Laminar forced convection in circular and flat ducts with wall axial conduction and external convection

An analytical solution is obtained for laminar forced convection in circular and flat ducts with the presence of axial duct wall conduction and external convection at the outer surface of the duct wall. The eigenvalues for the problem are determined using the solution for the constant temperature boundary condition. The heat transfer results depend on four nondimensional numbers. The wall and fluid temperatures depend strongly on the wall conductance parameter while the heat flux enhancement due to wall conduction is large at short distances from the duct inlet.     

反演高温腔体内壁面温度波动的辐射边界导热反问题

在工业生产中有很多情况需要获得高温腔体内壁温度波动,但在内壁面安装测温装置进行直接测量非常困难,一般通过测量外壁温度再进行反演计算间接获得。而已有反演计算方法未考虑高温壁面与周围环境之间的辐射传热,给反演计算结果带来一定误差,为此建立了考虑辐射边界条件的反演高温腔体内壁面温度波动的导热反问题数学模型,并构造了两组数值试验对数学模型的效果进行检验。计算结果表明,建立的数学模型能够很好的由高温腔体外壁面温度反演得到内壁面温度波动情况。     

Laminar tube flow heat transfer in non-newtonian fluids under arbitrary wall heat flux

Analytical solutions are developed for the wall temperature profile of a power law fluid in laminar flow in a circular tube. This profile is first developed for the boundary condition involving uniformly constant heat flux at the wall. This is next extended for the boundary condition involving an arbitrarily varying heat flux at the wall. The computed results are finally compared with measured values obtained from a horizontal recirculating flow experimental unit.     

Design of spherical vessels under steady-state thermal loading using thermo-elasto–plastic concept

Governing equilibrium equations of thick-walled spherical vessels made of material following linear strain hardening and subjected to a steady-state radial temperature gradient using elasto–plastic analysis are derived. By considering a maximum plastic radius and using the concept of thermal autofrettage for the strengthening mechanism, the optimum wall thickness of the vessel for a given temperature gradient across the wall thickness is obtained. Finally, in the case of thermal loading on a vessel, the effect of convective heat transfer on the optimum thickness is studied and a general formula for the optimum wall thickness and design graphs for several different cases are presented.     

Investigation of local heat transfer coefficients in plate heat exchangers with temperature oscillation IR thermography and CFD

A method for the measurement of local convective heat transfer coefficients from the outside of a heat-transferring wall has been developed. This method is contact-free and fluid independent, employing radiant heating by laser or halogen spotlights and an IR camera for surface temperature measurements; it allows for the rapid evaluation of the heat transfer coefficient distribution of sizable heat exchanger areas. The technique relies first on experimental data of the phase lag of the outer surface temperature response to periodic heating, and second on a simplified numerical model of the heat exchanger wall to compute the local heat transfer coefficients from the processed data. The IR temperature data processing includes an algorithm for temperature drift compensation, phase synchronization between the periodic heat flux and the measured temperatures, and Single Frequency Discrete Fourier Transformations. The ill-posed inverse heat conduction problem of deriving a surface map of heat transfer coefficients from the phase-lag data is solved with a complex number finite-difference method applied to the heat exchanger wall. The relation between the local and the mean heat transfer coefficients is illuminated, calculation procedures based on the thermal boundary conditions are given. The results from measurements on a plate heat exchanger are presented, along with measurements conducted on pipe flow for validation. The results show high-resolution surface maps of the heat transfer coefficients for a chevron-type plate for three turbulent Reynolds numbers, including a promising approach of visualizing the flow field of the entire plate. The area-integrated values agree well with literature data. CFD calculations with an SST and an EASM–RSM were carried out on a section of a PHE channel. A comparison with the measured data indicates the shortcomings of even advanced turbulence models for the prediction of heat transfer coefficients but confirms the advantages of EASM–RSM in complex flows.     

