Heat Transfer Regimes and Hysteresis in Porous Media Convection

2000 ◽  
Vol 123 (1) ◽  
pp. 145-156 ◽  
Author(s):  
Peter Vadasz
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Porous Media ◽  
Nusselt Number ◽  
Transition Point ◽  
Initial Conditions ◽  
Turbulent Convection ◽  
Practical Significance ◽  
Lorenz Equations ◽  
Adomian’S Decomposition Method

Results of an investigation of different heat transfer regimes in porous media convection are presented by using a truncated Galerkin representation of the governing equations that yields the familiar Lorenz equations for the variation of the amplitude in the time domain. The solution to this system is obtained analytically by using a weak non-linear analysis and computationally by using Adomian’s decomposition method. Expressions for the averaged Nusselt number are derived for steady, periodic, as well as weak-turbulent (temporal-chaotic) convection. The phenomenon of Hysteresis in the transition from steady to weak-turbulent convection, and backwards, is particularly investigated, identifying analytically its mechanism, which is confirmed by the computational results. While the post-transient chaotic solution in terms of the dependent variables is very sensitive to the initial conditions, the affinity of the averaged values of these variables to initial conditions is very weak. Therefore, long-term predictability of these averaged variables, and in particular the Nusselt number, becomes possible, a result of substantial practical significance. Actually, the only impact that the transition to chaos causes on the predicted results in terms of the averaged heat flux is a minor loss of accuracy. Therefore, the predictability of the results in the sense of the averaged heat flux is not significantly affected by the transition from steady to weak-turbulent convection. The transition point is shown to be very sensitive to a particular scaling of the equations, which leads the solution to an invariant value of steady-state for sub-transitional conditions, a result that affects the transition point in some cases.

Analysis of two approaches for an adiabatic boundary condition in porous media

2016 ◽  
Vol 26 (3/4) ◽  
pp. 977-998 ◽  
Author(s):  
Kun Yang ◽  
Xingwang You ◽  
Jiabing Wang ◽  
Kambiz Vafai
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Porous Media ◽  
Nusselt Number ◽  
Analytical Solutions ◽  
Channel Wall ◽  
Flux Distribution ◽  
Heat Flux Distribution ◽  
Content Type ◽  
Adiabatic Boundary Condition

Purpose – The purpose of this paper is to analyze two different approaches (Models A and B) for an adiabatic boundary condition at the wall of a channel filled with a porous medium. The analytical solutions for the velocity distribution, the fluid and solid phase temperature distributions are derived and compared with numerical solutions. The phenomenon of heat flux bifurcation for Model A is demonstrated. The effects of pertinent parameter C on the applicability of the Models A and B are discussed. Analytical solutions for the overall Nusselt number and the heat flux distribution at the channel wall are derived and the influence of pertinent parameters Da and k on the overall Nusselt number and the heat flux distribution is discussed. Design/methodology/approach – Two approaches (Models A and B) for an adiabatic boundary condition in porous media under local thermal non-equilibrium (LTNE) conditions are analyzed in this work. The analysis is applied to a microchannel which is modeled as a porous medium. Findings – The phenomenon of heat flux bifurcation at the wall for Model A is demonstrated. The effect of pertinent parameter C on the applicability of each model is discussed. Model A is applicable when C is relatively large and Model B is applicable when C is small. The heat flux distribution is obtained and the influence of Da and k is discussed. For Model A, ϕAfin increases and ϕAsub, ϕAcover decrease as Da decreases and k is held constant, ϕAsub increases and ϕAfin, ϕAcover decrease as k increases while Da is held constant; for Model B, ϕBfin increases and ϕBsub decreases either as Da decreases or k decreases. The overall Nusselt number is also obtained and the effect of Da and k is discussed: Nu increases as either Da or k decrease for both models. The overall Nusselt number for Model A is larger than that for Model B when Da is large, the overall Nusselt numbers for Models A and B are equivalent when Da is small. Research limitations/implications – Proper representation of the energy equation and the boundary conditions for heat transfer in porous media is very important. There are two different models for representing energy transfer in porous media: local thermal equilibrium (LTE) and LTNE. Although LTE model is more convenient to use, the LTE assumption is not valid when a substantial temperature difference exists between the solid and fluid phases. Practical implications – Fluid flow and convective heat transfer in porous media have many important applications such as thermal energy storage, nuclear waste repository, electronic cooling, geothermal energy extraction, petroleum processing and heat transfer enhancement. Social implications – This work has important fundamental implications. Originality/value – In this work the microchannel is modeled as an equivalent porous medium. The analytical solutions for the velocity distribution, the fluid and solid phase temperature distributions are obtained and compared with numerical solutions. The first type of heat flux bifurcation phenomenon, which indicates that the direction of the temperature gradient for the fluid and solid phases is different at the channel wall, occurs when Model A is utilized. The effect of pertinent parameter C on the applicability of the models is also discussed. The analytical solutions for the overall Nusselt number and the heat flux distribution at the channel wall are derived, and the effects of pertinent parameters Da and k on the overall Nusselt number and the heat flux distribution are discussed.

