Convective Heat Transfer Assignment 7 — Phase Change  
Question #1
A thin-wall copper pipe in which a cooling fluid flows is used to condensate steam. The steam incoming temperature is of $100^\circ$C, the pipe length is of 2 m, the pipe diameter is of 0.05 m, and the cooling fluid has the following properties:
$$ c=4000~{\rm J/kgK},~~~ k=0.5~{\rm W/m^\circ C},~~~ \rho=1000~{\rm kg/m^3},~~~ \mu=2.5\times 10^{-4}~{\rm kg/ms} $$ You conduct a first experiment in which the mass flow rate of the cooling fluid is of 1 kg/s, and the temperature of the cooling fluid entering the pipe is of $20^\circ$C. For a measured rate of condensation of the steam of 0.002 kg/s, and knowing that $T_{\rm sat}=100^\circ$C and $\Delta H_{\rm vap}=2260$ kJ/kg find $h_{\rm condensate}$.
Taking the latter into consideration, and assuming that $h_{\rm condensate}$ does not depend significantly on the cooling fluid inflow temperature and mass flow rate, estimate the bulk temperature at the exit should the temperature and mass flow rate of the cooling fluid entering the pipe be of $40^\circ$C and 0.01 kg/s, respectively.
Question #2
On a cold winter day, water in a river flows with an average velocity of 0.5 m/s. The bulk temperature of the river water is essentially constant at 0.2$^\circ$C. The air temperature is $-30^\circ$C. Under steady-state conditions, a continuous layer of ice forms on the top surface of the river. The depth of the river below the ice layer is 1 m, and its width is much larger than its depth. The convective heat transfer coefficient at the ice-air interface is $h_{\rm o}=40$ W/m$^2\cdot^\circ$C. The thermal conductivity of the ice may be assumed constant: $k_{\rm ice}=2.0~$W/m$\cdot ^\circ$C. The properties of the river water may be assumed constant at the following values: $$ \rho=1000~{\rm kg/m}^3~~~\\ \mu=8.6 \times 10^{-4}~{\rm kg/m\cdot s}~~~\\ k=0.60~{\rm W/m \cdot ^\circ C}~~~\\ c_p=4186~{\rm J/kg\cdot ^\circ C} $$ Estimate the thickness of the ice layer. Hints: (i) Water freezes at 0$^\circ$C; (ii) The hydraulic diameter of a channel is equal to twice its height.
Question #3
A boat in the antarctic is towing a small iceberg. The iceberg is at a temperature of $-30^\circ$C and is 10 m long, 10 m wide, and 2 m high. The iceberg has to be towed over a distance of 30 km. Knowing that the water temperature in the antarctic ocean is of $3^\circ$C, that the latent heat of melting for water is of 334 kJ/kg, that the water properties correspond to: $$ \rho_{\rm w}=1000~{\rm kg/m^3},~~~(c_p)_{\rm w}=4000~{\rm J/kgK}, ~~~k_{\rm w}=0.6~{\rm W/m^\circ C}, ~~~\mu_{\rm w}=10^{-3}~{\rm kg/ms} $$ and that the iceberg properties correspond to: $$ \rho_{\rm ice}=920~{\rm kg/m^3},~~~(c_p)_{\rm ice}=1900~{\rm J/kgK},~~~k_{\rm ice}=2.5~{\rm W/m^\circ C} $$ find out the percentage of the iceberg that would melt for the following two scenarios:
(a)  The boat speed is of 1.944 knots
(b)  The boat speed is of 0.1944 knot
Hints: (i) you can assume that there is no heat transfer between the iceberg and the air; (ii) 1 knot is equal to 1.852 km/hour.
1.  $192~{\rm W/{m^2}\,^\circ C}$, $54.6^\circ{\rm C}$.
2.  0.237 m.
3.  23.8%, 32.9%.
Due on Wednesday June 14th at 16:30. Do all three questions.
Note: I made a change to the formulation of Q1 on Tue Jun 13 at 11:15. Make sure you solve the latest Q1.
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