Modeling for thermal augmentation of turbulent flow in

Applied Thermal Engineering xxx (2016) xxx–xxx

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Modeling for thermal augmentation of turbulent flow in a circular tube fitted with twisted conical strip inserts Mahdi Pourramezan, Hossein Ajam ⇑ Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

h i g h l i g h t s

g r a p h i c a l a b s t r a c t

Using high Reynolds models,

turbulent flow and heat transfer are simulated by CFD. Scalable wall functions are more reliable than standard wall functions. Twisted conical strip inserts cause additional swirl and turbulence in the flow. f and Nu increase with lower twist angle, and smaller pitch but higher slant angle. The maximum values for Nu/Nu0, f/f0, and PEC are 3.5, 25, and 1.12, respectively.

a r t i c l e

i n f o

Article history: Received 31 October 2015 Accepted 3 March 2016 Available online xxxx Keywords: Turbulent flow CFD modeling Wall-functions Heat transfer Inserts

a b s t r a c t This paper investigates the reliable numerical simulation of thermo-hydraulic characteristics of turbulent flow in a circular tube fitted with twisted conical inserts. For this purpose, CFD was utilized to simulate considered three-dimensional cases with periodic boundary conditions. Initially, the formidable challenge of true modeling for the turbulent regime was investigated. Using high-Reynolds turbulence models, like k–e family, along with standard wall-functions, the results hardly become independent of the computational domain and by refining the near-wall mesh, the Nusselt number and the friction factor continually increase. Consequently, for completely fine grids, observed results have up to 120% discrepancy with accepted results of the plain tube. It is found that using scalable wall-functions entirely fixes this problem, and the simulated results are totally consistent with the results of the acceptable correlations. In addition, the results show that the Nusselt number and friction factor increase with the reduction of the pitch. Furthermore, increasing twist angle decreases Nu and f, while increasing slant angle increases them. Among the investigated cases, studying the PEC shows that higher pitches and slant angles result in higher values of PEC, and increasing twist angle decreases PEC. Moreover, nonstaggered alignments result in higher f, and Nu but lower PEC. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Due to the widespread applications of heat exchangers (HEXs) throughout various industrial fields, several methods and

⇑ Corresponding author. Tel.: +98 9151412282; fax: +98 5138763304. E-mail address: [email protected] (H. Ajam).

approaches have been proposed in order to enhance the performance of these devices. The other reason that attracts attention to heat exchangers is the expensive and limited energy resources, which has persuaded the engineers and researchers to put great effort into increasing the thermo-hydraulic efficiency of the heat exchangers. Among these efforts, enhancing the heat transfer rate on the tube side has always been highlighted for the improvement of

http://dx.doi.org/10.1016/j.applthermaleng.2016.03.029 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: M. Pourramezan, H. Ajam, Modeling for thermal augmentation of turbulent flow in a circular tube fitted with twisted conical strip inserts, Appl. Therm. Eng. (2016), http://dx.doi.org/10.1016/j.applthermaleng.2016.03.029

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M. Pourramezan, H. Ajam / Applied Thermal Engineering xxx (2016) xxx–xxx

Nomenclature D D/Dt f Gb Gk g h k L Nu p q‘‘ Re S Tm Tw U ^ u y+

tube diameter, m material derivative Darcy friction factor generation of turbulence kinetic energy due to buoyancy generation of turbulence kinetic energy due to the mean velocity gradients gravitational acceleration, m/s2 heat transfer coefficient, W/m2 K turbulence kinetic energy, J/kg or m2/s2 tube length, m average Nusselt number fluid pressure, N/m2 or Pa heat flux (W/m2) Reynolds number pitch, m mass-weighted average temperature average temperature of the tube wall velocity vector, m/s specific internal energy, kJ/kg dimensionless wall distance