NUMERICAL STUDY ON 3-D LIGHT AND HEAT TRANSPORT IN BIOLOGICAL TISSUES EMBEDDED WITH LARGE BLOOD VESSELS DURING LASER-INDUCED THERMOTHERAPY

Tissue vasculature plays an important role in the temperature responses of biological bodies subject to laser heating. For example, interfaces between blood vessel and its surrounding tissues may lead to reflection or absorption of the coming laser light. However, most of the previous efforts just treat this by considering a collective model. To date, little attention has been paid to the effect of a single blood vessel on tissue temperature prediction during laser-induced thermotherapy. To resolve this important issue in clinics, we propose to simultaneously solve the three-dimensional (3-D) light and heat transport in several typical tissue domains with either one single blood vessel or two countercurrent blood vessels running through. Both surface and intervenient laser irradiations are considered in these studies. The 3-D heat transfer and blood flow models are established to characterize the temperature transients over the whole area. Coupled equations for heat and blood flow in multiple regions are solved using the blocking-off method. In particular, the Monte Carlo method is introduced to calculate the light transport inside the tissues as well as the blood vessel. Theoretical algorithms to deal with the complex interfaces between the tissues and vessels, and the tissue–air interface, are given. The heat generation pattern due to absorption of laser light is thus obtained by Monte Carlo simulation and then adopted into the heat and flow transport equations to predict the 3-D temperature transients over the whole domain. It is demonstrated that without considering large-size blood vessels inside the tissues, a very different temperature response is induced when subject to the same laser heating. Detailed temperature developments for the aforementioned vessel configurations are comprehensively analyzed. Implementation of the laser irradiation pattern to the clinical practices is discussed. We also test the effects of the buoyancy-driven blood flow due to laser heating on the tissue temperature response. This study may raise new issues to evaluate the contribution of a single blood vessel in modeling laser–tissue interaction. Such information is expected to be critical for accurate treatment planning in clinics.     

Free boundary shape of a convectively cooled solidified region

The two-dimensional steady-state shape of a solidified region, such as a frost layer, was determined analytically for formation on a plate that is convectively cooled. The nonuniform shape of the layer is produced by exposure to a spatially nonuniform distribution of radiant energy. For high convective cooling the cooled wall approaches a uniform temperature, and an exact solution is obtained for the free boundary shape. For a lesser amount of convective cooling, the variation in temperature along the cooled boundary is treated by a boundary perturbation method. Some illustrative examples are given that show the effects of nonuniform heating and the magnitude of convective heat transfer at the cooled wall. Only one boundary condition is approximated by the perturbation solution ; all of the other boundary conditions are satisfied exactly. The calculated results given here were found to satisfy the approximate boundary condition within a very small error.     

Effects of an unsteady thermal boundary condition on forced convection heat transfer

A numerical analysis based on adjoint formulation of unsteady forced convection heat transfer is proposed to generally evaluate effects of the thermal boundary condition on the heat transfer characteristics. A numerical solution of the adjoint problem enables us to predict the heat transfer characteristics, such as the total heat transfer rate or the temperature at a specific location, when the thermal boundary conditions change arbitrarily with time. Moreover, using the numerical solution of the adjoint problem, we can obtain the optimal thermal boundary conditions in both time and space to maximize the heat transfer at any arbitrary time. Numerical solutions of the adjoint problem in a lid‐driven cavity are presented to illustrate the capability of the present method. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 31(3): 237–247, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10032     

Natural convection heat transfer in a circular enclosure

Natural convection heat transfer in a circular enclosure, one half of which was heated and the other half of which was cooled, was investigated experimentally, focusing on the effect of the inclination angle. The experiments were carried out with water. Flow and temperature field were visualized by using the aluminum and liquid-crystal suspension method. The results show that with downward heating the heat transfer coefficient increased as the inclination angle of the boundary between the heating wall and the cooling wall approached the vertical. But with upward heating, the heat transfer coefficient showed minimal change, exhibiting a small peak value when the inclination angle was γ ˜ –45°. The heat transfer coefficient of a flat circular enclosure was estimated from the circular enclosure's heat transfer coefficient. These results can be explained by the obtained flow and temperature fields. © 1999 Scripta Technica, Heat Trans Asian Res, 28(2): 152–163, 1999     

Heat Transfer Analysis for a Concentric Tube Heat Exchanger Including the Wall Axial Conduction

The effect of wall axial conduction on the heat transfer in a concentric tube heat exchanger is examined for the inner flow laminar flow regime. The procedure used for the current analysis combines the analytical solution for the inner fluid with a numerical approximation for the wall conduction and has the capability of handling the temperature variation for the outer fluid. Both parallel and counterflow cases are evaluated for the analysis, and results are presented in terms of the axial variations of fluids and wall temperatures. Effects of the heat capacity rate ratio of the fluids on the temperature variations and on the mean heat flux are also pointed out. The effect of the exchanger length is included for the analysis. It is concluded that the total heat transfer between the fluids is greatly influenced by the wall axial conduction for the counterflow arrangement and is not ignorable when the heat capacity rate ratio of fluids are smaller than unity.     

