Study of the analytical solution to the heat transfer problem and surface temperature in a semi-infinite body with a constant heat flux at the surface and an initial temperature distribution

In many practical cases, one heats a semi-infinite solid with a constant heat flux source. For such an unsteady heat transfer problem, if the body has a uniform initial temperature, the analytical solution has been given by Carslaw and Jaeger. The surface temperature of the semi-infinite body follows the $\sqrt t $ -rule, that is, the surface temperature changes in proportion to square root of heating time. But if, instead of the uniform initial temperature, the body has a temperature distribution at the beginning of heating, the analytical solution has not yet been developed. Analytical solutions to the same problem with an exponential or a linear initial temperature distribution are obtained in this paper. It is shown, that in the case of a linear initial temperature distribution the surface temperature also changes according to $\sqrt t $ -rule Approximating the initial temperature distribution near the surface by its tangent at the surface, it is found that the surface temperature within a short time after the start of heating should also satisfy the $\sqrt t $ -rule, in spite of an arbitrary initial temperature distribution. The experimental data support this argument. Furthermore, the constant heat flux can be calculated after relationship between the surface temperature and heating time according to the equation derived in this paper, if the initial temperature distribution or its first-order derivative at the surface is known.     

Thermoelastic state of a half-space having a thermally active crack parallel to its boundary

Using the boundary integral equation method, the problem of stationary heat conduction and thermoelasticity for a semi-infinite body with a crack parallel to its boundary is solved. Temperature or heat flow on the crack is prescribed. The body boundary is heat-insulated or is at zero temperature. The dependence of the stress intensity factor on the depth of occurrence of a circular crack at a constant temperature or under a constant heat flow is studied. In contrast to mechanical loading, thermal loading shows less SIF values than in an infinite body __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 46–54, April 2007.     

Investigation of a hypersonic viscous shock layer on rotating axisymmetric bodies in the presence of blowing

The hypersonic flow of a laminar stream of viscous compressible gas past blunt axisyrametric bodies rotating about the longitudinal axis is considered. It is assumed that gas blows from the surface of the body. The solution of the problem is obtained by a finite-difference method in a wide range of Reynolds numbers and blowing and rotation parameters. Some results of the calculations characterizing the effect of the rotation on the velocity and temperature profiles across the shock layer, on the friction and heat transfer coefficients, and the shock wave separation are given for the neighborhood of the stagnation point. For large Reynolds numbers and strong blowing an analytic solution of the problem is found in an approximation of two inviscid layers separated by a contact surface. The calculations are made for the flow past a sphere and a paraboloid and it is shown that in the presence of rotation the maximum of the heat flux is shifted from the stagnation point onto the side surface of the body. The dependence of the pressure distribution, the heat flux, and the friction coefficient is investigated for cases of constant and variable blowing over the contour of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–114, January–February, 1986.     

Steady accretion of an elastic body on a hard spherical surface and the notion of a four-dimensional reference space

Taking the cue from experiments on actin growth on spherical beads, we formulate and solve a model problem describing the accretion of an incompressible elastic solid on a rigid sphere due to attachment of diffusing free particles. One of the peculiar characteristics of this problem is that accretion takes place on the interior surface that separates the body from its support rather than on its exterior surface, and hence is responsible for stress accumulation. Simultaneously, ablation takes place at the outer surface where material is removed from the body. As the body grows, mechanical effects associated with the build-up of stress and strain energy slow down accretion and promote ablation. Eventually, the system reaches a point where internal accretion is balanced by external ablation. The present study is concerned with this stationary regime called “treadmilling”.The principal ingredients of our model are: a nonstandard choice of the reference configuration, which allows us to cope with the continually evolving material structure; and a driving force and a kinetic law for accretion/ablation that involves the difference in chemical potential, strain energy and the radial stress. By combining these ingredients we arrive at an algebraic system which governs the stationary treadmilling state. We establish the conditions under which this system has a solution and we show that this solution is unique. Moreover, by an asymptotic analysis we show that for small beads the thickness of the solid is proportional to the radius of the support and is strongly affected by the stiffness of the solid, whereas for large beads the stiffness of the solid is essentially irrelevant, the thickness being proportional to a characteristic length that depends on the parameters that govern diffusion and accretion kinetics.     

