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3 Pages

### EX1Problem02

Course: NE 400, Spring 2010
School: N.C. State
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Word Count: 528

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fuel A rod has a volumetric heat generation rate given by r 2 z q(r , z ) = q01 + 2 cos R He where is a known positive constant. a) A common axial flux shape that is used in PWR safety analysis is the 1.55 chopped cosine, where the 1.55 is the axial peaking factor Fz . Give a step by step solution procedure for determining the axial extrapolation distance that corresponds to this peaking factor....

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fuel A rod has a volumetric heat generation rate given by r 2 z q(r , z ) = q01 + 2 cos R He where is a known positive constant. a) A common axial flux shape that is used in PWR safety analysis is the 1.55 chopped cosine, where the 1.55 is the axial peaking factor Fz . Give a step by step solution procedure for determining the axial extrapolation distance that corresponds to this peaking factor. Give all equations. Integrals that can be evaluated analytically should be done so. If the solution requires iteration, it is sufficient to give the iteration equation, state the variable to be solved for iteratively and state "solve iteratively". Assuming the extrapolation distance is known from part a) and the additional known data given below, develop equations for: b) c) d) e) f) The average linear heat rate The maximum linear heat rate The average heat flux The maximum heat flux The total heat generated in the rod Again, integrals that can be evaluated analytically should be done so. If the solution requires iteration, it is sufficient to give the iteration equation, state the variable to be solved for iteratively and state "solve iteratively". Known Problem Data Volumetric Heat Generation Rate Amplitude Rod Height Rod Diameter Pellet Diameter SOLUTION The linear heat rate as a function of position is q ( z ) = q0 H Do D = 2R R q ( r , z )2rdr 0 q( z ) = R 0 r 2 z 2rdr q01 + 2 cos R He which has solution q( z z ) = q0 [2 + ]R 2 cos 24 14 244 3 He q0 The heat flux as a function of position is related to the linear heat rate by Heat Generation 20 1 q( z )Do = q( z ) such that the heat flux is q( z ) = z q0 cos H Do e For the given data a) Extrapolation Distance The axial peaking factor is defined as Fz q max q For the linear heat rate distribution given above, qmax = q0 such that Fz = 1 1 H z cos H dz -H / 2 e H /2 = 1 1 H e z sin H H e -H / 2 H /2 = 1 2 H e H sin 2H H e or 1 2 H e H = sin 2H Fz H e For x 2H e H 1 1 = sin (x ) Fz x which is a transcendental equation in x and can be solved iteratively. Given x, then x= H 2He He = H 2x which can be solved for the extrapolated height H e . Given H e = H + 2 , the extrapolation distance is = He - H 2 Heat Generation 20 2 b) Average Linear Heat Rate The average linear heat rate is q = 1 H q( z)dz = H q H /2 1 -H / 2 or q = 1 H z q0 cos H dz -H / 2 e H /2 which from above has solution q = q0 where q0 has been defined previously. 2 H e H sin 2H H e c) Maximum Linear Heat Rate From above qmax = q0 = q0 [2 + ]R 2 2 d) Average Heat Flux The average heat flux is related to the average linear heat rate through qDo = q such that q = q = q0 2 H e H sin 2H e Do Do H e) Maximum Heat Flux qmax = qmax q = 0 Do Do f) Total Heat Generated in the Rod q = H q = q0 2H e H sin 2H e Heat Generation 20 3
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