Convective Equation Research Articles

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.

Benchmarking Electrolytes for the Solid Oxide Electrolyzer Using a Finite Element Model

The demand for green hydrogen is increasing, as it is estimated to reduce ten percent of total global green-house-gas emissions from fossil fuel. The solid oxide electrolysis cell (SOEC) is an electrochemical energy-conversion device (EECD) that produces green hydrogen via steam electrolysis. It is preferred to other EECDs for clean hydrogen production owing to its high efficiency, robust kinetics, and lack of precious-metal requirements for cell construction. Herein, we report a Multiphysics model describing the transport phenomena in the SOEC. The governing equations used in the model include a thorough description of the electrode kinetics and of the behavior of the three electrode–electrolyte interfaces in the cell. For the first time, the effect of the scandium-doped zirconia (SCGZ), yttrium-stabilized zirconia (YSZ), and gadolinium-doped ceria (GDC) electrolytes was modeled at different temperatures and pressures. By linking the convection and diffusion equations with the Butler–Volmer at shorter scales, a true representation of the cell operation was simulated. Our models show a R2 value of over 0.996 in predicting the cell-polarization curves and electrochemical properties at the given operating conditions. The impedance of the SCGZ was 0.0240 Ohm.cm2. This value was two- and four-fold lower than the values of the YSZ and GDC, respectively. Furthermore, our theoretical findings of both the polarization data and the impedance were in good agreement with the experimental data.

Nonsimilar modeling analysis of Carreau–Yasuda mixed convective flow in a porous medium subjected to Soret and Dufour influences

AbstractThis nonsimilar convection study is about the flow of Carreau–Yasuda (CY) nanofluid model above a vertically extendible surface. Convection in a fluid‐filled permeable medium has given due consideration because of its relevance in a variety of applications, including insulation, relocation of water from geothermal reservoirs, storage of nuclear waste, renewable energy, mechanical engineering, and enhanced oil reservoir recovery. By virtue of linear stretching and buoyancy effects, flow in a stationary fluid is induced along a vertical porous surface. In x‐momentum equation, linear buoyancy in the context of temperature and concentration is taken into consideration. Modeling of energy expression is done in the presence of Dufour and Soret influences. Governing differential system describing convection equations is changed into nonlinear partial differential system (PDE) by implementing applicable nonsimilar transformations. By making use of analytical local nonsimilarity (LNS) technique and bvp4c (numerical finite difference‐based algorithm), the transformed dimensionless nonsimilar structure is simulated numerically. At the end, the alteration of important nondimensional numbers is studied on transport quantities such as temperature, concentration and velocity field. The repercussions of relevant parameters on drag coefficient, Nusselt number and Sherwood number have been tabulated. Numerical simulations of nonsimilar model suggests that the velocity profile reduces due to rise in the values of Weissenberg number, porosity and suction parameter. The temperature profile is increased in comparison with the higher estimates, Eckert, and Dufour numbers. Because of larger values of Soret and Prandtl number, an increase in concentration profile is seen. Friction coefficient and Nusselt number increases with respect to higher estimations of porosity parameter, Weissenberg number and Prandtl number respectively, whereas they decrease against Dufour and Eckert variations.

$\mathbb H^1$-random attractors for 2d stochastic convective Brinkman-Forchheimer equations in unbounded domains

$\mathbb H^1$-random attractors for 2d stochastic convective Brinkman-Forchheimer equations in unbounded domains

L1/LDG algorithm for time‐Caputo space‐Riesz fractional convection equation in two dimensions

This paper proposes an L1/LDG algorithm for a time–space fractional convection equation on a bounded rectangular domain . Here, the temporal partial derivative is in the sense of Caputo with order , and the spatial partial derivatives are of Riesz type with orders in ‐ and ‐directions, respectively. L1 approximation is utilized to evaluate the temporal Caputo derivative, while the local discontinuous Galerkin (LDG) method is adopted to deal with the spatial Riesz derivative. The rigorous numerical stability analysis and error estimate are presented which are supported by illustrative numerical examples.

