Assume that the water is originally at the freezing temperature Tf, that the cooling medium inside the tube surface is just below the freezing temperature at a uniform temperature Ts, and that ho is the total heat transfer coefficient between the cooling medium and the pipe-ice interface. Also show the thermal circuit.
Show that the dimensionless equation for ice formation at the outside of a tube of radius ro is
where
GIVEN
? Water freezing on the outside of a tube
? Tube radius = ro
? Cooling medium temperature = T ? (uniform)
? Heat transfer coefficient between the cooling medium and the pipe-ice interface = ho
FIND
? Draw the thermal circuit and show that the dimensionless equation for ice formation is as shown
above
ASSUMPTIONS
? Steady state
? The thermal capacitance of the ice layer is negligible
? T? is constant
? The water is at the freezing temperature, Tf
? The properties of the ice are uniform
SKETCH
The thermal circuit is shown below
where
The rate of heat transfer is given by
This is the heat flow rate which removes the latent heat of fusion necessary for freezing the ice as shown by Equation (9.34)
where pLf is the latent heat.
Combining these two equations
Expressing the above equation in terms of these dimensionless parameters:
Integrating
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