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

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

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 coefficient.

Helically Coiled Heat Exchangers 323

The coils with PCD 100 mm, 200 mm, 300 mm and 400 mm were analysed. In all these cases, the coil pitch and pipe diameter were kept at 30 mm and 20 mm respectively and the coils

The effect of PCD is to influence the centrifugal force on the moving fluid. This will in turn affect the secondary flows along the pipe cross section. As the PCD is increased, the effect of coil curvature on flow decreases and hence centrifugal forces play a lesser role in flow characteristics. For the coil with PCD=100 mm, the entrance effects are seen to be present up to an angle of 40o. While for PCDs 200, 300 and 400 this change to 20o, 10o and 6o respectively. For the case of coil with PCD=100 mm, the difference between Nusselt number at the inner and outer location in the fully developed heat transfer region is 200. As we move to coils of higher PCDs, this difference comes down and for a coil of PCD=400 mm, it

To correlate the average Nusselt numbers in the fully developed region *Nu* with pitch circle diameter of the coil, the dimensionless parameter curvature ratio *δ* (=*r/Rc*) is used. The

11.0 *Nu* 265.65

In this analysis, a helical coil with a pipe of inner diameter (*2r*) 20 mm and pitch circle diameter (*PCD*) of 300 mm was considered. Analyses were carried out by changing the coil pitch. Coil with pitch of (i) zero, (ii) 15 mm, (iii) 30 mm, (iv) 45 mm and (v) 60 mm were

When the coil pitch is zero, local Nusselt numbers at the top and bottom points on the periphery of a cross section are almost the same. As the coil pitch is increased, the difference between them also increases. This difference is caused by torsion experienced by the fluid. As the pitch increases, the torsional effect also increases. However, variation of local *Nu* for the coils with pitch of 45 and 60 mm are identical. Average values of Nusselt number in the

It is found that the *Nuavg* increases marginally with increase in pitch and almost insensitive to its further changes at higher pitches. The percentage increase, when the pitch is changed from 0 mm to 15 mm is about 1% and this value changes to 0.2% when the pitch is changed from 45 mm to 60 mm. For any engineering application, the tube pitch has to be higher than pipe diameter and in that range the changes in *Nuavg* due to changes in pitch are negligible.

H, mm 0 15 30 45 60 Nuavg 189.24 191.08 191.75 192.27 192.55

verifying the nature of the proposed correlation. The equation is found to give a good fit.

. The Nusselt number can be correlated to

, (6)

reduces to 134. Thus the effect of centrifugal force on heat transfer is evident.

**3.1.1 Influence of Pitch Circle Diameter (PCD)** 

consisted of two turns (Jayakumar et al., 2010a).

correlation proposed is of the form *<sup>n</sup> Nu C*

curvature ratio as,

analysed.

**3.1.2 Influence of coil pitch (H)** 

fully developed region is given in table 1.

Table 1. Average values of Nusselt number.

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

The boundary conditions and the discretisation schemes used in this analysis are same as those given in section 2.3. The following sub-sections consider influence of each of the coil parameters separately. In all of the cases, average of the Nusselt number in the fully developed heat transfer region (where the *Nu* remains almost constant, see Fig. 6) is used as the representative value. Study has been carried out using the CFD package FLUENT 6.3 (3D, double precision). Each of the runs takes about 10 hours on a Xenon 2.4 GHz computer with 2 GB RAM.

Fig. 7. Variation of *Nu* around the circumference at various cross section of the pipe (0o Inner, 90o bottom, 180o Outer and 270o top).

The boundary conditions and the discretisation schemes used in this analysis are same as those given in section 2.3. The following sub-sections consider influence of each of the coil parameters separately. In all of the cases, average of the Nusselt number in the fully developed heat transfer region (where the *Nu* remains almost constant, see Fig. 6) is used as the representative value. Study has been carried out using the CFD package FLUENT 6.3 (3D, double precision). Each of the runs takes about 10 hours on a Xenon 2.4 GHz computer

(a) θ=2o (b) θ=8o

(c) θ=30 o (d) θ=72o

(e) θ=210o (f) θ=380o

Fig. 7. Variation of *Nu* around the circumference at various cross section of the pipe

(0o Inner, 90o bottom, 180o Outer and 270o top).

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

with 2 GB RAM.

## **3.1.1 Influence of Pitch Circle Diameter (PCD)**

The coils with PCD 100 mm, 200 mm, 300 mm and 400 mm were analysed. In all these cases, the coil pitch and pipe diameter were kept at 30 mm and 20 mm respectively and the coils consisted of two turns (Jayakumar et al., 2010a).

The effect of PCD is to influence the centrifugal force on the moving fluid. This will in turn affect the secondary flows along the pipe cross section. As the PCD is increased, the effect of coil curvature on flow decreases and hence centrifugal forces play a lesser role in flow characteristics. For the coil with PCD=100 mm, the entrance effects are seen to be present up to an angle of 40o. While for PCDs 200, 300 and 400 this change to 20o, 10o and 6o respectively. For the case of coil with PCD=100 mm, the difference between Nusselt number at the inner and outer location in the fully developed heat transfer region is 200. As we move to coils of higher PCDs, this difference comes down and for a coil of PCD=400 mm, it reduces to 134. Thus the effect of centrifugal force on heat transfer is evident.

To correlate the average Nusselt numbers in the fully developed region *Nu* with pitch circle diameter of the coil, the dimensionless parameter curvature ratio *δ* (=*r/Rc*) is used. The correlation proposed is of the form *<sup>n</sup> Nu C* . The Nusselt number can be correlated to curvature ratio as,

$$Nu = 265.65 \left(\delta\right)^{0.11} \text{ \AA} \tag{6}$$

verifying the nature of the proposed correlation. The equation is found to give a good fit.
