Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 18
Текст из файла (страница 18)
(2001), culled from thework of Abramowitz and Stegun (1955). Maple V, Release 6.0 can also be used toBOOKCOMP, Inc. — John Wiley & Sons / Page 174 / 2nd Proofs / Heat Transfer Handbook / Bejan———0.74718pt PgVar———Normal PagePgEnds: TEX[174], (14)SPECIAL FUNCTIONSgenerate these tables. Figure 3.2 displays graphs of J0 (x), J1 (x), Y0 (x), and Y1 (x).These functions exhibit oscillatory behavior with amplitude decaying as x increases.Figure 3.3 provides plots of I0 (x), I1 (x), K0 (x), and K1 (x) as a function of x andthese functions exhibit monotonic behavior.1.5J0(x)1.0Bessel Function1234567891011121314151617181920212223242526272829303132333435363738394041424344451750.5J1(x)0.0Y0(x)[175], (15)⫺0.5Y1(x)⫺1.0⫺1.5Lines: 878 to 892———0.01.02.03.04.05.0x6.07.08.0Figure 3.2 Graphs of J0 (x), J1 (x), Y0 (x), and Y1 (x).9.010.00.714pt PgVar———Normal PagePgEnds: TEX[175], (15)Figure 3.3 Graphs of I0 (x), I1 (x), K0 (x), and K1 (x).BOOKCOMP, Inc.
— John Wiley & Sons / Page 175 / 2nd Proofs / Heat Transfer Handbook / Bejan176123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERThomson functions ber i (x), beii (x), ker i (x), and keii (x) arise in obtaining thereal and imaginary parts of the modified Bessel functions of imaginary argument.The subscripts i denote the order of the Thomson functions. Note that it is customary to omit the subscript when dealing with Thomson functions of zero order.Hence ber 0 (x), bei0 (x), ker 0 (x), and kei0 (x) are written as ber(x), bei(x), ker(x),and kei(x).
The Thomson functions are defined by√I0 (x ι) = ber(x) + ι bei(x)√K0 (x ι) = ker(x) + ι kei(x)(3.47)(3.48)withber(0) = 1bei(0) = 0ker(0) = ∞kei(0) = −∞(3.49)[176], (16)Expressions for the derivatives of the Thomson functions are1d[ber(x)] = √ [ber 1 (x) + bei1 (x)]dx2d1[bei(x)] = √ [bei1 (x) − ber 1 (x)]dx2d1[ker(x)] = √ [ker 1 (x) + kei1 (x)]dx2d1[kei(x)] = √ [kei1 (x) − ker 1 (x)]dx2Lines: 892 to 974(3.50)(3.51)(3.52)(3.53)Table 3.7 gives the values of ber(x), bei(x), ker(x), and kei(x) for 1 ≤ x ≤ 5,and the values of dber(x)/dx, dbei(x)/dx, dker(x)/dx, and dkei(x)/dx are provided for the same range of x values in Table 3.8.
