The results of an experimental study of hydraulic resistance of a laminar flow with air stream macrovortices in a long tube with inserted straight and twisted tapes are presented. It was found that there is a minimum in the relationship of the hydraulic resistance coefficient with the rotation pitch of the tape inserted into the tube. An empirical equation, recommended for use in a wide tape rotation pitch variation range, has been derived for calculating hydraulic resistance.

To intensify heat transfer processes in tubular apparatuses, straight and twisted tapes are inserted in the tubes [1,2,3]. When straight tapes are placed in the diametral plane of the tubes, semicircular channels are formed. In such channels with angular zones, an additional secondary flow in the transverse direction joins the main flow of the heat carrier streams in the longitudinal direction, which exerts an activating effect on the heat and pulse transfer processes. Insertion of twisted tapes ensures swirling of the flow of the medium, which leads to vortex formation, radical restructuring of the flow vis-à-vis its straight flow and, as a result, to further intensification of pulse and heat transfer processes.

Data on hydraulic resistance of flows of homogeneous media in tubes with tape inserts are scarce in the literature. Calculating equations obtained by various authors to determine the characteristics of hydraulic resistance are often not congruent. In this respect, primarily short channels with tube length to tube diameter ratio *l*/*d* ≤ 70, tapes with a relative twist pitch *s*/*d* ≤ 20 (*s* – absolute pitch corresponding to full 360° rotation of the tape and *d* – inner diameter of the tube), and streams with developed turbulent flow conditions were investigated.

In the region of laminar flow with macrovortices where the Reynolds number (Re) is greater than its critical value Re_{cr}, which is in accord with transition to turbulent conditions of intensification of heat transfer with swirling of the flow in the tubes by the twisted tape, is economically expedient [3]. In this case, the economic efficiency of intensification of heat transfer increases with increase of *s*/*d* value. In the region of developed turbulent flow (where Re > Re_{cr}), flow swirling by use of twisted tape becomes economically inexpedient.

Evidently, further investigations of hydraulic resistance during flow of various media in tubes having tape inserts should be aimed at economically preferable ranges of variation of the parameters that affect hydraulic resistance.

**Experimental Unit and Procedure of Investigation**

A horizontal corrosion-resistant steel tube with a diameter *d* = 14 ·10^{−3} m and having a hydraulic stabilization portion of length *l*_{h.s} = 0.765 m (*l*_{h.s}/*d* = 54) and a measurement portion of length 4 m was investigated in this work,

Air stream with a temperature of 25°C was injected into the tube. The static pressure *p* of the air stream was measured through a hole (diameter 0.8 ·10^{−3} m) drilled onto the lateral generatrix of the wall of the measurement portion in sections with coordinates along the portion length *x*_{0} = 0 m (starting section at the boundary with the hydraulic stabilization portion), *x*_{1} = 1 m, *x*_{3} = 3 m, and *x*4 = 4 m. The pressure loss Δ*p* of the air stream at various lengths Δ*x* of the section; and at the start and end of the measurement portion was determined from the difference of the measured pressures *p* in different sections of the measurement portion.

The error of measurement of the pressure *p* of the air stream and of the pressure difference Δ*p* was not greater than 2 Pa.

Corrosion-resistant steel tapes (with a technically smooth surface) with a thickness δ = 0.25 ·10^{−3} m and a 13.5 ·10^{−3} m width was inserted across the whole length of the measurement portion. The air stream was investigated with insertion into the tube of straight tape and of twisted tapes with relative pitches *s*/*d* = 67.0, 28.6, 19.0, and 13.6 (with careful control of uniformity of the twist pitch over the tape length). For all the studied inserts, the leading edge of the tape in the starting section of the measurement portion lay in the vertical-diametral plane of the tube. The tape in the measurement portion was fixed by bracing with a thin wire (at the start of the tube) and by tight connection (at the end of the tube).

The measurements were made under steady air stream flow conditions.

When air was blown through the empty tube without inserts it was found that the measurement results for all lengths of the measurement portion in the Reynolds number variation range (5−28)·10^{3} agree well with the Blasius law for the coefficient of hydraulic resistance of friction of the flow of the medium in the tube

The maximum deviation of the experimental frictional resistance coefficient values from those calculated by

Eq. (1) is not more than 6%. In the experiments with inserted tape, the hydraulic resistance coefficient was determined by the equation

where

Δ*p* – measured air stream pressure drop over the length of the portion Δ*x* ;

ρ – air density;

*d*_{e} = *d*(π*d* – 4*d*)/[(π*d* + 2(*d* – *d*)] – equivalent diameter of the channel;

*w* – mean flow rate referred to transverse (diametral) section of the tube.

