Placa De 9 Nodos

UNIVERSIDAD AUTONOMA DE YUCATAN FACULTAD DE INGENIERIA QUIMICA |
PROYECTO METODOS NUMERICOS |
PROBLEMA DE 9 NODOS (PLACA PLANA)MATERIA : METODOS NUMERICOS PROFESOR : MIGUEL ANGEL CAN EK |
JOSE EMMANUEL KU BONILLA |
8 DE MAYO 20012 |

Este problema de transferencia de calor esta basado en un ejercicio del libro de transferencia de calor de J.P. Holman que es acerca de losfenómenos de conducción del calor por placas planas en diferentes dimensiones plano (x y) y como para este tipo de problemas se utiliza el método de los nodos para poder hallar el calor en la placa bidimensional. |

Steady-state heat transfer was calculated in systems in which the temperature gradient and area could be expressed in terms of one space coordinate. We now wish to analyze the moregeneral case of two-dimensional heat flow. For steady state with no heat generation, the Laplace equation applies.

δ2 Tδx2+ δ2 Tδy2=0 1-1

Assuming constant thermal conductivity. The solution to this equation may be obtained by analytical, numerical, or graphical techniques. The objectiveof any heat-transfer analysis is usually to predict heat flow or the temperature that results from a certain heat flow. The solution to Equation (1-1) will give the temperature in a two-dimensional body as a function of the two independent space coordinates x and y. Then the heat flow in the x and y directions may be calculated from the Fourier equations

qx=KAxδTδx1-2

qy=KAyδTδy 1-3

These heat-flow quantities are directed either in the x direction or in the y direction. The total heat flow at any point in the material is the resultant of the qx and qy at thatpoint. Thus the total heat-flow vector is directed so that it is perpendicular to the lines of constant temperature in the material, as shown in Figure 1. So if the temperature distribution in the material is known, we may easily establish the heat flow.

Figure 1.Sketch showing the heat flow in two dimensions

Consider a two-dimensional body that is to be divided into equal increments in boththe x and y directions, as shown in Figure 2. The nodal points are designated as shown, the m locations indicating the x increment and the n locations indicating the y increment. We wish to establish the temperatures at any of these nodal points within the body, using Equation (1-1) as a governing condition. Finite differences are used to approximate differential increments in the temperature andspace coordinates; and the smaller we choose these finite increments, the more closely the true temperature distribution will be approximated.

Figure 2. Sketch illustrating nomenclature used in two dimensional numerical analysis of heat conduction.

The temperature gradients may be written as follows:
δTδxm+1/2,n≈Tm+1,n-Tm,n∆x
δTδxm-1/2,n≈Tm,n-Tm-1,n∆x
δTδym,n+1/2≈Tm,n+1-Tm,n∆yδTδym,n-1/2≈Tm,n-Tm,n-1∆y
δ2Tδx2m,n≈δTδxm+12,n-δTδxm-12,n∆x=Tm+1,n+Tm-1,n-2Tm,n(∆x)2
δ2Tδy2m,n≈δTδym,n+12,-δTδym,n-12,∆y=Tm,n+1+Tm,n-1,-2Tm,n(∆y)2
Thus the finite-difference approximation for Equation (1-1) becomes
Tm+1,n+Tm-1,n-2Tm,n(∆x)2+Tm,n+1+Tm,n-1,-2Tm,n(∆y)2=0
If ∆x = ∆y, then
Tm+1,n+Tm-1,n+Tm,n+1+Tm,n-1-4Tm,n=0 1-4
Since weare considering the case of constant thermal conductivity, the heat flows may all be expressed in terms of temperature differentials. Equation (1-4) states very simply that the net heat flow into any node is zero at steady-state conditions. In effect, the numerical finite-difference approach replaces the continuous temperature distribution by fictitious heat-conducting rods connected between...