Chapter 8: Internal Flow The fluid is confined by a surface – flow and heat transfer in tubes of different cross section Hydrodynamic consideration: Reynolds number: where um is the mean fluid velocity In a fully developed flow, the critical Reynolds number for the onset of turbulence is:

For laminar flow ( ), the hydrodynamic entry length : For turbulent flow, approximately, , independent of Reynolds number The mean velocity: For external flows, , free stream velocity – reference velocity For Internal flows, a mean velocity um is defined: For constant and , um is a constant If we know velocity distribution For incompressible flow in a circular tube,

Velocity profile in the fully developed region: Laminar flow, incompressible, constant property flow, circular tube Momentum equation in (r, x) coordinate system for = constant

Boundary conditions: (symmetry condition) with um Replace dp/dx Pressure gradient and friction factor in fully developed flow Moody friction factor:

For laminar flow, For turbulent flow – experiment results –Moody diagram (e/D: relative roughness)

Thermal Considerations The mean temperature Thermal energy transported by the fluid as it moves past the cross section The mean temperature is defined such that

Fully Developed Conditions: The heat transfer can also be calculated using Newton’s law of cooling: (external flow: ) Difference: may be constant, changes with x Fully Developed Conditions: Fully developed flow: , velocity profile does not change , the temperature profile T(r) is continuously changing with x Define a dimensionless temperature or the flow is said to be thermally fully developed

If we expand the above equation, and solve for Also

For uniform surface heat flux (if the tube wall were heated electrically, or the outer surface were uniformly irradiated) constant For constant surface temperature ( a phase change occurring at the outer surface--boiling or condensation)

The Energy Balance How the mean temperature Tm(x) varies with position along the tube How the total convection heat transfer qconv is related to difference in temperature at the inlet and outlet P: perimeter Not a distribution, just a energy balance, very general, apply to any flow structure (without change of phase)

Constant surface heat flux

Constant surface temperature or

If we integrate from the tube inlet to some axial position x within the tube, we can obtain similar but more general result: Reconsider the energy balance equation

: log mean temperature difference for total heat transfer rate (This is compared to arithmetic mean temperature difference: ) For heat transfer between the fluid flowing over a tube and the fluid passing through the tube, it is the temperature of an external fluid, rather than the tube surface temperature, which is fixed --- Heat exchangers

Laminar Flow in Circular Tubes: Thermal Analysis and Convection Correlations

The Fully developed Region (Thermal and Hydrodynamic) and circular tube, the energy equation is: For fully developed velocity Thermally fully developed with constant surface heat flux

Integrating twice: At At Find dTm/dx from this relation

For laminar flow, fully developed condition with a constant surface temperature for Ts = constant For turbulent flow, empirical correlations are used, see the table on the next page.