Question

Heat Transfer in a Steel Pan A medium sized steel pan is used to boil water on top of an electric range. Approximately 85% of the 1000 W produced by the heating unit is transferred nearly uniformly into the pot during use. Express the most appropriate form of the heat diffusion equation that describes the heat transfer through the bottom section of the pan. Include all relevant boundary conditions. The bottom section of the pan has a thickness of 0.5 cm and has a diameter of 20 cm. Justify the your choices. Clearly state all assumptions – boundary conditions, materials properties, etc. Please include a control volume sketch with you justification.

Answer 1

given
heat transfer in a steel pan
rate at which heat is provided by the heating unit = Q'
Q' = 1000 W

energy succesfully transferred to the bottom surface of the pan = q'
q' = 0.85Q' = 850 W

heat diffusion equation is tgiven by

in one D
q' = -k*AdT/dx

this problem can be treated as one d with teh circular bas eof the steel pan being a plane conductor wuith height in x direction
then rate of heat transfer through the pan is given by
q' = -k*pi*r^2*dT/dx
here k is thermal conductivity of the pan
now
q' = 850 W
r = D/2 = 10 cm = 0.1 m
t = 0.5 cm = 0.005 m
since the thickness is very low as comparedc to the diameter of the pan we will get uniform heat transfer through the pan and we can wasily ignore edge effects

thermal conductivity of steel, k = 36W/mK
hence

850/pi*0.1^2*36 = -dT/dx
dT = -751.56500*dx
integrating
T(x) = -751.565x - To
there To is temeperature at bottom of the pan and T is temepraure anywhere inside the pan where x ranges from 0 to thiocness t of the pan