Inverse Heat Conduction Problem Research Articles

Oil and Gas Structures: Forecasting the Fire Resistance of Steel Structures with Fire Protection under Hydrocarbon Fire Conditions

The hydrocarbon temperature–time curve is widely used instead of the standard curve to describe the temperature in the environment of structural surfaces exposed to fire in oil and gas chemical facilities and tunnels. This paper presents calculations of the ratio of time to reach critical temperatures at different nominal fire curves for steel structures such as bulkheads and columns with different types of fireproofing. The thermophysical properties of the fireproofing materials were obtained by solving the inverse heat conduction problem using computer simulation. It was found that the time interval for reaching critical temperatures in structures with different types of fireproofing in a hydrocarbon fire decreased, on average, by a factor of 1.2–1.7 compared to the results of standard fire tests. For example, for decks and bulkheads with mineral wool fireproofing, the K-factor of the ratio of the time for reaching the critical temperature of steel under the standard curve to the hydrocarbon curve was 1.30–1.62; for plaster, it was 1.56; for cement boards, it was 1.34; for non-combustible coatings, it was 1.38–2.0; and, for epoxy paints, it was 1.71. The recommended values of the K-factor for fire resistance up to 180 min (incl.) were 1.7 and, after 180 min, 1.2. The obtained dependencies would allow fireproofing manufacturers to predict the insulation thickness for expensive hydrocarbon fire experiments if the results of fire tests under standard (cellulosic) conditions are known.

An Inverse Problem for Estimating Spatially and Temporally Dependent Surface Heat Flux with Thermography Techniques

In this study, an inverse conjugate heat transfer problem is examined to estimate temporally and spatially the dependent unknown surface heat flux using thermography techniques with a thermal camera in a three-dimensional domain. Thermography techniques encompass a broad set of methods and procedures used for capturing and analyzing thermal data, while thermal cameras are specific tools used within those techniques to capture thermal images. In the present study, the interface conditions of the plate and air domains are obtained using perfect thermal contact conditions, and therefore we define the problem studied as an inverse conjugate heat transfer problem. Achieving the simultaneous solution of the continuity, Navier–Stokes, and energy equations within the air domain, alongside the heat conduction equation in the plate domain, presents a more intricate challenge compared to conventional inverse heat conduction problems. The validity of our inverse solutions was verified through numerical simulations, considering various inlet air velocities and plate thicknesses. Notably, it was found that due to the singularity of the gradient of the cost function at the final time point, the estimated results near the final time must be discarded, and exact measurements consistently produce accurate boundary heat fluxes under thin-plate conditions, with air velocity exhibiting no significant impact on the estimates. Additionally, an analysis of measurement errors and their influence on the inverse solutions was conducted. The numerical results conclusively demonstrated that the maximum error when estimating heat flux consistently remained below 3% and higher measurement noise resulted in the accuracy of the heat flux estimation decreasing. This underscores the inherent challenges associated with inverse problems and highlights the importance of obtaining accurate measurement data in the problem domain.

Parametric investigation of flow and heat transfer on the reflooding quenching of a cylinder rod in forced convection

A reflooding quenching boiling experiment in forced convection was conducted to investigate the flow and heat transfer performance of a FeCrAl cylindrical rod under various subcooling degrees and flow rates. Employing image processing and inverse heat conduction problem (IHCP) methods, the research provides insights into vapor film thickness, quenching front propagation, wall temperature cooling rate and minimum film boiling temperature. Key observations include the presence of numerous extremely small bubbles dispersing around the rod and a third quenching front appears in the middle of the rod under higher subcooling condition at 25 °C, attributed to a thinner vapor film, smaller bubbles resulting from film rupture and the inertia of fluid flow. Within the experimental range, the impact of the subcooling degree on the fluid flow and heat transfer performance during the quenching boiling process is more pronounced than that of the coolant flow rate. The evolution of the vapor film indicates a substantial reduction of 78.76 % in film thickness and a significant increase of 185.71 % in propagation velocity with an increase in subcooling degree, while an elevation in flow rate lead to a 29.84 % decrease in film thickness and a 45 % increase in propagation velocity. Heat transfer performance benefits from increased subcooling degrees throughout the entire quenching boiling process, whereas flow rate impacts the heat transfer performance of film boiling alone. Furthermore, by incorporating the impact of flow velocity, a predictive formula for Tmin is established with an error margin within ±5 %.

