Time-dependent Heat Flux Research Articles

The wall heat flux partitioning during the pool boiling of water on thin metallic foils

• We study nucleate boiling on substrates of small thermal capacity using IR camera. • Microlayer does not fully evaporate due to limited thermal capacity of the heater. • Microlayer evaporation and rewetting are less important compared to thick heaters. • Most energy is removed by convective effects created by bubble growth and departure. • We re-derived heat flux partitioning model to quantify each heat transfer mechanism. In this work, we studied the wall heat flux partitioning during the pool boiling of water on thin metallic surfaces. We conducted boiling experiments on surfaces where we engineered nucleation sites by nanosecond-fiber-laser texturing. These nucleation sites form triangular lattice patterns with different pitches. We measured the time-dependent temperature and heat flux distributions on the boiling surface using an infrared camera. We developed post-processing algorithms to measure, based on these distributions, all the fundamental boiling parameters used in heat flux partitioning models (e.g., nucleation site density, bubble wait and growth time, and bubble footprint radius) and the actual partitioning of the heat flux, i.e., how much heat is transferred by evaporation of the microlayer, rewetting of the surface, and convective effects. This work reveals that the mechanisms of heat transfer on substrates of small thermal capacity are very different compared to substrates of large thermal capacity. With water, the bubble microlayer typically does not dry out and the surface temperature at rewetting is practically the same as the rewetting fluid temperature. These effects limit the efficiency of microlayer evaporation and rewetting heat transfer. Instead, convective effects generated by the bubble growth process remove most of the energy from the heated surface. This behavior is captured by a heat flux partitioning model that we re-derived from first principles to describe the heat transfer mechanisms on substrate of small thermal capacity.

Thermoelastic Modeling of Time Fractional Heat conduction in Circular Disk with Internal Heat Generation

In this paper, thermoelastic modeling of a circular sector disk is done under the influence of fractional time theory of thermoelasticity to evaluate the changes in temperature flow, displacement and stress behavior subjected to internal heat generation within it. At the fixed edge a time dependent heat flux is applied while thermally insulation type boundary are maintained at upper part of surface as well as the lower part surface is kept at zero temperature. Also certain boundary surfaces at 0 are fixed as a function of temperatures field , 1 f 2 f which is function of respectively. Furthermore, method of Integral transformations is employed to determine the required solution of equation of heat conduction. The obtained expressions areof Bessel’s type and in form of series. For numerical analysis the pure aluminum circular sector disk considered and obtained results are plotted as shown in figures.

Successive droplet impingement onto heated surfaces of different wettabilities

New high-performance electronic components generate a massive amount of heat per unit of area, and spray cooling is a field of interest for this application. Spray cooling achieves higher heat transfer coefficients than commonly used forced air convection cooling systems. In this study, we performed multiple droplet impingement experiments over heated surfaces in the non-boiling regime, a scenario that is closer to the phenomena of spray cooling than individual droplet impact. This paper focuses on the effects of droplet interactions with surfaces exhibiting distinct wettabilities. A surface thermocouple was used concomitantly as the impact surface and the temperature sensor. The surface thermocouple was originally hydrophilic (uncoated), which later we treated with a low cost commercial superhydrophobic coating (coated). High-speed imaging showed the droplet motion during impact. We proposed a one-dimensional inverse heat conduction semi-analytical solution, adopting a time-dependent droplet heat flux as one of the boundary conditions to represent the droplet cooling effect. We compared the heat transfer performances of both uncoated and coated surfaces for different droplet velocities but same initial temperature. The results showed higher heat transfer rates on uncoated surface relative to the coated one. Also, higher droplet velocities showed higher heat transfer mainly due to the increase in the droplet spreading diameter in contact with the solid. Successive droplets showed lower heat flow for the uncoated surface because of the low thermal effusivity and thickness of the heated solid, where most of the heat is removed by the first droplet. The first droplet thermal resistance and damping effects for the subsequent droplets, reduced the overall cooling performance. On the other hand, for the coated surface we found that the consecutive droplets continued to transfer heat with a similar rate as the previous one, which is desirable for spray cooling. However, the reduced overall heat transfer performance for the coated surface can be explained by the air pockets within the coating structure.

