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Numerical simulation of fluid flow in an annular channel with outer transversally corrugated wall
International Journal of Heat and Mass Transfer 90 (2015) 743–751
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical simulation of fluid flow in an annular channel with outer transversally corrugated wall V.I. Artemov 1, K.B. Minko ⇑, G.G. Yan’kov 2 National Research University ‘‘Moscow Power Engineering Institute’’, Krasnokazarmennaya 14, Moscow 111250, Russia
a r t i c l e
i n f o
Article history: Received 11 March 2015 Received in revised form 7 June 2015 Accepted 6 July 2015
Keywords: Corrugated tube Pressure drop Numerical simulation High-temperature superconductivity cable Friction factor Hydraulic-resistance coefficient
a b s t r a c t This paper presents the results of a numerical simulation of liquid-nitrogen flow in a corrugated tube (cryostat) housing high-temperature superconductivity (HTS) cable. Two variants of cable location were considered: along the axis of the cryostat and on the bottom surface of the cryostat. Available data from the literature were used for verification of different turbulence models. The verification results of low-Reynolds-number turbulence models (k–e b k–x) and the algebraic LVEL model showed that these models of turbulence describe the friction factor (hydraulic-resistance coefficient) for the annuli quite accurately. However, experimental data for transversally corrugated channels were reproduced with much greater accuracy using the algebraic LVEL model and the k–x turbulence model. The dependencies of the friction factor from the Reynolds number for corrugated cryostat with HTS cable located concentrically for different values of corrugation pitch and corrugation depth were calculated. The friction factor of HTS cable located eccentrically was 20% lower than that of concentrically located HTS cable. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In recent decades, the phenomenon of high-temperature superconductivity (HTS) has been used to make significant progress in establishing efficient power transmission systems [1]. HTS cables deliver up to 10 times as much power as conventional electric power transmission cables. They are poised to help reduce grid congestion as well as installation and operating costs. The mass introduction of cryogenic HTS cables into power distribution networks is a very promising concept. Today, researchers and manufacturers worldwide are continuously developing and testing prototype HTS power lines [2–7]. Studies have been carried out both on laboratory facilities with short cables (tens of metres) and on large industrial facilities with long cables (several kilometres). Despite the unique design features of some units, the various types of HTS cables, the wide range of operating voltages and currents, and the different types of current (AC or DC), a common feature of these systems is the use of flexible corrugated cryostats for the thermal control system.3 ⇑ Corresponding author. Tel.: +7 (903) 552 23 48. E-mail addresses: [email protected] (V.I. Artemov), [email protected] (K.B. Minko), [email protected] (G.G. Yan’kov). 1 Tel.: +7 (903) 363 08 72. 2 Tel.: +7 (916) 659 28 11. 3 Ultralong transmission lines require the use of combining sections with smooth and corrugated cryostats [3,8]. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.020 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
The most important task for designing cryostats for HTS cables is to ensure an appropriate level of temperature and to avoid boiling the liquid nitrogen in the elements of construction. This is the case because the operating range of the liquid state of nitrogen is relatively narrow; at atmospheric pressure, it equals 14.2 K (with a freezing point of 63.0 K and a boiling point of 77.2 K). With increasing pressure, the specified range is extended (at two bars, the operating range is 20.6 K), but the increase in temperature leads to a significant reduction of the critical current. For example, paper [6] reported that tests on 500-m HTS cables, created during the Japanese super-ACE project, showed that the critical current value fell from 2290 A to 1570 A when the liquid nitrogen temperature was increased from 69 K to 77 K. For short cables, one can always choose a nitrogen flow rate in order to ensure that the temperature difference is not greater than a predetermined value, even with significant heat gain through the insulation. The resulting pressure losses are small (because of the short channel length), so the problem of the hydraulic resistance of the channel does not matter. With the increasing length of the lines, the situation changes. Even with good insulation (heat gain in practice not greater than 1–2 W/m), the total heat gain is great, and to prevent the boiling of nitrogen, its flow rate should be increased.4 Thus, a 4 For ultralong transmission lines under certain parameters, volumetric heat generation due to pressure forces work and exceed the convective removal of heat. In this case, a further increase in flow rate does not lead to a decrease in the temperature difference.