Determination of boundary conditions from the solution of the inverse heat conduction problem in the gas nitriding process

Two non-linear unsteady methods for solving the inverse heat conduction problem are discussed and compared in this paper. The first method is based on the finite element method, uses the variational calculus and the BFGS (Broyden-Fletcher–Goldfarb–Shanno) optimization algorithm. The second is an analytical and numerical method based on the approximation of the solution in the form of a linear combination of Chebyshev polynomials. In numerical tests, the stability of the temperature, heat flux density, and heat transfer coefficient obtained from both methods was analyzed. On the basis of experimental data, oscillations of the measured gas temperature and temperature in the component were analyzed. Oscillations of the gas temperature obtained from the experiment were taken into consideration during tests. The first method to solve the inverse problem was used to determine the boundary conditions for the entire gas nitriding process. As a result, stable values of temperature, heat flux density, and the heat transfer coefficient on the surface of the component under treatment in a real gas nitriding process, so far unpublished, were obtained.

A Bayesian Spatio-Temporal Modeling Approach to the Inverse Heat Conduction Problem

Abstract A wide range of science and engineering problems involve inferring the system behavior and physical models from indirect observations. These are known as inverse problems. Unlike forward problems (predictive models), inverse problems are difficult to solve due to ill-posedness arising from one or more of non-existence, non-uniqueness (existence of many solutions), or instability (small perturbations in the measurements emerging from noise could lead to large changes in the solution). Therefore, direct inversion of the inverse problem in complex real-world systems is impractical. Several modeling approaches have been developed and adopted in the literature. However, the existing approaches do not consider spatio-temporal relationships. Meanwhile, various real systems and processes exist whose inverse problems can be comfortably modeled using spatio-temporal modeling techniques. We present a spatio-temporal inverse model considering interaction effects for modeling this type of inverse problem. We adopted a Bayesian inference framework for the subsequent parameter estimation problem and developed a Gibbs sampling algorithm to sample from the posterior distribution of the parameter of interest. Finally, we evaluate the proposed methodology through case studies on a 1D heat simulation and pool boiling experiment, using multi-sensor thermocouple measurements to reconstruct heat flux. Compared to the conventional methods, the proposed method performs better and can be adopted for solving inverse heat conduction problems. Moreover, incorporating the spatio-temporal interaction term notably improved the model's performance.

Genetic Algorithm as the Solution of Non-Linear Inverse Heat Conduction Problems: a Novel Sequential Approach

Genetic Algorithm as the Solution of Non-Linear Inverse Heat Conduction Problems: a Novel Sequential Approach

Determination of transient heat transfer by cooling channel in high-pressure die casting using inverse method

Complex shapes of aluminum castings are typically manufactured during the short cycle process known as the high-pressure die casting (HPDC). High productivity is ensured by introducing die cooling through a system of channels, die inserts or jet coolers. Die cooling can also effectively help in reducing internal porosity in cast components. Accurate simulations based on sophisticated numerical models require accurate input data such as material properties, initial and boundary conditions. Although the heat is dominantly dissipated through die cooling, indicating the importance of knowing precise thermal boundary conditions, open literature lacks a detailed information about the spatial distribution of heat transfer coefficient. This study presents an inverse method to determine accurate heat transfer coefficients of a die insert based on temperature measurements in multiple points by 0.5 mm K-type thermocouples and a subsequent solution of the two-dimensional inverse heat conduction problem. The solver was built in the open-source CFD code OpenFOAM and the free library for nonlinear optimization NLopt. The results are presented for the commonly used 10 mm die insert with a hemispherical tip and coolant flow rates ranging from 100 l/h to 200 l/h. Heat transfer coefficients reach values well above 50 kW/m2K in the hemispherical tip, which is followed by a secondary peak and then a gradual drop to values around 1 kW/m2K further downstream.

