Finite difference formulation nodes

C Define these terms used in the finite difference formulation node, nodal network, volume element, nodal spacing, and difference equation. 

Note that the boundary conditions have no effect on the finite difference formulation of interior nodes of the medium. This is not surprising since the control volume used in the development of the formulation does not involve any part of the boundary. You may recall that the boundary conditions had no effect on the differential equation of heat conduction in the medium either. 

In finite difference formulation, the temperature is assumed to vary linearly between the nodes. 

FIGURE 5-14 Schematic for the finite difference formulation of the left boundary. node of a plane wall. 

One way of obtaining (he finite difference formulation of a node on an insulated boundary is to treat insulation as "zero heat flux and to write an energy balance, as done in Eq. 5 -23. Another and more practical way is to treat the node on an insulated boundary as an imerior node. Conceptually (his is done... 

Node 1 is an interior node, and the finite difference formulation at that node is obtained directly from Eq. 5-18 by setting m = 1 ... 

That is, the finite difference formulation of an interior node is obtained by adding the temperaiures of the four nearest neighbors of the node, subtructing four limes the temperature of the node itself and adding the heat generation lerm. It can also be expressed in this form, which is easy to remember ... 

The development of finite difference formulation of boundary nodes in two- (or three-) dimensional problems is similar to the development in the one-dimensional case discussed earlier. Again, the region is partitioned between the nodes by forming volume elements around the nodes, and an energy balance is written for each boundary node. Various boundary conditions can be handled as discussed for a plane wall, except that the volume elements ill the two-dimensional case involve heat transfer in the y-direction as well as the x-direction. Insulated surfaces can still be viewed as mirrors, and the... 

A) Node 8. This node is identical to node 7, and the finite difference formulation of this node can be obtained from that of node 7 by shifting the node numbers by 1 (i.e., replacing subscript m by rr + 1). It gives... 

Noie that in the case of no heal generation and t = 0.5, the explicit finite difference formulation for a general interior node reduces to T , = (T/,-1 +, )/2, which has the interesting interpretation that the temperature... 

To gain a better understanding of the stability criterion, consider the explicit finite difference formulation for an interior node of a plane wall (Eq. 5 47) for the case of no heat generation,... 

Node 2 is a boundary node subjected to convection, and the implicit finite difference formulation at that node can be obtained from this formulation by expressing the left side of the equation at time step / -4 1 instead of i as... 

The exterior surface of the Trombe v/ail is subjected to convection as well as to heat flux. The explicit finite difference formulation at that boundary is obtained hy writing an energy balance on the volume element represented by node 5,... 

This completes the finite difference formulation of the problem. Next we need to determine the upper lirnit of the time step Af from the stability criterion, which requires the coefficient of T in the expression (the primary coefficient) to be greater than or equal to zero for all nodes. The smallest primary coefficient In the nine equations here is the coefficient of ti in the expression, and thus the stability criterion (or this problem can be expressed as... 

The finite difference formulation of transieiii heat conduction problems is based on an energy balance that also accounts for tire variation of the energy content of the volume element during a time interval At. The heat transfer and heat generation terms are expressed at the previous time. step fin the explicit method, and at the new time step i I 1 in the implicit method. For a general node III, the finite difference formulations are expressed as... 

The finite difference formulation at node 0 at the left bound-aiy of a plane wall for steady one-dimensional heat conduction can be expres.sed as... 

Using the Tmile difference form of the first derivative (not the energy balance approach), obtain the finite difference formulation of the boundary nodes for the case of insulation at the left boundary (node 0) and radiation at the right boundary (node 5) with an emissiviiy of e and surrounding temperature of... 

S-14C How can a node on an insulated boundary be treated as an interior node in the finite difference formulation of a plane wall Explain. 

Consider steady heat conduction in a plane wall whose left. surface (node 0) is maintained at 30°C while the right surface (node 8) is subjected to a heat flux of 1200 W/mT Express the finite difference formulation of the boundary nodes 0 and 8 for the case of no heal generation. Also obtain the finite dif-... 

Consider steady one-dimensional beat conduction in a composite plane wall consisting of iwo layers A and B in perfect contact at the interface. The wall involves no heat generation. The nodal network of the medium consisl.s of nodes 0, 1 (at Ihe interface), and 2 with a uniform nodal spacing of A.x. Using the energy balance approach, obtain Ihe finite difference formulation of this problem for the case of insulation at the left... 

Consider steady one-dimensional heat conduction in a pin fin of constant diajneter D with constant thermal conductivity. The fm is losing heal by conveclion to the ambient air at T with a heat transfer coefficient of A. The nodal network of the fm consists of nodes 0 (at Ihe base), 1 (in the middle), and 2 (at the fin tip) with a uniform nodal spacing of Ax. Using the energy balance approach, obtain the finite difference formulation of (his problem to determine T, and T2 for Ihe case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in C. 

Consider a large plane wall of thickness L = 0.3 m, llietnial conductivity k = 2.5 W/m - C, and surface area A = 24 m. The left side of the wall is subjected to a heat flux of q o -- 350 W/m while the temperature at that surface i.s measured to be To = 60°C. Assuming steady oiic-dimensional heat transfer and taking the nodal spacing to be 6 cm, a) obtain the finite difference formulation for the six nodes and (f>) determine the temperature of the other surface of the wall by solving those equations. 

C 1 he explicit finite difference formulation of a general interior node for transient two-dimensional heat conduction is given by... 

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