Form Of Heat Transfer Equation Research Articles

Integral Form of the Heat Transfer Equation With Arbitrarily Moving Boundary and Arbitrary Heat Source

Abstract For the first time, an integral form of one-dimensional heat transfer equation in a semi-infinite domain with a boundary, moving arbitrarily in time, and a heat source, depending arbitrarily on time and space location, is obtained. The obtained integral equation relates time histories of the temperature and its gradient at the boundary of the domain with the temperature at any given point inside or at the boundary of the domain. In the latter case, it delivers closed form integral equation for the rate of boundary movement in nonlinear problems where the time history of boundary movement is one of problem unknowns. The obtained equation accounts explicitly for the presence of an arbitrary heat source in the domain, while other existing methods do not allow a closed integral formulation to be obtained in such a case. The equation may be used for an analytical investigation of several types of boundary value problems (BVPs), as well as for numerical solution of such problems. Particular cases of this equation with a trivial heat source are known to demonstrate chaotic behavior. It is expected that the same is true for some nontrivial heat source functions, and this conjecture will be explored in subsequent publications.

Numerical modeling of non-Fourier heat transfer and fluid flow during plasma arc welding of AISI 304 stainless steel

ABSTRACTThis paper is an attempt to study the evolution of temperature profiles and weld pool geometry during plasma arc welding (PAW) by solving the transient Navier–Stokes and Energy equations. The analysis for an AISI 304 stainless steel rectangular plate was carried out using a flexible written program in Fortran. Due to the low accuracy of the Fourier heat transfer equation for short times and large dimensions, a non-Fourier form of heat transfer equation was used. Gaussian heat source is considered as the heat source model. The fluid flow in the molten pool is of interest because it can change the temperature distribution in and around the molten zone. The governing equations for fluid flow were solved by the finite-volume method in which the SIMPLE method was utilized for pressure–velocity coupling. The effects of heat conduction, fluid flow, and force actions at the weld pool were considered. Thermo-physical properties such as thermal conductivity, specific heat, and dynamic viscosity vary as a function of temperature. There are two mechanisms involved which actively cause heat transfer to the surroundings: radiation and convection heat transfer. The numerical results are compared to the experimental data. The results corroborate that the weld pool thickness in the cross section of PAW and the time taken by molten metal to reach the end of thick metal are in good agreement with the experimental measurements. Finally, the results obtained from the assumed Fourier heat transfer are compared for the same study.

The Study of Cooling Process on Runout Table by Simulation Method

This research aims to achieving the effective cooling parameter on the runout table (ROT) of strip steel in hot rolling process. The 2-dimensional transient heat conduction is developed including the external force convection and heat source due to translational motion. The material property, boundary, and initial condition are defined and bounded to model geometry. The strip velocity, cooling water temperature, and external fluid velocity are chosen for the influent parameters during cooling process at ROT. To find the optimality of cooling operating requirement, simulation study is conducted throughout this research. To reach the objective of optimal cooling consumption at ROT, temperature distribution in the strip steel during cooling governs by the form of heat transfer equation. To solve 2-dimensional transient heat conduction by numerical methods, the backward difference formula (BDF) applies to discretization of partial differentiation equation (PDE). The parallel sparse direct linear solver (PARDISO) and conjugate gradients method are comparatively applied to computation in linear algebraic equation. The simulation studies are divided into 12 case studies with three variations subjected to cooling conditions at ROT. From simulation results, the range of such three variations can be identified in relation to economic cooling system and desired quality of products.