Rime- and glaze-ice accretion due to freezing rain falling vertically on a horizontal thermally insulated overhead line conductor

A theory of atmospheric icing due to freezing rain on an overhead line conductor (OHLC) is developed. The rain falls vertically on a horizontal OHLC that is thermally insulated. It is assumed that the collection efficiency of the accretion surface is unity and that this surface is in thermodynamic equilibrium with the environment.

For air temperature T_A 0°C and raindrop temperature T_D 0°C, the freezing rain accretes as rime ice, provided that the temperature of the ice surface T_l < 0°C. The evolution equation governing the mass transfer at the accretion surface is solved analytically, yielding the shape of the rime-ice surface. Equations governing the thermal state of the rime-ice deposit are also given. These determine the onset of wet growth or glaze accretion at the upper stagnation line during suitable environmental conditions.

For environmental conditions producing an ice surface at temperature T_l = 0°gC, the freezing accretes as glaze. Equations governing the heat and mass transfer at the surface determine the shape of the glaze surface and the downward viscous motion of the unfrozen water. For T_D < 0°C, glaze evolution equations are developed for T_A 0°C and T_A 0°C. Analytical solutions of these equations are obtained. In particular, when T_D < −T_A < 0°C, the evolution equation predicts a novel limiting growth that is triangular in shape. Further study of the mass and heat transfer conditions, in the neighborhood of this final stage of glaze accretion, shows that it is maintained in thermodynamic equilibrium with its warm air environment.     

Certain optimum forms of radiating bodies in supersonic gas flow

We solve two variational problems on determining the optimum form of a radiating body of given dimensions in a gas flow at high supersonic velocity with a laminar flow regime in the boundary layer. We consider the case in which two heat transfer processes are significant: convective heat transfer from the gas to the body and radiation from the body surface. The first problem involves finding the contour of the body which receives the minimal thermal flux. In the solution of the second problem we seek that form of the thermally isolated body for which its surface temperature will be minimal for given parameters of the approaching stream or for which the motion velocity will be maximal for a given wall temperature and flight altitude.     

The effect of the time structure of laser pulse on the temperature distribution in homogeneous body with coating

In the present paper an analytical solution of the transient heat conduction problem for the nonhomogeneous body, consisting of bulk foundation and thin coating of different material, is presented. This body is heated through the outer surface of coating by heat flux generated due to absorption of pulsed laser radiation. The dependence of nonstationary temperature on the thermophysical properties of the foundation and coating were studied.     

Planar solidification of a warm flowing-liquid under different boundary conditions

Planar solidification of a warm flowing liquid with the convective heat transfer to the growing solid layer, has been analysed for the boundary conditions of constant temperature, constant heat flux and convective heat flux at the surface respectively. The mathematical formulation of the problem resulted in a coupled set of two differential equations in temperature and solid thickness as function of position, time and the problem parameters. Analytical expressions for the temperature distribution within the growing solid layer, the rate of solidification and the solidification time are obtained. The perturbation techniques employed here is simple and straight forward in contrast with the earlier techniques. Good agreement between the experimental results and the present solutions is obtained for the convective heat flux boundary condition. The results of this analysis are useful in the design and analysis of experiments dealing with freezing/melting in one dimension. The role of the parameter Stefan number which is small for phase change materials, is discussed in context with the storage of thermal energy.     

Investigations of Heat and Mass Transfer for Thermal Protection Materials in a Long Flight

The problem of unsteady coupled heat and mass transfer in the course of motion of a spherically blunted conical body fabricated with the use of thermal protection materials is considered. Numerical integration is applied to study the characteristics of heat and mass transfer at constant stagnation parameters (Mach number 6, altitude 15 km, and flight time 600 s), which impose severe constraints on the choice of materials for thermal protection. It is demonstrated that the use of advanced ceramic materials ensures an admissible temperature regime and maintaining the initial geometry of the body, including its motion at an angle of attack.     

