Presence Of Internal Heat Research Articles

Heat and mass transfer analysis for boundary layer stagnation-point flow towards a heated porous stretching sheet with heat absorption/generation and suction/blowing

A similarity analysis is performed to investigate the structure of the boundary layer stagnation-point flow and heat transfer over a stretching sheet in a porous medium subject to suction/blowing and in the presence of internal heat generation/absorption. A scaling group of transformations is applied to get the invariants. Using the invariants, a third and a second order ordinary differential equations corresponding to the momentum and energy equations are obtained respectively. Boundary layer velocity and temperature profiles are determined numerically for various values of the ratio of free stream velocity and stretching velocity, the permeability parameter, suction/blowing parameter, heat source/sink parameter, Prandtl number. It is found that the horizontal velocity increases with the increasing value of the ratio of the free stream velocity ( ax) and the stretching velocity ( cx). The temperature decreases in this case. At a particular point of the porous stretching sheet, the non-dimensional fluid velocity decreases with the increase of the permeability of the porous medium and also with the increasing suction parameter when the free stream velocity is less than stretching velocity whereas fluid velocity increases with the increasing injection parameter. But when the free stream velocity is greater than the stretching velocity the opposite behaviour of horizontal velocity is noticed. The dimensionless temperature at a point of the sheet decreases due to suction but increases due to injection. The temperature at a point is found to decrease with the increasing Prandtl number.

Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption

The problem of steady, laminar, hydromagnetic heat transfer by mixed convection over a continuously stretching surface with power-law variation in the surface temperature or heat flux in the presence of internal heat generation/absorption effect is investigated. The surface velocity of the stretching surface was assumed to vary according to power-law. The surface is assumed to be permeable to allow for possible fluid wall blowing or suction. The governing non-similar equations are obtained by introducing a suitable transformation and are solved numerically. The obtained results are checked against previously published work on special cases of the problem and are found to be in excellent agreement. A parametric study is performed to illustrate the influence of the magnetic parameter, wall mass transfer coefficient, the coefficients of space-dependent and temperature-dependent internal heat generation/absorption, and the buoyancy force parameters on the fluid velocity and temperature profiles. Numerical data for the local skin-friction coefficient and the local Nusselt number have been tabulated for various parametric conditions.

Non-linear effects from variable thermal conductivity and mantle internal heating: implications for massive melting and secular cooling of the mantle

The temperature-dependence of the phonon portion of the thermal conductivity k( T, P) devised by Hofmeister [Science 283 1699-1706] decreases with temperature, the same as in the dependence of mantle viscosity. Such a functional relationship of ∂k/ ∂T<0, when coupled with internal heating would present a situation very conducive for positive feedback action. On the other hand, the photon dependence of the conductivity has a functional relationship of ∂k/ ∂T>0. Thus, there can be a tradeoff between the phonon and photon contributions in the conductivity in the presence of internal heating. We have conducted two-dimensional calculations of mantle convection up to a surface Rayleigh number of around five million and an internal heating of chondritic abundance, with the extended-Boussinesq approximation in which the dissipation number has been set to 0.47 and depth-dependent thermal expansivity, decreasing by a factor of 5 across the mantle. The value of the constant mantle viscosity and the amount of internal heating are varied. For an enhanced radiative contribution [J. Geophys. Res. 84 (B4) 1603-1610] the radiative component of the thermal conductivity can exceed the phonon contribution in the upper mantle. Our results show that in all cases with basal heating the average mantle temperature of the variable conductivity models are higher than those of the corresponding constant conductivity models. But the interior thermal difference between the two conductivity models decreases (1) with greater vigor of convection, (2) an increase of internal heating and (3) an increase in the radiative contribution to the conductivity. The interior mantle temperature is significantly hotter, more than 500 °C, than the constant conductivity model, for the k( T, P) model with the less enhanced radiative component [Science 283 1699–1706]. These results would suggest that some sort of massive melting in the young earth might have occurred with k( T, P) and that there should not be so much radioactivity in the lower mantle today without incurring the wrath of some melting. We have also studied the effects of k( T, P) on slowing down the mantle secular cooling process by monitoring the gradual decrease in mantle temperature following an imposition of an adiabatic boundary condition at the core-mantle boundary. A decay time of 3.6 Gy has been taken for the mantle radioactivity and we have varied the initial amount of radioactive heating from chondritic value to four times the chondritic value. A significant delay in the cooling process of at least 1–2 Gy is found for a surface Rayleigh number of between 5×10 6 to 5×10 7. The mantle temperature can be heated up by 300–400 °C for initial radiogenic heating value characteristic of the Archean. We find the strongest deviations from the constant conductivity case for a silicate model by Hofmeister [Science 283 1699-1706] and intermediate values for an enhanced radiative conductivity model comparable to the model of Shankland et al. [J. Geophys. Res. 84 (B4) 1603-1610]. Such high mantle temperatures maintained for a long time by variable thermal conductivity would have important consequences on the thermal and petrological evolution of the mantle.

