Internal Flow:General ConsiderationsChapter 8Sections 8.1 through 8.3

Entrance ConditionsEntrance 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?

Entrance Conditions (cont) Thermal boundary layer develops on surface of tube and thickens with increasing x. Isothermal core shrinks as boundary layer grows. Conditions are then said to be thermally fully developed. Is the temperature profile invariant with x in the fully developed region?

Entrance Conditions (cont) 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?

Mean QuantitiesThe Mean Velocity and Temperature

Mean Quantities (cont) Linkage of mean temperature to thermal energy transport associated with flow through a cross section:

Entry LengthsHydrodynamic and Thermal Entry Lengths

Entry Lengths (cont) Onset of turbulence occurs at a critical Reynolds number of For laminar flow, how do hydrodynamic and thermal entry lengths compare for a gas?An oil?A liquid metal?

Fully Developed FlowFully 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?

Fully Developed Flow (cont)Turbulent flow in a roughened circular tube:

Fully Developed Flow (cont)

Mean TemperatureDetermination of the Mean TemperatureDetermination 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,

Mean Temperature (cont) Special Case: Uniform Surface Heat FluxWhy does the surface temperature vary with x as shown in the figure?In principle, what value does Ts assume at x=0?Total heat rate:

Mean Temperature (cont) Special Case: Uniform Surface TemperatureIntegrating from x=0 to any downstream location,Overall Conditions:

Mean Temperature (cont) Special Case: Uniform External Fluid Temperature

Problem: Water Flow Through Pipe in FurnaceProblem 8.17: Estimate temperature of water emerging from a thin-walled tube heated by walls and air of a furnace. Inner and outerconvection coefficients are known.

Problem: Water Flow Through Pipe in Furnace (cont)SCHEMATIC:

Problem: Water Flow Through Pipe in Furnace (cont)