Heat transfer through plane walls

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 ... 

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 ... 

Note that the heat transfer area A is constant tor a plane wall, and the rate of heat transfer through a wall separating two mediums is equal to the temperature difference divided by the total thermal resistance between the mediums. Also note that (he thermal resistances are in series, and the equivalent thermal resistance is determined by simply adding the individual resistances, just like the electrical resistances connected in series. Thus, the electrical analogy still applies. We summarize this as the rate of steady heat transfer between two surfaces is equal to the temperature difference divided by the total thermal resistance behveen those Uvo surfaces. 

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 ... 

We have already discussed the overall heat-transfer coefficient in Sec. 2-4 with the heat transfer through the plane wall of Fig. 10-1 expressed as... 

Flfl. 10-1 Overall heat transfer through a plane wall. 

Analysis The bottom section of the pan has a large surface area relative to its thickness and can,be approximated as a large plane wall. Heat flux is applied to the bottom surface of the pan iiniforrtily, and the conditions on the Inner surface are also uniform. Therefore, we expect the heat transfer through the bottom section of the pan to be from the bottom surface toward the top, and heat transfer in this case can reasonably be approximated as being onedimensional. Taking the direction normal to the. bottom surface of the pan to be the x-axis, we will have T = T(x) during steady operation since the temper-, ature in this case will depend on Xonly. 

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 ... 

Then the rate of steady heat transfer through a plane wall, cylindrical layer, or spherical layer can be expressed as... 

The thenn.3l resistance network for heat transfer through a plane wall subjected to convection on both sides,... 

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. 

Consider a plane wail that consists of two layers (such as a brick wall with a layer of insulation). The rate of steady heat transfer through this two-layer composite wall can be expressed as (Fig. 3-9)... 

Steady heat transfer through multilayered cylindrical or spherical shells can be handled just like multUayered plane walls discussed earlier by simply add ing an additional resistance in series for each additional layer. For example, the steady heat transfer rale through the three-layered composite cylinder of length L shown in Fig. 3-26 with convection on both sides can be expressed as... 

The specified temperature boundary condition is the simplest boundary condition to deal with. For one-dimensional heat transfer through a plane wall of thickness L, the specified temperature boundary conditions on both the left and right surfaces can be expressed as (Fig. 5-13)... 

Consider a solid plane wall (medium B) of area A, thickness L, and density p. The wall is subjected on both sides to different concentrations of a species A to which it is permeable. The boundary surfaces at.t = 0 and x - L are located within the solid adjacent to the interfaces, and the mass fractions of A at those surfaces are maintained at and 2. respectively, at all times (Fig. 14-19). The mass fraction of species A in the wall varies in the. v-direction only and can be expressed as >v (.t). Therefore, mass transfer through the wall in this case can be modeled as steady and one-dimensional. Here we determine the rate of mass diffusion of species A through the wall using a similar approach to that used in Chapter 3 for heat conduction. 

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)... 

This equation is only precise for the case of a plane wall. For the technically important case of heat transfer through tubes (cylindrical geometry), the different geometry must be taken into account by a mean tube surface area for heat transfer (Equations 2.3.1-15 and 2.3.1-16 Eigure 2.3.1-lb) ... 

The Plane Wall. To calculate the heat-transfer rate through a plane wall, Fourier s law can be appHed directly. 

This important mechanism of heat transfer is now considered in more detail for the flow of heat through a plane wall of thickness x as shown in Figure 9.5. 

Planck s constant, 23 33 Planck spectrum, 23 2 Plan-Do-Check-Act (PDCA) model, 21 174 Plane-polarized light (PPL), 16 470, 476 Planetary rotation reactor, 22 154, 155 Planetary type mixers, 16 720 Plane wall, heat-transfer rate through,... 

This chapter is concerned with the prediction of the heat transfer rate from the wall of a duct to a fluid flowing through the duct, the flow in the duct being turbulent. The majority of the attention will be given to axi-symmetnc flow through pipes and two-dimensional flow through plane ducts, i.e , essentially to flow between parallel plates. These two types of flow are shown in Fig. 7.1. 

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. 

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. 

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... 

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. 

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). 

Equation 7.18 shows that unlike the rectilinear plane wall, the temperature distribution is not linear. The heat flux declines as r increases due to increasing area, rdSdz. Nonetheless, we can apply Fourier s law to this result to obtain an expression for the heat transfer rate through the wall of a pipe with length L or for that matter a rotary dmm cylinder of length L as... 

We will now consider the problem of calculating the heat transfer from a hot fluid to a composite plane of refractory wall and through an outer steel shell. 

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