Physics-informed graph convolutional neural network for modeling geometry-adaptive steady-state natural convection

This paper presents a novel deep learning-based surrogate model for steady-state natural convection problem with variable geometry. Traditional deep learning based surrogate models are more or less limited by the requirement for large amounts of training data, loss of accuracy due to pixelization of the original data, the low accurate prediction near the boundaries and low geometry adaptive capability. To overcome the above challenges, the proposed natural convection prediction framework is mainly composed of a physics-informed neural network (PINN) and a graph convolutional neural network (GCN), called natural convection prediction model based on physics-informed graph convolutional network (NCV-PIGN). The GCN serves as the prediction module, inferring and predicting natural convection phenomena by considering the interactions between unstructured nodes and their neighbor; the PINN incorporates the governing equations of natural convection into the loss function of the neural network, allowing the predictions from GCN to satisfy the constraints imposed by the physical laws. The advantages of this framework are twofold: the operation principles of the GCN better align with the development of the temperature field in real situations, and the embedding of physical information strengthens the model's understanding of the flow field, accurately describing the variations of temperature gradients at the boundary positions while reducing the model's reliance on training data. Finally, to validate the superiority of the NCV-PIGN, we analyze its geometric adaptability and accuracy of prediction using single and dual heat source cases. We compare the model's prediction results at different sampling point quantities and contrast them with those of purely data-driven models. The results demonstrate that the excellent geometric adaptability and prediction capability of the proposed model can be achieved with only 20 training data and once the fully trained the model can solve natural convection problems within 3 ms. The max and mean relative errors in predicting the temperature field are less than 2% and 0.4% for both single and dual heat source cases. Compared to the pure data-driven model, the proposed model has reduced the maximum error by 65.5% and the mean error by 72%. These results validate the effectiveness of the developed NCV-PIGN model, enabling better performance of the deep learning-based surrogate models for natural convection problems.

Constructal design of vertical multiscale triangular fins in natural convection

AbstractConstructal design of vertical multiscale triangular fins in natural convection is investigated in this paper. The design consists of two parts. The first part is for single‐scale triangular fins. The objective in the first design is to reach to the highest heat transfer density from the fins for three fin angles (15°, 30°, and 45°). The single‐scale fins are placed in a horizontal array and considered as isothermal fins. The degrees of freedom are the fin angle, and the fin‐to‐fin spacing. The constraint is the fin height. The second part is for multiscale fins where small fins are placed between the large fins which are optimized in the first part. In the second part, the angles of the large and small scales fins are kept constant at (15°). The optimal fin‐to‐fin spacing which is obtained in the first part is considered a constraint in the second part. The Rayleigh numbers in this design are (Ra = 103, 104, and 105). The two‐dimensional mass, momentum, and energy equations for natural convection are solved with the finite volume method. The results show that there is a benefit of placing the small‐scale fins where the percentage increase in the heat transfer density is (10.22%) at (Ra = 103), and (50.6%) at (Ra = 105) due to existence of the small fins between the large fins.

Physics-informed graph convolutional neural network for modeling fluid flow and heat convection

