Homework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is:

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1 . Homewok 3 MAE 8C Poblems, 5, 7, 0, 4, 5, 8, 3, 30, 3 fom Chpte 5, msh & Btt Point souces emit nuetons/sec t points,,, n 3 fin the flux cuent hlf wy between one sie of the tingle (blck ot). The flux fo point souce is: ϕ fom poblem. 4 The flux is then the sum of the point souces fluxes. ϕ 4 4 The fist tem is the flux fom points n which e equl the secon tem is the flux fom point 3. The cuent fom points n cncle so you only nee to clculte the cuent fom point 3. J D ϕ D 3 The cuent hs iection which is in the iection of the vecto fom point 3 to the point you e clculting the flux. 5. This cse in n infinite moeto so using the flux eqution At the cente of the sque the cuent is zeo ue to symety. The flux is simply the sum of the 4 point souces. ϕ i e D

2 The flux t point ii is gin the sum of the fluxes. e e ϕ ii D D The cuent fom ech point is just, J D ϕ e which is given equtions befoe The cuents fom point n cncel. ooking t the igm, the hoizonl potion of the cuents fom poitns 3 n 4 cncel. The sum of the veticl cuents with thei iection being own in the igm. Distnce fom point 3 to ii sme fo 4 to ii. J 3 e J 4 4 cuent fom point 3 n 4 t ii J e sin tn ( ) in the ownw iection 4 Whee the sine n tngent tems come fom foming ight tingle with points ii,, 3 n then using the tio of the sies to fin the ngle t point ii n then using tht ngle to fin the potion of the cuent in the ownw iection.

3 7. Point souce in n infinite moeto, ϕ e 4 D J D ϕ 4 e ) To fin the numbe of neutons pssing though sufce, fin the cuent t the sufce then integte ove the sufce e. J e 4 whee is the ius of the sphee which hs e 4 totl numbe of neutons e b) The numbe bsobe pe secon within the sphee is equl to Σ ϕ V in ou cse the volume integl is ove the volume of the sphee e Σ Σ ϕ V 4 e e 4 D D 0 c) Veify the continuity eqution, the numbe bobe the numbe leke equls the numbe pouce. e Σ e e Σ D Σ

4 0. Cuent equls zeo by symety, n the flux cn hve no spcil epenence ue to symety. Diffusion eqution ϕ ϕ D Eq (5.9) flux must be constnt ϕ C ϕ Σ plug into iffusion eqution, C C D D Σ 4. Agin using the iffusion eqution, ϕ ϕ D Thee is now non-unifomity in so the solutions cn vy in the iection. A genel solution to the iffusion eqution is, ϕ A sinh cosh B C Now using bouny conitions, the flux nees to be finite t 0 so B 0. Note tht the fist tem oes NOT go to infinity s goes to zeo, this cn be seen by expning sinh s n infinite seies. Agin we hve, C D C D Σ ϕ A sinh Σ Now using, ϕ( R ) 0 0 A sinh R ( R ) Σ A ( R ) Σ R sinh ϕ ( R ) Σ R sinh sinh Σ Σ sinh ( R ) R sinh

5 using, sinh( x) sinh( x) ϕ Σ sinh ( R ) sinh R b) J D ϕ D Σ sinh ( R ) sinh R ( R ) J D sinh R cosh sinh c) How mny neutons lek fom the sphee? JR ( ) 4 R ) D4 R ( R ) sinh R sinh R R cosh R R Avege pobbility tht souce neuton will leve the sphee is the numbe tht lek ivie by the totl numbe pouce D4 R ( R ) sinh R sinh R R cosh R R D4 R ( R ) sinh R sinh R R cosh R R

6 5. J D ϕ sin ϕ i A i R J DA i cos R R sin R R : 50cm ekge.cm30 5 cos( ) sin( ) 4R cms : neutons pe secon s R R.7cm0 6 cos( ) sin( ) 4R cms : neutons pe secon s R R 3.05cm0 6 cos( ) sin( ) 4R cms : neutons pe secon s R R 8. J D ϕ cos y ϕ T Acos x cos z Cn't figue out how to mke it til so just clling it. J D ϕ T x x y ϕ T y z ϕ T z whee x is the unit vecto in the x iection ) J DA cos x cos y cos z sin x cos x sin y z sin x y cos y cos z z b) Evlute J otte into x ht t x / n then integte fom y -/ to / n z -/ to /. J, y, z DA cos cos y cos z sin cos sin y z sin x y cos y cos z z

7 J, y, z x DA eking neutons cos y cos z DA x DA4 cos y y pe sie c) By symety, the totl numbe is just the nswe fom pt b) times 6. DA 4 3. ) b : cm N A : mol b) N A b cm s 6.05 gm mol gmy ecys pe secon y s cuies 30. N : 00 i : 0.. N n : 0.. N x : i T :.85cm Σ :.097 cm 8i cm N D :.6cm : 8cm : 0 8 cm s

8 ϕ : T D sinh ( x) T cosh T ϕ whee x is in units of metes n the flux is in units of neutons/cm^*s x Using Eq to clculte the chnge in the theml iffusion pmete ( ) T ( ρ, T) T ρo, T o ρ o ρ T T o m ssuming To T T : T ρ Tb : T ρ b Tc : T ρ c.838cm.849cm.85cm ρ : ρ b : ρ c : sinh ( x) T T ϕ : ϕb : D cosh T Tb D sinh ( x) Tb cosh Tb ϕc : T D sinh ( x) Tc cosh Tc This clucltion tkes into ccount the chnge in ensity ue to ing boic ci to wte. The smll concenttions esult in vey smll chnge in the flux.

9 ϕ ϕb ϕc x Clely this oesn't look like enough of n effect to mke this poblem wothwhile, I think the coect wy to o this poblem involves using the bsoption coss-section fo boic ci. I ws unble to fin this numbe.

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