Steady heat conduction plane walls

Consider steady heat conduction through a large plane wall of thickness Alv = L and area A, as shown in Fig. 1 22. The temperature difference across the wall is AT =7 — 7V Experiments have shown that the rale of heat transfer Q through the wall is doubled when the temperature difference AT" across the wall or the area A normal to the direction of heat transfer is doubled, but is halved when the wall lliickness L is doubled. Thus we conclude that the rate of heat conduction through a plane layer is proportional to the temperature difference across the layer and the heal transfer area, but is inversely proportional to the thickness of the layer. That is. 

Transient and steady heat conduction in a plane wall. 

Write an interactive computer program to calculate the heat transfer rate and the value of temperature anywhere in the medium for steady one-dimensional heat conduction in a plane wall whose thermal conductivity varies linearly as k(T) l o(l + pT) wliere the constants k(, and p are specified by the user for specified temperature boundary conditions. 

We stait this chapter with one-dimensional steady heat conduction in a plane wall, a cylinder, and a sphere, and develop relations for thennal resistances in these geometries. We also develop thermal resistance relations for convection and radiation conditions at the boundaries. Wc apply this concept to heat conduction problems in multilayer plane wails, cylinders, and spheres and generalize it to systems that involve heat transfer in two or three dimensions. We also discuss the thermal contact resislance and the overall heat transfer coefficient and develop relations for the critical radius of insulation for a cylinder and a sphere. Finally, we discuss steady heat transfer from finned surfaces and some complex geometries commonly encountered in practice through the use of conduction shape factors. 

Consider a plane wall of thickness L and average thermal conductivity k. The Isvo surfaces of the wall are maintained at constant temperatures of r, and T2. For one-dimensional steady heat conduction through the wall, we have 7(.v). Then Fourier s law of heat conduction for the wall can be expressed as... 

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-... 

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 ... 

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... 

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. 

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. 

SOLUTION A plane wall with specified surface temperatures is given. The variation of temperature and the rate of heat transfer are to be determined. Assumptions 1 Heat conduction is steady. 2 Heat conduction is one-dimensional since the wall is large relative to its thickness and the thermal... 

Consider steady one-dimensional heat conduction in a large plane wall of thickness I. and constant thermal conductivity Icwith no heat generation. Obtain expressions for the variation of temperature v/ithin the wall for the following pairs of boundary conditions (Fig. 2-44) ... 

Consider a large plane wall of thickness L = 0.06 m and thermal conductivity k = 1.2 W/m C in space. The wall is covered with white porcelain tiles that have an emissivity of e = 0.85 and a solar absorptivity of a = 0.26, as shown in Fig. 2-48. The inner surface of the v/all is maintained at Ti = 300 K at all times, while the outer surface Is exposed to solar radiation that is incident at a rate of 800 W/m. The outer surface is also losing heal by radiation to deep space at 0 K. Determine the temperature of the outer surface of the wall and the rate of heat transfer through the wall when steady operating conditions are reached. What would your response be if no solar radiation was incident on the surface ... 

Then the rate of steady heat transfer through a plane wall, cylindrical layer, or spherical layer for the case of variable thermal cuuductivity can be determined by replacing the constant thermal conductivity k in iiqs. 2-57, 2-59, and 2-61 by the expression (or value) from Eq. 2-75 ... 

We have mentioned earlier that in a plane wall the temperature varies linearly during steady one-dimensional heat conduction when the thermal conductivity is constant. But this is no longer the case when the thermal conductivity changes with temperature, even linearly, as shown in Fig. 2-63. 

Consider a plane wall of thickness L whose thermal conductivity varies linearly in a specified temperature range as k T) = kad + PT) where kg and p are constants. The wall surface at x = 0 is maintained at a constant temperature of 7i while the surface at r = (.is maintained at Tj, as shown in Fig. 2-64. Assuming steady one-dimensional heat transfer, obtain a relation for (a) the heat transfer rate through the wall and [b) the temperature distribution 7(x) in the wall. 

