Uniform lateral mass flux effect on natural convection of non-Newtonian fluids over a cone in porous media

In this paper, we have investigated a boundary layer analysis for uniform lateral mass flux effect on natural convection of non-Newtonian power-law fluids along an isothermal or isoflux vertical cone embedded in a porous medium. Numerical results for the dimensionless temperature profiles as well as the local Nusselt number are presented for the mass flux parameter, viscosity index n and geometry shape parameter λ. The local surface heat transfer increases for the case withdrawal of fluid, the increase of the value of λ. The local Nusselt number is found to be significantly affected by the surface mass flux than the viscosity index.     

Free convection boundary layers in a saturated porous medium with lateral mass flux

The effects of uniform lateral mass flux on the free convection boundary layer on a vertical wall in a saturated porous medium are considered. A series valid near the leading edge is derived and this is extended by a numerical solution of the full equations. Asymptotic expansions, valid at large distances along the plate, are derived in both the cases of withdrawal and injection of fluid. In the former case the boundary layer has constant thickness, while in the latter case there is a region of constant temperature next to the wall made up of fluid that has been injected through the wall, with an outer region where thermal diffusion is important.     

Natural convection of non-Newtonian fluids through permeable axisymmetric and two-dimensional bodies in a porous medium

The present work concerns the natural convection of non-Newtonian power-law fluids with or without yield stress over the permeable two-dimensional or axisymmetric bodies of arbitrary shape in a fluid-saturated porous medium. Using the fourth-order Runge-Kutta scheme method and shooting method we obtain the local non-similarity solutions. The parameters that include the dimensionless yield stress Ω, permeable constant c and power index n are studied, and the heat flux and the wall temperature are taken into consideration as variables. The local non-similarity solutions are found to be in excellent agreement with the exact solution. It is found that the results depend strongly on the values of the yield stress parameter, the wall temperature distributions, the lateral mass flux rate, and the heat flux at the boundary.     

Analytical and numerical solutions to a problem of convection in a porous media with lateral mass flux

We consider the problem of convection in a porous medium adjacent to a heated vertical porous plate. This problem has applications in the re-injection of hot water into a geothermal reservoir [1]. For large Rayleigh numbers, thermal boundary layers are formed and boundary layer theory is the obvious method of investigation. A similarity solution can be obtained when it is stipulated that the wall temperature and the lateral mass flux are power law functions of distance along the plate. In two particular cases, analytical solutions are found. In other cases, the profiles of the normalized velocity and temperature are obtained with accute accuracy using the “quasilinerization” method.     

The effect of uniform lateral mass flux on free convection about a vertical cone embedded in a saturated porous medium

The effect of uniform lateral mass flux on natural convection about a cone embedded in a saturated porous medium is numerically analyzed. The surface is maintained at a uniform wall temperature (UWT) or uniform heat flux (UHF). The transformed governing equations are solved by Keller box method. Numerical data for the dimensionless temperature profiles and the local Nusselt number are presented for a wide range of the mass flux parameter. In general, it has been found that the local surface heat transfer rate increases owing to suction of fluid. This trend reversed for blowing of fluid. The mass flux parameter is found to have a more pronounced effect on the local Nusselt number for the case of UWT than it does for the case of UHF.     

Heat and mass transfer in a cross-flow membrane-based enthalpy exchanger under naturally formed boundary conditions

Heat and mass transfer mechanisms in a cross-flow parallel plate membrane-based enthalpy exchanger for heat and moisture recovery from exhaust air streams are investigated. The flow is assumed laminar and hydrodynamically fully developed, but developing in thermal and concentration boundaries. Contrary to the traditional methods to assume a uniform temperature (concentration) or a uniform heat flux (mass flux) boundary condition, in this study, the real boundary conditions on the exchanger surfaces are obtained by the numerical solution of the coupled equations that govern the transfer of momentum, thermal energy, and moisture in the two cross-flow air streams and through the membrane. The naturally formed heat and mass boundary conditions are then used to calculate the local and mean Nusselt and Sherwood numbers along the cross-flow passages, in the developing region and thereafter. A comparison was made with those results under uniform temperature (concentration) and uniform heat flux (mass flux) boundary conditions, for rectangular ducts of various aspect ratios. An experiment is done to verify the prediction of outlet moisture content.     