Heat Transfer Analysis of Laminar Pulsating Flow in a Rectangular Channel Using Infrared Thermography

Abstract While numerous applied studies have successfully demonstrated the feasibility of unsteady cooling solutions, a consensus has yet to be reached on the local instantaneous conditions that result in heat transfer enhancement. This work aims to experimentally validate a recent analytical solution (on a local time-dependent basis) for the common flow condition of a fully developed incompressible pulsating flow in a uniformly heated vessel. The experimental setup is found to approximate the ideal constant heat flux boundary condition well, especially for the decoupled unsteady scenario where the amplitude of the most significant secondary contributions (capacitance and lateral conduction) amounts to 1.2% and 0.2% of the generated heat flux, respectively. Overall, the experimental measurements for temperature and heat flux oscillations are found to coincide well with a recent analytical solution to the energy equation by the authors. Furthermore, local time-dependent heat flux enhancements and degradations are observed to be qualitatively similar to those of wall shear stress from a previous study, suggesting that the thermal performance is indeed influenced by hydrodynamic behavior.

An Analytical Model for Transient Heat Transfer with a Time-Dependent Boundary in Solar- and Waste-Heat-Assisted Geothermal Borehole Systems: From Single to Multiple Boreholes

With the increasing engineering applications of geothermal borehole heat exchangers (BHEs), accurate and reliable mathematical models can help advance their thermal design and operations. In this study, an analytical model with a time-dependent heat flux boundary condition on the borehole wall is developed, capable of predicting the thermal performance of single, double, and multiple closed-loop BHEs, with an emphasis on solar- and waste-heat-assisted geothermal borehole systems (S-GBS and W-GBS) for energy storage. This analytical framework begins with a one-dimensional transient heat conduction problem subjected to a time-dependent heat flux for a single borehole. The single borehole scenario is then extended to multiple boreholes by exploiting lines of symmetry (or thermal superposition). A final expression of the temperature distribution along the center line is attained for single, double, and multiple boreholes, which is verified with a two-dimensional finite-element numerical model (less than 0.7% mean absolute deviation) and uses much lesser computational power and time. The analytical solution is also validated against a field-scale experiment from the literature regarding the borehole and ground temperatures at different time frames, with an absolute error below 6.3%. Further, the thermal performance of S-GBS and W-GBS is compared for a 3-by-3 borehole configuration using the analytical model to ensure its versatility in thermal energy storage. It is concluded that our proposed analytical framework can rapidly evaluate closed-loop geothermal BHEs, regardless of the numbers of boreholes and the type of the heat flux on the borehole wall.

A homogenization method to solve inverse Cauchy–Stefan problems for recovering non-smooth moving boundary, heat flux and initial value

In the paper, we solve two Stefan problems. The first problem recovers an unknown moving boundary by specifying the Cauchy boundary conditions on a fixed left-end. The second problem finds a time-dependent heat flux on the left-end, such that a desired moving boundary can be achieved. Then, we solve an inverse Cauchy-Stefan problem, using the over-specified Cauchy boundary conditions on a given moving boundary to recover the solution. Resorting on a homogenization function method, we recast these problems into the ones having homogeneous boundary and initial conditions. Consequently, the approximate solution is obtained by solving a linear system obtained from the collocation method in a reduced domain. For the first Stefan problem the moving boundary can be determined accurately, after solving a nonlinear equation at each discretized time. For the second Stefan problem, we can obtain the required boundary heat flux without needing of iteration. Numerical examples, including non-smooth ones, confirm that the novel methods are simple and robust against large noise. Moreover, the Stefan and inverse Cauchy-Stefan problems are solved without initial conditions.

