Steam in a heating system flows through tubes whose outer diameter is 5 cm and whose walls are maintained at a temperature of 130 ◦c. circular aluminum alloy 2024-t6 fins (k = 186 w/m · k, ηf = 0.97) of outer diameter 6 cm and constant thickness 1 mm are attached to the tube. the space between the fins is 3 mm, and thus there are 250 fins per meter length of the tube. heat is trans- ferred to the surrounding air at t[infinity] = 25 ◦c, with a heat transfer coefficient of 40 w/m2 · k.
determine the increase in heat transfer from the tube per meter of its length as a result of adding fins.

Newton's law of cooling states that the rate of change in the temperature t(t) of a body is proportional to the difference between the temperature of the medium m(t) and the temperature of the body. that is, startfraction dt over dt endfraction equals upper k left bracket upper m left parenthesis t right parenthesis minus upper t left parenthesis t right parenthesis right bracket , where k is a constant. let kequals0.04 left parenthesis min right parenthesis superscript negative 1 and the temperature of the medium be constant, m(t) font size decreased by 3 equivalent font size decreased by 3 294 kelvins. if the body is initially at 369 kelvins, use euler's method with hequals0.1 min to approximate the temperature of the body after (a) 30 minutes and (b) 60 minutes.