Reconstruction of internal heat source in biological tissue using parallel particle swarm optimization

With the rapid development of engineering thermophysics, researches on human biological heat transfer phenomena has gradually shifted from qualitative to quantitative. The heat distribution information of human lesions is of great value for disease analysis, diagnosis, and treatment. It is a typical inverse problem of heat conduction that deriving the distribution of internal heat sources from the temperature distribution on the body surface. Differing from traditional numerical methods for solving heat conduction, this paper transforms such an inverse problem of bio-heat transfer into a direct one, thereby avoiding complex boundary conditions and regularization processes. To noninvasively reconstruct the internal heat source and its corresponding 3D temperature field in biological tissue, the parallel particle swarm optimization (PPSO) is used in the simulation module, where the position P(x, y, z) of point heat source in biological tissue and its corresponding temperature T are set as the optimization variables. Under a certain optimized sample, one can obtain the simulated temperature distributing on the surface of the module, then subtract the simulated temperature from the measured temperature of the same surface which was measured using a thermal infrared imager. If the absolute value of the difference is smaller, it indicates that the current sample is closer to the true location and temperature of the heat source. When the values of optimization variables are determined, the corresponding 3D temperature field is also confirmed. The simulation results show the simulated position and temperature of the heat source are very approximate with those of the real experimental module. The method presented in this paper has enormous potential and promising prospects in clinical research and application, such as tumor hyperthermia, disease thermal diagnosis technology, etc.

Numerical Analysis of Convective Heat Transfer in Quenching Treatments of Boron Steel under Different Configurations of Immersed Water Jets and Its Effects on Microstructure

In this work, the effects of jet impact angle and water flow on the heat-transfer coefficient in boron steel probes were analyzed. Angles of 90°, 75° and 60° were used with stirring flows of 33 l·min−1, 25 l·min−1, 13 l·min−1 and 6 l·min−1. The aim consisted of determining the heat-extraction rates by analyzing the correlation programmed in the Ansys Fluent 2020R2 software when different cooling conditions are used, avoiding many experiments, and establishing quenching conditions free of surface defects on the workpiece. This process is currently used in heavy machinery, requiring high hardness and wear resistance. The fluid-dynamic field was validated using a scale physical model using the particle image velocimetry technique, PIV. In contrast, the thermal field was validated with transient state experiments solving the inverse heat conduction problem, IHCP. The results show that for high flows (33 l·min−1), the jets with an angle of 90° impact the entire surface of the piece, but their cooling rate is slower compared to the other angles, being 243.61 K·s−1, and 271.70 K·s−1, 329.56 K·s−1 for 75° and 60°, respectively. However, for low flows (6 l·min−1), the impact velocities are very similar for the three cases, promoting more homogeneous cooling rates of 58.47 K·s−1, 73.58 K·s−1 and 63.98 K s−1 for angles of 90°, 75° and 60°, respectively. Likewise, through the use of CCT diagrams, it was determined that regardless of the cooling rate, the final structure will always be a mixture of martensite–bainite due to the effect of boron as determined experimentally, which implies a more significant proportion of martensite at higher cooling rates.

Analysis of the effect of surface coating and texturization on the estimation of effective thermophysical properties

This article aims to investigate the influence of surface modifications on the effective thermophysical properties of textured carbon steel and coated hard metal. The novelty of this work lies in the simultaneous estimation of thermal diffusivity and thermal conductivity via measurements using one active heating surface and Bayesian inference to solve the inverse heat conduction problem. To achieve this, an experimental setup was developed where samples were partially heated on an active surface, and the temperature was measured at two different points on the surface to estimate the thermal properties. The method was applied to two different thermal models using the same set of experimental data. Thus, the thermophysical properties were determined with and without the Markov Chain Monte Carlo technique using the Metropolis-Hastings sampling algorithm. Both techniques proved suitable for estimating the thermophysical properties. It is noteworthy that texturization and coating did not significantly modify the effective thermal diffusivity of the analyzed samples. However, there were considerable changes in the effective thermal conductivity, with an increase of approximately 13 % in the textured carbon steel sample and a reduction of approximately 11 % in the coated hard metal sample. The expanded uncertainty of the estimated effective thermophysical properties was less than 14 %. These results reflect low dispersion and good accuracy of the experimental technique used.