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Nomenclature (dp/dz)0 the overall constant pressure gradient (Pa/m) d0 the minimum diameter of the transversally corrugated tube (m) de the characteristic dimension in the Reynolds number (m) din inlet diameter (m) dout outer diameter (m) e the eccentricity h corrugation depth (m) p pressure (Pa) pf the ‘‘periodic’’ pressure component (Pa) Re Reynolds number s corrugation pitch (m) u velocity vector (m/s) us the friction velocity (m/s) the average velocity (m/s) u yp the distance from the wall of the nodal points in the near-wall control volume (m)
reliable prediction of total pressure drop in cryogenic channels is very important because the pressure losses determine the distance between the pumping stations. In spite of the availability of a large quantity of experimental data on pressure drop in different systems, the information on pressure drop in corrugated channels remains very limited [9– 14]. One way to determine total pressure drop during fluid flow in cryogenic cryostat with HTS cable is the numerical simulation of such systems, using available data for verification. This paper presents the results of a verification of three turbulence models and the results of a numerical simulation of liquid-nitrogen flow in a corrugated tube housing HTS cable. Two variants of cable location were considered: along the axis of the cryostat and on the bottom surface of the cryostat. The main goal of the calculation was to determine the friction factor of fluid flow through corrugated pipe with HTS cable. Due to the fact that, for typical conditions, the change in temperature within the ‘‘periodic’’ section of HTS cables is negligible, the calculation can be made for liquid with constant
Fig. 1. Scheme of the annuli.
yþ MAX z ze zs Dl
Dp/Dz
the maximum value of the dimensionless distance from the wall of the nodal points in the near-wall control volume z coordinate (m) coordinate of reattachment point (m) coordinate of separation point (m) the distance between the centres of the inner and outer tubes (m) the average pressure gradient (Pa/m)
Greek symbols leff the effective dynamic viscosity (Pa s) m the kinematic viscosity (m2/s) n the Darcy friction factor q density (kg/m3) sw wall shear stress (Pa)
properties. Of course, in practice, during the calculation for hydraulic resistance of all systems, one must take into account the variability of liquid nitrogen properties as the temperature increases.
Fig. 2. Friction factor for concentric (e = 0) and eccentric (e = 0.5; 1.0) annuli; 1 – experimental data [20], 2 – LVEL model, 3 – k–e model, 4 – k–x model.
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2. Mathematical model The assumption of periodicity was applied to simulate fully developed fluid flow, which is realised in the main part of a
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corrugated cryostat with HTS cable. Three-dimensional Reynolds averaged Navier–Stokes equations were used to describe the flow in this system (in a Cartesian coordinate system):
div ðquÞ ¼ 0; @pf ; @x @pf ; div ðquuy leff ruy Þ ¼ @y @pf dp div ðquuz leff ruz Þ ¼ þ ; dz 0 @z div ðquux leff rux Þ ¼
ð1Þ
where q is density, kg/m3; pf is the ‘‘periodic’’ pressure component, Pa; ðdp=dzÞ0 is the overall constant pressure gradient, Pa/m; u ¼ ðux ; uy ; uz Þ is velocity, m/s; and leff is the effective dynamic coefficient of viscosity, Pa s. Pressure is divided into two components determined by the following relation:
pðx; y; zÞ ¼ pf ðx; y; zÞ þ
Fig. 3. Schematic illustration of liquid flow in the channel from [21].
Table 1 Pressure gradients and separation and reattachment points for wavy-wall channel. Parameters
Pressure gradient
Separation point, zs
Reattachment point, ze
Simulation, k–e model Simulation, k–x model Simulation, LVEL model DNS results [21] Experimental data [22]
0.0135 0.0107 0.0108 0.0157 –
0.131 0.123 0.123 0.14 0.22
0.596 0.678 0.744 0.59 0.58
dp z: dz 0
ð2Þ
In the numerical simulation of single-phase liquid flow, the main problem is selecting the model to describe the turbulent transport of momentum. In this paper, the following low-Reynolds-number turbulence models were used: the k–e model from [15,16], the k–x model from [17] with boundary conditions from [18], and the algebraic model proposed in [19] (the LVEL model). Verification of these turbulence models was carried out on several test problems: flow in concentric and eccentric annuli [20], flow in a wavy-wall channel [21,22], and flow in a corrugated channel [10–14]. The results presented in this paper were obtained using the author’s CFD code ANES [23], developed in the Department of Engineering Thermophysics, Moscow Power Engineering Institute. In this study, the CFD code ANES was used as a convenient and versatile software environment for the numerical solution of the conservation equations.
3. Testing of turbulence models 3.1. Flow in concentric and eccentric annuli Numerical simulation of flow in concentric and eccentric annuli with dimensions dout/din = 40/20 mm (Fig. 1) corresponding to the data of [20] was performed. Fig. 2 presents a comparison of the calculation results with the experimental data [20]. The characteristic dimension de = dout din in the Reynolds number was used. The Darcy friction factor n is given by
n¼
Dp=Dz d; 2 e 0:5qu
ð3Þ
is the where Dp=Dz is the average pressure gradient, Pa/m; and u average velocity, m/s. The eccentricity e is given by
e¼
2 along wall of wavy-wall Fig. 4. Distribution of wall shear stress normalised by qu channel: 1 – k–x model, 2 – k–e model, 3 – LVEL model, 4 –DNS results from [21].
2 Dl ; dout din
ð4Þ
where Dl is the distance between the centres of the inner and outer pipes (Fig. 1), m. The data presented in Fig. 2 shows that there are no significant differences among the results of calculations performed using the three different models of turbulence and that these results exhibit good general agreement with the experimental data [20].
