An Inefficient Fin Non-Dimensional Parameter to Measure Gas Temperatures Efficiently
- Created: Tuesday, 01 May 2012
This method provides a convenient sensing error guideline.
A gas containment vessel that is not in thermal equilibrium with the bulk gas can affect its temperature measurement. The physical nature of many gas dynamics experiments often makes the accurate measurement of temperature a challenge. The environment itself typically requires that the thermocouple be sheathed, both to protect the wires and hot junction of the instrument from their environment, and to provide a smooth, rigid surface for pressure sealing of the enclosure. However, that enclosure may also be much colder than the gas to be sensed, or vice-versa. Either way, the effect of such gradients is to potentially skew the temperature measurements themselves, since heat may then be conducted by the instrument.
Schematic of the Apparatus Used in the Rocket Throat Cooling Experiment. The three instruments measure throat temperature, gas temperature, and gas pressure (top to bottom). At left, the gas-sensing probe is nearly flush with the inner surface of the throat, corresponding with a low inefficient fin non-dimensional number. At right, the D (probe diameter) and L (distance past the enclosure wall) were changed to correspond to a non-dimensional number of 4.60." class="caption" align="right">Thermocouple designers traditionally address this problem by insulating the sheath from the thermocouple leads and hot junction as much as possible. The thermocouple leads are typically packed in a ceramic powder inside the sheath, protecting them somewhat from temperature gradients along the sheath, but there is no effective mechanism to shield the sheath from the enclosure body itself. Standard practice dictates that thermocouples be used in installations that do not present large thermal gradients along the probe. If this conduction dominates heat transfer near the tip of the probe, then temperature measurements may be expected to be skewed. While the same problem may be experienced in the measurement of temperature at various points within a solid in a gradient, it tends to be aggravated in the measurements of gas temperature, since heat transfer dependent on convection is often less efficient than conduction along the thermocouple.
The proposed solution is an inefficient fin thermocouple probe. Conventional wisdom suggests that in many experiments where gas flows through an enclosure (e.g., flow in pipe, manifold, nozzle, etc.), the thermocouple be introduced flush to the surface, so as not to interfere with the flow. In practice, however, many such experiments take place where the flow is already turbulent, so that a protruding thermocouple probe has a negligible effect on the flow characteristics. The key question then becomes just how far into the flow should a thermocouple protrude in order to properly sense the gas temperature at that point. Modeling the thermocouple as an “inefficient fin” directly addresses this question. The appropriate assumptions in this case are: one-dimensional conduction along the fin; steady-state, constant, and homogeneous thermal conductivity; negligible radiation; and a uniform, constant heat transfer coefficient over the probe surface. It is noted that the nature of the ceramic-filled probe makes the key assumption of homogeneous thermal conductivity that much more conservative.
Normally a mathematical expression is used to assess fin efficiency, i.e., how far from the fin base heat can be carried. In this case, however, the thermocouple probe should be designed to be an inefficient fin; that is, parameters should be chosen such that the temperature of the wall does not affect the temperature sensed at the tip of the probe. This inefficient fin parameter is then numerically equal to ln(l00/% error), where % error is computed with respect to the temperature difference between the wall and the fluid. A one-to-one correspondence between this parameter and the error in sensed temperature may thus be established. For example, for a maximum error of 5%, the non-dimensional parameter value is 3.00. For an error of 1%, the target parameter value becomes 4.60. This parameter dictates the minimum distance for a given probe.
This simple method provides a convenient guideline to maintain flow temperature sensing error within a predefined range, given a temperature mismatch between a gas and its surrounding walls. This approach was put to practice in such an experiment, where a hot rocket nozzle was cooled using a twophase fluid (where the fluid temperature may thus be verified, using the saturation pressure). The measured temperature in the cooling annulus showed good agreement with the method, and the thermocouple be came essentially insulated from the wall by setting the hot junction at a distance corresponding to the parameter value of 4.60.
This work was done by Patrick Lemieux, William Murray, Terry Cooke, and James Gerhardt of California Polytechnic State University for Dryden Flight Research Center. DRC-010-030