I have water flowing at 4 GPM at 2.5 feet per second through 50ft of 0.375 in. outer diameter tubing with 0.014 in. wall thickness. The water flows through copper tubing into the 50ft coil section, which is submerged in an adiabatic container of 22lbs of ice, and then out again. How do I set up the heat transfer equation to find the time it will take the ice to melt, as well as the temperature of the water as it leaves the adiabatic container? Thank you.
You need to calculate the heat flux through the pipe. Since the water gives away heat the temperature will drop as it flows through the 50ft coil section and the the heat flux depends on the temperature difference between the water in the pipe and the temperature of the ice on the outside of the pipe.
If you want to calculate it manually you could divide the coil into two sections. Outgoing water temperature from the first section is ingoing water temperature in the second section. Now, divide the section into four parts and do the calculation again. Continue to divide the coil section into more and more subsections until the difference in the outgoing temperature is neglectable.
Good luck!
It seems that you have a dynamic thermo-fluid system here. Your water temperature will depend on time and position. On a first approximation you could consider the pipe as a homogeneous volume, but for a best accuracy you will have to discretize it into N volumes.
For the thermal transfer part, it seems that the limiting transfer will be the forced convection between the water and the pipe. I think you can consider the inner wall temperature to be at 0°C. To take it further you can also model the resistive transfer of the wall.
To estimate the quantity of melted ice per time step, you can divide the heat loss by the fusion enthalpy of water.
So for N=1 the heat balance equation should look like that:
m_flow * h_in - hT * A *(T_water - T_ice) = m_flow * h_out
with :
- m_flow the mass flowrate of water
- h the specific enthalpy of the water
- hT the convective heat transfer coefficient
- A the surface of echange
- T the temperature of the water
Then you have the time for melting all the ice:
t= m_ice * deltaH_of_fusion / (hT * A *(T_water - T_ice))
with m_ice the mass of ice
P.S. I said before that it is a dynamic problem, but you can neglect the capacitive effect of the volume, so you can write a steady state equation.
The challenge in this problem is to calculate heat transfer coefficients for each section of tubing based on the flowing water local temperature and an assumed constant temperature in the bath. You basically have to compute the heat lost by the flowing fluid in each section then compute the temperature of the fluid entering the next section. Then compute the heat transfer in the next section and so on. You assume that the bath temperature is constant until you've transfered enough heat to melt all the ice.
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