扭带结构影响管内传热与熵产的研究进展

doi:10.16085/j.issn.1000-6613.2022-0158

摘要/Abstract

摘要：

随着工业技术不断发展，传统换热管的传热方式已经无法满足高热流密度下的热量输运要求。扭带插入物是一种能够有效提高换热管传热效率的强化传热元件，以其结构简单、加工容易的特点受到了很多学者的关注和研究。管内流体的传热性能及熵产往往作为评价换热管性能的重要参数，因此扭带结构与流动工质对这些参数的影响成为近年来研究的重点。本文主要综述了近十年来不同结构扭带对管内传热与熵产影响的研究进展。首先，将文献中研究的扭带按照几何结构进行分类，阐述和分析了不同类型扭带对换热管的传热、熵产以及综合性能的影响，试图找出几何结构与换热管传热性能以及熵产之间的联系。其次，介绍了扭带与纳米流体复合传热技术的研究进展。最后，归纳了研究人员为达到传热性能最大化以及熵产最小化而建立的传热和熵产模型，并对模型的优缺点进行了评价。

关键词: 扭带, 纳米流体, 强化传热, 综合性能, 熵产

Abstract:

With the continuous development of industrial technology, the heat transfer in traditional heat exchange tubes has been unable to meet the heat transport requirement under high heat flux. Twisted tape insert is a kind of heat transfer enhancement element which can effectively improve the heat transfer efficiency in heat exchange tube. It has been focused and studied widely by many scholars because of its simple structure and easy processing characteristics. The heat transfer performance and entropy production of fluid in the tube are generally used as important parameters to evaluate the performance of heat exchange tube. Therefore, the influences of twisted tape structure and working fluid on these parameters become the focus of research in recent years. This paper mainly reviews the research progress of the effects of different structural twisted tapes on heat transfer and entropy production in tubes in the past decade. Firstly, the twisted tapes used in the literature are classified according to the geometric structure, and the effects of different types of twisted tape on heat transfer, entropy generation and comprehensive performance of heat exchange tubes are expounded and analyzed, trying to find out the relationships between geometric structure and heat transfer performance as well as entropy generation in heat exchange tubes. Secondly, the research progress of composite heat transfer technology with twisted tape and nanofluid is introduced. Finally, the heat transfer and entropy generation models established by researchers to maximize heat transfer performance and minimize entropy generation are summarized, and the advantages and disadvantages of the models are evaluated.

Key words: twisted tape, nanofluids, heat transfer enhancement, comprehensive performance, entropy generation

中图分类号:

TK124

林清宇, 王祝, 冯振飞, 凌彪, 陈镇. 扭带结构影响管内传热与熵产的研究进展[J]. 化工进展, 2022, 41(11): 5709-5721.

LIN Qingyu, WANG Zhu, FENG Zhenfei, LING Biao, CHEN Zhen. Review progress on twisted tape structure for heat transfer and entropy generation in tube[J]. Chemical Industry and Engineering Progress, 2022, 41(11): 5709-5721.

图/表 8

图1 普通扭带几何模型[7]

图2 具有穿孔的V形切割扭带几何模型[13]

图3 具有穿孔挡板的扭带几何模型[15]

图4 普通扭带与交替轴扭带物理模型[23]

图5 圆管内插螺旋片几何模型[27]

图6 内插扭带管流场分布[31]

表1 内插扭带管内传热模型 (续表1)