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Table 2 Basic geometrical characteristics of corrugated pipes [10–14]. No.
Reference
Minimum diameter d0, m
Corrugation depth h, m
Corrugation pitch s, m
1 2 3 4 5 6 7 8
[10] [11] [11] [12] [12] [12] [13] [14]
0.01475 0.038 0.062 0.013 0.0197 0.0267 0.0127 0.04572
0.0015 0.003 0.003 0.0016 0.00235 0.00275 0.004 0.00254
0.00435 0.0063 0.0076 0.00512 0.00646 0.0072 0.00264 0.01016
3.2. Flow in a wavy-wall channel Modelling of the flow in a wavy-wall channel, which is shown in Fig. 3, was performed. Wavy-wall channels differ from corrugated channels in the following way: in wavy-wall channels, wave pitch is the same order as channel height s de, whereas in
corrugated channels, corrugation pitch is the same order as corrugation depth s h. Table 1 and Fig. 4 present a comparison of the numerical simu de =m ¼ 6760 with the lation results for de = 1, h = 0.1, and Re ¼ qu results of the direct numerical simulations (DNS) [21] and the experimental data [22]. In the numerical simulation, separation
Fig. 5. Friction factors of corrugated channels. Geometrical characteristics of the channel correspond to the position numbers in Table 2: a – No. 1, b – No. 2, c – No. 3, d – No. 4. 1 – experimental data [10–12], 2 – LVEL model, 3 – k–e model, 4 – k–x model.
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Fig. 6. Friction factors of corrugated channels. Geometrical characteristics of the channel correspond to the position numbers in Table 2: a – No. 5, b – No. 6, c – No. 7, d – No. 8. 1 – experimental data [12–14], 2 – LVEL model, 3 – k–e model, 4 – k–x model.
and reattachment points corresponded to the points at which the wall shear stress changed sign. It is interesting to note that the calculations using the k–e model correspond most closely to the DNS data. As for the coordinates of the separation point, none of the models led to values close to the experimental data [22]. 3.3. Flow in a corrugated tube There is a small amount of experimental data on friction factors for several samples of corrugated pipes [10–14]. Based on this data, verification of the turbulence models was performed. Table 2 shows the basic geometrical characteristics of corrugated pipes under consideration. It should be noted that during the presentation of experimental data in dimensionless form, the author of [14] used the average velocity corresponding to the maximum diameter of corrugated pipe. For this reason, the data of [14], presented for comparison in [13], are incorrect. This error is discussed in [24], which shows the correct comparison. The Reynolds num d0 =m, where u is the average velocity in ber is defined as Re ¼ qu
the minimum cross section of the channel, d0 is the inner (minimum) diameter of the corrugated channel. Figs. 5 and 6 show a comparison of the numerical simulation results with the experimental data [10–14]. Figs. 5 and 6 clearly show that the experimental friction-factor data for corrugated channels were reproduced with much greater accuracy using the algebraic LVEL model and the k–x turbulence model. The maximum deviation of the calculated results from the experimental data did not exceed 30%. A key reason for the deviation in the calculated values obtained by using different turbulence models is the fact that the total pressure drop during fluid flow in these channels is determined mainly by the pressure distribution along wall (in the flow separation zone), but not by wall shear stress. The contribution of wall shear stress to the total pressure drop did not exceed 5%. This fact explains to a certain extent the good agreement between the friction-factor values calculated using the k–x turbulence model and the experimental data. The ‘‘success’’ of the algebraic LVEL model is unexpected. It is noted that none of the models predicted the small increase of the friction factor with increasing Reynolds
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Fig. 8. Streamlines of fluid flow in corrugated channel No.1 [10] for a given pressure-gradient value (k–x model, (dp/dz)0 = 104 Pa/m).
Fig. 9. Dependence of friction factor n and Reynolds number Re from the maximum value of the dimensionless distance from the wall y+MAX of the nodal points in the near-wall control volume for corrugated channel No. 1 [10], k–x model (j, d – selected grid).
Fig. 7. Distribution of wall shear stress (a) and ‘‘periodic’’ pressure (b) along wall of corrugated channel No. 1 [10] for a given pressure-gradient value (k–x model, (dp/ dz)0 = 104 Pa/m).