This paper introduces a novel surrogate model for two-dimensional adaptive steady-state thermal convection fields based on deep learning technology. The proposed model aims to overcome limitations in traditional frameworks caused by network types, such as the requirement for extensive training data, accuracy loss due to pixelated preprocessing of original data, and inability to predict information near the boundaries with precision. We propose a new framework that consists primarily of a physical-informed neural network (PINN) and a graph convolutional neural network (GCN). The GCN serves as the prediction module and predicts thermal convection in the two-dimensional computational domain by considering the mutual influence between unstructured nodes and their neighbors. On the other hand, the PINN acts as the physical constraint module of the framework by embedding the control equation of thermal convection into the loss function of the neural network, ensuring that the inference and prediction results of the GCN comply with the constraints of the control equation. The advantages of this framework lie in two aspects. First, the computation mechanism of the GCN is more in line with the actual evolution of temperature fields. Second, the PINN enhances the cognitive ability of the surrogate model toward the convection field information. It accurately describes the changes of temperature gradient information at the boundary position and reduces the model's demand for training data. To validate the advantages of the proposed model, we gradually analyzed the model's geometric adaptability and predictive accuracy from the single cylinder case to the double cylinder case. We also investigated the impact of the number of sampling points on model training and compared the model's prediction results with those of a purely data-driven model. The results show that the proposed model exhibits good geometric adaptability and stability. With only 20 training data, the mean error of the proposed model in predicting the velocity and temperature field is less than 1% and 0.6% for the single cylinder, and less than 2% and 1% for the double cylinder case, while the mean error of the purely data-driven GCN model in predicting the velocity and temperature field is 9.4% and 6.4% for the double cylinder case. These findings demonstrate the effectiveness of the proposed physics-informed graph convolutional neural network, allowing for more accurate prediction of fluid flow and heat convection using surrogate model.

Global well-posedness and convergence to equilibrium for the Abels-Garcke-Grün model with nonlocal free energy

We investigate the nonlocal version of the Abels-Garcke-Grün (AGG) system, which describes the motion of a mixture of two viscous incompressible fluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard system characterized by concentration-dependent density and viscosity, and an additional flux term due to interface diffusion. In particular, the Cahn-Hilliard dynamics of the concentration (phase-field) is governed by the aggregation/diffusion competition of the nonlocal Helmholtz free energy with singular (logarithmic) potential and constant mobility. We first prove the existence of global strong solutions in general two-dimensional bounded domains and their uniqueness when the initial datum is strictly separated from the pure phases. The key points are a novel well-posedness result of strong solutions to the nonlocal convective Cahn-Hilliard equation with singular potential and constant mobility under minimal integral assumption on the incompressible velocity field, and a new two-dimensional interpolation estimate for the L4(Ω) control of the pressure in the stationary Stokes problem. Secondly, we show that any weak solution, whose existence was already known, is globally defined, enjoys the propagation of regularity and converges towards an equilibrium (i.e., a stationary solution) as t→∞. Furthermore, we demonstrate the uniqueness of strong solutions and their continuous dependence with respect to general (not necessarily separated) initial data in the case of matched densities and unmatched viscosities (i.e., the nonlocal model H with variable viscosity, singular potential and constant mobility). Finally, we provide a stability estimate between the strong solutions to the nonlocal AGG model and the nonlocal Model H in terms of the difference of densities.

Existence and regularity of strong solutions to a nonhomogeneous Kelvin-Voigt-Cahn-Hilliard system

A well-known diffuse interface model consists of the Navier-Stokes equations nonlinearly coupled with the convective Cahn-Hilliard equation. This system describes the evolution of an incompressible isothermal Newtonian mixture of binary fluids. In this article we investigate a variant of this model, which consists of the nonhomogeneous Kelvin-Voigt equations coupled with the Cahn-Hilliard equations. We prove the existence of global weak and strong solutions in a two and three dimensions. Furthermore, we prove some regularity results for the strong solutions and show their uniqueness in both two or three-dimensional bounded domains. Lastly, we also solve a problem of uniqueness of regular solutions that was left open by Giorgini and Temam (2020) [5].