研究者	工质	流动状态	参数	提出或建议使用的关联式	结论
Sowi等^［54］		湍流	LIPR=0.2~0.3，φ=0~3%，Re=4000~16000	$N u = 0.0029 R e 0.86 P r 0.4 (1 + φ) (0.15 1 + L I P R) 2.1$ $f = 720.2 R e - 0.94 (1 + φ) 0.16 (1 + L I P R) 2.1$	LIPR=0.25的扭带在10000雷诺数下，实现了最大的整体增强率
Dagdevir等^［55］	水	湍流	σ=1mm，W=17mm，Re=5217~22754	$N u p = 0.034299 R e 0.785361 P r 0.413538 (0.01 + P p / y) - 0.09157$ $N u d = 0.033199 R e 0.800351 P r 0.358743 (0.01 + P p / y) - 0.19756$	与带孔的和普通的扭结带相比，使用凹陷的扭结带在传热性能上有优势。节距比为0.25的凹陷扭结带在实验中显示出最好的传热性能
Sheikholeslami等^［56］	CuO	湍流	PR=1.5~3，BR=0.25~0.75，Re=5000~15000	$N u * = 0.34 + 0.42 × 10 - 4 P R - 0.26 B R + 0.28 R e * - 0.017 (P R) (B R) - 0.03 (P R) (R e ) + 0.29 (B R) (R e) + 3.7 × 10 - 3 (P R) 2 + 0.38 (B R) 2 - 0.18 (R e ) 2$ $f * = 1.03 - 0.43 P R + 0.78 B R - 0.06 R e * - 0.41 (P R) (B R) + 0.01 (P R) (R e ) - 0.26 (B R) R e + 0.12 (P R) 2 + 1.36 (B R) 2 - 0.04 (R e ) 2, 其中 N u * = 0.01 N u, R e * = 10 - 4 R e, f * = 10 f$	努塞尔数是宽度比和雷诺数的一个递增函数，和间距比是一个递减函数。摩擦系数随着扭率的减小而增加
Wongcharee等^［29］	CuO	层流	φ=0.3%~0.7%，Re=830~1990	交替轴 $N u = 0.026 R e - 0.927 P r 0.4 (1 + φ) 0.128$ $f = 4.487 R e - 0.297 (1 + φ) 0.101$ $η = 0.027 R e - 0.693 (1 + φ) 0.094$ 普通扭带 $N u = 0.005 R e 1.062 P r 0.4 (1 + φ) 0.112$ $f = 3.234 R e - 0.308 1 + φ 0.082$ $η = 0.006 R e 0.832 (1 + φ) 0.085$	Nu随着Re和纳米流体浓度的增加而增加。在Re=1990，体积分数0.7%的CuO纳米流体协同交替轴可以获得最大PEC为5.53
Oni等^［57］	水	湍流	w=18mm，σ=1mm，y=54mm，Re=5000~20000	$N u = 116.961 + R e 1.323 P r - 2.938 p w 0.296 y w - 2.645$ $f = 20.294 R e - 0.522 p w 0.151 y / w y / w - 1 1.462$	交替轴三角形切割的管子性能最好，其努塞尔数和摩擦系数分别是装有普通扭带管子的1.63~2.18倍和2.60~3.15倍，而其PEC最大值为1.43
Eiamsa-ard等^［29］	TiO₂	湍流	N=1~4，φ=0.07%~0.21%，Re=5400~15200	同向布置的多扭带 $N u = 0.103 R e 0.618 P r 0.4 (1 + N) 0.768 (1 + φ) 0.438$ $f = 0.437 R e - 0.233 (1 + N) 1.25 (1 + φ) 0.268$ $η = 3.597 R e - 0.178 (1 + N) 0.351 (1 + φ) 0.349$ 反向布置的多扭带 $N u = 0.104 R e 0.618 P r 0.4 (1 + N) 0.789 (1 + φ) 0.439$ $f = 0.468 R e - 0.234 (1 + N) 1.217 (1 + φ) 0.266$ $η = 3.529 R e - 0.178 (1 + N) 0.384 (1 + φ) 0.349$ 反向交叉布置的多扭带 $N u = 0.096 R e 0.618 P r 0.4 (1 + N) 0.834 (1 + φ) 0.468$ $f = 0.416 R e - 0.234 (1 + N) 1.221 (1 + φ) 0.262$ $η = 3.401 R e - 0.178 (1 + N) 0.427 (1 + φ) 0.381$	努塞尔数和摩擦系数都随着扭带数量的增加而变大。实验结果表明，反向排列传热性能更好