number. This effect is possibly due to the fact that in numerical simulation the channel wall was considered hydraulically smooth, whereas in fact it can be quite rough. Figs. 7 and 8 show the distributions of shear stress, pressure along the wall of the corrugated channel No. 1 [10], and the streamlines for a given pressure-gradient value ((dp/dz)0 = 104 Pa/m). The colour of streamlines corresponds to velocity magnitude. A separation occurs at z/s = 0.035, and a reattachment appears near (z/s = 0.88). The maximum of pressure on the wall is generated when the main stream collides with wall (z/s = 0.88). Processing of results for wall shear stress and pressure distribution on the wall showed that the contribution of
wall shear stress to the total pressure drop at this case equal 0.002%. 3.4. Choice of the grid parameters and the number of periodic segments in the computational domain Special calculations were carried out to determine the dependence of the solution on the number of nodes in the computational domain. For final calculation, grids were chosen such that further mesh step reduction led to less than 5% changes in the friction factor and the average velocity (flow rate and Reynolds number) at a given pressure gradient (dp/dz)0. It corresponded to 30,000–50,000 nodes in the control domain for two-dimension problems. To illustrate the effect of the grid on the calculation results, Fig. 9 shows the dependence of the friction factor n and Reynolds number Re
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Fig. 11. Scheme of the corrugated cryostat with HTS cable.
from the maximum value of the dimensionless distance from the wall yþ MAX ¼ ðus yp =mÞMAX of the nodal points in the near-wall control volume for the corrugated channel No. 1 [10] for a given pressure-gradient value ((dp/dz)0 = 104 Pa/m). Calculation of the friction velocity us was based at all points on the wall shear stress except for at separation zones, where it was modified in different ways depending on which turbulence models were used. The Reynolds number in Fig. 9 is defined by the inner (minimum) diameter of the corrugated channel. To control for the assumption of the solution periodicity, flow in the corrugated channel consisting of several periodic sections (two to five) was simulated. For example, Fig. 10 shows the distribution of shear stress and ‘‘periodic’’ pressure along the wall in the corrugated channel No.1 [10].
3.5. Numerical prediction of pressure drop in a corrugated cryostat with HTS cable
Fig. 10. Distribution of wall shear stress (a) and ‘‘periodic’’ pressure (b) along wall of corrugated channel No. 1 [10] for a given pressure-gradient value (k–x model, (dp/dz)0 = 104 Pa/m).
Table 3 Main parameters of the basic version of corrugated cryostat with HTS cable. Parameter
Value
Diameter of the cryogenic cable din, m Inner diameter of the corrugated channel d0, m Outer diameter of the corrugated channel dout, m Corrugation depth h, m Corrugation pitch s, m Hydraulic diameter de, m Corrugation pitch to hydraulic diameter ratio s/de Corrugation depth to hydraulic diameter ratio h/de Eccentricity e
0.04 0.064 0.07 0.003 0.0065 0.024 0.271 0.125 0.0
The corrugated channel whose characteristics are presented in Table 3 was chosen as a basic variant. Calculations were performed for the two locations of the cylindrical HTS cable in the cryostat (Fig. 11). The effect of the geometric characteristics on the friction factor of the corrugated channel was analysed in detail only for variant (a) in Fig. 11. The calculations were performed using the k–x turbulence model. The value de = do din was used to obtain the Reynolds number. Fig. 12 shows the dependence of the friction factor from the Reynolds number for different values of corrugation pitch s (a) and corrugation depth h (b), and Fig. 13 shows the dependence of the friction factor from corrugation pitch s and corrugation depth h for Reynolds number Re = 30,000. The results show that the friction factor increases with increasing ratios of corrugation depth and pitch to a characteristic size of the channel and the friction factor decreases with increasing the Reynolds number for fixed values of corrugation parameters. The slope of these curves is small. Fig. 14 shows the dependence of the friction factor for the base variant of a corrugated channel from the Reynolds number for the concentric (e = 0) and eccentric (e = 1) locations (Eq. (4)) of the HTS cable. It can be seen that the friction factor for the eccentric (e = 1)
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Fig. 13. Dependence of friction factor from corrugation pitch s and corrugation depth h for Reynolds number Re = 30,000.
Fig. 12. Dependence of friction factor from Reynolds number for different values of corrugation pitch s (a) and corrugation depth h (b) for concentric position of the HTS cable.
location of the HTS cable was 20% lower than that for the concentric (e = 0) location.
Fig. 14. Dependence of friction factor for base variant of corrugated channels from Reynolds number for concentric (e = 0) and eccentric (e = 1) locations of HTS cable.
4. Conclusion The verification results of low-Reynolds-number turbulence models (k–e and k–x) and of the algebraic LVEL model showed that these models of turbulence describe the friction factor for the annuli quite accurately. However, friction-factor data for transversally corrugated channels were reproduced with much greater accuracy using the algebraic LVEL model and the k–x turbulence model. It is noted that none of the models predicted the small increase of the friction factor with increasing Reynolds number. The dependencies of the friction factor from the Reynolds number for corrugated cryostat with the concentric location of HTS cable for corrugation pitch 0.06 < s/de < 0.54 and for corrugation
depth 0.06 < h/de < 0.3 were calculated. The friction factor for the eccentric (e = 1) location of the HTS cable was 20% lower than that for the concentric (e = 0) location. Conflict of interest None declared. Acknowledgements The authors extend sincere thanks for the financial support of the Russian Foundation for Basic Research (project 14-08-31619).
