Heat Transfer Handbook part 14 pps

BOOKCOMP, Inc. — John Wiley & Sons / Page 120 / 2nd Proofs / Heat Transfer Handbook / Bejan 120 THERMOPHYSICAL PROPERTIES OF FLUIDS AND MATERIALS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [120], (78) Lines: 2711 to 2734 ——— 4.29408pt PgVar ——— Normal Page PgEnds: T E X [120], (78) ρ m c p dT dt = λ ∇ 2 T (2.58) Grouping the material properties, the thermal diffusivity is defined as α D = λ/ρ m c p . Thus, the important thermophysical properties are α D , λ, ρ m , and c p . In general, these properties can be functions of direction, deformation, and temperature. Some crystalline elements, such as carbon, bismuth, and tin, have anisotropic thermal con- ductivities. Some polymers develop anisotropy after finite deformation (Choy et al., 1978; Broerman et al., 1999; Ortt et al., 2000). The temperature dependence of α is sometimes less strong than that of λ, which can simplify analytical solutions of conduction problems (Ozisik, 1980). For analyses of transient heat transfer, α D is the important parameter, while for analyses of steady heat transfer and boundary condi- tions of transient analyses, λ is required. Equation (2.58) is nonphysical because it predicts an infinite speed of propagation of temperature change, that is, a temperature change in one part of the body causes an immediate change in temperature throughout the body. Substituting the Maxwell– Cattaneo equation for Fourier’s equation yields 1 α D dT dt + 1 c 2 d 2 T dt 2 =∇ 2 T (2.59) where c has dimensions of velocity. If c is approximated as the speed of sound in the body, then for good conductors the ratio of the coefficients is α D /c = 10 −11 s (Parrott and Stuckes, 1975). Thus, except in rare circumstances, finite propagation speeds are important only for very short times. Joseph and Preziosi (1989, 1990) provide an extensive review of studies examining Maxwell–Cattaneo conduction. In general, analyses of thermal conduction assume that materials are rigid and incompressible. This is not strictly so. Relaxing this assumption requires the addition of the simultaneous solution of the balance of linear momentum and the addition of the stress work term (see, e.g., Day, 1985). The linear coefficient of thermal expansion may be defined as µ = (L − L 0 )/ [ L 0 (T − T 0 ) ] , where L is the length of the solid at its new temperature T , and L 0 is its length at the reference temperature T 0 .At room temperature, µ typically ranges from 0.6 × 10 −6 °C −1 for silicon carbide to 500 × 10 −6 °C −1 for rubber (Brown, 1967). The volumetric coefficient of thermal expansion µ v = (1 +µ) 3 ≈ (1 +3µ) relates the specific volume at T to the specific volume at T 0 . 2.3.2 Behavior of Thermophysical Properties of Solids The thermal conductivity and specific heat are defined above as continuum properties. Some indication of their behavior may be found by considering the molecular nature of materials. In solids, thermal transport properties result from molecular vibrations, and in electrical conductors, from electron transport. The vibrations of the molecules in a crystal lattice may be analyzed as harmonic oscillators and quantized (see, e.g., Reif, 1965 or Brown, 1967). These quanta are called phonons, and heat conduction BOOKCOMP, Inc. — John Wiley & Sons / Page 121 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMOPHYSICAL PROPERTIES OF SOLIDS 121 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [121], (79) Lines: 2734 to 2745 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [121], (79) can be pictured as the diffusion of phonons from a hotter region to a colder one. Transport of electrons dominates that of phonons in metals at moderate temperatures, which explains why their thermal conductivities are typically larger than those of dielectrics. Typical values of thermal conductivity for metallic elements at 300 K range from 23 W/m ·K for zirconium to 427 W/m ·K for silver, whereas