C Consider one-dimensional heat conduction through a large plane wall with no heat generation that is perfectly irtsu-lated on one side and is subjected to convection and radiation on the other side. It is claimed that under steady conditions, the temperaltire in a plane wall must be uniform (the same everywhere). Do you agree with this claim Why ... 

S2C It is stated that the temperature in a plane wall with constant thermal conductivity and no heat generation varies linearly during steady one-dimensional heat conduclion. Will this still be the case when the wall loses heat by radiation from its surfaces ... 

C Consider steady oac-dimeosional heal conduction in a plane wall, long cylinder, and sphere with constant thermal conductivity and no heat generation. Will the temperature in any of these mediums vary linearly Explain. 

C Consider steady one-dimensional heat conduction in a plane wall in which the thermal conductivity varies linearly. The error involved in heat transfer calculations by assuming constant thermal conductivity at the average temperature is (a) none, [b) small, or (c) signiricam. 

C The temperature of a plane wall during steady onedimensional heat conduction varies linearly when the thermal conductivity is constant. Is this still the ca.se when the thermal conductivity varies linearly with temperature ... 

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. 

Consider steady one-dimensional heat conduction through a plane wall, a cylindrical shell, and a spherical shell of unifonn thickness with constant thcimophysical propenies and no thermal energy generation. The geometry in which the variation of temperature in the direction of heal transfer will be linear is... 

Consider a large plane wall of thickness L, thermal conductivity k, and surface area A. The left surface of the wall is exposed to the ambient air at T. with a heat transfer coefficient of h while the right surface is insulated. The variation of temperature in (he wall fur steady one-dimensional heat conduction with no heat generation is... 

Now consider steady one-dimensional heat transfer through a plane wall of thickness L, area A, and thermal conductivity k that is exposed to convection on both sides to fluids at temperatures ro,i and T 2 with heat transfer coefficients /i and hj, respectively, as shown in Fig. 3-6. Assuming T i < < i> variatiqji of temperature will be as shown in the figure. Note that the temperature va es linearly in the wall, and a.symptotically approaches r , and J 2 die fluids we move away from the wall. 

In practice we often encounter plane walls that consist of several layers of different materials. The tbermal resistance concept can still be used to detennine the rate of steady heat transfer through such composite walls. As you may have already guessed, this is done by simply notiifg that the conduction resistance of each wall i.s IJkA connected in series, and using the electrical analogy. That is, by dividing the temperature difference between two surfaces at known temperatures by the total thermal resistance between them. 

Now consider steady one-dimensional heat conduction in a plane wall of thickness L with heat generation. The wall is subdivided into M sections of equal thickness Ax UM in the x-direction, separated by planes passing... 

To demonstrate the approach, again consider steady one-dimeiisional heal transfer in a plane wall of thickness L with heat generation. r) and constant conductivity k. The wall is now subdivided into M equal regions of thickness Ax = UM in the x-direction, and the divisions between the regions are selected as the nodes. Therefore, we have A/ + I nodes labeled 0, 1, 2,..., m, m,m + 1,... , A/, as shown in Figure 5-10. The. r-coordinate of any node m is simply x = nAx, and the temperature at that point is T x = Elements are formed by drawing vertical lines through (he midpoints between the nodes. Note that all interior elements represented by interior nodes are full-size elements (they have a thickness of A.t), whereas the two elements at the boundaries are half-sized. 

Recall that when temperature varies linearly, the steady rate of heat conduction across a plane wall of thickness L can be expressed as... 

Boundary conditions most commonly encountered in practice are the specified temperature, specified heat flux, convection, and radiation boundary conditions, and here we develop the finite difference fonnulations for them for the case of steady one-dimensional heat conduction in a plane wall of thickness L as an example. The node number at the left surface at. r = 0 is 0, and at the right surface at x = L it is M. Note that the width of Ihe volume element for either boundary node is Ax/2. 

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... 

S-M Consider steady one-dimensional 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, 4, and 5 with a uniform nodal spacing of A.r. 

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