Combined heat and mass transfer by hydromagnetic natural convection over a cone embedded in a non-Darcian porous medium with heat generation/absorption effects

 The problem of combined heat and mass transfer by natural convection over a permeable cone embedded in a uniform porous medium in the presence of an external magnetic field and internal heat generation or absorption effects is formulated. The cone surface is maintained at either constant temperature and constant concentration or uniform heat and mass fluxes. In addition, the cone surface is assumed permeable in order to allow for possible fluid wall suction or blowing. The resulting governing equations are non-dimensionalized and transformed into a non-similar form and then solved numerically by an implicit, iterative, finite-difference method. Comparisons with previously published work are performed and excellent agreement between the results is obtained. A parametric study of the physical parameters is conducted and a representative set of numerical results for the temperature and concentration profiles as well as the local Nusselt number and the Sherwood number is illustrated graphically to show special trends of the solutions. Received on 5 June 2000 / Published online: 29 November 2001     

Mixed Convection in a Darcy–Brinkman Porous Medium with a Constant Convective Thermal Boundary Condition

The characteristics of the boundary layer flow past a plane surface adjacent to a saturated Darcy–Brinkman porous medium are investigated in this paper. The flow is driven by an external free stream moving with constant velocity. The surface is heated with a convective boundary condition with constant heat transfer coefficient. The problem is non-similar and is investigated numerically by a finite difference method. The problem is governed by four non-dimensional parameters, that is, the convective Darcy number, the convective Grashof number, the Prandtl number, and the axial distance along the plate. The influence of these parameters on the results is investigated, and the results are presented in tables and figures. The Darcy term and the Grashof term in the momentum equation contradict each other and this contradiction makes the problem complicated. However, the wall shear stress and the wall temperature increase continuously along the plate and the wall temperature always tends to 1.     

Approximate solutions to the Stefan problem with internal heat generation

Using a quasi-static approach valid for Stefan numbers less than one, we derive approximate equations governing the movement of a phase change front for materials which generate internal heat. These models are applied for both constant surface temperature and constant surface heat flux boundary conditions, in cylindrical, spherical, plane wall and semi-infinite geometries. Exact solutions with the constant surface temperature condition are obtained for the steady-state solidification thickness using the cylinder, sphere, and plane wall geometries which show that the thickness depends on the inverse square root of the internal heat generation. Under constant surface heat flux conditions, closed form equations can be obtained for the three geometries. In the case of the semi-infinite wall, we show that for constant temperature and constant heat flux out of the wall conditions, the solidification layer grows then remelts.     

Temperature distributions in a periodically stratified layer with a vertical dug well

The stationary problems of heat conduction in a periodically stratified layer with a vertical cylindrical hole is considered. The lateral surface of the hole is kept at a given piece-wise constant temperature. The boundary planes at zero temperature are taken into account, and the ideal thermal contact between the composite constituents is assumed. The problem within the homogenized model with microlocal parameters is solved.     

Heat Transfer in Supersonic Low-Re Flow Past a Sphere and a Cylinder

Supersonic perfect-gas flow past a sphere and a cylinder at a constant surface temperature is investigated on the basis of a numerical solution of the Navier-Stokes equations. The Re-dependence of the Nusselt number and the recovery coefficient is calculated for the Reynolds numbers ranging from 1 to 1000 and Mach numbers 3 (cylinder) and 5 (sphere) and compared with the experimental data. The influence of slip and no-slip conditions imposed on the body surface on the heat transfer parameters and the base flow is analyzed.     

Free convection through conducting non-Newtonian fluids over a thin axisymmetric body

An analysis of free convection heat transfer in electrically conducting power law non-Newtonian fluid over a thin axisymmetric body of constant temperature is carried out. The uniform external magnetic field acts normally to the surface through the induced boundary layer. In view of the fact that most of the non-Newtonian fluids have large Prandtl number, the momentum equation is simplified. The equations of conservation of mass, momentum and energy which govern and describe the flow and heat transfer are solved numerically. The effect of the magnetic field on the velocity, temperature, the coefficient of friction and the Nusselt number are investigated. Numerical results are tabulated, presented graphically and discussed.     

