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Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering

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5.2 Empirical relations for natural-convective heat transfer        

5.2.1 Natural convection around a flat vertical surface         

Churchill and Chu provided the correlation for average heat transfer coefficient for natural convection for different ranges of Rayleigh number.

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering                    (5.7)
Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering       (5.8)

 

It should be noted that the eq. 5.7 and 5.8 are also applicable for an inclined surface upto less than inclination from the vertical plane.

The above relations can be used for the vertical cylinder if the boundary layer thickness is quite small as compared to the diameter of the cylinder. The criteria to use the above relation for vertical cylinder is,

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering                  (5.9)

where,  is the diameter and  is the height of the cylinder.


5.2.2 Natural convection around a horizontal cylinder          

Churchill and Chu has provided the following expression for natural-convective heat transfer.

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering                (5.10)

Condition of applicability of the eq.5.10:

5.2.3 Natural convection around a horizontal flat surface   

In the previous case of vertical flat surface, the principal body dimension was in-line with the gravity (i.e. vertical). Therefore, the flow produced by the free convection was parallel to the surface regardless of whether the surface was hotter or cooler compared to the bulk fluid around. However, in case of horizontal flat plate the flow pattern will be different and shown in fig. 5.3.

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering
Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering
 

Fig. 5.3: A representative flow pattern (natural convection) for (a) hot surface down, (b) hot surface up, (c) cold surface down, and (d) cold surface up

Thus from fig. 5.3 it is understood that there are in fact two cases (i) when the heated plate facing up or cooled plate facing down, and (ii) heated plate facing down or cooled plate facing up.

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering

 

where, Lc is characteristic length defined as below.

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering


5.2.4 Natural convection around sphere

Churchill proposed,

 

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering(5.14)

Condition for applicability: Pr ≥ 0.7; Ra ≤ 1011

 

5.2.5 Natural convection in enclosure

It is another class of problems for which there are many cases and their corresponding correlations are also available in the literature. Here two cases will be discussed, (i) in which a fluid is contained between two vertical plates separated by a distance d, (ii) the other where the fluid is in an annulus formed by two concentric horizontal cylinders.

In the case first, the plates are at different temperature, T1 and T2. Heat transfer will be from higher temperature (T1) to lower temperature (T2) through the fluid.

The corresponding Grashof number will be

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering

McGregor and Emery proposed the following correction for free convection heat transfer in a vertical rectangular enclosure, where the vertical walls are heated or cooled and the horizontal surfaces many be assumed adiabatic,

 

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering(5.15)

Applicability conditions for the above equation are,

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering

or,

 

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering                       (5.16)

Applicability conditions are,

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering

Here L/d is known as the aspect ratio.

At steady state condition, the heat flux (qx) is equal thus,

 

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering

where, kc(= Nuxk) is known as the apparent thermal conductivity.

In the second case the heat transfer is involved in the enclosure formed by two concentric cylinders in horizontal position, the correlation given by Raithby and Holland,

 

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering(5.17)

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering is the modified Rayleigh number given by,

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering

where, di and d0 are the outer and inner diameter of the inner and outer cylinders, respectively. The enclosure characteristic length l is defined as (d0 - di).

The applicability of the eq. 5.17 is 102Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering  > 107.

It should be noted that the rate of heat flow by natural convection per unit length is same as that through the annular cylindrical region having effective thermal conductivity ke for the case,

Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering

where, T1 and T2 are the temperatures of the inner and outer cylindrical walls, respectively.

The document Heat Transfer by Natural Convection - 2 Notes | Study Heat Transfer - Mechanical Engineering is a part of the Mechanical Engineering Course Heat Transfer.
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