The Nusselt number:
For heat transfer from a horizontal surface, the Nusselt number can be expressed as

where Dh is the hydraulic diameter, dT/dy is the vertical temperature gradient at the surface of the heated block, Tb is the temperature at the heated surface, and Tm is the mean temperature of the duct. Hydraulic diameter is a hypothetical length-scale which allows one to compare data from noncircular ducts. It is defined as

where A is the duct cross-sectional area and P is the distance around the duct perimeter. Note, that when the duct cross-section is circular, Dh becomes the diameter of that circle.

Estimating dT/dy

An estimate of the vertical temperature gradient at the surface of the heated block can obtained by measuring the vertical distance between adjacent interference fringes closest to the block. In this experiment, the temperature difference DTF between adjacent (say, dark) fringes is 2.8o K. The finite difference expression DTF/(y2-y1) approximates dT/dy. Note that when you carry out your measurements on the hologram, positions are reported in pixels (picture elements). Since you know the dimensions of the experimental apparatus, you can deduce the relationship between pixels and millimeters.

Estimating (Tb-Tm)

We need the value of this expression for the chosen vertical cross-section. The temperature Tb is unknown. But, if we encouch Tm in terms of Tb, Tb will disappear from the problem. Tm is the mean temperature along the vertical cross-section. The temperature of the ith fringe from the block along this vertical section is Tfi=Tb-i*DTF. (Note that temperatures decrease as you go farther from the block.) Using a finite differences, Tm may be approximated as

where n is the number of the farthest fringe (presumably at or near the upper surface of the duct). This is a "trapezoidal rule" finite difference approximation to Tm. If one expresses each Tfi in terms of Tb and DTF, Tm will consist of Tb plus other terms, thus providing for the cancelling of Tb in the expression (Tb-Tm).