= 6. (16 points) Derive the 1-D governing equation for heat transfer in spherical coordinates (radial...

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= 6. (16 points) Derive the 1-D governing equation for heat transfer in spherical coordinates (radial direction only). Ignore convection. The unit volume of a sphere is A?????? ?(?+A?) 34?? 3. Simplify the unit volume of a sphere before using it in the derivation and assume (A?)2 = (A?) 3 = 0. 3 4 4 A?2241??? = ?(?+A?) 3- ?? 3 3 3 4 ?(? 2+2?A?A? 2)(?+A?) ?? 3 A?224??? = 4 3 ?(? 3 + 2? 2A? + ?A? + ??A? + 2?A? 2 +A? 3 - ? 3) 4 4?224??? ?(3? 2A? +3?A? 2 +A? ) 3 Assume: A?2~0, A? 3~0 for small?? or small A? A?224222 4 = 3 ?(3? 2A?)=4?(? 2A?) A??h??? = 4?? 2 at ? = ??h??? 4?(?A?) 2 at? +A? Energy In = flux ? area? time = ?" A221???A? | = ? 4?? 2 A? | " Energy Out = flux ? area? time = ? ?+A? A ?????A? | 2+2 = ? ?+A? 4?(? +A?) 2 A? +A? ?+A? Generation heat? volume ? time = Q? ??h??? A? Q 4?(? 2A?) A? Storage = volume? density? specific heat? temperature = 4?(? 2A?)?? p4 AT = 6. (16 points) Derive the 1-D governing equation for heat transfer in spherical coordinates (radial direction only). Ignore convection. The unit volume of a sphere is A?????? ?(?+A?) 34?? 3. Simplify the unit volume of a sphere before using it in the derivation and assume (A?)2 = (A?) 3 = 0. 3 4 4 A?2241??? = ?(?+A?) 3- ?? 3 3 3 4 ?(? 2+2?A?A? 2)(?+A?) ?? 3 A?224??? = 4 3 ?(? 3 + 2? 2A? + ?A? + ??A? + 2?A? 2 +A? 3 - ? 3) 4 4?224??? ?(3? 2A? +3?A? 2 +A? ) 3 Assume: A?2~0, A? 3~0 for small?? or small A? A?224222 4 = 3 ?(3? 2A?)=4?(? 2A?) A??h??? = 4?? 2 at ? = ??h??? 4?(?A?) 2 at? +A? Energy In = flux ? area? time = ?" A221???A? | = ? 4?? 2 A? | " Energy Out = flux ? area? time = ? ?+A? A ?????A? | 2+2 = ? ?+A? 4?(? +A?) 2 A? +A? ?+A? Generation heat? volume ? time = Q? ??h??? A? Q 4?(? 2A?) A? Storage = volume? density? specific heat? temperature = 4?(? 2A?)?? p4 AT