Effect of Transverse Base Width Restraint on the Cracking Behavior of Massive Concrete

The effect of considering the third dimension in mass concrete members on its cracking behavior is investigated in this study. The investigation includes thermal and structural analyses of mass concrete structures. From thermal analysis, the actual temperature distribution throughout the mass concrete body was obtained due to the generation of heat as a result of cement hydration in addition to the ambient circumstances. This was performed via solving the differential equations of heat conduction and convection using the finite element method. The finite element method was also implemented in the structural analysis adopting the concept of initial strain problem. Drying shrinkage volume changes were calculated using the procedure suggested by ACI Committee 209 and inverted to equivalent temperature differences to be added algebraically to the temperature differences obtained from thermal analysis. Willam-Warnke model with five strength parameters is used in modeling of concrete material in which cracking and crushing behavior of concrete can be included. The ANSYS program was employed in a modified manner to perform the above analyses. A thick concrete slab of 1.5m in thickness and 10m in length was analyzed for different widths 2, 4, 8, and 10m to produce different aspect ratios (B/L) of 0.2, 0.4, 0.8, and 1.0 respectively. The results of the analyses show an increase in cracking tendency of mass concrete member as the aspect ratio of the same member is increased due to the effect of transverse base restraint. Accordingly, such effect cannot be ignored in the analysis of base restrained mass concrete structures subjected to temperature and drying shrinkage volume changes.

Application of a Partially Invariant Exact Solution of the Thermosolutal Convection Equations for Studying the Instability of an Evaporative Flow in a Channel Heated from Above

The characteristics of a stationary flow of a volatile liquid driven by a co-current gas flux in a flat horizontal mini-channel upon the non-zero transverse temperature drop are studied. We use an exact solution of the thermosolutal convection equations for describing the heat and mass transfer caused by the combined action of gas pumping, buoyancy, thermocapillarity and linear heating of the channel walls in a two-layer system. The influence of heating from above on the parameters of the ground state and the stability characteristics of the basic flow is explored using an example of the ethanol–air system. We evaluate the thresholds of the linear stability and select the most dangerous modes. Heating from above results in flow stabilization. Instability appears in the form of oscillatory cellular convective patterns.

Полулагранжевы аппроксимации оператора конвекции в симметричной форме

Рассмотрены два полулагранжевых численных метода для одномерного (по пространству) уравнения переноса с оператором в симметричной форме: эйлероволагранжев и лагранжево-эйлеров. Оба метода свободны от ограничения Куранта на соотношение шагов по времени и пространству. Причем во втором методе достигнут второй порядок аппроксимации для гладких решений и продемонстрировано отсутствие численной вязкости для разрывных решений. Purpose. The purpose of the study is the development and comparison of two numerical semi-Lagrangian methods with fulfillment of the conservation law at a discrete level. The approach is applied for the transport equation in the symmetric form, reflecting the law of conservation for the square of the transferred substance. The article presents the Euler – Lagrangian method, built on a rectangular difference grid that uses local values of characteristics to calculate the coefficients of difference equations. Lagrangian – Euler method is built on a spatial non-uniform grid obtained by crossing the characteristic trajectories of the equation with lines in time. Methodology. The integro-interpolation method is applied to derive approximations for the differential operator which allowed obtaining simple formulas connecting values of the grid function at the neighboring layers in time. Numerical calculations of characteristic trajectories are held by the Euler method or the Runge – Kutta method of the second order, depending on the required accuracy. Findings. Numerical methods with the mentioned properties are developed and numerically confirmed, convergence and discrete conservation laws for them are mathematically proved. The first order convergence for both time and space is proved for the Euler – Lagrange method. The second order convergence also in time and space is proved for the Lagrange – Euler method. Originality/value. The Euler – Lagrange and Lagrange – Euler methods for the numerical solution of the convection equation are developed. These methods induce differential conservation law at discrete level. The first and the second order of convergence correspondingly are mathematically proved for them. The Lagrange – Euler method has showed two improved aspects: firstly, it has greater order of convergence than the Euler – Lagrange one and secondly, it allows solving problems with the discontinuous solutions without smoothing them.

