Heat Flux History Research Articles

Open-loop temperature rise history control method with scanning electron beam as a programmable heat source

Temperature-rise history control technology is widely applied in rapid thermal process and material thermal assessment. We present an open-loop temperature rise history control method. The fundamental concept involves loading a heat flux history onto the sample’s surface, inversed from the target temperature rise history, to regulate the sample’s temperature through programming the energy flux history of the scanning electron beam. We reconstructed the temperature-rise history of the leading edge of an aircraft during the HIFIRE-5b flight test with a relative deviation of 1%. Additionally, we reconstructed a typical temperature rise curve of a silicon wafer during rapid thermal processing with a relative deviation of 1%. In theory, this temperature control method can be extended to two-dimensional and three-dimensional heat transfer processes to achieve transient control of sample surface/cross-section temperature field with one- or two-dimensional distribution characteristics. This research can provide a high-performance temperature control technology for a wide range of industrial fields, and lay a theoretical foundation for the development of related technologies based on temperature history analysis.

A Detailed Study of Boiling Phenomena in Forced Convective Quenching Experiments

A detailed study at the microscopic level of the transient behavior of boiling phenomena during quenching of an AISI 304 stainless steel, conical-end, cylindrical probe in flowing water at 60 °C was conducted using high-speed video recordings and cooling curve data acquisition. Two free-stream velocities (0.2 and 0.6 m/s) and two initial probe temperatures (850 and 950 °C) were investigated. It was complemented by video recordings at 60 fps to calculate the wetting front velocity. Surface heat flux histories at the thermocouple positions were estimated by solving the corresponding Inverse Heat Conduction Problem. As the water velocity increases the duration of the vapor film at each thermocouple position shortens and the difference among cooling curves at the thermocouple positions decreases. Increasing the initial temperature increases the duration of the vapor film stage. The wetting front velocity increased from 4 to 6 m/s. The time to reach the maximum surface heat flux at the position of the thermocouple closest to the probe tip ranged from 9.7 to 18 s. Undulations at the vapor-liquid interface that appear periodically and propagate in the direction of quenchant flow were observed early during the vapor film stage. Later, before the collapse of the vapor film, a kind of a shock wave was generated at the probe tip which modified the appearance of the film. Once the Leidenfrost temperature is reached, a wetting front (which consists of many small bubbles that coalesce rapidly in a very small area) is formed and travels in the direction of quenchant flow while fewer and larger bubbles nucleate and grow upstream. The nucleation, growth and detachment of the larger bubbles was studied; as the free-stream velocity decreases larger values of bubble maximum diameter and half-life time were observed, while the initial temperature has a marginal effect on these quantities.

Time-resolved heat flux determination from fast-responding temperature-sensitive paint measurement in a shock tunnel using transient in-situ calibration

Measurement of the global heat flux using fast-responding temperature-sensitive paint (fast TSP) in a shock tunnel is crucial to understanding complex aerothermal effects, but its accuracy is limited by the simplified inverse algorithm as well as the uncertainty in the thermal properties and the thickness of the TSP. In the present work, an in-situ transient calibration method is proposed to determine important parameters in one-dimensional double-layer semi-infinite heat conduction for TSP measurement via heat flux gauges. A fast-converging iterative scheme based on the Levenberg–Marquardt algorithm is used to search for the optimal parameters that minimize the difference between the transient experimental and simulated temperature histories. Furthermore, the heat flux history is determined using the impulse method according to the calibrated parameters. A fast-TSP measurement of 34° ramp flow was conducted at 75 kHz in a Mach-12.1 shock tunnel to validate the proposed method. The results show that the heat flux histories obtained from the TSP and the heat flux gauge results are in good agreement throughout the test period of 11 ms. Moreover, the use of multiple gauges in the calibration further improves the overall consistency of the heat flux amplitude. The in-situ transient calibration method effectively improves the accuracy of the measurement of time-resolved heat flux with relatively low hardware and computational costs.