typical values for dielectrics run from 0.12 W/m ·K for paper to almost 3.0 W/m ·Kfor granite and marble (Baehr and Stephan, 1998). Diamond is anomalous in that it has a thermal conductivity as high as 2310 W/m ·K at 300 K (Ho et al., 1974). As the temperature approaches absolute zero, both the thermal conductivity and the specific heat tend toward zero, in accordance with the third law of thermodynam- ics. In dielectrics, the change in conductivity decreases as 1/T 3 , while in metals it decreases as 1/T , owing to the transport of electrons. As the material warms, the conductivity usually reaches a maximum and then decreases with increasing temper- ature. The trend at higher temperatures is not universal, however, and the thermal conductivity may still increase with temperature for some materials that melt or de- compose before the maximum is reached. The molar specific heat of many simple materials reaches approximately 3R at moderate temperatures, in accordance with the observations of Dulong and Petit (see, e.g., Brown, 1967; Reif, 1965). Modeling of materials such as glasses and amorphous polymers is less complete (Kittel, 1996). Dashora (1994) examined the temperature dependence of the thermal diffusivity of elastomers, and Eiermann (1966) discussed a resistive network model of heat con- duction in amorphous polymers. Some material systems, such as heterogeneous polymers (Bigg, 1995), biological tissues (Chato, 1985), and composite materials (Dowding et al., 1996) have been modeled using apparent thermal properties. In fact, these material systems are com- posed of materials with different densities, thermal conductivities, and specific heats. Prediction of the performance of material systems that include continuous fibers of dispersed particles or voids within a matrix material is difficult. Measurement of ap- parent properties may then be more expedient and accurate for a given system. Cooper and Trezek (1971), for example, proposed correlations for the apparent conductivity of biological soft tissue as functions of the mass fractions of water, protein, and fat. Even so, tabulated and correlated values of the thermophysical properties of biolog- ical materials must be used with caution because of potential anisotropy, specimen- to-specimen variation, and changes due to denaturation of protein during heating. 2.3.3 Property Values of Solid Materials Unlike gases and liquids, there is no standard, reference-quality computer package for the calculation of thermophysical properties of solids. Perhaps the most extensive compilation of solid properties comes from the Center for Information and Numerical Data Synthesis and Analysis (CINDAS) at Purdue University, which was established by Yeram S. Touloukian (1981) as the Thermophysical Properties Research Center (TPRC). The reference materials produced by this group have been used extensively as a resource for this part of the chapter. Tables of typical values for metallic alloys and nonmetallic solids have been taken from the introductory text by Bejan (1993). BOOKCOMP, Inc. — John Wiley & Sons / Page 122 / 2nd Proofs / Heat Transfer Handbook / Bejan 122 THERMOPHYSICAL PROPERTIES OF FLUIDS AND MATERIALS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [122], (80) Lines: 2745 to 2759 ——— 0.0pt PgVar ——— Normal Page * PgEnds: Eject [122], (80) Tables 2.8, 2.9, and 2.10 list the properties of solids that have been grouped into categories of solid elements, solid metallic alloys, and miscellaneous nonmetallic solids. Unlike those of fluids, the properties of solids vary little with changes in pressure, and thus the tables in this section neglect the effects of pressure on solid property values. The thermal conductivities of elements are sensitive to even minute levels of im- purity, especially at low temperatures, although the purity of the samples measured is often omitted in reports of measurements. Even near room