Band Heater Heat Flux Characterization Using Inverse Heat Conduction Problem Models

Abstract An experimental apparatus was constructed to correlate water flowrate and temperature rise under an external band heater. Due to the physical characteristics of the band heater, its transient heating behavior is unknown. This paper investigates the application of inverse heat conduction problem (IHCP) methods to characterize the heat flux from the band heater. Three experiments with different heating times (5, 10, and 20 s) and no flowrate were conducted to measure the transient temperature under the 400 W band heater. Type-T thermocouples measure surface temperature at the centerline of the band heater. The experimental results are computed with five different heat conduction models. The models are chosen to identify how the heat flux response varies from a simplified to a realistic model. Additionally, the results of the experimental heat flux are compared to the manufacturer band heater data (58.9 kW/m2) for each model. The minimum time needed for the heater to fully energize the system is from 10 to 12 s. The residuals for each model are analyzed and used to evaluate the appropriateness of the five different models. The results show that the use of simpler models can achieve results similar to those of complex models, with less time and computational cost.

Investigation of operational limit of a pulsating heat pipe by estimating local heat transfer

How pulsating heat pipes (PHPs) reach their operational limit has not yet been fully understood. This study aims to provide a complete picture of the termination mechanism of the self-oscillation of vapor and liquid. Experimental studies on a 10-turn PHP with HFC-134a were conducted and the filling ratio (FR) was from 20% to 80%. The thermo-fluid behavior in the PHP was investigated by temperature measurements with a high-resolution and high-speed infrared camera and estimation of fluid-to-wall heat flux distributions by solving inverse heat conduction problems. The results suggested that the PHP, increasing heat load, reached the operational limit due to different mechanisms depending on the filling ratio: at a high FR (80%), the liquid volume ratio increased with the increase of the operating temperature, resulting in the compressed liquid phase. At a low FR (20%), when a large amount of heat was applied, the fluid in the evaporator dried out and became a superheated vapor. The PHP with an optimum FR (50%) transferred the maximum heat under the same evaporator temperature, as the fluid in the PHP was able to keep the saturated two-phase state until the evaporator temperature exceeded the critical temperature.

Fire Protection of Steel Structures of Oil and Gas Facilities: Multilayer, Removable, Non-Combustible Covers

Fire protection is required to protect metal structures of oil and gas facilities from fires. Such fire protection should provide high fire resistance limits: 60, 90, 120 and more minutes. Specialists of LLC “RPC PROMIZOL ” developed a multilayer, removable type of fire protection made of superfine basalt fibre and ceramic materials for operation in Arctic conditions. Five experimental studies were carried out in standard and hydrocarbon fire regimes. The fire protection effectiveness of the products for I20 beams without load was obtained: a 50 mm thick coating provided 130 min of a standard fire regime; a 15 mm thick coating provided 60 min. The 15 mm thick coating provided 30 min of a hydrocarbon fire regime and the 50 mm thick coating provided 93 min of a hydrocarbon fire regime. The I40 beam under a load of 19.9 tf showed an R243 for the standard fire regime. The coefficients of effective thermal conductivity and specific heat capacity of fire-retardant compositions were determined by solving the inverse heat conduction problem. The problem was solved by modelling using the QuickField 7.0 software package, which implements FEM. Modelling showed that for obtaining the fire resistance limit R120 under the standard fire regime for the sample steel structure from an I40 beam, it is enough to apply fire protection with a thickness of 25 mm instead of 50 mm, which agrees with the experimental data. For the hydrocarbon regime, it is predicted that R120 can be obtained at a thickness of 45 mm instead of 50 mm.