A new bootstrap technique to quantify uncertainty in estimates of ground surface temperature and ground heat flux histories from geothermal data

Abstract. Estimates of the past thermal state of the land surface are crucial to assess the magnitude of current anthropogenic climate change as well as to assess the ability of Earth System Models (ESMs) to forecast the evolution of the climate near the ground, which is not included in standard meteorological records. Subsurface temperature reacts to long-term changes in surface energy balance – from decadal to millennial time scales – thus constituting an important record of the dynamics of the climate system that contributes, with low-frequency information, to proxy-based paleoclimatic reconstructions. Broadly used techniques to retrieve past temperature and heat flux histories from subsurface temperature profiles based on a singular value decomposition (SVD) algorithm were able to provide robust global estimates for the last millennium, but the approaches used to derive the corresponding 95 % confidence interval were wrong from a statistical point of view in addition to being difficult to interpret. To alleviate the lack of a meaningful framework for estimating uncertainties in past temperature and heat flux histories at regional and global scales, we combine a new bootstrapping sampling strategy with the broadly used SVD algorithm and assess its performance against the original SVD technique and another technique based on generating perturbed parameter ensembles of inversions. The new bootstrap approach is able to reproduce the prescribed surface temperature series used to derive an artificial profile. Bootstrap results are also in agreement with the global mean surface temperature history and the global mean heat flux history retrieved in previous studies. Furthermore, the new bootstrap technique provides a meaningful uncertainty range for the inversion of large sets of subsurface temperature profiles. We suggest the use of this new approach particularly for aggregating results from a number of individual profiles, and to this end, we release the programs used to derive all inversions in this study as a suite of codes labeled CIBOR v1: Codes for Inverting BORholes, version 1.

A regularization method for inverse heat transfer problems using dynamic Bayesian networks with variable structure

In this paper, dynamic Bayesian networks (DBNs) were employed to solve one-dimensional inverse heat transfer problems (IHTPs) with temperature history data. Temperature-dependent conductivities and surface heat flux histories were discretized into a finite number of parameters and used as parent nodes in the DBN. Parameter sensitivity analyses were adopted to identify the dominant parameters at each time step. Non-dominant parameters were removed from the corresponding frames for the sparsification of the DBN, which reduces the number of parameters to be identified simultaneously. Thus, the structure of the DBN was dynamically optimized for the regularization of inverse problems. Numerical and experimental tests for IHTPs were conducted to validate the proposed method. Both thermal conductivities and surface heat flux were identified with relatively small error, even if considering the measurement error. Sensitivities of the remaining parameters increased significantly with the parameter identification process, and insensitive parameters in the initial DBN frames were also effectively identified in the subsequent frames. The accuracy of parameter identification can be improved, and the cost of the parameter sampling combination can be reduced. The proposed method is not subject to heat transfer problems and can be used for a wide range of applications.

Analytical solution of unsteady heat conduction in multilayer internal combustion engine walls

An analytical solution of the unsteady heat conduction problem in multilayer walls was developed and applied to study thermal barrier coatings for reciprocating internal combustion engines. The mathematical solution was derived using the matrix method and complex analysis/residue-calculus Laplace transform inversion techniques. The one-dimensional domain includes a time-varying heat flux on the combustion chamber surface, and a time-varying temperature on the backside coolant surface. These boundary conditions are simulated as the superposition of two adjacent triangular, unit-magnitude pulses, which gives a piece-wise linear representation of the applied flux or temperature history. The temperature at any location of interest is found as the convolution of the transfer function, which results from the inversion, with the discrete-time heat flux or backside temperature history. The transfer functions describe the exact response and only depend on multilayer architecture, therefore, they can be computed a priori. The triangular pulse method provides a more than two order of magnitude reduction in computation time compared to a finite difference solution at the same level of accuracy. A 104-fold speed increase relative to a finite difference solution was realized when using the Overlap-add convolution technique for a full drive cycle simulation, i.e., a long time record. The surface response or transfer function acts like a finite impulse response and can be viewed in the frequency domain to reveal complementary information about coating performance.

Heat transfer modeling within the microclimate between 3D human body and clothing: effects of ventilation openings and fire intensity

PurposeThe purpose of this study is to determine the effect of ventilation openings and fire intensity on heat transfer and fluid flow within the microclimate between 3D human body and clothing.Design/methodology/approachOn account of interaction effects of fire and ventilation openings on heat transfer process, a 3D transient computational fluid dynamics model considering the real shape of human body and clothing was developed. The model was validated by comparing heat flux history and distribution with experimental results. Heat transfer modes and fluid flow were investigated under three levels of fire intensity for the microclimate with ventilation openings and closures.FindingsTemperature distribution on skin surface with open microclimate was heavily depended on the heat transfer through ventilation openings. Higher temperature for the clothing with confined microclimate was affected by the position and direction of flames injection. The presence of openings contributed to the greater velocity at forearms, shanks and around neck, which enhanced the convective heat transfer within microclimate. Thermal radiation was the dominant heat transfer mode within the microclimate for garment with closures. On the contrary, convective heat transfer within microclimate for clothing with openings cannot be neglected.Practical implicationsThe findings provided fundamental supports for the ease and pattern design of the improved thermal protective systems, so as to realize the optimal thermal insulation of the microclimate on the garment level in the future.Originality/valueThe outcomes broaden the insights of results obtained from the mesoscale models. Different high skin temperature distribution and heat transfer modes caused by thermal environment and clothing structure provide basis for advanced thermal protective clothing design.