temperature, values quoted by different sources may vary by 15% or more. The values of thermal con- ductivity for all elements recommended by Ho et al. (1974), listed in Table 2.8, were derived from critical and comprehensive evaluations by CINDAS of the data available in the literature up to the early 1970s. Accuracy for the thermal conductivity values may vary from 2 to 20%, depending on the purity and temperature levels. The reader should refer to the original work for detailed information if requirements for accuracy are high. 2.3.4 Measuring Thermophysical Properties of Solids Measuring thermal conductivity and thermal diffusivity requires the development of experimental approximations of boundary value problems. Carslaw and Jaeger (1959) provide analytical solutions to many classical boundary value problems that have been useful in the measurement of thermal transport properties. Reviews of methods for measuring thermal transport properties may be found in Maglic et al. (1984, 1992), Shirtliffe and Tye (1985), Parrott and Stuckes (1975), and Jakob (1955). The American Society for Testing and Materials (ASTM) has established several standards for measuring transport properties, and the National Institute of Standards and Technology (NIST) can provide some standard reference materials (SRMs). As noted above, sources of error include impurities in the sample and variables omitted from the analysis, such as deformation of polymers (Choy et al., 1978; Greig and Sahota, 1978; Doss and Wright, 2000). New methods of measurement are constantly being developed to overcome such sources of error, and the results are published as appropriate. Thermal Conductivity Direct measurement of thermal conductivity has tradi- tionally used steady-state methods. For materials of moderate to high thermal con- ductivity (∼10 to 500 W/m ·K), axial heat flow, radial heat flow, and direct electrical heating methods are often used (Maglic et al., 1984). Materials of lower thermal con- ductivity are most commonly tested in using the guarded hot plate (thermal insulation materials) or hot wire methods; the latter is a transient method. These methods can provide high accuracy and simple data reduction but require a relatively long time to reach steady state. This reduces their suitability for measuring properties of a material that may change during measurement, such as biological tissues. Thermal conductiv- ity is sometimes determined via indirect methods by measuring the diffusivity and using the density and specific heat to calculate the conductivity. The 3ω technique (text continues on page 140) BOOKCOMP, Inc. — John Wiley & Sons / Page 123 / 2nd Proofs / Heat Transfer Handbook / Bejan 123 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [123], (81) Lines: 2759 to 2791 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [123], (81) TABLE 2.8 Thermophysical Properties of Solid Elements a Aluminum (Al) T 50 100 150 200 273 298 400 500 600 700 800 900 ρ 2736 2732 2726 2719 2706 2701 2681 2661 2639 2616 2591 2564 c p — 481 — 797 881 893 944 994 1044 1094 1144 1194 λ 1350 302 248 237 236 237 240 236 231 225 218 210 α D — 230 — 109 99.0 98.3 94.8 89.2 83.8 78.6 73.6 68.6 Antimony (Sb) T 50 100 150 200 273 298 400 400 600 700 800 900 ρ — 6724 6714 6704 6688 6683 6660 6636 6613 6590 6568 c p ————206208214220226232238244 λ 88.3 46.4 35.6 30.2 25.5 24.4 21.3 19.5 18.3 17.4 16.8 16.7 α D ————18.5 17.6 15.0 13.4 12.2 11.4 10.7 Beryllium (Be) T — 100 200 273 298 400 500 600 800 1000 1200 1400 ρ ————1850 1843 1835 1826 1807 1785 1763 1739 c p — 195 1109 1624 1764 2119 2317 2458 2671 2847 λ — 990 301 218 201 161 139 126 106 90.8 78.7 69.4 α D ————61.6 41.2 32.7 28.1 22.0 17.9 Bismuth (Bi) T 50 100 200 273 298 350 400 500 ρ 9872 9855 9817 9788 9778 9758 9738 9700 c p — 109 120 122 123 126 129 134 (continued) BOOKCOMP, Inc. — John Wiley & Sons / Page 124 / 2nd Proofs / Heat Transfer Handbook / Bejan 124 