Effects of throughflow and internal heat generation on the onset of convection in a fluid layer

The throughflow and internal heat generation effects on the onset of convection in an infinite horizontal fluid layer are investigated. The boundaries are considered to be rigid (however permeable) and perfectly conducting. The resulting eigenvalue problem is solved by using the Galerkin method, and the effects of various parameters in the stability results are analyzed. The results indicate that the stability of the system is significantly affected by both throughflow and internal heat generation in the fluid layer. The Prandtl number comes into play due to the presence of throughflow and it has a profound effect on the stability of the system. It is found that, in the presence of internal heating, throughflow in one direction supresses convection while throughflow in the other direction encourages it.

Mixing in the transition to strong turbulence in infinite Prandtl number thermal convection as applied to the Earth’s mantle

Recently there has developed a great deal of interest in the transition to hard turbulence in thermal convection. This transition and the corresponding hard turbulence regime has been the subject of laboratory and numerical experiments. Our study of convection at high Rayleigh numbers (Ra) has been motivated by the phenomenon of subsolidus mantle convection, whose Ra may range between 5×105 and O(108), depending on the still uncertain estimates of the lower-mantle viscosity. We have conducted a series of calculations at Ra spanning between 5×105 and 108 because we are interested in the transition from weak to strong turbulence in mantle convection and the effects this transition would have on mixing. We have employed a two-dimensional finite-element method, combined with an implicit, predictor–corrector, time-stepping scheme to advance the evolutionary temperature equation. The biharmonic equation for the streamfunction is solved exactly by a variational equation at each time step. We present direct numerical simulations of two-dimensional, high Ra thermal convection for constant property fluids with both base-heated and partially internally heated configurations in large-aspect-ratio boxes (up to 10). We will emphasize the importance of visualization of these strongly time-dependent results, as they present a formidable challenge in the management and analysis of the data. First, an adequate resolution of the boundary layer and mixing layer requires high resolution. In our large (10) aspect-ratio box runs we have employed around 27 000 unevenly spaced elements, resulting in about 105 unknowns per time step. Each run goes up to between 105 and 106 time steps in order to get many overturns of the dominant cells. We will present videos displaying the evolution of the physical fields, in particular the temperature and vorticity fields, which give a vivid portrayal of the mixing dynamics. Visualization is a handy medium for illustrating and also for discovering the richness of the mixing process, its multiple spatial and time scales in the transition from weak to strong turbulence. At Ra around 106, we have found that convection does not take place in a strictly cellular manner. Thermals emanate from the hot and cold boundary layers, which are superimposed on a large-scale type of circulation. These boundary layer instabilities enhance mixing of the interior. The fate of these instabilities is determined ultimately by the large-scale flow. With greater amounts of internal heating, the large-scale flow becomes smaller and mixing between distant regions is inhibited, but that between neighboring cells is enhanced by the changes of the flow pattern induced by internal heating. We have found a ‘‘mixing layer’’ above the thermal boundary layer in the hard-turbulent regime. At high Ra, between 107 and 108, we find breakdown of globally connected thermal plumes for base-heated convection. In this hard-turbulent regime the plumes become disconnected, ‘‘podlike’’ structures, thus inhibiting efficient vertical mixing. This disconnection of plumes in base-heated convection is to be regarded as a manifestation of the soft to hard turbulence transition. In the presence of internal heating and high Ra the descending cold boundary layers become dominant and serve to promote the interaction between the top and bottom, while the ascending plumes diminish in strength and disappear altogether. Mixing in the mantle thus influenced by the vigor in convection, the amount of internal heating, and the aspect ratio of the global configuration.

Nonlinear Convection in a Horizontal Layer with an Internal Heat Source

The problem of thermal convection is studied in a horizontal layer of fluid with an internal heat source which is restricted to vary linearly with the vertical variable. Square and hexagonal cells are found to be the only possible stable convection cells. Finite amplitude instability can occur for a range of the amplitude of convection. The presence of internal heating is found to be able to affect strongly the cell's size, the critical Rayleigh number, the heat transported by convection, the stability of the convective motion and the internal motion of the hexagonal cells.

A thermistor nomograph for a three-point fit

A description is given of a nomograph which has been designed to determine all thermistor three-point fit curves that are `parallel' to a desired curve. The nomograph first establishes the resistance to the minimum thermistor and the associated minimum network. If the parallel curve for this network is less than the desired curve, it can be realized. A higher resistance thermistor increases the parallel curve. Any such curve that is lower than the desired curve can be used. The difference in resistance is removed with a build-out resistor. The nomograph is designed for temperatures of 4°C (40°F), 32°C (90°F), and 60°C (140°F). These are equal intervals above and below 32°C. 32°C is somewhat higher than the usual room temperatures to allow for the presence of internal heating of the cabinet. The design is based on negligible self-heating of the thermistor.