Weakly non linear convection in a shallow horizontal cavity under uniform crossed heat fluxes

A weakly nonlinear stability analysis is developed for the unicellular natural convection pattern in a horizontal fluid layer heated by uniform crossed heat fluxes applied on the horizontal and vertical walls, the remaining walls being adiabatic. The critical Rayleigh number is determined, for different Prandtl numbers and heat flux ratios, from the analytical parallel approximation for the single cell flow. For the range of parameters considered, stable, supercritical bifurcations occur solely for longitudinal disturbances with three velocity components. Depending on the value of Pr and the heat flux ratio, the instability is oscillating and hydrodynamic at small Pr and thermal and steady for larger Pr. Computations of the Landau coefficients for the two interacting modes reveal that the hydrodynamic mode is the only stable mode at the codimension points.     

Linear stability of free convection in a vertical cavity heated by uniform heat fluxes

A linear stability analysis is performed of the free convection flow in a vertical cavity heated by uniform heat fluxes at the four walls. As the flow under such heating conditions is quasi-parallel, it is possible to derive analytical solutions for the basic flow which are then used to determine the critical Rayleigh numbers, solving the stability equations by a shooting method for different values of the governing parameters. Comparison of the stability analysis predictions with the numerical simulations shows the validity of the parallel flow approximation for cavity aspects ratios of 4, or higher.     

Multiple steady states for natural convection in a shallow porous cavity subject to uniform heat fluxes

Multiplicity of steady states in natural convection within a shallow porous cavity, exposed to uniform heat fluxes on all sides, is investigated. Analytical solution for the stream function and temperature fields, in the central region of the cavity, is obtained using a parallel flow assumption. It is demonstrated that multiple steady states are possible, some of which are unstable. A finite difference method is used to obtain numerical solutions of the full governing equations. A good agreement is observed between the analytical predictions and the numerical simulations.     

Bifurcation of convention flows in a rectangular cavity subjected to univorm heat fluxes

A study is made of free convection flow in a horizontal rectangular cavity submitted to a uniform heat flux at the horizontal as well as vertical walls. As the flow under such heating condition is quasi-parallel, it is possible to derive an equation for the flow amplitude having the canonical form π³ - δπ - 9!R^*/384³ = 0 where R^* is the Rayleigh number based on the heat flux applied at the side walls and δ is the reduced Rayleigh number based on the heat flux applied at the horizontal boundaries. Based on this equation, various diagrams were constructed to show the unfolding of the well-known pitchfork bifurcation.     

Double-pass flow heat transfer in a parallel-plate channel for improved device performance under uniform heat fluxes

A design of inserting in parallel an impermeable sheet to divide an open conduit into two subchannels for conducting double-pass laminar countercurrent operations under uniform heat fluxes, resulting in substantially improved the heat-transfer rate, has been designed and investigated theoretically by using an eigenfunction expansion in terms of power series for the homogeneous part and an asymptotic solution for the non-homogeneous part. The theoretical results of heat-transfer efficiency enhancement in double-pass parallel-plate heat exchangers are represented graphically and compared with those in the single-pass operations without an impermeable sheet inserted. The influence of the impermeable-sheet location on the heat-transfer efficiency enhancement as well as on the power consumption increment in double-pass operations has also been delineated.     

Mixed convection from a vertical plate to non-Newtonian fluids with uniform surface heat flux

This paper proposed a mixed-convection parameter ζ to analyze the steady laminar mixed convection heat transfer between non-Newtonian fluids and a vertical plate with constant wall heat flux. The obtained finite-difference solutions are uniformly valid over the entire range of mixed convection from the pure forced convection limit (ζ = 0) to pure free convection limit (ζ = 1). Typical velocity and temperature profiles in the boundary layer are presented. Furthermore, the variations of the local heat transfer rates as well as wall frictions along the plate are shown explicitly.     

Convection heat transfer coefficients for turbulent flow between parallel plates with unequal heat fluxes

This paper contains a semi-theoretical analysis of asymmetric heat transfer in fully developed two dimensional turbulent flow between parallel walls.

An extension of the analogy between the transfer of heat and the transfer of momentum (due to Mizushina), is used to determine the heat transfer coefficients for the respective boundaries. The results of the analysis are presented in the form of working formulae from which the relative magnitudes of the heat transfer coefficients may be determined. The formulae may be applied to practical engineering problems, but care is necessary because the assumptions made in the theory impose restrictions on the Prandtl and Reynolds number ranges for which the predictions are acceptable.

It is immediately obvious from the graphical presentation of the results for a particular fluid, that the heat flux ratio is an important additional parameter, and that cognizance of this should be taken in cases of non-uniform heating.

By its nature, the present analysis is preliminary, and an experimental programme has been devised to study the general problem of asymmetric heat transfer and to test the theory.

While channel flow has been considered, the results are thought to be valid for axisymmetric flow in annuli with small outside diameter/inside diameter ratios.     

Natural convection slip flow in a vertical microchannel heated at uniform heat flux

Numerical solutions for steady state developing natural convection flow in air, in vertical parallel-plate microchannels are accomplished. An asymmetric heating is considered and the walls are assumed to be at uniform heat flux. A first-order model is used for slip and jump boundary conditions and an analytical solution for the fully developed flow is also given. Results are performed for air, for the heat flux ratio in the 0.0–1.0 range, for Rayleigh, Ra, and Knudsen, Kn, numbers from 10^?1 to 8 × 10³ and from 0.0 to 0.10, respectively. The maximum mass flow rate is always obtained for the highest considered Kn value, whereas the average Nusselt number, Nu, increases for lower Ra (<10) and decreases for Ra > 100. Wall temperature profiles have the lowest values for highest considered Kn value at lower Ra, whereas for the developing flow, they present opposite trends. For developing flow, velocity profiles for asymmetric and symmetric heating are completely different. In developing flow velocity profiles along the wall present the highest increases for asymmetric heating and the highest considered Kn value.     

The development of laminar natural-convective flow in a vertical uniform heat flux duct

An account of a theoretical and experimental study of laminar natural-convective flow in heated vertical ducts is presented. The ducts are open-ended and circular in cross-section and their internal surfaces dissipate heat uniformly.

Temperature and velocity fields and the relationship between Nusselt and Rayleigh numbers were obtained by solving the governing equations by a step-by-step numerical technique. Two Rayleigh numbers are introduced, one expressed in terms of the uniform heat flux and the other in terms of the mean wall temperature. The influence that the Prandtl number has on the relationship between the Nusselt and Rayleigh numbers is discussed. Three inlet conditions were examined; they all gave the same Nusselt relationship at small Rayleigh numbers and the differences between the Nusselt relationships obtained at large Rayleigh numbers were only small.

Experimentally determined Nusselt numbers, with air as the convected fluid, agreed satisfactorily with the theoretical relationship.     

Numerical analysis of conjugate heat transfer in a partially open square cavity with a vertical heat source

Conjugate heat transfer in partially open square cavity with a vertical heat source has been numerically studied. The cavity has an opening on the top with several lengths and three different positions. The other walls of cavity were assumed adiabatic. The heat source was located on the bottom wall of cavity and it has got a width such as Printed Circuit Boards (PCB). Steady state heat transfer by laminar natural convection and conduction is studied numerically by solving two dimensional forms of governing equations with finite difference method. The results were reported for various governing parameters such as Rayleigh number (10³ ≤ Ra ≤ 10⁶), conductivity ratio, opening position, opening length, PCB distance and PCB height. The numerical results were discussed with streamlines, isotherms, Nusselt number and velocity profiles on x- and y-directions. It is found that ventilation position has a significant effect on heat transfer.     

Laminar free-convection in water with variable physical properties adjacent to a vertical plate with uniform heat flux

The steady laminar boundary layer flow of water along a vertical stationary uniform flux plate is studied. The working fluid is water whose density-temperature relationship is nonlinear at low temperatures and viscosity and thermal conductivity are functions of temperature. The results are obtained with the numerical solution of the boundary layer equations and cover the temperature range between 40 and 0 °C taking into account the temperature dependence of μ, k and ρ. Both upward and downward flow is considered. The variation of μ, k and ρ with temperature has a strong influence on the results.     

Laminar free-convection in glycerol with variable physical properties adjacent to a vertical plate with uniform heat flux

The steady laminar boundary layer flow of glycerol along a vertical stationary plate with uniform heat flux is studied in this paper. The density, thermal conductivity and heat capacity of this liquid are linear functions of temperature but dynamic viscosity is a strong, almost exponential, function of temperature. The results are obtained with the numerical solution of the boundary layer equations. Both upward flow (plate heating) and downward flow (plate cooling) is considered. The variation of μ with temperature has significant influence on wall heat transfer and much stronger influence on wall shear stress. It was also found that the similarity exponent, which is equal to 0.20 for the classical problem with constant properties, is lower than 0.20 in the upward flow and higher than 0.20 in the downward flow.     

Recycle effect on heat-transfer efficiency improvement in a double-pass parallel-plate heat exchanger under uniform wall fluxes

The recycle concept was applied to a double-pass parallel-plate heat exchanger under uniform wall fluxes to enhance the device performance improvement. The mathematical statement of the conjugated Graetz problem was developed theoretically and the analytical solution was obtained by using an eigenfunction expansion in terms of power series for the homogeneous part and an asymptotic solution for the inhomogeneous part. The theoretical predictions of wall temperature and average Nusselt numbers are represented graphically with Graetz number and impermeable-sheet location as parameters. A substantial heat-transfer efficiency improvement is achieved by employing such a recyclic device with suitable adjustments of the impermeable-sheet position and recycle ratio, instead of using the single-pass device. The results indicate that the recyclic operation can effectively enhance the heat transfer efficiency, especially when the recyclic device was operated in large Graetz number.     

Convection heat and mass transfer along a vertical heated plate with film evaporation in a non-Darcian porous medium

A numerical analysis has been carried about to study the heat and mass transfer of forced convection flow with liquid film evaporation in a saturated non-Darcian porous medium. Parametric analyses were conducted concerning the effects of the porosity ε, inlet liquid Reynolds number Re_l, inlet air Reynolds number Re_a on the heat and mass transfer performance. The results conclude that better heat and mass transfer performances are noticed for the system having a higher Re_a, a lower Re_l, and a higher ε. Re_l plays a more important role on the heat and mass transfer performance than Re_a and ε. For the case of ε = 0.4 and Re_a = 10,000, the increases of Nu and Sh for Re_l = 50 are about by 33.9% and 35.3% relative to the values for Re_l = 250.     

Heat-transfer efficiency improvement of double-pass concentric circular heat exchangers under uniform wall fluxes

The new design of the double-pass operation with external recycle is proposed to apply to enhance the heat-transfer efficiency of concentric circular heat exchangers under uniform wall fluxes. The mathematical formulation was developed theoretically and the analytical solutions were achieved by using the orthogonal expansion technique with the eigenfunctions expanding in terms of an extended power series. The theoretical predictions were represented graphically and compared with those obtained from the single-pass device without external recycle and the previous work [C.D. Ho, S.C. Yeh, Improvement in device performance on laminar counterflow concentric circular heat exchangers with uniform wall flux, International Journal of Heat and Mass Transfer 49 (2006) 202–2032] under the different flow pattern. The results show that a considerable heat-transfer efficiency improvement was obtained with a suitable adjustment of the subchannel thickness ratio and recycle ratio. The power consumption increment due to the device with external recycle was also discussed in this study.     

Laminar free convection from a vertical plate with uniform surface heat flux in chemically reacting systems

Two-dimensional steady laminar free convection from a vertical plate with uniform surface heat flux rate is studied in a gas where a reversible very fast reaction of dissociation A↔2B takes place at atmospheric pressure. The effective thermophysical properties of the gas in the interval of dissociation are evaluated and the governing boundary-layer equations are solved numerically by a finite-difference method with control volume formulation for a wide range of values of the independent variables which have a significant influence on the phenomenon. In the case of undisturbed fluid temperature T_∞ smaller than T_0.5, corresponding to a rate of dissociation α=0.5, three different heat transfer regimes, marked by two critical heat fluxes, may be distinguished as the surface heat flux rate increases. The theoretical results obtained for the critical heat fluxes as well as the coefficient of convection are expressed in terms of correlations among dimensionless parameters defined through the mixture effective properties.     

Soret effect on the boundary layer flow regime in a vertical porous enclosure subject to horizontal heat and mass fluxes

The combined thermo- and double-diffusive convection in a vertical tall porous cavity subject to horizontal heat and mass fluxes was investigated analytically and numerically using the Darcy model with the Boussinesq approximation. The investigation focused on the effect of Soret diffusion on the boundary layer flow regime. The governing parameters were the thermal Rayleigh number, R_T, the Lewis number, Le, the buoyancy ratio, N, the Soret parameter, M, which characterized the Soret effect, and the aspect ratio of the enclosure, A_r. The results demonstrated the existence of a boundary layer flow solution for which the Soret parameter had a strong effect on the heat and mass transfer characteristics. For M ≠ 1 and M ≠ −1/Le, the profiles of the vertical velocity component, v, temperature, T, and solute concentration, S, exhibited boundary layer behaviors at high Rayleigh numbers. Furthermore, as R_T increased, the temperature and solute concentration became vertically and linearly stratified in the core region of the enclosure. The thermo-diffusion effect on the boundary layer thickness, δ, was discussed for a wide range of the governing parameters. It was demonstrated analytically that the thickness of the boundary layer could either increase or decrease when the Soret parameter was varied, depending on the sign of the buoyancy ratio. The effect of R_T on the fluid flow properties and heat and mass transfer characteristics was also investigated.