**2.1 Property variation of the working fluid**

Implication of using values of transport and thermal properties of the hot and cold fluids as functions of temperature is investigated (Jayakumar et. al 2008a). From the analysis, it can be seen that an error Nusselt number is about 24% when the properties at ambient conditions are used.

#### **2.2 Data extraction**

The results of simulation are exported as a *CGNS (CFD General Notation System,* www.cgns.org) file. The fields exported are pressure, temperature, velocity magnitude, *x*, *y* and *z* velocities, viscosity, density, specific heat, thermal conductivity of the fluid; wall temperature and wall heat flux.

For post-processing a visualisation package *AnuVi* developed by Computer Division, BARC, India is used. *AnuVi* is a cross-platform CFD post processor and Scientific Visualization Framework and is built on top of the open source software like *Python (*www.python.org*)*, *Visualization Tool Kit (*VTK, www.vtk.org)*, WxWidgets* (www.wxwidgets.org) and *FFmpeg (*www.ffmpeg.org*)*. It can handle many standard file formats like *CGNS, PLOT3D, VTK, STL, OBJ, BYU* and *PLY* and has features to provide animation, extraction and derivation of data over many data components with advanced graphics (including shading, contouring, lighting and transparency). The package has features like Session Handling, Seamless Data integration, *Python* Language Scripting etc. Rendering is handled by *OpenGL* and can be accelerated with advanced graphics hardware. The feature of *Python* language scripting gives unlimited control to user which can be used for automation of data extraction and visualization.

For extraction of data and visualisation, the CGNS files are processed to create planes at desired spacing in the computational domain. Since the fluid properties are temperature dependent, the bulk fluid temperature at a cross section is evaluated using the relation,

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

As a representative case, coil of *PCD* = 200 mm and coil pitch of 30 mm is considered for discussion. Diameter of the pipe used in the coil is 20 mm. Boundary layer mesh was generated for the pipe fluid volume. Optimised grid after the grid independency studies was used in the analysis. Pressure velocity coupling was done using the SIMPLEC scheme. Momentum equations were discretised using QUICK scheme. Power Law scheme of discretisation is used for turbulent kinetic energy and dissipation rate equations. Convergence criterion used was 1.0e-5 for continuity, velocities, *k*, and *ε*. Temperature dependent properties as polynomial functions were used for water. For the energy equation third order QUICK discretisation scheme was employed. Convergence criterion for energy

Implication of using values of transport and thermal properties of the hot and cold fluids as functions of temperature is investigated (Jayakumar et. al 2008a). From the analysis, it can be seen that an error Nusselt number is about 24% when the properties at ambient

The results of simulation are exported as a *CGNS (CFD General Notation System,* www.cgns.org) file. The fields exported are pressure, temperature, velocity magnitude, *x*, *y* and *z* velocities, viscosity, density, specific heat, thermal conductivity of the fluid; wall

For post-processing a visualisation package *AnuVi* developed by Computer Division, BARC, India is used. *AnuVi* is a cross-platform CFD post processor and Scientific Visualization Framework and is built on top of the open source software like *Python (*www.python.org*)*, *Visualization Tool Kit (*VTK, www.vtk.org)*, WxWidgets* (www.wxwidgets.org) and *FFmpeg (*www.ffmpeg.org*)*. It can handle many standard file formats like *CGNS, PLOT3D, VTK, STL, OBJ, BYU* and *PLY* and has features to provide animation, extraction and derivation of data over many data components with advanced graphics (including shading, contouring, lighting and transparency). The package has features like Session Handling, Seamless Data integration, *Python* Language Scripting etc. Rendering is handled by *OpenGL* and can be accelerated with advanced graphics hardware. The feature of *Python* language scripting gives unlimited control to user which can be used for automation of data extraction and

For extraction of data and visualisation, the CGNS files are processed to create planes at desired spacing in the computational domain. Since the fluid properties are temperature dependent, the bulk fluid temperature at a cross section is evaluated using the relation,

**2. Heat transfer characteristics of single-phase flows** 

coefficient is recommended.

balance was 1.0e-07.

conditions are used.

**2.2 Data extraction** 

visualization.

temperature and wall heat flux.

**2.1 Property variation of the working fluid** 

$$T\_b = \frac{\int \mu \rho \mathbf{C}\_p T dA}{\int \mu \rho \mathbf{C}\_p dA},\tag{1}$$

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 cross section are calculated using the formula,

$$Nu\_{loc} = \frac{2r}{k} \left(\frac{q^\*}{T\_w - T\_b}\right). \tag{2}$$

The heat flux is calculated by, " *<sup>w</sup> q kTn* , where *n* is the normal direction.

