Hou-Cheng Huang

Finite Element Analysis for Heat Transfer

The use of Finite Elements for the modelling of heat transfer problems is here described for the first time. Applicable in Mechanical, Civil as well as in Materials engineering, it copes with heat transfer in virtually any medium including polymers and ceramics. The essential theory is covered and full implementational details given including two FORTRAN programs. HEAT2D, a program capable of steady and transient, linear and nonlinear analyses of diffusive ,and convective heat transfer and its adaptive version HADAPT are also presented with full instructions and documented examples. The complete source code for both programs and sample input data files are provided on a floppy disc included with the book. The programs can be run on any computer with a FORTRAN 77 compiler.

Temporal Discretisation for Heat Conduction

Problems where the temperature field at various points in the domain varies with time are referred to as transient problems, as opposed to steady state problems, where the temperature remains constant at a given point in the domain, for all times. In structural mechanics transient problems are analogous with dynamics and steady state problems are analogous with statics. The finite element discretisation discussed in the previous chapter was limited to the heat conduction equations without the term containing the temporal derivative. Although, most real life heat transfer problems are time-dependent, for many engineering problems it is sufficient to calculate a steady spatial temperature field. For example, the temperature field for electrical or mechanical machinery in operational conditions may be calculated as a steady state problem governed by the steady heat conduction equation and appropriate boundary conditions, using the procedure outlined in the previous chapter. However, there are other problems where the transient effects cannot be ignored. For example, it may be required to calculate the temperature field for machinery subjected to time-dependent or cyclic thermal loading. Other examples are phase change problems (solidification, melting etc.). For such problems the complete heat conduction equations including the temporal derivative term must be used. Therefore, a temporal discretisation of the transient heat conduction equations is required in addition to the spatial discretisation.

Finite Element Analysis for Transient Non-Newtonian Flow

In this chapter a finite element method for analysing transient non-Newtonian flow is presented. Examples of transient non-Newtonian flows occur widely in industry in many situations e.g. extrusion, spinning, injection and blow moulding. Numerical simulation of such problems is facilitated through the theory embodied in Computational Fluid Dynamics. This chapter provides some insight into how this may be achieved for the generalised non-Newtonian flows by employing finite element methods in a transient solution process. The method chosen here is a Taylor-Galerkin Pressure correction method [1] which belongs to a large class of methods using uncoupled or segregated approaches (where velocity and pressure variables are uncoupled, as opposed to the mixed methods discussed earlier). The basic idea for segregated solutions stems from the work of Chorin [2]. Two main variants of this method have matured simultaneously in finite element circles over the past decade. The first method preferred by Gresho et. al. [3] has come to be known as the fractional step method, an appellation given by Donea et. al. [4]. The second method is sometimes referred to (as done here) as the velocity correction method according to Kawahara and Ohmiya [5] (the Taylor-Galerkin pressure correction method used here is a variant of this). The essential difference between the two methods is that for the former the segregation (of velocity and pressure) is effected after the GFEM discretization of the differential equations, while for the latter, it happens at the differential equation stage. The segregated methods have also been identified as projection methods [2,6]. This can be seen by writing the Navier Stokes equations (6.4) and (6.5) as, n n

Steady Non-Newtonian Flow

frac{{partial {text{v}}}}{{partial t}} + nabla P = f({text{v}})

Adaptive Heat Transfer Analysis

n n(6.1) n nand n n

Finite Element Analysis of Non-Newtonian Flow

nabla cdot {rm v} = 0