气泡曳力系数模型分区研究

doi:10.11949/0438-1157.20190604

摘要/Abstract

摘要：

由曳力系数表征的曳力模型作为重要的相间力模型之一，被广泛应用于Euler-Euler方法和Euler-Lagrange方法下的连续相和离散相动量方程中。由于现有气泡曳力系数模型形式各异且适用范围有限，因此需要对已有模型进行充分地评价。考虑到已有曳力系数模型的适用范围和气泡变形的影响，参考各曳力系数模型采用的相关参数，建议基于Reynolds数Re和Weber数We分区选择最佳模型。将分区曳力系数模型、已有曳力模型与实验数据对比，发现分区曳力系数模型总体预测结果更符合实验测量值。将分区曳力系数模型应用至数值模拟中，可以更精确地追踪不同尺寸气泡的位置，使数值模拟结果更接近真实的物理情况。

关键词: 曳力模型, 曳力系数, 变形, 气泡运动

Abstract:

Drag force model which characterized by the drag coefficient, acting as a crucial interface force model, was generally applied into the momentum equation of continuous phase and dispersed phase in Euler-Euler approach and Euler-Lagrange approach. Previous drag coefficient models of bubbles are required to sufficiently estimated due to the different forms and limited applicability. Optimal model is selected considered the applicability of exist models and the influence of bubble shape deformation, which is partitioned and characterized by Reynolds number and Weber number according to the related parameters adopted in previous models. The drag coefficient predicted by the present model generally perform better than previous models by comparing with experimental data. A more physically and accurate predictions of bubble motions can be achieved by adopting the present model in numerical simulations to track the locations of various size of bubbles.

Key words: drag force model, drag coefficient, shape deformation, bubble motion

中图分类号:

TB 126

周毓佳, 赵陈儒, 薄涵亮. 气泡曳力系数模型分区研究[J]. 化工学报, 2019, 70(S2): 108-116.

Yujia ZHOU, Chenru ZHAO, Hanliang BO. Partitioned investigation of drag coefficient models of bubbles[J]. CIESC Journal, 2019, 70(S2): 108-116.

图/表 7

表1 常用曳力系数模型

Table 1 Common drag coefficient models

文献	适用范围	公式
[9]	全范围	$C D = 24 R e 1 + 0.15 R e 0.687 ? R e ≤ 1000 0.44 ? ? ? ? ? ? ? ? ? ? ? ? ? R e > 1000$
[10]	0.1≤Re≤100	$C D = 16 R e 1 + 8 R e + 12 1 + 3.315 R e 1 / 2 - 1$
[11]	Re≥200高黏体系	$C D = 48 R e G E 1 - 2.21 H E R e$
[4]	10^-2<Eo<10³ 10^-3<Re<10⁶ 10^-14<Mo<10⁷	$C D = m a x m i n 16 R e 1 + 0.15 R e 0.687, 48 R e, ? 83 E o E o + 4$
[12]	0.1≤Re≤100 9×10^-7≤Mo≤7	0.1≤Re<0.5 $C D = 8 R e 1 + 1 1 - 0.5 1 + 250 R e 5 - 2$ 0.5≤Re≤100 $C D = 16 R e 1 + R e 1.385 12 1 / 55 + 83 R e 4 / 3 M o 1 / 3 24 1 + M o 1 / 3 + R e 4 / 3 M o 1 / 3$
[6]	适用于球形气泡	$C D = 16 R e ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? R e < 1.5 14.9 R e 0.78 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1.5 ≤ R e < 80 48 R e 1 - 2.21 R e + 1.86 × 10 - 15 R e 4.756 ? ? 80 ≤ R e < 1500 2.61 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1500 ≤ R e$

表1 常用曳力系数模型

Table 1 Common drag coefficient models

文献	适用范围	公式
[9]	全范围	$C D = 24 R e 1 + 0.15 R e 0.687 ? R e ≤ 1000 0.44 ? ? ? ? ? ? ? ? ? ? ? ? ? R e > 1000$
[10]	0.1≤Re≤100	$C D = 16 R e 1 + 8 R e + 12 1 + 3.315 R e 1 / 2 - 1$
[11]	Re≥200高黏体系	$C D = 48 R e G E 1 - 2.21 H E R e$
[4]	10^-2<Eo<10³ 10^-3<Re<10⁶ 10^-14<Mo<10⁷	$C D = m a x m i n 16 R e 1 + 0.15 R e 0.687, 48 R e, ? 83 E o E o + 4$
[12]	0.1≤Re≤100 9×10^-7≤Mo≤7	0.1≤Re<0.5 $C D = 8 R e 1 + 1 1 - 0.5 1 + 250 R e 5 - 2$ 0.5≤Re≤100 $C D = 16 R e 1 + R e 1.385 12 1 / 55 + 83 R e 4 / 3 M o 1 / 3 24 1 + M o 1 / 3 + R e 4 / 3 M o 1 / 3$
[6]	适用于球形气泡	$C D = 16 R e ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? R e < 1.5 14.9 R e 0.78 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1.5 ≤ R e < 80 48 R e 1 - 2.21 R e + 1.86 × 10 - 15 R e 4.756 ? ? 80 ≤ R e < 1500 2.61 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1500 ≤ R e$

