Plane walls conduction

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... 

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... 

Figure 1.1 Conduction in a plane wall. In one dimension, Fourier s law becomes... 

Fourier s law is applied to conduction in a plane wall, as shown in Fig. 1.1. The heat flow Q x is the heat energy transfer in the x direction. 

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... 

Problem The thermal conductivity of a plane wall varies with... 

A plane wall, 0.15 m thick, internally generates heat at a rate of 6 x 104 W/m3. One side of the wall is insulated and the other side is exposed to an environment at 25°C. The heat transfer coefficient between the wall and the environment is 750 W/(m2.K). The thermal conductivity of the wall is 20 W/(m.K). Calculate the maximum temperature in the wall. 

FIG. 5. Variation of the dimensionless electrophoretic velocity U/Uq of a cylinder with distance parameter X in the vicinity of a plane wall (1) parallel to a dielectric plane, (2) perpendicular to a conducting wall. 

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... 

A plane wall is constructed of a material having a thermal conductivity that varies as the square of temperature according to the relation k = fc0(l + /3T1). Derive an expression for the heat transfer in such a wall. 

Derive equations that describe the temperature profiles for a plane wall, long hollow cylinder, and hollow sphere. Assume constant thermal conductivity, and temperature at the walls as Tt and 7 2. 

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

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

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. 

Plot the variation of temperature along the sample, and calculate the thermal conductivity of the sample material. Based on these temperature readings, do you iliinl steady operating conditions are established Are there any temperature readings that do nor appear right and should be discarded Also, discuss when and how the temperature profile in a plane wall will deviate from a straight line. 

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. 

Transient and steady heat conduction in a plane wall. 

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

Heat Conduction Equation in a Large Plane Wall... 

Noting that the area A is constant for a plane wall, the one-dimensional transient heal conduction equation in a plane wall becomes... 

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... 

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. 

Discussion The proper sign of the square root term (-E or -) is determined from the requirement that the temperature at any point within the medium must remain between Ti and T. This result explains why the temperature distribution in a plane wall is no longer a straight line when the thermal conductivity varies with temperature. 

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. 

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