Figure 3.4 displays graphs ofber(x), bei(x), ker(x), and kei(x).TABLE 3.7Functions ber(x), bei(x), ker(x), and kei(x)xber(x)1.001.502.002.503.003.504.004.505.000.984380.921070.751730.39997−0.22138−1.19360−2.56342−4.29909−6.23008bei(x)0.249570.557560.972291.457181.937592.283252.292691.686020.11603BOOKCOMP, Inc. — John Wiley & Sons / Page 176 / 2nd Proofs / Heat Transfer Handbook / Bejan———0.36082pt PgVarker(x)kei(x)0.286710.05293−0.04166−0.06969−0.06703−0.05264−0.03618−0.02200−0.01151−0.49499−0.33140−0.20240−0.11070−0.05112−0.016000.002200.009720.01119———Normal PagePgEnds: TEX[176], (16)177SPECIAL FUNCTIONS123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 3.8Functions d ber(x)/dx, d bei(x)/dx, d ker(x), dx, and d kei(x)/dxxd ber(x)/dxd bei(x)/dxd ker(x)/dxd kei(x)/dx1.001.502.002.503.003.504.004.505.00−0.06245−0.21001−0.49307−0.94358−1.56985−2.33606−3.13465−3.75368−3.845340.497400.730250.917010.998270.880480.43530−0.49114−2.05263−4.35414−0.69460−0.29418−0.10660−0.016930.021480.032990.031480.024810.017190.352370.295610.219810.148900.092040.050980.023910.00772−0.00082[177], (17)Lines: 974 to 1000———0.45917pt PgVar———Normal PagePgEnds: TEXFigure 3.4 Thomson functions.3.3.6[177], (17)Legendre FunctionsThe Legendre function, also known as the Legendre polynomial Pn (x), and the associated Legendre function of the first kind Pnm (x), are defined by1 dn 2(x − 1)n2n n! dx ndmPnm (x) = (1 − x 2 )m/2 m [Pn (x)]dxPn (x) =(3.54)(3.55)The Legendre function Qn (x) and the associated Legendre function of the secondkind, Qmn (x), are defined byQn (x) =(−1)n/2 · 2n [(n/2)!]2n!(n − 1)(n + 2) 3 (n − 1)(n − 3)(n + 2)(n + 4) 5× x−x +x − ···3!5!(n = even, |x| < 1)BOOKCOMP, Inc.
— John Wiley & Sons / Page 177 / 2nd Proofs / Heat Transfer Handbook / Bejan(3.56)178123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFER(−1)(n+1)/2 · 2n−1 ([(n − 1)/2]!)2Qn (x) =1 · 3 · 5···nn(n + 1) 2 n(n − 2)(n + 1)(n + 3) 4× 1−x −x − ···2!4!(n = odd, |x| < 1)2 m/2Qmn (x) = (1 − x )(3.57)md[Qn (x)]dx m(3.58)Several relationships involving Pn (x), Qn (x), Pnm (x), and Qmn (x) are useful inheat conduction analysis.
They arePn (−x) = (−1)n Pn (x)Pn+1 (x) =2n + 1nxPn (x) −Pn−1 (x)n+1n+1d d Pn+1 (x) −Pn−1 (x) = 2(n + 1)Pn (x)dxdxQn (x = ±1) = ∞Pnm (x) = 0Qmn (x(3.59)(3.60)Lines: 1000 to 1068(3.61)(3.62)(m > n)= ±1) = ∞(3.63)STEADY ONE-DIMENSIONAL CONDUCTIONIn this section we consider one-dimensional steady conduction in a plane wall, ahollow cylinder, and a hollow sphere. The objective is to develop expressions forthe temperature distribution and the rate of heat transfer. The concept of thermalresistance is utilized to extend the analysis to composite systems with convectionoccurring at the boundaries. Topics such as contact conductance, critical thickness ofinsulation, and the effect of uniform internal heat generation are also discussed.3.4.1Plane WallConsider a plane wall of thickness L made of material with a thermal conductivity k,as illustrated in Fig.
3.6. The temperatures at the two faces of the wall are fixed at Ts,1and Ts,2 with Ts,1 > Ts,2 . For steady conditions with no internal heat generation andconstant thermal conductivity, the appropriate form of the general heat conductionequation, eq. (3.4), isBOOKCOMP, Inc. — John Wiley & Sons / Page 178 / 2nd Proofs / Heat Transfer Handbook / Bejan———-4.87883pt PgVar———Normal PagePgEnds: TEX(3.64)The numerical values of the Legendre functions and their graphs can be generatedwith Maple V, Release 6.0.
