Discontinuous Heat Source Research Articles

A wavelet-based collocation technique to find the discontinuous heat source in inverse heat conduction problems

This paper is devoted to an inverse problem of determining discontinuous space-wise dependent heat source in a linear parabolic equation from the measurements at the final moment. In the existing literature, a considerably accurate solution to the inverse problems with an unknown space-wise dependent heat source is impossible without introducing any type of regularization method but here we have to determine the unknown discontinuous space-wise dependent heat source accurately using the Haar wavelet collocation method (HWCM) without applying the regularization technique. This HWCM is based on finite-difference and Haar wavelets approximation to the inverse problem. In contrast to other numerical techniques, in HWCM, we used Haar functions that create a well-conditioned system of algebraic equations. The proposed method is stable and convergent because the numerical solution converges to the exact solution without observing any difficulty. Finally, some numerical examples are presented to verify the validity of the HWCM for different cases of the source term.

A 3D analytical thermal model in grinding considering a periodic heat source under dry and wet conditions

Predicting the temperature distribution in the workpiece is of high importance to control and prevent thermally induced damages in grinding. Grinding force has an oscillating nature because of the discrete grit-workpiece interaction. The force fluctuation is more pronounced in macro level when the wheel vibrates due to run out or when segmented, grooved or pattern wheels are utilized. These fluctuations cause a discontinuous or a time-dependent heat source. In this study, the dynamic part of the heat source is included in an analytical thermal model as a new approach to understand detailed workpiece temperature behavior in grinding. Furthermore, heat convection from side surfaces of the workpiece was not included in previous analytical models and is formulated and discussed in this study for the first time. By considering proper boundary conditions an analytical solution for the workpiece temperature distribution by using the periodic heat source concept is proposed. The influential parameters are discussed theoretically, and it is shown that the heat source fluctuations cause an increase in maximum temperature compared to the case with a constant heat source. Furthermore, it is shown that the 2D solution is quite reliable and precise enough to predict the maximum temperature of the slab workpiece having a long width in both wet and dry conditions. Grinding experiments have been carried out to validate the model accuracy by embedding foil thermocouples in a split workpiece setup while using CBN electroplated and aluminum oxide wheels. Validation of discontinuous heat source model was investigated while the wheels have some run out. The effects of grinding conditions on the maximum temperature have been obtained experimentally and compared with the proposed model. The results show that considering a time-dependent heat source not only explains and predicts the cyclic nature of temperature but also improves the prediction of workpiece temperature accuracy.

Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient method

The inverse problem of identifying the unknown time-dependent heat source H(t) of the variable coefficient heat equation ut=(k(x)ux)x+F(x)H(t), with separable sources of the form F(x)H(t), from supplementary temperature measurement h(t)≔u(0,t) at the left end of the rod, is investigated. The Fourier method is employed to illustrate the comparison of spacewise (F(x)) and time-dependent (H(t) heat source identification problems. An explicit formula for the Fréchet gradient of the cost functional J(H)=‖u(0,⋅;H)−h‖L2(0,Tf)2 is derived via the unique solution of the appropriate adjoint problem. The Conjugate Gradient Algorithm, based on the gradient formula for the cost functional, is then proposed for numerical solution of the inverse source problem. The algorithm is examined through numerical examples related to reconstruction of continuous and discontinuous heat sources H(t), when heat is transferred through non-homogeneous as well as composite structures. Numerical analysis of the algorithm applied to the inverse source problem in typical classes of source functions is presented. Computational results, obtained for random noisy output data, show how the iteration number of the Conjugate Gradient Algorithm can be estimated. Based on these results it is shown that this iteration number plays a role of a regularization parameter. Numerical results illustrate bounds of applicability of the proposed algorithm, and also its efficiency and accuracy.

A Space-Time Meshless Method That Removes Numerical Oscillations When Solving PDEs with High Discontinuities

A space-time diffuse approximation method avoiding any secondary finite-difference time-stepping procedure, is presented to solve partial differential equations (PDEs) containing high discontinuities. This method is validated with a non-inner source heat equation. It is shown that its accuracy is improved when using a new proposed weight function. Furthermore, the application of the new weight function, respecting the principle of causality, removes some spurious oscillations appearing with high temporally discontinuous heat sources (such as Heaviside). Results, presented as isotherms or temperature fields, get closer to those obtained with an implicit first-order backward finite–difference scheme.

Numerical Analysis of the Sludge Drying Behaviors with Bottom Periodic Heating

The drying of sludge can reduce its mass and the volume and consequently the cost of storage, handling and transport. In this paper, a one-dimensional model is established for the sludge drying process and hot water from the solar energy is applied as a discontinuous heat source (periodic heating). Based on the simplified physical model, the thermal behavior of the sludge drying process is investigated by numerical method. The effects of the sludge depth, mixing time and heat transfer coefficient are analyzed. The simulation results show that the most important effecting factor is the mixing time. One hour mixing cycle can increase heat transfer rate to 430%, 130% than that of without mixing and two hours mixing cycle, respectively. On the other side, both the sludge thickness and the heat transfer coefficient have some effects. For the cases studied the 40cm thick sludge with 1 hour mixing time is an optimal option.

Identification of spacewise and time dependent source terms in 1D heat conduction equation from temperature measurement at a final time

Inverse problems of identifying the unknown spacewise and time dependent heat sources F(x) and H(t) of the variable coefficient heat conduction equation ut=(k(x)ux)x+F(x)H(t) from supplementary temperature measurement (uT(x)≔u(x,Tf)) at a given single instant of time Tf>0, are investigated. For both inverse source problems, defined to be as ISPF and ISPH respectively, explicit formulas for the Fréchet gradients of corresponding cost functionals are derived. Fourier analysis of these problems shows that although ISPF has a unique solution, ISPH may not have a unique solution. The conjugate gradient method (CGM) with the explicit gradient formula for the cost functional J1(F) is then applied for numerical solution of ISPF. New collocation algorithm, based on the piecewise linear approximation of the unknown source H(t), is proposed for the numerical solution of the integral equation corresponding to ISPH. The proposed two numerical algorithms are examined through numerical examples for reconstruction of continuous and discontinuous heat sources F(x) and H(t). Computational results, with noise free and noisy data, show efficiency and high accuracy of the proposed algorithms.

Calibration-Free Volume Flow Measurement Principle Based on Thermal Time-of-Flight (TToF)

For measurement problems with unknown fluids a new calibration-free volume flow measurement technique is developed. This measurement technique includes a discontinuous heat source associated with a minimum of two thermal sensors arranged downstream. The heat source is used to generate a signal code [dT/dt] by injection of local thermal pulses into the flow line. The thermal sensors detect the time-dependent thermal gradients in the fluid at different positions. Herewith, the measurement technique aims at the determination of the flow velocity by ToF, with the particular advantage of applying to measurements of any fluids with unknown properties. The volume flow is determined by integrating the flow profile.

Accuracy of a domain decomposition method for the recovering of discontinuous heat sources in metal sheet cutting

This paper describes a method for the retrieval of discontinuous heat sources involved in a metal cutting process. Two smoothing techniques were used in the present study in order to smooth noisy sensor data recorded by temperature sensors. The two smoothing techniques are a least squares polynomial fit and a Lagrangian smoothing. These data are then used to recover the heat source. It is found that the least squares polynomial provides over-smoothed results and the Lagrangian smoothing produces phyically acceptable results of the retrieved heat source.

An experimental study on natural convective heat transfer from a vertical wavy surface heated at convex/concave elements.

Natural convective heat transfer from a vertical wavy surface with discontinuous heating has been studied experimentally by considering the effect of heat conduction in unheated elements. The wavy surface is constructed with concave and convex semicircular cylinders and has discontinuous heat sources on concave or convex surfaces. The behavior of the flow and the temperature distributions in the boundary layer of surfaces have been discussed to clarify the effect of heat conduction in the unheated elements on local Nusselt number for the case of convex- or concave-heating. Empirical formulas have been derived to predict the average Nusselt number along the wavy surface.