**1. Introduction**

It has been widely reported in literature that heat transfer rates in helical coils are higher as compared to those in straight tubes. Due to the compact structure and high heat transfer coefficient, helical coil heat exchangers find extensive use in industrial applications such as power generation, nuclear industry, process plants, heat recovery systems, refrigeration, food industry, etc. (Abdulla 1994; Bai et al. 1999; Futagami and Aoyama 1988; Jensen and Bergles 1981; Patankar et al. 1974; Xin et al., 1996). Heat exchanger with helical coils is used for residual heat removal systems in islanded or barge mounted nuclear reactor systems, wherein nuclear energy is utilised for desalination of seawater (Manna et al., 1998). The performance of the residual heat removal system, which uses a helically coiled heat exchanger, for various process parameters was investigated by Jayakumar and Grover (1997). The work had been extended to find out the stability of operation of such a system when the barge on which it is mounted is moving (Jayakumar, 1999; Jayakumar et al., 2002).

### **1.1 Terminology of helically coiled pipes**

Fig. 1 gives the schematic of the helical coil. The pipe has an inner diameter *2r*. The coil diameter is represented by *2RC* (measured between the centres of the pipes). The distance between two adjacent turns, called pitch is *H*. The coil diameter is also called as pitch circle diameter (*PCD*). The ratio of pipe diameter to coil diameter (*r/Rc*) is called curvature ratio, . The ratio of pitch to developed length of one turn (*H/2 Rc*) is termed non-dimensional pitch, . Consider the projection of the coil on a plane passing through the axis of the coil. The angle, which projection of one turn of the coil makes with a plane perpendicular to the axis, is called the helix angle, . Consider any cross section of the pipe created by a plane passing through the coil axis. The side of pipe wall nearest to the coil axis is termed inner side of the coil and the farthest side is termed as outer side of the coil. Similar to Reynolds number for flow in pipes, Dean number is used to characterise the flow in a helical pipe.

#### **1.2 Review of single-phase flow and heat transfer**

Heat transfer and flow through a curved tube is comprehensively first reviewed by Berger et al. (1983) and subsequently by Shah and Joshi (1987). The latest review of flow and heat transfer characteristics is provided by Naphon & Wongwises (2006). The characteristics of

Helically Coiled Heat Exchangers 313

between turbulence emergence and curvature effects, the same Reynolds number flow may present an equal or even a lower hydraulic resistance in a curved channel than it does in a straight one. Apparently, the reducing effect of curvature on friction, due to the smoothing of turbulence emergence, equals, or even overcomes, the increasing effect due to the secondary flow. But in practical applications, due to layout and economic considerations, the value of torsion is never reaches an effect of destabilization of flow and hence reduction

Another important phenomena observed in helical tubes is the relamianrization. The fluid flow, which was originally turbulent, changes to laminar while flowing inside a helical pipe. This has been experimentally demonstrated by Sreenivasan and Strykowski (1983). The experiment was conducted using a pipe of diameter 19.1 mm wound to form a coil of 90 mm. In the experiment, dye streak introduced at two locations, viz., into the straight section upstream of the coil and into the fourth turn of the coil. It has been observed that the dye introduced in the straight section diffuses rapidly, indicating that the flow there is turbulent. While the dye injected into the fourth turn remains perfectly unruffled for a long distance,

It has been seen that, in helical pipes transition from laminar to turbulent flow regime takes place at a Reynolds number higher than that for a similar straight pipe. Correlations were proposed by Ito (1959), Schmidt (1967), Srinivasan (1970) et al., Janssen et al,. (1978) etc. Critical Reynolds number obtained from the above correlations for a range of curvature ratio from 0.01 to 0.25 is plotted in fig. 2. In the lower range of curvature ratios (*δ*<0.05), all of the correlations provide approximately the same value for the *Recr*. Correlations provided by Ito et al and Schmidt et al. gives almost equal values of *Recr* for the entire range of curvature ratios which is of practical interest and these correlations are used in the present

It has already been seen that the flow phenomena in curved tubes are much more complex than that in a straight tube. The pressure drop occurring in a helical tube is found to be

