Transient heat conduction plane walls

Transient Heat Conduction in a Plane Wall 313 Two-Dimensional Transient Heat Conduction 324... 

C Write down the one-dimensiotial transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation in its simplest form, and indicate what each variable represents. 

We start this chapter with the analysis of lumped systems in which the temperature of a body varies with time but remains uniform throughout at any time. Then we consider the variation of temperature with time as well as position for one-dimensional heat conduction problems such as those associated with a large plane wall, a long cylinder, a sphere, and a semi infinite medium using transient temperature charts and analytical solutions. Finally, we consider transient heat conduction in multidimensional systems by utilizing the product solution. 

TRANSIENT HEAT CONDUCTION IN LARGE PLANE WALLS, LONG CYLINDERS. AND SPHERES WITH SPATIAL EFFECTS... 

Therefore, the proper form of the dimensionless time is t atlL, which is called the Fonrier number Fo, and we recognize Bi = kJliL as the Biot number defined in Section 4—1. Then the formulation of the one-diniensional transient heat conduction problem in a plane wall can be expressed in nondimensional form as... 

This completes the analysts for the solution of one-dimensional transient heat conduction problem in a plane wall. Solutions in other geometries such as a long cylinder and a sphere can be determined using the same approach. The results for all three geometries arc summarized in Table 4—1. The solution for the plane wall is also applicable for a plane wall of thickness L whose left surface at, r = 0 is insulated and the right surface at.t = T. is subjected to convection since this is precisely the mathematical problem we solved. 

Transient Heat Conduction in Large Plane Walls,... 

S-73 Consider transient heat conduction in a plane wall whose left surface (node 0) i.s maintained at. >0°C while the tiglil surface (node 6) is subjeeted to a solar heal flux of 600 W/m. The wall is initially at a uniform temperature of 50°C. Express the explicit finite difference fomiulalion of the boundary nodes 0 and 6 for the case of no heal generation. Also, obtain die finite difference formulaiioti for the total amount of heat transfer at the left boundary during the first three lime steps. 

Consider transient heat conduction in a plane wall with variable heal generation and constant thermal conductivity. The nodal network of (he medium consists of nodes 0, 1, 2, 3, and 4 with a uniform nodal spacing of A.r. The wall is initially at a specified temperaWre. The temperature at the right bound ary (node 4) is specified. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary... 

Consider one dimensional transient heat conduction in a composite plane wall that consists of two layers A and ft with perfect contact at the interface. The wall involves no heat... 

On the other hand, an estimate of the response time of the slabs assembly can be easily taken as the time needed for the mid plane to undergo 99% of a sudden drop of the wall temperature. The solution of such transient heat conduction problem gives the characteristic time as (Carslaw et al., 1986, Bird et al., 1%0) ... 

Transient and steady heat conduction in a plane wall. 

The formulation of heat conduction problems for the determination of the one-dimj .nsional transient temperature distribution in a plane wall, a cylinder, or a sphefeTesults in a partial differential equation whose solution typically involves irtfinite series and transcendental equations, wliicli are inconvenient to use. Bijt the analytical soluliop provides valuable insight to the physical problem, hnd thus it is important to go through the steps involved. Below we demonstrate the solution procedure for the case of plane wall. 

Coefficients used in the one-term approximate solution of transient one-dimensional heat conduction in plane walls, cylinders, and spheres (B = hUk for a plane wall of thickness ZL, and Bi = hrjkfor a cylinder or sphere of radius r )... 

The transient temperature charts and analytical solutions presented earlier can be used to determine the temperature distribution and heat transfer in one-dimensioual heal conduction problems associated with a large plane wall, a... 

Consider transient one-dimensional heat conduction in a plane wall of thickness /. with heal generation e(x, l) that may vary with time and position and... 

I or example, in the case of transient one-dimensional heat conduction in a plane wall with specified surface temperatures, the explicit finite difference equations for all the nodes (which are interior nodes) are obtained from Eq. 5-47. The coefficient of TjJ, in the T expression is 1 - 2t, which is independent of the node number / , and thus the stability criterion for all nodes in this case is 1 — 2t s 0 or... 

C Consider transient onc-dimensional heat conduction in a plane wall that is to be solved by the explicit method. If both sides of the wall ate subjected to specified heat flux, express the stability criterion for this problem in its simplest form. 

Consider one-dimensional transient heal conduction in a plane wall with variable heat generation and variable... 

The diffusion coefficients in solids are typically very low (on the order of 10 to 10" mVs), and thus the diffusion process usually affects a thin layer at the surface. A solid can conveniently be treated as a semi-infinite medium during transient mass diffusion regardless of its size and shape when the penetration depth is small relative to the thickness of the solid. When this is not the case, solutions for one dimensional transient mass diffusion through a plane wall, cylinder, and sphere can be obtained from the solution.s of analogous heat conduction problems using the Heisler charts or one term solutions pieseiited in Chapter 4. 

Figure 3.2.19 Transient one-dimensional heat conduction in a large plane wall (thickness 21).

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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