Eiamsa-ard等^［58］	TiO₂	湍流	φ=0.07%~0.21%，y₀/y=1.5~2.5，σ=0.8mm，W=8mm，Re=5400~15200	$N u = 0.267 R e 0.617 P r 0.4 y 0 / y - 0.213 1 + φ 0.505$ $f = 2.057 R e - 0.234 y 0 / y - 0.311 1 + φ 0.886$ $η = 5.538 R e - 0.179 y 0 / y - 0.109 (1 + φ) 0.209$	努塞尔数、摩擦系数和热性能随着重叠旋流比的减少和TiO₂体积分数的增加而增加
Dalkılıç等^［59］	SiO₂+ 石墨/水	湍流	φ=0.5%~1%，l=0~42cm，L=2m，H/D=5，Re=3400~11000	$N u r e g = 0.00012 R e 1.148294 P r 1.596637 1 + φ 9.429766 l / L + 1 0.401864$ $f r e g = 0.646889 R e - 0.318925639 (1 + φ) 1.90706352 (l / L + 1) 7.158388067$	在高雷诺数下，努塞尔数随着混合纳米流体浓度的增加和扭带插入长度的增加而增加，而摩擦系数随着雷诺数的增加而减小
Sundar等^［49］		湍流	H=0.09~1.5m，D=0.018m，Re=9000~22500	$N u r e g = 0.3666 P r 0.4 R e 0.8304 (0.001 + φ) 0.04704 (1 + H D) 0.06281$ $f r e g = 2.068 R e - 0.433 (1 + φ) 0.01 (1 + H D) 0.004815$	在雷诺数为10000和22000的情况下，使用体积分数为0.5%的纳米流体的摩擦系数是水的1.096倍和1.2657倍，传热系数也分别提高了22.76%和33.3%
Xiong等^［60］	CuO	湍流	b=5~15mm，σ=1mm，Re=5000~15000	$N u = 127.06 + 49.76 R e * + 1.52 b + 0.057 b R e * + 1.64 b 2$ $f = 0.44 - 0.73 R e * + 0.15 b - 0.029 b R e * + 0.065 b 2$	扭带宽度的增加会导致更强的旋涡，这有助于对流，所以Nu随着b的增加而增加
研究者	工质	流动状态	参数	提出或建议使用的关联式	结论
Seemawute等^［61］	水	湍流	Re=5000~20000	边缘切割和交替轴 $N u = 0.422 R e 0.544 P r 0.4 w W - 0.148$ $f = 59.08 R e - 0.615 w W - 0.18$ 边缘切割 $N u = 0.126 R e 0.658 P r 0.4 w W - 0.101$ $f = 18.82 R e - 0.556 w W - 0.193$	相同条件下，边缘切割和交替轴的PEC最高，可以达到1.25
Wongcharee等^［62］	水	层流	y/W=3~5，Re=830~1990	交替轴 $N u = 0.032 R e 0.986 P r 0.4 (y W) - 0.594$ $f = 12.886 R e - 0.304 (y w) - 0.896$ 普通扭带 $N u = 0.005 R e 1.139 P r 0.4 (y W) - 0.521$ $f = 6.559 R e - 0.303 (y W) - 0.674$	层流下，相较于普通扭带，交替轴传热性能要高得多。在扭率为3和雷诺数830的情况下，使用交替轴的最大PEC为5.25
Maddah等^［63］	Al₂O₃	湍流	Re=5000~21000，GPR=0.6~2，W=18mm	$N u = 0.056 R e 0.72 P r 0.4 (1 + π φ) 2.75 (1 + π 2 T R) 1.1 G P R - 0.75$ $f = 0.375 R e - 0.24 (1 + 3 π φ) 0.6 (1 + π T R 1.4) G P R - 0.35$
Dong等^［18］	Al₂O₃	湍流	Re=19322~64407	$N u = 0.602 R e 0.367 P r 1.28 (1 + φ) 6.49$ $f = 0.630 R e - 0.281 (1 + φ) 3.84$	研究氧化铝纳米颗粒和扭带对管内热性能的综合影响，建立新的传热模型