Three-dimensional chemically nonequilibrium viscous shock layer on a catalytic surface with allowance for associated heat transfer

A three-dimensional flow of dissociating air past blunt bodies is investigated in the framework of the thin viscous shock layer theory. Multicomponent diffusion and homogeneous chemical reactions, including dissociation, recombination, and exchange reactions, are taken into account. The generalized Rankine-Kugoniot conditions are specified on the shock wave and the conditions which take into account the heterogeneous catalytic reactions, on the surface of the body. The viscous shock layer equations are solved together with the heat equations inside the coating, which is carbon with a deposited thin film of SiO₂, or quartz. The case of a thermally insulated surface is also considered. The problem for the case of the motion of a body along the re-entry trajectory into Earth's atmosphere is investigated numerically. The temperature of the surface and the heat flux toward it are given as a dependence on the height (tine) of the flight for different cases of the specification of the catalytic reactions. It is shown that the difference between the heat fluxes towards the thermally insulated surface and the fluxes toward the heat-conducting surface in the neighborhood of the stagnation point is of the order of 6–12% for all the cases considered. This makes it possible to decouple the solution of the problem of heat conduction in the body.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 140–146, November–December, 1985.deceased     

Radiative and convective heat transfer in a plane layer of absorbent curtain

An examination is made of the thermal state of a plane layer of gray gas injected into a turbulent stream of high temperature gas flowing over a permeable flat plate.Similarity-type formulations of problems are encountered both in examination of flow near a stagnation point, and also in analysis of the lifting of the boundary layer by intense injection through a porous plate [1]. The examination described relates to the following idealized formulation of the problem (Fig. la).In a plane layer of gray absorbing medium, formed by plane-parallel diffusely radiating surfaces (1-porous plate; 2-boundary of high temperature turbulent gas stream), heat transfer is accomplished by radiation and convection of the layer normal to the surfaces and by molecular heat conduction. All the physical and optical properties of the medium and of the boundary surfaces are assumed to be constant, independent of temperature.The temperature of the wetted surface of the specimen and also that of the fictitious surface determining the upper bound of the lift-off region, are given.Also assumed given is the velocity of the injected medium, which is constant throughout the entire lift-off layer. This idealization appreciably facilitates our examination without in principle changing its features.A very simplified examination of this problem was given in [2]. The special case of a medium with low optical thickness was examined in [3,4].The problem was examined in [5] under the assumption that molecular heat conduction in the medium is negligibly small.In the formulation considered the generalized energy equation has the form     

Mathematical simulation of mixed convection in a vertical cylindrical vessel

Mixed laminar convection in a vertical cylindrical vessel is investigated. A flow model is developed and the temperature distribution in the core is obtained for a given specific heat flux on the lateral surface. The results of the analytic simulation are compared with numerical solutions of the problem for a specific heat flux which is constant on the lateral wall or varies linearly with height. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 9–17, November–December, 1998.     

Thermographic validation of a novel,laminate body,analytical heat conduction model

The two-region fin model captures the heat spreading behaviour in multilayered composite bodies (i.e., laminates), heated only over a small part of their domains (finite heat source), where there is an inner layer that has a substantial capacity for heat conduction parallel to the heat exchange surface (convection cooling). This resulting heat conduction behaviour improves the overall heat transfer process when compared to heat conduction in homogeneous bodies. Long-term heat storage using supercooling salt hydrate phase change materials, stovetop cookware, and electronics cooling applications could all benefit from this kind of heat-spreading in laminates. Experiments using laminate films reclaimed from post-consumer Tetra Brik cartons were conducted with thin rectangular and circular heaters to confirm the laminate body, steady-state, heat conduction behaviour predicted by the two-region fin model. Medium to high accuracy experimental validation of the two-region fin model was achieved in Cartesian and cylindrical coordinates for forced external convection and natural convection, the latter for Cartesian only. These were conducted using constant heat flux finite heat source temperature profiles that were measured by infrared thermography. This validation is also deemed valid for constant temperature heat sources.