The pressure losses due to narrowing and widening of the air stream at the inlet and outlet of the measurement portion upon use of tape and the losses due to energy consumption for swirling motion of the air stream (determinable by the equation [1] Δ*p*_{e} = ρ(*w*π*d*/*s*)^{2}/4) for the conditions of the experiments in total were a fraction of one percent of the measured Δ*p* values and were ignored in experimental data processing because they were insignificant.

**Experimental Results and Their Analysis**

The results of the experiments with insertion of straight tape into the tube are plotted in Figs. 1 and 2 as the functions ξ_{t}/ξ = *f* (Re_{e}), where ξ_{t} – experimental value of the hydraulic resistance coefficient, ξ – value of the coefficient calculated by Eq. (1) at the same Reynolds number Re = Re_{e} = *wd*_{e}/ν (ν – kinematic air viscosity coefficient).

In the studied range of the values Re_{e} = 3·10^{3}−15 ·10^{3}, the function ξ_{t}/ξ = *f* (Re_{e}) is diminishing and bears an asymptotic nature. The ξ_{t}/ξ values differ for the starting and end parts of the measurement portion with the length Δ*x* = 1 m (Δ*x*/*d* = 71.4). In the end part of the measurement portion, the hydraulic resistance is higher than in its starting part (Fig. 1). At Re_{e} > 1·104 , the quantity ξ_{t} /ξ in the starting part of the measurement portion becomes self-modeling with respect to the Re_{e} number and is equal to unity, which agrees with the experimental data of other works [4].

For the whole measurement portion with a length of 4 m (Δ*x*/*d* = 285.6), the function ξ_{t}/ξ = *f* (Re_{e}) (Fig. 2) practically coincides (the difference is not more than 4%, which is within the measurement error range) with the same function for the end part of the measurement portion (Fig. 1, line 2).

Based on the obtained data it can be suggested that the transverse secondary flows in the air stream in the starting part of the measurement portion over the length Δ*x*/*d* = 71.4 are inadequately developed and do not significantly affect the characteristics of the hydraulic resistance of the flow. At Δ*x*/*d* = 71.4, as the secondary flows develop, their influence becomes considerable and manifests itself in the increased hydraulic resistance of the air stream in the tube with inserted straight tape. The increase of the ξ_{t}/ξ value with the decrease of the Re_{e} value from 12 ·10^{3} to 3·10^{3} can be attributed to the influence on the hydraulic resistance of retarded laminar regions of the air stream in the angular areas of the channel. These regions occupy a larger part of the transverse section of the canal, the smaller the Re_{e}.

By approximating the obtained experimental data for the flow in the tube with inserted straight tape at Re_{e} = (3−12)·10^{3} the following equation was derived:

where *A* = 1.05 and *B* = 5 ·10^{6} at Δ*x*/*d* = 285.6; *A* = 1.0 and *B* = 3·10^{6} for Δ*x*/*d* = 71.4.

No differences in the Δ*p*/Δ*x* values were observed in the starting part of the measurement portion (Δ*x* = 1 m) and in the end part of the measurement portion (Δ*x* = 1 m) when twisted tape was inserted over the length of the measurement portion of the tube. Evidently, swirled air stream with use of twisted tape stabilizes over a smaller length than with inserted straight tape, and over the relative length Δ*x*/*d* = 71.4 is fully stabilized. This conclusion agrees with the results of numerical calculations [5], according to which the longitudinal

gradient of the pressure of the swirling flow becomes constant after the portion with a length roughly equaling the pitch s of the tape. In the experimental work [6], the length of the hydraulic stabilization portion was estimated to be *l*_{h.s} ≈ 22*d*_{e}.

The experimental values of the ratio Δ*p*/Δ*x* as a function of Re_{e} at different values of the ratio s/d are plotted in Fig. 3.

The obtained data (in comparison with the experimental data of other authors) are plotted in ξ_{t} – Re_{e} coordinates in Fig. 4 and in ξ_{t}/ξ – *s*/*d* coordinates in Fig. 5.

