Transient freezing of liquids in forced flow inside convectively cooled tubes

The aim of this investigation is to study the characteristics of solidification of a liquid flowing through a convectively cooled pipe under different flow situations. A mathematical model is developed by establishing an energy balance. The amount of heat transferred from the liquid layer to the freeze-front in a tube section is directly proportional to the convective heat transfer coefficient and the difference between the average temperature of the fluid at the section and the freezing point. The limiting conditions for the commencement of the solidification process are stated. The formulated set of conjugated heat transfer equations is analysed for both laminar and turbulent flows. The governing equations are solved numerically for a specific range of parameters and their characteristics of the process discussed.     

TRANSIENT THERMOVTSCOELASTIC RESPONSES: STABILITY AND ACCURACY OF FINITE-ELEMENT SOLUTIONS

This paper is concerned with the stability and accuracy of finite-element analyses of transient thermoviscoelasticity subjected to irreversible thermodynamic processes. The study includes the use of an incremental free energy that incorporates internal state variables in terms of a discretized generalized Maxwell model. Numerical integration is then performed, with past histories being updated in each discretized time step. The effects of thermomechanical coupling, internal dissipation, boundary conditions, and various temporal operators are studied with respect to spectral radii, phase errors, numerical damping, etc. A stability criterion predicted by spectral radii for combined equations of motion and heat conduction confirms the consequent numerical results for transient response according to the estimated degree of stability. Stable solutions generally lead to convergence to the exact solution, but with lower rates of convergence than those of linear-uncoupled equations. For simplicity, the numerical examples deal with one-dimensional structures.     

Discretized pressure Poisson algorithm for the steady incompressible flow on a nonstaggered grid

In the aspect of numerical methods for incompressible flow problems, there are two different algorithms: semi-implicit method for pressure-linked equations (SIMPLE) series algorithms and the pressure Poisson algorithm. This paper introduced a new discretized pressure Poisson algorithm for the steady incompressible flow based on a nonstaggered grid. Compared with the SIMPLE series algorithms, this paper did not introduce three correction variables. So, there is no need to implement the guess-and-correct procedure for the calculation of pressure and velocity. Compared with the pressure Poisson algorithm, there is no need to calculate unsteady Navier–Stokes equations for steady problems in the new discretized pressure Poisson algorithm. Meanwhile, as the finite volume method and cell-centered grid are used, the governing equation for pressure is obtained from the continuity equation and the boundary conditions for pressure are easily obtained. This new discretized pressure Poisson algorithm was tested at the lid-driven cavity flow problem on a nonstaggered grid and the results are also reliable.     

Thermal propagation in solids due to surface laser pulsation and oscillation

Most studies of heat transfer pertaining to a planar medium subjected to time-varying and spatially-decaying laser incidence along with external surface cooling are based on the diffusion theory (parabolic heat conduction equation), an approximation that implies a non-physical infinite speed of energy transport. In this study, temperature distributions within one-dimensional plates subjected to the aforementioned heating and cooling conditions, but with thermal energy propagation at a finite speed, are presented. Incident energy that is partially absorbed at the external surface is transferred through the plate by conduction, while the remaining energy is convectively cooled to the environment. The present investigation will examine three different time characteristics of the incident heat sources which include a continuously operating, a pulsed and an oscillatory laser source. The temperature results were obtained by using a simple and concise finite-difference algorithm based on the Godunov scheme, a method developed for the solution of resulting characteristic equations that govern thermal wave propagations within the solid.     

Free boundary shape of a convectively cooled solidified region

The two-dimensional steady-state shape of a solidified region, such as a frost layer, was determined analytically for formation on a plate that is convectively cooled. The nonuniform shape of the layer is produced by exposure to a spatially nonuniform distribution of radiant energy. For high convective cooling the cooled wall approaches a uniform temperature, and an exact solution is obtained for the free boundary shape. For a lesser amount of convective cooling, the variation in temperature along the cooled boundary is treated by a boundary perturbation method. Some illustrative examples are given that show the effects of nonuniform heating and the magnitude of convective heat transfer at the cooled wall. Only one boundary condition is approximated by the perturbation solution ; all of the other boundary conditions are satisfied exactly. The calculated results given here were found to satisfy the approximate boundary condition within a very small error.     