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [124], (82) Lines: 2791 to 2819 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [124], (82) TABLE 2.8 Thermophysical Properties of Solid Elements a (Continued) Bismuth (Bi) (Continued) λ 32.6 16.5 9.69 8.20 7.89 7.39 7.04 6.63 α D — 15.4 8.2 6.9 6.5 6.0 5.6 5.1 Boron (B) T 50 100 150 200 273 298 400 600 800 1000 1100 1300 ρ —————2500 2496 2488 2479 2470 2465 2455 c p — — — — 1061 1104 1277 1618 1958 2299 2469 λ 404 190 93.5 55.1 31.8 27.4 16.8 10.6 9.60 9.85 10.1 α D —————9.95.32.62.01.71.7 Cadmium (Cd) T 50 100 200 273 298 373 400 473 500 573 ρ — 8799 8724 8666 8646 8585 8562 8499 8475 8405 c p — — — 228 231 237 240 247 249 256 λ 120 103 99.3 97.5 96.9 95.3 94.7 92.8 92.0 89.1 α D — — — 49.3 48.6 46.7 46.1 44.3 43.6 41.4 Calcium (Ca) T — 100 200 273 298 350 400 500 573 600 800 1000 ρ — 1568 1559 1552 1549 1544 1539 1528 1520 1517 c p — — — 649 658 676 693 728 754 763 774 λ — — 222 206 201 193 188 181 179 178 153 128 α D — — — 204 197 185 176 163 156 154 BOOKCOMP, Inc. — John Wiley & Sons / Page 125 / 2nd Proofs / Heat Transfer Handbook / Bejan 125 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [125], (83) Lines: 2819 to 2849 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [125], (83) Carbon (C) T 50 100 150 200 273 298 400 600 800 1000 1500 2500 ρ (amorphous) — — — — — 1950 1948 1944 1940 1936 1924 1892 c p (graphite) — — — — 633 744 1040 1364 1595 1800 λ (amorphous) 0.377 0.668 0.938 1.18 1.50 1.59 1.89 2.19 2.37 2.53 3.48 Cerium (Ce) T 50 100 150 200 273 298 400 500 600 700 800 1000 ρ — — — — — 6899 6888 6875 6861 6846 6831 6795 c p — — — — 186 187 190 194 198 201 205 λ 3.79 6.00 7.66 9.00 10.8 11.3 13.3 15.0 16.5 18.0 19.3 21.8 α D — — — — — 8.8 10.1 11.3 12.2 13.1 13.8 Cesium (Cs) T 50 100 150 200 273 298 ρ 1962 1944 1926 1907 1880 1871 c p — — — — 218 232 λ 44.7 39.7 37.8 36.8 36.1 35.9 α D — — — — 88.0 82.5 Chromium (Cr) T 50 100 200 273 298 400 600 800 1000 1200 1500 1800 ρ — — — — 7139 7124 7086 7042 6995 6944 6860 6761 c p — 190 382 454 460 484 532 579 627 674 746 817 λ 317 159 111 96.5 93.9 90.9 80.7 71.3 65.4 61.9 57.2 52.6 α D — — — — 28.6 26.3 21.4 17.5 14.9 13.2 11.2 9.5 (continued) BOOKCOMP, Inc. — John Wiley & Sons / Page 126 / 2nd Proofs / Heat Transfer Handbook / Bejan 126 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [126], (84) Lines: 2849 to 2880 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [126], (84) TABLE 2.8 Thermophysical Properties of Solid Elements a (Continued) Cobalt (Co) T 50 100 200 273 298 400 600 800 1000 1200 1400 1700 ρ — 8919 8892 8869 8860 8823 8744 8642 8561 8475 c p — 234 376 428 434 458 505 553 600 647 694 765 λ 299 167 122 105 100 85.4 67.4 58.2 52.1 49.3 41.7 43.0 α D — 80.0 36.5 27.7 26.0 21.1 15.3 12.2 10.1 8.99 Copper (Cu) T 50 100 200 273 298 400 600 800 1000 1100 1200 1300 ρ — 9009 8973 8942 8931 8884 8788 8686 8576 8519 8458 8396 c p — 254 357 384 387 397 416 435 454 464 474 483 λ 1250 482 413 403 401 393 379 366 352 346 339 332 α D — 210.6 128.9 117.2 116.1 111.5 103.7 96.8 90.3 87.5 84.6 81.8 Gold (Au) T 50 100 200 273 298 400 600 800 1000 1100 1200 1300 ρ — 19,030 18,950 18,900 18,880 18,790 18,620 18,440 18,250 18,140 18,030 17,920 c p — 109 124 128 128 131 138 144 150 153 156 159 λ 421 327 323 319 318 311 298 284 270 262 255 247 α D — 157.7 137.4 132.4 131.3 125.9 116.4 107.2 98.8 94.5 90.7 86.7 Iron (Fe) T 50 100 200 273 298 400 600 800 1000 1200 1400 1600 ρ 7918 7913 7895 7876 7869 7838 7772 7699 7624 7632 7527 7424 c p — 216 384 440 452 501 596 692 787 629 629 629 λ 405 134 94.0 83.5 80.4 69.5 54.7 43.3 32.8 28.3 31.2 33.0 α D — 78.4 31.0 24.1 22.6 17.7 11.8 8.1 5.5 5.9 6.6 7.1 BOOKCOMP, Inc. — John Wiley & Sons / Page 127 / 2nd Proofs / Heat Transfer Handbook / Bejan 127 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [127], (85) Lines: 2880 to 2911 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [127], (85) Lead (Pb) T 50 100 200 273 298 400 500 600 ρ — 11,520 11,430 11,360 11,340 11,230 11,130 11,010 c p — 118 125 128 129 133 137 141 λ 43.6 39.7 36.7 35.6 35.3 34.0 32.8 31.4 α D — 29.2 25.7 24.5 24.2 22.8 21.5 20.2 Lithium (Li) T 50 100 200 273 298 300 350 400 450 ρ 547 546 541 535 534 533 530 526 521 c p — — — 3373 3644 3666 4208 4751 5293 λ 235 104 90.1 85.9 84.8 84.7 82.8 80.4 α D — — — 47.6 43.6 43.3 37.1 32.2 Magnesium (Mg) T 50 100 200 273 298 400 500 600 700 800 900 ρ — 1761 1751 1742 1739 1726 1711 1696 1680 1663 1645 c p — 648 929 973 1004 1086 1135 1172 1204 1232 1259 λ 465 169 159 157 156 153 151 149 147 146 145 α D — 148.1 97.7 92.6 89.3 81.6 77.7 74.9 72.7 71.2 70.0 Manganese (Mn) T 50 100 200 273 298 400 600 800 900 1000 ρ — — 7474 7440 7428 7376 7257 7109 7020 c p — — — 442 456 514 628 741 198 855 λ 4.06 5.79 7.17 7.68 7.81 α D — — — 2.3 2.3 (continued) BOOKCOMP, Inc. — John Wiley & Sons / Page 128 / 2nd Proofs / Heat Transfer Handbook / Bejan 128 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [128], (86) Lines: 2911 to 2942 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [128], (86) TABLE 2.8 Thermophysical Properties of Solid Elements a (Continued) Nickel (Ni) T 50 100 200 273 298 400 600 800 1000 1200 1500 1700 ρ — — — — 8898 8860 8779 8694 8606 8516 8372 c p — 232 383 428 440 486 577 550 563 576 595 608 λ 400 164 107 94.1 90.0 80.2 65.6 67.6 71.8 76.2 82.6 α D — — — — 23.2 18.6 12.9 14.1 14.8 15.5 16.6 Platinum (Pt) T 50 100 200 273 298 400 600 800 1000 1400 1800 1900 ρ — 21,500 21,500 21,460 21,450 21,390 21,270 21,140 21,010 20,720 20,400 20,310 c p — 101 127 134 134 137 142 147 152 162 172 λ 109 77.5 72.6 71.7 71.6 71.8 73.2 75.6 78.7 87.1 96.1 97.8 α D — 35.6 26.6 25.0 24.8 24.5 24.3 24.4 24.7 26.0 27.4 Potassium (K) T 50 100 200 273 298 300 323 336 ρ 911 904 888 874 869 869 864 861 c p — — — 723 738 739 753 760 λ 112 107 104 104 103 102 100 99 α D — — — 165 160 160 154 150 Silicon (Si) T 50 100 200 273 298 400 600 800 1000 1200 1400 1600 ρ 2422 2422 2421 2420 2420 2418 2413 2407 2401 2394 2388 2381 c p — — — 678 713 798 868 905 932 λ 2680 884 264 168 149 98.9 61.9 42.2 31.2 25.7 23.5 22.1 α D — — — 102.3 86.4 51.3 29.5 19.4 13.9 BOOKCOMP, Inc. — John Wiley & Sons / Page 129 / 2nd Proofs / Heat Transfer Handbook / Bejan 129 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [129], (87) Lines: 2942 to 2973 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [129], (87) Silver (Ag) T 50 100 200 273 298 400 500 600 700 800 1000 1200 ρ — — 10,550 10,500 10,490 10,430 10,360 10,300 10,230 10,160 10,010 9855 c p — 187 225 233 235 240 246 252 258 264 275 287 λ 700 444 430 429 429 425 419 412 404 396 379 361 α D — — 181 175 174 169 164 159 153 148 137 128 Sodium (Na) T 50 100 200 273 298 300 371 ρ — 1007 990 975 970 970 955 c p — — — 1178 1202 1204 1274 λ 158 136 142 142 142 141 132 α D — — — 124 122 121 109 Tantalum (Ta) T 50 100 200 273 298 400 600 800 1000 1500 2000 3000 ρ — — — — 16,600 16,570 16,500 16,430 16,360 16,180 15,970 15,340 c p — — — 143 143 146 150 155 160 λ 72.0 59.2 57.5 57.4 57.5 57.8 58.6 59.4 60.2 62.2 64.1 66.6 α D — — — — 24.1 23.9 23.6 23.3 23.1 Tin (Sn) T 50 100 200 273 298 400 500 ρ — 5815 5783 5757 7307 7255 7199 c p — 189 214 224 228 246 263 λ 115 85.3 73.3 68.2 66.8 62.2 59.6 α D — 77.6 59.2 52.8 40.0 34.9 31.5 (continued) . 149 147 146 145 α D — 148 .1 97.7 92.6 89.3 81.6 77.7 74.9 72.7 71.2 70.0 Manganese (Mn) T 50 100 200 273 298 400 600 800 900 1000 ρ — — 7474 7440 7428 7376 7257 7109 7020 c p — — — 442 456 514. density and specific heat to calculate the conductivity. The 3ω technique (text continues on page 140 ) BOOKCOMP, Inc. — John Wiley & Sons / Page 123 / 2nd Proofs / Heat Transfer Handbook / Bejan 123 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [123],. 26.3 21.4 17.5 14. 9 13.2 11.2 9.5 (continued) BOOKCOMP, Inc. — John Wiley & Sons / Page 126 / 2nd Proofs / Heat Transfer Handbook / Bejan 126 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [126],