Periodicals, Inc.     

Optimal paths for minimizing entransy dissipation during heat transfer processes with generalized radiative heat transfer law

A common of finite-time heat transfer processes between high- and low-temperature sides with generalized radiative heat transfer law [q ∝ Δ(Tⁿ)] is studied in this paper. In general, the minimization of entropy generation in heat transfer processes is taken as the optimization objective. A new physical quantity, entransy, has been identified as a basis for optimizing heat transfer processes in terms of the analogy between heat and electrical conduction recently. Heat transfer analyses show that the entransy of an object describes its heat transfer ability, as the electrical energy in a capacitor describes its charge transfer ability. Entransy dissipation occurs during heat transfer processes, as a measure of the heat transfer irreversibility with the dissipation related thermal resistance. Under the condition of fixed heat load, the optimal configurations of hot and cold fluid temperatures for minimizing entransy dissipation are derived by using optimal control theory. The condition corresponding to the minimum entransy dissipation strategy with Newtonian heat transfer law (n = 1) is that corresponding to a constant heat flux rate, while the condition corresponding to the minimum entransy dissipation strategy with the linear phenomenological heat transfer law (n = −1) is that corresponding to a constant ratio of hot to cold fluid temperatures. Numerical examples for special cases with Newtonian, linear phenomenological and radiative heat transfer law (n = 4) are provided, and the obtained results are also compared with the conventional strategies of constant heat flux rate and constant hot fluid (reservoir) temperature operations and optimal strategies for minimizing entropy generation. Moreover, the effects of heat load changes on the optimal hot and fluid temperature configurations are also analyzed.     

Numerical study of melting in an enclosure with discrete protruding heat sources

In order to explore the capability of a solid–liquid phase change material (PCM) for cooling electronic or heat storage applications, melting of a PCM in a vertical rectangular enclosure was studied. Three protruding generating heat sources are attached on one of the vertical walls of the enclosure, and generating heat at a constant and uniform volumetric rate. The horizontal walls are adiabatic. The power generated in heat sources is dissipated in PCM (n-eicosane with the melting temperature, T_m = 36 °C) that filled the rectangular enclosure. The advantage of using PCM is that it is able to absorb high amount of heat generated by heat sources due to its relatively high energy density. To investigate the thermal behaviour and thermal performance of the proposed system, a mathematical model based on the mass, momentum and energy conservation equations was developed. The governing equations are next discretised using a control volume approach in a staggered mesh and a pressure correction equation method is employed for the pressure–velocity coupling. The PCM energy equation is solved using the enthalpy method. The solid regions (wall and heat sources) are treated as fluid regions with infinite viscosity and the thermal coupling between solid and fluid regions is taken into account using the harmonic mean of the thermal conductivity method. The dimensionless independent parameters that govern the thermal behaviour of the system were next identified. After validating the proposed mathematical model against experimental data, a numerical investigation was next conducted in order to examine the thermal behaviour of the system by analyzing the flow structure and the heat transfer during the melting process, for a given values of governing parameters.     

Regularity for the evolution of p-harmonic maps

This paper presents our study of regularity for p-harmonic map heat flows. We devise a monotonicity-type formula of scaled energy and establish a criterion for a uniform regularity estimate for regular p-harmonic map heat flows. As application we show the small data global in the time existence of regular p-harmonic map heat flow.     

Numerical modelling of EHD effects on heat transfer and bubble shapes of nucleate boiling

In this article, mathematical and numerical models are developed to study pure electrohydrodynamic (EHD) effects on heat transfer and bubble shapes when an initial bubble attached to a superheated horizontal wall in nucleate boiling. In the modelling of EHD effects on heat transfer, an undeformed bubble is considered; the electric body force and Joule heat are added to the momentum and energy equations; governing equations for heat, fluid flow and electric fields are coupled numerically and solved using a non-orthogonal body-fitted mesh system with necessary interfacial treatments at the gas–liquid boundary. While, to study the pure effect of EHD on the deformation of the bubble, the evaluation of a deformable bubble without heat transfer is simulated by volume of fluid (VOF) method based on an axial symmetric Cartesian coordinate system. The simulations indicate that EHD can effectively enhance heat transfer rate of nucleate boiling by influencing the motion of the ring vortex around the bubble and that bubble can be elongated due to the pull in axial direction and push in the negative radial direction by the electric field force.     

Duality on gradient estimates and Wasserstein controls

We establish a duality between Lp-Wasserstein control and Lq-gradient estimate in a general framework. Our result extends a known result for a heat flow on a Riemannian manifold. Especially, we can derive a Wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound. By applying our result to a subelliptic heat flow on a Lie group, we obtain a coupling of heat distributions which carries a good control of their relative distance.     

A fundamental solution of the equations of thermoelastic bending of orthotropic plates

We construct a fundamental solution of the equations of thermoelastic bending of orthotropic plates under symmetric heat transfer and a linear heart distribution over the thickness. On the lateral surfaces convective heat transfer with a surrounding medium occurs in accordance with Newton's law. The effect of the orthotropy parameters and heat transfer on the magnitude of the bending moment and torque is studied numerically. Two figures. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 87–92.     

