Distribution of Friction Heat Between a Stationary Pin and Rotating Ring

The analysis of the heat distribution between a stationary pin and rotating ring was considered. To solution of the governing quasi-stationary heat conductivity equation the finite Fourier transform was used. The convective cooling from outer and internal surface of the ring as the boundary conditions were considered. The ring surface temperature, the average temperature on the ring surface and the heat distribution coefficient for the studied system were determined. The numerical results for the temperatures and heat distribution coefficient which demonstrated the effects of the Biot number and internal radius of the ring on them, were presented.     

Heat flux sensor with minimal impact on boundary conditions

A technique for determining the heat transfer on the far surface of a wall based on measuring the heat flux and temperature on the near wall is presented. Although heat transfer measurements have previously been used to augment temperature measurements in inverse heat conduction methods, the sensors used alter the heat flow through the surface, disturbing the very quantity that is desired to be measured. The ideal sensor would not alter the boundary condition that would exist were the sensor not present. The innovation of this technique in that it has minimal impact on the wall boundary condition. Since the sensor is placed on the surface of the wall, no alteration of the wall is needed. The theoretical basis for the experimental technique as well as experimental results showing the heat flux sensor performance is presented.     

Axially symmetric thermal stress of an external circular crack under general thermal conditions

Summary The paper presents a solution for the linear thermoelastic problem of determining axisymmetric stress and displacement fields in an isotropic elastic solid of infinite extent weakened by an external circular crack under general mechanical loadings and general thermal conditions. The mechanical loadings and thermal conditions applied on the crack faces are axisymmetric, being non-symmetric about the crack plane. In similar lines of [7], equations of equilibrium of an elastic solid conducting heat have been solved using Hankel transforms and Abel operators of the first kind. Expressions for stress, displacement, temperature and heat flux functions are obtained in terms of Abel transforms of the first kind of the jumps of stress, displacement, temperature and heat flux at the crack plane. Two types of thermal conditions, that is, general surface temperatures and general heat flux on faces of the crack are considered. In both the cases, closed form solutions have been obtained for the unknown functions solving Abel type of integral equations. Explicit expressions for stresses, displacements, temperature fields, stress intensity factors have been obtained. Two special cases of thermal conditions in which: (i) crack faces are subjected to constant non-symmetric temperatures over a circular ring area, (ii) crack faces are subjected to constant non-symmetric heat flux over a circular ring area, have been considered. In some special cases, results have been compared with those from the literature.     

Influence of the velocity gradient on the stagnation point heating in hypersonic flow

In a number of experimental and numerical publications a deviation has been found between the measured or computed stagnation point heat flux and that given by the theory of Fay and Riddell. Since the formula of Fay and Riddell is used in many applications to yield a reference heat flux for experiments performed in wind tunnels, for flight testing and numerical simulations, it is important that this reference heat flux is as accurate as possible. There are some shortcomings in experiments and numerical simulations which are responsible in some part for the deviations observed. But, as will be shown in the present paper, there is also a shortcoming on the theoretical side which plays a major role in the deviation between the theoretical and experimental/numerical stagnation point heat fluxes. This is caused by the method used so far to determine the tangential velocity gradient at the stagnation point. This value is important for the stagnation point heat flux, which so far has been determined by a simple Newtonian flow model. In the present paper a new expression for the tangential velocity gradient is derived, which is based on a more realistic flow model. An integral method is used to solve the conservation equations and, for the stagnation point, yields an explicit solution of the tangential velocity gradient. The solution achieved is also valid for high temperature flows with real gas effects. A comparison of numerical and experimental results shows good agreement with the stagnation point heat flux according to the theory of Fay and Riddell, if the tangential velocity gradient is determined by the new theory presented in this paper.This article was processed by the author using theLAT_EX style filepljour2 from Springer-Verlag.     

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.     

Application of Fractional Derivatives in Thermal Analysis of Disk Brakes

This paper presents a Fractional Derivative Approach for thermal analysis of disk brakes. In this research, the problem is idealized as one-dimensional. The formulation developed contains fractional semi integral and derivative expressions, which provide an easy approach to compute friction surface temperature and heat flux as functions of time. Given the heat flux, the formulation provides a means to compute the surface temperature, and given the surface temperature, it provides a means to compute surface heat flux. A least square method is presented to smooth out the temperature curve and eliminate/reduce the effect of statistical variations in temperature due to measurement errors. It is shown that the integer power series approach to consider simple polynomials for least square purposes can lead to significant error. In contrast, the polynomials considered here contain fractional power terms. The formulation is extended to account for convective heat loss from the side surfaces. Using a simulated experiment, it is also shown that the present formulation predicts accurate values for the surface heat flux. Results of this study compare well with analytical and experimental results.     

