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Research Papers: Forced Convection

Explicit Inversion of Stodola’s Area-Mach Number Equation

[+] Author and Article Information
Joseph Majdalani

Mechanical, Aerospace and Biomedical Engineering Department, University of Tennessee Space Institute, Tullahoma, TN 37388maji@utsi.edu

Brian A. Maicke

Mechanical, Aerospace and Biomedical Engineering Department, University of Tennessee Space Institute, Tullahoma, TN 37388

J. Heat Transfer 133(7), 071702 (Mar 30, 2011) (7 pages) doi:10.1115/1.4002596 History: Received May 26, 2009; Revised September 11, 2010; Published March 30, 2011; Online March 30, 2011

Stodola’s area-Mach number relation is one of the most widely used expressions in compressible flow analysis. From academe to aeropropulsion, it has found utility in the design and performance characterization of numerous propulsion systems; these include rockets, gas turbines, microcombustors, and microthrusters. In this study, we derive a closed-form approximation for the inverted and more commonly used solution relating performance directly to the nozzle area ratio. The inverted expression provides a computationally efficient alternative to solutions based on traditional lookup tables or root finding. Here, both subsonic and supersonic Mach numbers are obtained explicitly as a function of the area ratio and the ratio of specific heats. The corresponding recursive formulations enable us to specify the desired solution to any level of precision. In closing, a dual verification is achieved using a computational fluid dynamics simulation of a typical nozzle and through Bosley’s formal approach. The latter is intended to confirm the truncation error entailed in our approximations. In this process, both asymptotic and numerical solutions are compared for the Mach number and temperature distributions throughout the nozzle.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 2

Comparison between numerics and asymptotics in both subsonic and supersonic regimes for γ=1.4; the selected area in (a) is magnified in (b) to showcase the degree of agreement at increasing asymptotic orders as ε→1

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Figure 1

Sensitivity of the pressures and temperatures to the nozzle expansion ratio at γ=1.4

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Figure 3

CFD nozzle geometry and mesh selection (a) showing enlarged inset in (b)

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Figure 4

Cold flow simulation results depicting (a) the velocity vector distribution, (b) the Mach number contours, and (c) a comparison of the wall, centerline, and average CFD predictions to the present three-term analytical approximation over the length of the nozzle. Here, air is used with TC=300 K and γ=1.4.

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Figure 5

Cold flow simulation results depicting (a) the thermal map in K and (b) a comparison of the wall, centerline, and average CFD predictions to the present three-term analytical approximation of the temperature over the length of the nozzle. Here, air is used with TC=300 K and γ=1.4.

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Figure 6

Hot flow simulation results depicting (a) the nozzle thermal map in K and (b) a comparison of the wall, centerline, and average CFD predictions to the present three-term analytical approximation of the temperature over the length of the nozzle. Here, air is used with TC=2200 K and γ=1.4.

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Figure 7

Normalized heat flux obtained from Stodola’s equation, the present approximation, and CFD predictions using the hot flow simulation; the spatial evolution of Stodola’s Mach number is also shown on the right

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Figure 8

Asymptotic error entailed in supersonic En and subsonic E¯n shown for γ=1.4

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