On Simplified Models for the Rate- and Time-Dependent Performance of Stratified Thermal Storage

2007 ◽  
Vol 129 (3) ◽  
pp. 214-222 ◽  
Author(s):  
Ying Ji ◽  
K. O. Homan
Keyword(s):  
Entropy Generation ◽  
Peclet Number ◽  
Volume Fraction ◽  
Thermal Storage ◽  
Time Dependent ◽  
Simplified Models ◽  
Péclet Number ◽  
Thermal Mixing ◽  
Discharge Volume ◽  
Cumulative Entropy

In direct sensible thermal storage systems, both the energy discharging and charging processes are inherently time-dependent as well as rate-dependent. Simplified models which depict the characteristics of this transient process are therefore crucial to the sizing and rating of the storage devices. In this paper, existing models which represent three distinct classes of models for thermal storage behavior are recast into a common formulation and used to predict the variations of discharge volume fraction, thermal mixing factor, and entropy generation. For each of the models considered, the parametric dependence of key performance measures is shown to be expressible in terms of a Peclet number and a Froude number or temperature difference ratio. The thermal mixing factor for each of the models is reasonably well described by a power law fit with Fr2Pe for the convection-dominated portion of the operating range. For the uniform and nonuniform diffusivity models examined, there is shown to be a Peclet number which maximizes the discharge volume fraction. In addition, the cumulative entropy generation from the simplified models is compared with the ideally-stratified and the fully-mixed limits. Of the models considered, only the nonuniform diffusivity model exhibits an optimal Peclet number at which the cumulative entropy generation is minimized. For each of the other models examined, the cumulative entropy generation varies monotonically with Peclet number.

Magnetohydrodynamics and Soret Effects on Bioconvection in a Porous Medium Saturated With a Nanofluid Containing Gyrotactic Microorganisms

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
S. Shaw ◽  
P. Sibanda ◽  
A. Sutradhar ◽  
P. V. S. N. Murthy
Keyword(s):  
Porous Medium ◽  
Differential Equations ◽  
Sherwood Number ◽  
Peclet Number ◽  
Volume Fraction ◽  
Soret Effect ◽  
Magnetic Field Effect ◽  
Lewis Number ◽  
Solute Concentration ◽  
Péclet Number

We investigate the bioconvection of gyrotactic microorganism near the boundary layer region of an inclined semi infinite permeable plate embedded in a porous medium filled with a water-based nanofluid containing motile microorganisms. The model for the nanofluid incorporates Brownian motion, thermophoresis, also Soret effect and magnetic field effect are considered in the study. The governing partial differential equations for momentum, heat, solute concentration, nanoparticle volume fraction, and microorganism conservation are reduced to a set of nonlinear ordinary differential equations using similarity transformations and solved numerically. The effects of the bioconvection parameters on the thermal, solutal, nanoparticle concentration, and the density of the micro-organisms are analyzed. A comparative analysis of our results with previously reported results in the literature is given. Some interesting phenomena are observed for the local Nusselt and Sherwood number. It is shown that the Péclet number and the bioconvection Rayleigh number highly influence the local Nusselt and Sherwood numbers. For Péclet numbers less than 1, the local Nusselt and Sherwood number increase with the bioconvection Lewis number. However, both the heat and mass transfer rates decrease with bioconvection Lewis number for higher values of the Péclet number.

Accurate Boundary Element Solutions for Highly Convective Unsteady Heat Flows

2005 ◽  
Vol 127 (10) ◽  
pp. 1138-1150 ◽  
Author(s):  
M. M. Grigoriev ◽  
G. F. Dargush
Keyword(s):  
Boundary Element ◽  
Peclet Number ◽  
Iterative Solvers ◽  
Heat Diffusion ◽  
Time Dependent ◽  
Time Interval ◽  
Spatial Integration ◽  
Péclet Number ◽  
Heat Flows ◽  
Analytic Integration

Several recently developed boundary element formulations for time-dependent convective heat diffusion appear to provide very efficient computational tools for transient linear heat flows. More importantly, these new approaches hold much promise for the numerical solution of related nonlinear problems, e.g., Navier–Stokes flows. However, the robustness of these methods has not been examined, particularly for high Peclet number regimes. Here, we focus on these regimes for two-dimensional problems and develop the necessary temporal and spatial integration strategies. The algorithm takes advantage of the nature of the time-dependent convective kernels, and combines analytic integration over the singular portion of the time interval with numerical integration over the remaining nonsingular portion. Furthermore, the character of the kernels lets us define an influence domain and then localize the surface and volume integrations only within this domain. We show that the localization of the convective kernels becomes more prominent as the Peclet number of the flow increases. This leads to increasing sparsity and in most cases improved conditioning of the global matrix. Thus, iterative solvers become the primary choice. We consider two representative example problems of heat propagation, and perform numerical investigations of the accuracy and stability of the proposed higher-order boundary element formulations for Peclet numbers up to 105.