Unsteady heat conduction with phase change applied to a novel thermal plug and abandonment process

The continuous exploration of oil wells has increased the demand for plug and abandonment procedures. Current techniques used for well plugging involve the cementing process, which is extremely expensive and difficult to perform. Aiming to overcome these challenges, a new approach in this area has been investigated. This novel technology, named here as Thermal Plug and Abandonment (TP&A), proposes a chemical mixture that would be introduced through the production tube or the steel casing. Then, this mixture is ignited and the exothermic reaction generates enough heat to melt the wellbore components. After the cooling stage, the solidified mass composed by these components and the products of the reaction will serve as a seal to plug the well. Under such circumstances, this work aims to investigate this new technology assuming a thermite mixture that is introduced through the steel casing. For that, a numerical analysis is employed to investigate the heat conduction and the phase change through the oil well structure, which is modelled as a two-dimensional axisymmetric domain. The heat generated by the reaction is approached as a spatial and time-dependent heat flux profile that was estimated based on experiments found in literature. The thermal behavior is assessed to discover if the heat generated by the reaction is enough to form a plug composed by all components found in an oil well environment. It was found that temperatures are high enough to melt most part of the steel and a significant amount of the cement layer. The investigation also found that temperatures heavily drop through the cement layer, which avoids any notable melting of the cap rock. Finally, the heat fluxes’ profiles were increased, and it was found that a TP&A procedure that accounts for a seal composed by only the thermite products and the melted steel layer might be a more practical approach.

Unsteady convective heat transfer from a flat plate with heat flux that varies in space and time

Convective heat transfer due to laminar fluid flow past a flat plate is a standard problem in heat transfer. While constant heat flux or temperature along the plate is often assumed for solving such problems, there may be several practical scenarios where the heat flux along the plate varies as a function of both space and time. Developing an analytical solution for the resulting plate temperature distribution is important for understanding and optimizing the thermal performance of such systems. While some work exists on analyzing problems with time-dependent or space-dependent heat flux, there is a lack of work on the general problem where the heat flux is a function of both space and time. This paper presents a solution for this problem by solving the integral form of the energy equation, along with the use of fourth-order Karman-Pohlhausen polynomials for velocity and temperature distributions in the momentum and thermal boundary layers. A non-linear, first order, hyperbolic partial differential equation for the plate temperature is derived in response to the time- and space-varying plate heat flux. This equation is shown to agree well with results from past work for several special cases. Numerical solutions for the generalized equation are presented. Based on this approach, the plate temperature distribution is predicted for several heat flux profiles that may be of interest in practical applications. Results from this work improve our understanding of unsteady convective heat transfer, and contribute towards modeling and optimization of practical engineering systems where such phenomena occur.

Reconstruction of the heat transfer coefficients and heat fluxes in heat conduction problems

This paper studies the reconstruction of the spatial-dependent-heat transfer coefficient γ(x) and- heat flux q(x), using the Levenberg–Marquardt (L–M) connected with the surrogate functional method and the modified conjugate gradient method (MCGM) (problem I). In addition to studying the reconstruction of γ(x) and space–time-dependent heat flux q(x,t) using the MCGM (problem II). The numerical results obtained by the L–M and MCGM demonstrate the efficiency, accuracy, and robustness of the proposed methods. The obtained results are compared with those obtained from an inverse approach connected with COMSOL Multiphysics under the same conditions. Finally, a conclusion and some remarks are presented.

Simultaneous reconstruction of thermal boundary condition and physical properties of participating medium

A hybrid optimization technique, which combines decentralized fuzzy inference method (DFIM) and sequential quadratic programming (SQP) algorithm, is developed to reconstruct the thermal boundary condition and physical properties of participating medium simultaneously. The coupled radiation-conduction heat transfer in the medium is solved by finite volume method in combination with discrete ordinate method. The reconstruction task is formulated as an inverse problem which is solved by the hybrid DFIM-SQP technique from the knowledge of surface temperature and exit radiative intensity. Retrieval results demonstrate that the time-dependent heat flux, absorption coefficient, scattering coefficient and thermal conductivity of participating medium can be accurately retrieved by the present methodology. The proposed DFIM-SQP technique is more accurate and efficient in solving the simultaneous reconstruction problem than DFIM, conjugate gradient method, SQP, particle swarm optimization and krill herd algorithms.

