Heat and Mass Transfer 2(2+0)

Lesson 13. Dimensional analysis of free and forced convection

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 10⁵.

   

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 10⁵,

   

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