Problem 3: Ordinary Differential Equations
A straight fin of uniform rectangular cross section (0.5 mm x 100 mm) with a length (L) of 5 cm is attached to a base surface of temperature 110°C (Tb). The surface of the fin is exposed to a cooling fluid at 20°C (Tf) with a convection heat transfer coefficient (h) of 15 W/(m².K). The conductivity (k) of the fin material is 400 W/(m.K).
(a) Plot the temperature profile along the length of the fin, i.e., T(x), and
(b) Calculate the temperature at x = 2.5 cm.
(c) Determine the heat transferred (q) by the fin.
From energy balance, the following differential equation is obtained:
d²T/dx² - hP/(kAc) * (T - Tf) = 0
where P is the fin perimeter, Ac is the fin cross-sectional area and x is the distance measured from the base. The equation has the following boundary conditions:
• The temperature at the base of fin (x = 0): T(0) = Tb
• The rate at which energy is transferred to the fluid by convection from the tip (x = L) must equal the rate at which energy reaches the tip by conduction through the fin:
hAc(T(L) - Tf) = -kAc * dT/dx|_{x=L}
The heat transferred by the fin can be calculated from the following
q = -kAc * dT/dx|_{x=0}