An Investigation of Heat Transfer in a Cavity Flow in the Noncontinuum Regime

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Chariton Christou ◽  
S. Kokou Dadzie

Volume diffusion (or bi-velocity) continuum model offers an alternative modification to the standard Navier–Stokes for simulating rarefied gas flows. According to this continuum model, at higher Knudsen numbers the contribution of molecular spatial stochasticity increases. In this paper, we study a microcavity heat transfer problem as it provides an excellent test for new continuum flow equations. Simulations are carried out for Knudsen numbers within the slip and higher transition flow regimes where nonlocal-equilibrium and rarefaction effects dominate. We contrast the predictions by a Navier–Stokes model corrected by volume diffusion flux in its constitutive equations to that of the direct simulation Monte Carlo (DSMC) method and the standard Navier–Stokes model. The results show improvement in the Navier–Stokes prediction for the high Knudsen numbers. The new model exhibits proper Knudsen boundary layer in the temperature and velocity fields.

2013 ◽  
Vol 13 (5) ◽  
pp. 1330-1356 ◽  
Author(s):  
G. H. Tang ◽  
G. X. Zhai ◽  
W. Q. Tao ◽  
X. J. Gu ◽  
D. R. Emerson

AbstractGases in microfluidic structures or devices are often in a non-equilibrium state. The conventional thermodynamic models for fluids and heat transfer break down and the Navier-Stokes-Fourier equations are no longer accurate or valid. In this paper, the extended thermodynamic approach is employed to study the rarefied gas flow in microstructures, including the heat transfer between a parallel channel andpressure-driven Poiseuille flows through a parallel microchannel andcircular microtube. The gas flow characteristics are studied and it is shown that the heat transfer in the non-equilibrium state no longer obeys the Fourier gradient transport law. In addition, the bimodal distribution of streamwise and spanwise velocity and temperature through a long circular microtube is captured for the first time.


Author(s):  
Edimilson J. Braga ◽  
Marcelo J. S. de Lemos

This work compares two different approaches for obtaining numerical solutions for laminar natural convection within a square cavity, which is filled by a fixed amount of a solid conducting material. The first model considered, namely, porous-continuum model, is based on the assumption that the solid and the fluid phases are seen as the same medium, over which volume-averaged transport equations apply. Secondly, a continuum model is considered to solve the momentum equations for the fluid phase that would resemble a conjugate heat transfer problem in both the solid and the void space. In the continuum model, the solid phase is composed of square obstacles, equally spaced within the cavity. In both models, governing equations are numerically solved using the finite volume method. The average Nusselt number at the hot wall, obtained from the porous-continuum model, for several Darcy numbers, are compared with those obtained with the second approach, namely the continuum model, with different number of obstacles. When comparing the two methodologies, this study shows that the average Nusselt number calculated for each approach for the same Ram differs between each other and that this discrepancy increases as the Darcy number decreases, in the porous-continuum model, or the number of blocks increases and their size decreases, in the continuum model. A correlation is suggested to modify the macroscopic thermal expansion coefficient in order to match the average Nusselt numbers calculated by the two models for Ram = const = 104 and Da ranging from 1.2060×10−4 to 1.


Author(s):  
Rémy Fransen ◽  
Nicolas Gourdain ◽  
Laurent Y. M. Gicquel

This work focuses on numerical simulations of flows in blade internal cooling system. Large Eddy Simulation (LES) and Reynolds-Averaged Navier Stokes (RANS) approaches are compared in a typical blade cooling related problem. The case is a straight rib-roughened channel with high blockage ratio, computed and compared for both a periodic and full spatial domains. The configuration was measured at the Von Karman Institute (VKI) using Particle Image Velocimetry (PIV) in near gas turbine operating conditions. Results show that RANS models used fail to predict the full evolution of the flow within the channels where massive separation and large scale unsteady features are evidenced. In contrast LES succeeds in reproducing these complex flow motions and both mean and fluctuating components are clearly improved in the channels and in the near wall region. Periodic computations are gauged against the spatial computational domain and results on the heat transfer problem are addressed.


