3. ( 40 points) A flat solid plate made of an aluminum alloy is initially at TI=425C, and then it is submerged in a quiescent fluid by suspending the plate vertically in air at constant bulk temperature T=20.0C. The plate is cooled until it reaches Tf=55C. The vertical plate is square with H=275 mm length per side by W=25mm thick. The alloy properties at Ts=0.5(Tt+Tf)=240C are density of 2770kg/m3, specific heat capacity of 875J/kgK, and thermal conductivity of 177W/mK. The fluid air properties at Tfilm=0.5(Ts+T)=130C are density of 0.871kg/m3, specific heat capacity of 1010J/kgK, viscosity 2.30105kg/ms and thermal conductivity of 0.0338W/mK. The air thermal expansion coefficient has an empirical correlation =1/Tfilm (in Kelvin). (a) Calculate the Rayleigh number for this system. (b) From the attached tables, find the appropriate Nu correlation for this system. Determine the symbolic expression and the value of the average heat transfer coefficient h using the correlation. (c) Now suppose that we add a fan and now the air flows downward (in the direction of g ) with velocity 0.015m/s. Calculate the Reynolds number for this system. From the attached tables, find the appropriate Nu correlation for this system. Determine the symbolic expression and the value of the average heat transfer coefficient h using the correlation. (d) Using the lumped capacitance approximation, determine the expression for the temperature of the plate T(t) (e) Compare the rates of free and forced convection cooling by plotting a graph of the plate temperature (in C ) versus cooling time (in hr) for both the free convection h in part (b) and the forced convection h in part (d). Hint: Make sure your plot is correctly labeled. You must include the equation and data table that you are plotting. You are encouraged to use Matlab to generate the plot and include your script in your answer. (f) How long will it take to cool the plate to Tf=55C by free and forced convection, respectively? TABLE 8.2 Nusselt number for fully developed laminar flow in a circular tube annulus with one surface insulated and the other at constant temperature TABLE 7.7 Summary of convection heat Iransfer correlations for external flow a.b "Correlations in this table pertain to isothermal surfaces; for special cases involviing an unheated starting length or a uniform surface heat flux. see Section 7.2 .4 or 7.2 .5 'When the heat and mass transfer analogy is applicable, the corresponding mass rransfer correlations may be obtained by replacing Nu and Pr by Sh and Sc, respectively. "The temperuture listed under "Conditions" is the temperature at which properties should be evaluated. "For tube banks and packed beds, properties are evaluated at the average fluid temperature, T=(T1+T0)/2. 14