Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 17
Текст из файла (страница 17)
. .)(3.14)(3.15)xwithi 0 erfc(x) = erfc(x) and22i −1 erfc(x) = √ e−xπ(3.16)[168], (8)The first two repeated integrals are12i erfc(x) = √ e−x − x erfc(x)π1 22i 2 erfc(x) =1 + 2x 2 erfc(x) − √ xe−x4πLines: 443 to 598(3.17)(3.18)Table 3.1 lists the values of erf(x), d erf(x)/dx, d 2 erf(x)/dx 2 , and d 3 erf(x)/dx 3for values of x from 0 to 3 in increments of 0.10. Table 3.2 lists the values oferfc(x), i erfc(x), i 2 erfc(x), and i 3 erfc(x) for the same values of x.
Both tableswere generated using Maple V (Release 6.0).3.3.2Gamma FunctionThe gamma function, denoted by Γ(x), provides a generalization of the factorial n!to the case where n is not an integer. It is defined by the Euler integral (Andrews,1992): ∞t x−1 e−t dt(x > 0)(3.19)Γ(x) =0and has the propertyΓ(x + 1) = xΓ(x)(3.20)which for integral values of x (denoted by n) becomesΓ(n + 1) = n!(3.21)Table 3.3 gives values of Γ(x) for values of x from 1.0 through 2.0.
These valueswere generated using Maple V, Release 6.0.BOOKCOMP, Inc. — John Wiley & Sons / Page 168 / 2nd Proofs / Heat Transfer Handbook / Bejan———0.81136pt PgVar———Normal PagePgEnds: TEX[168], (8)SPECIAL FUNCTIONS123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 3.1169Values of erf(x), d erf(x)/dx, d 2 erf(x)/dx 2 , and d 3 erf(x)/dx 3xerf(x)d erf(x)/dxd 2 erf(x)/dx 2d 3 erf(x)/dx 30.000.100.200.300.400.500.600.700.800.901.001.101.201.301.401.501.601.701.801.902.002.102.202.302.402.502.602.702.802.903.000.000000.112460.222700.328630.428390.520500.603860.677800.742100.796910.842700.880210.910310.934010.952290.966110.976350.983790.989090.992790.995320.997020.998140.998860.999310.999590.999760.999870.999920.999960.999971.128381.117151.084131.031260.961540.878780.787240.691270.594990.501970.415110.336480.267340.208210.158940.118930.087230.062710.044190.030520.020670.013720.008920.005690.003560.002180.001310.76992 × 10−30.44421 × 10−30.25121 × 10−30.13925 × 10−30.00000−0.22343−0.43365−0.61876−0.76923−0.87878−0.94469−0.96778−0.95198−0.90354−0.83201−0.74026−0.64163−0.54134−0.44504−0.35679−0.27913−0.21322−0.15909−0.11599−0.08267−0.05761−0.03926−0.02617−0.01707−0.01089−0.00680−0.00416−0.00249−0.00146−0.83552 × 10−3−2.25676−2.18962−1.99481−1.69127−1.30770−0.87878−0.44086−0.027650.333190.622440.830210.955601.005210.991070.928220.832510.718770.599520.484340.379730.289340.214510.154890.109000.074810.050100.032750.020910.013050.007950.00473The incomplete gamma function is defined by the integral (Andrews, 1992) ∞Γ(a, x) =t a−1 e−t dt(3.22)xValues of Γ(1.2, x) for 0 ≤ x ≤ 1 generated using Maple V, Release 6.0 are given inTable 3.4.3.3.3Beta FunctionsThe beta function, denoted by B(x,y), is defined by 1B(x,y) =(1 − t)x−1 t y−1 dt0BOOKCOMP, Inc.