overall performance of HEXs [1–5]. In addition to this traditional manner, which focuses on heat transfer enhancement in the tube boundary layer, many researchers have proposed a newer concept of augmented heat transfer in the core flow. Because of the convenience of manufacturing and the ability to retrofit existing devices, this concept has become popular in the industry, and, subsequently, several types of inserts have been suggested to be fitted in the tubes. Twisted tape [6–8], spiral spring [9], porous media [10–12], conical strip [13], screw-tape [14], wire-coil [15], etc., are some instances of various kinds of inserts have been proposed and investigated in the literature. The task of properly evaluating the merit of a proposed augmentation technique is, perhaps, as important as developing and applying the technique. For this purpose, several studies have been conducted, both numerically [16–18] and experimentally [19–22], on tubes with inserts for laminar and turbulent flows. The main objective of using the inserts is to increase the Nusselt number, but, unfortunately, it entails a huge increase in the friction factor, and, as a result, a bigger pressure drop through the flow. Comparing these two consequences, various criteria have been proposed in the literature [23–25]. One of the well-known performance evaluation criteria (PEC) which comprehensively evaluates the overall thermo-hydraulic performance of different enhancement techniques is represented below [23,26]:

PEC ¼

Nu=Nu0 ðf =f 0 Þ

1=3

ð1Þ

Considering the given equation for PEC, it is obvious that this parameter is more sensitive to the Nusselt number than the friction factor. In addition, a true calculation of the Nusselt number strongly depends on the proper modeling of the domain at the regions close to the walls, especially the wall which receives the heat flux. When the flow is laminar, one can consider an initial grid, and by refining the mesh, the results will finally be independent of the computational domain and, in most cases, the results are acceptable. However, since there are several layers with totally different characteristics in the boundary layer of a turbulent flow [27], the accuracy of the numerical results is strongly dependent

Greek symbols slant angle, deg b pressure gradient per unit length, Pa/m Dp pressure drop (p1 p2), N/m2 or Pa e turbulence dissipation rate, J/kg s or m2/s3 h twist angle, deg k thermal conductivity, W/m K l dynamic viscosity, kg/m s or Pa s m kinematic viscosity, m2/s q fluid density, kg/m3 r turbulent Prandtl number s shear stress, Pa U viscous-dissipation function, W/m3 or kg/m s3 r vector differential operator, m1

a

Subscripts 0 plain tube in inlet m mean value out outlet t turbulent w wall

on the near-wall modeling, which significantly impacts the fidelity of solutions, inasmuch as walls are the main source of mean vorticity and turbulence. There are several turbulence models with different near wall treatments. Considering that k–e models [28–30] provide acceptable accuracy and are computationally less expensive, they have been widely used in industrial applications. However, Since these models assume that the flow is totally turbulent, they are not valid for the near wall region, where there are laminar layers, called the ‘‘viscous sublayer” and buffer layers, which must be treated with another approach [27]. To overcome this deficiency, semi-empirical formulas called ‘‘wall functions” are used to bridge the viscosity-affected region between the wall and the fully-turbulent region. The use of wall functions obviates the need to modify the turbulence models to account for the presence of the wall [31]. In the case of using wall functions, one of the most important parameters, which should be considered, is the dimensionless wall distance (y+), which is given below [32]:

yþ ¼

qus y l

ð2Þ

where us is the friction velocity and defined as (sw/q)1/2. The value of y+ is up to 5, in viscous sublayer, and is up to 60, in buffer layer. In most previous studies, related to heat transfer augmentation in turbulent flow, the standard wall functions [33] have been used for modeling the near wall treatment. The main shortcoming of standard wall functions is their strong dependence on the location of the mesh point closest to the wall, where the wall functions are applied. The first mesh point must be located out of the viscous sublayer and, consequently, the y+ value must be higher than 10. y+ values of below 15 will gradually result in unbounded errors in wall shear stress, and wall heat transfer [31,34]. Since ignoring the importance of y+ value and wall function selection can lead to enormous error in the simulated results, the validity of the results of many numerical studies, which modeled turbulent flow, and ignored the importance of y+ value, ought to be reconsidered.



The objective of the present paper is to investigate the results of the numerical simulation of turbulent flow through a circular tube for two different wall functions. In addition, a new configuration of inserts, called twisted conical strip insert, which is a combination of conical strip inserts and twisted tape inserts, is proposed. However, in order to investigate the performance of the inserts, this paper studies the effects of the pitch, slant angle, and twist angle on the thermo-hydraulic characteristics of the flow, including the Nusselt number, friction factor and performance evaluation criteria (PEC) for both staggered and non-staggered alignments.