Viscous dissipation and thermoconvective instabilities in a horizontal porous channel heated from below

A linear stability analysis of the basic uniform flow in a horizontal porous channel with a rectangular cross section is carried out. The thermal boundary conditions at the impermeable channel walls are: uniform incoming heat flux at the bottom wall, uniform temperature at the top wall, adiabatic lateral walls. Thermoconvective instabilities are caused by the incoming heat flux at the bottom wall and by the internal viscous heating. Linear stability against transverse or longitudinal roll disturbances is investigated either analytically by a power series formulation and numerically by a fourth order Runge-Kutta method. The special cases of a negligible effect of viscous dissipation and of a vanishing incoming heat flux at the bottom wall are discussed. The analysis of these special cases reveals that each possible cause of the convective rolls, bottom heating and viscous heating, can be the unique cause of the instability under appropriate conditions. In all the cases examined, transverse rolls form the preferred mode of instability.     

Inverse estimation of spatially and temporally varying heating boundary conditions of a two-dimensional object

In many dynamic heat transfer situations, the temperature at the heated boundary is not directly measurable and can be obtained by solving an inverse heat conduction problem (IHCP) based on measured temperature or/and heat flux at the accessible boundary. In this study, IHCP in a two-dimensional rectangular object is solved by using the conjugate gradient method (CGM) with temperature and heat flux measured at the boundary opposite to the heated boundary. The inverse problem is formulated in such a way that the heat flux at heated boundary is chosen as the unknown function to be recovered, and the temperature at the heated boundary is computed as a byproduct of the IHCP solution. The measurement data, i.e., the temperature and heat flux at the opposite boundary, are obtained by numerically solving a direct problem where the heated boundary of the object is subjected to spatially and temporally varying heat flux. The robustness of the formulated IHCP algorithm is tested for different profiles of heat fluxes along with different random errors of the measured heat flux at the opposite boundary. The effects of the uncertainties of the thermophysical properties and back-surface temperature measurement on inverse solutions are also examined.     

Numerical study of the interactions and merge of multiple bubbles during convective boiling in micro channels

Multi bubbles interaction and merger in a micro-channel flow boiling has been numerically studied. Effects of mass flux (56, 112, 200, and 335 kg/m² 1 s), wall heat flux (5, 10, and 15 kW/m²) and saturated temperature (300.15 and 303.15 K) are investigated. The coupled level set and volume of fluid (CLSVOF) method and non-equilibrium phase model are implemented to capture the two-phase interface, and the lateral merger process. It is found that the whole transition process can be divided to three sub-stages: sliding, merger, and post-merger. The evaporation rate is much higher in the first two stages due to the boundary layer effects in. Both the mass flux and heat flux affect bubble growth. Specifically, the bubble growth rate increase with the increase of heat flux, or the decrease of mass flux.     

Conjugate natural convection about a vertical cylindrical fin with lateral mass flux in a saturated porous medium

The problem of conjugate natural convection about a vertical cylindrical fin with uniform lateral mass flux in a fluid-saturated porous medium has been studied numerically. Solutions based on the third level of truncation are obtained by the local nonsimilarity method. The effects of the surface mass flux, the conjugate convection-conduction parameter, and the surface curvature on fin temperature distribution, local heat transfer coefficient, local heat flux, average heat transfer coefficient, and total heat transfer rate are presented. A comparison with finite-difference solutions for the case of constant wall temperature was made, and found in a good agreement.     

A one-fluid,two-dimensional flow simulation model for a kettle reboiler

A one-fluid, or algebraic slip, model has been developed to simulate two-dimensional, two-phase flow in a kettle reboiler. The model uses boundary conditions that allow for a change in flow pattern from bubbly to intermittent flow at a critical superficial gas velocity, as has been observed experimentally. The model is based on established correlations for void fraction and for the force on the fluid by the tubes. It is validated against pressure drop measurements taken over a range of heat fluxes from a kettle reboiler boiling R113 and n-pentane at atmospheric pressure.The model predicts that the flow pattern transition causes a reduction in vertical mass flux, and that the reduction is larger when the transition occurs at a lower level. Before transition, the frequently-used, one-dimensional model and the one-fluid model are shown to predict similar heat-transfer rates because similar magnitudes of mass flux are predicted. After transition, the one-dimensional model significantly over-predicts the mass fluxes. The average heat-transfer coefficient predicted by the one-fluid model is consequently about 10% lower. The one-fluid model shows that tube dryout can be expected at much lower heat fluxes than previously thought and that the fluid kinetic energy available to induce tube vibrations is significantly smaller.     