Correlations for estimation of pressure drop was proposed by Ito (1959), Srinivasan et al. (1968), Tarbell & Samuels (1973), Ruffel (1974), Xin et al. (1997), Ju et al. (2001), Guo et al. (2001) etc. Ali (2001) and Naphon, & Wongwises (2006) has consolidated correlations for

Heat transfer in helical coils has been experimentally investigated by Seban & McLaughlin (1963) both for laminar and turbulent flow regimes for flow of water with constant wall flux BC. Roger & Mayhew (1964) studied heat transfer to fluid flowing inside a helical pipe which was heated by steam. Mori and Nakayama (1967a) investigated forced convective heat transfer in turbulent regime for wall heat flux boundary condition. Variation of physical properties with temperature changes were not taken into account in their work.

in value of critical Reynolds number.

**1.2.2 Critical Reynolds number** 

work for determination of flow regime.

**1.2.3 Pressure drop in single-phase flow** 

**1.2.4 Heat transfer in single-phase flow** 

higher than that for straight tubes for the same flow rate.

estimation of pressure drop for flow through helical pipes.

indicating the laminar state of the flow in the helical coil.

Fig. 1. Basic geometry of a helical pipe.

flow, pressure drop and heat transfer have been reported by many investigators. The heat transfer enhancement in helical coil systems is reported by Prabhanjan et al. (2004), Berger et al. (1983), Janssen & Hoogendoorn (1978) and Ruthven (1971). Condensing heat transfer and pressure drop of refrigerant R 134A in helicoidal (helical double pipe heat exchanger) is experimentally investigated by Kang et al. (2000). The effect of torsion on the flow in a helical tube of circular cross-section is experimentally investigated by Yamamoto et al. (1995) for a range of Reynolds numbers from about 500 to 20000. Study of fluid flow through curved tubes are of interest to the medical community since many arteries are curved (Zabielski, & Mestel, 1998a; Zabielski, & Mestel, 1998b).

#### **1.2.1 Laminar-turbulent transition**

The curved shape of the tube causes the flowing fluid to experience centrifugal force. The extent of centrifugal force experienced depends on the local axial velocity of the fluid particle and the radius of curvature of the coil. The fluid particles flowing at the core of the pipe have higher velocities than those flowing near to the pipe wall. Thus the fluid particles flowing close to the tube wall experience a lower centrifugal force than the fluid particles flowing in the tube core. This causes the fluid from the core region to be pushed towards the outer wall (away from the coil axis). This stream bifurcates at the wall and drives the fluid towards the inner wall along the tube periphery, causing generation of counter-rotating vortices called secondary flows. The secondary flows produce additional transport of the fluid over the cross section of the pipe. This additional convective transport increases both the heat transfer and the pressure drop when compared to that in a straight tube.

It has been found that the effect of coil curvature is to suppress turbulent fluctuations arising in the flowing fluid and smoothing the emergence of turbulence. Thus it increases the value of the Reynolds number required to attain a fully turbulent flow, as compared to that of a straight pipe. The above effect of turbulent fluctuations suppression enhances as the curvature ratio increases. Torsion, on the other hand, is found to destabilize the flow, reducing the Reynolds number at which turbulence emerges. It may impart a Reynolds number for transition to turbulent, close to or even lower than the ones characteristic of straight pipe flow. The above destabilizing effect first increases, as torsion increases, reaches a maximum and then decreases with further increase in torsion. Due to the interaction

flow, pressure drop and heat transfer have been reported by many investigators. The heat transfer enhancement in helical coil systems is reported by Prabhanjan et al. (2004), Berger et al. (1983), Janssen & Hoogendoorn (1978) and Ruthven (1971). Condensing heat transfer and pressure drop of refrigerant R 134A in helicoidal (helical double pipe heat exchanger) is experimentally investigated by Kang et al. (2000). The effect of torsion on the flow in a helical tube of circular cross-section is experimentally investigated by Yamamoto et al. (1995) for a range of Reynolds numbers from about 500 to 20000. Study of fluid flow through curved tubes are of interest to the medical community since many arteries are curved