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In the present investigation, the inner diameter of the tube, and the diameter of the connecting rod are considered 20 mm and 1 mm, respectively. The pitch varies by 10 mm, 20 mm and 30 mm; the slant angle varies by 15°, 30°, and 45°, and the twist angle varies by 5°, 20°, and 35°, while the geometry angle for all cases is 90°. The thickness of the inserts is 1 mm at the tip and 0.5 mm at the base, where the inserts are attached to the connecting rod. In addition, in both staggered and non-staggered alignments, after each pitch, inserts rotate around the connecting rod 90° clockwise (in left view). 2.2. Governing equations

2. Numerical modeling 2.1. Description of the insert geometry

The numerical simulation involves solving the basic conservation equations. These equations are:

As shown in Fig. 1, twisted conical strip insert, which is investigated during this study, have four main parameters, including; 1. Pitch (S), which is the horizontal distance between two successive blades. 2. Twist angle (h), which is the angle between the connecting rod axis and the twist axis (the axis around which a blade revolves). 3. Slant angle (a), which is the angle between each blade and its twist axis. 4. Geometry angle, which is the revolve angle of each blade around the twist axis.

Mass conservation or continuity equation:

@q þ r ðqUÞ ¼ 0 @t

ð3Þ

Momentum conservation equation:

q

DU ¼ qg rp þ r s Dt

ð4Þ

Energy conservation equation:

^ Du q þ pðr UÞ ¼ r ðkrTÞ þ U Dt

ð5Þ

Fig. 1. Geometry and schematics of the inserts from different views. S = 20 mm, h = 35°, a = 45°.


4


Both RNG [29] and the realizable [30] k–e models were used to simulate turbulent flow in the plain (smooth) tube, but only the realizable model was used for enhanced tube. This model has two equations for turbulence kinetic energy (k) and turbulence dissipation rate (e):

@ @ @ ðqkÞ þ ðqkuj Þ ¼ @t @xj @xj

lþ

lt @k þ Gk þ Gb qe Y M þ Sk rk @xj

lt @ e re @xj e2 pffiffiffiffiffiffi þ C 1e Gb þ Se þ qC 1 Se qC 2 k þ ve

@ @ @ ðqeÞ þ ðqeuj Þ ¼ @t @xj @xj

lþ

ð6Þ

ð7Þ

where

h i pffiffiffiffiffiffiffiffiffiffiffiffi g C 1 ¼ max 0:43; gþ5 ; g ¼ S ke ; S ¼ 2Sij Sij ; @u i ; C 1e ¼ 1:44; C 2 ¼ 1:9; rk ¼ 1:0; Sij ¼ 12 @xj þ @u @x i

j

re ¼ 1:2

ð8Þ

2.3. Computational grid and boundary conditions In this study the commercial software, ANSYS FLUENT 14.5 [35] which solves the Reynolds-Averaged Navier–Stokes (RANS) equations, was used to simulate hydrodynamic and thermal fields.

The time-averaged RANS equations are used because of the complex nature of the present flow and large range of turbulence scales. A double-precision, implicit, pressure-based, steady-state, three-dimensional solver is used for the calculations. Furthermore, turbulence is treated with the realizable k–e model. The realizable k–e model is a relatively recent development widely used for engineering applications, which yields better performance in many industrial turbulent flows, and also likely to provide superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation to other k–e models [36]. Moreover, the pressure–velocity coupling was modeled with the semi-implicit method for the pressurelinked equations (SIMPLE) scheme. The energy, momentum, turbulent kinetic energy, and turbulent dissipation rate equations are solved with second-order upwind discretization and the gradients on the faces are evaluated with the node-based method. Since the physical geometry of the problem and the expected patterns of the flow and thermal solutions have a periodically repeating nature, in order to have fully-developed flow and heat transfer, and reduce the computational load, the problem was modeled as a periodic zone. For this purpose, one cycle of inserts, for both staggered and non-staggered alignments, was modeled and periodic boundary condition was implemented at the inlet and outlet of the tube. It means that the inlet and outlet no longer

Fig. 2. Computational grid for staggered alignment with pitch = 10 mm (right side) and non-staggered alignment with pitch = 20 mm (left side). h = 20°, a = 45°.


