Steady-State Conduction—One Dimension

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. 

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This is the steady-state conduction solution in one dimension for rectangular coordinates which shows that the temperature profile is linear in a slab of a given material. 

In reality, heat is conducted in all three spatial dimensions. While specific building simulation codes can model the transient and steady-state two-dimensional temperature distribution in building structures using finite-difference or finite-elements methods, conduction is normally modeled one-... 

A refinement of the above approach is to treat the composite as a quasihomo-geneous system with specific electronic/ionic conductivities and gas diffusion properties, with the electrochemical reaction distributed uniformly over its volume. The problem has been solved in one dimension by Costamagna et al. [1998], but then-solution appUes to the steady state and is not relevant to the present discussion, which concentrates on the frequency domain. Furthermore, intuition would suggest that a one-dimensional model would not completely describe a system as complex as that depicted in Figure 4.1.14. 

In 1855 Adolf Pick drew an analogy between the heat transfer phenomena by conduction and those of mass transfer. Assuming a steady state, the first diffusion theory postulates that the flux of the diffusing substance through a unit area is directly proportional to the concentration gradient measured in a direction normal to this surface unit. In one spatial dimension, this relationship can be written as ... 

For a standard business card, in the vertical length dimension, determine the steady burning rate (g/s) for one side of the card saturated with ethanol. Only the ethanol bums. Show your analysis and all assumptions. This is a calculation, not an experimental determination, though experiments can be conducted. State all data and sources used. You will have to make approximations and estimates for quantities in your analysis. 

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