Author(s):  
Holger Martin

In 1969, S. G. Brush and C. W. F. Everitt published a historical review, that was reprinted as subchapter 5.5 Maxwell, Osborne Reynolds, and the radiometer, in Stephen G. Brush’s famous book The Kind of Motion We Call Heat. This review covers the history of the explanation of the forces acting on the vanes of Crookes radiometer up to the end of the 19th century. The forces moving the vanes in Crookes radiometer (which are not due to radiation pressure, as initially believed by Crookes and Maxwell) have been recognized as thermal effects of the remaining gas by Reynolds — from his experimental and theoretical work on Thermal Transpiration and Impulsion, in 1879 — and by the development of the differential equations describing Thermal Creeping Flow, induced by tangential stresses due to a temperature gradient on a solid surface by Maxwell, earlier in the same year, 1879. These fundamental physical laws have not yet made their way into the majority of textbooks of heat transfer and fluid mechanics so far. A literature research about the terms of Thermal Transpiration and Thermal Creeping Flow, in connection with the radiometer forces, resulted in a large number of interesting papers; not only the original ones as mentioned in subchapter 5.5 of Brush’s book, but many more in the earlier twentieth century, by Martin Knudsen, Wilhelm Westphal, Albert Einstein, Theodor Sexl, Paul Epstein and others. The forces as calculated from free molecular flow (by Knudsen), increase linearly with pressure, while the forces from Maxwell’s Thermal Creeping Flow decrease with pressure. In an intermediate range of pressures, depending on the characteristic geometrical dimensions of flow channels or radiometer vanes, an appropriate interpolation between these two kinds of forces, as suggested by Wilhelm Westphal and later by G. Hettner, goes through a maximum. Albert Einstein’s approximate solution of the problem happens to give the order of magnitude of the forces in the maximum range. A comprehensive formula and a graph of the these forces versus pressure combines all the relevant theories by Knudsen (1910), Einstein (1924), Maxwell (1879) (and Hettner (1926), Sexl (1928), and Epstein (1929) who found mathematical solutions for Maxwells creeping flow equations for non-isothermal spheres and circular discs, which are important for thermophoresis and for the radiometer). The mechanism of Thermal Creeping Flow will become of increasing interest in micro- and submicro-channels in various new applications, so it ought to be known to every graduate student of heat transfer in the future. That’s one of the reasons why some authors have recently questioned the validity of the classical Navier-Stokes, Fourier, and Fick equations: Dieter Straub (1996) published a book on an Alternative Mathematical Theory of Non-equilibrium Phenomena. Howard Brenner (since 2005) wrote a number of papers, like Navier-Stokes, revisited, and Bi-velocity hydrodynamics, explicitly pointing to the forces acting on the vanes of the lightmill, to thermophoresis and related phenomena. Franz Durst (since 2006) also developed modifications of the classical Navier-Stokes equations. So, Reynolds, Maxwell, and the radiometer may finally have initiated a revision of the fundamental equations of thermofluiddynamics and heat- and mass transfer.


2021 ◽  
Vol 22 (4) ◽  
Author(s):  
Damian Goik ◽  
Krzysztof Banaś ◽  
Jan Bielański ◽  
Kazimierz Chłoń

We describe an approach for efficient solution of large scale convective heat transfer problems, formulated as coupled unsteady heat conduction and incompressible fluid flow equations. The original problem is discretized in time using classical implicit methods, while stabilized finite elements are used for space discretization. The algorithm employed for the discretization of the fluid flow problem uses Picard's iterations to solve the arising nonlinear equations. Both problems, heat transfer and Navier-Stokes quations, give rise to large sparse systems of linear equations. The systems are solved using iterative GMRES solver with suitable preconditioning. For the incompressible flow equations we employ a special preconditioner based on algebraic multigrid (AMG) technique. The paper presents algorithmic and implementation details of the solution procedure, which is suitably tuned, especially for ill conditioned systems arising from discretizations of incompressible Navier-Stokes equations. We describe parallel implementation of the solver using MPI and elements of PETSC library. The scalability of the solver is favourably compared with other methods such as direct solvers and standard GMRES method with ILU preconditioning.  


2017 ◽  
Vol 23 (3) ◽  
pp. 519-540
Author(s):  
Mahdi Boukrouche ◽  
Imane Boussetouan ◽  
Laetitia Paoli

We consider an unsteady non-isothermal incompressible fluid flow. We model heat conduction with Cattaneo’s law instead of the commonly used Fourier’s law, in order to overcome the physical paradox of infinite propagation speed. We assume that the fluid viscosity depends on the temperature, while the thermal capacity depends on the velocity field. The problem is thus described by a Navier–Stokes system coupled with the hyperbolic heat equation. Furthermore, we consider non-standard boundary conditions with Tresca’s friction law on a part of the boundary. By using a time-splitting technique, we construct a sequence of decoupled approximate problems and we prove the convergence of the corresponding approximate solutions, leading to an existence theorem for the coupled fluid flow/heat transfer problem. Finally, we present some numerical results.


Author(s):  
Chaolei Zhang ◽  
Yongsheng Lian

Air circulation and temperature distribution inside a domestic refrigerator chamber are two important factors in refrigerator design. They are critical for food quality control and energy saving and are affected by natural/forced convection, radiation and layout of the stored food. Knowledge about the actual air flow and temperature distributions inside a refrigerator is required to improve temperature homogeneity and reduce energy consumption. In present work we numerically study the air circulation and the heat transfer phenomena in a domestic frost-free refrigerator. The inner compartment, the evaporator and the outside thermal insulation foam are considered. The conjugate heat transfer problem is studied by solving the unsteady laminar Navier-Stokes equations using a finite volume method. The Boussinesq approximation is used to model the natural convection. The discrete ordinate method is adopted to take into account the radiation heat transfer between the cold back evaporator and warm surfaces to further understand the impact of radiation. The accuracy of the numerical methods is verified through grid sensitivity analysis and comparison with available numerical and experimental data. Comparisons are made with and without radiation. Our simulations show that radiation significantly changes the temperature distribution and air circulation pattern. The effects of shelf and food stored on the temperature distribution and air circulation are also studied by comparing three configurations: empty refrigerator, empty refrigerator with shelves and loaded refrigerator with food.


2009 ◽  
Vol 30 (4) ◽  
pp. 282-291
Author(s):  
Caio S. Scherer ◽  
Liliane B. Barichello

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 680 ◽  
Author(s):  
Alexander Beckmann ◽  
Anirudh Rana ◽  
Manuel Torrilhon ◽  
Henning Struchtrup

Due to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier–Stokes–Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed.


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