Development of experimental methodology for estimation of thermo-physical properties of engineering materials using inverse method

In the present article, a novel experimental methodology is proposed for the simultaneous estimation of unknown thermal conductivity (k) and specific heat capacity (cp) of engineering materials. The experimental method involves radiative heating of the test sample and measuring its temperature response at two different locations. Proper arrangements have been made to realize one-dimensional heat conduction within the test sample. The one-dimensional heat conduction equation with convection-radiation heat loss boundary condition is considered as the forward model as it best mimics the experimental heat transfer. The inverse problem is formulated as a parameter estimation problem, and solved using Levenberg-Marquardt algorithm. First, numerical estimations were carried out with synthetic temperature data. It was found that the time-dependent heat flux boundary condition is more suitable to estimate the properties of higher thermal conductivity materials. Then, real-time temperature measurements were carried out on Stainless Steel (SS), Mild Steel (MS), Brass (BS), Aluminium (Al), and Copper (Cu) test samples. The estimated k of SS, MS, BS, Al, and Cu in (W/(m K)) is 14.03, 54.63, 103.21, 221.53 and 379.89, respectively. And, the estimated cp of SS, MS, BS, Al, and Cu in (J/(kg K)) is 500.44, 460.84, 395.01, 939.58, and 398.43 respectively. The estimated thermal properties are also validated against experimentally measured values obtained using the Modified Transient Plane Source (MTPS) sensor. It was found that the estimated k and cp in case of SS has a deviation of 7% and 4%, respectively.

Inverse Estimation of Time-Dependent Heat Flux in Stagnation Region of Annular Jet on a Cylinder Using Levenberg–Marquardt Method

All solving methods available in literature are formulated for direct solution of stagnation point flow and its heat transfer impinging on the surfaces with known boundary conditions. In this paper for the first time, a numerical code based on Levenberg–Marquardt method is presented for solving the inverse heat transfer problem of an annular jet on a cylinder and estimating the time-dependent heat flux using temperature distribution at a specific point. For this purpose, the numerical solution of the dimensionless temperature and the convective heat transfer in a radial incompressible flow on a cylinder rod is carried out as a direct problem. Using similarity variable and appropriate transformations, momentum and energy equations are converted into Semi-similar equations. The heat flux is then estimated by applying the Levenberg–Marquardt approach. This technique is an iterative approach based on minimizing the least-square summation of the error values, the error being the difference between the estimated and measured temperatures. Results of the inverse analysis indicate that the Levenberg–Marquardt algorithm is an efficient and acceptably stable technique for estimating heat flux. This method also exhibits considerable stability for noisy input data. The maximum value of the sensitivity coefficient is related to the estimation of exponential heat flux and its value is 0.1619 also the minimum value of the sensitivity coefficient is 5.6210-6 which is related to the triangular heat flux. The results show that the parameter estimation error in calculating the triangular and trapezoidal heat flux is greater than the exponential and sinus–cosines heat flux because the maximum value of RMS error is obtained for these two cases, which are 0.481 and 0.489, respectively the reason for the increase in the errors in estimating these functions is the existence of points where the first derivative of the function does not exist.

Semi-derivative integral method to transient heat conduction time-dependent heat flux boundary conditions

Transient heat conduction in semi-infinite medium with a time dependent heat flux as boundary condition has been solved by a semi-derivative integral-balance method. Two versions boundary fluxes have been considered: power-law and exponential.

2D linear finite element simulation of laser metal heating for digital twins

In the context of laser-based additive manufacturing, the thermal behavior of the substrate is relevant to define process parameters vis-à-vis piece quality. The existing literature focuses on two process variables: (a) lumped laser power and (b) process speed. However, this literature does not consider other variables, such as those related to the laser power distribution. To fill this vacuum, this manuscript includes the laser power spatial distributions (Gaussian, uniform circular and uniform rectangular) in addition to (a) and (b) above in 2D linear substrate heating simulations. The laser energy is modeled as a time dependent heat flux boundary condition on top of the domain. The total laser delivered power was identical for all spatial distributions. The results show that the laser intensity spatial distribution strongly affects the maximum temperature, and the depth and width of the heat affected zone. These 2D finite element simulations prove to be good options for digital twin based design environments, due to their simplicity and reasonable temperature error, compared to non-linear analysis (considered as ground truth for this case). Future publications address non-linear finite element simulations of the laser heating process (including convection and radiation and temperature dependent substrate properties).

