Prob lem 1 Jnve stigple ihsubation e enhuel Haere s wheaher Nadus for shene Fa Kus purpok foluan. the waed h sechan u.3F ot lext 5/me ho tiine Huot it fallan hollaw mehuad hqune except Sphese Use We Seme is eped Lo Thent yau text book memot rom is Hhe hene Critical Thickness of Insulation for a Cylinder 13 2C In Fig. 13.2-3, a layer of insulation is installed around the outside of a cylinder whose radius te fixed and with a length L. The cylinder has a high thermal conductivity and the inner tem- perature T at point r outside the cylinder is fixed. An example is the case where the cylinder is a metal pipe with saturated steam inside. The outer surface of the insulation at T, is exposed to an environment at To where convective heat transfer occurs. It is not obvious if adding more insulation with a thermal conductivity of k will decrease the heat-transfer rate. At steady state, the heat-transfer rate q through the cylinder and the insulation equals the rate of convection from the surface: q=h,A(T, - T o) (13.2-23) As insulation is added, the outside area, which is A 2Tr2L, increases, but T2 decreases. However, it is not apparent whether q increases or decreases. To determine this, an similar to Eq. (13.2-5), with the resistance of the insulation represented by Eq. (13.1-11), is written using the two resistances: equation 2T LT-T,) In (r,) (13.2-24) 1 + k To determine the effect of the thickness of insulation on q, we take the derivative of q with respect to 2, equate this result to zero, and obtain the following for maximum heat flow: -27L (T, -T,) 1r,k-1/h In (5 Ir) rh dq dr2 - 0 (13.2-25) 1 k Solving, k (13.2-26) ()= ho where (r2)er is the value of the critical radius when the heat-transfer rate is a maximum. Hence, if the outer radius r2 is less than the critical value, adding more insulation will actually increase the heat-transfer rate a. Also, if the outer radius is greater than the critical, adding more insulation will decrease the heat-transfer rate. Using values of k and h, typically encountered, the critical radius is only a few mm. As a result, adding insulation on small elec- trical wires could increase the heat loss. Adding insulation to large pipes decreases the heat- transfer rate. Figure 13.2-3, Critical radius for insulation of a cylinder or a pipe. insulation T1 q -T2 TO