Helical and coiling spirals have been long used as heat money changers in power and chemical procedures. This paper shall present the construct of coiling cones to heighten heat and mass transportation, besides to supply a better infinite use than the ordinary spirals. Helical and coiling spirals are known to hold better heat and mass transportation than consecutive tubings, that ‘s attributed to the secondary flow superimposed on the primary flow, known as Dean Vortex.

The Dean Number depicting the dean whirl is a map of Reynolds Number and the square root of the curvature ratio, so changing the curvature ratio for the same spiral would change the Dean Number. Numerical probe based on the commercial CFD package fluent is used to analyze the consequence of altering the structural parametric quantities ( taper angle of the coiling spiral, pitch and the base radius of curvature ) on the Nusselt Number and heat transportation coefficient.

Six chief spirals holding pipe diameters of 10 and 12.5 millimeter and base radius of curvature of 50, 75 and 100 millimeters were used in the probe. It was found that as the taper angle increases both Nusselt Number and the heat transportation coefficient additions, besides the pitch at the assorted taper angles was found to hold influence on Nusselt Number and the heat transportation coefficient. A MATLAB codification was built to cipher the Nusselt Number at each spiral turn so cipher its norm based on empirical correlativity of Manlapaz and Churchill for ordinary coiling spirals, the CFD simulation consequences were found acceptable when compared with the Matlab consequences.

## Introduction

Helical spirals have been long and widely used as heat money changers in power, petrochemical, HVAC, chemical and many other industrial procedures. Helical and coiling spirals are known to hold better heat and mass transportation compared to straight tubings, the ground for that is the formation of a secondary flow superimposed on the primary flow, known as Dean Vortex [ 1 ] . The Dean Vortex was foremost observed by Eustice ; so legion surveies have been reported on the flow Fieldss that arise in curving pipes ( Dean, White, Hawthorne, Horlock, Barua, Austin and Seader ) [ 2 ] . The first effort to mathematically depict the flow in a coiled tubing was made by Dean, he found that the secondary flow induced in curving pipes ( Dean Vortex ) is a map of Reynolds Number and the curvature ratio, the Dean Number is widely used to qualify the flow in curving tubings:

De = Re * ( 1 )

It has been widely observed that the flow indoors coiled tubings remains in the syrupy government up to a much higher Reynolds Number than that for consecutive tubings Srinivasan et Al. [ 1 ] . The curvature-induced coiling whirls ( Dean Vortex ) tend to stamp down the oncoming of turbulency and hold passage. The critical Reynolds Number which describes the passage from laminar to turbulent flow is given by any correlativities ; the undermentioned correlativity is given by Srinivasan et Al. [ 1 ] :

Recr = 2100 * ( 1+12 ) ( 2 )

Dennis and Ng [ 3 ] numerically studied laminal flow through a curving tubing utilizing a finite difference method with accent on two versus four whirl flow conditions. They ran simulations in the Dean scope of 96 to 5000. The four whirl solutions would merely look for a Dean figure greater than 956. Dennis and Riley [ 4 ] developed an analytical solution for the to the full developed laminar flow for high Dean Numbers. Though they could non happen a complete solution to the job, they stated that there is strong grounds that at high Dean Numbers the flow develops into an inviscid nucleus with a syrupy boundary bed at the pipe wall.

The consequence of pitch on heat transportation and force per unit area bead was studied by Austin and Soliman [ 5 ] for the instance of uniform wall heat flux. The consequences showed important pitch effects on both the clash factor and the Nusselt Number at low Reynolds Numbers, though these effects weakened as the Reynolds figure increased. The writers suggested that these pitch effects are due to free convection, and therefore lessening as the forced convection becomes more dominant at higher Reynolds Numbers. The consequence of the pitch on the Nusselt Number in the laminar flow of helicoidal pipes was besides investigated by Yang et al [ 6 ] Numerical consequences for to the full developed flow with a finite pitch showed that the temperature gradient on one side of the pipe will increase with increasing tortuosity ; nevertheless, the temperature gradient on the antonym will diminish. Overall, the Nusselt Number somewhat decreases with increasing tortuosity for low Prandtl Numbers, but significantly decreases with larger Prandtl Numbers. On the other manus Germano [ 7 ] introduced an extraneous co-ordinate system to analyze the consequence of tortuosity and curvature on the flow in a coiling pipe. In the consequences of the disturbance method indicated that the tortuosity had a 2nd order consequence and curvature had a first order consequence on the flow. Further surveies by Tuttle [ 8 ] indicated that the frame of mention ( coordinate system ) determines if the tortuosity consequence is foremost or 2nd order.