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical simulation of fluid flow in an annular channel with outer transversally corrugated wall V.I. Artemov 1, K.B. Minko ⇑, G.G. Yan’kov 2 National Research University ‘‘Moscow Power Engineering Institute’’, Krasnokazarmennaya 14, Moscow 111250, Russia
a r t i c l e
i n f o
Article history: Received 11 March 2015 Received in revised form 7 June 2015 Accepted 6 July 2015
Keywords: Corrugated tube Pressure drop Numerical simulation High-temperature superconductivity cable Friction factor Hydraulic-resistance coefficient
a b s t r a c t This paper presents the results of a numerical simulation of liquid-nitrogen flow in a corrugated tube (cryostat) housing high-temperature superconductivity (HTS) cable. Two variants of cable location were considered: along the axis of the cryostat and on the bottom surface of the cryostat. Available data from the literature were used for verification of different turbulence models. The verification results of low-Reynolds-number turbulence models (k–e b k–x) and the algebraic LVEL model showed that these models of turbulence describe the friction factor (hydraulic-resistance coefficient) for the annuli quite accurately. However, experimental data for transversally corrugated channels were reproduced with much greater accuracy using the algebraic LVEL model and the k–x turbulence model. The dependencies of the friction factor from the Reynolds number for corrugated cryostat with HTS cable located concentrically for different values of corrugation pitch and corrugation depth were calculated. The friction factor of HTS cable located eccentrically was 20% lower than that of concentrically located HTS cable. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In recent decades, the phenomenon of high-temperature superconductivity (HTS) has been used to make significant progress in establishing efficient power transmission systems [1]. HTS cables deliver up to 10 times as much power as conventional electric power transmission cables. They are poised to help reduce grid congestion as well as installation and operating costs. The mass introduction of cryogenic HTS cables into power distribution networks is a very promising concept. Today, researchers and manufacturers worldwide are continuously developing and testing prototype HTS power lines [2–7]. Studies have been carried out both on laboratory facilities with short cables (tens of metres) and on large industrial facilities with long cables (several kilometres). Despite the unique design features of some units, the various types of HTS cables, the wide range of operating voltages and currents, and the different types of current (AC or DC), a common feature of these systems is the use of flexible corrugated cryostats for the thermal control system.3 ⇑ Corresponding author. Tel.: +7 (903) 552 23 48. E-mail addresses: [email protected] (V.I. Artemov), [email protected] (K.B. Minko), [email protected] (G.G. Yan’kov). 1 Tel.: +7 (903) 363 08 72. 2 Tel.: +7 (916) 659 28 11. 3 Ultralong transmission lines require the use of combining sections with smooth and corrugated cryostats [3,8]. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.020 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
The most important task for designing cryostats for HTS cables is to ensure an appropriate level of temperature and to avoid boiling the liquid nitrogen in the elements of construction. This is the case because the operating range of the liquid state of nitrogen is relatively narrow; at atmospheric pressure, it equals 14.2 K (with a freezing point of 63.0 K and a boiling point of 77.2 K). With increasing pressure, the specified range is extended (at two bars, the operating range is 20.6 K), but the increase in temperature leads to a significant reduction of the critical current. For example, paper [6] reported that tests on 500-m HTS cables, created during the Japanese super-ACE project, showed that the critical current value fell from 2290 A to 1570 A when the liquid nitrogen temperature was increased from 69 K to 77 K. For short cables, one can always choose a nitrogen flow rate in order to ensure that the temperature difference is not greater than a predetermined value, even with significant heat gain through the insulation. The resulting pressure losses are small (because of the short channel length), so the problem of the hydraulic resistance of the channel does not matter. With the increasing length of the lines, the situation changes. Even with good insulation (heat gain in practice not greater than 1–2 W/m), the total heat gain is great, and to prevent the boiling of nitrogen, its flow rate should be increased.4 Thus, a 4 For ultralong transmission lines under certain parameters, volumetric heat generation due to pressure forces work and exceed the convective removal of heat. In this case, a further increase in flow rate does not lead to a decrease in the temperature difference.
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Nomenclature (dp/dz)0 the overall constant pressure gradient (Pa/m) d0 the minimum diameter of the transversally corrugated tube (m) de the characteristic dimension in the Reynolds number (m) din inlet diameter (m) dout outer diameter (m) e the eccentricity h corrugation depth (m) p pressure (Pa) pf the ‘‘periodic’’ pressure component (Pa) Re Reynolds number s corrugation pitch (m) u velocity vector (m/s) us the friction velocity (m/s) the average velocity (m/s) u yp the distance from the wall of the nodal points in the near-wall control volume (m)
reliable prediction of total pressure drop in cryogenic channels is very important because the pressure losses determine the distance between the pumping stations. In spite of the availability of a large quantity of experimental data on pressure drop in different systems, the information on pressure drop in corrugated channels remains very limited [9– 14]. One way to determine total pressure drop during fluid flow in cryogenic cryostat with HTS cable is the numerical simulation of such systems, using available data for verification. This paper presents the results of a verification of three turbulence models and the results of a numerical simulation of liquid-nitrogen flow in a corrugated tube housing HTS cable. Two variants of cable location were considered: along the axis of the cryostat and on the bottom surface of the cryostat. The main goal of the calculation was to determine the friction factor of fluid flow through corrugated pipe with HTS cable. Due to the fact that, for typical conditions, the change in temperature within the ‘‘periodic’’ section of HTS cables is negligible, the calculation can be made for liquid with constant
Fig. 1. Scheme of the annuli.
