Applications of Steady-State Conduction One Dimension:
INTRODUCTION:
We now wish to examine the applications of Fourier’s law of heat conduction to calculation of heat flow in some simple one-dimensional systems. Several different physical shapes may fall in the category of one-dimensional systems: cylindrical and spherical systems are one-dimensional when the temperature in the body is a function only of radial distance and is independent of azimuth angle or axial distance. In some two-dimensional problems the effect of a second-space coordinate may be so small as to justify its neglect, and the multidimensional heat-flow problem may be approximated with a one-dimensional analysis. In these cases the differential equations are simplified, and we are led to a much easier solution as a result of this simplification.
THE PLANE WALL
First consider the plane wall where a direct application of Fourier’s law may be made. Integration yields
(47)
when the thermal conductivity is considered constant. The wall thickness is ?x, and T1 and T2 are the wall-face temperatures. If the thermal conductivity varies with temperature according to some linear relation k =k0(1+?T ), the resultant equation for the heat flow is
(48)
If more than one material is present, as in the multilayer wall shown in Figure 47, the analysis would proceed as follows: The temperature gradients in the three materials are shown, and the heat flow may be written
(49)
Note that the heat flow must be the same through all sections.
Figure 13 One-dimensional heat transfer through a composite wall and electrical analog.
Solving these three equations simultaneously, the heat flow is written
(50)
At this point we retrace our development slightly to introduce a different conceptual viewpoint for Fourier’s law. The heat-transfer rate may be considered as a flow, and the combination of thermal conductivity, thickness of material, and area as a resistance to this flow. The temperature is the potential, or driving, function for the heat flow, and the Fourier equation may be written
a relation quite like Ohm’s law in electric-circuit theory. In Equation (47) the thermal resistance is _x/kA, and in Equation (49) it is the sum of the three terms in the denominator. We should expect this situation in Equation (49) because the three walls side by side act as three thermal resistances in series. The one-dimensional heat-flow equation for this type of problem may be written
(51)
Where the Rth are the thermal resistances of the various materials. The units for the thermal resistance are ?C/W or ?F ? h/Btu.
For Series and parallel one-dimensional heat transfer through a composite wall and electrical analog. in the building industry to use a term called the R value, which is defined as
The units for R are ?C ?m2/W or ?F ? ft2 ? h/Btu. Note that this differs from the thermal resistance concept discussed above in that a heat flow per unit area is used.
Example: let . 10
An exterior wall of a house may be approximated by a 4-in layer of common brick [k = 0.7 W/m. ?C] followed by a 1.5-in layer of gypsum plaster [k =0.48 W/m. ?C]. What thickness of loosely packed rock-wool insulation [k =0.065 W/m. ?C]. should be added to reduce the heat loss (or gain) through the wall by 80 percent?
RADIAL SYSTEMS
Cylinders
Consider a long cylinder of inside radius ri, outside radius ro, and length L, such as the one shown in Figure 14.We expose this cylinder to a temperature differential Ti ?To and ask what the heat flow will be. For a cylinder with length very large compared to diameter, it may be assumed that the heat flows only in a radial direction, so that the only space coordinate needed to specify the system is r. Again, Fourier’s law is used by inserting the proper area relation. The area for heat flow in the cylindrical system is
So that the Fourier law is written
or (52)
Figure 14 One-dimensional heat flow through a hollow cylinder and electrical analog.
Figure 15 One-dimensional heat flow through multiple cylindrical sections and electrical analog.
For the multiple cylindrical sections and electrical analog with the boundary conditions
The solution to Equation (52) is
(53)
and the thermal resistance in this case is
The thermal-resistance concept may be used for multiple-layer cylindrical walls just as it was used for plane walls. For the three-layer system shown in Figure 15 the solution is
(54)
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