Transient Conduction: Spatial Effects and the Role of Analytical Solutions

Transcription

1 Transent Cnductn: Spatal Effects and the Rle f Analytcal Slutns

2 Slutn t the Heat Equatn fr a Plane Wall wth Symmetrcal Cnvectn Cndtns If the lumped capactance apprxmatn can nt be made, cnsderatn must be gven t spatal, as well as tempral, varatns n temperature durng the transent prcess. Fr a plane wall wth symmetrcal cnvectn cndtns and cnstant prpertes, the heat equatn and ntal/bundary cndtns are: 2T 1 T = x2 α t T x,0 T T x x= 0 (5.26) = (5.27) = 0 T k = h T( L, t) T x x= L Exstence f seven ndependent varables: T T x,, t T, T, k, α, h (5.28) (5.29) = (5.30) Hw may the functnal dependence be smplfed?

3 Nn-dmensnalzatn f Heat Equatn and Intal/Bundary Cndtns: Dmensnless temperature dfference: θ * x Dmensnless crdnate: x * L αt Dmensnless tme: t * F L 2 F the Furer Number θ = θ T T T T hl The Bt Number: B k Exact Slutn: θ * = f x *, F, B sld cs * = C nexp n 2 F nx * n= 1 θ ζ ζ C = 4snζ n ζ tanζ 2ζ + sn 2 = ( ζ ) n n n n n B (5.39a) (5.39b,c) See Appendx B.3 fr frst fur rts (egenvalues ζ,..., ζ 1 4 ) f Eq. (5.39c)

4 The One-Term Apprxmatn ( F > 0.2) : Varatn f mdplane temperature (x * = 0) wth tme F : θ ( T T ) ( T T ) C exp ( ζ F) * Table 5.1 C and ζ as a functn f B 1 1 Varatn f temperature wth lcatn (x * ) and tme F : θ = θ cs ζ ( x ) * * 1 * Change n thermal energy strage wth tme: Δ E = Q st (5.41) (5.40b) (5.43a) snζ 1 Q = Q 1 * θ (5.46) ζ 1 Q = ρc T T (5.44) Can the fregng results be used fr a plane wall that s well nsulated n ne sde and cnvectvely heated r cled n the ther? Can the fregng results be used f an sthermal cndtn Ts T s nstantaneusly mpsed n bth surfaces f a plane wall r n ne surface f a wall whse ther surface s well nsulated?

5 Graphcal Representatn f the One-Term Apprxmatn The Hesler Charts Mdplane Temperature:

6 Temperature Dstrbutn: Change n Thermal Energy Strage:

7 Radal Systems Lng Rds r Spheres Heated r Cled by Cnvectn. B = hr / k F= αt/ r 2 One-Term Apprxmatns: Lng Rd: Eqs. (5.49) and (5.51) Sphere: Eqs. (5.50) and (5.52) C, ζ Table Graphcal Representatns: Lng Rd: Fgs. D.4 D.6 Sphere: Fgs. D.7 D.9

8 The Sem-Infnte Sld A sld that s ntally f unfrm temperature T and s assumed t extend t nfnty frm a surface at whch thermal cndtns are altered. Specal Cases: Case 1: Change n Surface Temperature (T s ) ( 0, ) (,0) T t = T T x = T s T x, t Ts x = erf T T 2 αt (5.57) s q = s ( T ) k T s παt (5.58)

9 Case 2: Unfrm Heat Flux ( q = q ) 1 2 α π 2 2 q t/ x T( x, t) T = exp k 4α t q x x erfc k 2 αt (5.59) s Case 3: Cnvectn Heat Transfer, ht T k = h T T t x x= 0 (, ) ( 0, ) T x t T x = erfc T T 2 αt 2 hx h αt x h αt exp + erfc k k αt k (5.60)

10 Multdmensnal Effects Slutns fr multdmensnal transent cnductn can ften be expressed as a prduct f related ne-dmensnal slutns fr a plane wall, P(x,t), an nfnte cylnder, C(r,t), and/r a sem-nfnte sld, S(x,t). See Equatns (5.64) t (5.66) and Fg Cnsder superpstn f slutns fr tw-dmensnal cnductn n a shrt cylnder: (,, ) T r x t T T T = (, ) (, ) P x t x C r t (, ) T x t T T r,t T = x T T T T Plane Wall Infnte Cylnder