yþ MAX z ze zs Dl
Dp/Dz
the maximum value of the dimensionless distance from the wall of the nodal points in the near-wall control volume z coordinate (m) coordinate of reattachment point (m) coordinate of separation point (m) the distance between the centres of the inner and outer tubes (m) the average pressure gradient (Pa/m)
Greek symbols leff the effective dynamic viscosity (Pa s) m the kinematic viscosity (m2/s) n the Darcy friction factor q density (kg/m3) sw wall shear stress (Pa)
properties. Of course, in practice, during the calculation for hydraulic resistance of all systems, one must take into account the variability of liquid nitrogen properties as the temperature increases.
Fig. 2. Friction factor for concentric (e = 0) and eccentric (e = 0.5; 1.0) annuli; 1 – experimental data [20], 2 – LVEL model, 3 – k–e model, 4 – k–x model.
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2. Mathematical model The assumption of periodicity was applied to simulate fully developed fluid flow, which is realised in the main part of a
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corrugated cryostat with HTS cable. Three-dimensional Reynolds averaged Navier–Stokes equations were used to describe the flow in this system (in a Cartesian coordinate system):
div ðquÞ ¼ 0; @pf ; @x @pf ; div ðquuy leff ruy Þ ¼ @y @pf dp div ðquuz leff ruz Þ ¼ þ ; dz 0 @z div ðquux leff rux Þ ¼
ð1Þ
where q is density, kg/m3; pf is the ‘‘periodic’’ pressure component, Pa; ðdp=dzÞ0 is the overall constant pressure gradient, Pa/m; u ¼ ðux ; uy ; uz Þ is velocity, m/s; and leff is the effective dynamic coefficient of viscosity, Pa s. Pressure is divided into two components determined by the following relation:
pðx; y; zÞ ¼ pf ðx; y; zÞ þ
Fig. 3. Schematic illustration of liquid flow in the channel from [21].
Table 1 Pressure gradients and separation and reattachment points for wavy-wall channel. Parameters
Pressure gradient
Separation point, zs
Reattachment point, ze
Simulation, k–e model Simulation, k–x model Simulation, LVEL model DNS results [21] Experimental data [22]
0.0135 0.0107 0.0108 0.0157 –
0.131 0.123 0.123 0.14 0.22
0.596 0.678 0.744 0.59 0.58
dp z: dz 0
ð2Þ
In the numerical simulation of single-phase liquid flow, the main problem is selecting the model to describe the turbulent transport of momentum. In this paper, the following low-Reynolds-number turbulence models were used: the k–e model from [15,16], the k–x model from [17] with boundary conditions from [18], and the algebraic model proposed in [19] (the LVEL model). Verification of these turbulence models was carried out on several test problems: flow in concentric and eccentric annuli [20], flow in a wavy-wall channel [21,22], and flow in a corrugated channel [10–14]. The results presented in this paper were obtained using the author’s CFD code ANES [23], developed in the Department of Engineering Thermophysics, Moscow Power Engineering Institute. In this study, the CFD code ANES was used as a convenient and versatile software environment for the numerical solution of the conservation equations.
3. Testing of turbulence models 3.1. Flow in concentric and eccentric annuli Numerical simulation of flow in concentric and eccentric annuli with dimensions dout/din = 40/20 mm (Fig. 1) corresponding to the data of [20] was performed. Fig. 2 presents a comparison of the calculation results with the experimental data [20]. The characteristic dimension de = dout din in the Reynolds number was used. The Darcy friction factor n is given by
n¼
Dp=Dz d; 2 e 0:5qu
ð3Þ
is the where Dp=Dz is the average pressure gradient, Pa/m; and u average velocity, m/s. The eccentricity e is given by
e¼
2 along wall of wavy-wall Fig. 4. Distribution of wall shear stress normalised by qu channel: 1 – k–x model, 2 – k–e model, 3 – LVEL model, 4 –DNS results from [21].
2 Dl ; dout din
ð4Þ
where Dl is the distance between the centres of the inner and outer pipes (Fig. 1), m. The data presented in Fig. 2 shows that there are no significant differences among the results of calculations performed using the three different models of turbulence and that these results exhibit good general agreement with the experimental data [20].