Second Law Aspects of Simplified Models for Sensible Thermal Storage

2000 ◽  
Author(s):  
K. O. Homan
Keyword(s):  
Numerical Model ◽  
Entropy Generation ◽  
Thermal Storage ◽  
Heat Fluxes ◽  
Analytical Models ◽  
Storage Devices ◽  
Thermal Mixing ◽  
Thermal Layer ◽  
Temperature Liquid ◽  
The One

Abstract This paper presents results for entropy generation during the inflow of a low temperature stream into a sensible thermal storage vessel initially filled with a uniformly high temperature liquid. The level of internal entropy generation due to thermal mixing between the cold and hot liquid corresponds to losses in the usable fraction of the stored volume and therefore decreased efficiency. Empirically, the observed behavior of sensible storage devices spans the range of nearly mixed to well stratified. In this investigation, analytical models for these two limits, the fully mixed and ideally stratified conditions, are used to bound the entropy generation levels of the observed behaviors. A numerical model for stratified storage systems based on the one-dimensional convective energy equation which accounts for aspects of the observed thermal mixing is then examined in relation to the afore-mentioned limits. The results show that even at moderate throughflow rates, the fully mixed and ideally stratified limits are separated by orders of magnitude in terms of entropy generated. The empirically-based numerical model exhibits mixing levels midway between these two limits and thereby underscores the potential for significant improvements in efficiency. Examination of the numerical model shows the crucial importance of resolving the evolution of the interior thermal layer and the boundary heat fluxes in computing the entropy generation.

The Effects of Forced Convection on the Power Dissipation of Constant-Temperature Thermal Conductivity Sensors

1997 ◽  
Vol 119 (1) ◽  
pp. 30-37 ◽  
Author(s):  
Y. Huang ◽  
H. H. Bau
Keyword(s):  
Thermal Conductivity ◽  
Forced Convection ◽  
Constant Temperature ◽  
Power Dissipation ◽  
Peclet Number ◽  
Asymptotic Theory ◽  
Time Dependent ◽  
Péclet Number ◽  
Theoretical Predictions ◽  
On Line

The effect of forced convection on the power dissipation of cylindrical and planar, constant temperature, thermal conductivity detectors (TCDs) is investigated theoretically. Such detectors can be used either for on-line continuous sensing of fluid thermal conductivity or for determining the sample concentrations in gas chromatography. A low Peclet number, asymptotic theory is constructed to correlate the TCD’s power dissipation with the Peclet number and to explain experimental observations. Subsequently, the effect of convection on the TCD’s power dissipation is calculated numerically for both time-independent and time-dependent flows. The theoretical predictions are compared with experimental observations.

Mixing of the fluid phase in slowly sheared particle suspensions of cylinders

2017 ◽  
Vol 818 ◽  
pp. 807-837 ◽  
Author(s):  
Kjetil Thøgersen ◽  
Marcin Dabrowski
Keyword(s):  
Diffusion Coefficient ◽  
Fractal Dimension ◽  
Peclet Number ◽  
Volume Fraction ◽  
Fluid Phase ◽  
Particle Suspensions ◽  
Péclet Number ◽  
Self Diffusion ◽  
Wide Range ◽  
Self Diffusion Coefficient

We introduce a finite element model for neutrally buoyant particle suspensions of cylinders at zero Reynolds number and infinite Péclet number in the purely hydrodynamic limit, which allows us to access a high-accuracy fluid velocity field at any time during the simulation. We use the diffusive strip method to characterize the development of the concentration field in the fluid phase of sheared suspensions from initial thin filaments, and characterize the structures that form with their fractal dimension. We find that the growth of the fractal dimension of the filaments scales with the increase of mean square displacement in the fluid phase. Further, we measure the concentration distribution of tracers in the fluid phase, as well as the shear-induced self-diffusion coefficient in both the solid phase and the fluid phase. We demonstrate that the shear-induced self-diffusion coefficient is slightly larger in the fluid phase at infinite Péclet number. Finally, we investigate enhanced mass diffusivity in the fluid phase by systematic measurements of the shear-induced self-diffusion coefficient in the fluid phase for a wide range of fluid tracer Péclet numbers. We find that the functional dependence $D_{s}/D=1+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6FC}}Pe^{\unicode[STIX]{x1D701}}$ (where $D_{s}$ is the shear-induced self-diffusion coefficient, $D$ is the molecular diffusivity and $\unicode[STIX]{x1D719}$ is the particle volume fraction) fits the observations fairly well. We measure the constants $\unicode[STIX]{x1D6FD}=2.98\pm 0.39$, $\unicode[STIX]{x1D6FC}=1.61\pm 0.26$ and $\unicode[STIX]{x1D701}=0.900\pm 0.031$.