Based on the measurement results, the dependence of the specific pressure loss of the swirled air stream on the Reynolds number (Fig. 3) corresponds to the proportional relationship Δ*p*/Δ*x* ∼ \( {\operatorname{Re}}_{\mathrm{e}}^{1.65} \) which agrees with the results of experimental investigation of vigorously swirled air streams [8] where the influence of the *s*/*d* value on the Reynolds number exponent was found to be weak.

Based on the data in [8], the value of the exponent varies from 1.615 to 1.65 with increase of *s*/*d* from 5 to 12.

Figures 4 and 5 illustrate sharp increase of hydraulic resistance of the swirled air stream with increase of *s*/*d* value in the *s*/*d* < 20 region. Note that there is a considerable scatter of experimental data of various authors in this s/d value range.

It is evident from Fig. 5 that in the *s*/*d* = 25–40 range the hydraulic resistance of a swirled stream has a minimum, which is associated with the following factors. It is known that when the flow of the medium swirls upon use of twisted tape inserted into the tube, the level of turbulence of the flow is lower, the greater the degree of swirling. Here, alongside the decrease of the *s*/*d* value decreases the power consumption for creation and maintenance of turbulent formations in the swirled flow, which is expressed in reduction of hydraulic resistance. But, on the other hand, decrease of *s*/*d* value facilitates development of transverse secondary flows [4], which causes increase of resistance to movement of the swirled flow. The influence of secondary flows on the flow velocity field and the frictional resistance is greater, the lower the *s*/*d* value. It can be suggested that even though both factors exert influence simultaneously, for the *s*/*d* > 30 region the first one is more dominant (decrease of turbulence) and for the *s*/*d* < 30 region, the second one is dominant (development of secondary flows).

The obtained experimental data can be approximated by the equation

which was obtained in the region of laminar flow with macrovortices at Re_{e} ≥ 5 ·10^{3} for the studied range of values of the ratio *s*/*d* = 13.6–67.0.

Extrapolation of Eq. (4) to the *s*/*d* < 15 region (Fig. 5, dashed segment of curve 13) indicates satisfactory agreement of calculations by Eq. (4) with the experimental data of other authors. Equation (4) can be used to get good asymptotic approximation to the limiting case of straight tape inserted into the tube when *s*/*d* = ∞. At Re_{e} = 1·10^{4}, for example, in calculation by Eq. (4) for the case with use of straight tape ξ_{t} = 0.0342 and by (3) obtained for the case where straight tape is used, ξ_{t} = 0.0348.

Thus, Eq. (4) is a sufficiently adequate universal variant of description of hydraulic resistance for flow of a homogeneous medium through a tube containing straight and twisted tape inserts in a wide range of variation of tape rotation pitch *s*/*d* = 10–∞ and in the Reynolds number variation range Re_{e} = 5 ·10^{3}−Re_{e.cr}, where Ree.cr is the critical Reynolds number corresponding to the boundary of transition of laminar mode of flow of a medium having macrovortices to turbulent mode. According to the data in [8], Re_{e.cr} = (15−16)·10^{3} in all degrees of flow swirl.

## Conclusion

Based on the conducted experimental investigations of hydraulic resistance during laminar flow of an air stream having macrovortices through a tube containing straight and twisted tape, it is concluded that the dependence of the resistance coefficient on the Reynolds number is much greater in comparison with the flow of a stream through a tube without inserts.

Hydraulic stabilization of a flow in a tube with straight tape insert occurs in a portion of longer length than in an empty tube and takes place over a length *l*/*d* > 70.

Hydraulic resistance of a stabilized flow in a straight channel with a semicircular cross section is 9% (and more) than in a round tube.

In a tube with twisted tape insert, hydraulic resistance has a minimum in the *s*/*d* = 25–40 range.

Equation (4), which generalizes experimental data, was derived for calculating hydraulic resistance. This equation is recommended for use in a wide range of variation of the pitch of twist of tapes inserted in tubes.

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Translated from Khimicheskoe i Neftegazovoe Mashinostroenie, Vol. 56, No. 8, pp. 7−10, August, 2020.

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Pechenegov, Y.Y. Hydraulic Resistance of Flows in Tubes with Inserted Straight and Twisted Tapes.
*Chem Petrol Eng* **56, **609–615 (2020). https://doi.org/10.1007/s10556-020-00817-5

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### Keywords

- coefficient of hydraulic resistance
- swirled flow in tube
- straight tape
- twisted tape
- hydrodynamic stabilization
- minimum resistance
- generalized equation