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CFD study of heat transfer for oscillating ﬂow in helically coiled tube heat-exchanger Changzhao Pan a,b , Yuan Zhou a , Junjie Wang a,∗ a b

Key Laboratory of Cryogenics, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China

a r t i c l e

a b s t r a c t

i n f o

Article history: Received 17 November 2013 Received in revised form 1 July 2014 Accepted 2 July 2014 Available online 10 July 2014 Keywords: Helically coiled CFD Heat transfer correlation Friction factor correlation Uniform design Field synergy principle

The heat transfer and pressure drop for oscillating ﬂow in helically coiled tube heat-exchanger were numerically investigated based on the Navier–Stokes equations. The correlation of the average Nussel number and average friction factor were proposed considering the frequency and the inlet velocity. The oscillating ﬂow heat transfer problems are inﬂuenced by many factors. Hence we need an easy way to reduce the numbers of simulation or experiment. Therefore, the method of uniform design was adopted and the feasibility of this method was veriﬁed. The ﬁeld synergy principle was used to explain the heat transfer enhancement of oscillating ﬂow in helically coiled tube heat-exchanger. The result shows that the smaller the volume average ﬁeld synergy angle in the helically coiled tube, the better the rate of heat transfer. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Helically coiled tube heat-exchanger is widely used in industry such as chemical plants, power generation plants, nuclear industry, food processing, cryogenic and medical equipment, etc. The steady ﬂow in coiled tube has been studied by many researchers. Dean (1927, 1928) studied the curved channel ﬂow and clariﬁed the enhancement of heat transfer in curved channel. The centrifugal force in curved tube pushes the ﬂuid particles toward the core region and produces a secondary ﬂow ﬁeld, which can enhance the heat transfer in the curved tube. The Dean number was used to characterize the ﬂow of Newtonian ﬂuids in helically coiled tube. The ﬂuid ﬂow and heat transfer in curved tube was ﬁrstly reviewed by Berger et al. (1983) and Shah and Joshi (1987). The latest review of helically coiled tube was presented by Naphon and Wongwises (2006). Prabhanjan et al. (2004), Nandakumar and Masliyah (1982) and Shah and Joshi (1987) used the boundary condition of constant wall temperature and constant heat ﬂux to investigate the heat transfer coefﬁcient of helically coiled tube. Experimental and numerical studies of the double pipe helically heat exchanger have been investigated by Rennie and Raghavan (2005, 2006a,b). Jayakumar et al. (2008) studied the ﬂuid-to-ﬂuid

∗ Corresponding author. Tel.: +86 10 82543758; fax: +86 10 62564050. E-mail addresses: [email protected], [email protected], [email protected] (J. Wang). http://dx.doi.org/10.1016/j.compchemeng.2014.07.001 0098-1354/© 2014 Elsevier Ltd. All rights reserved.

helically coiled tube heat exchanger by experimental and numerical method. Jayakumar et al. (2010) also brought out clearly the variation of local Nusselt number along the length and circumference of a helical tube. The hydrodynamics and heat transfer in tube-in-tube helical heat exchanger has been studied by Kumar et al. (2006). The oscillatory ﬂow in helically coiled tube heat exchanger is widely used in Stirling engines or cryocoolers such as VM cryocooler (Zhou, 1984; Kuosa et al., 2012; Simon and Seume, 1988). Oscillatory heat transfer in the helical coiled tube plays a key role in the heat driven cryocooler which works between three temperature resources. The intermediate resource of this type cryocooler will be kept a constant temperature by using liquid nitrogen or mixrefrigerants (Zhou, 1984; Zhou et al., 2012) to obtain the higher cooling power. Helical coiled tube heat exchanger is usually used because of its high performance in heat transfer. However, the heat transfer correlations of oscillating ﬂow have not been published until recent years (Kuosa et al., 2012). Several literatures have studied the nature of oscillatory ﬂows and presented the pressure drop and heat transfer coefﬁcient in Stirling engine heat exchangers (Simon and Seume, 1988; Tew and Geng, 1992). Bouvier et al. (2005) described an experiment to study the oscillating ﬂow inside a cylindrical tube and shown the difﬁculty of deﬁning a dimensionless heat ﬂux density to model local heat transfer in oscillating ﬂow. Kuosa et al. (2012) studied the tubular heaters and coolers in Stirling engines. Guo et al. (2002) studied the transient convective heat transfer in a helical tube under pressure drop type oscillations, f = 0. 05–0.003 Hz.

