Time-varying Heat Transfer Coefficient Research Articles

An inverse front tracking method for modeling solidification process in a smelting furnace

An inverse process based on the conjugate gradient and adjoint problem is developed for predicting the time-varying bank thickness during the solidification/melting process in a smelting furnace. The direct problem rest on a single-phase moving boundary model to simulate the solid bank and refractory brick, simultaneously. The time-varying heat flux at solid/liquid interface, initial bank thickness and time-varying heat transfer coefficient at outer surface of the furnace are considered as the unknowns. Unlike the methods developed so far based on the enthalpy model, the current method does not require any prior information regarding the thermophysical properties of the liquid-phase and complex physical phenomena that occur in it. This improves the uniqueness of the solution that is inherent in the ill-posed inverse methods. To avoid the so-called “inverse crime”, a two-phase enthalpy method is used to generate the simulated measurement. These measurements are used in the present single-phase method for reconstruction of bank history. The model has shown good performance for non-intrusive temperature obtained from the outer surface of the furnace. The inverse model has been validated through an experiment conducted in a metallurgical reactor. A reasonable agreement between the estimated and measured molten salt bank indicates the existence and convergence of the inverse solution.

Inverse boundary problem in estimating heat transfer coefficient of a round pulsating bubbly jet: design of experiment

ABSTRACT An inverse algorithm is developed to estimate the transient convective coefficient distribution of a pulsating bubbly jet impinging on a cylindrical thermal mass. Cooling of the thermal mass is simulated by solving the two-dimensional transient heat equation in a cylindrical coordinate system using the finite difference method. The sum of squared differences between calculated and measured temperature data is the objective functional. As this is a nonlinear IHCP, the adjoint method is employed to derive the gradient components of the objective function, needed for the implementation of CGM. The procedure is carried out for six gas Reynolds numbers changing continuously through ten seconds of the jet impingement. The inverse scheme is validated using noiseless temperature data. The accuracy and convergence rate of the algorithm is then assessed by simulating the cooling process of thermal mass with four different materials. Moreover, the effects of sensors’ location, as well as initial guess on the estimation, are evaluated. Time-varying heat transfer coefficients are then estimated in the presence of Gaussian noise, having four standard deviations. Results demonstrate that, despite the estimation is affected by the noise level, the adjoint method is most efficient and rapidly convergent for the thermal mass with moderate thermal conductivity having a noisy temperature field of 1 .

Inverse estimation of time-varying heat transfer coefficients for a hollow cylinder by using self-learning particle swarm optimization

Abstract An inverse methodology is proposed to estimate a time-varying heat transfer coefficient (HTC) for a hollow cylinder with time-dependent boundary conditions of different kinds on inner and outer surfaces. The temperatures at both the inner surface and the interior domain are measured for the hollow cylinder, while the time history of HTC of the outer surface will be inversely determined. This work first expressed the unknown function of HTC in a general form with unknown coefficients, and then regarded these unknown coefficients as the estimated parameters which can be randomly searched and found by the self-learning particle swarm optimization (SLPSO) method. The objective function which wants to be minimized was found with the absolute errors between the measured and estimated temperatures at several measurement times. If the objective function converges toward the null, the inverse solution of the estimated HTC will be found eventually. From numerical experiments, when the function of HTC with exponential type is performed, the unknown coefficients of the HTC function can be accurately estimated. On the contrary, when the function of HTC with a general type is conducted, the unknown coefficients of HTC are poorly estimated. However, the estimated coefficients of an HTC function with the general type can be regarded as the equivalent coefficients for the real function of HTC.

Determination of quasi-static and residual stresses in the course of the thermoplastic hardening in a boundary layer of the material

In order to induce residual compressive stresses in the boundary layer of a steel plate heated to temperature T0 = 600 °C, a method of jet water cooling with time-varying heat transfer coefficients is considered. By solving the corresponding heat conduction problem, it is shown that the maximum temperature difference between the surface and the center of a 2 mm thick plate is 105 °C. It is observed within 0.02 s since the start of cooling, the time of its complete cooling is equal to 0,32 s. Calculations of temperature stresses using the finite element method showed that temperature tensile stresses occur on the outer surface of the plate. They may reach 26 kg/mm2 which exceeds the yield strength of this material. As a result of plastic deformation, the thermal tensile stress relieving occurs in the course of cooling of the plate surface layer; when it is completely cooled, the residual compressive stresses 21.8kg/mm2. (on the surface of the plate) are induced in this layer (0,25 mm depth).

