Fluid Mechanics and Heat Transfer

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对流扩散反应方程的cfl条件

对流扩散反应方程是描述物质传输过程中同时考虑了对流、扩散和反应的数学模型。在数值计算中，为了确保计算结果的准确性和稳定性，需要满足CFL( Courant-Friedrichs-Lewy)条件，该条件是一种数值稳定性条件，能够控制时间步长的选取。

CFL条件的提出

CFL条件是由Richard Courant、Kurt Friedrichs和Hans Lewy在1928年提出的。他们发现，在求解偏微分方程的数值计算中，存在一个与物理问题无关的数值稳定性条件，即CFL条件。当时间步长超过CFL条件时，数值解就会出现不稳定、震荡以及计算结果不准确等问题。

CFL条件的定义

CFL条件是根据对流速度、网格尺寸和扩散系数之间的关系来定义的。在对流扩散反应方程中，对流项的影响取决于对流速度，而扩散项的影响取决于网格尺寸和扩散系数。CFL条件的定义如下：

CFL条件 = 对流速度 × 时间步长 / 网格尺寸 ≤ 1

其中，对流速度是描述物质在流动中传输的速度，时间步长是数值计算中的时间间隔，网格尺寸是用来离散化空间的单元大小。

CFL条件的意义 CFL条件的意义在于保证数值计算的稳定性。当满足CFL条件时，数值解才能保持稳定，不会发散或者出现震荡现象。否则，如果时间步长选取过大或网格尺寸选取过小，会导致计算结果不准确，甚至影响到计算的收敛性。

满足CFL条件的选择

为了满足CFL条件，需要合理选择时间步长和网格尺寸。一般来说，时间步长与网格尺寸的比值需要小于或等于对流速度，即：

时间步长 / 网格尺寸 ≤ 对流速度 / 扩散系数

这样可以确保数值计算的稳定性和准确性。当网格尺寸变小时，要相应减小时间步长，以保持CFL条件的满足。

总结

对流扩散反应方程的CFL条件是一种数值稳定性条件，可以有效控制数值计算的稳定性和准确性。合理选择时间步长和网格尺寸，以满足CFL条件，是进行对流扩散反应方程数值计算的重要一步。只有在满足CFL条件的情况下，才能得到可靠的数值解，准确地反映实际物理过程的特征。

EFFECT OF 000P-2509/791030I432SIS02 0010

FLUID DISPERSION COEFFICIENTS ON PARTICLE-TO-FLUID HEAT TRANSFER COEFFICIENTS IN PACKED BEDS CORRELATION OF NUSSELT NUMBERS N WAKAO,t S KAGUEI and T FUNAZKRI School of Engmeermg, Yokohama National Umverslty, Hodgaya-ku, Yokohama, Japan 240 (Recerued 24 February 1978. accepted 17 May 1978) Abstract-The pubhshed heat transfer data obtamed from steady and nonsteady measurements are corrected for the axial thud thermal dispersion coefficient values proposed by Wakao[ I] The corrected data m the range of Reynolds number from 15 to 8500 are correlated by the analogous form of the mass correlation proposed by Wakao and Funazkn[Z] Nu=2+ 1 1 Pr”)Reo6 This work for particle-to-flmd heat transfer 1s an ex- tenston of that reported by Wakao and Funazkn[2] m which the pubhshed particle-to-flmd mass transfer data were corrected for axial fluld dlsperslon coefficients Slmdarly to the data collectlon for mass transfer, we confine ourselves to the heat transfer measurements which satisfy the followmg condltlons 1 Particles m bed bemg all actlve 2 Number of particle layers m heat transfer bed being greater than two Table 1 summarizes the expenmentai condltlons of the collected measurements The data are shown as Nu vs Re m Fig 1 ConsIderable scattering and anomalous decrease m Nusselt number are seen at low Reynolds numbers In fact, the anomaly of Nusselt numbers has been a subject of dlscusslon for long However, not all the experimental mvestigators have supported the anomaly of Nusselt numbers From frequency response measurements, Llttman and Shva[3], and Gunn and De Souza[4] have estimated, respectively, from their own experiments the llmltmg Nusselt numbers of about 0 4 and 10 Some theoretical studies have also been made on this subject There IS, of course, no exact theory which describes the transport phenomena m packed beds Different theoretlcal estimates have given different hml- tmg Nusselt numbers Just hke the expenmental data, the two completely dIvergent conclusions have been drawn from the estimates Cormsh[S] indicates, assummg an analogy between electrostatics and heat, that the heat transfer coefficient m dense system of particles IS considerably smaller than that for a single sphere m an mfirute medium Kunu and Suzukl[6] pomt out that flow channeling m the bed IS the reason for the anomaly Nelson and Galloway [7] indicate that the anomaly IS explamed by a renewal of fluld tTo whom correspondence should be made element surrounding each particle Schlunder [8] shows, on the assumption that packed bed IS a bundle of parallel capillaries, that transfer coefficients decrease with a decrease of flow rate at lower Reynolds numbers On the other hand, Pfeffer and Happel[9] obtam, by applymg a free surface model, a lrmitmg Nusselt number of about 13 at the bed void fraction 0 4 From an analysis of steady transfer m a stagnant fluid m a concentrlc hollow sphere, Mlyauchl[lO] shows a hmltmg Nusselt number of about 18 at the bed void fraction 0 4 Stiren- sen and Stewart[ll] study a creepmg flow through a cubic array of spheres to find a hmltmg Nusselt number of about 3 9 Wakao et al [l, 12-141, however, find the defect of the fundamental equations responsible for the anomalous decrease m Nusselt number at lower Reynolds numbers REVIEW AND CORRECTI ON OF THE DATA OBTAINED FROM STEADY MEASUREMENTS Sdmultaneous heat and mass transfer study evaporatron of water and drfiusron-controlled chemrcal reactton on part&e surface The mass transfer data obtained from the stmultaneous heat and mass transfer studies have been used for the data correlation m the mass transfer paper[2] The cor- responding heat transfer data are, therefore, used m the present paper The data are those obtatned from the measurements of evaporation of water by Hougen et al [15, 161, Hurt[l7], Galloway et al [18], Thodos et al [19-231 and Bradshaw and Myers[24], and those determmed from the catalytic decomposltlon of hydrogen peroxide on metal spheres by Satterfield and Resmck[25] The measurements were made with sohd particles havmg codstant surface temperatures throughout the beds In the studies the system was described as dTF h,a U~+CbC.FU-~- T,.)=O 325 326 N WAKAO et al Kunu and Smith [271 C Cybulskl e/a/ C281 K Bradshow et al C461 L Hacdley and Hsggs C453 (Hougen eta/ Cl5. l-63 D Elchharn and White C301 E Llttman et o/ 1291 M F Gum and De Sauza 141 G Balokrlshnan and Pet C323 i Hurt Cl71 Galloway et a/ Cl81 Thadas et a/ 119-231 Brodshaw and Myers 1241 H Bhattacharyya and Pel m31 N Llndouer C471 I Glosw and Thodas 1351 0 Eaumslster and Bennett C3ll J Satterfleld and Resruck C251 P Turner et 01 C39. 401 A Liquid-Sdld B-P Gas-S&d

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