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National Advisory Committee for Aeronautics, Technical Notes - Experimental Determination of the Time Constants and Nusselt Numbers for Bare Wire Thermocouples in High velocity Air Streams and Analytic Approximation of Conduction and Radiation Errors

Performance evaluation and. control of jet engines, as well as the
fundamental study of the related combustion phenomena, depend consider—
ably on a knowledge of the steady and variable temperatures of the gas
within the engine. At present, these teweratures are most commonly
measured with thermocouples. The measurement accuracy is generally
dependent on the temperature level, the gas velocity, the temperature of
the surrounding walls, and the thermocouple construction. The design of
a thermocouple for any particular application represents a compromise
among the contradictory factors of accuracy, life, ruggedness, and
rapidity of response to changes as they are influenced by conduction and
radiation losses, partial adiabatic recovery, erosion, size, and con-
vective heat—transfer rate with the moving gas.

Whether a thermocouple is designed to respond rapidly to tempera-
ture changes or to measure accurately the steady-state temperature of
the gas, the controlling factors are the same, although the orders of
importance may be different. Although this report is concerned prin-
cipally with speed of response, consideration of the related steady-
state accuracy is included.

Limited data have been available for estimating time constants and
these data are for certain thermocouple designs and for a small range of
operating conditions. Data presented in reference 1 are time constants
of bare—wire and radiation-shielded thermocouples at Reynolds numbers
(based on wire diameter) from about 15 to 900 and Mach numbers from 0.05
to 0.14. The work was performed in exhaust gases at 10000 F, at gas
velocities up to 250 feet per second, and with a step change of approxi-
mately 700° F. For a given thermocouple, the time constant was found to
vary with the 0.5 power of the Reynolds number. Additional response-
rate data on very fine wires for a Reynolds number range of 5 to 540 and
a Mach number range of 0.02 to 0.1 are presented in reference 2.

As shown in reference 2 and in other references, the study of response
rates can be tantamount to the measurement of Nusselt number. The rela-
tion between Reynolds number and Nusselt number, and therefore response
rate, for cylinders in cross flow is represented in a compilation of
data in reference 3 for a Reynolds number range of 0.1 to 250,000 and
mach numbers up to 0.1. As gas velocity approaches the sonic value, it
is conceivable that compressibility effects, as represented by the mach
number, may influence the Husselt number. This influence is shown by
the data presented in reference 4. These data were obtained in the
Reynolds number range of 80 to 550 and mach number range of 0.4 to 2.2.