Promoting or Suppressing Transition to Weak Turbulence in Porous Media Convection via Feedback Control

2011 ◽  
Author(s):  
Peter Vadasz
Keyword(s):  
Porous Media ◽  
Feedback Control ◽  
Transition Point ◽  
Initial Conditions ◽  
Practical Significance ◽  
Weak Turbulence ◽  
Simple Transformation ◽  
Transition To Chaos ◽  
Main Effect ◽  
Transformation Of Variables

A method of using feedback control to promote or suppress the transition to chaos in porous media convection is demonstrated in this paper. Feedback control is used in the present paper to provide a comparison between an analytical expression for the transition point to chaos and numerical results. In addition it is shown that such a feedback control can be applied as an excellent practical means for controlling (suppressing or promoting) chaos by using a Magyari transformation. The latter shows that the controlled model can be transformed into the uncontrolled one via a simple transformation of variables implying that the main effect the feedback control has on the solution is equivalent to altering the initial conditions. The theoretical and practical significance of such an equivalent alteration of the initial conditions is presented and discussed.

A New Nusselt Number for Complicated Configurations

2013 ◽  
Author(s):  
Tom I-Ping Shih ◽  
Srisudarshan Krishna Sathyanarayanan
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Local Heat ◽  
Negative Slope ◽  
Test Problems ◽  
Rectangular Duct ◽  
Circular Duct ◽  
Flow And Heat Transfer ◽  
Local Heat Flux

Convective heat transfer over surfaces is generally presented in the form of the heat-transfer coefficient (h) or its nondimensional form, the Nusselt number (Nu). Both require the specification of the free-stream temperature (Too) or the bulk (Tb) temperature, which are clearly defined only for simple configurations. For complicated configurations with flow separation and multiple temperature streams, the physical significance of Too and Tb becomes unclear. In addition, their use could cause the local h to approach positive or negative infinity if Too or Tb is nearly the same as the local wall temperature (Twall). In this paper, a new Nusselt number, referred to as the SCS number, is proposed, that provides information on the local heat flux but does not use h and hence by-passes the need to define Too or Tb. CFD analysis based on steady RANS with the shear-stress transport model is used to compare and contrast the SCS number with Nu for two test problems: (1) compressible flow and heat transfer in a straight duct with a circular cross section and (2) compressible flow and heat transfer in a high-aspect ratio rectangular duct with a staggered array of pin fins. Parameters examined include: Reynolds number at the duct inlet (3,000 to 15,000 for the circular duct and 15,000 and 150,000 for the rectangular duct), wall temperature (Twall = 373 K to 1473 K for the circular duct and 313 K and 1,173 K for the rectangular duct), and distance from of the inlet of the duct (up to 100D for the circular duct and up to 156D for the rectangular duct). For the circular duct, Nu was found to decrease rapidly from the duct inlet until reaching a minimum and then to rise until reaching a nearly constant value in the “fully” developed region if the wall is heating the gas. If the wall is cooling the gas, then Nu has a constant positive slope in the “fully” developed region. The location of the minimum in Nu and where Nu becomes nearly constant in value or in slope are strong functions of Twall. For the SCS number, the decrease from the duct inlet is monotonic with a negative slope, whether the wall is heating or cooling the gas. Also, different SCS curves for different Twall approach each other as the distance from the inlet increases. For the rectangular duct, Nu tends to oscillate about a constant value in the pin-fin region, whereas SCS tends to oscillate about a line with a negative slope. For both test problems, the variation of SCS is not more complicated than Nu, but SCS yields the local heat flux without need for Tb, a parameter that is hard to define and measure for complicated problems.