Kalb and Seader [ 9 ] numerically studied the heat transportation in coiling spirals in instance of unvarying heat flux utilizing an extraneous toroidal co-ordinate system. They have found that for Prandtl Numbers greater than 0.7, it was shown that the local Nusselt Number in the country of the inner wall was ever less than that of a consecutive tubing, and increasing less as the Dean Number is increased till it reached a confining value. The local Nusselt Numbers on the outer wall continued to increase with increasing Dean Number. Fully developed laminar flow and heat transportation was studied numerically by Zapryanov et Al. [ 10 ] utilizing a method of fractional stairss for a broad scope of Dean ( 10 to 7000 ) and Prandtl ( 0.005 to 2000 ) Numberss. Their work focused on the instance of changeless wall temperature and showed that the Nusselt figure increased with increasing Prandtl Numberss, even for instances at the same Dean figure.

Spiral spirals have received small attending compared to coiling spirals, though the reported consequences of coiling spirals show better public presentation than coiling 1s. Figueiredo and Raimundo [ 12 ] by experimentation investigated the thermic response of a hot-water shop and the thermic discharge features from heat money changer spirals placed indoors. The classical cylindrical spiral and the level coiling spiral were investigated. The consequences indicated that the efficiency of level coiling spiral was higher than that of a cylindrical one. The consequences from comparing between the theoretical account and experiments were in good understanding. Naphon and Suwagrai [ 13 ] studied the Effect of curvature ratios on the heat transportation in the horizontal spirally coiled tubings both by experimentation and numerically, they have found that due to the centrifugal force, the Nusselt figure and force per unit area bead obtained from the spirally coiled tubing are 1.49, 1.50 times higher than those from the consecutive tubing, severally.

Helical cone spirals have even received lower attending than coiling spirals, merely really few research workers have investigated the capablenesss of these spirals due to the complexness of the construction, it was difficult to look into it both numerically and by experimentation. Yan Ke et Al. [ 14 ] have investigated the coiling cone tubing bundles both numerically and still some foregoing experiments, the writers found that the cone angle has a important consequence on heightening the heat transportation coefficient, besides they ‘ve found that the pitch has about no consequence on the heat transportation.

The purpose of this paper is to further numerically look into the consequence of the taper angle on Nusselt Number and the heat transportation coefficient for coiling cone spirals, besides to further look into the consequence of the pitch on the heat transportation for these spirals. Finally seek to optimise the coiling cone spirals and supply rule preparation for it.

## Numeric Simulation

Helical Cone Coil Geometry

The Geometry of the coiling cone tubing is shown in Fig.1 ; both the curvature and tortuosity are variable along the tubing. The bottom radius of curvature is donated ( R ) , the pipe diameter ( a ) , the coiling pitch as ( P ) , the consecutive tallness ( H ) and eventually the inclined tallness ( I ) . For a consecutive coiling spiral the tallness ( H ) will be equal to ( I ) but when altering the disposition angle ( ? ) , the tallness of the spiral ( I ) will alter in conformity to that angle, while maintaining ( H ) invariable.

## Figure: Coiling Coil Geometry

Three bottom radii of curvatures ( R ) were used 50, 75 and 100 millimeter, besides two pipe diameters ( a ) were used 10 and 12.5 millimeter. So as to maintain the tallness of the spiral ( H ) invariable, the tallness ( I ) which changes with regard to the taper angle ( ? ) was proposed. It should be noted that the coiling spiral spiral is chiefly optimized to be used as a capacitor ( dehumidifier ) for a solar HDH desalinization unit.

Simulation Model

The laminar flow in the coiling spiral spiral is simulated utilizing the commercial CFD package Fluent. In the simulation of the laminar fluid flow, the flow and force per unit area equations were solved with SIMPLEC algorithm, which is one of the three widely, used speed force per unit area matching algorithm in Fluent. The Second Order Upwind algorithm was employed in the discretization of the equations because of its truth and iterating efficiency. The parametric quantities of laminar fluid flow theoretical account were in conformity with the default values of the CFD package:

Purf = 0.3 Murf = 0.7 ( 3 )

Where, the Purf and Murf severally denote the Under Relaxation Factor of force per unit area and impulse of the fluid flow inside the tubing during the iterating of the computation.