表1 内插扭带管内传热模型 (续表1)

研究者	工质	流动状态	参数	提出或建议使用的关联式	结论
Sowi等^［54］		湍流	LIPR=0.2~0.3，φ=0~3%，Re=4000~16000	$N u = 0.0029 R e 0.86 P r 0.4 (1 + φ) (0.15 1 + L I P R) 2.1$ $f = 720.2 R e - 0.94 (1 + φ) 0.16 (1 + L I P R) 2.1$	LIPR=0.25的扭带在10000雷诺数下，实现了最大的整体增强率
Dagdevir等^［55］	水	湍流	σ=1mm，W=17mm，Re=5217~22754	$N u p = 0.034299 R e 0.785361 P r 0.413538 (0.01 + P p / y) - 0.09157$ $N u d = 0.033199 R e 0.800351 P r 0.358743 (0.01 + P p / y) - 0.19756$	与带孔的和普通的扭结带相比，使用凹陷的扭结带在传热性能上有优势。节距比为0.25的凹陷扭结带在实验中显示出最好的传热性能
Sheikholeslami等^［56］	CuO	湍流	PR=1.5~3，BR=0.25~0.75，Re=5000~15000	$N u * = 0.34 + 0.42 × 10 - 4 P R - 0.26 B R + 0.28 R e * - 0.017 (P R) (B R) - 0.03 (P R) (R e ) + 0.29 (B R) (R e) + 3.7 × 10 - 3 (P R) 2 + 0.38 (B R) 2 - 0.18 (R e ) 2$ $f * = 1.03 - 0.43 P R + 0.78 B R - 0.06 R e * - 0.41 (P R) (B R) + 0.01 (P R) (R e ) - 0.26 (B R) R e + 0.12 (P R) 2 + 1.36 (B R) 2 - 0.04 (R e ) 2, 其中 N u * = 0.01 N u, R e * = 10 - 4 R e, f * = 10 f$	努塞尔数是宽度比和雷诺数的一个递增函数，和间距比是一个递减函数。摩擦系数随着扭率的减小而增加
Wongcharee等^［29］	CuO	层流	φ=0.3%~0.7%，Re=830~1990	交替轴 $N u = 0.026 R e - 0.927 P r 0.4 (1 + φ) 0.128$ $f = 4.487 R e - 0.297 (1 + φ) 0.101$ $η = 0.027 R e - 0.693 (1 + φ) 0.094$ 普通扭带 $N u = 0.005 R e 1.062 P r 0.4 (1 + φ) 0.112$ $f = 3.234 R e - 0.308 1 + φ 0.082$ $η = 0.006 R e 0.832 (1 + φ) 0.085$	Nu随着Re和纳米流体浓度的增加而增加。在Re=1990，体积分数0.7%的CuO纳米流体协同交替轴可以获得最大PEC为5.53
Oni等^［57］	水	湍流	w=18mm，σ=1mm，y=54mm，Re=5000~20000	$N u = 116.961 + R e 1.323 P r - 2.938 p w 0.296 y w - 2.645$ $f = 20.294 R e - 0.522 p w 0.151 y / w y / w - 1 1.462$	交替轴三角形切割的管子性能最好，其努塞尔数和摩擦系数分别是装有普通扭带管子的1.63~2.18倍和2.60~3.15倍，而其PEC最大值为1.43
Eiamsa-ard等^［29］	TiO₂	湍流	N=1~4，φ=0.07%~0.21%，Re=5400~15200	同向布置的多扭带 $N u = 0.103 R e 0.618 P r 0.4 (1 + N) 0.768 (1 + φ) 0.438$ $f = 0.437 R e - 0.233 (1 + N) 1.25 (1 + φ) 0.268$ $η = 3.597 R e - 0.178 (1 + N) 0.351 (1 + φ) 0.349$ 反向布置的多扭带 $N u = 0.104 R e 0.618 P r 0.4 (1 + N) 0.789 (1 + φ) 0.439$ $f = 0.468 R e - 0.234 (1 + N) 1.217 (1 + φ) 0.266$ $η = 3.529 R e - 0.178 (1 + N) 0.384 (1 + φ) 0.349$ 反向交叉布置的多扭带 $N u = 0.096 R e 0.618 P r 0.4 (1 + N) 0.834 (1 + φ) 0.468$ $f = 0.416 R e - 0.234 (1 + N) 1.221 (1 + φ) 0.262$ $η = 3.401 R e - 0.178 (1 + N) 0.427 (1 + φ) 0.381$	努塞尔数和摩擦系数都随着扭带数量的增加而变大。实验结果表明，反向排列传热性能更好
