User:Shikha dhurve

Derivation

The Nusselt number may be obtained by a non dimensional analysis of the Fourier's law since it is equal to the dimensionless temperature gradient at the surface: , where q is the heat flux, k is the thermal conductivity and T the fluid temperature. Indeed if:, and we arrive at : then we define : so the equation become : By integrating over the surface of the body: , where [edit]Empirical Correlations

[edit]Free convection [edit]Free convection at a vertical wall Cited[1] as coming from Churchill and Chu:

[edit]Free convection from horizontal plates If the characteristic length is defined

where is the surface area of the plate and  is its perimeter, then for the top surface of a hot object in a colder environment or bottom surface of a cold object in a hotter environment[1]

And for the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment[1]

[edit]Flat plate in laminar flow The local Nusselt number for laminar flow over a flat plate is given by[2]

[edit]Flat plate in turbulent flow The local Nusselt number for turbulent flow over a flat plate is given by[2]

[edit]Forced convection in turbulent pipe flow [edit]Gnielinski correlation Gnielinski is a correlation for turbulent flow in tubes:[2]

where f is the Darcy friction factor that can either be obtained from the Moody chart or for smooth tubes from correlation developed by Petukhov[2]:

The Gnielinski Correlation is valid for[2]:

[edit]Dittus-Boelter equation The Dittus-Boelter equation (for turbulent flow) is an explicit function for calculating the Nusselt number. It is easy to solve but is less accurate when there is a large temperature difference across the fluid. It is tailored to smooth tubes, so use for rough tubes (most commercial applications) is cautioned. The Dittus-Boelter equation is:

where: is the inside diameter of the circular duct is the Prandtl number for heating of the fluid, and for cooling of the fluid.[1] The Dittus-Boelter equation is valid for [3]

Example The Dittus-Boelter equation is a good approximation where temperature differences between bulk fluid and heat transfer surface are minimal, avoiding equation complexity and iterative solving. Taking water with a bulk fluid average temperature of 20 °C, viscosity 10.07×10¯⁴ Pa·s and a heat transfer surface temperature of 40 °C (viscosity 6.96×10¯⁴, a viscosity correction factor for can be obtained as 1.45. This increases to 3.57 with a heat transfer surface temperature of 100 °C (viscosity 2.82×10¯⁴ Pa·s), making a significant difference to the Nusselt number and the heat transfer coefficient. [edit]Sieder-Tate correlation The Sieder-Tate correlation for turbulent flow is an implicit function, as it analyzes the system as a nonlinear boundary value problem. The Sieder-Tate result can be more accurate as it takes into account the change in viscosity ( and ) due to temperature change between the bulk fluid average temperature and the heat transfer surface temperature, respectively. The Sieder-Tate correlation is normally solved by an iterative process, as the viscosity factor will change as the Nusselt number changes.[4] [1] where: is the fluid viscosity at the bulk fluid temperature is the fluid viscosity at the heat-transfer boundary surface temperature The Sieder-Tate correlation is valid for[1]

[edit]Forced convection in fully developed laminar pipe flow For fully developed internal laminar flow, the Nusselt numbers are constant-valued. The values depend on the hydraulic diameter. For internal Flow:

where: Dh = Hydraulic diameter kf = thermal conductivity of the fluid h = convective heat transfer coefficient [edit]Convection with uniform surface heat flux for circular tubes From Incropera & DeWitt,[5]

[edit]Convection with uniform surface temperature for circular tubes For the case of constant surface temperature,[5]