Interdisciplinary Applied Mathematics

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4.1.2 Adiabatic Compressible Flows — Fanno Theory


In this section we use the Fanno flow theory to analyze compressible adiabatic gas flows. Most of the microsystem components are fabricated using silicon as the base material. Since silicon is a good heat conductor, the isothermal flow assumption presented in the previous section is more relevant for many silicon-based microsystem components. However, for insulator materials, such as glass, approximate adiabatic flow conditions may be valid. Therefore, study of compressible adiabatic gas flows may be useful in applications that utilize insulator materials.


We will compare the experimental data in (Harley, 1993; Harley et al., 1995), obtained under almost isothermal flow conditions, with the adiabatic Fanno flow theory. Alt hough the isothermal and adiabatic flows show significant differences for high Mach number flows, Fanno flow analysis can predict the general trends of the experimental results, enabling approximate analysis for many engineering applications.


In the experiments of (Harley et al., 1995), about an order of magnitude pressure drop is imposed through a microchannel of length 10.9 mm. Since the temperature was almost constant, about an order of magnitude change in the inlet to outlet density ratio was observed. This case corresponds to severe compressibility effects, which may eventually result in a choked flow. Such large pressure drops and even flow choking may become relevant in analysis of micronozzles for aerospace applications (see Section 6.6).


The approximation is based on one-dimensional adiabatic constant-area flow, i.e., the so-called Fanno flow, conditions. The experimental data reported in (Harley, 1993), are summarized in Table 4.2. The Fanno flow equations can be derived by using a friction factor (/ = у Jq fdx) aver-

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