The effect of the top and bottom wall temperatures on the laminar natural convection in an air-filled square cavity

The effect of the top and bottom wall temperatures on the natural convection heat transfer characteristics in an air-filled square cavity driven by a difference in the vertical wall temperatures was investigated by measuring the temperature distributions along the heated vertical wall and visualizing the flow patterns in the cavity. The experiments were performed at a horizontal Grashof number of 1.9 × 10⁸. Increasing the top wall temperature resulted in a separated flow region on the top wall, which caused a secondary flow between the separated flow and the boundary layer on the heated vertical wall. This secondary flow had a significant effect on the heat transfer in this region. Changes in the top and bottom wall temperatures changed the temperature gradient and the average temperature of the air outside the thermal boundary layers in the cavity. The local heat transfer along much of the heated vertical wall could be correlated by Nu = C · Ra^0.32, but the constant C increased when the average of the top and bottom wall temperatures increased.     

Thermal boundary condition effects on forced convection heat transfer

We propose a numerical solution of an adjoint problem of forced convection heat transfer to evaluate the mean heat transfer characteristics under arbitrary thermal boundary conditions. Using the numerical solution of the adjoint problem under the Dirichlet condition, which can be computed by slightly modifying a conventional heat transfer code, we obtain an influence function of local surface temperature on total heat transfer. As a result, the total heat transfer for arbitrary surface temperature distributions can be calculated by the influence function. Similarly, using the numerical solution of the adjoint problem under the Neumann condition, we can also obtain an influence function of the local heat flux on the mean surface temperature. The influence functions for a circular cylinder and for an in-line square rod array are presented to illustrate the capability of this method. © 1999 Scripta Technica, Heat Trans Asian Res, 28(3): 227–238, 1999     

Study of heat and mass transfer in MmNi4.6Al0.4 during desorption of hydrogen

In this paper, a numerical investigation of two-dimensional coupled heat and mass transfer during desorption of hydrogen in a cylindrical metal hydride reactor containing MmNi_6.4Al_0.4 is presented. By considering the variation in heat transfer fluid temperature along the axial direction (variable wall temperature boundary condition), the changes in hydride bed temperature at different axial locations are presented. The effect of variable wall temperature boundary condition on hydrogen desorption rate for different hot fluid temperatures and hydride bed thicknesses is investigated. The rate of hydrogen desorption at different hot fluid temperatures showed good agreement with the experimental data reported in the literature. As the desorption progresses, the change in heat transfer fluid temperature along the axial direction is found to decrease with time and becomes unchanged at the end of the process. The effect of variable wall temperature boundary condition on desorption time is found to be significant for the hydride bed thicknesses of about 7.5 mm and more. For a given bed thickness of 17.5 mm, the maximum difference in desorption time between variable wall and constant wall temperature convective boundary conditions is about 375 s at 303 K.     

The Influence of Subsurface Temperature Measurements on the Determination of Transient Wall-Side Boundary Conditions: An Analytical Tool

An analytical approximation for the two-dimensional heat conduction problem with temporally and spatially oscillating thermal boundary conditions is presented. In addition, simplified expressions for the damping and phase shift of the temperature signal between the surface and a “measurement” position inside the wall are derived and their validity in dependency on the inverse Fourier number is shown. The solution algorithm is implemented in a MATLAB program and made available for other researchers. In two case studies, the analytical solution is applied in order to evaluate the ability of experimental systems to measure instantaneous and local heat transfer coefficients as well as temperatures on a boundary.     

Approximate analytic solution of heat conduction in hollow semi-spheres flying at hypersonic speed

Heat transfer in blunt noses of hypersonic vehicles with coolant inside can be approximately considered as heat conduction in hollow semi-sphere with aerodynamic heating on the outer boundary and enhanced cooling on the inner boundary. Theoretical investigations of temperature field in hollow semi-spheres were carried out by solving the two-dimensional axsymmetric conduction equation, which could be transformed into Legendre equation when the separation of variables is applied. However, for such a semi-sphere flying at hypersonic speed, the distribution of heat transfer rates as an outer boundary condition is so complex that the integration in the Legendre solution is nearly impossible to be completed. In this paper, a 4th order Legendre polynomial, derived by the method of undetermined coefficients, was adopted to approach the local similarity solution of hypersonic aerodynamic heating and simplify the integration process, by which an approximate solution could be set up for the temperature field. The approximate solution is also validated by comparing the analytical results with data from numerical simulations, in which the conduction equation is solved with the improved Richardson scheme. Both analytical and numerical results are compared to each other and match quite well.     

