A solar-heating water system is being considered that consists of a rectangular tank full of water under pressure (so that it does not boil) insulated on 5 of its 6 faces. The tank is buried with the un-insulated face uppermost, exposed to the sun. The exposed face is coated with a high emissivity coating, ensuring that the emissivity of the exposed face is effectively 1.0. The depth of the tank is d. The tank is made of thin- walled steel so that its mass is negligible by comparison with the mass of contained water. The water in the tank is continuously stirred, so that it may be considered to be of uniform temperature throughout. Heat is lost by convection and radiation at the exposed face. Let: T be the temperature (in K) of water in the tank, To be the ambient temperature (in K), assumed to be constant, h be the convection heat transfer coefficient (in WmK) for convection losses, assumed to be constant, 0 = 5.67x10* WmK* be the Stefan-Boltzmann constant, Po be the density of water (in kgm) assumed to be constant, c. be the specific heat capacity of water (in Jkg-'K ) assumed to be constant, Q, be the heat flux from the sun (in Wm2) at the exposed surface of the tank. . Under the assumptions above, conservation of energy implies that the water temperature T' is given by the solution of the nonlinear IVP 1 dT dt -{6, +h(T. -T)+0(T* T)}, 7(0)=T. P.dc (1.0) Task 1 For the following calculation, take T. = 288 K. If Q, is constant, the maximum temperature of the water is given by the solution (for 1) of the equation Q; +h(T.-T)+(T* -T4)=0. -T0T (1.1) Let the solution of Eqn. (1.1), corresponding to the maximum temperature, be denoted by Tmx (a) Design a spreadsheet in MS-Excel to implement the Newton-Raphson method to solve Eqn. (1.1) using the initial guess T, given by T, =T, + , h and then use it to complete the table below taking h = 7 Wm+K' (the first row is included for you so that you can check that you are on the correct track): Qs / Wm2 To/K Tmax/K 250 323.71 307.25 500 750 1000 1250 1500 Table 1. Tmx for h= 7 Wm?! Quote your solutions to 2 d.p. and include extracts of your spreadsheet with explanations of the formulae that you used. [20 marks]