The curved shape of the tube causes the flowing fluid to experience centrifugal force. The extent of centrifugal force experienced depends on the local axial velocity of the fluid particle and the radius of curvature of the coil. The fluid particles flowing at the core of the pipe have higher velocities than those flowing near to the pipe wall. Thus the fluid particles flowing close to the tube wall experience a lower centrifugal force than the fluid particles flowing in the tube core. This causes the fluid from the core region to be pushed towards the outer wall (away from the coil axis). This stream bifurcates at the wall and drives the fluid towards the inner wall along the tube periphery, causing generation of counter-rotating vortices called secondary flows. The secondary flows produce additional transport of the fluid over the cross section of the pipe. This additional convective transport increases both

It has been found that the effect of coil curvature is to suppress turbulent fluctuations arising in the flowing fluid and smoothing the emergence of turbulence. Thus it increases the value of the Reynolds number required to attain a fully turbulent flow, as compared to that of a straight pipe. The above effect of turbulent fluctuations suppression enhances as the curvature ratio increases. Torsion, on the other hand, is found to destabilize the flow, reducing the Reynolds number at which turbulence emerges. It may impart a Reynolds number for transition to turbulent, close to or even lower than the ones characteristic of straight pipe flow. The above destabilizing effect first increases, as torsion increases, reaches a maximum and then decreases with further increase in torsion. Due to the interaction

the heat transfer and the pressure drop when compared to that in a straight tube.

Fig. 1. Basic geometry of a helical pipe.

**1.2.1 Laminar-turbulent transition** 

(Zabielski, & Mestel, 1998a; Zabielski, & Mestel, 1998b).

between turbulence emergence and curvature effects, the same Reynolds number flow may present an equal or even a lower hydraulic resistance in a curved channel than it does in a straight one. Apparently, the reducing effect of curvature on friction, due to the smoothing of turbulence emergence, equals, or even overcomes, the increasing effect due to the secondary flow. But in practical applications, due to layout and economic considerations, the value of torsion is never reaches an effect of destabilization of flow and hence reduction in value of critical Reynolds number.

Another important phenomena observed in helical tubes is the relamianrization. The fluid flow, which was originally turbulent, changes to laminar while flowing inside a helical pipe. This has been experimentally demonstrated by Sreenivasan and Strykowski (1983). The experiment was conducted using a pipe of diameter 19.1 mm wound to form a coil of 90 mm. In the experiment, dye streak introduced at two locations, viz., into the straight section upstream of the coil and into the fourth turn of the coil. It has been observed that the dye introduced in the straight section diffuses rapidly, indicating that the flow there is turbulent. While the dye injected into the fourth turn remains perfectly unruffled for a long distance, indicating the laminar state of the flow in the helical coil.

#### **1.2.2 Critical Reynolds number**

It has been seen that, in helical pipes transition from laminar to turbulent flow regime takes place at a Reynolds number higher than that for a similar straight pipe. Correlations were proposed by Ito (1959), Schmidt (1967), Srinivasan (1970) et al., Janssen et al,. (1978) etc. Critical Reynolds number obtained from the above correlations for a range of curvature ratio from 0.01 to 0.25 is plotted in fig. 2. In the lower range of curvature ratios (*δ*<0.05), all of the correlations provide approximately the same value for the *Recr*. Correlations provided by Ito et al and Schmidt et al. gives almost equal values of *Recr* for the entire range of curvature ratios which is of practical interest and these correlations are used in the present work for determination of flow regime.

#### **1.2.3 Pressure drop in single-phase flow**

It has already been seen that the flow phenomena in curved tubes are much more complex than that in a straight tube. The pressure drop occurring in a helical tube is found to be higher than that for straight tubes for the same flow rate.

Correlations for estimation of pressure drop was proposed by Ito (1959), Srinivasan et al. (1968), Tarbell & Samuels (1973), Ruffel (1974), Xin et al. (1997), Ju et al. (2001), Guo et al. (2001) etc. Ali (2001) and Naphon, & Wongwises (2006) has consolidated correlations for estimation of pressure drop for flow through helical pipes.

#### **1.2.4 Heat transfer in single-phase flow**

Heat transfer in helical coils has been experimentally investigated by Seban & McLaughlin (1963) both for laminar and turbulent flow regimes for flow of water with constant wall flux BC. Roger & Mayhew (1964) studied heat transfer to fluid flowing inside a helical pipe which was heated by steam. Mori and Nakayama (1967a) investigated forced convective heat transfer in turbulent regime for wall heat flux boundary condition. Variation of physical properties with temperature changes were not taken into account in their work.