Magnetophotothermal interaction in a rotating solid cylinder of semiconductor silicone material with time dependent heat flow

A mathematical model linking thermoelasticity to photothermal experiments is proposed with the consideration of the photothermal effect. The system equations for coupled plasma, heat conduction with phase-lags (PLs), and motion equations are introduced and solved by using the Laplace transform technique. The photothermal, thermal, and elastic waves in a rotating solid cylinder of semiconductor material are analyzed with the proposed model. The cylinder surface is constrained and subjected to a time-dependent pulse heat flux. The sensitivity of the physical fields for the angular velocity, PLs, and thermal vibration parameters is investigated. In addition, the effects of the effective parameters on the physical quantities are graphically illustrated and discussed in detail.

Inverse analysis of the time-dependent heat flux in stagnation point flow of incompressible fluid impinging on a cylinder with uniform surface suction-blowing using Levenberg–Marquardt method

ABSTRACT In this paper for the first time, a numerical code based on Levenberg–Marquardt method is presented for solving inverse heat transfer problem of axisymmetric stagnation flow impinging on a cylinder with uniform transpiration and to estimate the time-dependent heat flux using temperature distribution at a point. The effect of noisy data on the final result is studied. The maximum value of the sensitivity coefficient is related to estimation of exponential heat flux and its value is 0.1718 also the minimum value of the sensitivity coefficient is 7.50 × 10−6 which is related to the triangular heat flux. The results show that the parameter estimation error in calculating the triangular and trapezoidal heat flux is greater than the exponential and sinus–cosines heat flux because the maximum value of RMS error is obtained for these two cases, which are 0.458 and 0.492, respectively the reason for the increase in error in estimating these functions is the existence of points where the first derivative of the function does not exist. This method also exhibits considerable stability for noisy input data. In most cases, surface blowing decreases the prediction accuracy by removing the boundary layers from the surface whereas suction acts vice versa.

A freezing problem with varying thermal coefficients and convective boundary condition

This study investigates a one-dimensional Stefan problem by taking variable specific heat and thermal conductivity which depends on temperature. Time dependent heat flux is also assumed at the boundary x = 0 in the problem. The similarity solution for the problem is constructed when $$ \alpha = \beta $$ and $$ p = q = 1 $$ . The uniqueness and existence of the similarity solution to the problem have been also established. The problem is solved approximately for all $$ \alpha $$ and $$ \beta $$ with the aid of the shifted Chebyshev tau method. To check the precision of the proposed approximate solution, we have made its comparisons with the obtained exact solution. The comparisons done are shown by tables which sufficiently agree with the exact solution. The effects of the parameters on the movement of interface and temperature profiles are also carried out.

Simultaneous estimation of internal temperature field and boundary time-dependent heat flux in absorbing and scattering media using the unscented Kalman smoothing technique

Based on the future measured temperature, the Rauch-Tung-Striebel smoothing technique coupled with unscented Kalman filter is introduced to resolve the nonlinear inverse radiation-conduction heat transfer problem, which is nonlinear because of specific heat and thermal conductivity with temperature and existence of the radiative heat transfer. For the direct process, the finite volume method is applied to calculate the energy equation and radiative transfer equation is computed by the discrete ordinate method. Based on the measurement temperature signals of the right surface, the left surface time-dependent heat flux and interior temperature in absorbing and scattering medium are retrieved in near real time by using the proposed method. The different number of the future measured temperature, measurement noise, and process noise covariance are applied to examine the feasibility and reliability of the present algorithm. All retrieval results show that the Rauch-Tung-Striebel smoothing technique coupled with unscented Kalman filter is accurate and stable. Compared to the unscented Kalman filter, the accuracy of the retrieval temperature distribution is improved significantly, the deviation and oscillation of the retrieval time-dependent heat flux are also decreased substantially.

