Conduction through a plane wall

HEAT TRANSFER BY CONDUCTION 9.3.1. Conduction through a plane wall... 

For conduction through a plane wall of area A and thickness x, eqn. (82) may be integrated to give... 

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... 

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

Discussion This is the same result obtained earlier. Note that heat conduction through a plane wall with specified surface temperatures can be determined directly and easily without utilizing the thermal resistance concept. However, the thermal resistance concept serves as a valuable tool in more complex heat transfec problems, as you will see in the following examples. Also, the units W/m "f and W/m K for thermal conductivity are equivalent, and thus inter changeable. This is also the case for °C and K for temperature differences. 

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. 

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 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. 

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. 

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. 

Steady nondirectional heat transfer through a plane wall of thickness x and area A is represented in Fig. 4A. Assuming that thermal conductivity does not change with temperature, the temperature gradient is linear and equal to (Ti - T2)/x, where Ti is the temperature of the hot face and T2 the temperature of the cool face. Eq. (21) then becomes Eq. (22)... 

A control volume drawn around a plane wall with three layers is shown in Fig. 1.2. Three different materials, M, N and P, of different thicknesses, AxM, AxN and AxP, make up the three layers. The thermal conductivities of the three substances are kM, kN and kp respectively. By the conservation of energy, the heat conducted through each of the three layers have to be equal. Fourier s law for this control volume gives... 

Hear conduction through a large plane Wall of thickness Ax and area A. 

One-dimensional heat conduction through a volume element in a large plane wall. 

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 ... 

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... 

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. 

CoWsider a plane wall of thickness 2L initially at a uniform temperature of T , as shown in Fig. 4—1 In. At lime t = 0, the wall is immersed in a fluid at temperature 7 and is subjected to convection heal transfer from both sides with a convection coefficient of h. The height and the widlh of the wall are large relative to its thickness, and thus heat conduction in the wall can be approximated to be one-dimensional. Also, there is thermal symmetry about the inidplane passing through.x = 0, and thus the temperature distribution must be symmetrical about tlie midplane. Therefore, the value of temperature at any -.T value in - A "S. t 0 at any time t must be equal to the value at f-.r in 0 X Z, at the same time. This means we can formulate and solve the heat conduction problem in the positive half domain O x L, and then apply the solution to the other half. 

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. 

Many practical mass transfer problems involve the diffusion of a species through a plane-parallel medium that does not involve any homogeneous chemical reactions under one-dimensional steady conditions. Such mass transfer problems are analogous to the steady one-dimensioiial heat conduction problems in a plane wall with no heal generation and can be analyzed similarly. In fact, many of the relations developed in Chapter 3 can be used for mass transfer by replacing temperature by mass (or molar) fraction, thermal conductivity by pD g (or CD ), and heat flux by mass (or molar) flux (Table 14-8). 

Fig. 4 Conduction of heat through (A) a plane wall and (B) a composite wall.

Overall Heat Transfer through Plane Walls and Cylindrical Shells Surrounded by Fluids The combination of conduction and convection may be characterized by the thermal transmittance Uh- For a plane wall surrounded by fluids of different temperature (Figure 3.2.16) we have ... 

Introduction of the thermal conductibility a (= XI pmo Cp). which represents how fast heat is transported through a material, yields the common form of the Fourier s second law for a one-dimensional heat transfer in a 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... 

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... 

The conductive heat loss per unit volume from a plane flame in a circular tube of diameter D can be estimated by the following simple reasoning. Consider an Eulerian element of gas of length dx in the tube whose walls are maintained at the temperature Tq. The energy per second conducted to the walls from this element is the product of the thermal conductivity A, a mean temperature gradient (T 7 )/(D/2), and the wall area nDdx. The rate of heat loss per unit volume is then obtained through division by the volume of the element dxnD I4 ... 

Figure 4.39 gives another example, taken from Nature. The Corynebacterium considered here has more or less spherical, homodisperse cells, with diameters of 1.1 and 0.8 pm for the longer and shorter axis. Such cells are fascinating model colloids. The relaxation frequency, Ae and AK behave as expected for colloidal particles with a finite conduction behind the shear plane, which, in this case, is caused by the ions in the cell wall. As in the previous example, the data were analyzed with (4.8.30 and 31(, using (4.6.56] for Du. The three curves, drawn through the measuring points, refer to this interpretation with the values of indicated. The conductivity of the cell wall exceeds that of the bulk... 

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 ... 

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