Reprint of: Surface temperature of a multi-layer thermal barrier coated wall subject to an unsteady heat flux

A one-dimensional analytical method of determining the surface temperature of a thermal barrier coated plane wall subjected to an arbitrarily time-varying heat flux was developed. The method allows fast computation, enabling it to be embedded in combustion simulation software to correctly calculate wall heat transfer, which is important for reciprocating engine applications with thin, low thermal inertia coatings that have large temperature swings. The method relies on superposition of the response to successive step changes in heat flux, obtained by discretizing the heat flux history, to compile the solution. The embedded problem of a step-change in heat flux is addressed using the one-dimensional heat diffusion equation in conjunction with the matrix method in the Laplace domain, assuming the lateral conduction is neglected. For a two-layer wall (one coating layer on top of the metal wall) in the limit where kρc of the coating is small relative to that for the wall material, an analytical inversion to the time domain can be derived. For modest values of this parameter or for more than one coating layer, the residue theorem is invoked to invert to the time domain, with the poles and residues found numerically. The analytical inversion, however, provided the impetus for defining a set of non-dimensional parameters that uniquely characterize the surface temperature swing for arbitrary periodic heat flux to the wall. Non-dimensional results are given for a sinusoidially varying surface heat flux.

Surface temperature of a multi-layer thermal barrier coated wall subject to an unsteady heat flux

A one-dimensional analytical method of determining the surface temperature of a thermal barrier coated plane wall subjected to an arbitrarily time-varying heat flux was developed. The method allows fast computation, enabling it to be embedded in combustion simulation software to correctly calculate wall heat transfer, which is important for reciprocating engine applications with thin, low thermal inertia coatings that have large temperature swings. The method relies on superposition of the response to successive step changes in heat flux, obtained by discretizing the heat flux history, to compile the solution. The embedded problem of a step-change in heat flux is addressed using the one-dimensional heat diffusion equation in conjunction with the matrix method in the Laplace domain, assuming the lateral conduction is neglected. For a two-layer wall (one coating layer on top of the metal wall) in the limit where kρc of the coating is small relative to that for the wall material, an analytical inversion to the time domain can be derived. For modest values of this parameter or for more than one coating layer, the residue theorem is invoked to invert to the time domain, with the poles and residues found numerically. The analytical inversion, however, provided the impetus for defining a set of non-dimensional parameters that uniquely characterize the surface temperature swing for arbitrary periodic heat flux to the wall. Non-dimensional results are given for a sinusoidially varying surface heat flux.

An Analytical Approach for Calculating Instantaneous Multilayer-Coated Wall Surface Temperature in an Engine

<div class="section abstract"><div class="htmlview paragraph">Thermal swing coatings that have low volumetric heat capacity and low thermal conductivity are attractive because they have the potential to significantly reduce heat transfer to the combustion chamber walls. This paper presents an analytical method for determining the exact solution of the time-resolved wall temperature during the engine cycle for any number of coating layers and properties using the Laplace transformed heat diffusion equation. The method relies only on material properties and the past heat flux history, and represents the exact solution of the heat diffusion equation. The analytical nature of the solution enables fast computation and, therefore, application to system-level optimization calculations. The model relies on an assumption of one-dimensional heat flow, and constant material properties. The major advantage of this approach compared to the standard finite difference approach, wherein the wall is finely discretized, is that there is no approximation and the accuracy is guaranteed, <i>i.e.,</i> it does not depend on nodal density. Results are presented for both a quasi-steady operating condition and for an engine transient event. In the latter case a 200-fold reduction in computation time relative to a finite difference code was realized.</div></div>

Real-time estimation of time-dependent heat flux for 3D finite domain employing thermal mode and recursive least square deconvolution

This paper presents a novel surface heat flux estimation method for the 3D finite domain experiencing dynamic convection and surface heat flux. The thermal mode parameter is adopted to produce a mathematical relation between the surface heat flux history function and the transient temperature response vector with modal superposition method and to locate specific measurement point which can represent transient temperature change of the domain. The direct solution is obtained using the modal superposition method with thermal modes and modal time constants representing thermal characteristics of domain, and the proposed surface heat flux estimation method is derived by the inverse solution applying the novel recursive least square deconvolution (RLSD). The proposed method is validated by numerical tests and experiments that include two main advanced points: employment of the dynamic convection condition and implementation of efficient computing for real-time estimation. As a result, the root mean square error (RMSE) of numerical tests show the results within 2% for the maximum magnitude of hypothetical heat flux for all cases. Also the estimation result from experiment presents the RMSE within 2.5% for the maximum magnitude of measured heat flux history, achieved in only 49 s of computing time for long-term estimation of 100 min.