Analytical Solutions of Heat-Conduction Problems with Time-Varying Heat-Transfer Coefficients

Analytical Solutions of Heat-Conduction Problems with Time-Varying Heat-Transfer Coefficients

Heat transfer augmentation in unsteady conjugate thermal systems – Part II: Applications

Many energy conversion and other thermal-fluid systems exhibit unsteady convective heat exchange. In such systems, generic spatiotemporal variations in the flow give rise to variations in the heat flux for a given fluid–solid temperature difference, which can be interpreted as spatiotemporal fluctuations of the instantaneous heat transfer coefficient. These variations can lead to unsteady conjugate heat transfer, in which the exchanged heat flux arises from an interaction between the bulk fluid temperature and the temperature in the solid. Further, the nonlinear coupling between the fluctuating temperature differences and the heat transfer coefficient can lead to an effect we refer to as augmentation, which quantitatively describes the ability of a particular arrangement to have a different time-mean heat flux from the product between the mean heat transfer coefficient and the mean temperature difference across the fluid. It is important to be able to understand and to model in a simple framework the effects of the material properties, the geometry and the character of the heat transfer coefficient on the thermal response of the fluid–solid system, and consequently to predict the overall heat transfer performance of these systems.This paper, which follows on from its predecessor [1], is concerned with the phenomenon of augmentation in simple, one-dimensional, unsteady and conjugate fluid–solid systems. A simple semi-analytical one-dimensional model of heat transfer with a time-varying heat transfer coefficient, which was presented in Mathie and Markides [1], is applied herein to two different paradigm problems. Such a model can be an important tool in the design of improved heat exchangers and thermal insulation, through for example, the novel selection of materials to exploit these augmentation effects. The first flow considered is a thin, wavy fluid film flowing over a heated plate. This film flow exhibits a periodic fluctuation in the heat transfer coefficient, that is linked to the wavy interfacial deformations of free surface of the liquid film. The second flow considered concerns the heat transfer behind a backwards-facing step, which exhibits broadband fluctuations in the heat transfer coefficient due to the flow separation and turbulence behind the step. The model predictions of the augmentation ratio for these problems are also compared to direct measurements from each case. Good agreement is observed with the experimental results for the global heat transfer trends. In both cases the augmentation ratio was negative, reflecting a reduction in time-averaged heat transfer. For the backwards-facing step flow a low magnitude of augmentation ratio was observed, however, the thin film flows exhibited augmentation ratios of as high as 10%.

Heat transfer augmentation in unsteady conjugate thermal systems – Part I: Semi-analytical 1-D framework

This paper is concerned with the phenomenon of heat transfer augmentation in one-dimensional unsteady and conjugate fluid–solid systems. The overriding purpose is to propose a simple framework for the description of the effect of unsteadiness on the overall heat exchange performance of these systems, leading to the improved understanding and prediction of related processes. An analytical model is developed that describes the thermal interaction between the solid and the fluid domains with the use of a time-varying heat transfer coefficient. Augmentation is a non-linear effect that arises from an interaction between fluctuations in the heat transfer coefficient and fluctuations in the fluid–solid temperatures. It describes the difference between the time-averaged heat transfer rate, and the multiple of the time-averaged heat transfer coefficient and the time-averaged temperature difference across the fluid. It is found that the degree of augmentation can be defined in terms of key independent problem variables, including: a time-averaged Biot number, a dimensionless solid thickness (normalised by an unsteady thermal diffusion lengthscale), a heat transfer coefficient fluctuation intensity (amplitude normalised by the mean), and a heat capacity ratio between the fluid and solid domains. The model is used to produce regime maps that describe the range of conditions in which augmentation effects are exhibited. Such maps can be used in the design of improved heat exchangers or thermal insulation, for example through the novel selection of materials that can exploit these augmentation effects. Cases are considered for which the bulk fluid temperature is fixed, and for which the bulk fluid temperature is allowed to respond to the solid, both in thermally developing and fully developed flows. Generally the augmentation effect is found to be negative, reflecting a reduction in the heat exchange capability. However, regions of positive augmentation are uncovered in thermally developing convective flows, which has important implications for heat exchanger design.

Inward and outward solidification of cylindrical castings: The role of the metal/mold heat transfer coefficient

The numerical simulation of cylindrical castings can only produce reliable information if the thermal boundary conditions, such as the metal/mold heat transfer coefficient (h), are known accurately. Despite the importance of h for cylindrical shaped castings, information available in the literature is meager. In the present study, an extensive experimental study has been developed encompassing horizontal and vertical cylindrical shaped castings. Alloys having quite different thermal properties and freezing ranges have been selected for the experimental study. An inverse heat conduction method supported by temperature measurements at known locations inside the cylindrical heat conducting bodies has been used to derive time-varying heat transfer coefficients in both concave and convex mold/alloy surfaces (inward and outward solidification, respectively). It is shown that the h(t) profiles are given by an expression of the form h = at±m, where (−) refers to the inward solidification and (+) to the outward solidification against an inner mold.

INVERSE HEAT CONDUCTION PROBLEM OF ESTIMATING TIME-VARYING HEAT TRANSFER COEFFICIENT

An inverse heat conduction problem is solved by using Alifanov's iterative regularization method to estimate the time-varying heat transfer coefficient of forced-convective flow boiling over the outer surface of a heater tube. The estimation is based on transient temperature measurements taken by a thermocouple on the inner surface of the circular tube on which the flow boiling occurs. The inverse problem is solved as an optimization problem in which a squared residue functional is minimized with the conjugate gradient method. A sensitivity problem is solved to determine the step size in the direction of descent, and an adjoint problem is solved to determine the gradient of the functional. No prior information is used in the functional form of the heat transfer coefficient variation with time. Numerical experiments are performed to test the proposed mathematical model of the measurement of flow boiling heat transfer coefficient. Effects of the time scale of the heat transfer coefficient variation, the mea...

INVERSE HEAT CONDUCTION PROBLEM OF ESTIMATING TIME-VARYING HEAT TRANSFER COEFFICIENT

An inverse heat conduction problem is solved by using Alifanov's iterative regularization method to estimate the time-varying heat transfer coefficient of forced-convective flow boiling over the outer surface of a heater tube. The estimation is based on transient temperature measurements taken by a thermocouple on the inner surface of the circular tube on which the flow boiling occurs. The inverse problem is solved as an optimization problem in which a squared residue functional is minimized with the conjugate gradient method. A sensitivity problem is solved to determine the step size in the direction of descent, and an adjoint problem is solved to determine the gradient of the functional. No prior information is used in the functional form of the heat transfer coefficient variation with time. Numerical experiments are performed to test the proposed mathematical model of the measurement of flow boiling heat transfer coefficient. Effects of the time scale of the heat transfer coefficient variation, the measurement errors, and the data acquisition rates on the inverse solution are investigated. We found that for a given experimental configuration there is a physical limitation on the time scale of the heat transfer coefficient variation that can be estimated by the inverse analysis.

Compact Simulation Model of Non-air Conditioned Room and Estimation of Its Thermal Characteristics under Winter Conditions.

This study investigates thermal behavior of an occupied room exposed to winter weather conditions by means of quasi-steady-state model of the room. The space is not air-conditioned during the winter season, i.e. there is no humidity control and thermal performance of heaters maintains desired indoor air temperature. To obtain pure thermodynamic characteristics of the space, the model was set up and a series of tests was carried out in order to validate the mathematical model and find the thermal characteristics of the enclosure, which could be used directly for direct control techniques (DDC) if applied. The build -up model's method imposes boundary conditions onto room-faced surroundings walls and couples Fourier equations' thermal network employing modified Biot number. It also enables us to simulate an hourly energy use as well as heat flux calculus based on sub-hourly time steps. The model features a water equivalent content of the room, time-varying heat transfer coefficients, and boundary conditions onto surrounding walls, respectively. As a full dynamic method it accounts for thermal influence of outdoor climate conditions, occupancy use and other internal sources of thermal energy. The paper emphasizes selection of system boundaries and practical calculus of radiation heat exchanges without claiming shape factor algebra. The numerical results of temperature transient time responses obtained from mathematical model are well in accordance with those obtained from measurement in the actual room.

Time-varying heat transfer coefficients between tube-shaped casting and metal mold

An experimental and numerical study were made on the time-varying heat transfer coefficient h( t) between a tube-shaped casting and metal molds. One dimensional treatment was adopted in analyzing the heat flows between the casting and the inner and the outer mold. The sequential function specification method was employed to solve the nonlinear inverse heat conduction problem. In order to investigate the different behavior of h( t) for different alloys, casting experiments were carried out with three Al-base alloys and pure Al having different types of solidification behavior. It was found that the temperature change of the outer mold showed a normal heating and cooling curve. However, that of the inner mold was unusual especially for the alloys with a wide solidification range, i.e. the temperature increases first rapidly, then halts for a while and then increases again showing finally a regular heating and cooing curve. The resulting heat transfer coefficient at the interface to the inner mold h i( t) decreases temporarily and then increases, while the one at the interface to the outer mold h o( t) decreases monotonously to a quasi steady state. The abnormal heat transfer phenomenon at the inner interface for the alloys with a wide solidification range was concluded to be caused by a slight movement of the semi-solid inner wall at the inside of the tube-shaped casting due to the solidification contraction of the casting freezing in a mushy type.

