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

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

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

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:

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:

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