图1 常用曳力系数模型随Re的变化趋势

Fig.1 Common drag coefficient models as function of Reynolds number

图2 纯净系统、污染系统曳力系数对比

Fig.2 Drag coefficient comparison between pure systems and contaminated systems

表2 用于分区的曳力系数模型

Table 2 Drag coefficient model for partition

文献	公式
[13]	$C D = 16 R e 1 + 8 / 15 χ - 1 + 0.015 3 G E - 2 R e 1 + 0.015 R e + 8 R e + 12 1 + 3.315 H E G E R e 1 / 2 - 1 1 + 0.3 S r 5 / 2 ? ? ? ? ?$
[16]	$C D = C D, M e i 2 + C D E o 2$ ， $C D, M e i = 16 R e 1 + 8 R e + 12 1 + 3.315 R e 1 / 2 - 1$ ， $C D E o = 4 E o E o + 9.5$
[17]	Re>100, We>5， $C D, s e p a r a t e d = C D, W e → 0 + 2.5 t a n h 0.2 W e - 1.5 C D, W e → ∞ - C D, W e → 0$ Re<100， $C D = C D, W e → 0 + t a n h 0.021 W e 1.6 C D, W e → ∞ - C D, W e → 0$ 3<We<5， $C D ≈ C D, M o o r e, C D, s e p a r a t e d$
[4]	$C D = m a x m i n 16 R e 1 + 0.15 R e 0.687, 48 R e, ? 83 E o E o + 4$
[18]	$C D = 83 E o E o + 4$

表2 用于分区的曳力系数模型

Table 2 Drag coefficient model for partition

文献	公式
[13]	$C D = 16 R e 1 + 8 / 15 χ - 1 + 0.015 3 G E - 2 R e 1 + 0.015 R e + 8 R e + 12 1 + 3.315 H E G E R e 1 / 2 - 1 1 + 0.3 S r 5 / 2 ? ? ? ? ?$
[16]	$C D = C D, M e i 2 + C D E o 2$ ， $C D, M e i = 16 R e 1 + 8 R e + 12 1 + 3.315 R e 1 / 2 - 1$ ， $C D E o = 4 E o E o + 9.5$
[17]	Re>100, We>5， $C D, s e p a r a t e d = C D, W e → 0 + 2.5 t a n h 0.2 W e - 1.5 C D, W e → ∞ - C D, W e → 0$ Re<100， $C D = C D, W e → 0 + t a n h 0.021 W e 1.6 C D, W e → ∞ - C D, W e → 0$ 3<We<5， $C D ≈ C D, M o o r e, C D, s e p a r a t e d$
[4]	$C D = m a x m i n 16 R e 1 + 0.15 R e 0.687, 48 R e, ? 83 E o E o + 4$
[18]	$C D = 83 E o E o + 4$

图3 分区曳力系数模型We-Re图谱

Fig.3 Partitioned drag coefficient model based on We and Re

表3 曳力模型和可用实验数据的平均误差

Table 3 Mean error between drag coefficient models and available experimental data

文献	液相	Mo	Tomiyama 误差/%	分区模型误差/%
[24]	glycerol water	2.52×10^-4	5.5599	5.9768
	isoamyl alcohol	8.03×10^-7	13.1066	12.5934
	toluene	9.52×10^-11	11.6916	3.6500
[25]	mineral oil	1.45×10^-2	10.0721	10.6080
	turpentine	2.41×10^-9	22.6868	16.1017
	varsol	4.3×10^-10	14.2942	9.7976
	water	2.6×10^-11	14.4201	19.0249
[26]	glycerol water	1.15×10^-4	11.2001	9.3108
[27]	water	1.26×10^-11	11.8615	26.2564
[28]	water	5.03×10^-4	32.0094	9.0646
[29]	water	2.01×10^-11	28.0055	14.1608
[30]	water	2.01×10^-11	18.6085	6.8536
平均误差			16.1264	11.9499

图4 曳力系数模型与实验数据对比

Fig.4 Comparisons between drag coefficient models and experimental data

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