Correction for effect of temperature-dependent diffusivity on temperature-sensitive-paint heat-flux measurement

A simple method is developed for correcting the effect of the temperature-dependent thermal diffusivity on temperature-sensitive-paint (TSP) heat flux measurements in hypersonic wind tunnels. The relation between the heat flux obtained using the constant thermal diffusivity and the heat flux obtained using the temperature-dependent diffusivity is discussed based on an analysis of the nonlinear heat conduction equation. The non-dimensional correction factor is introduced and modeled as an approximate function of the measured surface temperature for the monotonically increasing heat flux histories, which can be determined in situ using the data at several selected points. The corrected heat flux field is obtained by multiplying the correction factor field with the heat flux field obtained using the analytical inverse heat transfer method for constant thermal properties. This method is validated through simulations.

Mathematical Modeling of the Thermal Response of Laboratory-Scale Probes

Abstract Experiments for cooling curve analysis of laboratory-scale quenched probes produce a wetting front that advances at a finite velocity along the longitudinal axis of the probe as cooling progresses. The wetting front velocity depends on probe material and geometry as well as the type of quenchant, its temperature, and agitation. Because of the wetting front, the mathematical model of the direct heat conduction problem to compute the evolution of the thermal field within the probe may be characterized as a moving front boundary problem. In this work, cooling curves at three subsurface locations along the length of conical-end cylindrical probes, fabricated with AISI 304 stainless steel, during quenching from 850°C with water at 60°C flowing parallel to the probe’s longitudinal axis were measured. A fourth thermocouple located at the geometrical center of the probe was used to validate the model. Two values of free-stream water velocity (0.2 and 0.6 m/s) were studied. As a first approximation, the measured cooling curves were used to estimate the corresponding surface heat flux histories by solving three separate 1-D inverse heat conduction problems. Then, the thermal field evolution within the probe was computed by solving a 2-D axis-symmetrical heat transfer model. The boundary condition at the probe surface was modeled with a composite mapping of the surface heat flux histories. The surface heat flux history estimated for the lower section of the cylindrical part of the probe was applied as a boundary condition for the probe tip. It was found that the cooling curve at the lowest thermocouple position was overestimated; thus, the surface heat flux history for the probe tip was divided by 1.2 to correctly model the thermal response. The model was successfully validated by comparing measured and computed cooling curves for the thermocouple located at the probe’s geometrical center.

Influence of horizontal projection on upward flame spread over XPS thermal insulation material

SummaryExperimental methods and theoretical analysis are employed to investigate the effects of horizontal projection on upward flame spread over extruded polystyrene thermal insulation material of building façade. The average flame height (Hf) (or area S) drops exponentially as dimensionless projection width (w*) rises, and dropping rates are more significant for wider projection. The decrease of flame area is more remarkable than that of flame height. There exists a low temperature zone which is surrounded by the horizontal projection, the flame, and the façade. The average of maximum temperature ( ) of zone below the projection first decreases and then increases as projection width (w) rises. The temperature dropping rate increases exponentially with w*. of the zone adjoining the surface of extruded polystyrene sample above the projection decreases linearly with w*. For wider projection (w ≥ 10 cm), the average temperature above it first increases and then drops as height rises. The heat flux history could be divided into 4 stages, ie, ignition stage, smoke heat transfer stage, flame heat transfer stage, and attenuation stage. The average heat flux ( ) at each stage decreases as w rises. The maximum value is observed at the third stage, where increases linearly with S.

Temperature and heat flux changes at the base of Laurentide ice sheet inferred from geothermal data (evidence from province of Alberta, Canada)

Using a previously published temperature log of the 2363-m-deep borehole Hunt well (Alberta, Canada) and the results of its previous interpretation, the new reconstructions of ground surface temperature and surface heat flux histories for the last 30 ka have been obtained. Two ways to adjust the timescale of geothermal reconstructions are discussed, namely the traditional method based on the a priori data on thermal diffusivity value, and the alternative one including the orbital tuning of the surface heat flux and the Earth’s insolation changes. It is shown that the second approach provides better agreement between geothermal reconstructions and proxy evidences of deglaciation chronology in the studied region.