11 Prblem 5.66: Chargng a thermal energy strage system cnsstng f a packed bed f Pyrex spheres. KNOWN: Dameter, densty, specfc heat and thermal cnductvty f Pyrex spheres n packed bed thermal energy strage system. Cnvectn ceffcent and nlet gas temperature. FIND: Tme requred fr sphere t acqure 90% f maxmum pssble thermal energy and the crrespndng center and surface temperatures. SCHEMATIC:

12 ASSUMPTIONS: (1) One-dmensnal radal cnductn n sphere, (2) Neglgble heat transfer t r frm a sphere by radatn r cnductn due t cntact wth adjnng spheres, (3) Cnstant prpertes. ANALYSIS: Wth B h(r /3)/k = 75 W/m 2 K (0.0125m)/1.4 W/m K = 0.67, the lumped capactance methd s napprprate and the apprxmate (ne-term) slutn fr ne-dmensnal transent cnductn n a sphere s used t btan the desred results. T btan the requred tme, the specfed chargng requrement ( Q/ Q = 0.9) must frst be used t btan the dmensnless center temperature, θ *. Frm Eq. (5.52), 3 ζ1 Q 1 ( ζ1) ζ1 cs ( ζ1) Q θ = 3sn Wth B hr /k = 2.01, ζ and C frm Table 5.1. Hence, 3 ( ) θ = = =

13 Frm Eq. (5.50c), the crrespndng tme s 2 r θ t = ln 2 αζ 1 C α = k / ρc = 1.4 W / m K / 2225 kg /m 835J/kg K = m /s, 2 ( m) ln( 0.155/1.48) m /s ( 2.03) t = = 1,020s Frm the defntn f θ *, the center temperature s T = T T T = 300 C 42.7 C = C g, g, The surface temperature at the tme f nterest may be btaned frm Eq. (5.50b) wth r = 1, ( ζ ) 1 θ sn Ts = Tg, + ( T Tg,) = 300 C 275 C = C ζ Is use f the ne-term apprxmatn apprprate?

14 Prblem: 5.82: Use f radatn heat transfer frm hgh ntensty lamps ( q 4 2 s = 10 W/m ) fr a prescrbed duratn (t=30 mn) t assess ablty f frewall t meet safety standards crrespndng t maxmum allwable temperatures at the heated (frnt) and unheated (back) surfaces. KNOWN: Thckness, ntal temperature and thermphyscal prpertes f cncrete frewall. Incdent radant flux and duratn f radant heatng. Maxmum allwable surface temperatures at the end f heatng. FIND: If maxmum allwable temperatures are exceeded. SCHEMATIC:

15 ASSUMPTIONS: (1) One-dmensnal cnductn n wall, (2) Valdty f semnfnte medum apprxmatn, (3) Neglgble cnvectn and radatve exchange wth the surrundngs at the rradated surface, (4) Neglgble heat transfer frm the back surface, (5) Cnstant prpertes. ANALYSIS: The thermal respnse f the wall s descrbed by Eq. (5.59) ( α π) 1/2 2 x 2q t/ q x x T( x, t) = T + exp erfc k 4α t k 2 α t 7 2 where, α = k/ ρc = m /s and fr p ( α π) 1/2 t = 30 mn = 1800s, 2q t / / k = K. Hence, at x = 0, T ( 0,30 mn) = 25 C C = C < 325 C 2 At 1/2 x = 0.25m, x / 4αt = 12.54, q x / k = 1, 786K, and x / 2 αt = Hence, 6 T 0.25m, 30 mn = 25 C C C ~ 0 25 C

16 Bth requrements are met. Is the assumptn f a sem-nfnte sld fr a plane wall f fnte thckness apprprate under the fregng cndtns? COMMENTS: The fregng analyss may r may nt be cnservatve, snce heat transfer at the rradated surface due t cnvectn and net radatn exchange wth the envrnment has been neglected. If the emssvty f the surface and the temperature f the surrundngs are assumed t be ε = 1 and T sur = 298K, radatn exchange at T s = C wuld be q = εσ T T = 6,080 W/m K, rad s sur whch s sgnfcant (~ 60% f the prescrbed radatn). Hwever, under actual cndtns, the wall wuld lkely be expsed t cmbustn gases and adjnng walls at elevated temperatures.