3506:
3494:
34:
757:
2373:
3521:
Open cavities are defined as the regions formed between adjacent fins and stand for the essential promoters of nucleate boiling or condensation. These cavities are usually utilized to extract heat from a variety of heat generating bodies. From 2004 until now, many researchers have been motivated to
2524:
2818:
1044:
1582:
2932:
879:
3219:
is the fin cross-sectional area at the base. Fin performance can also be characterized by fin efficiency. This is the ratio of the fin heat transfer rate to the heat transfer rate of the fin if the entire fin were at the base temperature,
1834:
573:
2201:
1168:
2618:
3333:
in this equation is equal to the surface area of the fin. The fin efficiency will always be less than one, as assuming the temperature throughout the fin is at the base temperature would increase the heat transfer rate.
1672:
3181:
2165:
3419:
3302:
344:
562:
250:
1961:
1896:
A similar approach can be used to find the constants of integration for the remaining cases. For the second case, the tip is assumed to be insulated, or in other words to have a heat flux of zero. Therefore,
2065:
437:
2379:
2697:
2683:
2994:
1415:
3045:
1285:
891:
1228:
1443:) conditions, however: the tip can be exposed to convective heat transfer, insulated, held at a constant temperature, or so far away from the base as to reach the ambient temperature.
1452:
2824:
2009:
3485:
3091:
467:
752:{\displaystyle -kA_{c}\left.\left({\frac {dT}{dx}}\right)\right\vert _{x+dx}=-kA_{c}\left.\left({\frac {dT}{dx}}\right)\right\vert _{x}+Ph\left(T-T_{\infty }\right)dx.}
376:
is the cross-sectional area of the differential element. Furthermore, the convective heat flux can be determined via the definition of the heat transfer coefficient h,
768:
3217:
2368:{\displaystyle {\frac {\theta }{\theta _{b}}}={\frac {\cosh {m(L-x)}+\left({\frac {h}{mk}}\right)\sinh {m(L-x)}}{\cosh {mL}+\left({\frac {h}{mk}}\right)\sinh {mL}}}}
1683:
1057:
If the fin has a constant cross-section along its length, the area and perimeter are constant and the differential equation for temperature is greatly simplified to
3449:
3331:
1891:
1864:
1339:
1312:
374:
89:
to change the first two options. Thus, adding a fin to an object, increases the surface area and can sometimes be an economical solution to heat transfer problems.
1441:
1063:
2538:
469:
is the temperature of the surroundings. The differential convective heat flux can then be determined from the perimeter of the fin cross-section P,
3505:
1593:
3099:
3493:
2076:
3758:
3343:
3226:
2070:
Finally, we can use the temperature distribution and
Fourier's law at the base of the fin to determine the overall rate of heat transfer,
3057:
Fin performance can be described in three different ways. The first is fin effectiveness. It is the ratio of the fin heat transfer rate (
261:
475:
147:
1903:
141:
With these assumptions, conservation of energy can be used to create an energy balance for a differential cross section of the fin:
2519:{\displaystyle {\sqrt {hPkA_{c}}}\theta _{b}{\frac {\sinh {mL}+{\frac {h}{mk}}\cosh {mL}}{\cosh {mL}+{\frac {h}{mk}}\sinh {mL}}}}
2020:
762:
Rearranging this equation and using the definition of the derivative yields the following differential equation for temperature,
2813:{\displaystyle {\frac {\theta }{\theta _{b}}}={\frac {{\frac {\theta _{L}}{\theta _{b}}}\sinh {mx}+\sinh {m(L-x)}}{\sinh {mL}}}}
3595:
Lorenzini, G.; Biserni, C.; Rocha, L.A.O. (2011). "Geometric optimization of isothermal cavities according to Bejan's theory".
3487:
is the sum of the heat transfer from the unfinned base area and all of the fins. This is the efficiency for an array of fins.
2624:
2946:
3643:
1352:
49:
are surfaces that extend from an object to increase the rate of heat transfer to or from the environment by increasing
382:
1039:{\displaystyle kA_{c}{\frac {d^{2}T}{dx^{2}}}+k{\frac {dA_{c}}{dx}}{\frac {dT}{dx}}-Ph\left(T-T_{\infty }\right)=0.}
3000:
1577:{\displaystyle hA_{c}\left(T(L)-T_{\infty }\right)=-kA_{c}\left.\left({\frac {dT}{dx}}\right)\right\vert _{x=L},}
1233:
74:
2927:{\displaystyle {\sqrt {hPkA_{c}}}\theta _{b}{\frac {\cosh {mL}-{\frac {\theta _{L}}{\theta _{b}}}}{\sinh {mL}}}}
1176:
2014:
For the fourth and final case, the fin is assumed to be infinitely long. Therefore, the boundary condition is:
1972:
3777:
1446:
For the first case, the second boundary condition is that there is free convection at the tip. Therefore,
1966:
For the third case, the temperature at the tip is held constant. Therefore, the boundary condition is:
3701:
3454:
3060:
3750:
105:
82:
70:
3668:
3734:
3550:
874:{\displaystyle k{\frac {d}{dx}}\left(A_{c}{\frac {dT}{dx}}\right)-Ph\left(T-T_{\infty }\right)=0}
445:
116:
To create a tractable equation for the heat transfer of a fin, many assumptions need to be made:
1829:{\displaystyle h\left(C_{1}e^{mL}+C_{2}e^{-mL}\right)=km\left(C_{2}e^{-mL}-C_{1}e^{mL}\right).}
3754:
3717:
3692:
Hill, R.; Veghte, J. (1976). "Jackrabbit ears: surface temperatures and vascular responses".
