Notice that we need the diffusive coefficients at the interfaces of the elemental volumes Eq. (4.11) and (4.12). Therefore, when these coefficients are variables inside the physical domain, we must evaluate an average to the interface. |
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 Figure 6.1 – Interface between elemental volumes. |
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The use of a linear diffusivity variation, between points P and E, may imply incorrect approximations, particularly in such cases that occur abrupt changes in conductivity. An example of this is found when we are dealing with composite materials. In this section, we will discuss the thermal conductivity average (k) used in csfl-lib-1.0 and CFD Studio to calculate properties at the interfaces, and for all other diffusive terms - Gf - this software uses the same averages. We should remark that it is not the exact value of local diffusivity our first goal. The main objective is to obtain a good approximation of the diffusive flux at the interface. From this one-dimensional heat equation, applied to figure (6.1), we have: |
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(6.1) |
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So we should obtain an expression to ke that gives us a suitable ke flux. Considering kP the conductivity of the elemental volume P and kE the conductivity of the volume E, we obtain the following expression: |
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(6.2) |
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| So, combining (6.1) and (6.2), we find: | |||
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(6.3) |
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In which: |
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(6.4) |
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And considering the generality of the coordinates, we have: |
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(6.5a) |
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(6.5b) |
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The equation 6.3 is the average between the diffusivity coefficients at the interfaces used in CFD Sinflow Library. We can see that when we make fe = 0,5, i. e., the interface e is equidistant to P e E, and we have: |
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(6.6) |
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This is the harmonic mean of kPand kE. We may obtain the average at the north face by an analogy, so we have: |
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(6.7) |
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And: |
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(6.8) |
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And considering the generality of the coordinates, we have: |
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(6.9a) |
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(6.9b) |
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The evaluation of the metrics used in (6.5) and (6.9) shall be discussed in Chapter 11. |
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