Conjugate gradient method with regularization in estimating mold surface heat flux during continuous casting

The reconstruction of the boundary heat flux between solidifying steel and water-cooled mold wall using measured temperature data is recognized as an inverse heat conduction problem (IHCP). A Tikhonov spatial regularization approach to enhance the smoothness of the Conjugate Gradient Method (CGMr) was proposed which incorporates three different orders of spatial regularization: zeroth, first, and second order. The optimal regularization parameter was selected using a modified L-curve method. The accuracy of the CGMr is investigated with Case 1: a triangular time-spatial variation heat flux, and Case 2: a step change in the form of rectangular variation heat flux. The effects of measurement noise, grid points and time-step size are investigated. The results show that the minimum relative error (eRMS) of the predicted Case 1 heat flux is 1.98%, 7.64%, and 8.02% for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 2 heat flux are 3.82%, 13.42%, and 14.91%. Subsequently, CGMr algorithm is applied to calculate the heat flux in a mold simulator experiment. By comparing the relationship between heat fluxes reconstructed by CGMr after different iteration numbers, it is observed that the recovered heat flux of 2.14 MW/m2 with zeroth regularization remains highly stable.

Determination of interfacial heat transfer coefficient at the frozen sand mold casting process of ZL101 alloy

Compared to the resin sand mold casting process, frozen casting is more environmentally friendly, providing a better working environment and enhanced supercooling degree. The interfacial heat transfer coefficient (IHTC) between frozen sand mold and metal is an important parameter that significantly influences the final mechanical properties and microstructure of the castings. This paper solved the inverse heat conduction problem using the finite difference method (FDM). In addition, the conjugate gradient method (CGM) was adopted to calculate the temperature distribution and heat flux in the molten metal. At the same time, the particle swarm optimization algorithm (PSO) was used in temperature distribution determination in frozen sand mold. The interfacial heat transfer coefficient (IHTC) was estimated during the solidification of ZL101. The results showed a good agreement between calculated and experimental data, obtaining accurate casting interface temperature Tm, frozen sand mold interface temperature Ts, heat flux q, and IHTC. The analysis of the IHTC variation revealed a water content value within the range of 4 wt.% to 5 wt.% resulted in IHTC in two types of interpretation, called ‘fluctuation type’ and ‘turning type’.

Monitoring internal heat fluxes on Pulsating Heat Pipes using Kalman filter – Numerical and experimental results

The present study proposes an approach to monitor the wall-to-fluid internal heat flux on Pulsating Heat Pipes (PHP), which provides meaningful information about the efficiency and limiting operation conditions of such devices. In this sense, the current work adopts the Kalman Filter (KF) to estimate the heat flux between the working fluid and the internal tube wall along the adiabatic section of a PHP device by solving an inverse heat conduction problem. The KF is a well-known tracking technique, and therefore the current study is attractive due to the possibility of real-time evaluation of internal heat fluxes, a prospect that is extremely difficult to realize by standard whole domain inverse problem techniques. A high-speed and high-resolution infrared camera performs the temperature acquisition on the external wall of the PHP device. Two mathematical heat conduction models are considered: complete and thin-wall models. The complete model determines the temperature and heat flux evolution over time, and the thin-wall model is used to correct the random walk parameter on the KF formulation. Since the time step needed to the KF evolution model is smaller than the measurements acquisition interval, a matrix manipulation is used during the prediction step, reducing the number of mathematical operations. Hence, instead of predicting the states until reaching the measurement times, a proper mathematical manipulation reduces several mathematical operations to a single one. Then, the KF estimation capability is evaluated by considering synthetic data with different noise levels and different heat flux values. Subsequently, real experimental data are employed, and the results are compared with the ones obtained via a standard whole domain technique. The results obtained with the KF presented good visual agreement with the exact solution. Once the KF presents a delayed response, which is a characteristic of some filtering techniques, the errors with synthetic data were calculated offsetting the delay of the estimation in time in order to make a fair comparison, showing a maximum estimation error lower than 36 % considering the highest noise level. The synthetic data considered 20 s of time experiment, and the KF was able to estimate the PHP internal heat flux in less than 12 s, including the smoothing temperature step, showing fast and accurate results. For the real experimental data, the agreement between the KF and a whole domain technique is highly commendable, demonstrating a deviation of 25.5 % between both techniques. Moreover, the KF required significantly less computational time. The proposed methodology for heat flux estimation on PHPs is promising to evaluate their thermal behavior in real-time due to a low CPU time and computational memory required, as well as its simplicity to be implemented, possibly on embarked devices. Such information may therefore provide a thermal indicator of the correct functioning of PHPs in real time applications.