Application of Fractional Derivatives in Thermal Analysis of Disk Brakes

This paper presents a Fractional Derivative Approach for thermal analysis of disk brakes. In this research, the problem is idealized as one-dimensional. The formulation developed contains fractional semi integral and derivative expressions, which provide an easy approach to compute friction surface temperature and heat flux as functions of time. Given the heat flux, the formulation provides a means to compute the surface temperature, and given the surface temperature, it provides a means to compute surface heat flux. A least square method is presented to smooth out the temperature curve and eliminate/reduce the effect of statistical variations in temperature due to measurement errors. It is shown that the integer power series approach to consider simple polynomials for least square purposes can lead to significant error. In contrast, the polynomials considered here contain fractional power terms. The formulation is extended to account for convective heat loss from the side surfaces. Using a simulated experiment, it is also shown that the present formulation predicts accurate values for the surface heat flux. Results of this study compare well with analytical and experimental results.     

Retrieving the heat transfer coefficient for jet impingement from transient temperature measurements

Algorithm of retrieving the heat transfer coefficient (HTC) from transient temperature measurements is presented. The unknown distributions of two types of boundary conditions: the temperature and heat flux are parameterized using a small number of user defined functions. The solutions of the direct heat conduction problems with known boundary temperature and flux are expressed as a superposition of auxiliary temperature fields multiplied by unknown parameters. Inverse problem is formulated as a least squares fit of calculated and measured temperatures and is cast in a form of a sum of two objective functions. The first results originates from an inverse problem for retrieving the boundary temperature the second comes from the inverse problem for reproducing the boundary heat flux. The final form of the objective function is obtained by enforcing constant in time value of the heat transfer coefficient. This approach leads to substantial regularization of the results, when compared with the standard technique, where HTC is calculated from separately reconstructed temperature and heat flux on the boundary. The validation of the numerical procedure is carried out by reconstructing a known distribution of the HTC using simulated measurements laden by stochastic error. The proposed approach is also used to reconstruct the distribution of the HTC in a physical experiment of heating a cylindrical sample using an impinging jet.     

Numerical nonlinear inverse problem of determining wall heat flux

The inverse problem of determining time-variable surface heat flux in a plane wall, with constant or temperature dependent thermal properties, is numerically studied. Different kinds of incident heat flux, including rectangular waveform, are assumed. The solution is numerically solved as a function estimation problem, so that no a priori information for the functional waveforms of the unknown heat flux is needed. In all cases, a solution in the form of a piece-wise function is used to approach the incident flux. Transient temperature measurements at the boundary, from the solution of the direct problem, served as the simulated experimental data needed as input for the inverse analysis. Both direct and inverse heat conduction problems are solved using the network simulation method. The solution is obtained step-by-step by minimising the classical functional that compares the above input data with those obtained from the solution of the inverse problem. A straight line of variable slope and length is used for each one of the stretches of the desired solution. The influence of random error, number of functional terms and the effect of sensor location are studied. In all cases, the results closely agree with the solution.     

Experimental study on condensation heat transfer of steam on vertical titanium plates with different surface energies

Visual experiments were employed to investigate heat transfer characteristics of steam on vertical titanium plates with/without surface modifications for different surface energies. Stable dropwise condensation and filmwise condensation were achieved on two surface modification titanium plates, respectively. Dropwise and rivulet filmwise co-existing condensation form of steam was observed on unmodified titanium surfaces. With increase in the surface subcooling, the ratio of area (η) covered by drops decreased and departure diameter of droplets increased, resulting in a decrease in condensation heat transfer coefficient. Condensation heat transfer coefficient decreased sharply with the values of η decreasing when the fraction of the surface area covered by drops was greater than that covered by rivulets. Otherwise, the value of η had little effect on the heat transfer performance. Based on the experimental phenomena observed, the heat flux through the surface was proposed to express as the sum of the heat flux through the dropwise region and rivulet filmwise region. The heat flux through the whole surface was the weighted mean value of the two regions mentioned above. The model presented explains the gradual change of heat transfer coefficient for transition condensation with the ratio of area covered by drops. The simulation results agreed well with the present experimental data when the subcooling temperature is lower than 10 °C.     