Numerical Modeling on the Performance of the Wind Tower Integrated into the Storied Building For Reducing Electrical Energy Consumption For Spaces Cooling

Natural convection in integrated wind tower into a high storied building is numerically investigated by solving natural convection equations with the Boussinesq approximation. The present investigation deals with velocity and temperature distribution in a wind tower integrated into a high building divided into three stories; in which the upper horizontal heated wall is vented by a fresh air jet entranced by the wind tower and hot waste air exits from different outlet openings. The purpose of this study is to evaluate the Performance of integrated wind tower into the storied building, thus providing a wind and buoyancy factors for induced natural ventilation to drive air flow through the floors. Numerical solutions of the Navier Stokes equations and energy equation have been solved by the Thomas’ algorithm. Solutions are presented for various geometrical aspect ratios and for different values of Grashof and Reynolds numbers. The results are presented in terms of streamlines, isotherms, velocity, and heat transfer intensity versus the governing control parameters in detail.

Passive thermal management strategy for cooling multiple portable electronic components: Hybrid nanoparticles enhanced phase change materials as an innovative solution

Modern electronics technologies have given great interest to phase change materials (PCMs). The present work is in line with this philosophy and deals with the effectiveness of a PCM-based heat sink and nanoparticles in insertion for thermal management of several electronic components. Present numerical study explores the solid-liquid phase change process of n-eicosane in an enclosure considered for cooling of electronics in the presence of one, two, and three protruding electronic components. To improve thermal management capability, four types of nanoparticles, namely, silver (Ag), magnesium oxide (MgO), multi-walled carbon nanotube (MWCNT), and silicon dioxide (SiO2), are inserted into the n-eicosane by single and hybrid ways at volume fractions ranging from 0 % to 4%, and by introducing a lateral fin playing the role of a motherboard. It should be noted that this study is interested in analysing the combined effects of the volume fraction of inserted nanoparticles ranging from 0 % to 4%, the type of inserted nanoparticles, and the number of electronic components installed at the heat sink on the overall heat transfer and flow structure of the fluid PCM. For this purpose, a mathematical model has been developed for the formulation of the governing equations of the conjugate natural convection. The enthalpy-porosity technique is applied for modelling the phase change process. Results reveal a significant relationship between the melting time and the operating temperature of the electronic component. A long phase change process is characterized by a high utilization of latent heat and subsequently the heat rejection becomes efficient. Hence, the longer melting times result in reduced operating temperatures. By insertion of SiO2-MWCNT hybrid nanoparticles, the melting time is increased by approximately 92 % compared to pure PCM, and the plateau temperature is mitigated by 11 °C by using one component in the enclosure. Furthermore, by increasing the number of electronic components, the flow structure, thermal field and melting front become more uniform. Thus, it can be recommended to use three electronic components with a lower heat generation rate instead of using one electronic component with three times higher heat dissipation in a PCM-based cooling system.

Pre-training strategy for solving evolution equations based on physics-informed neural networks

The physics informed neural network (PINN) is a promising method for solving time-evolution partial differential equations (PDEs). However, the standard PINN method may fail to solve the PDEs with strongly nonlinear characteristics or those with high-frequency solutions. To address this problem, we propose a novel method named pre-training PINN (PT-PINN) which can improve the convergence and accuracy of the standard PINN method by combining with the resampling strategy and the existing optimizer combination technique. The PT-PINN method transforms the difficult problem on the entire time domain to relatively simple problems defined on small subdomains. The neural network trained on small subdomains provides the neural network initialization and extra supervised learning data for the problems on larger subdomains or on the entire time-domain. Numerical experiments show that the PT-PINN successfully solves the evolution PDEs with strong non-linearity and/or high frequency solutions, including the strongly nonlinear heat equation, the Allen-Cahn equation, the convection equation with high-frequency solutions and so on. The PT-PINN method is a reliable and competitive method for solving the time-evolution PDEs.

Experimental comparison and numerical study of flow and heat transfer characteristics of the pipe rows in heat pipe heat exchanger

Lube oil cooling is a very important heat transfer process on board, and the use of liquid-liquid heat pipe heat exchangers (HPHE) is easier to maintain and save energy, and does not have the risk of leakage. In this work, the heat transfer and flow in the heat pipe row of liquid-liquid HPHE is analyzed. The numerical model of HPHE based on the thermal resistance network of heat pipe is constructed and verified with the actual experimental data, and its maximum deviation is 8.1%. Results show that increasing the working fluid temperature in the evaporative section improved the operating performance of the heat pipe inside the HPHE, with the maximum equivalent thermal conductivity reaching 7112 W·m−1·K−1 at 70 °C. Furthermore, increasing the hot fluid flow rate improved the heat transfer rate of the HPHE more than increasing the cold fluid inlet flow rate. The convective heat transfer performance equations were fitted to the HPHE, with reliable accuracy within 5% obtained for both the forced convection correlations in the condensing and evaporating sections. The pipe row effect was observed to have some variability, with the periodic variation of the heat transfer coefficient per pipe row being more pronounced. The heat transfer coefficient of the pipe row at the outlet decreased by 15.5%, and the heat transfer coefficient of the pipe row at the baffle gap was also about 15% lower than that of the middle pipe row.

Unconditional stability of first and second orders implicit/explicit schemes for the natural convection equations

In this paper, we consider and analyze the stability of three implicit/explicit (IMEX) schemes for the time-dependent natural convection problem, these considered numerical schemes contain the first order backward Euler scheme, second order Crank-Nicolson IMEX scheme and BDF2-AB2 combination. All numerical schemes deal with the linear terms implicitly and the nonlinear terms explicitly. Then the original nonlinear problem is split into two linearized subproblems with constant coefficient matrixes, which reduces the computational complexity and saves a lot of storage. The biggest feature of this paper is to establish the unconditional stability of three IMEX schemes for incompressible flows, which improves and supplements the published theoretical findings. Finally, some numerical examples are provided to show the performances of the considered numerical schemes.

An unconditionally energy-stable scheme for the convective heat transfer equation

PurposeThis paper aims to present an unconditionally energy-stable scheme for approximating the convective heat transfer equation.Design/methodology/approachThe scheme stems from the generalized positive auxiliary variable (gPAV) idea and exploits a special treatment for the convection term. The original convection term is replaced by its linear approximation plus a correction term, which is under the control of an auxiliary variable. The scheme entails the computation of two temperature fields within each time step, and the linear algebraic system resulting from the discretization involves a coefficient matrix that is updated periodically. This auxiliary variable is given by a well-defined explicit formula that guarantees the positivity of its computed value.FindingsCompared with the semi-implicit scheme and the gPAV-based scheme without the treatment on the convection term, the current scheme can provide an expanded accuracy range and achieve more accurate simulations at large (or fairly large) time step sizes. Extensive numerical experiments have been presented to demonstrate the accuracy and stability performance of the scheme developed herein.Originality/valueThis study shows the unconditional discrete energy stability property of the current scheme, irrespective of the time step sizes.

Finite element analysis on heat convection effect on air temperature distribution in a heated room

Using the finite element method to solve heat transfer problem in solid medium has been popular since last century, but heat convection effect in fluid medium makes the problem cumbersome to solve efficiently. Thermal transfer calculation of fluid medium is complicated, because solving heat convection and Navier-Stokes equation is hard. By making proper assumptions, intricate equations can be skipped, and it is possible to give rough estimate of heat transfer result through a simple and efficient calculation. Which is significant for design and verifiction of heating system for physicists and engineers. The study aims to solve the temperature distribution of a certain heat system, using finite elment method and with given assumpted air circumfluence. In the calculation, the room domain is subdivided into finite elments, and the boundary condition is given according to the heat influx and emmision, air circumfluence is introduced to simulate the effect of convection. The simplified model illustrated that the finite element method is effective in giving quantitative result of heat transfer calculation. It also qualitatively showcased the significance of heat convection, when the result is compared to calculation without any air flow.