Numerical Model of Simultaneous Multi-Regime Boiling Quenching of Metals

This work presents a heat transfer and boiling model that computes the evolution of the temperature field in a representative steel workpiece quenched from 850 or 930 °C by immersion in water flowing at average velocities of 0.2 or 0.6 m/s, respectively. Under these conditions, all three boiling regimes were present during cooling: stable vapor film, nucleate boiling, and single-phase convection. The model was based on the numerical solution of the heat conduction equation coupled to the solution of the energy and momentum equations for water. The mixture phase approach was adopted using the Lee model to compute the rates of water evaporation–condensation. Heat flux at the wall was calculated for all regimes using a single semi-mechanistic model. Therefore, the evolution of boiling regimes at every position on the wall surface was automatically determined. Predictions were validated using laboratory results, namely: (a) videorecording the upward motion of the wetting front along the workpiece wall surface; and (b) cooling curves obtained with embedded thermocouples in the steel probe. Wall heat flux calculations were used to determine the importance of the simultaneous presence of all three boiling regimes on the heat flux distribution. It was found that this simultaneous presence leads to high heat flux variations that should be avoided in production lines. In addition, it was determined that the corresponding inverse heat conduction problem to estimate the active heat transfer boundary condition must be set-up for 2D heat flow.

Identifying the Heat Source in Radially Symmetry and Axis-Symmetry Problems

This paper solves the inverse source problem of heat conduction in which the source term only varies with time. The application of the discrete regularization method, a kind of effective radial symmetry and axisymmetric heat conduction problem source identification that does not involve the grid integral numerical method, is put forward. Taking the fundamental solution as the fundamental function, the classical Tikhonov regularization method combined with the L-curve criterion is used to select the appropriate regularization parameters, so the problem is transformed into a class of ill-conditioned linear algebraic equations to solve with an optimal solution. Several numerical examples of inverse source problems are given. Simultaneously, a few numerical examples of inverse source problems are given, and the effectiveness and superiority of the method is shown by the results.

Modeling of the thermophysical properties of molding materials by solving the inverse heat conductivity problem

Modeling of the thermophysical properties of molding materials by solving the inverse heat conductivity problem

Vat photopolymerization additive manufacturing process modeling: a thermal-chemical coupling approach informed by in-situ and ex-situ characterization data

Vat photopolymerization (VPP) is one of the most widely used additive manufacturing methods. The VPP process temperature and material curing reaction interplay with each other to critically determine the final product quality. Insights about the time-varying process temperature and degree of conversion (DoC) is desired for VPP process control but difficult to attain due to lacking effective operando characterization technologies. This work reports a new method to create a thermal-chemical model of the VPP process by solving an inverse heat conduction problem (IHCP) based on in-situ observable temperature measurement to estimate the chemistry reaction-induced heat source that is a function of DoC. Ex-situ photo differential scanning calorimetry (Photo-DSC) characterization is used to initialize the chemistry reaction model parameters so that DoC can be calculated. Specifically, vat substrate temperature is measured using an in-situ infrared thermal camera and used as input to solve an IHCP for estimating exothermic heat generation rate for the internal heat generation component at the curing part. Overall, the newly developed VPP modeling framework combines an IHCP that is optimized by in-situ thermal monitoring with a chemical reaction heat generation and conduction model that is educated by Photo-DSC characterization. The model predictions of temperature and DoC are experimentally validated by comparing against in-situ temperature measurement and ex-situ spectroscopy measurement of prints at different exposure times.