5


exist and there is one boundary with equal values of velocity, turbulence characteristics, etc., for both sides of the domain. Although the values of temperature and pressure at each point continually change during successive cycles, it should be emphasized that both fields have periodic natures, and their profiles, at upstream and downstream, are similar but not equal. To implement the periodic boundary condition, mesh structure of the inlet and outlet faces must be coupled to guarantee the coincidence of the position of all nodes at the both sides of the tube. In this situation, since the pressure at each point decreases in each cycle, the local pressure can be decomposed into two parts: ~ðx; y; zÞ, and a linearly-varying component, a periodic component, p bx, where b is the pressure gradient per unit length, and x is the periodic direction. When the mass flow rate is known, the value of b is not known a priori, and the solution must be iterated until the defined mass flow rate is achieved in the computational model. After conver~ðx; y; zÞ as the pressure gence of the solution, since FLUENT gives p distribution, the real pressure can be obtained by the following relation:

~ðx; y; zÞ þ bx pðx; y; zÞ ¼ p

ð9Þ

As already mentioned, one cycle of the insert is modeled for each case. Therefore, the length of the domain depends on the pitch value of the insert. Since the pitch (S) varies by 10 mm,

Fig. 3. Comparisons between the numerical results for (f) and the results of Filonenko correlation [37] (y+ = 5). Fig. 5. Variation of the friction factor with respect to the Reynolds number. (a) Pitch varying, h = 20°, a = 45°. (b) Twist angle varying, S = 20 mm, a = 45°. (c) Slant angle varying, S = 20 mm, h = 20°.

Table 1 Values of cf, ch, m and n for Eqs. (14) and (15).

Fig. 4. Comparisons between the numerical results for (Nu) and the results of Dittus–Boelter correlation [38] (y+ = 5).

cf

m

ch

n

Staggered alignment

P = 10 P = 30 h=5 h = 35 a = 15 a = 45

0.8923 0.3972 0.9039 0.4469 0.3727 0.4842

0.093 0.102 0.077 0.14 0.22 0.087

0.0483 0.0394 0.0418 0.0289 0.0367 0.0463

0.7908 0.7932 0.8129 0.8098 0.7502 0.7825

Non-staggered alignment

P = 10 P = 30 h=5 h = 35 a = 15 a = 45

1.7334 0.8408 1.7758 0.622 0.3998 0.9775

0.101 0.092 0.039 0.128 0.202 0.084

0.0541 0.0566 0.0485 0.0348 0.034 0.045

0.7954 0.7689 0.8358 0.7968 0.7601 0.8081

0.3646

0.265

0.0204

0.8

Plain tube


6


Fig. 6. Velocity distribution for Re = 20,000, S = 20 mm and a = 45°; (a) h = 5°, staggered alignment. (b) h = 35°, staggered alignment. (c) h = 5°, non-staggered alignment. (d) h = 35°, non-staggered alignment.

20 mm, and 30 mm, the domain length, for staggered alignment, varies by 40 mm, 80 mm, and 120 mm, respectively, while the length of the domain for non-staggered alignment is half of these values. Conducting investigations to ensure the independence of the simulated results from the grid size, we meshed the tube and insert into tetrahedral control volumes (cells), including two computational grid zones, insert zone and tube zone, for both staggered and non-staggered alignments. For each cycle, coarse, moderate, and fine, grids were adopted, and the temperature and velocity profiles at inlet and middle cross sections were studied. The results showed a reasonable independence for the moderate grid, and the finer grid presented no significant change in the temperature and

velocity distributions. For staggered alignment, the number of control volumes, which resulted in independence was 400,000 CVs, 650,000 CVs and 950,000 CVs for cycle length of 40 mm, 80 mm, and 120 mm, respectively. Since the length of the domain for non-staggered alignment is half of the length of staggered alignment, the number of cells for non-staggered alignments are roughly half of the above values. Furthermore, relatively high grid densities are employed in the near wall regions, for both inserts, and tube walls, where the highest gradients of temperature and velocity were expected to occur (see Fig. 2). In the present investigation, the material of the insert is aluminum and air is considered as the working fluid, which is assumed a Newtonian and incompressible fluid. Since the

Fig. 7. Streamlines for Re = 20,000, S = 20 mm and h = 20°; (a) a = 15°, staggered alignment. (b) a = 30°, staggered alignment; (c) a = 15°, non-staggered alignment; (d) a = 30°, non-staggered alignment.