Eiamsa-ard等^［58］	TiO₂	湍流	φ=0.07%~0.21%，y₀/y=1.5~2.5，σ=0.8mm，W=8mm，Re=5400~15200	$N u = 0.267 R e 0.617 P r 0.4 y 0 / y - 0.213 1 + φ 0.505$ $f = 2.057 R e - 0.234 y 0 / y - 0.311 1 + φ 0.886$ $η = 5.538 R e - 0.179 y 0 / y - 0.109 (1 + φ) 0.209$	努塞尔数、摩擦系数和热性能随着重叠旋流比的减少和TiO₂体积分数的增加而增加
Dalkılıç等^［59］	SiO₂+ 石墨/水	湍流	φ=0.5%~1%，l=0~42cm，L=2m，H/D=5，Re=3400~11000	$N u r e g = 0.00012 R e 1.148294 P r 1.596637 1 + φ 9.429766 l / L + 1 0.401864$ $f r e g = 0.646889 R e - 0.318925639 (1 + φ) 1.90706352 (l / L + 1) 7.158388067$	在高雷诺数下，努塞尔数随着混合纳米流体浓度的增加和扭带插入长度的增加而增加，而摩擦系数随着雷诺数的增加而减小
Sundar等^［49］		湍流	H=0.09~1.5m，D=0.018m，Re=9000~22500	$N u r e g = 0.3666 P r 0.4 R e 0.8304 (0.001 + φ) 0.04704 (1 + H D) 0.06281$ $f r e g = 2.068 R e - 0.433 (1 + φ) 0.01 (1 + H D) 0.004815$	在雷诺数为10000和22000的情况下，使用体积分数为0.5%的纳米流体的摩擦系数是水的1.096倍和1.2657倍，传热系数也分别提高了22.76%和33.3%
Xiong等^［60］	CuO	湍流	b=5~15mm，σ=1mm，Re=5000~15000	$N u = 127.06 + 49.76 R e * + 1.52 b + 0.057 b R e * + 1.64 b 2$ $f = 0.44 - 0.73 R e * + 0.15 b - 0.029 b R e * + 0.065 b 2$	扭带宽度的增加会导致更强的旋涡，这有助于对流，所以Nu随着b的增加而增加
研究者	工质	流动状态	参数	提出或建议使用的关联式	结论
Seemawute等^［61］	水	湍流	Re=5000~20000	边缘切割和交替轴 $N u = 0.422 R e 0.544 P r 0.4 w W - 0.148$ $f = 59.08 R e - 0.615 w W - 0.18$ 边缘切割 $N u = 0.126 R e 0.658 P r 0.4 w W - 0.101$ $f = 18.82 R e - 0.556 w W - 0.193$	相同条件下，边缘切割和交替轴的PEC最高，可以达到1.25
Wongcharee等^［62］	水	层流	y/W=3~5，Re=830~1990	交替轴 $N u = 0.032 R e 0.986 P r 0.4 (y W) - 0.594$ $f = 12.886 R e - 0.304 (y w) - 0.896$ 普通扭带 $N u = 0.005 R e 1.139 P r 0.4 (y W) - 0.521$ $f = 6.559 R e - 0.303 (y W) - 0.674$	层流下，相较于普通扭带，交替轴传热性能要高得多。在扭率为3和雷诺数830的情况下，使用交替轴的最大PEC为5.25
Maddah等^［63］	Al₂O₃	湍流	Re=5000~21000，GPR=0.6~2，W=18mm	$N u = 0.056 R e 0.72 P r 0.4 (1 + π φ) 2.75 (1 + π 2 T R) 1.1 G P R - 0.75$ $f = 0.375 R e - 0.24 (1 + 3 π φ) 0.6 (1 + π T R 1.4) G P R - 0.35$
Dong等^［18］	Al₂O₃	湍流	Re=19322~64407	$N u = 0.602 R e 0.367 P r 1.28 (1 + φ) 6.49$ $f = 0.630 R e - 0.281 (1 + φ) 3.84$	研究氧化铝纳米颗粒和扭带对管内热性能的综合影响，建立新的传热模型

