Question

The 7-mm-thick base plate of an iron is made of an aluminum alloy (? = 2,700 kgm-3; c = 950 Jkg-1K-1; k = 200 Wm-1K-1; ? = 0.8), and has outer surface area of 0.04 m2, perimeter of 0.7 m, and maximum length of 0.3 m. An electrical resistance heater applies a uniform heat flux of 12.5 kWm-2 to the inner surface of the base plate when the iron is turned on. The outer surface of the base plate is exposed to 25 ?C ambient air and exchanges radiation with large surroundings at 25 ?C. Estimate the time required for the base plate to reach 150 ?C from an initial temperature of 25 ?C. Use the NDSolve command in Mathematica for numerical integration of the governing differential equation.

Answer 1

here we have to assume that heat is lost to the air by only ratiation there is no convective heat transfer between surface and surrounding

weight of the aluminium alloy = 2700*0.04*0.007= 0.756 kg

emmisivity of allow () = 0.8

applying the heat balance

Heat input - Heat output = Heat accumulated .................................(1)

heat in by heating coil = 12500*0.04 watt

= 500 watt or J/sec

heat out by radiation = watt

= 0.04*0.8*5.67*10^-8(T⁴₁ - 298⁴) watt

heat accumulated =

=0.756*950*(dT/dt)

where temprature(T) is in kelvin and time(t) is in seconds

putting these values in the equation (1) and integrating the equation by putting tha limits of integration

t = 93.058 seconds