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C. Pan et al. / Computers and Chemical Engineering 69 (2014) 59–65

Nomenclature A Cf cp De f h H n m Nu Pr q r Rc Re t T u Va

area (m2 ) friction factor speciﬁc heat (J kg−1 ) Dean number frequency (Hz) heat transfer coefﬁcient (W/m2 K) coil pitch (m) unit vector along outward normal the number of turns Nusselt number Prandtl number heat ﬂux (W/m2 ) pipe radius (m) pitch circle radius (m) Reynolds number cycle time (s) temperature (K) velocity (ms−1 ) Valensi number

Greek symbols thermal conductivity (W/m K) density (kg/m3 ) kinematic viscosity (m2 /s) pressure drop (Pa) p dynamic viscosity (Pa s) ˚ dissipation function Subscripts A amplitude working gas f w wall in tube inlet maximum max osc oscillatory ﬂow Superscript – average

Fig. 1. The schematic of helically coiled tube.

cooling power (Zhou, 1984; Rule and Qvale, 1969). So the diameter of tube should be small enough. According to the structure of Zhou’s (1984) cryocooler, the diameter of the tube is chosen 2r = 3 mm, and the diameter of the coil is 2Rc = 103 mm, the coil pitch is H = 22 mm. A dimensionless parameter can be used to characterize the ﬂow of Newtonian ﬂuids in helically coiled tube, which can be expressed in Dean number (Dean, 1928; Jayakumar et al., 2008):

De = Re

2. Dimensionless similarity criterions The schematic of helically coiled tube used in this paper is shown in Fig. 1. The simulation is mainly used in VM cryocooler. In this cryocooler, the empty volume should be reduced to obtain the larger

(1)

where, Re is the Reynolds number. In oscillatory ﬂows, Reynolds number can be characterized as (Gül and Akpinar, 2007): uA · 2r

Remax =

(2)

where, uA the amplitude of the velocity in oscillatory ﬂow, 2r is the characteristic length of pipe, and is the kinematic viscosity. Similar to Reynolds number, Valensi number is another dimensionless number that is important in oscillatory ﬂows. The Valensi number, Va is deﬁned as, 2f · (2r)2

Va =

(3)

where f is frequency, so the Va number reﬂects the frequency of oscillatory ﬂows. The average Nusselt number and friction factor are deﬁned as follows: Nuosc =

Most of the investigations involved either helical coiled tube or oscillating ﬂow. However, the oscillating ﬂow in the helical coiled tube requires a comprehensive study. This article is dedicated to obtain the correlations of heat transfer and pressure drop for oscillating ﬂow in the helical coiled tube by using CFD method. Large numbers of simulation are needed in order to obtain the interaction of velocity and the frequency f, which both inﬂuence the ﬂow characteristics in this case. The other factors, such as pitch and the number of turns, will increase the numbers of simulation signiﬁcantly. Therefore, it needs an easy way to reduce the times of simulation or experiments in multi factors heat transfer problems. The uniform design method is an experimental design method and can reduce the times of simulation or experiments signiﬁcantly. The present work used this method to study the heat transfer and veriﬁed its feasibility. And then, the enhancement mechanism of oscillating ﬂow in the helical coiled tube was investigated by the viewpoint of ﬁeld synergy principle (Guo et al., 1998, 2005).

r Rc

Cf =

h · 2r

(4)

2 · 2r · p u2A L

(5)

where is thermal conductivity, is density, h is average heat transfer coefﬁcient, which can be evaluated by: h=