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Table 2 Basic geometrical characteristics of corrugated pipes [10–14]. No.
Reference
Minimum diameter d0, m
Corrugation depth h, m
Corrugation pitch s, m
1 2 3 4 5 6 7 8
[10] [11] [11] [12] [12] [12] [13] [14]
0.01475 0.038 0.062 0.013 0.0197 0.0267 0.0127 0.04572
0.0015 0.003 0.003 0.0016 0.00235 0.00275 0.004 0.00254
0.00435 0.0063 0.0076 0.00512 0.00646 0.0072 0.00264 0.01016
3.2. Flow in a wavy-wall channel Modelling of the flow in a wavy-wall channel, which is shown in Fig. 3, was performed. Wavy-wall channels differ from corrugated channels in the following way: in wavy-wall channels, wave pitch is the same order as channel height s de, whereas in
corrugated channels, corrugation pitch is the same order as corrugation depth s h. Table 1 and Fig. 4 present a comparison of the numerical simu de =m ¼ 6760 with the lation results for de = 1, h = 0.1, and Re ¼ qu results of the direct numerical simulations (DNS) [21] and the experimental data [22]. In the numerical simulation, separation
Fig. 5. Friction factors of corrugated channels. Geometrical characteristics of the channel correspond to the position numbers in Table 2: a – No. 1, b – No. 2, c – No. 3, d – No. 4. 1 – experimental data [10–12], 2 – LVEL model, 3 – k–e model, 4 – k–x model.
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Fig. 6. Friction factors of corrugated channels. Geometrical characteristics of the channel correspond to the position numbers in Table 2: a – No. 5, b – No. 6, c – No. 7, d – No. 8. 1 – experimental data [12–14], 2 – LVEL model, 3 – k–e model, 4 – k–x model.
and reattachment points corresponded to the points at which the wall shear stress changed sign. It is interesting to note that the calculations using the k–e model correspond most closely to the DNS data. As for the coordinates of the separation point, none of the models led to values close to the experimental data [22]. 3.3. Flow in a corrugated tube There is a small amount of experimental data on friction factors for several samples of corrugated pipes [10–14]. Based on this data, verification of the turbulence models was performed. Table 2 shows the basic geometrical characteristics of corrugated pipes under consideration. It should be noted that during the presentation of experimental data in dimensionless form, the author of [14] used the average velocity corresponding to the maximum diameter of corrugated pipe. For this reason, the data of [14], presented for comparison in [13], are incorrect. This error is discussed in [24], which shows the correct comparison. The Reynolds num d0 =m, where u is the average velocity in ber is defined as Re ¼ qu
the minimum cross section of the channel, d0 is the inner (minimum) diameter of the corrugated channel. Figs. 5 and 6 show a comparison of the numerical simulation results with the experimental data [10–14]. Figs. 5 and 6 clearly show that the experimental friction-factor data for corrugated channels were reproduced with much greater accuracy using the algebraic LVEL model and the k–x turbulence model. The maximum deviation of the calculated results from the experimental data did not exceed 30%. A key reason for the deviation in the calculated values obtained by using different turbulence models is the fact that the total pressure drop during fluid flow in these channels is determined mainly by the pressure distribution along wall (in the flow separation zone), but not by wall shear stress. The contribution of wall shear stress to the total pressure drop did not exceed 5%. This fact explains to a certain extent the good agreement between the friction-factor values calculated using the k–x turbulence model and the experimental data. The ‘‘success’’ of the algebraic LVEL model is unexpected. It is noted that none of the models predicted the small increase of the friction factor with increasing Reynolds
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Fig. 8. Streamlines of fluid flow in corrugated channel No.1 [10] for a given pressure-gradient value (k–x model, (dp/dz)0 = 104 Pa/m).
Fig. 9. Dependence of friction factor n and Reynolds number Re from the maximum value of the dimensionless distance from the wall y+MAX of the nodal points in the near-wall control volume for corrugated channel No. 1 [10], k–x model (j, d – selected grid).
Fig. 7. Distribution of wall shear stress (a) and ‘‘periodic’’ pressure (b) along wall of corrugated channel No. 1 [10] for a given pressure-gradient value (k–x model, (dp/ dz)0 = 104 Pa/m).