7

temperature difference in fluid is less than 50 K, all the properties of the fluid and solid are assumed to be constant. The Momentum boundaries of no-slip and no-penetration are set for all walls. The constant heat flux was imposed on the tube wall, and the value of the heat flux varies proportionally with the Reynolds number, in order to have relatively constant values of outlet temperature for all cases. In addition, the specified mass flow rate of the fluid changes in such a way that the Reynolds number changes from 5000 to 45,000, which ensures that the tubular flow is in the turbulent regime. Moreover, the insert wall, which is the interface between solid (insert zone) and fluid (tube zone), is thermally coupled to solve the conduction heat transfer in the inserts. The convergence criteria were set as the relative residuals of continuity equation to be of the order of 107, while the residuals of the other variables were of the order of 108 or less. Furthermore, sufficient attention was paid to the trend of the residuals variation to ensure actual convergence. 2.4. Calculation methods for Nu, f, and PEC When the convergence is achieved, velocity and temperature distributions are known, and thermo-hydraulic parameters can be obtained. For the fluid flow in tubes, the friction factor is described by Darcy friction factor (f), which can be calculated as follows:

f ¼

2bD

qU 2m

ð10Þ

where b is the mean pressure gradient which is defined as b = Dp/L; and Um is the cross-sectional area-weighted average of the fluid velocity, which is constant, due to continuity. For the purpose of calculating heat transfer between the tube wall and fluid, convection heat transfer coefficient (h) can be obtained from the below relation:

h¼

q00 T in;m þ T out;m where T m ¼ Tw Tm 2

ð11Þ

Finally, the Nusselt number (Nu) can be calculated as follows:

Nu ¼

hD k

ð12Þ

In addition, the value of the performance evaluation criterion (PEC) can be obtained from Eq. (1). It should be mentioned that both enhanced and plain tubes are identical in the crosssectional, and heat transfer areas used for calculating the average fluid velocity, and the convective heat transfer coefficient, respectively [37].

Fig. 8. Variation of the Nusselt number with the Reynolds number. (a) Pitch varying, h = 20°, a = 45°. (b) Twist angle varying, S = 20 mm, a = 45°. (c) Slant angle varying, S = 20 mm, h = 20°.

3. Results and discussion 3.1. Validation of the model In order to validate our simulation method, the results of two famous correlations, Filonenko correlation [38] for friction factor, and Dittus–Boelter correlation [39] for the Nusselt number, were compared to the numerical results of realizable and RNG k–e models for a plain tube. From Figs. 3 and 4, it is observed that, when the near wall region is treated by standard wall functions, the numerical results have a huge (up to 205% for f and up to 201% for Nu) discrepancy with the results of the correlations. Although this error is smaller for realizable k–e model (173% for f and 152% for Nu), the results are still unacceptable. Moreover, in the case of using standard wall functions, using finer grids, boundary mesh or mesh adaption, which causes refinement of the grid in the wall normal direction,

deteriorates the results, and mesh independence becomes impossible. In fact, this is the main shortcoming of the standard wall functions. When the first layer of the grid approaches the wall and y+ becomes smaller than 10, the first points locate in viscous sublayer [34]. As a result, it leads to use of the turbulence model at near wall regions, although it is known that the high Reynolds models, like k–e family, are not valid at near wall regions, in which the flow is laminar. Therefore, when these models are used, smaller y+ values cause false near wall modeling, which causes the mentioned error for wall shear stress (f) and heat transfer (Nu). In order to avoid this problem, scalable wall functions approach was proposed by Grotjans and Menter [40], which forces the usage of the log law in conjunction with the standard wall functions approach. This is achieved by introducing a limiter in the y+ calculations [31]:


8

yþ ¼ Max yþ ; yþlimit


ð13Þ

where y+limit = 11.25. It means when y+ > 11.25, the grid should be refined by the user to improve the accuracy of the results and also to achieve mesh independence. On the other hand, when y+ < 11.25, all grid layers with y+ values below the limit are treated by standard wall functions. Comparing the numerical results obtained by using the scalable wall functions with the correlation results, verifies that the scalable wall functions are the better choice than the standard wall functions (see Figs. 3 and 4). In addition, although both RNG and realizable models have roughly the same results for the friction factor, the results of the realizable k–e model for the Nusselt number are more accurate than RNG model. Therefore, realizable model is selected for our investigations with the inserts, and the near wall region was treated with the scalable wall function.

3.2. Flow field Fig. 5 shows the variation of the friction factor with respect to the Reynolds number for different pitches (S), twist angles (h), and slant angles (a). Clearly, the behavior of the friction factor in all cases is completely consistent with that of the smooth tube, as an increase in the Reynolds number causes a decrease in the friction factor, and in high Reynolds numbers, the friction factor will become relatively less dependent on the Reynolds number. Furthermore, when the other parameters are the same, the friction factor of the non-staggered alignments is always higher than that of the staggered alignments. As the pitch decreases, Fig. 5a shows that the friction factor increases, and this increase is more severe for smaller pitches. Moreover, the effect of the alignment type is more evident for lower pitches. This trend is also seen in Fig. 5b as lower twist angles result in higher friction factors. In this situation, the maximum value of the friction factor belongs to the nonstaggered alignment with the lowest twist angle (5°), which is about 25 times higher than that of the plain tube. It should be mentioned that, as the twist angle approaches zero, the effect of the twist vanishes and the inserts become conical strip inserts. Investigating the effect of the slant angle in Fig. 5c shows that the higher slant angles cause higher friction factors. Comparing the effect of these three parameters (S, h, and a), it is evident that the twist angle has the greatest impact on the friction factor. Fitting the

numerical results into an equation, we found that the friction factor for all cases varies as follows:

f ¼ cf Rem

ð14Þ

The values of the constants, cf and m are shown in Table 1. Fig. 6 shows the velocity distribution in cross-sectional planes for two different twist angles (5° and 35°). Contrary to the velocity profile of the plain tube, in which the maximum velocity occurs at the center, it is clear that the maximum velocity of the enhanced tubes, in all cases, occurs close to the tube wall. Moreover, it can be observed that the maximum velocity and also disturbance of the flow are higher in non-staggered alignments. Furthermore, as the twist angle decreases, the blades get more distance from the connecting rod, and cover a larger cross-sectional area, which increases the pressure drop. In this situation, dead flow zones were observed between the insert blades and the connecting rod, which result in the increase in maximum velocity, due to continuity (see Fig. 6a and c). These semi-stagnant zones also generate internal vortices, which totally disturb the flow and dissipate the energy. Additionally, for lower twist angles, more of the tip of the insert places close to the tube wall, which plays the critical role of disturbing the boundary layer and could be another cause of the higher pressure drop. Fig. 7 shows the streamlines for two different slant angles (15° and 30°). Again, the maximum velocity and also disturbance of the flow are higher in non-staggered alignment. Clearly, for the lower slant angle, the flow is relatively axial (see Fig. 7a and c), but the higher slant angle generates a completely swirling flow within the tube (see Fig. 7b and d). For the non-staggered alignment, the generated swirl is more intense and the stream lines have smaller pitches, which also cause a higher pressure drop. It should be noted that for non-staggered alignments, in both Figs. 6 and 7, two cycles are plotted, in order to have a better comparison. 3.3. Heat transfer Fig. 8 shows the variation of the Nusselt number with respect to the Reynolds number. For all cases, the results show that the trend of the Nusselt variation is the same as that of the plain tube, and the Nusselt number increases by the increase of the Reynolds number. Furthermore, the effects of the investigated parameters (S, h, and a) are more significant in higher Reynolds numbers. When the other parameters are the same, non-staggered alignments have

Fig. 9. Temperature distribution for Re = 20,000, S = 20 mm and a = 45°; (a) h = 5°, staggered alignment. (b) h = 35°, staggered alignment. (c) h = 5°, non-staggered alignment. (d) h = 35°, non-staggered alignment.



