Convective Heat Transfer Assignment 8 — 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.
 06.07.17
 Question #2
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.
 Question #3
You are walking on a frozen lake that has a radius of 50 m and you wish to determine the thickness of the ice layer without drilling through the ice. Remembering your heat transfer class, you rather decide to measure the temperature of the ice surface and the temperature of the air far from the ice. Knowing that there is no wind, that the air temperature far from the surface is of $-20^\circ$C, that the temperature of the ice touching the air is of $-10^\circ$C, find the thickness of the ice. You can use the following thermophysical data for ice, liquid water, and air:
 Property Ice Liquid water Air $c_p$, kJ/kgK 2 4.2 1 $k$, W/mK 2.3 0.6 0.02 $\mu$, kg/ms $10^{-3}$ $10^{-5}$ $\rho$, kg/m$^3$ 920 1000 1.2
 05.30.18
 1. $192~{\rm W/{m^2}\,^\circ C}$, $54.6^\circ{\rm C}$. 2. 23.8%, 32.9%. 3. 4.69 m.
 $\pi$ 