Steady heat conduction 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. 

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 steady heat conduction through the walls of a house during a winter day. We know that heat is continuously lost to the outdoors through the wall. Wc intuitively feel that heat transfer through the wall i.s in the normal direction to the wall surface, and no significant heat transfer takes place in the wall in other directions CFig. 3-1). 

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 ... 

I FIGURE 11.27 Heat conduction through an external wall. The temperature distribution over die wall thickness is linear only under steady-state conditions. 

This similarity was established in [2] by consideration of the second-order differential equations of diffusion and heat conduction. Under the assumptions made about the coefficient of diffusion and thermal diffusivity, similarity of the fields, and therefore constant enthalpy, in the case of gas combustion occur throughout the space this is the case not only in the steady problem, but in any non-steady problem as well. It is only necessary that there not be any heat loss by radiation or heat transfer to the vessel walls and that there be no additional (other than the chemical reaction) sources of energy. These conditions relate to the combustion of powders and EM as well, and were tacitly accounted for by us when we wrote the equations where the corresponding terms were absent. 

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

Solution This flow is z-axisymmetric. We, thus, select a cylindrical coordinate system, and make the following simplifying assumptions Newtonian and incompressible fluid with constant thermophysical properties no slip at the wall of the orifice die steady-state fully developed laminar flow adiabatic boundaries and negligible of heat conduction. 

FIG. 5-4 Thermal circuit for Example 1. Steady-state conduction in a furnace wall with heat losses from the outside surface by convection (hc) and radiation (hR) to the surroundings at temperature Tsur. The thermal conductivities kD, kg, and ks are constant, and there are no sources in the wall. The heat flux q has units of W/m2. 

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) ... 

Discussion The last solution represents a family of straight lines whose slope is - (j(//r. Physically, this problem corresponds to requiring the rate of heat supplied to the vral at x = 0 be equal to the rate ol heat removal from the other side of the wall at x L. But this is a consequence of the heat conduction through the wall being steady, and thus the second boundary condition does not provide any new information. So it Is not surprising that the solution of this problem is not unique. The three cases discussed above are summarized in Fig. 2-45. 

Note that the outer surface temperature turned out to be lower than the inner surface temperature. Therefore, the heat transfer through the wall is toward the outside despite the absorption of solar radiation by the outer surface. Knowing both the inner and outer surface temperatures of the wall, the steady rate of heat conduction through the v/all can be determined from... 

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. 

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 ... 

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 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... 

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. 

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... 

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. 

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