Solve using MATLAB

Consider a mathematical model for heat transfer in solids which is submerged in fluids,

eq. (1)

where is the temperature gradient between the temperature of the solid and the surrounding constant ambient temperature in kelvin K (1K = 273+^oC), is material constant, and A is contact surface area. Suppose a heated material sample at 80^oC is being cooled in an oven maintaining the constant temperature of 5^oC. The material constants and contacting surface area are given to be and .

a) Write down the state equation (1) in terms of state variable such that it can be used for computing the response of the solid .

b) Find the numerical response using ODE45 and compare the response with those obtain analytically in Lab 1 by plotting them on a single plot together. Summarize your observation related to this comparison plot.

Choose for plots where, is the time constant for cooling.

c)

Find the numerical response using ode45 and plot the responses for two cases. Case I: When the material parameter changes in a range of and Case II: When the surface area changes in a range of .

Based on the response plots, summarize, what is the effect of material constants and surface areas on the response and what combination of and , can be chosen based on these plots to cool down this material sample faster?

Choose for plots where, is the time constant for cooling.