— John Wiley & Sons / Page 169 / 2nd Proofs / Heat Transfer Handbook / Bejan(3.23)[169], (9)Lines: 598 to 618———0.85535pt PgVar———Normal Page* PgEnds: Eject[169], (9)170123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERTABLE 3.2Values of erfc(x), i erfc(x), i 2 erfc(x), and i 3 erfc(x)xerfc(x)i erfc(x)i 2 erfc(x)i 3 erfc(x)0.000.100.200.300.400.500.600.700.800.901.001.101.201.301.401.501.601.701.801.902.002.102.202.302.402.502.602.702.802.903.001.000000.887540.777300.671370.571610.479500.396140.322200.257900.203090.157300.119790.089690.065990.047710.033890.023650.016210.010910.007210.004680.002980.001860.001140.68851 × 10−30.40695 × 10−30.23603 × 10−30.13433 × 10−30.75013 × 10−40.41098 × 10−40.22090 × 10−40.564190.469820.386610.314220.252130.199640.155940.120100.091170.068200.050250.036470.026050.018310.012670.008620.005770.003800.002460.001560.97802 × 10−30.60095 × 10−30.36282 × 10−30.21520 × 10−30.12539 × 10−30.71762 × 10−40.40336 × 10−40.22264 × 10−40.12067 × 10−40.64216 × 10−50.33503 × 10−50.250000.198390.155660.120710.092480.069960.052260.038520.028010.020080.014200.009890.006790.004590.003060.002010.001300.82298 × 10−30.51449 × 10−30.31642 × 10−30.19141 × 10−30.11387 × 10−30.66614 × 10−40.38311 × 10−40.21659 × 10−40.12035 × 10−40.65724 × 10−50.35268 × 10−50.18595 × 10−50.96315 × 10−60.49007 × 10−60.094030.071690.054060.040300.029690.021610.015540.011030.007730.005340.003640.002450.001620.001060.68381 × 10−30.43386 × 10−30.27114 × 10−30.16686 ×10−30.10110 ×10−30.60301 ×10−40.35396 × 10−40.20445 × 10−40.11619 × 10−40.64951 × 10−50.35711 × 10−50.19308 × 10−50.10265 × 10−50.53654 × 10−60.27567 × 10−60.13922 × 10−60.69101 × 10−7The beta function is related to the gamma function:B(x,y) =Γ(x)Γ(y)Γ(x + y)(x > 0, y > 0)(3.24)has the symmetry propertyB(x,y) = B(y,x)(3.25)and for nonnegative integers,B(m,n) =(m − 1)!(n − 1)!(m + n − 1)!BOOKCOMP, Inc.
— John Wiley & Sons / Page 170 / 2nd Proofs / Heat Transfer Handbook / Bejanm, n nonnegative integers(3.26)[170], (10)Lines: 618 to 706———-0.49988pt PgVar———Normal Page* PgEnds: Eject[170], (10)SPECIAL FUNCTIONS123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 3.3171Gamma FunctionxΓ(x)xΓ(x)xΓ(x)xΓ(x)1.001.011.021.031.041.051.061.071.081.091.101.111.121.131.141.151.161.171.181.191.201.211.221.231.241.000000.994330.988840.983550.978440.973500.968740.964150.959730.955460.951350.947400.943590.939930.936420.933040.929800.926700.923730.920890.918170.915580.913110.910750.908521.251.261.271.281.291.301.311.321.331.341.351.361.371.381.391.401.411.421.431.441.451.461.471.481.490.906400.904400.902500.900720.899040.897470.896000.894640.893380.892220.891150.890180.889310.888540.887850.887260.886760.886360.886040.885810.885660.885600.885630.885750.885951.501.511.521.531.541.551.561.571.581.591.601.611.621.631.641.651.661.671.681.691.701.711.721.731.740.886230.886590.887040.887570.888180.888870.889640.890490.891420.892430.893520.894680.895920.897240.898640.900120.901670.903300.905000.906780.908640.910570.912580.914670.916831.751.761.771.781.791.801.811.821.831.841.851.861.871.881.891.901.911.921.931.941.951.961.971.981.992.000.919060.921370.923760.926230.928770.931380.934080.936850.939690.942610.945610.948690.951840.955070.958380.961770.965230.968770.972400.976100.979880.983740.987680.991710.995811.00000TABLE 3.4 Incomplete GammaFunction, Γ(a, x), a = 1.2xΓ(a, x)0.000.100.200.300.400.500.600.700.800.901.000.918170.868360.809690.750740.693660.639320.588130.540240.495640.454260.41597BOOKCOMP, Inc.