1/t 1/t

t 0

t 0

qdt

(Tf − Tw )dt

(6)

where q is heat ﬂux, t is cycle time, Tf is the temperature of working gas, Tw is the temperature of the tube. p is average pressure drop, which can be calculated by: 1 t

p =

t

pdt

(7)

0

L is the length of the coiled tube, which can be calculated by:

L=m

H 2 + (2RC )2

(8)

where m is the number of turns, H is the coil pitch, the case of this simulation is m = 1, so L = 315 mm. According to Latzko’s (1921) research, the entrance length of tube can be obtain by

x D

entry

= 0.623Re0.25

(9)

C. Pan et al. / Computers and Chemical Engineering 69 (2014) 59–65 Table 1 The geometric data of simulation.

Table 2 Boundary conditions.

Variable

Size

Variable

Inlet

Wall

Outlet

2r 2Rc H m

3 mm 103 mm 22 mm 1

u

vA sin(2ft)

v

0 0 100 K ∂P/∂n = 0

0 0 0 77 K ∂P/∂n = 0

∂u/∂n = 0 ∂v/∂n = 0 ∂w/∂n = 0 ∂T/∂n = 0

w T P

Because the Re number in this article is in the range of 5000–17,500, the entrance length is no more than 22 mm, which is only 7% of the total length. So the entry effect is negligible. Summary of the geometric data is given in Table 1.

3.1. Mathematical model Helium is used as the working ﬂuid in the present work. The inlet velocity is changing with time by using the user deﬁned function (UDF):

vin = vA sin(2ft)

Velocity

3 4 5 6 7 8 9 10

Frequency 1

5

10

15

20

25

30

35

1 9 17 25 33 41 49 57

2 10 18 26 34 42 50 58

3 11 19 27 35 43 51 59

4 12 20 28 36 44 52 60

5 13 21 29 37 45 53 61

6 14 22 30 38 46 54 62

7 15 23 31 39 47 55 63

8 16 24 32 40 48 56 64

(10)

where vA = 3–10 m/s, f = 1–35 Hz. Because the critical dimensionless value is 400–780 (Merkli and Thomann, 1975; Hino et al., 1976; Zhao and Cheng, 1996; Tang et al., 2013) (ˇcri = 2Remax /Va0.5 ), the most points of simulation are in the range of turbulence ﬂow except several low velocity and high frequency points. The mass, momentum and energy equations for the working gas (helium) were solved by Fluent 6.3

∇ · (V ) = 0

(11)

T ∂V 2 + ∇ [V · (V )] = −∇ p − ∇ [(∇ · V )] + ∇ · [(∇ V ) ] 3 ∂t

+ ∇ · [(∇ V )]

cp

1e6 Pa

Table 3 The simulation points by using full factors method (64 times).

3. Numerical method

61

3.2. Full factorial design and uniform design (12)

∂T ∂p + cp V · ∇ T = ∇ · [(∇ T )] + + V · ∇p + ˚ ∂t ∂t

The dependency of grid was studied before the actual analysis. The grids used in this study are refer to the article (Jayakumar et al., 2008), Four structured grids are texted to get grid-independent solution, 0.2 mm, 0.25 mm, 0.3 mm, 0.35 mm. Because the mass and energy errors do not further reducing when the number of grid further reﬁnement, the 0.3 mm grid (1.025e8 cells m−3 ) was chosen in simulation.

(13)

where T is temperature, is density, p is pressure, cp is speciﬁc heat and is dynamic viscosity. ˚ is the dissipation function, and its expression can be referred in Rohsenow’s article. The boundary conditions are: the temperature of working gas is constant 100 K at inlet. UDF is hooked in Fluent to set boundary condition or modify equations. We use it to guide the inlet velocity to change sinusoidally (Eq. (10)). The static pressure outlet boundary condition is adopted and the relative pressure is set as zero; tube wall is the constant temperature condition 77 K, which is usual used the liquid nitrogen to keep this constant temperature in VM cryocooler (Zhou, 1984; Zhou et al., 2012). The initial condition is: the velocity in the entire ﬂow ﬁeld is zero; the temperature is set as 77 K and the pressure is set as 106 Pa. Because the temperature difference is not large, the constant properties of mean temperature are used. At the inlet a turbulent intensity of 4% and hydraulic diameter of the largest size eddy, which is taken as 0.3 times pipe inner diameter (Jayakumar et al., 2010), are speciﬁed. Summary of the boundary conditions is given in Table 2. The commercial CFD code Fluent 6.3 (Fluent, 2006) is used for the numerical solution. The simulation is done as transient processes and PISO algorithm is employed. Momentum equations are discredited using the second upwind scheme. The k–e RNG turbulence model is used in computation. The convergence tolerance criterions for energy equations are 1 × 10−6 and the others are 1 × 10−3 . A computer with 2.0 GHz processor and 2.0 GB RAM is used and it will take several hours to get the result of half-cycle.

Uniform design method, based on quasi-Monte Carlo method, was proposed by Fang et al. (2000), Fang (1980) and Wang and Fang (1981). It has been successfully used in many industrial and scientiﬁc experiments (Fang, 1980; Chan and Lo, 2004; Liu and Chan, 2004), but it never used in heat transfer area. By comparing the result to full factorial will verify the feasibility of the uniform design method The results of simulation need to be processed by the multiple regression method for calculating the correlation of heat transfer coefﬁcient and pressure drop factor. The factors of the maximum velocity and frequency mainly inﬂuence the ﬂow and heat transfer. The method of full factorial method can consider those two factors compressively, but it needs n2 times simulations, where n is the number of levels of each factor, as shown in Table 3. For using the UD method to analysis the heat transfer problems, the following steps are necessary: (1) choose a parameter search domain and determine a suitable number of levels for each parameter. (2) Choose a suitable UD table to accommodate the number of parameters and levels. The UD table can be obtained in article (Kaitai, 1994). (3) From the UD table, randomly determine the run order of simulation, and then study the character of ﬂow and heat transfer. (4) Fit the simulation’s result and obtain the correction of Nu and Cf . Uniform design (UD) method seeks design points that are uniformly scattered on domain (Fang et al., 2000). Using this method can effectively reduce the number of tests. For 3 factors and 7 levels’ test, using the full factorial method will simulate 73 times, but using the uniform design method only need simulate 7 times. Uniform design method use the Un (ns ) table to design the simulation, n is the number of simulation, s is the number of factors. In this paper, we use U8 (82 ) table to design the simulation points, as shown in Table 4.

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Table 4 The simulation points by using uniform design method. Velocity (m/s)

Frequency (Hz)

3 4 5 6 7 8 9 10

15 35 10 30 5 25 1 20

4. Result and discussion 4.1. Model validation Fig. 2 shows the heat transfer coefﬁcient of straight pipe and curved tube in steady and oscillating ﬂow. The correlation of Dittus and Boelter is used in the straight tube, and it shows that the Nu number in oscillatory curved tube ﬂow is higher than steady straight tube ﬂow, but with the increase of Re number the difference of two ﬂows become smaller. The correlation of Kalb and Seader (1972) and Shchukin (1969) apply for the turbulence ﬂow in curved tube. The result of Kalb and Seader (1972) is suitable for De = 80–1200, so when the Re > 7000 this result is not correct. The result of Shchukin (1969) shows that with the increase of Re number the difference between straight and curved ﬂows become larger, which is not consistent with the result of Guo et al. (2002). The correlation of Guo et al. (2002) used in unsteady oscillatory ﬂow in curved tube, but the frequency is very low, f = 0.05–0.003. The line in Fig. 2 used Gül’s (Gül and Akpinar, 2007) correlation for f = 1. The result shows that the present result is consistent with Chen’s when the Re number is small. The result of Gül and Akpinar (2007) used in unsteady oscillatory ﬂow in straight tube (f = 10). The result of Gül’s and present keeps the same trends. Therefore, the result of present work is reasonable. Fig. 3 shows the friction factor of straight pipe and curved tube in steady and oscillating ﬂow. The correlation of Mishra and Gupta (1979) is used in the curved tube, and it shows that the friction factor in curved tube is larger than straight tube. The result Gül and Akpinar (2007) is used for the oscillatory ﬂow in straight tube, and it shows that as the increasing of frequency the friction factor will increase linearly. Comparing to the friction factor of straight tube, the friction factor of curved tube (present work) is larger. With the

Fig. 2. The heat transfer coefﬁcient of straight pipe and curved tube in steady and oscillating ﬂow.

Fig. 3. The friction factor of straight pipe and curved tube in steady and oscillating ﬂow.

Fig. 4. The time-varying Cf and Nu in the half cycle (Demax = 1492.773 and f = 5 Hz).

increasing Re number the difference of two tubes become smaller, and the trend is consistent with the straight tube. 4.1.1. The correlation of heat transfer coefﬁcient and pressure drop factor Fig. 4 shows the time-varying Cf and Nu number in the case of Demax = 1492.773 and f = 5 Hz. The heat transfer ﬁrst decreases then increases at the beginning of one cycle. It is because that the changing rate of the temperature difference and heat ﬂux is different. The largest pressure drop is not appearing in the middle of one case. It is because that the deceleration of helium will increase the pressure drop. Fig. 5 shows the change of Nu number and Cf corresponding to Demax number and Va number. The Nu number increases with the increase of Demax number and Va number, but the inﬂuence of Demax number is signiﬁcant. However, the inﬂuence of Demax number and Va number in friction factor are different. The friction factor increases with the increase of Va number but decrease with the Demax number, as shown in Fig. 5(c) and (d). The physical signiﬁcance of Va is the ratio of the time scale of viscous penetration to the oscillation period (Tang et al., 2013, 2014). The larger value of Va number means the shallower viscous penetrated, which means the more signiﬁcant characteristics of oscillatory ﬂow and the larger pressure drop. The Demax number shows the ratio of inertial and

C. Pan et al. / Computers and Chemical Engineering 69 (2014) 59–65

63

Fig. 5. The changing of Nu number and Cf corresponding to De number and Va number. Table 5 Comparison of full factorial design and uniform design. Correlation of Nu and Cf

Adj. R-square

Nuosc = 2.83514Va0.01646

0.964

Cf = 0.1195Va

0.994

De 0.50096 0.3 s Pr 10

−1.01987 0.93841 De

Full factorial design

10

0.56388 0.3 Nuosc = 2.03593Va0.01441 De Pr

10−1.00825 0.96618 De

Uniform design

heat transfer, the uniform design method is a better choice to be used for analysis.

Cf = 0.09354Va

10

0.993 (F = 8725.8) 0.993 (F = 979.7)

viscous forces. The larger value of Demax number means the weaker of viscous force, which leads the smaller friction factor. In summary, the oscillating ﬂow can enhance heat transfer at a certain extent, but it increases pressure drop signiﬁcantly. As mentioned previously in (Guo et al., 2002; Tang et al., 2013, 2014), the time-averaged heat transfer coefﬁcient of helical coils with oscillatory ﬂow can be expressed in following form: Nuosc = aVab Cf = dVae

De c 10

De g 10

Pr 0.3

(14)

(15)

Using the multiple regression analysis of Origin 8.0 (OriginLab, 2014), the coefﬁcient in Eqs. (14) and (15) can be got. Table 5 lists the results of those two methods. It can be seen that using the uniform design method can obtain the similar correlation to full factorial design (the error less than 5%). Uniform design method can be used in heat exchanger, which can greatly reduce the number of experiments and get accurate results. When the experimental conditions are difﬁcult and multiple factors inﬂuence the ﬂow and

4.1.2. Some possible applications As mentioned in Section 4.2, the uniform design method is a powerful method for heat transfer problems inﬂuenced by multi factors, which can reduce the times of simulations or experiments. When considering the inﬂuence of pitches (H) and loops (m) in the helically coiled tube heat-exchanger, the characteristic parameter will increase to 4. If one characteristic parameter chooses 8 levels test, the times of simulation will be 84 for full factorial design. However, the uniform design method only just needs 8 times to obtain the correlation by using U8 (84 ) table to design. For other multi-factors heat transfer problems, such as multistructural parameters and multi-operating parameters problems, the uniform design method is a good choice. 4.2. Field synergy principle For a deep understanding of the enhancement heat transfer in oscillating ﬂow in helically coiled tube heat-exchanger, Fig. 6 shows the temperature contours and the velocity at a select plane (x–y plane, the 90◦ plane along the length coil), which contains the changing of temperature and velocity in a half cycle. At the initial moment of simulation (t = 0.01 s), the velocity is too small to reveal the effect of centrifugal force. The effect of centrifugal force becomes signiﬁcant when the velocity increases, and the core region of ﬂow ﬁeld move to the outside. That increases the gradient of outside temperature and decreases that of the inside. So at this moment, the enhancement of heat transfer reﬂected in the outside of helically coiled tube. At the end of the half cycle, the secondary ﬂow ﬁeld is build and a circulatory motion is produced, which is

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C. Pan et al. / Computers and Chemical Engineering 69 (2014) 59–65

Fig. 6. The temperature contours and z-velocity line at x–y plane at various times (Demax = 1492.773 and f = 5 Hz, the 90◦ plane along the length coil).

The best heat transfer occurs in around 0.03 s, and then the heat transfer reduces continuously. The time average ﬁeld synergy angle is 86.84◦ . We also calculate the ﬁeld synergy angle in steady ﬂow by using FLUENT 6.3, which is 88.97◦ . Therefore, the heat transfer of the oscillatory ﬂow in helically coiled tube is stronger than steady ﬂow. 5. Conclusion

Fig. 7. The volume average ﬁeld synergy angle in the half cycle.

also the heat transfer enhancement mechanism of steady curved tube (Dean, 1927). The ﬁeld synergy principle, which has been successfully used to explain the heat transfer enhancement in the tubes ﬁtted with helical screw-tape inserts and curved square channel (Guo et al., 2010, 2011) is employed to analysis the helically coiled tube. The ﬁeld synergy principle is that the convective heat transfer rate depends both on the magnitudes and the synergy of the temperature gradient and the velocity vector at the same conditions (Guo et al., 1998, 2005). The intersection level of the synergy between the temperature gradient and the velocity vector can be expressed as (OriginLab, 2014; Guo et al., 2010)

= arccos

· ∇T U ∇T | |U||

In the present work, the heat transfer and pressure drop of oscillating ﬂow in helically coiled tube heat-exchanger have been investigated by using the commercial CFD code Fluent. There are two factors, frequency and velocity, inﬂuencing the ﬂow in oscillating helically coiled tube heat-exchanger. The full factorial design and the uniform design method have been used to obtain the correlations respectively. The correlation for the calculations of average heat transfer coefﬁcients and average pressure drop were proposed, shown in Table 5. The results have veriﬁed the feasibility of the uniform design method for multi-factors heat transfer problems, which can reduce the times of experiment effectively. The ﬁeld synergy principle was used to explain the heat transfer enhancement in oscillating ﬂow in helically coiled tube heatexchanger. The result shows that the volume average ﬁeld synergy angle is less 2.45% in the oscillating ﬂow, which is 86.84 in oscillating ﬂow and 88.97 in the steady ﬂow. So the volume average ﬁeld synergy angle can reﬂect the level of heat transfer in oscillatory ﬂow. The smaller angle means the better heat transfer. Acknowledgements This work was supported by the National Natural Science Foundation of China (Foundation Nos. 51276188 and 51176198), the project of State Grid (KJ-2012-627).

(16)

Fig. 7 shows the volume average ﬁeld synergy angle in the half cycle, Demax = 1492.773 and f = 5 Hz. The ﬁgure shows that at the beginning of one cycle the volume average ﬁeld synergy angle becomes small as the velocity increasing, which means the enhancement of heat transfer. Then the effect of centrifugal force becomes signiﬁcant. Enhancement of heat transfer at the outside and reduction of heat transfer at the inside of helically coiled tube make the overall heat transfer reduced at ﬁrst and then enhanced.

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