Helically Coiled Heat Exchangers 315

drop and void fraction measurement for two-phase counter current flow of gas and liquid in a helical coil. They compared the results with Lockhart-Martinelli correlation, Dukler's correlation and Hughmark's correlation and suggested that Lockhart-Martinelli parameter could be modified to obtain a better correlation. In their later work, Stepanek & Kasturi (1972) proposed correlations for void fraction and pressure drop in terms of new correlating parameters. Flow of air-water mixture through a helically coiled tube was studied by Whalley (1980) and the flow pattern transition between stratified and annular flow was examined. Rangacharyulu and Davies (1984) experimentally studied pressure drop and hold-up for counter-current upward flow of air-liquid system through copper coils. They proposed a new correlation for two-phase frictional pressure drop based on the modified Lockhart-Martinelli parameter. Flow of two-phase air-water mixture in helically coiled tube was studied by Watanabe et al. (1993). They found out the thickness of water film on the

Czop et al. (1994) carried out experiments on water-SF6 flow through a helically coiled tube of 19.8 mm id with 1170 mm coil diameter. It has been observed that the two-phase pressure drops are very much different from those calculated with Lockhart-Martinelli correlation but are in fairly good agreement with the Chisholm correlation. Awwad et al. (1995) carried out experimental investigations of air-water two-phase flow in horizontal helicoidal pipes. They have found that the pressure drop multiplier is strongly related to superficial velocities of air and water. The helix angle has almost no effect on pressure drop, even though coil diameter has certain effects at low flow rates. Xin et al. (1996) measured the pressure drop and void fraction for an air-water mixture flowing through vertical helicoidal pipes. In their later work, Xin et al. (1997) investigated the effect of coil geometries and flow rates of air and water on two-phase flow pressure drop in annular vertical and horizontal helical pipes. It has been observed that unlike two-phase flow through straight pipes, the pressure drop multipliers for helical pipes are dependent on the flow rates in addition to the Martinelli

Experimental investigations of oil-water-air three phase flows were carried out by Chen & Guo (1999) with an objective to separate gas-oil-water mixture. Murai et al. (2006) have experimentally studied the nature of flow patterns for flow of air-water mixture in a helically coiled tube. They established the effect of centrifugal acceleration on the flow regime map and brought out the spatial and temporal flow structure distribution. Jayakumar et al. (2010b) has reported numerical investigation of heat transfer to two-phase air-water mixture flowing through helical pipes. In that work, the variation of phasic velocity, temperature and void fraction at various cross-sections along the length of tube are presented. Influence of the coil parameters and inlet void fraction in heat transfer is also

The chapter is organised as follows: Detailed characteristics and physics of fluid flow and heat transfer to single-phase water flowing through helical pipes are presented in next section. In the section 3, influences of various coil parameters on heat transfer for different boundary conditions are analysed. The results are used for generation of correlations to estimate the average and local values of Nusselt numbers. Nature of variation of Nusselt number at various positions along wall periphery is discussed in section 4. The generalised

results are converted into an equation for estimation of local Nusselt number.

wall of the coil at different points around the circumference experimentally.

parameter.

discussed in that paper.

**1.4 Outline of the chapter** 

Fig. 2. Critical Reynolds number predicted by various correlations.

Mori and Nakayama (1967b) subsequently studied heat transfer under constant wall temperature boundary condition for the same helical coils. They had observed that the Nusselt number is remarkably affected by a secondary flow due to curvature. They had stated that the same formula used for estimation of heat transfer rates in wall flux boundary conditions can be used for the wall temperature boundary condition as well. Heat transfer and pressure drop in helical pipes was studied by Yildiz et al. (1997).

CFD study of helically coiled double pipe heat exchangers for laminar flow situations were carried out by Rennie and Raghavan (2005, 2006a). They have modelled the heat transfer from hot fluid to cold fluid using the CFD package PHOENICS 3.3 and found out the overall heat transfer coefficients for counter current and parallel flows. Pressure drop and heat transfer in tube-in-tube helical heat exchanger under turbulent flow conditions was studied by Vimal Kumar et al. (2006) using the CFD package FLUENT 6. However, no correlation for estimation of Nu was given in these papers.