— John Wiley & Sons / Page 171 / 2nd Proofs / Heat Transfer Handbook / Bejan[171], (11)Lines: 706 to 706———0.1901pt PgVar———Normal PagePgEnds: TEX[171], (11)172123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERTABLE 3.5 Incomplete BetaFunction, Bt (0.3, 0.5)xBt (0.3, 0.5)0.000.100.200.300.400.500.600.700.800.901.000.000000.648020.941071.186761.415841.642841.879202.138752.445632.863674.55444[172], (12)Lines: 706 to 808The incomplete beta function, Bt (x,y), is defined by tBt (x,y) =(1 − t)x−1 t y−1 dt———0.82916pt PgVar(3.27)0Values of Bt (0.3, 0.5) for the range 0 ≤ t ≤ 1 generated using Maple V, Release6.0 are given in Table 3.5.———Normal PagePgEnds: TEX[172], (12)3.3.4Exponential Integral FunctionThe exponential integral function E1 (x) or −Ei (−x) for a real argument x is definedby ∞ −teE1 (x) or − Ei (−x) =dt(3.28)txand has the following properties:E1 (0) = ∞E1 (∞) = 0dE1 (x)e−x=−dxx(3.29)As indicated by the entries in Table 3.6, the function decreases monotonically fromthe value E1 (0) = ∞ to E1 (∞) = 0 as x is varied from 0 to ∞.3.3.5Bessel FunctionsBessel functions of the first kind of order n and argument x, denoted by Jn (x), andBessel functions of the second kind of order n and argument x, denoted by Yn (x), aredefined, respectively, by the infinite seriesBOOKCOMP, Inc.
— John Wiley & Sons / Page 172 / 2nd Proofs / Heat Transfer Handbook / BejanSPECIAL FUNCTIONS123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 3.6173Exponential Integral FunctionxE1 (x)xE1 (x)0.000.010.020.030.040.050.060.070.080.090.100.150.200.300.400.500.600.70∞4.037933.354712.959122.681262.467902.295312.150842.026941.918741.822921.464461.222650.905680.702380.559770.454380.373770.800.901.001.101.201.301.401.501.601.701.801.902.002.202.402.602.803.000.310600.260180.219380.185990.158410.135450.116220.100020.086310.074650.064710.056200.048900.037190.028440.021850.016860.01305∞(−1)m (x/2)2m+nJn (x) =m! Γ(m + n + 1)m=0[173], (13)Lines: 808 to 827———0.58415pt PgVar———Normal Page* PgEnds: Eject(3.30)[173], (13)andYn (x) =Jn (x) cos nπ − J−n (x)sin nπ(n = 0, 1, 2, .
. .)(3.31a)orYn (x) = limm→nJm (x) cos mπ − J−m (x)sin mπ(n = 0, 1, 2, . . .)(3.31b)Numerous recurrence relationships involving the Bessel functions are available(Andrews, 1992). Some that are relevant in this chapter areJ−n (x) = (−1)n Jn (x)nndJn (x)= Jn−1 (x) − Jn (x) = Jn (x) − Jn+1 (x)dxxxd nx Jn (x) = x n Jn−1 (x)dxd −nx Jn (x) = −x −n Jn+1 (x)dxBOOKCOMP, Inc. — John Wiley & Sons / Page 173 / 2nd Proofs / Heat Transfer Handbook / Bejan(3.32)(3.33)(3.34)(3.35)174123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERThe relations given by eqs. (3.32)–(3.35) apply to the Bessel functions of the secondkind when the J ’s are replaced by Y ’s.Modified Bessel functions of the first kind of order n and argument x, denoted byIn (x), and modified Bessel functions of the second kind of order n and argument x,denoted by Kn (x), are defined, respectively, by the infinite seriesIn (x) =∞(x/2)2m+nm! Γ(m + n + 1)m=0(3.36)andKn (x) =πI−n (x) − In (x)2 sin nπ(n = 0, 1, 2, .
. .)(3.37a)[174], (14)orKn (x) = limm→nπI−m (x) − Im (x)2 sin nπ(n = 0, 1, 2, . . .)(3.37b)Lines: 827 to 878In (x) and Kn (x) are real and positive when n > −1 and x > 0.A selected few of the numerous recurrence relationships involving the modifiedBessel functions areIn (x) = (ι)−n Jn (ιx)(3.38)I−n (x) = (ι)n J−n (ιx)(3.39)nndIn (x)= In−1 (x) − In (x) = In (x) + In+1 (x)dxxxd nx In (x) = x n In−1 (x)dxd −nx In (x) = x −n In+1 (x)dxK−n (x) = Kn (x)(n = 0, 1, 2, 3, . . .)dKn (x)nn= Kn (x) − Kn+1 (x) = −Kn−1 (x) − Kn (x)dxxxd nx Kn (x) = −x n Kn−1 (x)dxd −nx Kn (x) = −x −n Kn+1 (x)dx(3.40)(3.41)(3.42)(3.43)(3.44)(3.45)(3.46)Fairly extensive tables for the Bessel functions and modified Bessel functions oforders 1 and 2 and those of fractional order I−1/3 (x), I−2/3 (x), I1/3 (x), and I2/3 (x)in the range 0 ≤ x ≤ 5 with a refined interval are given in Kern and Kraus (1972),and polynomial approximations are given by Kraus et al.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
