## Lesson 13. Dimensional analysis of free and forced convection

Empirical Relations for Free and Forced Convection

1.      Free Convection:

In case of free convection, heat transfer coefficient or Nusselt is expressed as

Nua = h L/ k = f (Gr , Pr)

Where

Nua is average Nusselt Number

Gr is Grashoff number

Pr is Prandtl Number

A)          For Vertical Plates and Cylinders

where

L is Characteristic length and it is the height of the plate or cylinder

ha  is average heat transfer coefficient.

Gr is Grashoff Number

B)          Horizontal Cylinders

Where

L is Characteristic length and in this case it is the diameter of the cylinder

ha  is average heat transfer coefficient.

Gr is Grashoff Number

C)          Horizontal Square or Circular Plates

•  For horizontal hot surface facing upward or cold surface facing downward.

• For horizontal hot surface facing downward or cold surface facing upward.

Where

L is Characteristic length and in case of square plate it is the side of the plate

L is Characteristic length and in case of circular plate it is the diameter

ha  is average heat transfer coefficient.

Gr is Grashoff Number

The properties of the fluid should be calculated at the temperature

Where Ts = Plate surface temperature     Tf = Fluid temperature.

2.      Forced Convection

In case of forced convection, heat transfer coefficient or Nusselt is expressed as

Nux= f (x*, Rex , Pr)

Subscript ‘x’ has been added to emphasize our interest in conditions at a particular location on the surface.

Where

Nux is local Nusselt Number

Rex is local Reynolds Number

Nua = f (ReL , Pr)

Subscript ‘a’ indicates an average distance from x*= 0 to the location of interest.

Where,

Nua is average Nusselt Number

ReL is Reynolds number at the location of interest

(A)  Flow of fluid over a flat surface at constant temperature

• For laminar flow over flat plate which is valid for ReL < 5 x 105.

where  ha  is average heat transfer coefficient.

hx is the local heat transfer coefficient.

• If  the flow condition on the flat plate is partly laminar and partly turbulent then for

i) Only Laminar region

where,            ha  is average heat transfer coefficient.

hx is the local heat transfer coefficient.

ii) Only Turbulent region, which is valid for ReL > 5 x 105,

where  ha  is average heat transfer coefficient.

hx is the local heat transfer coefficient.

iii) Both Laminar and Turbulent region (mixed flow)

Where TS is plate surface temperature

Tf is fluid temperature

(B)  Fluid is flowing inside the tube or through the annulus

Where Ti and To are the inlet and outlet temperatures of the fluid and

Ts is surface temperature of the tube.

Characteristic Length or Equivalent Diameter (Lc or De):

Equivalent diameter is usually expressed by the following equation

Where Ac = Cross-sectional Area      and   P = Perimeter.

So for circular tube De = D (inner diameter of the tube). The equivalent diameter is also known as characteristic length. The characteristic lengths of a few geometries are given below.

1)         The fluid is flowing through a rectangular duct as shown in Figure 1, then

2)         If the fluid is flowing through the annulus as shown in Figure 2, then

3)         If the fluid is flowing through the annulus as shown in Figure 3, then

4)         If the fluid is flowing through the annulus as shown in Figure 4, then

Last modified: Thursday, 20 March 2014, 10:04 AM