Goering et al. (1997) has studied fully developed laminar convective heat transfer in curved pipes to investigate the dual influence of curvature and buoyancy. Direct numerical study on influence of curvature and torsion on turbulent flow in a helical pipe has been provided by Hüttl and Friedrich (2000). Later Hüttl and Friedrich (2001) have conducted a DNS study to bring out the details of the secondary flow in such systems. Recently Jayakumar et al. (2008a) have developed a correlation for estimation of inside heat transfer coefficient for flow of single-phase water through helically coiled heat exchangers. The correlation, which is validated against experiments, is applicable to a specific configuration of helical coil.

#### **1.3 Pressure drop and heat transfer for air-water two-phase flow**

Akagawa et al. (1971) measured pressure drop for two-phase gas liquid flow in helically coiled tubes for different curvature ratios. Kasturi & Stepanek (1972) carried out pressure

Mori and Nakayama (1967b) subsequently studied heat transfer under constant wall temperature boundary condition for the same helical coils. They had observed that the Nusselt number is remarkably affected by a secondary flow due to curvature. They had stated that the same formula used for estimation of heat transfer rates in wall flux boundary conditions can be used for the wall temperature boundary condition as well. Heat transfer

CFD study of helically coiled double pipe heat exchangers for laminar flow situations were carried out by Rennie and Raghavan (2005, 2006a). They have modelled the heat transfer from hot fluid to cold fluid using the CFD package PHOENICS 3.3 and found out the overall heat transfer coefficients for counter current and parallel flows. Pressure drop and heat transfer in tube-in-tube helical heat exchanger under turbulent flow conditions was studied by Vimal Kumar et al. (2006) using the CFD package FLUENT 6. However, no correlation

Goering et al. (1997) has studied fully developed laminar convective heat transfer in curved pipes to investigate the dual influence of curvature and buoyancy. Direct numerical study on influence of curvature and torsion on turbulent flow in a helical pipe has been provided by Hüttl and Friedrich (2000). Later Hüttl and Friedrich (2001) have conducted a DNS study to bring out the details of the secondary flow in such systems. Recently Jayakumar et al. (2008a) have developed a correlation for estimation of inside heat transfer coefficient for flow of single-phase water through helically coiled heat exchangers. The correlation, which is validated against experiments, is applicable to a specific configuration of helical coil.

Akagawa et al. (1971) measured pressure drop for two-phase gas liquid flow in helically coiled tubes for different curvature ratios. Kasturi & Stepanek (1972) carried out pressure

Fig. 2. Critical Reynolds number predicted by various correlations.

and pressure drop in helical pipes was studied by Yildiz et al. (1997).

**1.3 Pressure drop and heat transfer for air-water two-phase flow** 

for estimation of Nu was given in these papers.

drop and void fraction measurement for two-phase counter current flow of gas and liquid in a helical coil. They compared the results with Lockhart-Martinelli correlation, Dukler's correlation and Hughmark's correlation and suggested that Lockhart-Martinelli parameter could be modified to obtain a better correlation. In their later work, Stepanek & Kasturi (1972) proposed correlations for void fraction and pressure drop in terms of new correlating parameters. Flow of air-water mixture through a helically coiled tube was studied by Whalley (1980) and the flow pattern transition between stratified and annular flow was examined. Rangacharyulu and Davies (1984) experimentally studied pressure drop and hold-up for counter-current upward flow of air-liquid system through copper coils. They proposed a new correlation for two-phase frictional pressure drop based on the modified Lockhart-Martinelli parameter. Flow of two-phase air-water mixture in helically coiled tube was studied by Watanabe et al. (1993). They found out the thickness of water film on the wall of the coil at different points around the circumference experimentally.