Modeling of thermal phenomena in single laser beam and laser-arc hybrid welding processes using projection method

This paper concerns mathematical and numerical modeling of thermal phenomena accompanying single laser and laser-arc hybrid butt welding of steel sheets. Coupled heat transfer and fluid flow in the fusion zone were described respectively by transient heat transfer equation and Navier–Stokes equation. Laser beam and electric arc heat sources were modeled using different heat source power distributions. Latent heat associated with the material’s state changes, buoyancy forces and liquid material flow through a porous medium were taken into account in considerations. Differential governing equations were numerically solved using projection method combined with finite volume method. Elaborated solution algorithm was implemented into computer solver used for simulation of heat transfer and fluid flow during welding. The geometry of the weld and heat affected zone as well as cooling rates were estimated on the basis of numerically obtained temperature field.     

Probabilistic representations of solutions to the heat equation

In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if ϕ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition ϕ, is given by the convolution of ϕ with the heat kernel (Gaussian density). Our results also extend the probabilistic representation of solutions of the heat equation to initial conditions that are arbitrary tempered distributions.     

完备流形上带位势的热核估计

在完备非紧流形上获得了关于带位势热方程正解的梯度估计；接着，利用测地线的技巧获得了Harnack不等式；进一步，建立了两个积分不等式，综合Harnack不等式获得了热核的上下界；最后，利用函数的结果来控制p形式的热核。     

Thermal instability of a horizontal layer of micropolar fluid with heat source

The thermal instability of a horizontal layer of micropolar fluid which loses heat throughout its volume at a constant rate has been considered. The influence of the various micropolar fluid parameters on the onset of convection have been analysed. It is found that heat source and heat sink have the same destabilising effect in micropolar fluid. It is observed that the horizontal dimension of the cells remains insensitive to the changes in the micropolar fluid parameters and also to the heat source parameterQ except forQ values near zero, where the change is drastic. Further, it is observed that though the vertical component of velocity and the curl of microrotation do not vanish anywhere between the two boundaries forQ=0, they vanish at a point nearer to the lower boundary even for a small change in theQ value.     

Uniqueness of the finite-difference solution of the inverse problem of heat conduction

The article examines the problem of determination of the coefficients of heat conduction and heat capacity from a system of difference equations for the equation of heat conduction with additional information about the solution of the difference problem. Uniqueness of the solution of the inverse problem is proved.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach Matematicheskoi Fiziki, pp. 22–29, 1993.     

The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5], [6], [7], [8] it was assumed that the convective heat transfer parameter h_f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h_f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.     

On the heat flow of equation of surfaces of constant mean curvatures

We consider the initial and boundary value problem of heat flow of equation of surfaces of constant mean curvatures. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solutions to the heat flow that converges to zero in H ¹ exponentially as time goes to infinity.     

Numerical analysis of the thermal behaviour of a shell-and-tube heat storage unit using phase change materials

This work presents a numerical study of a latent heat storage unit (LHSU) consisting of a shell-and-tube. The shell space is filled with two phase change materials (PCMs), P116 and n-octadecane, with different melting temperatures (50 °C and 27.7 °C, respectively). A heat transfer fluid (HTF: water) flows by forced convection through the inner tube, and transfers the heat to PCMs. In order to compare the thermal performances of the latent heat storage unit using two phase change materials (LHSU2) and a single PCM (LHSU1), a mathematical model based on the conservation energy equations was developed and validated with experimental data. Several numerical investigations were conducted in order to examine the impact of the key parameters: the HTF inlet temperature (ranges from 50 to 60 °C), the mass flow rate of the HTF and the proportion mass of PCMs, on the thermal performances of the latent heat storage units using two PCMs and a single PCM, during charging process (melting). This parametric study provides guidelines for system thermal performance and design optimization.     

Determination of the coefficient in the stationary nonlinear equation of heat conduction

The article considers the problem of determining the solution-dependent coefficient of heat conductivity in a stationary nonlinear equation of heat conduction containing a parameter. Additional information for the determination of heat conductivity is provided by a function dependent on a parameter, which is obtained by solving a boundary-value problem. A uniqueness theorem is proved for the inverse problem.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 13–17, 1993.     

Analysis of performance of an iron-bath reactor using computational fluid dynamics

The performance of an iron-bath reactor has been studied using a comprehensive numerical model that combines a computational fluid dynamics approach for the gas phase and a heat and mass balance model for the bath. The model calculates:

• •coal, ore, flux and oxygen consumption;
• •post-combustion ratio (PCR);
• •heat-transfer efficiency (HTE);
• •off-gas temperature and composition;
• •heat transfer and chemical reactions between gas and iron and slag droplets; and
• •heat transfer between gas and bath, refractories and lance.

The model was validated with data reported by the Nippon Steel Corporation for a 100 t pilot plant, and the calculated and measured data are in good agreement. Modelling results showed that the dominant mechanisms of heat transfer from the gas to the bath are radiation to the slag surface and convection heat transfer to droplets.