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.
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
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.
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