Czop et al. (1994) carried out experiments on water-SF6 flow through a helically coiled tube of 19.8 mm id with 1170 mm coil diameter. It has been observed that the two-phase pressure drops are very much different from those calculated with Lockhart-Martinelli correlation but are in fairly good agreement with the Chisholm correlation. Awwad et al. (1995) carried out experimental investigations of air-water two-phase flow in horizontal helicoidal pipes. They have found that the pressure drop multiplier is strongly related to superficial velocities of air and water. The helix angle has almost no effect on pressure drop, even though coil diameter has certain effects at low flow rates. Xin et al. (1996) measured the pressure drop and void fraction for an air-water mixture flowing through vertical helicoidal pipes. In their later work, Xin et al. (1997) investigated the effect of coil geometries and flow rates of air and water on two-phase flow pressure drop in annular vertical and horizontal helical pipes. It has been observed that unlike two-phase flow through straight pipes, the pressure drop multipliers for helical pipes are dependent on the flow rates in addition to the Martinelli parameter.

Experimental investigations of oil-water-air three phase flows were carried out by Chen & Guo (1999) with an objective to separate gas-oil-water mixture. Murai et al. (2006) have experimentally studied the nature of flow patterns for flow of air-water mixture in a helically coiled tube. They established the effect of centrifugal acceleration on the flow regime map and brought out the spatial and temporal flow structure distribution. Jayakumar et al. (2010b) has reported numerical investigation of heat transfer to two-phase air-water mixture flowing through helical pipes. In that work, the variation of phasic velocity, temperature and void fraction at various cross-sections along the length of tube are presented. Influence of the coil parameters and inlet void fraction in heat transfer is also discussed in that paper.

#### **1.4 Outline of the chapter**

The chapter is organised as follows: Detailed characteristics and physics of fluid flow and heat transfer to single-phase water flowing through helical pipes are presented in next section. In the section 3, influences of various coil parameters on heat transfer for different boundary conditions are analysed. The results are used for generation of correlations to estimate the average and local values of Nusselt numbers. Nature of variation of Nusselt number at various positions along wall periphery is discussed in section 4. The generalised results are converted into an equation for estimation of local Nusselt number.

Helically Coiled Heat Exchangers 317

*b*

*loc*

heat flux in that cross-section. Hence, the mean Nusselt number is evaluated by;

*<sup>r</sup> <sup>q</sup> Nu*

As used by Lin and Ebadian (1997), average *Nu* at a cross section may be estimated by,

1 2 *Nu Nu d av* 

2

0

But this does not ensure that the Nusselt number so estimated is representative of the total

<sup>2</sup> *<sup>m</sup> av*

*<sup>r</sup> <sup>q</sup> Nu*

2

0 2

*m*

data extraction is available in Jayakumar (2009) and Jayakumar et al., (2010a).

constant wall heat flux boundary condition is discussed in the following sections.

flux at a given cross-section and is evaluated using eqns. 4 and 5.

0

*T*

cross section are calculated using the formula,

The heat flux is calculated by, "

Here, *Tw m*, and " *qm* are evaluated by the formula,

where,  *p*

*u C dA* 

Here *dA* is an elemental area of the pipe cross-section (see figure 5.1(b)). The wall temperatures at four locations (inner, outer, top and bottom of the pipe) in a cross section are also extracted. Using these data, values of local Nusselt number at four locations at that

" 2

*kT T* 

*w b*

 

"

,

*m wm b*

*kT T*

 *A d*

the wall to which the parameter is associated to. Thus the *Nuav* is based on the average heat

The above sets of operations are repeated at successive planes to cover the entire length of the pipe. All of the above processing have been done using Python scripts which runs on top of the AnuVi package. Various programs required to generate the cut planes etc was written in c++ programming language. MATLAB® has been extensively used for processing of the raw data, generation of 2D plots and for regression analysis. More details about the

The results of analysis carried out with constant wall temperature boundary condition and

*A d*

= *k, Tw or q"* as the case may be. Here *∆A* is the area of elemental ring located along

*<sup>w</sup> q kTn* , where *n* is the normal direction.

*u C TdA*

*p*

, (1)

. (3)

. (4)

. (5)

. (2)

Sections 5 deal with analysis of two-phase flows through helical pipe. Details of numerical modelling employing the two-fluid model and validation are given. Factors influencing two-phase heat transfer are analysed and a correlation to estimate the heat transfer coefficient is recommended.
