Plane walls energy balance

Consider the control volume bounded by the planes 1, 2, A-A, and the wall as shown in Fig. 5-8. It is assumed that the thermal boundary layer is thinner than the hydrodynamic boundary layer, as shown. The wall temperature is 7 ,., the free-stream temperature is Tx, and the heat given up to the fluid over the length dx is dqw. We wish to make the energy balance... 

Consider a thin element of thickness Ax in a large plane wall, as shown in Fig. 2-13. Assume the density of the wall is p, the specific heat is c, and the area of the wall normal to the direction of heat transfer is A. An energy balance on this thin element during a small lime interval Af can be expressed as... 

Starting with an energy balance on a rectangular volume element, derive the one-dimensional transient heal conduction equation for a plane wall with constant thermal conductivity and no heat generation. 

Above we have developed a general relation for obtaining the finite difference equalion for each interior node of a plane wall. This relation is not applicable to the nodes on the boundaries, however, since it requires the presence of nodes on both sides of the node under consideration, and a boundary node does not have a neighbor ing node on at least one side, Therefore, we need to obtain the finite difference equations of boundary nodes separately. This is best done by applying an energy balance on the volume elements of boundary nodes. 

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... 

Consider steady heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0, 1,2, 3, and 4 with a uniform nodal spacing of A.t. Using the energy balance approach, obtain the finite difference formulation of the boundary nodes for the case of uniform heat flux qa at the left boundary (node 0) and convection at the right boundary (node 4) with a conveclion coefficient of h and an ambient temperature of T. ... 

The heating of a viscous fluid in laminar flow in a tube of radius R (diameter, D) will now be considered. Prior to the entry plane z < 0), the fluid temperature is uniform at Tf for z > 0, the temperature of the fluid will vary in both radial and axial directions as a result of heat transfer at the tube wall. A thermal energy balance will first be made on a differential fluid element to derive the basic governing equation for heat transfer. The solution of this equation for the power-law and the Bingham plastic models will then be presented. 

For a transient heat transfer process, for example, for heating up or cooling down a body, we have to consider the variation of temperature with time as well as with position. For a large plane wall of thickness 21, the heat conduction perpendicular to the (almost infinite) area A of the plate is one-dimensional. To derive the respective differential equation, we use the energy balance for a small slice with thickness Ax and volume A Ax (Figure 3.2.19) ... 

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