3608:
3189:
3742:
3709:
3604:
3427:
3309:
1869:
1842:
1317:
1290:
352:
3738:
101:
54:
3743:
2174:
Temperature distribution and heat transfer rate for fins of uniform cross sectional area
1420:
885:
the derivative on the left can be expanded to the most general form of the fin equation,
3705:
1163:{\displaystyle {\frac {d^{2}T}{dx^{2}}}={\frac {hP}{kA_{c}}}\left(T-T_{\infty }\right).}
3542:
3771:
3093:) to the heat transfer rate of the object if it had no fin. The formula for this is:
42:
3337:
The third way fin performance can be described is with overall surface efficiency,
2613:{\displaystyle {\frac {\theta }{\theta _{b}}}={\frac {\cosh {m(L-x)}}{\cosh {mL}}}}
120:
78:
567:
The equation of energy conservation can now be expressed in terms of temperature,
3669:"VII.H.4 Development of a Thermal and Water Management System for PEM Fuel Cells"
3647:
3546:
1049:
The cross-sectional area, perimeter, and temperature can all be functions of x.
66:
3558:
3554:
86:
58:
50:
1667:{\displaystyle h\theta (L)=-k\left.{\frac {d\theta }{dx}}\right\vert _{x=L}.}
3579:
3176:{\displaystyle \epsilon _{f}={\frac {{\dot {Q}}_{f}}{hA_{c,b}\theta _{b}}},}
93:
62:
3713:
17:
2160:{\displaystyle {\dot {Q}}_{\text{total}}={\sqrt {hPkA_{c}}}(C_{2}-C_{1}).}
1349:
The base of the fin is typically set to a constant reference temperature,
3721:
3538:
3531:
3553:. Nature has also taken advantage of the phenomena of fins; the ears of
65:
of an object determines the amount of heat it transfers. Increasing the
2170:
The results of the solution process are summarized in the table below.
97:
3414:{\displaystyle \eta _{o}={\frac {{\dot {Q}}_{t}}{hA_{t}\theta _{b}}},}
3297:{\displaystyle \eta _{f}={\frac {{\dot {Q}}_{f}}{hA_{f}\theta _{b}}}.}
33:
3561:
act as fins to release heat from the blood that flows through them.
1893:
to find the temperature distribution, which is in the table below.
339:{\displaystyle {\dot {Q}}(x)=-kA_{c}\left({\frac {dT}{dx}}\right),}
557:{\displaystyle d{\dot {Q}}_{conv}=Ph\left(T-T_{\infty }\right)dx.}
245:{\displaystyle {\dot {Q}}(x+dx)={\dot {Q}}(x)+d{\dot {Q}}_{conv}.}
32:
1956:{\displaystyle \left.{\frac {d\theta }{dx}}\right\vert _{x=L}=0.}
3530:
Fins are most commonly used in heat exchanging devices such as
81:
of the object increases the heat transfer. Sometimes it is not
3535:
28:
1341:
can now be found by applying the proper boundary conditions.
2060:{\displaystyle \lim _{L\rightarrow \infty }\theta _{L}=0\,}
1909:
1677:
The two boundary conditions can now be combined to produce
1623:
1525:
665:
595:
3622:
125:
Constant material properties (independent of temperature)
3749:(6 ed.). New York: John Wiley & Sons. pp.
2678:{\displaystyle {\sqrt {hPkA_{c}}}\theta _{b}\tanh {mL}}
3511:
Aluminium heat sink with high efficiency cooling fins.
2989:{\displaystyle {\frac {\theta }{\theta _{b}}}=e^{-mx}}
3457:
3430:
3346:
3312:
3229:
3192:
3102:
3063:
3003:
2949:
2827:
2700:
2627:
2541:
2382:
2204:
2079:
2023:
1975:
1906:
1872:
1845:
1686:
1596:
1455:
1423:
1355:
1320:
1293:
1236:
1179:
1066:
894:
771:
576:
478:
448:
385:
355:
264:
150:
3499:
Aluminium heat sink with low efficiency cooling fins
3578:Lienhard, John H. IV; Lienhard, John H. V. (2019).