Phase-change front prediction by measuring the wall temperature on which solidification occurs

Experimental data for the liquid-solid interface position as a function of time and the wall temperature of the convectively cooled tube on which freezing occurs are obtained and compared with two theoretical predictions. These comparisons show that approximate values of the phase-change front can be estimated by measuring the surface temperature on which freezing occurs and using the data in a simple formula derived on the basis of a quasi-steady state assumption. For more accurate predictions, however, a numerical procedure based on the optimization technique is needed.     

Cooling system optimisation of turbine guide vane

This paper discuses the problem of cooling system optimisation within a gas turbine vane. The analysis involves the optimisation of size and location of internal cooling passages within the vane. Cooling is provided with ten circular passages and heat is transported only convectively. The task is approached in 3D configuration. Each passage is fed with cooling air of constant parameters at the inlet. Also a constant pressure drop is assumed along the passage length. The thermal boundary conditions in passages varied with diameter and local vane temperature (passage wall temperature). The analysis is performed by means of the evolutionary approach for the optimisation task and FEM for the heat transfer predictions within the component. The optimisation is realised with genetic algorithm where two ways of individual (cooling system candidate) representations are used and juxtaposed. These are the classical binary representation of the design variables and the floating point form. The results show some potential stored in a vane cooling system. Appropriate passage distribution makes it possible to improve the operation condition for highly loaded thermal components. Also the comparison between binary versus floating point representation of the design variables show some superiority of the latter.     

Computation of Turbulent Natural Convection in a Rectangular Cavity with the Lattice Boltzmann Method

A numerical study of a turbulent natural convection in a rectangular cavity with the lattice Boltzmann method (LBM) is presented. The primary emphasis of the present study is placed on investigation of accuracy and numerical stability of the LBM for the turbulent natural-convection flow. A HYBRID method in which the thermal equation is solved by the conventional Reynolds-averaged Navier-Stokes equation (RANS) method while the conservation of mass and momentum equations are resolved by the LBM is employed in the present study. The elliptic-relaxation model is employed for the turbulence model and the turbulent heat fluxes are treated by the algebraic flux model. All the governing equations are discretized on a cell-centered, nonuniform grid using the finite-volume method. The convection terms are treated by a second-order central-difference scheme with the deferred correction method to ensure accuracy and stability of solutions. The present LBM is applied to the prediction of a turbulent natural convection in a rectangular cavity and the computed results are compared with the experimental data commonly used for the validation of turbulence models and those by the conventional finite-volume method. It is shown that the LBM with the present HYBRID thermal model predicts mean velocity components and turbulent quantities which are as good as those by the conventional finite-volume method. It is also found that the accuracy and stability of the solution is significantly affected by the treatment of the convection term, especially near the wall.     

Ångström methods applied to simultaneous measurements of thermal diffusivity and heat transfer coefficients: Part 1, theory

We present a method for simultaneous measurement of thermal diffusivity and local heat transfer coefficient based on a theory originally stated by Ångström. We apply a sinusoidally varying thermal flux incident on one face of a one-dimensional specimen and convectively cool its opposite face. This results in a sinusoidally varying temperature on the cooled face with a measurable phase lag between the incident and transmitted waves that depends upon the material properties and the heat transfer coefficient.     

Optimization of a magnetohydrodynamic flow based on the entropy generation minimization method

The entropy generation minimization method is applied to the optimization of a magnetohydrodynamic flow between two infinite parallel walls of finite electrical conductivity. The aim is to minimize the total energy loss due to irreversibilities produced by heat conduction, viscosity and Joule dissipation. The analytic solutions for the velocity, temperature and electric current density fields are used to calculate the global entropy generation rate explicitly in terms of the dimensionless parameters of the problem. It is shown that the entropy generation reaches a minimum when the walls that contain the flow are cooled convectively in an asymmetric way. In order to get a deeper insight of the physical process, a detailed analysis of the different dissipation sources is carried out.     

MULTIDISCIPLINARY FLOW/THERMAL AND INDUCED STRESSES IN CONVECTIVELY COOLED STRUCTURES

A multidisciplinary finite element methodology with stabilizing features to prevent undue oscillatory solution behavior for the velocities, pressure, and temperature fields and which subsequently permits computations of the resulting thermal loads for the associated stress analysis is described for convectively cooled structures subjected to high-intensity localized heating. Of particular interest are the influences of coolants that serve to cool the structure whose exposed skin is permitted to radiate to outer space. Of the three coolants investigated - namely, liquid hydrogen, water, and liquid sodium - it is observed that the liquid sodium serves as an effective coolant that is consistent with past related studies. The resulting thermally induced stresses arising from the assumption of elastic and a materially nonlinear elastoplastic model are also evaluated. The nonlinear model seems more realistic because of the situations encountered at high temperatures and as expected yields lower values of the stresses. Illustrative examples of a flat skin structure and a curved skin geometry representing the cowl leading edge are analyzed for the flow/thermal and induced stresses.     

Mathematical Model for Predicting Solidification and Cooling of Steel Inside Molds and in Air

A two-dimensional mathematical model is presented to describe the solidification and cooling of liquid steel. The liquid steel is poured into a mold to obtain a solid mass of desired shape, called an ingot. After cooling of the steel in the mold for some time, the mold is removed. Then the leftover ingot mass is cooled in air. This article is concerned with the above process. Nevertheless, the technique can be very applicable to other processes such as continuous casting.

Partial differential equations describing the process have been discretized using control-volume (or finite-volume) technique. The discretization equations obtained are of tridiagonal matrix form, which have been solved using the well-known tridiagonal matrix algorithm (TDMA) and the alternate direction implicit (ADI) solver. The model has been validated by measuring surface temperatures of molds and ingots using an infrared thermo-Vision scanner. This is then used to compute temperature distribution and solidification status of the ingot as a function of time and type of ingot.     

A MESHLESS FORMULATION FOR THREE-DIMENSIONAL LAMINAR NATURAL CONVECTION

A diffuse approximation method (DAM) for three-dimensional, incompressible, viscous fluid flow is presented. The method works directly with primitive variables. The discretized equations are solved using a first-order-in-time, implicit projection algorithm. The proposed method is verified by applying it to the steady three-dimensional differentially heated cubic cavity. The results are compared to those of numerical and experimental investigations in the literature.     

Thermally nonlinear generalized thermoelasticity of a layer

This article addresses a clarification study on the thermally nonlinear Fourier/ non-Fourier dynamic coupled (generalized) thermoelasticity. Based on the Maxwell-Cattaneo’s heat conduction law and the infinitesimal theory of thermoelasticity, governing equations for the thermally nonlinear small deformation type of generalized thermoelasticity are derived. The Bubnov–Galerkin scheme is implemented for spatial discretization. The spatially discretized equations are directly discretized in time domain using the fully damped Newmark method. The Newton–Raphson procedure is used to linearize the finite element equations. The layers are exposed to a thermal shock, so that the displacement, temperature, and stress waves propagate in layers. The effects of the time evolution, thermoelastic coupling, and thermal relaxation time on the response of the layers are investigated. Results reveal the significance of the thermally nonlinear analysis of generalized thermoelasticity for the conditions where large temperature elevations exist.     

A Novel Volume-of-Solid-Based Immersed-Boundary Method for Viscous Flow with a Moving Rigid Boundary

The present study adopts a volume-of-solid (VOS)-based immersed-boundary method (IBM) to simulate viscous incompressible flows interacting with moving solids using a numerical model based on the Navier-Stokes equations. The flow equations, with velocity–pressure variables, are discretized by a finite–element method on a nonuniform Cartesian grid, with the solutions obtained using a decoupled numerical scheme. Geometries featuring flexible solid obstacles in the flow are embedded in the Cartesian grid, with special treatments inside the embedded cell to ensure the accuracy of the solutions in the cut cells. In order to satisfy the no-slip condition of the body surface, a volume fraction is estimated to calculate the discretized body force inside the cut cell. More reasonable results for flow problems, including flows past a non-control/control circular cylinder and two cylinders moving against each other, are obtained by the present method. The time histories of drag and lift coefficients, as well as the vortex shedding frequencies, are extensively examined to demonstrate that the proposed method can be suitably combined with the fractional-step algorithm. Moreover, the temporal variations of velocity and vorticity fields are presented to demonstrate the capability of the present formulation in solving flow problems involving complex geometries, and the significance of the solid body forces on the flows.     