The commercial package Fluent uses both Navier – Stocks equation, continuity equation and the energy equation in the solution, the equations are solved for laminar, steady and 3D flow, and these equations are as follow:

Navier – Stockss equation:

u + V + tungsten = – + ? ( + + ) ( 4 )

u + V + tungsten = – + ? ( + + ) ( 5 )

u + V + tungsten = – + ? ( + + ) ( 6 )

The continuity equation:

+ + = 0 ( 7 )

The energy equation:

? cp ( u + V + tungsten ) = K ( + + ) ( 8 )

The 2nd measure was to do mathematical theoretical account confirmation, and as stated antecedently, really few experiments and mathematical simulations have been conducted on coiling cone tubings. In order to verify the truth of the mathematical theoretical account we are look intoing, the finite component theoretical account for the round cross sectional country made by Yan Ke et Al [ 14 ] has been used in the confirmation. Unstructured, non-uniform grid systems are used to discretize the chief government equations. The sweep grids were used to discretize the whole volume of the coiling spiral. The changeless temperature and non-slip boundary conditions were applied. The consequences of the mathematical theoretical account were found in understanding with the consequences of Yan Ke et Al. [ 14 ] , in the instance of round cross subdivision.

## Consequences and Discussion

Taper Angle

Twenty two theoretical accounts were used to analyze the consequence of the taper angle on the heat transportation coefficient and Nusselt Number. Table 1 shows the inside informations of these theoretical accounts. To hold a better apprehension for the consequences, each bottom radius of curvature ( R ) and pipe diameter ( a ) will be discussed individually, so a comparing between them will be made to hold a complete apprehension for the consequence of the taper angle on each instance, besides to cognize how to optimise each instance.

## Table: Coiling Coil Details

R ( millimeter )

a ( millimeter )

Rhenium

U ( m/s )

?

I ( millimeter )

70

8

1595

0.1

0, 20, 40

25, 26.6, 32.64

12

2392

0, 25, 40

25, 27.58, 32.64

80

8

1595

0, 20, 50, 60

25, 26.6, 38.9, 50

12

2392

0, 20, 50, 70

25, 26.6, 38.9, 73.1

90

8

1595

0, 25, 45, 60

25, 27.58, 35.4, 50

12

2392

0, 20, 40, 70

25, 26.6, 32.64, 73.1

For R = 70, 80, 90 millimeter and a = 8 millimeter, it can be clearly seen from Fig. ( 2 ) , Fig. ( 4 ) and Fig. ( 6 ) , That the heat transportation coefficient has increased with increasing the taper angle ( ? ) , Fig. ( 3 ) , Fig. ( 5 ) and Fig. ( 7 ) Show that Nusselt Number increases with increasing the taper angle excessively, while the surface country of heat transportation decreases with increasing the taper angle, taking to a lessening in the infinite required for installing and stuff used in fabrication. It should be noted that the addition in the Nusselt Number and heat transportation coefficient is logic, as the Nusselt Number varies straight with the Dean Number ( De ) which varies straight with the curvature ratio ( a / R ) . So, as ( R ) decreases when increasing the taper angle, the curvature ration additions. Finally, Fig. ( 8, a ) , ( 8, B ) , ( 8, degree Celsius ) represents the angles ( 0, 20, 40 ) severally for R = 70, these figures show the alteration in the speed due to the alteration in the taper angle, and as it can be seen that the speed increases which means that Reynolds Number is increasing and therefore the Dean Number. Besides, it could be noted that the centre of the chief flow is shifted towards the outwards of the pipe ( Dean Vortex ) . A multinomial curve adjustment is made to cognize the regulating equation for Nusselt Number and the heat transportation coefficient with the taper angle ( ? ) .

For R = 70, 80, 90 millimeter and a = 12 millimeter, it can besides be clearly seen from Fig. ( 9 ) , Fig ( 11 ) and Fig. ( 13 ) that the heat transportation coefficient additions with increasing the taper angle, while Fig. ( 10 ) , Fig. ( 12 ) and Fig. ( 14 ) shows that the Nusselt Number increases with increasing the taper angle. Fig. ( 15, a ) , ( 15, B ) , ( 15, degree Celsius ) represents the angles ( 0, 25, 40 ) severally for R = 70, these figures show that the centre of the chief flow is shifted towards the outwards of the pipe ( Dean Vortex ) . A multinomial curve adjustment is made to cognize the regulating equation for Nusselt Number and the heat transportation coefficient with the taper angle ( ? ) .

It can be clearly seen from the following curves that both the heat transportation coefficient and Nusselt Number increases when increasing the taper angle for any bottom radius of curvature ( R ) and any pipe diameter ( a ) .

