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Module 1. Basic Concepts, Conductive Heat Transfer...

Module 2. Convection

Module 3. Radiation

Module 4. Heat Exchangers

Module 5. Mass Transfer

## 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

Nu_{a} = h L/ k = f (Gr , Pr)

Where

Nu_{a} 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

h_{a} 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

h_{a} 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

h_{a} is average heat transfer coefficient.

Gr is Grashoff Number

The properties of the fluid should be calculated at the temperature

Where T_{s} = Plate surface temperature T_{f} = Fluid temperature.

**2. ****Forced Convection**

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

Nu_{x}= f (x^{*}, Re_{x} , Pr)

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

Where

Nu_{x} is local Nusselt Number

Re_{x} is local Reynolds Number

Nu_{a} = f (Re_{L} , Pr)

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

Where,

Nu_{a} is average Nusselt Number

Re_{L }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 Re

_{L}< 5 x 10^{5}.

where h_{a} is average heat transfer coefficient.

h_{x} 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, h_{a} is average heat transfer coefficient.

h_{x} is the local heat transfer coefficient.

**ii) Only Turbulent region, which is valid for Re _{L} > 5 x 10^{5},**

where h_{a} is average heat transfer coefficient.

h_{x} is the local heat transfer coefficient.

**iii) Both Laminar and Turbulent region (mixed flow)**

Where T_{S }is plate surface temperature

T_{f} is fluid temperature

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

Where T_{i} and T_{o} are the inlet and outlet temperatures of the fluid and

T_{s }is surface temperature of the tube.

**Characteristic Length or Equivalent Diameter (L _{c} or D_{e}):**

Equivalent diameter is usually expressed by the following equation

Where A_{c} = Cross-sectional Area and P = Perimeter.

So for circular tube D_{e} = 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**