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

$$N\mu\_{av} = \frac{1}{2\pi} \int\_0^{2\pi} \left( N\mu\_{\phi} \right) d\phi \,. \tag{3}$$

But this does not ensure that the Nusselt number so estimated is representative of the total heat flux in that cross-section. Hence, the mean Nusselt number is evaluated by;

$$N\mu\_{av} = \frac{2r}{k\_m} \left(\frac{q\_m}{T\_{av,m} - T\_b}\right). \tag{4}$$

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

$$\rho\_m = \frac{\int\_0^{2\pi} \left(\rho \Delta A\right) d\phi}{\int\_0^0 \left(\Delta A\right) d\phi}.\tag{5}$$

where, = *k, Tw or q"* as the case may be. Here *∆A* is the area of elemental ring located along the wall to which the parameter is associated to. Thus the *Nuav* is based on the average heat flux at a given cross-section and is evaluated using eqns. 4 and 5.

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

The results of analysis carried out with constant wall temperature boundary condition and constant wall heat flux boundary condition is discussed in the following sections.

Helically Coiled Heat Exchangers 319

Fig. 3. Velocity (m s-1) contours at various planes along the length of the coil.

Fig. 4. Temperature (K) contours at various planes along the length of the coil.

The Nusselt numbers at the top and bottom side of the cross sections show prominent periodic behaviour in the developing region of the pipe. Oscillatory motion of fluid particles, as observed in fig. 5, influences heat transfer around the periphery. The fluctuational behaviour of the Nusselt number has been reported by other investigators as well (Lin and Ebadian, 1999; Liu, 1992; Patankar et al., 1974). In the later regions of the coil, the Nusselt number at the top and bottom differ only marginally. Figure 6 also shows average values of Nusselt number (eqn. (4)) along the length of the pipe. It is found that it

of the coil is the lowest.

leads to higher heat transfer coefficients. In a similar way, the Nu along the inner periphery

## **2.3 Analysis with constant wall temperature boundary condition**

In this analysis, hot water at 330 K at a specified velocity of 0.8 ms-1 is entering the helical pipe at the top, where an inlet velocity boundary condition is specified. The flow velocity is such that the flow regime is turbulent. The fluid is made to cool down as it flows along the tube by specifying a wall temperature of 300 K. Temperature dependent values of fluid properties are used in this analysis. At the pipe wall, for the energy equation, a Dirichlet boundary condition and for momentum and pressure equations homogenous Neumann boundary condition are specified. At the outlet, a pressure outlet boundary is enforced.

Fig. 3 shows an overview of velocity contours at various sections along the length of the coil. The planes are identified by the angle (*θ*) which is the angle that the plane makes with the plane passing through the pipe inlet. In fig. 5 the first plane shown on the top is at 10o from the inlet (i.e., *θ*=10o) and the subsequent planes are 10o apart. Up to an angle of *θ*=35o, the velocity profile at a cross section is found to be symmetric. Subsequently, this uniform velocity pattern changes to a pattern with a high velocity region located at the outer side of the coil. This behaviour is seen predominantly by *θ*=45o and continues to develop. It can be seen that by *θ*=135o, the high velocity region is present only in outer half cross-section. Area of high velocity region further reduces as the flow gets developed and covers approximately 1 3 rd of the flow area by *θ*=240o. No significant change in flow pattern is observed downstream.

Temperature distribution at various planes along the length of coil is shown in fig. 4. At the inlet, temperature is uniform across the cross section. Since the wall is maintained at a lower temperature, the fluid cools down as it flows through the coil. Up to an angle of 20o, heat transfer is uniform along the periphery. In contrast to heat transfer in a straight tube, high temperature regions are seen on the outer side of the coil. This phenomena is predominant from the plane at angle *θ*=50o. This trend continues to develop and by 150o, clearly three regions viz., high temperature (327-330 K) at the outer side of the coil, intermediate temperature (321 to 324 K) at the centre and low temperature (311 to 314 K) on the inner side of the coil, are visible. As the fluid flows down the pipe, this temperature profile gets developed and the area of high temperature region decreases and by *θ* = 360o, a fully developed temperature profile is attained and the fluid continues to lose heat due to the lower wall temperature.

As the fluid flows through the helical coil, fluid particles undergo rotational motion. The fluid particles also undergo movement from inner side of the coil to the outer side and viceversa. Fig. 5 shows particle trace for 10 fluid particles which are located along a line parallel to the X axis at the pipe inlet. It can be noted that these fluid particles are taking various trajectories and also move with different velocities. The particles, which were forming a line to begin with, are found to be totally scattered at the pipe exit. It can be clearly seen that the high velocity region oscillates as the fluid flows along the helical pipe. This causes fluctuations in the values of Nusselt number.

