Answer Happy

dr N t The ring element has volume V = 2trdrt. The energy balance on this differential volume can be stated as "thermal energy conducted into V = (thermal energy conducted out of 7) + (energy lost to surroundings by convection)." This can be expressed mathematically by ring element ro that + dT dT - kA - KA hĄ (T-T.) dr dr (conduction in) (conduction out) (convection loss) where the conduction terms are expressed by Fourier's law and the convection loss is given by Newton's law of cooling. (We skip over some of the heat transfer details.) Parameters are defined by A = 2Art and Ac = 2(2Ardr), k = thermal conductivity, and h = konvective heat • TO transfer coefficient. Substituting in the area Pipe parameters and rearranging gives = Fin - dᎢ dr Iruder dr dT dr 2hr (T - Tv) = 0 tk In the limit dr → 0 with u = T - T. (To is constant) and m = (2h/tk)1/2, this relation becomes 1d r dr du - dr -mu=0, which is modified Bessel's equation of order zero. The solution is expressible by use of To(mr) and Ko(mr) called modified Bessel functions of order 0 of the first and second kind: u(mr) = ci Io(mr) + c2 Ko(mr). Let's consider a specific problem with boundary conditions of specified temperatures at ri and ro being u(mri) = 100 and u(mro) = 10, respectively. With the general solution and the boundary conditions, determine the coefficients cị and ca, and plot the profile of u with respect to r for m = 0.1, 0.5 and 1.0.