Turbulent liquid film evaporation in a partially heated wall along a vertical tube

In this numerical study, the evaporative heat and mass transfer of a turbulent falling liquid film in a finite vertical tube are investigated. The liquid film flows in the tube's inner wall, whose outer wall is partially subject to thermal flux. Here, different configurations corresponding to thermal flux imposed on different external surface wall percentages are examined. External face zones where the heat flux is not applied are maintained insulated. The nonlinear set of parabolic mass, momentum, energy, and mass fraction conservation equations combined with boundary and interfacial conditions are treated numerically using implicit finite difference procedure. For falling liquid film analysis, an adapted Van Driest turbulence model is used. For the present work, it is supposed that gas flows in a laminar regime. We examine in this paper the impact of the percentage of heated surface area on flows as well as on heat and mass transfer. Obtained results for a partially heated wall are compared with those produced for an entirely heated wall.     

Dynamic behaviour of a boundary layer with condensation along a flat plate: comparison with suction

The dynamical behaviour of laminar and turbulent boundary layers inside which a condensation phenomena exists, has been experimentally and numerically studied. The temperature difference between the exchange cold wall and the saturated air-steam flow at atmospheric pressure and moderate temperature ( 0–90°C) does not exceed 20°C. Evaluation of the mass flux at the wall in such a case, allows the prediction of the ‘equivalent suction rate’ in a sucked boundary layer for the same external conditions but without any temperature difference. The comparison between the two phenomena reveals interesting similarities and some differences.     

Mass transport regimes in a laminar boundary layer with suction--parallel flow and high Peclet number

Mass transport in a boundary layer with suction was studied for a parallel flow, laminar regime and high Peclet number. Mass transport mechanisms involved were analyzed and the respective mass transport fluxes were quantified by numerical methods. According to the magnitude of the convective fluxes, mass transport regimes were established. A simple, but accurate equation was deduced to identify the dominant convective flux and the transport regime. This identification only requires measurable variables combined in dimensionless groups. The accuracy of the equation was proved through the numerical solution of the governing flow and mass transport equations. The concentration field inside the mass boundary layer and the concentration polarization level at the permeable surface are intrinsically related with the dominant convective flux. A simple equation was deduced relating the concentration polarization level at the permeable surface and the parameter , which characterizes the transport regime.     

An approach for determining the liquid water distribution in a liquid-feed direct methanol fuel cell

In determining the liquid water distribution in the anode (or the cathode) diffusion medium of a liquid-feed direct methanol fuel cell (DMFC) with a conventional two-phase mass transport model, a current-independent liquid saturation boundary condition at the interface between the anode flow channel and diffusion layer (DL) (or at the interface between the cathode flow channel and cathode DL) needs to be assumed. The numerical results resulting from such a boundary condition cannot realistically reveal the liquid distribution in the porous region, as the liquid saturation at the interface between the flow channel and DL varies with current density. In this work, we propose a simple theoretical approach that is combined with the in situ measured water-crossover flux in the DMFC to determine the liquid saturation in the anode catalyst layer (CL) and in the cathode CL. The determined liquid saturation in the anode CL (or in the cathode CL) can then be used as a known boundary condition to determine the water distribution in the anode DL (or in the cathode DL) with a two-phase mass transport model. The numerical results show that the water distribution becomes much more realistic than those predicted with the assumed boundary condition at the interface between the flow channel and DL.     