higher Nusselt numbers than those of the staggered alignments. As Fig. 8a shows, for both staggered and non-staggered alignments, lower pitches result in higher Nusselt numbers, and the alignment type is more effective for the moderate pitch value (20 mm). Fig. 8b shows the effect of the twist angle variation. It is found that the lower twist angles result in higher Nusselt numbers, and the effect of the alignment type is more considerable for lower twist angles. From Fig. 8c, it is evident that the higher slant angles cause higher Nusselt numbers, and the effect of the alignment type is more significant in higher slant angles. The same as the previous discussion, the twist angle is the most effective parameter, which influences the Nusselt number. In this situation, the maximum value of the Nusselt number is 378, which is about 3.5 times higher than that of the plain tube, and is corresponding to the lowest twist angle. The same as the previous section, the Nusselt number varies as follows:

Nu ¼ ch Ren

9

ð15Þ

The values of the constants, ch and n are shown in Table 1. Fig. 9 shows the temperature distribution in cross-sectional and horizontal planes for two different twist angles (5° and 35°). Considering cross-sectional planes, it is found that the wall temperature for higher twist angle, in both alignments, is higher than that of the lower twist angle. On the other hand, considering the temperature distribution on horizontal planes, it is clear that the average temperature at the outlet of the tube with lower twist angle insert is higher than that of the average temperature at the outlet of the tube with higher twist angle insert. Refer to Eq. (11), both lower wall temperature and higher outlet temperature result in higher Nusselt number, which is totally consistent with our observations in Fig. 8b. 3.4. Performance evaluation criteria (PEC)

Fig. 10. Variation of PEC with the Reynolds number. (a) Pitch varying, h = 20°, a = 45°. (b) Twist angle varying, S = 20 mm, a = 45°. (c) Slant angle varying, S = 20 mm, h = 20°.

In the previous sections, we discussed the effects of the three parameters (S, h, and a) on thermo-hydraulic characteristics of a tube fitted with twisted conical strip inserts. It has been found that the heat transfer enhancement caused by the inserts is followed by a considerable pressure drop. In the present section, using Eq. (1), performance evaluation criterion (PEC) is calculated, and the effects of the parameters (S, h, and a) on PEC are investigated. Considering Fig. 10, which shows the variation of PEC with respect to the Reynolds number, increasing the Reynolds number decreases the PEC. It means that, as the velocity increases the friction factor grows much faster than the Nusselt number. However, for high Reynolds numbers (Re > 45,000) the PEC will be independent of the Reynolds variation. These observations are consistent with the experimental and numerical results for two other types of inserts (double spiral spring and porous plane inserts), which are reported in [9,10], respectively. According to the previous sections, although non-staggered alignment enhances the heat transfer, and, consequently, results in higher Nusselt number, it also causes the more additional pressure drop. Therefore, the PEC of staggered alignments is higher than that of non-staggered ones, except for the lowest twist angle (h = 5°). From Fig. 10a, it can be observed that the smaller pitches result in lower PEC values, and the effect of the alignment type on PEC is more obvious for higher pitches. Considering Fig. 10b, for staggered alignments, the lowest twist angle has the lowest thermohydraulic performance, while for non-staggered alignments, the highest PEC belongs to the lowest twist angle. In this situation, the effect of the alignment type is more considerable for higher twist angles. Studying the effects of the variation of slant angle, as Fig. 10c shows, for both staggered and non-staggered alignments, the inserts with the higher slant angle result in higher PEC, and the effect of the alignment is more significant for lower slant angles. It is noteworthy that through all investigated cases, the maximum achieved value for PEC is 1.12. Considering the experimental and numerical results for turbulent flow through the tubes fitted with two other types of inserts (double spiral spring and porous plane inserts), which are reported in [9,10], for heat transfer augmentation with inserts in turbulent flow, PEC hardly exceeds 1.5, which is reasonably consistent with our results for PEC in the present study. As previously mentioned, when the twist angle approaches zero, the twisted conical strip inserts become conical strip inserts. Therefore, we can qualitatively compare our results for the lowest twist angle (h = 5°) to the results for conical strip inserts presented in [13]. It should be considered that although for laminar flow in tubes fitted with inserts, PEC values higher than 1.5 are also reported in the literature ([41,16,10]), Since the friction factor of