Variation of local Nusselt number along the length of the tube is presented in fig. 6. The X axis of the figure is the angle of the plane, starting from the pipe inlet. It is found that the Nusselt number on the outer side of the coil is higher than those at any other location at that cross-section. Due to the centrifugal forces, the velocity in the outer region is higher and this

In this analysis, hot water at 330 K at a specified velocity of 0.8 ms-1 is entering the helical pipe at the top, where an inlet velocity boundary condition is specified. The flow velocity is such that the flow regime is turbulent. The fluid is made to cool down as it flows along the tube by specifying a wall temperature of 300 K. Temperature dependent values of fluid properties are used in this analysis. At the pipe wall, for the energy equation, a Dirichlet boundary condition and for momentum and pressure equations homogenous Neumann boundary condition are specified. At the outlet, a pressure outlet boundary is enforced.

Fig. 3 shows an overview of velocity contours at various sections along the length of the coil. The planes are identified by the angle (*θ*) which is the angle that the plane makes with the plane passing through the pipe inlet. In fig. 5 the first plane shown on the top is at 10o from the inlet (i.e., *θ*=10o) and the subsequent planes are 10o apart. Up to an angle of *θ*=35o, the velocity profile at a cross section is found to be symmetric. Subsequently, this uniform velocity pattern changes to a pattern with a high velocity region located at the outer side of the coil. This behaviour is seen predominantly by *θ*=45o and continues to develop. It can be seen that by *θ*=135o, the high velocity region is present only in outer half cross-section. Area of high velocity region further reduces as the flow gets developed and covers approximately

rd of the flow area by *θ*=240o. No significant change in flow pattern is observed

Temperature distribution at various planes along the length of coil is shown in fig. 4. At the inlet, temperature is uniform across the cross section. Since the wall is maintained at a lower temperature, the fluid cools down as it flows through the coil. Up to an angle of 20o, heat transfer is uniform along the periphery. In contrast to heat transfer in a straight tube, high temperature regions are seen on the outer side of the coil. This phenomena is predominant from the plane at angle *θ*=50o. This trend continues to develop and by 150o, clearly three regions viz., high temperature (327-330 K) at the outer side of the coil, intermediate temperature (321 to 324 K) at the centre and low temperature (311 to 314 K) on the inner side of the coil, are visible. As the fluid flows down the pipe, this temperature profile gets developed and the area of high temperature region decreases and by *θ* = 360o, a fully developed temperature profile is attained and the fluid continues to lose heat due to the

As the fluid flows through the helical coil, fluid particles undergo rotational motion. The fluid particles also undergo movement from inner side of the coil to the outer side and viceversa. Fig. 5 shows particle trace for 10 fluid particles which are located along a line parallel to the X axis at the pipe inlet. It can be noted that these fluid particles are taking various trajectories and also move with different velocities. The particles, which were forming a line to begin with, are found to be totally scattered at the pipe exit. It can be clearly seen that the high velocity region oscillates as the fluid flows along the helical pipe. This causes

Variation of local Nusselt number along the length of the tube is presented in fig. 6. The X axis of the figure is the angle of the plane, starting from the pipe inlet. It is found that the Nusselt number on the outer side of the coil is higher than those at any other location at that cross-section. Due to the centrifugal forces, the velocity in the outer region is higher and this

**2.3 Analysis with constant wall temperature boundary condition** 

1 3

downstream.

lower wall temperature.

fluctuations in the values of Nusselt number.

Fig. 3. Velocity (m s-1) contours at various planes along the length of the coil.

leads to higher heat transfer coefficients. In a similar way, the Nu along the inner periphery of the coil is the lowest.

Fig. 4. Temperature (K) contours at various planes along the length of the coil.

The Nusselt numbers at the top and bottom side of the cross sections show prominent periodic behaviour in the developing region of the pipe. Oscillatory motion of fluid particles, as observed in fig. 5, influences heat transfer around the periphery. The fluctuational behaviour of the Nusselt number has been reported by other investigators as well (Lin and Ebadian, 1999; Liu, 1992; Patankar et al., 1974). In the later regions of the coil, the Nusselt number at the top and bottom differ only marginally. Figure 6 also shows average values of Nusselt number (eqn. (4)) along the length of the pipe. It is found that it

Helically Coiled Heat Exchangers 321

attains an almost constant value by 300o. This constant value is used in the developing correlations for estimation of Nusselt number. Apart from the centrifugal action, buoyancy effects will move the hot particles upward and then downward as it loses heat (as can be seen in fig 5). This up and down movements together with centrifugal and inertial forces will lead to an overall spiral movement of the fluid. This may be attributed to the periodic behaviour of Nusselt number at top and bottom sides of the cross sections along the length