Fast-converging steady-state heat conduction in a rectangular parallelepiped

A Green's function approach for precisely computing the temperature and the three components of the heat flux in a rectangular parallelepiped is presented. Each face of the parallelepiped may have a different, but spatially uniform, boundary condition. Uniform volume energy generation is also treated. Three types of boundary conditions are included: type 1, a specified temperature; type 2, a specified flux; or type 3, a specified convection boundary condition. A general form of the Green's function covering all three types of boundary conditions is given. An algorithm is presented to obtain the temperature and flux at high accuracy with a minimal number of calculations for points in the interior as well as on any of the faces. Heat flux on type 1 boundaries, impossible to evaluate with traditional Fourier series, is found by factoring out lower-dimensional solutions. A numerical example is given. This research and resulting computer program was part of a code verification project for Sandia National Laboratories.     

Heat and mass diffusion fluxes on a permeable wall with foreign-gas blowing

In the present paper, we consider the variation of heat and mass diffusion fluxes on a permeable plate with blowing of a foreign gas into the boundary layer, the fluxes being considered as functions of the permeability parameter, varied through variation of blowing intensity, free-stream velocity, or longitudinal coordinate. It is shown that at a fixed distance from the leading edge of the plate one can, varying the value of blowing intensity while preserving the uniformity of the blowing over the plate length, obtain a non-monotonic variation of the wall heat and mass diffusion fluxes. In contrast to the heat and mass diffusion fluxes, the shear stress always monotonically decreases with increasing the blowing intensity. Similar to the shear stress, on increase of permeability parameter achieved through changing either the free-stream velocity or the longitudinal coordinate the heat flux and the mass diffusion flux both show a monotonic reduction. Using the integral relations of boundary-layer theory, we have derived simple analytical expressions allowing determination of the maximum values of the heat and mass diffusion fluxes in laminar and turbulent flow regimes. The obtained analytical relations were verified by performed numerical simulations.     

Mass flux at ignition in reduced pressure environments

Ignition of solid combustible materials can occur at atmospheric pressures lower than standard either in high altitude environments or inside pressurized vehicles such as aircraft and spacecraft. NASA’s latest space exploration vehicles have a cabin atmosphere of reduced pressure and increased oxygen concentration. Recent piloted ignition experiments indicate that ignition times are reduced under these environmental conditions compared to normal atmospheric conditions, suggesting that the critical mass flux at ignition may also be reduced. Both effects may result in an increased fire risk of combustible solid materials in reduced pressure environments that warrant further investigation. As a result, a series of experiments are conducted to explicitly measure fuel mass flux at ignition and ignition delay time as a function of ambient pressure for the piloted ignition of PMMA under external radiant heating. Experimental findings reveal that ignition time and the fuel mass flux at ignition decrease when ambient pressure is lowered, proving with the latter what earlier authors had inferred. It is concluded that the reduced pressure environment results in smaller convective heat losses from the heated material to the surroundings, allowing for the material to heat more rapidly and pyrolyze faster. It is also proposed that a lower mass flux of volatiles is required to reach the lean flammability limit of the gases near the pilot at reduced pressures, due mainly to a reduced oxygen concentration, an enlarged boundary layer, and a thicker fuel species profile.     

Numerical analysis for heat shield ablation law in transpiration cooling system

The transpiration cooling control system of the thermal protective shield with surface ablation is a nonlinear control system of distributed parameter with moving boundary.As far as the boundary conditions the third king and a one-dimensional incompressible coolant flow under constant transpiration mass flux are concerned,this paper transforms the definite solution problem into the second-type Volterra integral equations,and applies them directly to compute the shield ablation law.Investigation of the system‘s control by transpiration mass flux,ablation amount and ablation velocity varying with time,time of the ablation‘s beginning and ending,etc.,are presented.     

Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate

The combined free convection boundary layer flow with thermal radiation and mass transfer past a permeable vertical plate is studied when the plate moves in its own plane. The plate is maintained at a uniform temperature with uniform species concentration and the fluid is considered to be gray, absorbing–emitting. The coupled unsteady non-linear momentum, energy and concentration equations governing the problem is obtained and made similar by introducing a time-dependent length scale. The similarity equations are solved numerically using superposition method. The resulting velocity, temperature and concentration distributions are shown graphically for different values of parameters entering into the problem. The numerical values of the local wall shear stress, local surface heat and mass flux are shown in tabular form.