3479:
3443:
3413:
3325:
3296:
3211:
3175:
3085:
3039:
2988:
2926:
2812:
2677:
2612:
2518:
2367:
2159:
2059:
2003:
1955:
1885:
1858:
1828:
1666:
1576:
1435:
1410:{\displaystyle \theta _{b}(x=0)=T_{b}-T_{\infty }}
1409:
1333:
1306:
1279:
1222:
1162:
1038:
873:
751:
556:
461:
431:
368:
338:
244:
3549:. They are also used in newer technology such as
2025:
432:{\displaystyle q''=h\left(T-T_{\infty }\right),}
3597:International Journal of Heat and Mass Transfer
1839:This equation can be solved for the constants
3040:{\displaystyle {\sqrt {hPkA_{c}}}\theta _{b}}
8:
1417:. There are four commonly possible fin tip (
92:One-piece finned heat sinks are produced by
3522:search for the optimal design of cavities.
1280:{\displaystyle \theta (x)=T(x)-T_{\infty }}
3737:; DeWitt, David P.; Bergman, Theodore L.;
1223:{\displaystyle m^{2}={\frac {hP}{kA_{c}}}}
137:Uniform convection across the surface area
3471:
3460:
3459:
3456:
3435:
3429:
3399:
3389:
3375:
3364:
3363:
3360:
3351:
3345:
3317:
3311:
3282:
3272:
3258:
3247:
3246:
3243:
3234:
3228:
3197:
3191:
3161:
3145:
3131:
3120:
3119:
3116:
3107:
3101:
3077:
3066:
3065:
3062:
3031:
3019:
3004:
3002:
2974:
2959:
2950:
2948:
2913:
2897:
2887:
2881:
2870:
2861:
2855:
2843:
2828:
2826:
2799:
2771:
2754:
2740:
2730:
2724:
2721:
2710:
2701:
2699:
2667:
2655:
2643:
2628:
2626:
2599:
2571:
2562:
2551:
2542:
2540:
2505:
2484:
2473:
2457:
2436:
2425:
2416:
2410:
2398:
2383:
2381:
2354:
2329:
2314:
2286:
2261:
2234:
2225:
2214:
2205:
2203:
2145:
2132:
2117:
2102:
2093:
2082:
2081:
2078:
2056:
2044:
2028:
2022:
1995:
1974:
1935:
1911:
1905:
1877:
1871:
1850:
1844:
1809:
1799:
1780:
1770:
1735:
1725:
1709:
1699:
1685:
1649:
1625:
1595:
1559:
1531:
1517:
1493:
1463:
1454:
1422:
1401:
1388:
1360:
1354:
1325:
1319:
1298:
1292:
1271:
1235:
1211:
1193:
1184:
1178:
1146:
1122:
1104:
1092:
1074:
1067:
1065:
1019:
975:
958:
948:
933:
915:
908:
902:
893:
854:
805:
799:
775:
770:
729:
699:
671:
657:
629:
601:
587:
575:
534:
495:
484:
483:
477:
453:
447:
415:
384:
360:
354:
309:
299:
266:
265:
263:
224:
213:
212:
185:
184:
152:
151:
149:
3609:10.1016/j.ijheatmasstransfer.2011.04.042
2172:
3584:(5th ed.). Mineola, NY: Dover Pub.
3570:
3489:
3745:Fundamentals of Heat and Mass Transfer
2004:{\displaystyle \theta (L)=\theta _{L}}
3644:"The Design of Chart Heat Exchangers"
7:
69:gradient between the object and the
3623:"Radiator Fin Machine or Machinery"
2035:
1494:
1402:
1272:
1147:
1020:
855:
730:
535:
454:
416:
25:
3504:
3492:
3480:{\displaystyle {\dot {Q}}_{t}}
3086:{\displaystyle {\dot {Q}}_{f}}
2787:
2775:
2587:
2575:
2302:
2290:
2250:
2238:
2151:
2125:
2032:
1985:
1979:
1609:
1603:
1483:
1477:
1378:
1366:
1261:
1255:
1246:
1240:
283:
277:
202:
196:
178:
163:
1:
1053:Uniform cross-sectional area
134:Uniform cross-sectional area
73:, increasing the convection
462:{\displaystyle T_{\infty }}
128:No internal heat generation
3794:
255:Fourier’s law states that
131:One-dimensional conduction
26:
75:heat transfer coefficient
3581:A Heat Transfer Textbook
3517:Inverted fins (cavities)
2196:Convection heat transfer
2185:Temperature distribution
3646:. Chart. Archived from
3625:. FinTool International
3212:{\displaystyle A_{c,b}}
2188:Fin heat transfer rate
3714:10.1126/science.982027
3481:
3451:is the total area and
3445:
3415:
3327:
3298:
3213:
3177:
3087:
3041:
2990:
2928:
2814:
2679:
2614:
2520:
2369:
2161:
2061:
2005:
1957:
1887:
1860:
1830:
1668:
1578:
1437:
1411:
1335:
1308:
1281:
1224:
1164:
1040:
875:
753:
558:
463:
433:
370:
340:
246:
38:
3482:
3446:
3444:{\displaystyle A_{t}}
3416:
3328:
3326:{\displaystyle A_{f}}
3299:
3214:
3178:
3088:
3042:
2991:
2929:
2815:
2680:
2615:
2521:
2370:
2162:
2062:
2006:
1958:
1888:
1886:{\displaystyle C_{2}}
1861:
1859:{\displaystyle C_{1}}
1831:
1669:
1579:
1438:
1412:
1336:
1334:{\displaystyle C_{2}}
1309:
1307:{\displaystyle C_{1}}
1282:
1225:
1165:
1041:
876:
754:
559:
464:
434:
371:
369:{\displaystyle A_{c}}
341:
247:
36:
3603:(17–18): 3868–3873.