表2 内插扭带管内熵产模型

研究者	扭带类型	公式	主要研究内容
Shafee等^［40］	双螺旋扭带	$X d = 6183.5 - 5907 R e * - 433.65 b + 428.9 b R e * + 37.5 b 2$	利用数值模拟建立了一个关于双螺旋扭带㶲损失分布的预测模型
Sheikholeslami等^［68］	交替轴扭带	$η Ⅱ = 0.74 + 1.12 × 10 - 3 P R + 7.2 × 10 - 5 B R + 0.018 R e + 5.4 × 10 - 4 P R B R + 1.43 × 10 - 3 P R R e - 1.48 × 10 - 3 R e B R - 5.3 × 10 - 3 R e 2$ $X d = 35.06 + 0.23 P R - 0.4 B R - 5.24 R e + 0.21 P R B R - 0.42 P R R e + 0.46 R e B R + 2.54 R e 2$	基于㶲损失和第二定律建立纳米流体湍流状态下的预测模型
Sheikholeslami等^［69］	双螺旋扭带	$S g e n, f = 6.8 × 10 - 3 + 5.7 × 10 - 3 R e * + 2.16 × 10 - 3 b + 1.68 × 10 - 3 b R e * + 6.67 × 10 - 4 b 2$ $S g e n, t h = 20.74 - 19.83 R e * - 1.46 b + 1.44 b R e * + 0.13 b 2$	在均匀热流的影响下，建立了双螺旋扭带的熵产模型，能够预测管内传热熵产的流动熵产
Sheikholeslami等^［70］	组合式扭带	$X d = 5578.65 - 4881.27 R e * - 71.69 b + 85.13 b R e * - 70.09 b 2$	在湍流状态下，建立了组合式扭带的熵产模型，能够预测Re和扭带宽度对㶲损失的影响
Sheikholeslami等^［71］	组合式扭带	$S g e n, f = 0.02 + 0.018 R e * + 4.9 × 10 - 3 b + 4.12 × 10 - 3 b R e * + 1.31 × 10 - 3 b 2$ $S g e n, t h = 20.31 - 16.41 R e * - 0.31 b + 0.29 b R e * - 0.17 b 2$	利用模拟数据建立了组合扭带熵产预测模型，对管内的流动熵产和传热熵产进行预测
Li等^［72］	螺旋扭带	$S g e n, f = 1.19 - 0.32 P R + 0.68 B R + 1.04 R e * - 0.021 (P R) (B R) - 0.013 (P R) (R e ) + 0.57 (B R) (R e ) + 0.09 (B R) 2$ $S g e n, t h = 30.17 + 16.2 P R + 0.79 B R - 27.5 R e * + 1.14 (P R) (B R) - 13.25 (P R) (R e ) + 1.54 (B R) (R e ) + 3.61 (B R) 2$	基于有限体积法建立了内插螺旋扭带管内熵产预测模型
Sheikholeslami等^［73］	螺旋扭带	$B e = 0.86 + 0.041 P R - 0.069 B R - 0.14 R e * + 0.038 (R e ) (P R) - 0.066 (R e ) (B R) + 0.02 (B R) (P R)$ $X d = 10120.3 + 4544.2 P R - 411.67 B R - 7863.8 R e * - 142.5 (P R) (B R) - 3876.16 (P R) (R e ) + 280.34 (B R) (R e )$	基于内插新型螺旋扭带圆管的流动，建立一种预测管内㶲损失和贝扬数的模型

