A mathematical model represents the dynamic response of a pressure sensor that (a) comprises a transducer connected to a pressure-sensing port via a pneumatic tube and (b) is operated under flow conditions like those expected to be encountered on advanced hypersonic aerospace vehicles operating at high altitudes. The model is applicable to both continuum and rarefied flows, including flows accompanied by large gradients of temperature.

Heretofore, the influences of rarefied-flow phenomena on the responses of such a pressure sensor were not well understood. In the presence of hypersonic flow, not only are measurements affected by nominal spectral distortion and acoustical resonance, but in addition, a large temperature gradient induced by boundary-layer heating can induce a molecular transpiration effect, in which gas molecules adjacent to the tube wall creep from the cold end of the tube to the hot end. Furthermore, at low pressure typical of operation at very high altitude, the gas in the tube becomes so rarefied that the flow slips at the tube wall. The present mathematical model was developed to enable evaluation of the dynamic response (the frequency response) under these flow conditions.

The model was derived from the equations of energy, continuity, momentum, and state. Effects of rarefied flow are represented by letting fluid elements slip at the tube wall; this is opposed to the classical "no-slip"condition used in continuum flow mechanics. For flows characterized by values of the Knudsen number between 0.01 and 1.0, molecular and continuum flow effects are important. Under these conditions, free-stream flow away from the wall boundary is identical to continuum flow; however, at the wall, the fluid elements do not stick to wall as they would in continuum flow. Instead, fluid elements slip along the wall -- hence the name "slip-flow" regime.

Under slip-flow conditions, the fluid velocity at the wall boundary can be decomposed into two parts: the slip velocity and the thermomolecular"creep" velocity. The slip velocity is proportional to the shear stress at the wall and is a result of reduced viscosity in a rarified fluid. The creep velocity is proportional to the longitudinal temperature gradient and inversely proportional to the local pressure. Kinetic theory predicts that gas molecules originating in the hotter region of the tube -- having kinetic energies greater than those of molecules originating from the colder region -- recoil more strongly than do the gas molecules from the colder region. As a result, the gas acquires a tangential momentum from the colder toward the hotter region. This net momentum gain causes the gas molecules at the wall to creep from the cold to the hot end of the tube.

To balance this creep, gas molecules in the center of the tube must migrate toward the colder end of the tube. The result of this "opposing flow equilibrium" is the establishment of a static pressure gradient such that cold region of the tube reads lower than the hot region and there is no net cross-sectional flow in the tube. At the wall boundary, the velocity is the sum of the slip and creep velocities. Other than this modification, the classical equations of fluid motion apply in this flow regime.

The equations of motion are linearized by use of small perturbations. The energy equation is decoupled from the momentum and continuity equations, assuming that the longitudinal wave expansion process within the tube is polytropic; that is, behaves according to a simple energy submodel that relates pressure, temperature, and density. The assumption of polytropic flow makes it possible to decouple the energy equation from the equations of momentum and continuity, without loss of generality. The boundary-value equations are radially averaged and solved with the help of the assumption that gas properties remain constant along the length of the tube. The resulting fundamental solution is used as a building block for complex situations in which fluid properties and tubing geometries vary longitudinally. The problem is solved recursively, starting at the transducer end and working toward the surface end of the tube. Using recursive formulas, solutions for arbitrary geometries and longitudinal temperature profiles are constructed.

The steady-state behavior of the model is analyzed by applying the final-value theorem to the recursive equation. The resulting expression is nondimensionalized and written as a function of the Knudsen number. Since it is extremely difficult to conduct controlled dynamic experiments under rarefied-flow conditions, the steady-state analysis is extremely important because it is the only feasible means of evaluating the range of validity of the slip-flow assumptions used in deriving the model.

Steady-state-response tests were performed under rarefied-flow conditions to verify the slip-flow assumptions used in deriving the boundary-value equations, and to establish a regime of validity for the model; the results (see figure) showed that the model is valid for Knudsen numbers up to about 0.6, and is valid for most of the slip-flow regime.

The model contributes to understanding of flow behavior at the limits of the continuum flow regime. It will enable instrumentation designers to evaluate the responses of pneumatic systems over a wide range of flow conditions in a general and unified way, without having to resort to *ad hoc* or special-case models. Potential nonaerospace applications of the model include prediction of the behaviors of fluids in micromachined systems, wherein the mean free paths of the working fluids are of the order of channel widths; and prediction of flows of highly viscous fluids for which Knudsen numbers can be sizeable under non-rarefied-flow conditions.

*This work was done by Stephen A. Whitmore of*Dryden Flight Research Center* and Brian Petersen of the University of California, Los Angeles. For further information, access the Technical Support Package (TSP) free on-line at www.techbriefs.com Or Click Here: DRC-9534.*

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This article first appeared in the January, 1999 issue of *NASA Tech Briefs* Magazine.

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