Oscillatory Heat Transfer in a Pipe Subjected to a Laminar Reciprocating Flow

1996 ◽  
Vol 118 (3) ◽  
pp. 592-597 ◽  
Author(s):  
T. S. Zhao ◽  
P. Cheng
Keyword(s):  
Heat Transfer ◽  
Experimental Data ◽  
Heat Flux ◽  
Nusselt Number ◽  
Numerical Study ◽  
Temperature Cycle ◽  
Uniform Heat Flux ◽  
Fluid Temperature ◽  
Cycle Space ◽  
Reciprocating Flow

An experimental and numerical study has been carried out for laminar forced convection in a long pipe heated by uniform heat flux and subjected to a reciprocating flow of air. Transient fluid temperature variations in the two mixing chambers connected to both ends of the heated section were measured. These measurements were used as the thermal boundary conditions for the numerical simulation of the hydrodynamically and thermally developing reciprocating flow in the heated pipe. The coupled governing equations for time-dependent convective heat transfer in the fluid flow and conduction in the wall of the heated tube were solved numerically. The numerical results for time-resolved centerline fuid temperature, cycle-averaged wall temperature, and the space-cycle averaged Nusselt number are shown to be in good agreement with the experimental data. Based on the experimental data, a correlation equation is obtained for the cycle-space averaged Nusselt number in terms of appropriate dimensionless parameters for a laminar reciprocating flow of air in a long pipe with constant heat flux.

Heat Transfer and Flow Friction Characteristics for Compact Cold Plates

2003 ◽  
Vol 125 (1) ◽  
pp. 104-113 ◽  
Author(s):  
Chang-Yuan Liu ◽  
Ying-Huei Hung
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Thermal Resistance ◽  
Mass Flow Rate ◽  
Mass Flow ◽  
Flow Rate ◽  
Average Nusselt Number ◽  
Cold Plate ◽  
Air Mass

Both experimental and theoretical investigations on the heat transfer and flow friction characteristics of compact cold plates have been performed. From the results, the local and average temperature rises on the cold plate surface increase with increasing chip heat flux or decreasing air mass flow rate. Besides, the effect of chip heat flux on the thermal resistance of cold plate is insignificant; while the thermal resistance of cold plate decreases with increasing air mass flow rate. Three empirical correlations of thermal resistance in terms of air mass flow rate with a power of −0.228 are presented. As for average Nusselt number, the effect of chip heat flux on the average Nusselt number is insignificant; while the average Nusselt number of the cold plate increases with increasing Reynolds number. An empirical relationship between Nu¯cp and Re can be correlated. In the flow frictional aspect, the overall pressure drop of the cold plate increases with increasing air mass flow rate; while it is insignificantly affected by chip heat flux. An empirical correlation of the overall pressure drop in terms of air mass flow rate with a power of 1.265 is presented. Finally, both heat transfer performance factor “j” and pumping power factor “f” decrease with increasing Reynolds number in a power of 0.805; while they are independent of chip heat flux. The Colburn analogy can be adequately employed in the study.

Hall Effects on Unsteady Magnetohydrodynamic Flow of a Nanofluid Past an Oscillatory Vertical Rotating Flat Plate Embedded in Porous Media

2021 ◽  
Vol 10 (2) ◽  
pp. 259-269
Author(s):  
M. Veera Krishna ◽  
N. Ameer Ahamad ◽  
Ali J. Chamkha
Keyword(s):  
Heat Transfer ◽  
Porous Media ◽  
Nusselt Number ◽  
Friction Coefficient ◽  
Local Nusselt Number ◽  
Hall Current ◽  
Skin Friction Coefficient ◽  
Engineering Industry ◽  
Hall Effects ◽  
The Impact

In the current investigative paper, the impact of Hall current on an unsteady magnetohydrodynamic liberated convection revolving flow of a nanofluid restricted with a uniform absorbent medium over an oscillatory moving vertical smooth plate with convective as well as diffusive frontier conditions has been reviewed. The non-dimensionalized governing differential equations by the appropriate frontier conditions are resolved by the perturbations technique. The impacts of the physical constants on the flow as well as the heat transfer features are displayed graphically and analyzed for Cu as well as Al2O3 nanoparticles. For the engineering industry, the skin friction coefficient, local Nusselt number, along with the Sherwood’s number are examined numerically in detail.

Dependence of Heat Transfer Coefficient on Porous Structure in Porous Media

2008 ◽  
Author(s):  
Akira Matsui ◽  
Kazuhisa Yuki ◽  
Hidetoshi Hashizume
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Porous Media ◽  
Heat Transfer Coefficient ◽  
Transfer Coefficient ◽  
High Performance ◽  
Heat Removal ◽  
High Heat ◽  
Metal Foams ◽  
High Heat Flux

Detailed heat transfer characteristics of particle-sintered porous media and metal foams are evaluated to specify the important structural parameters suitable for high heat removal. The porous media used in this experiment are particle-sintered porous media made of bronze and SUS316L, and metal foams made of copper and nickel. Cooling water flows into the porous medium opposite to heat flux input loaded by a plasma arcjet. The result indicates that the bronze-particle porous medium of 100μm in pore size shows the highest performance and achieves heat transfer coefficient of 0.035MW/m2K at inlet heat flux 4.6MW/m2. Compared with the heat transfer performance of copper fiber-sintered porous media, the bronze particlesintered ones give lower heat transfer coefficient. However, the stable cooling conditions that the heat transfer coefficient does not depend on the flow velocity, were confirmed even at heat flux of 4.6MW/m2 in case of the bronze particle-sintered media, while not in the case of the copper-fiber sintered media. This signifies the possibility that the bronze-particle sintered media enable much higher heat flux removal of over 10MW/m2, which could be caused by higher permeability of the particle-sintered pore structures. Porous media with high permeability provide high performance of vapor evacuation, which leads to more stable heat removal even under extremely high heat flux. On the other hand, the heat transfer coefficient of the metal foams becomes lower because of the lower capillary and fin effects caused by too high porosity and low effective thermal conductivity. It is concluded that the pore structure having high performance of vapor evacuation as well as the high capillary and high fin effects is appropriate for extremely high heat flux removal of over 10MW/m2.

Transient Internal Forced Convection Under Arbitrary Time-Dependent Heat Flux

2013 ◽  
Author(s):  
M. Fakoor-Pakdaman ◽  
M. Andisheh-Tadbir ◽  
Majid Bahrami
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Forced Convection ◽  
Analytical Model ◽  
Time Dependent ◽  
Bulk Temperature ◽  
Fluid Temperature ◽  
Flux Boundary Condition ◽  
Arbitrary Time

A new all-time model is developed to predict transient laminar forced convection heat transfer inside a circular tube under arbitrary time-dependent heat flux. Slug flow condition is assumed for the velocity profile inside the tube. The solution to the time-dependent energy equation for a step heat flux boundary condition is generalized for arbitrary time variations in surface heat flux using a Duhamel’s integral technique. A cyclic time-dependent heat flux is considered and new compact closed-form relationships are proposed to predict: i) fluid temperature distribution inside the tube ii) fluid bulk temperature and iii) the Nusselt number. A new definition, cyclic fully-developed Nusselt number, is introduced and it is shown that in the thermally fully-developed region the Nusselt number is not a function of axial location, but it varies with time and the angular frequency of the imposed heat flux. Optimum conditions are found which maximize the heat transfer rate of the unsteady laminar forced-convective tube flow. We also performed an independent numerical simulation using ANSYS to validate the present analytical model. The comparison between the numerical and the present analytical model shows great agreement; a maximum relative difference less than 5.3%.

scholarly journals Analytical Analysis of Heat Transfer and Entropy Generation in a Tube Filled with Double-Layer Porous Media

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1214 ◽  
Author(s):  
Kun Yang ◽  
Wei Huang ◽  
Xin Li ◽  
Jiabing Wang
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Nusselt Number ◽  
Double Layer ◽  
Entropy Generation ◽  
Single Layer ◽  
Entropy Generation Rate ◽  
Total Entropy ◽  
Interfacial Radius

The heat transfer and entropy generation in a tube filled with double-layer porous media are analytically investigated. The wall of the tube is subjected to a constant heat flux. The Darcy-Brinkman model is utilized to describe the fluid flow, and the local thermal non-equilibrium model is employed to establish the energy equations. The solutions of the temperature and velocity distributions are analytically derived and validated in limiting case. The analytical solutions of the local and total entropy generation, as well as the Nusselt number, are further derived to analyze the performance of heat transfer and irreversibility of the tube. The influences of the Darcy number, the Biot number, the dimensionless interfacial radius, and the thermal conductivity ratio, on flow and heat transfer are discussed. The results indicate, for the first time, that the Nusselt number for the tube filled with double-layer porous media can be larger than that for the tube filled with single layer porous medium, while the total entropy generation rate for the tube filled with double-layer porous media can be less than that for the tube filled with single layer porous medium. And the dimensionless interfacial radius corresponding to the maximum value of the Nusselt number is different from that corresponding to the minimum value of the total entropy generation rate.

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