表2 内插扭带管内熵产模型

研究者	扭带类型	公式	主要研究内容
Shafee等^［40］	双螺旋扭带	$X d = 6183.5 - 5907 R e * - 433.65 b + 428.9 b R e * + 37.5 b 2$	利用数值模拟建立了一个关于双螺旋扭带㶲损失分布的预测模型
Sheikholeslami等^［68］	交替轴扭带	$η Ⅱ = 0.74 + 1.12 × 10 - 3 P R + 7.2 × 10 - 5 B R + 0.018 R e + 5.4 × 10 - 4 P R B R + 1.43 × 10 - 3 P R R e - 1.48 × 10 - 3 R e B R - 5.3 × 10 - 3 R e 2$ $X d = 35.06 + 0.23 P R - 0.4 B R - 5.24 R e + 0.21 P R B R - 0.42 P R R e + 0.46 R e B R + 2.54 R e 2$	基于㶲损失和第二定律建立纳米流体湍流状态下的预测模型
Sheikholeslami等^［69］	双螺旋扭带	$S g e n, f = 6.8 × 10 - 3 + 5.7 × 10 - 3 R e * + 2.16 × 10 - 3 b + 1.68 × 10 - 3 b R e * + 6.67 × 10 - 4 b 2$ $S g e n, t h = 20.74 - 19.83 R e * - 1.46 b + 1.44 b R e * + 0.13 b 2$	在均匀热流的影响下，建立了双螺旋扭带的熵产模型，能够预测管内传热熵产的流动熵产
Sheikholeslami等^［70］	组合式扭带	$X d = 5578.65 - 4881.27 R e * - 71.69 b + 85.13 b R e * - 70.09 b 2$	在湍流状态下，建立了组合式扭带的熵产模型，能够预测Re和扭带宽度对㶲损失的影响
Sheikholeslami等^［71］	组合式扭带	$S g e n, f = 0.02 + 0.018 R e * + 4.9 × 10 - 3 b + 4.12 × 10 - 3 b R e * + 1.31 × 10 - 3 b 2$ $S g e n, t h = 20.31 - 16.41 R e * - 0.31 b + 0.29 b R e * - 0.17 b 2$	利用模拟数据建立了组合扭带熵产预测模型，对管内的流动熵产和传热熵产进行预测
Li等^［72］	螺旋扭带	$S g e n, f = 1.19 - 0.32 P R + 0.68 B R + 1.04 R e * - 0.021 (P R) (B R) - 0.013 (P R) (R e ) + 0.57 (B R) (R e ) + 0.09 (B R) 2$ $S g e n, t h = 30.17 + 16.2 P R + 0.79 B R - 27.5 R e * + 1.14 (P R) (B R) - 13.25 (P R) (R e ) + 1.54 (B R) (R e ) + 3.61 (B R) 2$	基于有限体积法建立了内插螺旋扭带管内熵产预测模型
Sheikholeslami等^［73］	螺旋扭带	$B e = 0.86 + 0.041 P R - 0.069 B R - 0.14 R e * + 0.038 (R e ) (P R) - 0.066 (R e ) (B R) + 0.02 (B R) (P R)$ $X d = 10120.3 + 4544.2 P R - 411.67 B R - 7863.8 R e * - 142.5 (P R) (B R) - 3876.16 (P R) (R e ) + 280.34 (B R) (R e )$	基于内插新型螺旋扭带圆管的流动，建立一种预测管内㶲损失和贝扬数的模型

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