1. Internal Flow: General Considerations

2. Entrance Conditions Must distinguish between entrance and fully developed regions. Hydrodynamic Effects: Assume laminar flow with uniform velocity profile at inlet of a circular tube. – Velocity boundary layer develops on surface of tube and thickens with increasing x. – Inviscid region of uniform velocity shrinks as boundary layer grows.  Does the centerline velocity change with increasing x? If so, how does it change? – Subsequent to boundary layer merger at the centerline, the velocity profile becomes parabolic and invariant with x. The flow is then said to be hydrodynamically fully developed.  How would the fully developed velocity profile differ for turbulent flow?

3. Thermal Effects: Assume laminar flow with uniform temperature,, at inlet of circular tube with uniform surface temperature,, or heat flux,. – Thermal boundary layer develops on surface of tube and thickens with increasing x. – Isothermal core shrinks as boundary layer grows. – Subsequent to boundary layer merger, dimensionless forms of the temperature profile become independent of x. Conditions are then said to be thermally fully developed.  Is the temperature profile invariant with x in the fully developed region?

4.  For uniform surface temperature, what may be said about the change in the temperature profile with increasing x?  For uniform surface heat flux, what may be said about the change in the temperature profile with increasing x?  How do temperature profiles differ for laminar and turbulent flow?

5. The Mean Velocity and Temperature Absence of well-defined free stream conditions, as in external flow, and hence a reference velocity or temperature, dictates the use of a cross- sectional mean velocity and temperature for internal flow. Linkage of mean velocity to mass flow rate: For incompressible flow in a circular tube of radius,

6. Linkage of mean temperature to thermal energy transport associated with flow through a cross section: For incompressible, constant-property flow in a circular tube, Newton’s Law of Cooling for the Local Heat Flux: What is the essential difference between use of for internal flow and for external flow?

7. Hydrodynamic and Thermal Entry Lengths Entry lengths depend on whether the flow is laminar or turbulent, which, in turn, depend on Reynolds number. The hydraulic diameter is defined as in which case, For a circular tube,

8. – Onset of turbulence occurs at a critical Reynolds number of – Fully turbulent conditions exist for Hydrodynamic Entry Length Thermal Entry Length For laminar flow, how do hydrodynamic and thermal entry lengths compare for a gas? An oil? A liquid metal?

9. Fully Developed Conditions Assuming steady flow and constant properties, hydrodynamic conditions, including the velocity profile, are invariant in the fully developed region. What may be said about the variation of the mean velocity with distance from the tube entrance for steady, constant property flow? The pressure drop may be determined from knowledge of the friction factor f, where, Laminar flow in a circular tube: Turbulent flow in a smooth circular tube:

10. Turbulent flow in a roughened circular tube: Pressure drop for fully developed flow from x 1 to x 2 : and power requirement

11. Requirement for fully developed thermal conditions: Effect on the local convection coefficient: Hence, assuming constant properties, Variation of h in entrance and fully developed regions:

12. Determination of the Mean Temperature Determination of is an essential feature of an internal flow analysis. Determination begins with an energy balance for a differential control volume. Why is the second equality in the foregoing expression considered to be approximate? Integrating from the tube inlet to outlet,

13. A differential equation from which may be determined is obtained by substituting for Special Case: Uniform Surface Heat Flux Why does the surface temperature vary with x as shown in the figure? In principle, what value does T s assume at x=0? Total heat rate:

14. Special Case: Uniform Surface Temperature Integrating from x=0 to any downstream location, Overall Conditions:

15. Special Case: Uniform External Fluid Temperature Note: Replacement of by T s,o if outer surface temperature is uniform.

16. Problem 8.17: Estimate temperature of water emerging from a thin-walled tube heated by walls and air of a furnace. Inner and outer convection coefficients are known.

17. SCHEMATIC: