J. Cent. South Univ. (2012) 19: 471-476

DOI: 10.1007/s11771-012-1027-0

Fluid flow and heat transfer characteristics of spiral coiled tube: Effects of Reynolds number and curvature ratio

YOO Geun-jong, CHOI Hoon-ki, DONG Wa-ryong

School of Mechatronics, Changwon National University, Changwon 641-773, Korea

? Central South University Press and Springer-Verlag Berlin Heidelberg 2012

Abstract: Numerical analysis was performed to investigate flow and heat transfer characteristics in spiral coiled tube heat exchanger. Radius of curvature of the spiral coiled tube was gradually increased as total rotating angle reached 12π. As the varying radius of curvature became a dominant flow parameter, three-dimensional flow analysis was performed to this flow together with different Reynolds numbers while constant wall heat flux condition was set in thermal field. From the analysis, centrifugal force due to curvature effect is found to have significant role in behavior of pressure drop and heat transfer. The centrifugal force enhances pressure drop and heat transfer to have generally higher values in the spiral coiled tube than those in the straight tube. Even then, friction factor and Nusselt number are found to follow the proportionality with square root of the Dean number. Individual effect of flow parameters of Reynolds number and curvature ratio was investigated and effect of Reynolds number is found to be stronger than that of curvature effect.

Key words: spiral coiled tube heat exchanger; Reynolds number; curvature ratio; secondary flow; friction factor; Nusselt number

1 Introduction

One of special features of the coiled tube flow could be recognized as secondary flow which was formed with centrifugal force due to curvature effect. The secondary flow has tendency to increase mixing activity to enhance heat and mass transfer. The most important physical factors determining characteristics of the coiled tube flow are hydraulic radius of the coiled tube cross section (r_h), flow velocity and radius of curvature (p). DEAN [1] suggested non-dimensional variable, Dean number (κ), by combining Reynolds number (Re) and curvature ratio (δ_h(=r_hR)) as

In heat exchangers, pressure drop for fluid flow and coefficient of heat transfer are generally known as the most important design factors. These design factors are mainly affected by Re in heat exchangers composed with straight tubes. However, the curvature ratio is another important factor for performance of heat exchangers with the curved tubes and eventually κ is introduced to express its effect. Types of coiled tubes could be classified as helical and spiral ones along with different curvature characteristics. The helical coiled tube had constant radius of curvature while the spiral coiled tube had varying radius of curvature. Since the helical coiled tube has fixed curvature ratio, κ is determined by Re only. Similar to the straight pipe flow, flow in the helical coiled tube could be fully developed hydraulically and thermally with certain entrance length. Under constant heat flux boundary condition, mean temperature yields linearly varying distribution along downstream in this flow.

Most of research activities on the coiled tube  focused on the helical type. Early studies [2-4] were done for single helical coiled tube with thermal effect to find characteristics of flow and heat transfer. Recent studies [5-8] were extended to investigate heat exchangers which included the helical coiled tube in their cell to find the pressure drop and heat transfer characteristics. SHOKOUHMAND et al [9] further extended their study for the shell and coiled tube heat exchanger with varying mass flow rate. Throughout these previous studies, common conclusion was drawn that the friction factor representing the pressure drop and heat transfer Nu was proportionally varied with  consistently. Experimental investigation for two-phase flow inside the helical coiled heat exchanger was also carried out by SANTINI et al [10]. JAYAKUMAR et al [11] presented Nu correlated with κ, the constant heat flux Pr and the constant wall temperature conditions in the helical coiled tube with varying fluid properties with temperature. GHORBANI et al [12] performed an experimental investigation on the mixed convection heat transfer effect in a coil-in-shell heat exchanger.

For the spiral coiled tube, flow inside the spiral coiled tube continuously developed without reaching fully developed condition due to the varying centrifugal force. For this type of flow, κ still could be a measure to express flow and heat transfer characteristics. However, it is still with some difficulties to identify effects of Re or curvature ratio.

Flow and heat transfer studies on the spiral coiled tube [13-14] suggested overall heat transfer coefficient as a function of mass flow rate and curvature. NAPHON [15] also investigated flow and heat transfer characteristics in the horizontal spiral coil tube. The pressure drop per unit length and Nu obtained from the spiral coil tube were found to be 1.50 and 1.49 times higher than those for the straight tube, respectively. However, their heat transfer coefficient had limitation to have only local explanation for varying radius of curvature effect.

Eventually, more detailed investigation was required for the spiral coiled tube. For better understanding of flow and heat transfer characteristics, detailed analyses were performed to find velocity distribution, pressure drop and heat transfer with different Reynolds numbers and curvature ratios, and their physical significances were presented. Individual effects of Re and δ_h on the flow were also assessed.

2 Numerical analysis

2.1 Shape of spiral coiled tube

Schematic of the spiral coiled tube is shown in   Fig. 1 and its radius of curvature is increased according to Eq. (1). Value of coefficient a is set as 0.05 and total rotation angle θ is 12π. Also inlet (I) and outlet (O) radii of curvature are defined as R_I and R_O, respectively.

                                   (1)

To identify the effect of varying radii of curvature on flow characteristics clearly, fully developed flow was introduced at inlet of the spiral coiled tube. This flow condition was achieved by placing the helical coiled tube of R_I in front of the inlet of the spiral coiled tube. The same feature of R_O was also placed at the outlet of the spiral coiled tube to exclude downstream effect for numerical analysis.

Fig. 1 Schematic of spiral coiled tube

2.2 Governing equations

Due to geometric effect of the spiral coiled tube, the secondary flow was expected to have different flow characteristics. Critical Re for turbulent flow was now modified from constant value of 2.3×10³ to Eq. (2), as DRAVID et al [16] suggested to include curvature effect:

                 (2)

Once geometric specifications of the spiral coiled tube for current study were applied to Eq. (2), the critical R_e was obtained in the range of 5.3×10³cr<9.3×10³. Further application of the flow conditions for this work yielded Re as 200

1) Continuity equation

                                 (3)

2) Momentum equation

         (4)

3) Energy equation

                    (5)

2.3 Boundary conditions

Boundary conditions for flow field were composed with constant mass flow rate at inlet, Neumann condition at outlet, and no-slip conditions at the boundary walls. Constant wall heat flux of =1 000 W/m² was set as the boundary condition for thermal field. Fluid inside the coiled tube was water.

Fluid properties usually vary with temperature for low speed flow and Eq. (6) is representing functional expression of their variation. Values of coefficients in   Eq. (6) for different properties are listed in Table 1.

                   (6)

2.4 Numerical analysis

The governing equations were discretized with the second order upwind scheme. The SIMPLE algorithm [17] was adopted to connect momentum equation and continuity equation through pressure correction terms. The set of discretized difference equations were solved with Fluent V6.3.26 software [18].

Table 1 Coefficients of property function

3 Results and discussion

3.1 Characteristics of flow field

For identifying flow characteristics, upfront analysis was performed for cold flow with constant flow properties by excluding heat transfer effect. Inside the coiled tube (as shown in Fig.1), analysis shows that centrifugal force is applied to shift fluid from inner boundary (inner) to outer boundary (outer) along mid-plane in cross section. Magnitude of the centrifugal force is known to be proportional to the square of mean mainstream velocity and inversely proportional to the radius of curvature. To have force balance with the centrifugal force, pressure gradient is formed across tube cross section and resulting iso-pressure distribution is formed, as shown in Fig. 2.

Fig. 2 Contours of iso-pressure distribution in cross section

In any certain cross section, boundary layer has low velocity due to viscous effect and flow velocity in circumferential direction, and v_θ has the same characteristics. The low velocity yields small centrifugal force in the boundary layer compared to center of cross section. Meanwhile, pressure gradient is formed from outside to center of curvature, as shown in Fig. 2. Therefore, small centrifugal force and high pressure eventually push fluid from outer to inner to satisfy mass conservation. Along the mid-plane, however, flow velocity has usually high value to have large centrifugal force, and this yields fluid moving to outer boundary from inner side. As a result, the secondary flow is formed in the cross section to have two-symmetric circulating cells as shown in Fig. 3. Since the secondary flow is formed mainly due to the centrifugal force, its strength is also determined following the same proportionality with the centrifugal force, i.e., square of the flow velocity (or R_e) and inverse of the radius of curvature. The spiral coiled tube has increased radius of curvature along downstream, thus the secondary flow strength is reduced as it goes to downstream and this trend is also shown in Fig. 3.

Fig. 3 Velocity vectors of secondary flow for Re=1 000:     (a) θ=3π; (b) θ=9π

In the straight tube, the maximum velocity location appears at the center line. However, it is shifted to outer region in the coiled tube due to extra body force as considered. Revised velocity distribution increases velocity gradient and pressure gradient in the outer region to have the secondary flow. The secondary flow increases fluid transfer across cross section, which eventually enhances energy transfer. Due to this nature, it is important to evaluate the effect of the secondary flow. This can be done with momentum shifting parameter,  M_x [19]:

                        (7)

where M_x is defined as degree of momentum shifting using momentum and moment of momentum balance. Figure 4 shows geometric relationship with some definitions. Parameter ?A_i represents infinitesimal area; x_i is x-direction distance of infinitesimal section from center of cross section; r_i is radius of cross section.

Fig. 4 Geometry for definition of momentum shifting

Since momentum shifting is resulted from the secondary flow, the momentum shifting parameter, M_x  depends on Re and the radius of curvature directly.  Figure 5 presents variation of M_x along downstream (increasing radius of curvature) for different Re. As it goes to downstream from inlet (θ=0) to outlet (θ=12π) of the spiral coiled tube, the radius of curvature is increased to have smaller centrifugal force yielding reduction of M_x. This trend is consistent even for different values of Re. On the other hand, increasing Re increases the centrifugal force to have higher value of M_x at the fixed downstream location. The effect of Re appears more apparent than curvature effect since the centrifugal force is proportional to square of the flow velocity while it is inversely proportional to the radius of curvature.

The spiral coiled tube is widely used in heat exchangers, and the most important design parameters for the heat exchangers are pressure drop and heat transfer capacity. To find the pressure drop effect in the coiled tube, common non-dimensional friction factor is adopted as

                             (8)

where ?p is pressure difference between inlet and outlet, d_h is hydraulic diameter of cross section, L is tube length from inlet (I) to outlet (O), and  is cross-sectional mean flow velocity. The pressure drop in the tube flow is affected by surface friction. In the straight tube, its slope is constant and inversely proportional to Re for laminar fully developed flow expressed as

                                    (9)

Fig. 5 Variation of momentum shifting parameter with streamwise velocity along spiral coiled tube from inlet to outlet

The coiled tube has even thinner boundary layer due to momentum shifting, which results in higher pressure drop compared to the straight tube. To find its trend, ratio of friction factors of the coiled tube to the straight tube, f_c/f_s, is summarized in Fig. 6. In this figure, f_c/f_s varies along downstream from inlet (I) to outlet (O) with different Reynolds numbers while horizontal axis represents square root of the Dean number () including the curvature effect. All the values of f_c/f_s are larger than one, showing that friction factor of the coiled tube is larger than that of the straight tube. These values also increase along with the increase in Re and κ, indicating the increase of similar trend in flow velocity and curvature ratio. For the same Re, f_c/f_s is increased with the Dean number increasing. Dean number is increased with the increase of curvature ratio, which means the decrease of the radius of curvature. Therefore, inverse proportionality of the radius of curvature on flow characteristics is consistently kept. For the same Dean number, f_c/f_s is increased as Re increases and its increasing trend is higher compared to that of constant Re case. Increasing trend of f_c/f_s, therefore, shows rather strong dependency on Re than curvature ratio due to the difference in proportional order of two parameters. To compare the characteristics of friction factor variations for the spiral coiled tube and helical coiled tube, the values of f_c/f_s are extracted at mid-plane between inlet (I) and outlet (O) to have fixed radius of curvature, as shown in Fig. 6. The line reveals linear proportionality with square root of the Dean number() for the case with the same radius of curvature and shows the same tendency of friction factor variation with the helical coiled tube flow as reported in various studies [8].

Fig. 6 Comparison of friction factor ratios for different Re and curvature ratios

3.2 Characteristics of thermal field

Once flow is fully developed in circular straight tube, the mean fluid temperature determined by Eq. (10) has linear distribution along downstream direction (L_s):

                         (10)

The mean surface temperature also retains the same gradient with that of the mean fluid temperature but has constant difference in magnitude as

                                (11)

With constant heat flux boundary condition, mean fluid temperature in the spiral coiled tube and mean surface temperature are presented along downstream direction in Fig. 7.

In this figure, mean temperatures of fluid and surface show same linear variation trends with those for the straight tube. The difference of temperatures, (T_s-T_m), is also maintained as nearly constant. This indicates temperature distribution in the spiral coiled tube has common characteristics with the straight tube.

Heat transfer performance of heat exchanger is often explained with mean Nu. Figure 8 shows mean Nu distribution in the spiral coiled tube,after normalizing with mean Nu in the straight tube,   for different mass flow rates and κ. It is well known that  has constant value of 4.36 in the straight tube under constant heat flux boundary condition once flow is fully developed. From Fig. 8,  is consistently found to have higher value compared to . This can be explained with existence of the secondary flow in the spiral coiled tube, which increases mixing effect enhancing heat transfer. The value of  is even larger for higher mass flow rate, which confirms the effect of the secondary flow since mixing effect is increased with higher momentum. With fixed mass flow rate (fixed Re), ratio of Nusselt numbers,  has higher value in inlet region (I) of high curvature ratio than in outlet region (O) of low curvature ratio. This indicates that curvature ratio is another factor to determine mixing effect. Figure 8 also reveals rapidly increasing rate of  for higher mass flow rate as square root of the Dean number () increases. This can be explained with mixed nature of  with Re, curvature ratio and difference in their varying order. Similar to ratio of friction factors, ratio of Nu is again extracted at mid-plane between inlet (I) and outlet (O) to have fixed radius of curvature, as shown by midline line in Fig. 8.

Fig. 7 Mean fluid and surface temperature distributions along downstream direction (=0.872 kg/s)

Fig. 8 Comparison of Nu ratio

The line has linear distribution with square root of the Dean number (κ^1/2), which indicates proportionality with Re or the flow velocity. This indicates that  for the spiral coiled tube is found to have the same proportionality characteristics with the helical coiled tube as reported in the literature.

4 Conclusions

1) The cross sectional velocity distributions of the main flow and the secondary flow show similar trends for both of the spiral and helical coiled tubes.

2) The friction factor and Nu follow similar proportions with Re and square root of the Dean number (κ^1/2) in both of the coiled tubes.

3) For both design parameters of the friction factor and Nu, Re is found to have stronger effect compared to the curvature ratio.

Nomenclature

References

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(Edited DENG Lü-xiang)

Foundation item: Work supported by the Second Stage of Brain Korea 21 Projects, Korea

Received date: 2011-05-24; Accepted date: 2011-10-10

Corresponding author: CHOI Hoon-ki, Associate professor, PhD; Tel: +82-55-213-3628; E-mail: hkchoi@changwon.ac.kr