Table 3.9 lists the values of P0 (x) through P4 (x) for therange −1 ≤ x ≤ 1, and a plot of these functions appears in Fig. 3.5.3.4[178], (18)[178], (18)STEADY ONE-DIMENSIONAL CONDUCTION123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 3.9x−1.00−0.80−0.60−0.40−0.200.000.200.400.600.801.00179Numerical Values of Pn (x)P0 (x)1.000001.000001.000001.000001.000001.000001.000001.000001.000001.000001.00000P1 (x)P2 (x)P3 (x)P4 (x)−1.00000−0.80000−0.60000−0.40000−0.200000.000000.200000.400000.600000.800001.000001.000000.460000.04000−0.26000−0.44000−0.50000−0.44000−0.260000.040000.460001.00000−1.00000−0.080000.360000.440000.280000.00000−0.28000−0.44000−0.360000.080001.000001.00000−0.23300−0.40800−0.113000.232000.375000.23200−0.11300−0.40800−0.233001.00000[179], (19)Lines: 1068 to 10681———0.01505pt PgVarP (2,x)P (1,x)P (3,x)P (5,x)⫺1———Normal PagePgEnds: TEX⫺0.50.5[179], (19)P (4,x)0⫺0.5⫺1Figure 3.5 Legendre polynomials.BOOKCOMP, Inc.
— John Wiley & Sons / Page 179 / 2nd Proofs / Heat Transfer Handbook / Bejan0.5x1180123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFER[180], (20)Lines: 1068 to 1105———0.74109pt PgVarFigure 3.6 One-dimensional conduction through a plane wall.d 2T=0dx 2———Normal PagePgEnds: TEX(3.65)[180], (20)with the boundary conditions expressed asT (x = 0) = Ts,1andT (x = L) = Ts,2(3.66)Integration of eq. (3.65) with subsequent application of the boundary conditions ofeq.
(3.66) gives the linear temperature distributionT = Ts,1 + (Ts,2 − Ts,1 )xL(3.67)and application of Fourier’s law givesq=kA(Ts,1 − Ts,2 )L(3.68)where A is the wall area normal to the direction of heat transfer.3.4.2Hollow CylinderFigure 3.7 shows a hollow cylinder of inside radius r1 , outside radius r2 , length L,and thermal conductivity k. The inside and outside surfaces are maintained at constanttemperatures Ts,1 and Ts,2 , respectively with Ts,1 > Ts,2 . For steady-state conductionBOOKCOMP, Inc.
— John Wiley & Sons / Page 180 / 2nd Proofs / Heat Transfer Handbook / BejanSTEADY ONE-DIMENSIONAL CONDUCTION123456789101112131415161718192021222324252627282930313233343536373839404142434445181LkTs,1r1r2qTs,2[181], (21)Figure 3.7 Radial conduction through a hollow cylinder.Lines: 1105 to 1146———in the radial direction with no internal heat generation and constant thermal conduc2.92108pttivity, the appropriate form of the general heat conduction equation, eq.
(3.5), is———Normal PagedTdr=0(3.69)* PgEnds: Ejectdrdrwith the boundary conditions expressed asT (r = r1 ) = Ts,1and[181], (21)T (r = r2 ) = Ts,2(3.70)Following the same procedure as that used for the plane wall will give the temperaturedistributionT = Ts,1 +Ts,1 − Ts,2rlnln(r1 /r2 )r1(3.71)and the heat flowq=3.4.32πkL(Tx,1 − Ts,2 )ln(r2 /r1 )(3.72)Hollow SphereThe description pertaining to the hollow cylinder also applies to the hollow sphereof Fig. 3.8 except that the length L is no longer relevant. The applicable form of eq.(3.6) isdTdr2=0(3.73)drdrBOOKCOMP, Inc. — John Wiley & Sons / Page 181 / 2nd Proofs / Heat Transfer Handbook / BejanPgVar182123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFER[182], (22)Figure 3.8 Radial conduction through a hollow sphere.Lines: 1146 to 1188———4.84712pt PgVarwith the boundary conditions expressed asT (r = r1 ) = Ts,1andT (r = r2 ) = Ts,2The expressions for the temperature distribution and heat flow are1Ts,1 − Ts,21T = Ts,1 +−1/r2 − 1/r1 r1rq=4πk(Ts,1 − Ts,2 )1/r1 − 1/r2———Long Page(3.74) * PgEnds: Eject[182], (22)(3.75)(3.76)3.4.4 Thermal ResistanceThermal resistance is defined as the ratio of the temperature difference to the associated rate of heat transfer.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