Noise limits on ITER plasma vertical stabilization system imposed by tungsten divertor monoblock thermal fatigue

Abstract Tokamak plasmas with vertically elongated cross-sections are unstable due to vertical displacements. Feedback stabilization of the ITER plasma current centre will be performed using the speed of its vertical displacements, dZ/dt, as input. Inevitable noise in the diagnostic signal for dZ/dt leads to “noisy” components in the feedback voltage and current in the stabilizing coils. This leads to noisy components in the vertical position of the plasma current centre and the divertor strike points. For burning plasma conditions on ITER, these strike point displacements may be a concern for thermal fatigue at the water cooling interface of the tungsten monoblocks constituting the divertor targets. A study is presented here in which this concern is examined for the first time. The baseline 15 MA scenario (fusion power of 500 MW with a 500 s flattop and fusion power gain, QDT = 10) is simulated with the DINA code, assuming low frequency noise in the dZ/dt diagnostic signal. The noise has uniform spectrum with a given root mean square (RMS) value ( =0.6 ms−1 or 0.2 ms−1) in the frequency band (0, 1 kHz). The results of these DINA simulations are combined with dissipative divertor plasma solutions obtained with the SOLPS-ITER plasma boundary code to provide a time dependent divertor target heat flux density profile. The latter is then imposed on a finite element model of the target monoblocks to assess the temporal evolution of the 3D temperature field in the block, including the Cu-W and Cu-CuCrZr joints at the cooling interface. These joints represent the points of the Cu and CuCrZr materials that see the largest temperature changes and are thus at greatest risk of failure. To evaluate an acceptance criterion on the non-cyclic and non-uniform thermal loads at the joints, an approach is developed which combines a Rainflow counting technique with Palmgren-Miner’s rule for fatigue accumulation. Analysis of the temperature evolution at the Cu-W joint shows that the RMS value of noise ∼0.6 ms−1 is unacceptable from the point of view of thermal fatigue for the expected total exposure time under burning plasma conditions that the first ITER divertor must survive. The lower value ( ∼0.2 ms−1) is found to be acceptable. A subsequent parametric study concludes that ≲ 0.44 ms−1 is just consistent with fatigue lifetime for the prescribed divertor power flux density profile, but should be kept lower than this to allow for some margin. Regarding the monoblock surface temperature, the natural power spreading caused by the separatrix movements is found to be beneficial from the point of view of recrystallization. For the level of noise in the dZ/dt diagnostic required to stay below joint fatigue limits, the average surface temperature on the most loaded monoblocks is reduced by ∼100 °C compared to the case with stationary strike points and the amount of time spent at this temperature at any one block by over 80 %.

The boiling crisis of water under exponentially escalating heat inputs in subcooled flow boiling at atmospheric pressure

We investigated the boiling crisis of water under exponentially escalating heat inputs using a specially-designed experimental apparatus featuring high-speed infrared thermography to measure the time-dependent temperature and heat flux distributions on the boiling surface, and high-speed video diagnostics to image the bubble growth process. We conducted flow boiling experiments with water at atmospheric pressure on a plate-type heater installed in a 3 × 1 cm2 cross section channel. We tested various values of flow velocity (corresponding to Reynolds numbers from 0 to 35,000), water subcooling (from 10 to 75°C) and rate of power rise (corresponding to exponential power escalation periods, τ, from 1.5 to 500 ms). At long periods critical heat flux (CHF) is independent of the power escalation period and the boiling processes appear to be physically similar to steady state. As the period decreases, the CHF values tend to increase following an asymptotic 1/τ trend. The mechanism that determines this trend depends on subcooling. For high subcooling, CHF monotonically increases as the period decreases, and the DNB occurs through a fully developed nucleate boiling process. By contrast, for low subcooling, CHF first increases as observed for high subcoolings, then decreases, and finally increases again as the period decreases. We observe that such non-monotonic transition is due to a change in the boiling crisis mechanism. Specifically, for very short periods and low subcooling, the boiling crisis happens during the growth of the very first generation of bubbles, which never detach from the heated surface, shortly after the onset of nucleate boiling. In such cases, fully-developed boiling is not achieved.