Water fills a rectangular chamber that has width X2-Xo, height Hand width W. On the sides perpendicular to the x axis (see the figure below), the temperature is maintained at To which is below the freezing point of water (for example -20°C) and at the opposite end at a temperature T2, for which the temperature water, kW is greater than the freezing point, e.g. 10°C. The Temperature freezing point of water is T(x) Tm. The other four sides T2=100 are all insulated so heat only transports in the x Tm=0°C direction. At steady state, part of the water in the To=-2006 chamber is frozen and part is liquid. The boundary between them is denoted as the position x=X1. The - X object is to calculate X1-Xo . The thermal conductivity of the ice is k' and that of the water is kW

(i) Write out the energy conservation equation in each region in terms of QxCE and water. Integrate to obtain the flux distributions. 

(ii) Relate the two constants in the solutions by the flux condition at the boundary between the ice and water so that only one constant remains. 

(iii) Use Fourier's law of heat conduction to obtain the temperature distributions in the two regions, Tice (x) and water(x) in terms of two more constants. Use the boundary conditions at xo (temperature To), X1 (continuity of temperature) and X2 (temperature T2) to obtain the constants 

(iv) Solve for the position of the ice/water boundary, X1-Xo from the integrated temperature solutions and the requirement of the temperature at the boundary. 

(v) From the continuity of flux at the ice/water boundary, and the ctivities of ice and water, in which region is the slope of the temperature field the largest. Does the figure depict the slopes correctly?

ice, ki water, kW Temperature T(x) T2 100 C Tm=0 H To=-2000 X, X2 N