Correlations for INTERNAL CONVECTION P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi An Essential Part of Exchanging Heat……..

Integrating from x=0 (T m = T m,i ) to x = L (T m = T m,o ): Constant Surface Heat Flux : Heating of Fluid

Constant Surface Temperature heating or cooling T x T x

T x T x

Define Log Mean Temperature Difference :

Constant Surface Temperature heating or cooling T x T x

The above expression requires knowledge of the exit temperature, which is only known if the heat transfer rate is known and vice versa. An alternate equation can be derived which eliminates the outlet temperature. We already Know

System Thermal Resistance:

Constant wall temperature : Thermally Developed Flow Boundary conditions: For hydrodynamically developed flow:

This problem has been solved by Bhatti (1985): Where,

Convection correlations: laminar flow in circular tubes 1. The fully developed region for constant surface heat flux for constant surface temperature Note: the thermal conductivity k should be evaluated at average T m

Thermally developing, hydrodynamically developed laminar flow (Re < 2300) Constant wall temperature: Constant wall heat flux:

Simultaneously developing laminar flow (Re < 2300) Constant wall temperature: Constant wall heat flux: which is valid over the range 0.7 < Pr < 7

Fully developed turbulent and transition flow (Re > 2300) Constant wall Temperature: Where Constant wall temperature: For fluids with Pr > 0.7 correlation for constant wall heat flux can be used with negligible error.

Convection correlations: turbulent flow in circular tubes A lot of empirical correlations are available. For smooth tubes and fully developed flow. For rough tubes, coefficient increases with wall roughness. For fully developed flows f is friction factor.

Heat Transfer in Entry Length A general expression for the ratio of the local heat transfer Coefficient to the fully developed value is

Variable properties Wall temperature T s or T w Fluid temperature T b (mean bulk temp) For small changes T i or T o may also be used For example there may be a large radial temperature gradient in circular duct. At what temperature properties are evaluated matters. There may be a need for temperature correction in correlations. Indices cp and vp correspond to constant and variable properties.

Some properties are strong functions of temperature. Convention: –For liquids lump all property variations to  (dynamic viscosity). Sometimes variations are lumped to Pr. –For gases use temperature dependence directly (everything depends on T) Fluids: Gases: where n and m depends on the case.

Turbulent Liquid Flow in Ducts Petukhov reviewed the status of heat transfer in fully developed turbulent pipe flow with both constant and variable physical properties. Validity range: 10 4 < Re b < 5 x 10 6, 2 < Pr b < 140 and 0.08 <  w /  b ) < 40 

Turbulent Gas Flow in Ducts Petukhov & popov reviewed the status of heat transfer in fully developed turbulent pipe flow with both constant and variable physical properties. Validity range: 10 4 < Re b < 4.3 x 10 6 and 0.37 <  w /  b ) < 3.1 for air  Validity range: 10 4 < Re b < 5.8 x 10 6 and 0.37 <  w /  b ) < 3.7 for hydrogen.

Noncircular Tubes: Correlations For noncircular cross-sections, define an effective diameter, known as the hydraulic diameter: Use the correlations for circular cross-sections.

Selecting the right correlation Calculate Re and check the flow regime (laminar or turbulent) Calculate hydrodynamic entrance length (x fd,h or L he ) to see whether the flow is hydrodynamically fully developed. (fully developed flow vs. developing) Calculate thermal entrance length (x fd,t or L te ) to determine whether the flow is thermally fully developed. We need to find average heat transfer coefficient to use in U calculation in place of h i or h o. Average Nusselt number can be obtained from an appropriate correlation. Nu = f(Re, Pr) We need to determine some properties and plug them into the correlation. These properties are generally either evaluated at mean (bulk) fluid temperature or at wall temperature. Each correlation should also specify this.

Heat transfer enhancement Enhancement Increase the convection coefficient Introduce surface roughness to enhance turbulence. Induce swirl. Increase the convection surface area Longitudinal fins, spiral fins or ribs.

Heat Transfer Enhancement using Inserts

Heat transfer enhancement :Coiling Helically coiled tube Without inducing turbulence or additional heat transfer surface area. Secondary flow