1. Internal Convection: Overview
Internal Flows: Thermal Effects
assume laminar flow with uniform inlet temperature Ti
two common boundary conditions
uniform heat flux
uniform surface temperature
simplest internal flow: laminar flow in circular tube
thermal boundary layer thickens on surface of tube with increasing x
isothermal core shrinks as boundary layer grows.
subsequent to boundary layer merger at the centerline, dimensionless forms of the temperature profile become invariant with x  thermally fully developed

2. Internal Convection: Mean Conditions
External flow reference conditions: free stream values u∞,T∞
Internal flow reference conditions: cross-sectional mean values um, Tm
Mean Velocity
- can be tied directly to the mass flow rate (a consequence of mass conservation)
cross-sectional flow area
- for incompressible flow in circular tubes of radius ro

3. Internal Convection: Mean Conditions
Mean Temperature
- can be tied directly to the thermal energy transport  amount of energy advected by the flow (carried by the flow)
cross-sectional flow area
- for incompressible, constant property flow in circular tubes of radius ro
- local heat flux defined by Newton’s Law of Cooling

4. Internal Convection: Entry Lengths
depends on laminar or turbulent condition  entry length a function of Reynolds number
for internal flow the characteristic length is the hydraulic diameter
cross-sectional flow area Ac perimeter P
hydraulic diameter:
Reynolds number:
Laminar/Turbulent Conditions
onset of turbulence (critical Reynolds number):
fully turbulent conditions:
Hydrodynamic Entry Length
laminar:
turbulent:
Thermal Entry Length
laminar:
turbulent:

5. Internal Convection: Fully Developed
Assuming steady flow and constant properties, hydrodynamic conditions, including the velocity profile, are invariant in the fully developed region.
The pressure drop may be determined from knowledge of the friction factor, f
laminar (analytical):
turbulent (empirical):
Pressure drop from x1 to x2 for fully developed flow:
The pump power required to overcome the resistance to flow from this pressure drop is:

6. Internal Convection: Fully Developed
Assuming steady flow and constant properties, relative thermal conditions, including the non-dimensional temperature profile, are invariant in the fully developed region.
The effect of the thermally fully developed conditions on the convection coefficient
variation of h
entrance region
fully developed region
assuming constant properties