Theory of transient experimental techniques for surface heat transfer

The paper describes the unified mathematical procedures of transient methods for measuring surface heat transfer rates. Three heat flux gauges are discussed: thin film, thick-wall gauges placed on semi-infinite substrates and thin-skin calorimeters. The aim of this paper is to present a method for a simple and accurate determination of the time-varying heat transfer coefficient (or heat flux) given an accurate temperature history of the body at a selected point beneath the surface. The interior temperature measurements are converted into local instantaneous heat transfer coefficients by solving the inverse heat conduction problem for the gauge. The effect of the inaccuracies in the measurement of the interior temperature was eliminated by cubic spline smoothing or digital filtering of the raw interior temperature data prior to using it in the inverse heat conduction analyses. General case closed form equations for instantaneous surface heat flux, or heat transfer coefficient, are developed.

Local heat transfer coefficients on the circumference of a tube in a gas fluidized bed

A heated horizontal heat transfer tube was installed 14.8 cm above the distributor plate in a square fluid bed measuring 30.5 × 30.5 cm. Four different Geldart B sized particle beds were used (sand of two different distributions, an abrasive and glass beads) and the bed was fluidized with cold air. The tube was instrumented with surface thermocouples around half of the tube circumference and with differential pressure ports that can be used to infer bubble presence. Numerical execution of the transient conduction equation for the tube allowed the local time-varying heat transfer coefficient to be extracted. Data confirm the presence of the stagnant zone on top of the tube associated with low superficial velocities. Auto-correlation of thermocouple data revealed bubble frequencies and the cross-correlation of thermal and pressure events confirmed the relationship between the bubbles and the heat transfer events. In keeping with the notion of a “Packet renewal” heat transfer model, the average heat transfer coefficient was found to vary in sympathy with the root-mean square amplitude of the transient heat transfer coefficient.

Wall-response factors to estimate the conduction heat gains with time-varying thermal coefficients

A simple methodology for solving time-varying problems of transient heat flow through building envelopes is presented. In forming a typical model to evaluate the conduction heat gains, a sol-air heat instead of the sol-air temperature is defined and two related thermal-response factors are introduced. To deal with the time-dependent system properties, the response factors are normalized by their mean values of static gain, and the actual values of static gain are thus shifted to the thermal inputs. This procedure allows us to estimate the building thermal loads of a linear dynamic model with built-in, time-varying heat-transfer coefficients.

A Method for Analyzing Heat Conduction With High-Frequency Periodic Boundary Conditions

A special method is developed for calculating the steady periodic temperature solution in solid bodies with high-frequency boundary conditions. The numerical difficulty associated with steep gradients and rapid temperature variation near the boundary is addressed by confining all transient temperatures to a narrow boundary layer of constant depth. The depth of the layer is specified in advance and depends only on the period of the boundary disturbance and the thermal diffusivity of the material. The transient solution in the surface layer is represented by a polynomial in its transverse coordinate, with time-varying coefficients determined by a Galerkin method. This solution is coupled with the steady interior solution by imposing continuity of temperature and time-averaged heat flux at the interface. Although the method is sufficiently general to handle nonlinear boundary conditions, it turns out to be particularly useful in the important case of a time-varying heat transfer coefficient. In the latter case, it is possible to decouple the solution process and determine the solution in the transient surface layer separately from the solution in the steady interior. This reduces the effort of determining the complete steady periodic solution to little more than a routine steady analysis. Comparison with an exact solution shows that the polynomial representation for the transient solution in the surface layer converges very rapidly with increasing order. Moreover, the solution at the surface turns out to be relatively insensitive to the choice of the layer depth as long as it is greater than a certain minimum value. An application to permanent mold casting is given, illustrating both the utility and accuracy of the method in a practical context.

Temperature distribution in plates and infinite prismatic bodies of complex cross section for a time-varying heat-transfer coefficient

We present a new method for solving heat-conduction problems with a time-varying heat-transfer coefficient in domains of complex shape, and cite numerical results for two problems.

Thermal conductivity of a two-layer wall for a time-varying heat-transfer coefficient and ambient temperature

An analytical solution of the problem of the thermal conductivity of a two-layer wall is presented. The solution is obtained by using a special series expansion of the Green's function.