3455:
3428:
3344:
3310:
3227:
3190:
3100:
3061:
3001:
2947:
2825:
2698:
2692:Constant Temperature
2625:
2539:
2380:
2202:
2077:
2021:
1973:
1904:
1870:
1843:
1684:
1594:
1587:which simplifies to
1453:
1421:
1353:
1318:
1291:
1234:
1177:
1064:
892:
769:
574:
476:
446:
383:
353:
262:
148:
77:, or increasing the
37:Some finned elements
3739:Lavine, Adrienne S.
3706:1976Sci...194..436H
3551:hydrogen fuel cells
2941:Infinite Fin Length
2182:Tip condition (x=L)
2175:
1436:{\displaystyle x=L}
3534:in cars, computer
3477:
3441:
3411:
3323:
3294:
3209:
3173:
3083:
3037:
2986:
2924:
2810:
2675:
2610:
2516:
2365:
2173:
2157:
2057:
2039:
2001:
1953:
1883:
1856:
1826:
1664:
1574:
1433:
1407:
1331:
1304:
1277:
1220:
1160:
1036:
871:
749:
554:
459:
429:
366:
336:
242:
39:
3760:978-0-471-45728-2
3700:(4263): 436–438.
3468:
3406:
3372:
3289:
3255:
3168:
3128:
3074:
3050:
3049:
3025:
2965:
2922:
2903:
2849:
2808:
2746:
2716:
2649:
2608:
2557:
2514:
2497:
2449:
2404:
2363:
2342:
2274:
2220:
2123:
2096:
2090:
2024:
1929:
1643:
1549:
1218:
1129:
1099:
993:
973:
940:
823:
788:
689:
619:
492:
327:
274:
221:
193:
160:
16:(Redirected from
3785:
3764:
3748:
3735:Incropera, Frank
3726:
3725:
3689:
3683:
3682:
3680:
3679:
3674:. Guillermo Pont
3673:
3665:
3659:
3658:
3656:
3655:
3640:
3634:
3633:
3631:
3630:
3619:
3613:
3612:
3592:
3586:
3585:
3575:
3508:
3496:
3486:
3484:
3483:
3478:
3476:
3475:
3470:
3469:
3461:
3450:
3448:
3447:
3442:
3440:
3439:
3420:
3418:
3417:
3412:
3407:
3405:
3404:
3403:
3394:
3393:
3380:
3379:
3374:
3373:
3365:
3361:
3356:
3355:
3332:
3330:
3329:
3324:
3322:
3321:
3303:
3301:
3300:
3295:
3290:
3288:
3287:
3286:
3277:
3276:
3263:
3262:
3257:
3256:
3248:
3244:
3239:
3238:
3218:
3216:
3215:
3210:
3208:
3207:
3182:
3180:
3179:
3174:
3169:
3167:
3166:
3165:
3156:
3155:
3136:
3135:
3130:
3129:
3121:
3117:
3112:
3111:
3092:
3090:
3089:
3084:
3082:
3081:
3076:
3075:
3067:
3046:
3044:
3043:
3038:
3036:
3035:
3026:
3024:
3023:
3005:
2995:
2993:
2992:
2987:
2985:
2984:
2966:
2964:
2963:
2951:
2933:
2931:
2930:
2925:
2923:
2921:
2920:
2905:
2904:
2902:
2901:
2892:
2891:
2882:
2877:
2862:
2860:
2859:
2850:
2848:
2847:
2829:
2819:
2817:
2816:
2811:
2809:
2807:
2806:
2791:
2790:
2761:
2747:
2745:
2744:
2735:
2734:
2725:
2722:
2717:
2715:
2714:
2702:
2684:
2682:
2681:
2676:
2674:
2660:
2659:
2650:
2648:
2647:
2629:
2619:
2617:
2616:
2611:
2609:
2607:
2606:
2591:
2590:
2563:
2558:
2556:
2555:
2543:
2525:
2523:
2522:
2517:
2515:
2513:
2512:
2498:
2496:
2485:
2480:
2465:
2464:
2450:
2448:
2437:
2432:
2417:
2415:
2414:
2405:
2403:
2402:
2384:
2374:
2372:
2371:
2366:
2364:
2362:
2361:
2347:
2343:
2341:
2330:
2321:
2306:
2305:
2279:
2275:
2273:
2262:
2253:
2226:
2221:
2219:
2218:
2206:
2176:
2166:
2164:
2163:
2158:
2150:
2149:
2137:
2136:
2124:
2122:
2121:
2103:
2098:
2097:
2094:
2092:
2091:
2083:
2066:
2064:
2063:
2058:
2049:
2048:
2038:
2010:
2008:
2007:
2002:
2000:
1999:
1962:
1960:
1959:
1954:
1946:
1945:
1934:
1930:
1928:
1920:
1912:
1892:
1890:
1889:
1884:
1882:
1881:
1865:
1863:
1862:
1857:
1855:
1854:
1835:
1833:
1832:
1827:
1822:
1818:
1817:
1816:
1804:
1803:
1791:
1790:
1775:
1774:
1751:
1747:
1746:
1745:
1730:
1729:
1717:
1716:
1704:
1703:
1673:
1671:
1670:
1665:
1660:
1659:
1648:
1644:
1642:
1634:
1626:
1583:
1581:
1580:
1575:
1570:
1569:
1558:
1554:
1550:
1548:
1540:
1532:
1522:
1521:
1503:
1499:
1498:
1497:
1468:
1467:
1442:
1440:
1439:
1434:
1416:
1414:
1413:
1408:
1406:
1405:
1393:
1392:
1365:
1364:
1340:
1338:
1337:
1332:
1330:
1329:
1313:
1311:
1310:
1305:
1303:
1302:
1287:. The constants
1286:
1284:
1283:
1278:
1276:
1275:
1229:
1227:
1226:
1221:
1219:
1217:
1216:
1215:
1202:
1194:
1189:
1188:
1169:
1167:
1166:
1161:
1156:
1152:
1151:
1150:
1130:
1128:
1127:
1126:
1113:
1105:
1100:
1098:
1097:
1096:
1083:
1079:
1078:
1068:
1045:
1043:
1042:
1037:
1029:
1025:
1024:
1023:
994:
992:
984:
976:
974:
972:
964:
963:
962:
949:
941:
939:
938:
937:
924:
920:
919:
909:
907:
906:
880:
878:
877:
872:
864:
860:
859:
858:
829:
825:
824:
822:
814:
806:
804:
803:
789:
787:
776:
758:
756:
755:
750:
739:
735:
734:
733:
704:
703:
698:
694:
690:
688:
680:
672:
662:
661:
643:
642:
628:
624:
620:
618:
610:
602:
592:
591:
563:
561:
560:
555:
544:
540:
539:
538:
509:
508:
494:
493:
485:
468:
466:
465:
460:
458:
457:
438:
436:
435:
430:
425:
421:
420:
419:
393:
375:
373:
372:
367:
365:
364:
345:
343:
342:
337:
332:
328:
326:
318:
310:
304:
303:
276:
275:
267:
251:
249:
248:
243:
238:
237:
223:
222:
214:
195:
194:
186:
162:
161:
153:
53:. The amount of
41:In the study of
21:
3793:
3792:
3788:
3787:
3786:
3784:
3783:
3782:
3768:
3767:
3761:
3733:
3730:
3729:
3691:
3690:
3686:
3677:
3675:
3671:
3667:
3666:
3662:
3653:
3651:
3642:
3641:
3637:
3628:
3626:
3621:
3620:
3616:
3594:
3593:
3589:
3577:
3576:
3572:
3567:
3543:heat exchangers
3528:
3519:
3512:
3509:
3500:
3497:
3458:
3453:
3452:
3431:
3426:
3425:
3395:
3385:
3381:
3362:
3347:
3342:
3341:
3313:
3308:
3307:
3278:
3268:
3264:
3245:
3230:
3225:
3224:
3193:
3188:
3187:
3157:
3141:
3137:
3118:
3103:
3098:
3097:
3064:
3059:
3058:
3055:
3027:
3015:
2999:
2998:
2970:
2955:
2945:
2944:
2906:
2893:
2883:
2863:
2851:
2839:
2823:
2822:
2792:
2736:
2726:
2723:
2706:
2696:
2695:
2651:
2639:
2623:
2622:
2592:
2564:
2547:
2537:
2536:
2489:
2466:
2441:
2418:
2406:
2394:
2378:
2377:
2334:
2325:
2307:
2266:
2257:
2227:
2210:
2200:
2199:
2141:
2128:
2113:
2080:
2075:
2074:
2040:
2019:
2018:
1991:
1971:
1970:
1921:
1913:
1908:
1907:
1902:
1901:
1873:
1868:
1867:
1846:
1841:
1840:
1805:
1795:
1776:
1766:
1765:
1761:
1731:
1721:
1705:
1695:
1694:
1690:
1682:
1681:
1635:
1627:
1622:
1621:
1592:
1591:
1541:
1533:
1527:
1524:
1523:
1513:
1489:
1473:
1469:
1459:
1451:
1450:
1419:
1418:
1397:
1384:
1356:
1351:
1350:
1347:
1321:
1316:
1315:
1294:
1289:
1288:
1267:
1232:
1231:
1207:
1203:
1195:
1180:
1175:
1174:
1142:
1135:
1131:
1118:
1114:
1106:
1088:
1084:
1070:
1069:
1062:
1061:
1055:
1015:
1008:
1004:
985:
977:
965:
954:
950:
929:
925:
911:
910:
898:
890:
889:
850:
843:
839:
815:
807:
795:
794:
790:
780:
767:
766:
725:
718:
714:
681:
673:
667:
664:
663:
653:
611:
603:
597:
594:
593:
583:
572:
571:
530:
523:
519:
482:
474:
473:
449:
444:
443:
411:
404:
400:
386:
381:
380:
356:
351:
350:
319:
311:
305:
295:
260:
259:
211:
146:
145:
114:
31:
23:
22:
15:
12:
11:
5:
3791:
3789:
3781:
3780:
3770:
3769:
3766:
3765:
3759:
3728:
3727:
3684:
3660:
3635:
3614:
3587:
3569:
3568:
3566:
3563:
3527:
3524:
3518:
3515:
3514:
3513:
3510:
3503:
3501:
3498:
3491:
3474:
3467:
3464:
3438:
3434:
3422:
3421:
3410:
3402:
3398:
3392:
3388:
3384:
3378:
3371:
3368:
3359:
3354:
3350:
3320:
3316:
3305:
3304:
3293:
3285:
3281:
3275:
3271:
3267:
3261:
3254:
3251:
3242:
3237:
3233:
3206:
3203:
3200:
3196:
3184:
3183:
3172:
3164:
3160:
3154:
3151:
3148:
3144:
3140:
3134:
3127:
3124:
3115:
3110:
3106:
3080:
3073:
3070:
3054:
3051:
3048:
3047:
3034:
3030:
3022:
3018:
3014:
3011:
3008:
2996:
2983:
2980:
2977:
2973:
2969:
2962:
2958:
2954:
2942:
2939:
2935:
2934:
2919:
2916:
2912:
2909:
2900:
2896:
2890:
2886:
2880:
2876:
2873:
2869:
2866:
2858:
2854:
2846:
2842:
2838:
2835:
2832:
2820:
2805:
2802:
2798:
2795:
2789:
2786:
2783:
2780:
2777:
2774:
2770:
2767:
2764:
2760:
2757:
2753:
2750:
2743:
2739:
2733:
2729:
2720:
2713:
2709:
2705:
2693:
2690:
2686:
2685:
2673:
2670:
2666:
2663:
2658:
2654:
2646:
2642:
2638:
2635:
2632:
2620:
2605:
2602:
2598:
2595:
2589:
2586:
2583:
2580:
2577:
2574:
2570:
2567:
2561:
2554:
2550:
2546:
2534:
2531:
2527:
2526:
2511:
2508:
2504:
2501:
2495:
2492:
2488:
2483:
2479:
2476:
2472:
2469:
2463:
2460:
2456:
2453:
2447:
2444:
2440:
2435:
2431:
2428:
2424:
2421:
2413:
2409:
2401:
2397:
2393:
2390:
2387:
2375:
2360:
2357:
2353:
2350:
2346:
2340:
2337:
2333:
2328:
2324:
2320:
2317:
2313:
2310:
2304:
2301:
2298:
2295:
2292:
2289:
2285:
2282:
2278:
2272:
2269:
2265:
2260:
2256:
2252:
2249:
2246:
2243:
2240:
2237:
2233:
2230:
2224:
2217:
2213:
2209:
2197:
2194:
2190:
2189:
2186:
2183:
2180:
2168:
2167:
2156:
2153:
2148:
2144:
2140:
2135:
2131:
2127:
2120:
2116:
2112:
2109:
2106:
2101:
2089:
2086:
2068:
2067:
2055:
2052:
2047:
2043:
2037:
2034:
2031:
2027:
2012:
2011:
1998:
1994:
1990:
1987:
1984:
1981:
1978:
1964:
1963:
1952:
1949:
1944:
1941:
1938:
1933:
1927:
1924:
1919:
1916:
1910:
1880:
1876:
1853:
1849:
1837:
1836:
1825:
1821:
1815:
1812:
1808:
1802:
1798:
1794:
1789:
1786:
1783:
1779:
1773:
1769:
1764:
1760:
1757:
1754:
1750:
1744:
1741:
1738:
1734:
1728:
1724:
1720:
1715:
1712:
1708:
1702:
1698:
1693:
1689:
1675:
1674:
1663:
1658:
1655:
1652:
1647:
1641:
1638:
1633:
1630:
1624:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1585:
1584:
1573:
1568:
1565:
1562:
1557:
1553:
1547:
1544:
1539:
1536:
1530:
1526:
1520:
1516:
1512:
1509:
1506:
1502:
1496:
1492:
1488:
1485:
1482:
1479:
1476:
1472:
1466:
1462:
1458:
1432:
1429:
1426:
1404:
1400:
1396:
1391:
1387:
1383:
1380:
1377:
1374:
1371:
1368:
1363:
1359:
1346:
1343:
1328:
1324:
1301:
1297:
1274:
1270:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1214:
1210:
1206:
1201:
1198:
1192:
1187:
1183:
1171:
1170:
1159:
1155:
1149:
1145:
1141:
1138:
1134:
1125:
1121:
1117:
1112:
1109:
1103:
1095:
1091:
1087:
1082:
1077:
1073:
1054:
1051:
1047:
1046:
1035:
1032:
1028:
1022:
1018:
1014:
1011:
1007:
1003:
1000:
997:
991:
988:
983:
980:
971:
968:
961:
957:
953:
947:
944:
936:
932:
928:
923:
918:
914:
905:
901:
897:
883:
882:
870:
867:
863:
857:
853:
849:
846:
842:
838:
835:
832:
828:
821:
818:
813:
810:
802:
798:
793:
786:
783:
779:
774:
760:
759:
748:
745:
742:
738:
732:
728:
724:
721:
717:
713:
710:
707:
702:
697:
693:
687:
684:
679:
676:
670:
666:
660:
656:
652:
649:
646:
641:
638:
635:
632:
627:
623:
617:
614:
609:
606:
600:
596:
590:
586:
582:
579:
565:
564:
553:
550:
547:
543:
537:
533:
529:
526:
522:
518:
515:
512:
507:
504:
501:
498:
491:
488:
481:
456:
452:
440:
439:
428:
424:
418:
414:
410:
407:
403:
399:
396:
392:
389:
363:
359:
347:
346:
335:
331:
325:
322:
317:
314:
308:
302:
298:
294:
291:
288:
285:
282:
279:
273:
270:
253:
252:
241:
236:
233:
230:
227:
220:
217:
210:
207:
204:
201:
198:
192:
189:
183:
180:
177:
174:
171:
168:
165:
159:
156:
139:
138:
135:
132:
129:
126:
123:
113:
110:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3790:
3779:
3778:Heat transfer
3776:
3775:
3773:
3762:
3756:
3752:
3747:
3746:
3740:
3736:
3732:
3731:
3723:
3719:
3715:
3711:
3707:
3703:
3699:
3695:
3688:
3685:
3670:
3664:
3661:
3650:on 2006-10-11
3649:
3645:
3639:
3636:
3624:
3618:
3615:
3610:
3606:
3602:
3598:
3591:
3588:
3583:
3582:
3574:
3571:
3564:
3562:
3560:
3556:
3552:
3548:
3544:
3540:
3537:
3533:
3525:
3523:
3516:
3507:
3502:
3495:
3490:
3488:
3472:
3465:
3462:
3436:
3432:
3408:
3400:
3396:
3390:
3386:
3382:
3376:
3369:
3366:
3357:
3352:
3348:
3340:
3339:
3338:
3335:
3318:
3314:
3291:
3283:
3279:
3273:
3269:
3265:
3259:
3252:
3249:
3240:
3235:
3231:
3223:
3222:
3221:
3204:
3201:
3198:
3194:
3170:
3162:
3158:
3152:
3149:
3146:
3142:
3138:
3132:
3125:
3122:
3113:
3108:
3104:
3096:
3095:
3094:
3078:
3071:
3068:
3052:
3032:
3028:
3020:
3016:
3012:
3009:
3006:
2997:
2981:
2978:
2975:
2971:
2967:
2960:
2956:
2952:
2943:
2940:
2937:
2936:
2917:
2914:
2910:
2907:
2898:
2894:
2888:
2884:
2878:
2874:
2871:
2867:
2864:
2856:
2852:
2844:
2840:
2836:
2833:
2830:
2821:
2803:
2800:
2796:
2793:
2784:
2781:
2778:
2772:
2768:
2765:
2762:
2758:
2755:
2751:
2748:
2741:
2737:
2731:
2727:
2718:
2711:
2707:
2703:
2694:
2691:
2688:
2687:
2671:
2668:
2664:
2661:
2656:
2652:
2644:
2640:
2636:
2633:
2630:
2621:
2603:
2600:
2596:
2593:
2584:
2581:
2578:
2572:
2568:
2565:
2559:
2552:
2548:
2544:
2535:
2532:
2529:
2528:
2509:
2506:
2502:
2499:
2493:
2490:
2486:
2481:
2477:
2474:
2470:
2467:
2461:
2458:
2454:
2451:
2445:
2442:
2438:
2433:
2429:
2426:
2422:
2419:
2411:
2407:
2399:
2395:
2391:
2388:
2385:
2376:
2358:
2355:
2351:
2348:
2344:
2338:
2335:
2331:
2326:
2322:
2318:
2315:
2311:
2308:
2299:
2296:
2293:
2287:
2283:
2280:
2276:
2270:
2267:
2263:
2258:
2254:
2247:
2244:
2241:
2235:
2231:
2228:
2222:
2215:
2211:
2207:
2198:
2195:
2192:
2191:
2187:
2184:
2181:
2178:
2177:
2171:
2154:
2146:
2142:
2138:
2133:
2129:
2118:
2114:
2110:
2107:
2104:
2099:
2087:
2084:
2073:
2072:
2071:
2053:
2050:
2045:
2041:
2029:
2017:
2016:
2015:
1996:
1992:
1988:
1982:
1976:
1969:
1968:
1967:
1950:
1947:
1942:
1939:
1936:
1931:
1925:
1922:
1917:
1914:
1900:
1899:
1898:
1894:
1878:
1874:
1851:
1847:
1823:
1819:
1813:
1810:
1806:
1800:
1796:
1792:
1787:
1784:
1781:
1777:
1771:
1767:
1762:
1758:
1755:
1752:
1748:
1742:
1739:
1736:
1732:
1726:
1722:
1718:
1713:
1710:
1706:
1700:
1696:
1691:
1687:
1680:
1679:
1678:
1661:
1656:
1653:
1650:
1645:
1639:
1636:
1631:
1628:
1618:
1615:
1612:
1606:
1600:
1597:
1590:
1589:
1588:
1571:
1566:
1563:
1560:
1555:
1551:
1545:
1542:
1537:
1534:
1528:
1518:
1514:
1510:
1507:
1504:
1500:
1490:
1486:
1480:
1474:
1470:
1464:
1460:
1456:
1449:
1448:
1447:
1444:
1430:
1427:
1424:
1398:
1394:
1389:
1385:
1381:
1375:
1372:
1369:
1361:
1357:
1344:
1342:
1326:
1322:
1299:
1295:
1268:
1264:
1258:
1252:
1249:
1243:
1237:
1212:
1208:
1204:
1199:
1196:
1190:
1185:
1181:
1157:
1153:
1143:
1139:
1136:
1132:
1123:
1119:
1115:
1110:
1107:
1101:
1093:
1089:
1085:
1080:
1075:
1071:
1060:
1059:
1058:
1052:
1050:
1033:
1030:
1026:
1016:
1012:
1009:
1005:
1001:
998:
995:
989:
986:
981:
978:
969:
966:
959:
955:
951:
945:
942:
934:
930:
926:
921:
916:
912:
903:
899:
895:
888:
887:
886:
868:
865:
861:
851:
847:
844:
840:
836:
833:
830:
826:
819:
816:
811:
808:
800:
796:
791:
784:
781:
777:
772:
765:
764:
763:
746:
743:
740:
736:
726:
722:
719:
715:
711:
708:
705:
700:
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43:heat transfer
35:
30:
19:
3744:
3697:
3693:
3687:
3676:. Retrieved
3663:
3652:. Retrieved
3648:the original
3638:
3627:. Retrieved
3617:
3600:
3596:
3590:
3580:
3573:
3559:fennec foxes
3547:power plants
3529:
3520:
3423:
3336:
3306:
3185:
3056:
2169:
2069:
2013:
1965:
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1838:
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1445:
1348:
1172:
1056:
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884:
761:
566:
441:
348:
254:
140:
121:Steady state
115:
112:General case
91:
79:surface area
46:
40:
3555:jackrabbits
3053:Performance
71:environment
67:temperature
18:Cooling fin
3678:2006-09-17
3654:2006-09-16
3629:2006-09-18
3565:References
87:economical
59:convection
55:conduction
51:convection
27:See also:
3539:heatsinks
3532:radiators
3466:˙
3397:θ
3370:˙
3349:η
3280:θ
3253:˙
3232:η
3159:θ
3126:˙
3105:ϵ
3072:˙
3029:θ
2976:−
2957:θ
2953:θ
2911:
2895:θ
2885:θ
2879:−
2868:
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2797:
2782:−
2769:
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2738:θ
2728:θ
2708:θ
2704:θ
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2569:
2549:θ
2545:θ
2533:Adiabatic
2503:
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2423:
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2352:
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2033:→
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1977:θ
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1632:θ
1616:−
1601:θ
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1345:Solutions
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158:˙
94:extrusion
63:radiation
3772:Category
3741:(2007).
391:″
83:feasible
3702:Bibcode
3694:Science
106:milling
102:skiving
98:casting
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3722:982027
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3424:where
3186:where
1173:where
442:where
349:where
3672:(PDF)
2095:total
104:, or
61:, or
3755:ISBN
3718:PMID
3557:and
3526:Uses
2908:sinh
2865:cosh
2794:sinh
2766:sinh
2749:sinh
2662:tanh
2594:cosh
2566:cosh
2500:sinh
2468:cosh
2452:cosh
2420:sinh
2349:sinh
2309:cosh
2281:sinh
2229:cosh
2179:Case
1866:and
1314:and
1230:and
47:fins
3710:doi
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3605:doi
3545:in
3536:CPU
2026:lim
85:or
29:Fin
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