## Figure: The consequence of the taper angle on the temperature and heat transportation coefficient ( R = 70 )

## Figure: The consequence of the taper angle on the Nusselt Number and country ( R = 70 )

## Figure: The consequence of the taper angle on the temperature and heat transportation coefficient ( R = 80 )

## Figure: The consequence of the taper angle on the country and Nusselt Number ( R = 80 )

## Figure: The consequence of the taper angle on the temperature and heat transportation coefficient ( R = 90 )

## Figure: The consequence of the taper angle on the country and Nusselt Number ( R = 90 )

## Figure: The consequence of the taper angle on the speed profile

## Figure: The consequence of the taper angle on the temperature and heat transportation coefficient ( R = 70 )

## Figure: The consequence of the taper angle on the country and Nusselt Number ( R = 70 )

## Figure: The consequence of the taper angle on the temperature and heat transportation coefficient ( R = 80 )

## Figure: The consequence of the taper angle on the country and Nusselt Number ( R = 80 )

## Figure: The consequence of the taper angle on the temperature and heat transportation coefficient ( R = 90 )

## Figure: The consequence of the taper angle on the country and Nusselt Number ( R = 90 )

## Figure: The consequence of the taper angle on the speed profile

Pitch

The consequence of the pitch fluctuation on the heat transportation coefficient and Nusselt Number will be studied in this subdivision utilizing eight mathematical theoretical accounts. As it has been discussed in the old subdivision, that as the taper angle increases the heat transportation coefficient and the Nusselt Number addition, so the two spirals that will be used in the simulation will hold taper angle peers to sixty. The building parametric quantities of the two spirals could be found in table 2. The tallness was kept changeless so the figure of bends was altering when altering the pitch.

It can be clearly seen in Fig. ( 16 ) and Fig. ( 17 ) That both the heat transportation coefficient and the Nusselt Number addition when increasing the pitch. That besides seems to be corroborating to the fact that as the pitch additions while maintaining the tallness invariable for coiling cone coils the curvature ratio ( a / R ) additions, which leads to an additions in Dean Number and so Nusselt Number. A multinomial curve adjustment is made to cognize the regulating equations.

## Table: Construction Parameters for the Two Coils

R ( millimeter )

a ( millimeter )

H ( millimeter )

U ( m/s )

Rhenium

P ( millimeter )

?

100

10

11

0.1

997

50,60,70,80

60

12.5

1246

## Figure: The consequence of pitch fluctuation on Nusselt Number and heat transportation coefficient ( a = 10 )

## Figure: The consequence of pitch fluctuation on Nusselt Number and heat transportation coefficient ( a = 12.5 )

Consequences Comparison with the MATLAB Code

A comparing between the MATLAB codification, which was built based on the experimental equation of Manlapaz and Churchill for ordinary coiling spirals subjected to constant wall temperature and the CFD consequences will be discussed in this subdivision, the ground for this comparing is to see whether these equations could be used for the coiling cone spirals or non. To use the equations on the coiling cone spiral, the equation will be calculated for every spiral bend so an mean value for the Nusselt Number will be evaluated. The comparing will be on one of the old consequences merely, the theoretical account will be the R = 5 and a = 10. From Fig. ( 18 ) it could be seen that the equation could be used till taper angle peers to forty but after that mistake increases significantly.

## Figure: Nusselt Numbet from the experimental equation Vs. Nusselt Number from the CFD simulation

## Decision

The heat transportation coefficient and Nusselt Number was found to increase when increasing the taper angle of the coiling cone spiral, the coiling cone spiral was found to hold lower country and that leads to a better infinite use in industrial applications and better stuff use in its fabrication. The coiling spiral pitch was found to be effectual when altering the taper angle, for both the Nusselt figure and the heat transportation coefficient.

Future experiments will be carried out to verify these mathematical consequences, and to analyze the consequence of both the taper angle and the pitch on both the heat transportation coefficient and the Nusselt Number.

## Terminology

a: Pipe radius ( millimeter ) .

Hydrogen: Coiling spiral tallness ( millimeter ) .

H: Heat reassign coefficient ( w/m2 K ) .

I: Inclined tallness ( millimeter ) .

Murf: Relaxation factor of impulse.

Phosphorus: Coiling Pitch ( millimeter ) .

Purf: Relaxation factor of force per unit area.

Roentgen: Coil radius of curvature ( millimeter ) .

Thymine: Temperature.

U: Inlet speed ( m/s ) .

Nu: Nusselt Number.

De: Dean Number.

Grecian Symbol

? : Taper angle.

? : Density.