CO-LOCATED EQUAL-ORDER CONTROL-VOLUME FINITE-ELEMENT METHOD FOR MULTIDIMENSIONAL,INCOMPRESSIBLE. FLUID FLOW -PART I: FORMULATION

A co-located equal-order control-volume-based finite-element method (CVFEW) for two- and three-dimensional, incompressible, viscous fluid flow is presented. The method works directly with the primitive variables. Triangular elements and polygonal control volumes, and tetrahedral elements and polyhedral control volumes are used to discretize the calculation domains in two- and three-dimensional problems, respectively. Two available flow-oriented upwind schemes (FLO and FLOS) and a novel mass-weighted skew upwind scheme (MAW) are investigated. In each dement, the velocity components in the mass flux terms are interpolated by special functions that prevent the generation of spurious pressure oscillations. The discretized equations are solved using an iterative sequential variable adjustment algorithm. Verification of the proposed CVFEM is presented in a companion article.     

A Multidomain Chebyshev Pseudo-Spectral Method for Fluid Flow and Heat Transfer from Square Cylinders

A simple multidomain Chebyshev pseudo-spectral method is developed for two-dimensional fluid flow and heat transfer over square cylinders. The incompressible Navier-Stokes equations with primitive variables are discretized in several subdomains of the computational domain. The velocities and pressure are discretized with the same order of Chebyshev polynomials, i.e., the P_N-P_N method. The Projection method is applied in coupling the pressure with the velocity. The present method is first validated by benchmark problems of natural convection in a square cavity. Then the method based on multidomains is applied to simulate fluid flow and heat transfer from square cylinders. The numerical results agree well with the existing results.     

Implementation of boundary conditions in pressure-based finite volume methods on unstructured grids

In this study, the implementation of boundary conditions for the Navier–Stokes and the energy equations, including the pressure and pressure correction equations, are presented in the context of finite volume formulation on cell-centered, colocated unstructured grids. The implementation of boundary conditions is formulated in terms of the contribution of boundary face of a cell to the coefficients of the discretized equation for either Dirichlet- or Neumann-type boundary conditions. Open boundaries through which the flow is not fully developed are also considered. In this case, a data reconstruction method is proposed for finding the boundary values of the variables at the correction stage. The validity of implementations is checked by comparing the results with some well-known benchmark problems.     

Numerical simulation and experimental validation of a helical double-pipe vertical condenser

A predictive model is developed to describe heat transfer and fluid dynamic behavior of a helical double-pipe vertical condenser used in an absorption heat transformer integrated to a water purification process. The condenser uses water as working fluid connected in countercurrent. Heat transfer by conduction in the internal tube wall is considered; in addition the change of phase is carried out into the internal tube. The dynamic model considers equations of continuity, momentum and energy in each flow. The discretized governing equations are coupled using an implicit step by step method. Comparison of the numerical simulation over range of experimental data presented in the heat device is applied to validate the model developed. The model is also evaluated of form dynamic to determine the principal operation variables that affect the condenser with the main objective to optimize and control the system. A variation of mass flow rate in the internal pipe induces important changes on the heat flux that the pressure and temperature.     

Entropy analysis of unsteady MHD three-dimensional flow of Williamson nanofluid over a convectively heated stretching sheet

This article addresses an investigation of the entropy analysis of Williamson nanofluid flow in the presence of gyrotactic microorganisms by considering variable viscosity and thermal conductivity over a convectively heated bidirectionally stretchable surface. Heat and mass transfer phenomena have been incorporated by taking into account the thermal radiation, heat source or sink, viscous dissipation, Brownian motion, and thermophoretic effects. The representing equations are nonlinear coupled partial differential equations and these equations are shaped into a set of ordinary differential equations via a suitable similarity transformation. The arising set of ordinary differential equations was then worked out by adopting a well-known scheme, namely the shooting method along with the Runge-Kutta-Felberge integration technique. The effects of flow and heat transfer controlling parameters on the solution variables are depicted and analyzed through the graphical presentation. The survey finds that magnifying viscosity parameter, Weissenberg number representing the non-Newtonian Williamson parameter cause to retard the velocity field in both the directions and thermal conductivity parameter causes to reduce fluid temperature. The study also recognizes that enhancing magnetic parameters and thermal conductivity parameters slow down the heat transfer rate. The entropy production of the system is estimated through the Bejan number. It is noticeable that the Bejan number is eminently dependent on the heat generation parameter, thermal radiation parameter, viscosity parameter, thermal conductivity parameter, and Biot number. The skillful accomplishment of the present heat and mass transfer system is achieved through the exteriorized choice of the pertinent parameters.