Dispersion in fixed beds

1985 ◽  
Vol 154 ◽  
pp. 399-427 ◽  
Author(s):  
Donald L. Koch ◽  
John F. Brady
Keyword(s):  
Peclet Number ◽  
Molecular Diffusion ◽  
Volume Fraction ◽  
Effective Diffusivity ◽  
Ensemble Averaging ◽  
Péclet Number ◽  
Physical Mechanisms ◽  
Molecular Diffusivity ◽  
Fixed Beds ◽  
Effective Diffusivities

A macroscopic equation of mass conservation is obtained by ensemble-averaging the basic conservation laws in a porous medium. In the long-time limit this ‘macro-transport’ equation takes the form of a macroscopic Fick's law with a constant effective diffusivity tensor. An asymptotic analysis in low volume fraction of the effective diffusivity in a bed of fixed spheres is carried out for all values of the Péclet number ℙ = Ua/Df, where U is the average velocity through the bed. a is the particle radius and Df is the molecular diffusivity of the solute in the fluid. Several physical mechanisms causing dispersion are revealed by this analysis. The stochastic velocity fluctuations induced in the fluid by the randomly positioned bed particles give rise to a convectively driven contribution to dispersion. At high Péclet numbers, this convective dispersion mechanism is purely mechanical, and the resulting effective diffusivities are independent of molecular diffusion and grow linearly with ℙ. The region of zero velocity in and near the bed particles gives rise to non-mechanical dispersion mechanisms that dominate the longitudinal diffusivity at very high Péclet numbers. One such mechanism involves the retention of the diffusing species in permeable particles, from which it can escape only by molecular diffusion, leading to a diffusion coefficient that grows as ℙ2. Even if the bed particles are impermeable, non-mechanical contributions that grow as ℙ ln ℙ and ℙ2 at high ℙ arise from a diffusive boundary layer near the solid surfaces and from regions of closed streamlines respectively. The results for the longitudinal and transverse effective diffusivities as functions of the Péclet number are summarized in tabular form in §6. Because the same physical mechanisms promote dispersion in dilute and dense fixed beds, the predicted Péclet-number dependences of the effective diffusivities are applicable to all porous media. The theoretical predictions are compared with experiments in densely packed beds of impermeable particles, and the agreement is shown to be remarkably good.

Improving the Efficiency of the Nodal Integral Method With the Portable, Extensible Toolkit for Scientific Computation

2002 ◽  
Author(s):  
Allen J. Toreja ◽  
Rizwan-Uddin
Keyword(s):  
Diffusion Equation ◽  
Peclet Number ◽  
Integral Method ◽  
Time Dependent ◽  
Scientific Computation ◽  
Péclet Number ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Nodal Integral Method ◽  
Successive Over Relaxation

An existing implementation of the nodal integral method for the time-dependent convection-diffusion equation is modified to incorporate various PETSc (Portable, Extensible Toolkit for Scientific Computation) solver and preconditioner routines. In the modified implementation, the default iterative Gauss-Seidel solver is replaced with one of the following PETSc iterative linear solver routines: Generalized Minimal Residuals, Stabilized Biconjugate Gradients, or Transpose-Free Quasi-Minimal Residuals. For each solver, a Jacobi or a Successive Over-Relaxation preconditioner is used. Two sample problems, one with a low Peclet number and one with a high Peclet number, are solved using the new implementation. In all the cases tested, the new implementation with the PETSc solver routines outperforms the original Gauss-Seidel implementation. Moreover, the PETSc Stabilized Biconjugate Gradients routine performs the best on the two sample problems leading to CPU times that are less than half the CPU times of the original implementation.

Efficient Boundary Element Methods for the Time-Dependent Convective Diffusion Equation

2003 ◽  
Author(s):  
G. F. Dargush ◽  
M. M. Grigoriev
Keyword(s):  
Boundary Element ◽  
Peclet Number ◽  
Convective Diffusion ◽  
Fundamental Solutions ◽  
Higher Order ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Time Dependent ◽  
Péclet Number

Higher-order boundary element methods (BEM) are presented for time-dependent convective diffusion in two dimensions. The time-dependent convective diffusion free-space fundamental solutions originally proposed by Carslaw and Jaeger are used to obtain the boundary integral formulation. Boundary element method solutions up to the Peclet number 106 are obtained for an example problem of unsteady convection-diffusion that possesses an exact solution. We investigate the convergence rate and accuracy of the higher-order boundary element formulations. An extremely high accuracy of the BEM solutions for highly convective flows is demonstrated. Moreover, it is shown that the use of time-dependent convective kernels provides an automatic upwinding for the entire range of Peclet numbers and also leads to very efficient algorithms as the Peclet number increases.

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