10


turbulent flow is smaller than that of laminar flow, using inserts in turbulent flow increases the friction factor several times and considering the nature of PEC, Eq. (1), these values (higher than 1.5) are very hard to achieve for turbulent flow. According to the present study, it is evident that the maximum PEC for the lowest twist angle is about one (1.022), and decreasing the twist angle results in lower PEC values. On the other hand, the PEC values reported in [13] are higher than 1.7, and even values higher than 2 were reported. As it was discussed in the previous sections, this error can be caused by the difficulties with the turbulent flow modeling, and violation of the allowed values of y+, which is one of the consequences of fine grids, while the standard wall functions are selected as the near wall treatment approach.

4. Conclusions Turbulence flow modeling with high-Reynolds turbulent models needs huge attention to the allowed value of y+. Neglecting the importance of the wall functions and y+ could lead to a false increase in friction factor and heat transfer. In comparison with the standard wall functions, the concept of the scalable wallfunctions is a more reliable choice, which ensures the proper modeling of the near wall treatment. Furthermore, realizable k–e model is suggested to be utilized to obtain more precise results. Considering the enhanced tubes, we found that the maximal friction factor (f) increases by 25 times, while the Nusselt number (Nu) is augmented by 3.5 times, comparing to those of the plain tube. Furthermore, f and Nu increase with decreasing the pitch and twist angle, while reduction of the slant angle causes the reduction in f and Nu. Moreover, PEC decreases with increasing Reynolds number or slant angle, but the reduction of the pitch results in lower PEC. Among the investigated parameters, the twist angle has the most impact on Nu and f. In addition, non-staggered alignments cause higher f and Nu but lower PEC. The effect of the alignment type on f and Nu is more considerable for lower pitches, lower twist angles, and higher slant angles. Acknowledgements The authors are grateful to the management and staff at HighPerformance Computing (HPC) center of Ferdowsi University of Mashhad for their great cooperation in this work. References [1] S.O. Akansu, Heat transfers and pressure drops for porous-ring turbulators in a circular pipe, Appl. Energy 83 (2006) 280–298. [2] G. Tanda, Effect of rib spacing on heat transfer and friction in a rectangular channel with 45° angled rib turbulators on one/two walls, Int. J. Heat Mass Transf. 54 (2011) 1081–1090. [3] F. Satta, D. Simoni, G. Tanda, Experimental investigation of flow and heat transfer in a rectangular channel with 45° angled ribs on one/two walls, Exp. Thermal Fluid Sci. 37 (2012) 46–56. [4] Z. Li, J. Lu, G. Tang, Q. Liu, Y. Wu, Effects of rib geometries and property variations on heat transfer to supercritical water in internally ribbed tubes, Appl. Therm. Eng. 78 (2015) 303–314. [5] N. Zheng, W. Liu, Z. Liu, P. Liu, F. Shan, A numerical study on heat transfer enhancement and the flow structure in a heat exchanger tube with discrete double inclined ribs, Appl. Therm. Eng. 90 (2015) 232–241. [6] Y. Wang, M. Hou, X. Deng, L. Li, C. Huang, H. Huang, G. Zhang, C. Chen, W. Huang, Configuration optimization of regularly spaced short-length twisted tape in a circular tube to enhance turbulent heat transfer using CFD modeling, Appl. Therm. Eng. 31 (2011) 1141–1149. [7] P. Promvonge, S. Pethkool, M. Pimsarn, C. Thianpong, Heat transfer augmentation in a helical-ribbed tube with double twisted tape inserts, Int. Commun. Heat Mass Transfer 39 (2012) 953–959. [8] S. Eiamsa-ard, P. Somkleang, C. Nuntadusit, C. Thianpong, Heat transfer enhancement in tube by inserting uniform/non-uniform twisted-tapes with alternate axes: effect of rotated-axis length, Appl. Therm. Eng. 54 (2013) 289– 309.

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Modeling for thermal augmentation of turbulent flow in

Modeling for thermal augmentation of turbulent flow in

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