number. This effect is possibly due to the fact that in numerical simulation the channel wall was considered hydraulically smooth, whereas in fact it can be quite rough. Figs. 7 and 8 show the distributions of shear stress, pressure along the wall of the corrugated channel No. 1 [10], and the streamlines for a given pressure-gradient value ((dp/dz)0 = 104 Pa/m). The colour of streamlines corresponds to velocity magnitude. A separation occurs at z/s = 0.035, and a reattachment appears near (z/s = 0.88). The maximum of pressure on the wall is generated when the main stream collides with wall (z/s = 0.88). Processing of results for wall shear stress and pressure distribution on the wall showed that the contribution of
wall shear stress to the total pressure drop at this case equal 0.002%. 3.4. Choice of the grid parameters and the number of periodic segments in the computational domain Special calculations were carried out to determine the dependence of the solution on the number of nodes in the computational domain. For final calculation, grids were chosen such that further mesh step reduction led to less than 5% changes in the friction factor and the average velocity (flow rate and Reynolds number) at a given pressure gradient (dp/dz)0. It corresponded to 30,000–50,000 nodes in the control domain for two-dimension problems. To illustrate the effect of the grid on the calculation results, Fig. 9 shows the dependence of the friction factor n and Reynolds number Re
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Fig. 11. Scheme of the corrugated cryostat with HTS cable.
from the maximum value of the dimensionless distance from the wall yþ MAX ¼ ðus yp =mÞMAX of the nodal points in the near-wall control volume for the corrugated channel No. 1 [10] for a given pressure-gradient value ((dp/dz)0 = 104 Pa/m). Calculation of the friction velocity us was based at all points on the wall shear stress except for at separation zones, where it was modified in different ways depending on which turbulence models were used. The Reynolds number in Fig. 9 is defined by the inner (minimum) diameter of the corrugated channel. To control for the assumption of the solution periodicity, flow in the corrugated channel consisting of several periodic sections (two to five) was simulated. For example, Fig. 10 shows the distribution of shear stress and ‘‘periodic’’ pressure along the wall in the corrugated channel No.1 [10].
3.5. Numerical prediction of pressure drop in a corrugated cryostat with HTS cable
Fig. 10. Distribution of wall shear stress (a) and ‘‘periodic’’ pressure (b) along wall of corrugated channel No. 1 [10] for a given pressure-gradient value (k–x model, (dp/dz)0 = 104 Pa/m).
Table 3 Main parameters of the basic version of corrugated cryostat with HTS cable. Parameter
Value
Diameter of the cryogenic cable din, m Inner diameter of the corrugated channel d0, m Outer diameter of the corrugated channel dout, m Corrugation depth h, m Corrugation pitch s, m Hydraulic diameter de, m Corrugation pitch to hydraulic diameter ratio s/de Corrugation depth to hydraulic diameter ratio h/de Eccentricity e
0.04 0.064 0.07 0.003 0.0065 0.024 0.271 0.125 0.0
The corrugated channel whose characteristics are presented in Table 3 was chosen as a basic variant. Calculations were performed for the two locations of the cylindrical HTS cable in the cryostat (Fig. 11). The effect of the geometric characteristics on the friction factor of the corrugated channel was analysed in detail only for variant (a) in Fig. 11. The calculations were performed using the k–x turbulence model. The value de = do din was used to obtain the Reynolds number. Fig. 12 shows the dependence of the friction factor from the Reynolds number for different values of corrugation pitch s (a) and corrugation depth h (b), and Fig. 13 shows the dependence of the friction factor from corrugation pitch s and corrugation depth h for Reynolds number Re = 30,000. The results show that the friction factor increases with increasing ratios of corrugation depth and pitch to a characteristic size of the channel and the friction factor decreases with increasing the Reynolds number for fixed values of corrugation parameters. The slope of these curves is small. Fig. 14 shows the dependence of the friction factor for the base variant of a corrugated channel from the Reynolds number for the concentric (e = 0) and eccentric (e = 1) locations (Eq. (4)) of the HTS cable. It can be seen that the friction factor for the eccentric (e = 1)
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Fig. 13. Dependence of friction factor from corrugation pitch s and corrugation depth h for Reynolds number Re = 30,000.
Fig. 12. Dependence of friction factor from Reynolds number for different values of corrugation pitch s (a) and corrugation depth h (b) for concentric position of the HTS cable.
location of the HTS cable was 20% lower than that for the concentric (e = 0) location.
Fig. 14. Dependence of friction factor for base variant of corrugated channels from Reynolds number for concentric (e = 0) and eccentric (e = 1) locations of HTS cable.
4. Conclusion The verification results of low-Reynolds-number turbulence models (k–e and k–x) and of the algebraic LVEL model showed that these models of turbulence describe the friction factor for the annuli quite accurately. However, friction-factor data for transversally corrugated channels were reproduced with much greater accuracy using the algebraic LVEL model and the k–x turbulence model. It is noted that none of the models predicted the small increase of the friction factor with increasing Reynolds number. The dependencies of the friction factor from the Reynolds number for corrugated cryostat with the concentric location of HTS cable for corrugation pitch 0.06 < s/de < 0.54 and for corrugation
depth 0.06 < h/de < 0.3 were calculated. The friction factor for the eccentric (e = 1) location of the HTS cable was 20% lower than that for the concentric (e = 0) location. Conflict of interest None declared. Acknowledgements The authors extend sincere thanks for the financial support of the Russian Foundation for Basic Research (project 14-08-31619).
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