Derive an equation of the form hc = f(T, D, U) for turbulent flow of water through a long tube in the temperature range between 20° and 100°C.

Question

GIVEN

• Turbulent water flow through a long tube

• Water temperature range (Tb) = 20°C to 100°C FIND

• An expression of the form h c= f(T, D, U)

ASSUMPTIONS

• Steady state

• Variation of properties with temperature can be approximated with a power law

• Fully developed flow

• Water is being heated

SKETCH



PROPERTIES AND CONSTANTS

for water

Answer 1

Applying the Dittus-Boelter expression for the Nusselt number



To put this in the required form, the fluid properties must be expressed as a function of temperature.

Assuming the power law variation



where A and n are constant evaluated from the property values.

For density





Solving these simultaneously



Applying a similar analysis for the remaining properties yields the following relationships





Substituting these into the expression for the heat transfer coefficient



COMMENTS

Note that in equations of the type derived, the coefficient has definite dimensions. Hence, the use of

such equations is limited to the conditions specified and are not recommended.