Heat and mass transfer studies on plate fin-and-elliptical tube type metal hydride reactors

In the present work a two-dimensional transient model to study the heat and mass transfer characteristics of plate fin-and-elliptical tube type metal hydride reactors is presented. The relevant governing equations are solved numerically. The heat transfer coefficient and pressure drop on external fin side are estimated using available correlations. Effects of external fluid flow rate and temperature on the fin-and-tube metal hydride reactor are studied. Results show that the use of elliptical metal hydride tubes in place of the standard circular tubes give rise to compact systems in addition to considerably lower fan power consumption, with very little change in the average heat and hydrogen transfer rates. Even though the performance of the reactor depends very much on the fin-and-tube arrangement, for all the arrangements considered here, the reactors with elliptical tubes were found to perform better in terms of compactness, weight and fan power consumption. Considering the aspects of mean hydride bed thickness, tube wall thickness, tube mass, compactness, heat and hydrogen transfer rates and fan power consumption, elliptical tubes of 0.6 eccentricity appear to offer the best solution for the given application.     

A two-dimensional inverse heat conduction problem in estimating the fluid temperature in a pipeline

An inverse heat conduction problem (IHCP) was investigated in the two-dimensional section of a pipe elbow with thermal stratification to estimate the unknown transient fluid temperatures near the inner wall of the pipeline. An inverse algorithm based on the conjugate gradient method (CGM) was proposed to solve the IHCP using temperature measurements on the outer wall. In order to examine the accuracy of estimations, some comparisons have been made in this case. The temperatures obtained from the solution of the direct heat conduction problem (DHCP) using the finite element method (FEM) were pseudo-experimental input data on the outer wall for the IHCP. Comparisons of the estimated fluid temperatures with experimental fluid temperatures near the inner wall showed that the IHCP could accurately capture the actual temperature in form of the frequency of the temperature fluctuations. The analysis also showed that the IHCP needed at least 13 measurement points for the average absolute error to be dramatically reduced for the present IHCP with 37 nodes on each half of the pipe wall.     

Buoyant Darcy flow driven by a horizontal temperature gradient: A linear stability analysis

The linear stability of a fluid saturated porous layer bounded by two parallel impermeable plane walls is investigated. The lower wall is subject to a uniform heat flux, while the upper wall is subject to a linearly varying temperature in a horizontal direction. Two parameters govern the onset of convection in the porous layer: the vertical Darcy–Rayleigh number, and the horizontal Darcy–Rayleigh number. The objective of this study is to obtain the onset conditions for the instability of the basic parallel flow in the layer. The governing balance equations are written in a dimensionless form and solved on assuming oblique roll disturbances, arbitrarily oriented in the horizontal plane. Mathematically, this leads to a system of two ordinary differential equations to be solved as an eigenvalue problem. The solution, carried out numerically, provides the neutral stability condition. The numerical solution is performed by employing a procedure based on the sixth-order Runge–Kutta method and on the shooting method for satisfying the boundary conditions at the upper boundary wall.     

Non-axisymmetric forced and free flow in a vertical circular duct

The fully developed laminar mixed convection in a vertical circular duct is studied analytically, with reference to non-axisymmetric boundary conditions such that the fluid temperature does not change along the axial direction. The Boussinesq approximation is applied by taking the average temperature in a duct section as the reference fluid temperature. The dimensionless momentum and energy balance equations are solved by employing Fourier series expansions of the temperature and the velocity fields. The solution shows that the temperature field is not influenced by the velocity distribution and that the Fanning friction factor is not affected by buoyancy. On the other hand, the velocity field is strongly influenced by the buoyancy forces and may display flow reversal phenomena. Two special cases are studied in detail: a duct with a sinusoidal wall temperature distribution; a duct subjected to an external convection heat transfer with two environments having different reference temperatures.     

Wall thickness optimization of thick-walled spherical vessel using thermo-elasto-plastic concept

A study of thick-walled spherical vessels under steady-state radial temperature gradients using elasto-plastic analysis is reported. By considering a maximum plastic radius and using the thermal autofrettage method for the strengthening mechanism, the optimum wall thickness of the vessel for a given temperature gradient across the vessel is obtained. Finally, in the case of thermal loading on a vessel, the effect of convective heat transfer on the optimum thickness is considered, and a general formula for the optimum thickness and design graphs for several different cases are presented.