One-Dimensional, Steady-State. State Conduction with Thermal Energy Generation

Transcription

1 One-Dimensional, Steady-State State Conduction with Themal Enegy Geneation

2 Implications of Enegy Geneation Involves a local (volumetic) souce of themal enegy due to convesion fom anothe fom of enegy in a conducting medium. The souce may be unifomly distibuted, as in the convesion fom electical to themal enegy (Ohmic heating): E g q = = I R e (3.38) o it may be non-unifomly distibuted, as in the absoption of adiation passing though a semi-tanspaent medium. Fo a plane wall, q exp( α x) Geneation affects the tempeatue distibution in the medium and causes the heat ate to vay with location, theeby pecluding inclusion of the medium in a themal cicuit.

3 The Plane Wall Conside one-dimensional, steady-state conduction in a plane wall of constant k, unifom geneation, and asymmetic suface conditions: Heat Equation: d dt d T q k q + = 0 + = 0 dx dx dx k (3.39) Is the heat flux q independent of x? Geneal Solution: T x = q/k x + C x+ C What is the fom of the tempeatue distibution fo q = 0? q > 0? q < 0? How does the tempeatue distibution change with inceasing q?

4 Symmetic Suface Conditions o One Suface Insulated: What is the tempeatue gadient at the centeline o the insulated suface? Why does the magnitude of the tempeatue gadient incease with inceasing x? Tempeatue Distibution: T x ql x = Ts k + L How do we detemine? T s Oveall enegy balance on the wall 0 ha T T + q A L = T s s s s ql = T + h Eout + Eg = 0 (3.4) How do we detemine the heat ate at x = L? (3.46)

5 Cylindical (Tube) Wall Radial Systems Spheical Wall (Shell) Solid Cylinde (Cicula Rod) Solid Sphee Heat Equations: Cylindical d dt k + q = 0 d d d d Spheical + = k dt d q 0

6 Solution fo Unifom Geneation in a Solid Sphee of Constant k with Convection Cooling: Tempeatue Distibution k T dt d 3 dt q C d = 3 + = q C + 6k C = 0 = = 0 C 0 T = T C = T + o s s T o 6k o q = + T o q 6k s Suface Tempeatue Oveall enegy balance: Eout + E g = 0 E T = T + s 3h O fom a suface enegy balance: in Eout = 0 qo q = q cond o conv T = T + s qo 3h A summay of tempeatue distibutions is povided in Appendix C fo plane, cylindical and spheical walls, as well as fo solid cylindes and sphees. Note how bounday conditions ae specified and how they ae used to obtain suface tempeatues.

7 Poblem 3.9 Themal conditions in a gas-cooled nuclea eacto with a tubula thoium fuel od and a concentic gaphite sheath: (a) Assessment of themal integity 8 3 fo a geneation ate of q =0 W/m. (b) Evaluation of tempeatue distibutions in the thoium and gaphite 8 8 fo geneation ates in the ange. 0 q 5x0 Schematic: Assumptions: () Steady-state conditions, () One-dimensional conduction, (3) Constant popeties, (4) Negligible contact esistance, (5) Negligible adiation, (6) Adiabatic suface at. Popeties: Table A., Thoium: T 000 K ; Table A., Gaphite: T 300 K. mp mp

8 Analysis: (a) The oute suface tempeatue of the fuel, T, may be detemined fom the ate equation whee T T q = Rtot n ( 3/ ) Rtot = + = m K/W πk π h g 3 The heat ate may be detemined by applying an enegy balance to a contol suface about the fuel element, E out = E o, pe unit length, E out = E g g Since the inteio suface of the element is essentially adiabatic, it follows that q = qπ = 7,907 W/m Hence, T = qr + T = 7,907 W/m( m K/W) + 600K = 93K tot With zeo heat flux at the inne suface of the fuel element, Eq. C.4 yields T T K K K K q q = + n < = + = 4k k t t

9 Since T and T ae well below the melting points of thoium and gaphite, the pescibed opeating condition is acceptable. (b) The solution fo the tempeatue distibution in a cylindical wall with geneation is q 4k t T = T + t + q n / ( T T) 4k n / t ( ) ( ) (C.) Bounday conditions at and ae used to detemine T and T. : q 0 = = = q k + T T q 4 k t n / = : = U T T q k + T T 4kt q n / (C.4) (C.7) ( ) ( π ) tot tot U = A R = R (3.3)

10 The following esults ae obtained fo tempeatue distibutions in the gaphite. 500 Tempeatue, T(K) qdot = 5E8 qdot = 3E8 qdot = E8 Radial location in fuel, (m) 8 3 Opeation at q = 5x0 W/m is clealy unacceptable since the melting point of thoium would be exceeded. To pevent softening of the mateial, which would occu 8 3 below the melting point, the eacto should not be opeated much above q = 3x0 W/m. The small adial tempeatue gadients ae attibutable to the lage value of. k t

11 Using the value of T q = ( 3 ) ( ) π k T T g n / 3 fom the foegoing solution and computing T 3 fom the suface condition, (3.7) the tempeatue distibution in the gaphite is T T 3 T g ( ) = n + T n ( / 3 ) 3 3 (3.6) 500 Tempeatue, T(K) Radial location in gaphite, (m) qdot = 5E8 qdot = 3E8 qdot = E8 8 3 Opeation at q = 5x0 W/m is poblematic fo the gaphite. Lage tempeatue gadients ae due to the small value of. k g

12 Comments: (i) What effect would a contact esistance at the thoium/gaphite inteface have on tempeatues in the fuel element and on the maximum allowable value of? to the schematic, whee might adiation effects be significant? q What would be the influence of such effect on tempeatues in the fuel element and the maximum allowable value of? (ii) Refeing q