Natural convection with evaporation in a vertical cylindrical cavity under the effect of temperature-dependent surface tension

The effect of surface tension on laminar natural convection in a vertical cylindrical cavity filled with a weak evaporating liquid has been analyzed numerically. The cylindrical enclosure is insulated at the bottom, heated by a constant heat flux from the side, and cooled by a non-uniform evaporative heat flux from the top free surface having temperature-dependent surface tension. Governing equations with corresponding boundary conditions formulated in dimensionless stream function, vorticity, and temperature have been solved by finite difference method of the second-order accuracy. The influence of Rayleigh number, Marangoni number, and aspect ratio on the liquid flow and heat transfer has been studied. Obtained results have revealed that the heat transfer rate at free surface decreases with Marangoni number and increases with Rayleigh number, while the average temperature inside the cavity has an opposite behavior; namely, it growths with Marangoni number and reduces with Rayleigh number.     

Numerical simulation of laminar natural convection in a laterally heated vertical cylindrical enclosure: application to crystal growth

Laminar natural convection has been studied in a laterally heated vertical cylindrical enclosure with a free insulated surface and a centrally located constant temperature wall at the top. These conditions are a simplification of the conditions existing in a Czochralski crystal pulling system. The laminar, axisymmetric flow of a Newtonian, constant physical properties fluid under Boussinesq’s approximation has been considered. Governing equations in primitive variable form are solved numerically by control volume method. SIMPLE algorithm due to Patankar has been used for the numerical simulation. The effects of the constant wall heat flux boundary condition at the side wall have been investigated whereas the bottom wall is considered to be insulated. Streamlines and isotherms are presented for various Rayleigh numbers and Prandtl numbers. Heat flux vectors through the melt are plotted for selected cases. The axial velocity and temperature variations at different horizontal sections of the crucible have been presented graphically to explain the transport processes inside the crucible. It has been observed that in case of low Pr and high Ra, flow separation occurs at the vertical wall of the crucible which leads to an oscillatory flow as Ra increases. The investigation has been extended to the oscillatory regime of flow in the zone of supercritical Rayleigh numbers and some unsteady results are also presented. Finally a heat transfer correlation has been developed for steady-state case.     

Thermal Dispersion Effects on Combined Convection in Non-Newtonian Fluids Along a Nonisothermal Vertical Plate in a Porous Medium

The method of non-similarity solution is used to study the influence of thermal dispersion on combined convection from vertical surfaces in a porous medium saturated with a power-law type non-Newtonian fluid. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. The transformed conservation laws are solved numerically for the case of variable surface heat flux conditions. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented.     

Efficient reconstruction of local heat fluxes in pool boiling experiments by goal-oriented adaptive mesh refinement

In this paper, we consider the efficient estimation of local boiling heat fluxes from transient temperature measurements in the heater close to the heater surface. For accurate prediction, heat flux estimation is formulated as a transient three-dimensional (3D) inverse heat conduction problem (IHCP). This inverse problem is ill-posed and cannot be treated straightforwardly by established numerical methods. In order to obtain a regularized stable solution, a large-scale time-dependent PDE-constrained optimization problem has to be solved and an appropriate stopping criterion for the termination of the iterative solution process has to be chosen. Since the boiling heat flux is non-uniformly distributed on the heater surface due to the strong local activity of the boiling process, the use of a fixed uniform spatial discretization is not efficient. Instead, an adaptive mesh refinement strategy can be used to obtain an appropriate discretization which significantly reduces the total computational effort. In this work, we present an automatic algorithm incorporating an adaptive mesh refinement via a heat flux-based a-posteriori error estimation technique. The suggested algorithm can cope with both spatially point-wise or highly resolved temperature observations efficiently. It is applied to real measurement data obtained from two different types of pool boiling experiments. The numerical results show that the computational effort can be reduced significantly for given estimation quality. This adaptive IHCP solution technique can be also viewed as an efficient soft sensor to deduce unmeasurable local boiling heat fluxes.     

Heat transfer in pool boiling of ammonia/water mixture

The nonazeotropic binary mixtures such as, methanol/water, ethanol/water and ammonia/water, have variable boiling and dew points, depending on the combination of substance and those mass fraction. It is expected to have a higher performance as a result of decreasing the thermodynamically irreversible loss, when there is a relevant mass fraction. Therefore, ammonia/water mixture is expected to use as working fluid in small temperature difference power generation cycles and absorption refrigeration cycles. However, few experiments were carried out for measuring heat transfer coefficient for ammonia/water mixture in the world. An experimental study has been carried out to measure boiling heat transfer coefficient of an ammonia/water mixture on a horizontal heated surface at low pressure of 0.2, 0.4 and 0.7 MPa and at low mass fraction of 0 < C < 0.27 and at high pressure 0.7, 1.0 and 1.5 MPa and at mass fraction of 0.5 < C < 1.0 and at heat flux under critical heat flux the heat transfer coefficient are compared with existing correlations prediction and a revised correlation can be proposed to predict them well.     

Mass transfer effects on the non-Newtonian fluids past a vertical plate embedded in a porous medium with non-uniform surface heat flux

 The present study is devoted to investigate the influences of mass transfer on buoyancy induced flow over vertical flat plate embedded in a non-Newtonian fluid saturated porous medium. The Ostwald–de Waele power-law model is used to characterize the non-Newtonian fluid behavior. Similarity solution for the transformed governing equations is obtained with prescribed variable surface heat flux. Numerical results for the details of the velocity, temperature and concentration profiles are shown on graphs. Excess surface temperature as well as concentration gradient at the wall associated with heat flux distributions, which are entered in tables, have been presented for different values of the power-law index n, buoyancy ration B and the exponent λ as well as Lewis number Le. Received on 26 April 2000     

Free convection from a vertical permeable circular cone with non-uniform surface heat flux

 In this paper, the problem of laminar free convection from a vertical permeable circular cone maintained with non-uniform surface heat flux is considered. The governing boundary layer equations are reduced non-similar boundary layer equations with surface heat flux proportional to x ⁿ (where x is the distance measured from the leading edge). The solutions of the reduced equations are obtained by using three distinct solution methodologies; namely, (i) perturbation solution for small transpiration parameter, ξ, (ii) asymptotic solution for large ξ, and (iii) the finite difference solutions for all ξ. The solutions are presented in terms of local skin-friction and local Nusselt number for smaller values of Prandtl number and heat flux gradient and are displayed in tabular form as well as graphically. Effects of pertinent parameters on velocity and temperature profiles are also shown graphically. Solutions obtained by finite difference method are also compared with the perturbation solutions for small and large ξ and found to be in excellent agreement. Received on 1 October 1999     

The extended Graetz problem with piecewise constant wall heat flux for laminar and turbulent flows inside concentric annuli

In the present paper, the heat transfer characteristics in the thermal entrance region of concentric annuli have been analysed for laminar and turbulent internal flow. Axial heat conduction effects in the fluid have been taken into account. The present paper shows an exact analytical solution for the problem of a piecewise uniform wall heat flux. The obtained analytical solution for the extended Graetz problem is as simple and efficient to compute as the related solution of the parabolic problem. The obtained results show the effect of axial heat conduction in the fluid for a semi-infinite heated section as well as for a finite length of the heated section. It is shown, that for a finite length of the heated section, axial heat conduction effects might be important even for higher Peclet number.     

Thermal dispersion effects on non-Darcy natural convection over horizontal plate with surface mass flux

Summary The effect of surface mass flux on the non-Darcy natural convection over a horizontal flat plate in a saturated porous medium is studied using similarity solution technique. Forchheimer extension is considered in the flow equations. The suction/injection velocity distribution has been assumed to have power function form Bx ^l , similar to that of the wall temperature distribution Ax ⁿ , where x is the distance from the leading edge. The thermal diffusivity coefficient has been assumed to be the sum of the molecular diffusivity and the dynamic diffusivity due to mechanical dispersion. The dynamic diffusivity is assumed to vary linearly with the velocity component in the x direction, i.e. along the hot wall. For the problem of constant heat flux from the surface (n=1/2), similarity solution is possible when the exponent l takes the value −1/2. Results indicate that the boundary layer thickness decreases whereas the heat transfer rate increases as the mass flux parameter passes from the injection domain to the suction domain. The increase in the thermal dispersion parameter is observed to favor the heat transfer by reducing the boundary layer thickness. The combined effect of thermal dispersion and fluid suction/injection on the heat transfer rate is discussed. Received 7 December 1995; accepted for publication 7 January 1997