Internal low Reynolds number turbulent heat transfer

The results of a semi-analytical and experimental study of adiabatic tube flow and an analytical and experimental investigation of the thermal entrance region for gas flowing through electrically heated circular tubes are presented. Emphasis is placed on the low Reynolds number turbulent flow regime--defined as fully turbulent flow at bulk Reynolds numbers from 3,000 to about 15,000.
Adiabatic air velocity and friction data and localized heat transfer measurements for air and helium, at low heating rates, are presented for this range.

The adiabatic data were obtained in a 1.61 inch 1D tube for flow at bulk Reynolds numbers from 3,000 to 15,000. A continuous, Reynolds number-dependent, profile is developed from the data by using a modification of Reichardts' wall and middle law eddy diffusivity expressions. The velocity profile satisfies continuity. It is valid for all Reynolds numbers in excess of 3,000 for which the flow is fully turbulent and the Blasius friction factor expression is valid.

The thermal entrance problem for a fully developed velocity profile is solved analytically by the method of Sparrow, Hallman, and Siegel. The solution is based on the profile developed from the velocity study. Tabular values of the eigenvalues and normalized Nusselt numbers for gases
are presented for it range of Reynolds numbers from 3,000 to 50,000. The axial variation of Nusselt number is found to be correlated by

[Nu/Nu_(∞)] = 1 + 0.8 (1+ 70,000 Re^(-3/2)) ((x/D)^(-1))

to within ± 5 per cent for x/D ≥2. The fully developed value agrees with the Dittus - Boelter correlation,

Nu_(∞) = 0.021 Re^(0.8) Pr^(0.4)

For the eigenvalues, λ^(2)_(n), and the associated constants, A_(n), correlations of the form

λ^(2)_(n) =A_(1,n) Re^(-b_(1,n) + C_(1,n) Re^(-d_(1,n))
A_(n) = -A_(2,n) Re^(-b_(2,n) + C_(2,n) Re^(-d_(2,n))

are obtained. The coefficients and powers are presented in tabular form.

Heat transfer data are presented, primarily for helium, for the low Reynolds number turbulent range. A one-quarter inch, resistively heated, vertical, circular tube was used for the study. The data cover an axial range from 1.2 to 96 diameters; wall-to-bulk temperature ratios vary from 1 to 1.4. In the low Reynolds number turbulent regime, these data clearly support the present analytical
solution rather than the prediction obtained by applying the eddy diffusivity distribution used by Sparrow, Hallman, and Siegel.