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Scale Analysis of Combined Natural and Forced Convection (Mixed Convection)

Introduction

Look around any modern building and you will see that the air inventory of each room is replenished continuously or intermittently by, in most cases, a central air-conditioning system. This means that in the vicinity of every heated wall or cooled window, the room air reservoir is actually in motion: the reservoir is forced into and out of the room by an external agent (the fan in the ventilation system). Depending on the strength of this forced circulation, the heat transfer from the wall to the room air may be ruled by either , forced convection, or a combination of natural and forced convection.There are many ways in which these two mechanisms can interact, as there are many ways in which the reservoir fluid may move relative to the direction of buoyant flow near the wall. Think of the heated wall jet rising on the outer surface of a flat solar collector in wintertime and how this wall jet will be affected by the changing wind direction and velocity. Due to the diversity of the natural– forced convection interaction, it is impossible to treat this subject fully; however, it is instructive to study one simple configuration and to experience the power and cost-effectiveness of pure scaling arguments. Let us consider the heat transfer from a vertical heated wall (T0) to an isothermal fluid reservoir moving upward (T∞, U∞), that is, in the same direction as the natural wall jet present when U∞ = 0.

From a heat transfer standpoint, the key question is: Under what conditions is the combined natural– forced phenomenon characterized (approximately) by the scales of pure natural convection, and conversely, under what conditions is it characterized by the scales of pure forced convection? In other words, what is the criterion for the transition from one convection mechanism to another?

The magnitude of different flows in the environment decides the dominant type of convection:

• Assisting flow: The buoyant motion is in the same direction as the forced motion.

• Opposing flow: The buoyant motion is in the opposite direction to the forced motion.

• Transverse flow: The buoyant motion is perpendicular to the forced motion.

In assisting and transverse flows, buoyancy enhances the associated with pure forced convection. On the other hand, opposing flows decrease the rate of heat transfer.

The Nusselt number, under combined natural and forced convection conditions, sums the contributions of natural and forced convection in assisting flows and subtracts them in opposing flows:

$Nu_{combined}=({Nu_{forced}}^{n}+-{Nu_{natural}}^{n})$

• Depending on the strength of this forced circulation, the heat transfer from the wall to the room air may be ruled by either natural convection or forced convection or a combination of natural and forced convection.
• There are many ways in which these two mechanisms can interact, as there are many ways in which the reservoir fluid may move relative to the direction of buoyant flow near the wall.

To decide the dominant convection type, we need to know the in the case of forced and natural convection. The type of convection mechanism is decided by the smaller of the two distances, $(\delta T)_{NC}$ or $(\delta T)_{FC}$, because the wall will leak heat to the nearest heat sink (or because currents seek and construct paths of greater access or faster mixing).

$(\delta T)_{NC}<(\delta T)_{FC}$

                                                    Natural Convection

$(\delta T)_{NC}>(\delta T)_{FC}$

                                                     Forced Convection

We shall discuss the above for fluids with Pr > 1 as well as those with Pr < 1.

Case 1: For Pr>1

If the mechanism is natural convection, the thermal distance between the entities is of order

$(\delta T)_{NC}\sim y\,{\text{Ra}}^{-1/4}\quad ({\text{for }}{\text{Pr}}>1)$

as reservoir fluid supplies the buoyant wall jet of thermal boundary layer thickness $(\delta T)_{NC}$.

If the mechanism is forced convection, the wall and the reservoir are separated by a thermal length of order

$(\delta T)_{FC}\sim y\,{\text{Re}}^{-1/2}\,y\,{\text{Pr}}^{-1/3}\quad ({\text{for }}{\text{Pr}}>1)$

Now,

${\frac {(\delta T)_{NC}}{(\delta T)_{NC}}}$=${\frac {{\text{Ra}}_{y}^{1/4}}{{\text{Re}}_{y}^{1/2}{\text{Pr}}^{1/3}}}$

If the above fraction is greater than 1, then natural convection dominates. If the above fraction is lesser than 1, then forced convection dominates.

Case 2: For Pr<1

If the mechanism is natural convection, the thermal distance between the heat-exchanging entities is of order

$\delta _{T}\sim H\left({\text{Ra}}_{H}\Pr \right)^{-1/4}$

as reservoir fluid supplies the buoyant wall jet of thermal boundary layer thickness $(\delta T)_{NC}$.

If the mechanism is forced convection, the wall and the reservoir are separated by a thermal length of order

${\frac {\delta _{T}}{L}}\sim {\text{Pe}}_{L}^{-1/2}\sim {\text{Pr}}^{-1/2}{\text{Re}}_{L}^{-1/2}$

In other words, for Pr > 1 fluids

${\frac {{\text{Ra}}_{y}^{1/4}}{{\text{Re}}_{y}^{1/2}{\text{Pr}}^{1/3}}}$ ^{> O(1)} Natural Convection

${\frac {{\text{Ra}}_{y}^{1/4}}{{\text{Re}}_{y}^{1/2}{\text{Pr}}^{1/3}}}$ <O(1) Forced Convection

${\frac {G_{r}y}{{\text{Re}}^{2}y}}={\frac {{\text{Ra}}^{1/4}y}{{\text{Re}}^{1/2}y\,{\text{Pr}}^{1/3}}}\cdot 4\,{\text{Pr}}^{1/3}$

${\frac {{\text{Bo}}_{y}^{1/4}}{{\text{Pe}}_{y}^{1/2}}}$ > O(1) Natural Covection

${\frac {{\text{Bo}}_{y}^{1/4}}{{\text{Pe}}_{y}^{1/2}}}$

$G_{r}y\sim 10^{9}\quad (10^{-3}\leq {\text{Pr}}\leq 10^{3})$

$R_{a}y\sim 10^{9}\,{\text{Pr}}\quad (10^{-3}\leq {\text{Pr}}\leq 10^{3})$

References

Bejan, A. (1995). *Convection Heat Transfer*. Wiley.

Incropera, F.P., & DeWitt, D.P. (2002). *Fundamentals of Heat and Mass Transfer*. Wiley.

Kays, W.M., Crawford, M.E., & Weigand, B. (2005). *Convective Heat and Mass Transfer*. McGraw-Hill.

Ni, X., et al. (2023). "Numerical analysis of mixed convection phenomena in heat transfer." *Journal of Heat Transfer*.

Chen, L., et al. (2022). "Multi-scale analysis of surface heterogeneity." *International Journal of Heat and Mass Transfer*.