# Interdisciplinary Applied Mathematics

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However, if the micronozzles are planar, fabricated by deep reactive ion etching    (DRIE)    techniques    (see Figure    6.31),    then    the    thrust    scales    also

linearly with the characteristic dimensions, and so does the Reynolds number. Therefore, reducing the thrust by a factor of 100 will also reduce the

Reynolds number by a factor of 100, and this will result in large inefficiencies. For an extruded (two-dimensional) nozzle, such as the DRIE-fabricated micronozzle, we can write the thrust in terms of the Reynolds number as follows:

Ft ~ MVexit ~ Re^HVexit,

where Re = jjj, and F[ is the height of the nozzle (large dimension). In such nozzles we can increase the stagnation pressure p0 in order to increase the Reynolds number. To maintain a constant value of the thrust, according to the above equation the dimensions of the nozzle should be proportionally decreased. The net gain is higher efficiency, which means that the amount of propellant needed for a given mission decreases, and this leads to a reduction of system mass. This approach increases significantly the ratio of thrust to propulsion system mass with a reduction in scales.

With regard to the operating regime, the high-pressure DRIE-fabricated micronozzle of (Bayt, 1999; Bayt and Breuer, 2000), is in the Reynolds number range of order 1000, and the Knudsen number is less than 0.03. The corresponding thrust levels are of order 10 mN. Lower-level thrusts would require low chamber pressures, which would lead to lower Reynolds number and higher Knudsen number. For thrust levels of 1 mN, for example, the Reynolds number range is 100 to 1000 and the Knudsen number can have values of order 1. In the following, we examine such effects and the resulting performance of micronozzles.

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