Natural and mixed convection boundary layers on vertical heated walls (B)

Underlying Flow Regime 3-07 © copyright ERCOFTAC 2004

Description

Introduction

The effects of buoyancy on heat transfer can be predicted readily for laminar flows. When the flow is in the upward direction past a heated surface (or downwards past a cooled surface) heat transfer is enhanced, whereas in the opposite cases heat transfer is impaired. These influences are the consequence of the distortion of the velocity field.

However, the situation is different with turbulent flows, as heat transfer is dictated by changes in turbulent diffusion. In configurations with forced and free convection aligned, local heat transfer coefficients significantly lower than those for forced flow alone can result. In contrast, for downward flow in heated tubes buoyancy forces cause a general enhancement of the turbulent diffusion properties of the flow, with the result that wall temperature distributions are well behaved and heat transfer coefficients are higher than those for forced flow alone. Eventually as the free convection component becomes more and more dominant, heat transfer for upward flow also becomes enhanced and in the limit the heat transfer coefficients for the two cases are the same.

This UFR is of particular relevance to British Energy’s and Magnox Electric’s Application Challenges.

Review of UFR studies and choice of test case

Studies of Petukhov and Nolde (1959), Petukhov and Strigin (1968), Herbert and Sterns (1968,1972), Jackson and Fewster (1977), Rouai (1987), Buyukalaca (1993) performed experiments with descending water flow in heated tubes. Jackson et al (2000) considered downward water flow in a heated annulus.

Studies of mixed convection heat transfer to air in a vertical tube with opposing forced and free components were made by Eckert and Diaguila (1954), Khosla, Hoffman and Pollock (1974), Brown and Gauvin (1965,1966) and Axcell (1975). Easby (1978) studied nitrogen at pressures up to 4 bar. Fewster (1976) conducted a programme of experiments on buoyancy-opposed mixed convection using supercritical pressure CO₂. In general, an enhancement of heat transfer relative to that for forced convection under corresponding conditions was reported.

Buyukalaca (1993) has produced a review of buoyancy-opposed flow for liquids such as water, oil, etc. He analysed the earlier data and his own experimental data using the semi-empirical ideas of Jackson and Hall (1979), and achieved an accurate correlation for buoyancy-opposed turbulent mixed convection flow in a vertical pipe.

Data for air under buoyancy-opposed flow conditions have been obtained by Eckert and Diaguila (1954), Khosla, Hoffman and Pollock (1974) and Axcell (1975). However, they all used tubes of short length to diameter ratio in their studies, and their data may be significantly complicated by entrance effects. Li (1994) has considered this flow, and his experiment does not suffer from the same complications as those of the earlier work.

Experiments for air flowing upwards in a heated vertical tube were made by Steiner (1971), Byrne and Ejiogu (1971), Carr, Connor and Buhr (1973), Perkins and McEligot (1975), Polyakov and Shindin (1986), Vilemas, Poskas and Kaupas (1992), Li (1994).

Buoyancy Parameters

This section discusses parameters that can be used to characterise the phenomena associated with vertical flows and heated tubes.

In a given mixed convection system, the Nusselt number is a function of Gr, Re, Pr, x/D and Tw/Tb. Although the general trend of heat transfer may be described by such a parameter, none of the available buoyancy parameters correlate the heat transfer data perfectly because of the complicated behaviour of developing turbulent mixed convection in the buoyancy aided condition. Jackson et al (1989) obtained the following equation

{\frac {\text{Nu}}{{\text{Nu}}_{f}}}={\left[1\pm 2.4\times {10}^{4}{\frac {G_{r}^{*}}{{\text{Re}}^{3.35}{\text{Pr}}^{0.9}}}{\left[{\frac {\text{Nu}}{{\text{Nu}}_{f}}}\right]}^{-2}\right]}^{0.46}

The negative sign refers to the buoyancy-aided case and the positive to the buoyancy-opposed one. Gr^* is the Grashof number based on wall heat flux, i.e.

In Yu (1991) and Li (1994), the buoyancy parameter B₀ is introduced:

Computational Studies of Turbulent Mixed Convection in a Vertical Tube

Various types of turbulence closures have been used, including (a) prescribed eddy viscosity models, (b) algebraic turbulent viscosity models, (c) one and (d) two equation transport models and (e) stress transport models. Both high and low Reynolds number forms of the k-ε model have been tried. When the high Reynolds number model has been used wall functions have been employed. The simpler models (a)-(c) do not provide satisfactory predictions. The standard k-ε model with traditional wall functions offer an improvement but there are significant discrepancies. Generally, the enhancement or reduction in heat transfer is under-estimated.

Cotton and Jackson (1987) carried out a study of turbulent mixed convection in vertical tubes using the low Reynolds number k-ε model of Launder and Sharma (1974). This model reproduced to good accuracy both the experimental heat transfer and field profiles for the air data of Carr, Connor and Buhr (1983). Li (1994) found a similar result for his experiments of upward and downward air flow in a heated tube.

Choice of UFR

The experiments described by Li (1994) are chosen as a test case. He undertook experiments of upward and downward air flow in a heated tube for various values of the Reynolds number and buoyancy parameter. This case has been chosen because (a) the tests were constructed to learn the lessons of previous experiments; for example, the test section is long enough to remove flow development effects, and (b) because it was a well constructed test.

Contributors: Mike Rabbitt - British Energy