Heat conduction equation plane wall

Thermal Conductivity of Laminar Composites. In the case of laminar composites or layered materials (cf. Figure 1.74), the thermal conductance can be modeled as heat flow through plane walls in a series, as shown in Figure 4.36. At steady state, the heat flux through each wall in the x direction must be the same, qx, resulting in a different temperature gradient across each wall. Equation (4.2) then becomes... 

Heat Conduction Equation in a Large Plane Wall... 

FIGURE 2-14 The simpUficaiion of the onedimensional heat conduction equation in a plane wall for the case of constant condticliviiy for steady conduction with no hear generation. 

An examination of the one dimensional tiansient heat conduction equations for the plane wall, cylinder, and sphere reveals that all tlu-ee equations can be expressed in a compact form as... 

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. 

Overall conductance (overall heat-transfer coefficient) n. In heat-transfer engineering, the reciprocal of the total thermal resistance, for heat flow through plane walls or tube walls. It is defined by the equation U = q/AAT, where q is the rate of heat flow through (and normal to) the surface of area A, and AT is the fall in temperature through the layer in the direction of q. This is a modification of Fomier s law, invented to deal conveniently with heat flow through stagnant fluid films adjacent... 

Overall Conductance n (overall heat-transfer coefficient) In heat-transfer engineering, the reciprocal of the total Thermal Resistance, for heat flow through plane walls or tube walls. It is defined by the equation ... 

Consider the plane wall with uniformly distributed heat sources shown in Fig. 2-8. The thickness of the wall in the x direction is 2L, and it is assumed that the dimensions in the other directions are sufficiently large that the heat flow may be considered as one-dimensional. The heat generated per unit volume is q, and we assume that the thermal conductivity does not vary with temperature. This situation might be produced in a practical situation by passing a current through an electrically conducting material. From Chap. 1, the differential equation which governs the heat flow is... 

We start this chapter with a description of steady, unsteady, and multidimensional heat conduction. Then we derive the differential equation that governs heat conduction in a large plane wall, a long cylinder, and a sphere, and generalize the results to three-dimensional cases in rectangular, cylindrical, and spherical coordinates. Following a discussion of the boundary conditions, we present tlie formulation of heat conduction problems and their solutions. Finally, we consider lieat conduction problems with variable thermal conductivity. 

Consider a 20-cm-ihick large concrete plane wall k 0.77 V/in °C) subjected to convection on both sides with r, = 27"C and A, = 5 W/m °C on the inside, and = 8°C and A2 = 12 W/m °C on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, [a) express the differential equations and the boundary conditions for steady one-dimensional heal conduction through the wall, (A) obtain a relation for the variation of temperature in the wall by solving the differential equation, and (c) evaluate the temperatures at the inner and outer surfaces of the wall. 

Equation 3-3 for heat conduction through a plane wall can be rearranged as... 

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. 

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... 

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. 

The temperature field of the solid phase can change due to heat conduction or due to heat transfer from the fluid phase. For the fluid, heat conduction does not have to be taken into account since the flow domain consists of disconnected channels but convective heat transfer due to the flow velocity as well as heat transfer from the solid is included. In the x-y plane no conduction within the fluid without intermediate transfer to the solid walls can occur. The reason for not including heat conduction in the z-direction is the fact that in most cases of practical relevance it is negligible when compared to convective transport. Following the derivation of Hardt and Baier [46], the fluid and solid temperature fields Tf and f are obtained as solutions of the equations... 

The equation for the one-dimensional steady heat conduction is Fourier s first law, Eq. (3.1.53). For a plane wall with thickness d and a temperature Tj on one side and a lower temperature T2 on the opposite side we obtain ... 

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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