Module 3. Dimensional analysis and heat transfer
Lesson 15
HEAT TRANSFER BY FREE CONVECTION
15.1 Introduction
The relationship between dimensionless groups are usually presented as power functions e.g. Nu = CRe^{n} × Pr^{m}, where c, n and m are abstract constants. Relationships of this type are purely empirical. They are employed only within the limits of independent variable fixed by experiment. They are not to be extrapolated for larger or smaller values of the argument.
Let us see how such relationships are practically found.
Consider Nu = CRe^{n}, Graphically this function can be expressed as a straight line on a log – log paper. (Fig. 15.1)
i.e. log Nu = Log C + n × log Re
Y = A + nx, denoting log Nu by Y, log Re by X and log c by A
n = tan φ, where φ is the angle the straight line makes with x – axis.
The constant ‘c’ is found from the relationship
holds for any point of the curve, if the test data is a curve, replace it by broken lines. Find c and n for different regions of broken line.
Experimental studies of heat transfer in the air moving within a tube have made possible the establishment of the following generalized relationship.
Nu_{f} = 0.018 Re^{0.80}
Heat transfer by free convection from different solids to fluid.
Relationship between the dimensionless groups can be represented by following power function.
Nu_{m} = where m stand for mean
The vales of C and n are different for different conditions and are functions of independent variable (Gr ×Pr).
Table 15.1

C 
n 
Flow Conditions 
1 × 10^{3} to 5 × 10^{+2} 
1.18 
1/8 
Transient conditions 
5 × 10^{+2}  2 × 10^{7} 
0.54 
¼ 
Flow in laminar 
2 × 10^{7}  1 × 10^{13} 
0.135 
1/3 
Flow becomes turbulent 
When (Gr ×Pr)_{m} < 1, Nu_{m} = 0.5 and remains constant, which means i.e. process of heating or cooling is fully determined by the thermal conductivity of the surroundings. This holds true only for heat transfer in film type conditions.
The above formula is applicable to any liquids and gases at P_{r} ≥ 0.7 and to bodies of any shape and size.
If the liquids and gases are free flow in horizontal tubes, we can use
For
air, formula is simplified as ,
this is convenient for practical calculations.