A new transient method for determining thermal properties of wall sections

This investigation outlines a straight-forward and low cost methodology for determining thermal properties of wall structures. The method eliminates the need to produce a step change boundary condition, and the error inherent in the departure from a step change that finite properties necessarily impose. The transient technique involves an experimental component whereby a high temperature thermal ramp boundary condition is applied to one wall face with the other exposed to the cooler ambient surroundings. Temperature and heat flux sensors are installed to monitor the transient heating behaviour at the wall faces. The measured transient wall temperature profiles are subsequently imposed as boundary equations to Fourier's equation in such a way that the analytic solution can provide a prediction of the transient surface heat flux. With this, the thermal diffusivity is estimated by using the effective thermal diffusivity of the wall material as a tuning parameter to regression fit the predicted and measured heat flux histories. Additionally, the steady solution facilitates the approximation of the effective thermal conductivity when used in conjunction with the steady surface temperature and heat flux measurements. To illustrate the technique, a test was performed on a 900mm×900mm×120mm thick solid concrete wall section. The effective thermal diffusivity was determined to be 7.2×10−7m2/s with corresponding effective thermal conductivity of 1.64W/mK and specific heat of 0.99kJ/kgK. Each property value is within the range of published literature for concrete.

Observing investigation of boiling characteristics around CHF for the downward facing heating surface

This paper investigates the boiling heat transfer characteristics near the critical heat flux (CHF) for the downward facing heating via the experimental observation and measurements. These boiling processes can be examined thoroughly by combining the heat flux histories, boiling curves, and the snapshots obtained from the high-speed video record. The fluctuation behaviors in the heat flux/temperature histories and boiling curves for the downward facing heating are mainly resulted from the iterative processes of bubble attachment/detachment on the heating surface for smaller oscillation in the nucleate boiling as well as the small bubble coalescence to an oblong bubble/this flat bubble film break-up into small bubbles for higher oscillation in the transition boiling. In addition, there are cyclic processes from the nucleate boiling regime to the transition boiling regime and back to the nucleate boiling regime. Several high peaks are revealed in the boiling curve and the averaged value of these peaks is defined as the CHF in this paper. The average CHF obtained for the pool boiling case in this study is close to those from the previous works. The injection flow effect on the enhancement of boiling heat transfer is also revealed in the measured data, especially for the smaller gap distance.

Propagation of nonlinear thermoelastic waves in porous media within the theory of heat conduction with memory: physical derivation and exact solutions

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid‐saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time‐fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time‐fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.

A Hybrid Procedure for the Sequential Estimation of Surface Heat Flux From Measurements of Surface Temperature

The sequential estimation of surface heat flux from discrete and noisy data of surface temperature is an ill-posed problem. From Duhamel's theorem and Fourier's law and using a stabilization technique based on the sequential application of the ordinary least squares (SOLS), we obtain a relatively simple but effective method. As the SOLS method uses a least squares fit over r-future and r-past temperatures, this method can be compared with the well-known function specification method (FSM). FSM is a more general method, but the numerical validations considered in this study reveal that the SOLS method gives better results. It is also shown that SOLS cannot be guided by the residual principle and consequently cannot be used in a practical scope. The ultimate goal of this study is to obtain a reliable estimation of heat flux history in an on-line industrial process where the tuneable parameter should be a time-variable r-value and it must be automatically updated from the experimental data. This requires that (i) the corresponding SOLS algorithm must be rewritten in recursive form, (ii) the classical definition of the residual principle is rewritten in recursive form, and (iii) the estimates are obtained from a hybrid procedure based on SOLS and FSM.

Experimentally investigating heat transfer and boiling characteristics for the downward facing wall heating

This paper experimentally investigates the heat transfer and boiling characteristics for the downward facing wall heating since this heating design has been gradually adopted in the industry, especially in the energy and heat transfer fields. The pool boiling and forced boiling cases are considered. These results include temperature/heat flux histories, snapshots of bubble sliding/breakup/coalescence characteristics, relationship of Nusselt (Nu) number vs Rayleigh (Ra) number for the natural convection, and boiling curves in the nucleate boiling regime. Based on the comparisons of heat flux histories and boiling curves, the heat transfer can be enhanced for the forced boiling case due to bubble disturbance by the injection flow. More oscillations in the temperature history and boiling curve are also revealed in the forced boiling case than the pool boiling case. Under the present experimental conditions, appropriate correlations can be derived using the least-square-error regression method. The exponent of fitting curve for the forced boiling is higher than that for the pool boiling, which quantitatively confirms enhancement of heat transfer by the forced boiling under the downward facing heating condition. In addition, the snapshots of boiling characteristics are also provided, clearly revealing the bubble dynamic behaviors. These results are seldom shown in the previous researches.