1. A spherical shell (Fig. 1} has inner and outer radii R1 and R2 . A hole is made in the shell at the north pole by cutting out the conical segment in the region 0 i 93 31 . A similar hole is made at the south pole by removing the portion [31 Elli 5'1: 3'1 . The surface 3: 91 is kept at temperature T2 T1 1 and the surface at 8 = (trEll] is held at T 2T2 . Find the steady state temperature distribution, using the heat conduction equation. \"-2- .1! "'I' All IImI'I i-u-r. '. \"will.\" in |.I' .I a .lllplu Fig. 1 2. Temperature distribution in an embedded sphere. A sphere of radius R and thermal conductivity k1 , is embedded in an infinite solid of thermal conductivity k0 . The center of the sphere is located at the origin of coordinates, and there is a constant temperature gradient A in the positive 2 direction far from the sphere. The temperature at the center of the sphere is T . The steady state temperature distributions in the sphere TL , and in the surrounding medium TD have been shown to be: 3k Tir,3!T= 0 *AerecosrgREan 1 kl+2k T ir BlTG 1 klku r 3 *AamcosQRrE a 2 0 ' ' _ k1+2k0 F ' q ' (a) What are the partial differential equations that must be satisfied by Eqs. 1 and 2? (b) Write down the boundary conditions that apply at r=R. (c) Showr that T1 and To satisfy their respective partial differential equations in (a). {(1) Show that Eqs. 1 and 2 satisfy the boundary conditions in (b). 3. Heat is generated in a spherical fuel element according to the relationship: r 3 cramp(Tu a T.vhere q is the volumetric heat generation rate, kWImE' , and J" is the radius of the sphere. Develop the equation that expresses the temperature difference between the center of the sphere and its surface. Assume T: T1 at r=0, T=T2 at r= r . 4. An electrical system employs fuses that are cylindrical in shape and have lengths of 0.5 cm and diameters of 0.1 mm. Air, at 30 C, surrounds a fuse with a surface coefficient of 10 mez . The fuse material melts at 900 C. Assuming all heat transfer to be from the fuse surface, estimate the time it will take for the fuse to blow after a current of 3 A ows through it. Pertinent properties of the fuse material are: Resistance: 0.2 9. k: 20 mez-k