The mathematical expression for the theoretical temperature distribution on the surface of a finite-length heated cylinder buried in a permeable flow field was derived in 1995. It was then realized that if the heat flux out of the cylinder is uniform over the surface of the cylinder, then the temperature distribution on the surface of the cylinder will vary as a function of the direction and magnitude of the ground water flow velocity past the cylinder. That analysis assumes a long, thin cylinder
(L/ R > ~ 10) buried in intimate contact with an infinite, saturated, porous medium whose thermal and hydraulic properties are homogeneous and isotropic, and is valid for Peclet numbers up to order 1.
Gravitational effects are neglected, meaning that natural convection resulting from heating the water around the probe is assumed to be insignificant. This is valid as long as forced convection is more important than the natural convection induced by warming the water around the probe.
The graph at right illustrates the temperature, calculated according to the stated assumptions, as a function of azimuth for a probe buried in an horizontal flow field where the groundwater is flowing in a direction 90 degrees from the reference direction on the probe. In the absence of any horizontal flow past the device, the temperature on the surface of the probe is independent of azimuth. If there is a significant horizontal component to the flow velocity, the temperature varies approximately as the cosine of the azimuth with the downstream side of the probe being warmer than the upstream side.
The formula requires that T be the temperature at position x,z on the surface of the probe; x is the angular distance in the horizontal plane from the reference direction to the point on the surface of the cylinder where the temperature is observed; z is the distance in the vertical direction from the midpoint of the probe to the point on the probe surface where the temperature is observed, made dimensionless by dividing by the half-length of the probe, L; R is the radius of the probe and delta is the half length of the heated section of the probe, made dimensionless by dividing by L.
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