It may be noted that commercial CFD codes may not provide the value of bulk fluid temperature at different cross-section for estimation of the local Nusselt numbers. The user may be able to specify only a single value of bulk fluid temperature for the entire computational domain. This can lead to estimation of incorrect values of Nusselt numbers. Variation of local values of Nusselt number along the periphery of the pipe wall at various locations of the pipe are shown in Fig. 7. In these figures, the angle *θ* refers to the angle which the current plane makes with the inlet plane. In the initial length of the pipe, up to an angle *θ*=10o, marginally higher rates of heat transfer is observed at the upper side of the pipe. Due to gravity effect, the hotter fluid will be present at the top and this result in higher values of Nusselt number at that location. As the flow gets developed, when the effect of centrifugal forces becomes appreciable, region of higher heat transfer shifts from angle 270o to 180o i.e. from the upper side of the pipe to outer side of the coil. This shift gets completed by *θ*=76o. It is observed that up to an angle of 140o, the percentage of circumference, which has a higher value of *Nu* is predominant. This percentage decreases and by *θ*=430o onwards this region is so low that the average Nusselt number starts decreasing. Bai et al. (1999) has provided a figure showing ratio of local Nusselt number to average Nusselt number (only at 8 angular locations around the periphery) for turbulent heat transfer in a horizontally oriented helical coil. They have also obtained a similar pattern in the fully developed region as the one presented here.

A correlation for estimation of inside heat transfer coefficient for flow of single-phase water through helically coiled heat exchangers is presented in previous section. (Jayakumar et al., 2008a). The correlation, which is validated against experiments, is applicable to the specific configuration of helical coil, since the research work was limited only to changes in flow rate of the streams. This section deals with the analysis of various configurations of helical coils. After establishing influence of the coil parameters, correlations for prediction of average Nusselt number have been developed. Subsequently correlation to predict the local values

CFD simulations are carried out by varying coil parameters such as (i) pitch circle diameter, (ii) tube pitch and (iii) pipe diameter and their influence on heat transfer has been studied. Helical coils of different configurations have been analysed for this purpose. The results of these computations (where temperature dependant fluid properties are used) are used for developing unified correlations for estimation of inside heat transfer coefficient for flow of single-phase water through helical coils. Since a large data set is considered, the correlation will be applicable to a wide range of coil configurations and Dean numbers. Analysis has been carried out with both constant wall temperature and constant wall heat flux boundary conditions in order to establish influence of the boundary condition on heat transfer

**3. Correlations for estimation of average Nusselt number** 

of Nusselt number as a function of angular location is presented.

of the pipe.

coefficient.

Fig. 5. Trace of fluid particles which are parallel to X axis at the inlet.

Fig. 6. Variation of Nusselt Number along the length coil.

Fig. 5. Trace of fluid particles which are parallel to X axis at the inlet.

Fig. 6. Variation of Nusselt Number along the length coil.

attains an almost constant value by 300o. This constant value is used in the developing correlations for estimation of Nusselt number. Apart from the centrifugal action, buoyancy effects will move the hot particles upward and then downward as it loses heat (as can be seen in fig 5). This up and down movements together with centrifugal and inertial forces will lead to an overall spiral movement of the fluid. This may be attributed to the periodic behaviour of Nusselt number at top and bottom sides of the cross sections along the length of the pipe.

It may be noted that commercial CFD codes may not provide the value of bulk fluid temperature at different cross-section for estimation of the local Nusselt numbers. The user may be able to specify only a single value of bulk fluid temperature for the entire computational domain. This can lead to estimation of incorrect values of Nusselt numbers.

Variation of local values of Nusselt number along the periphery of the pipe wall at various locations of the pipe are shown in Fig. 7. In these figures, the angle *θ* refers to the angle which the current plane makes with the inlet plane. In the initial length of the pipe, up to an angle *θ*=10o, marginally higher rates of heat transfer is observed at the upper side of the pipe. Due to gravity effect, the hotter fluid will be present at the top and this result in higher values of Nusselt number at that location. As the flow gets developed, when the effect of centrifugal forces becomes appreciable, region of higher heat transfer shifts from angle 270o to 180o i.e. from the upper side of the pipe to outer side of the coil. This shift gets completed by *θ*=76o. It is observed that up to an angle of 140o, the percentage of circumference, which has a higher value of *Nu* is predominant. This percentage decreases and by *θ*=430o onwards this region is so low that the average Nusselt number starts decreasing. Bai et al. (1999) has provided a figure showing ratio of local Nusselt number to average Nusselt number (only at 8 angular locations around the periphery) for turbulent heat transfer in a horizontally oriented helical coil. They have also obtained a similar pattern in the fully developed region as the one presented here.
