# Heat Exchangers In Power And Chemical Processes Biology Essay

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Helical and spiral coils have been long used as heat exchangers in power and chemical processes. This paper shall introduce the concept of helical cones to enhance heat and mass transfer, also to provide a better space utilization than the ordinary coils. Helical and spiral coils are known to have better heat and mass transfer than straight tubes, that's attributed to the secondary flow superimposed on the primary flow, known as Dean Vortex.

The Dean Number describing the dean vortex is a function of Reynolds Number and the square root of the curvature ratio, so varying the curvature ratio for the same coil would vary the Dean Number. Numerical investigation based on the commercial CFD software fluent is used to study the effect of changing the structural parameters (taper angle of the helical coil, pitch and the base radius of curvature) on the Nusselt Number and heat transfer coefficient.

Six main coils having pipe diameters of 10 and 12.5 mm and base radius of curvature of 50, 75 and 100 mm were used in the investigation. It was found that as the taper angle increases both Nusselt Number and the heat transfer coefficient increases, also the pitch at the various taper angles was found to have influence on Nusselt Number and the heat transfer coefficient. A MATLAB code was built to calculate the Nusselt Number at each coil turn then calculate its average based on empirical correlation of Manlapaz and Churchill for ordinary helical coils, the CFD simulation results were found acceptable when compared with the Matlab results.

## Introduction

Helical coils have been long and widely used as heat exchangers in power, petrochemical, HVAC, chemical and many other industrial processes. Helical and spiral coils are known to have better heat and mass transfer compared to straight tubes, the reason for that is the formation of a secondary flow superimposed on the primary flow, known as Dean Vortex [1]. The Dean Vortex was first observed by Eustice; then numerous studies have been reported on the flow fields that arise in curved pipes (Dean, White, Hawthorne, Horlock, Barua, Austin and Seader)[2]. The first attempt to mathematically describe the flow in a coiled tube was made by Dean, he found that the secondary flow induced in curved pipes (Dean Vortex) is a function of Reynolds Number and the curvature ratio, the Dean Number is widely used to characterize the flow in curved tubes:

De = Re * (1)

It has been widely observed that the flow inside coiled tubes remains in the viscous regime up to a much higher Reynolds Number than that for straight tubes Srinivasan et al. [1]. The curvature-induced helical vortices (Dean Vortex) tend to suppress the onset of turbulence and delay transition. The critical Reynolds Number which describes the transition from laminar to turbulent flow is given by any correlations; the following correlation is given by Srinivasan et al.[1]:

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

Dennis and Ng [3] numerically studied laminar flow through a curved tube using a finite difference method with emphasis on two versus four vortex flow conditions. They ran simulations in the Dean range of 96 to 5000. The four vortex solutions would only appear for a Dean number greater than 956. Dennis and Riley [4] developed an analytical solution for the fully developed laminar flow for high Dean Numbers. Though they could not find a complete solution to the problem, they stated that there is strong evidence that at high Dean Numbers the flow develops into an inviscid core with a viscous boundary layer at the pipe wall.

The effect of pitch on heat transfer and pressure drop was studied by Austin and Soliman [5] for the case of uniform wall heat flux. The results showed significant pitch effects on both the friction factor and the Nusselt Number at low Reynolds Numbers, though these effects weakened as the Reynolds number increased. The authors suggested that these pitch effects are due to free convection, and thus decrease as the forced convection becomes more dominant at higher Reynolds Numbers. The effect of the pitch on the Nusselt Number in the laminar flow of helicoidal pipes was also investigated by Yang et al [6] Numerical results for fully developed flow with a finite pitch showed that the temperature gradient on one side of the pipe will increase with increasing torsion; however, the temperature gradient on the opposite will decrease. Overall, the Nusselt Number slightly decreases with increasing torsion for low Prandtl Numbers, but significantly decreases with larger Prandtl Numbers. On the other hand Germano [7] introduced an orthogonal coordinate system to study the effect of torsion and curvature on the flow in a helical pipe. In the results of the perturbation method indicated that the torsion had a second order effect and curvature had a first order effect on the flow. Further studies by Tuttle [8] indicated that the frame of reference (coordinate system) determines if the torsion effect is first or second order.

Kalb and Seader [9] numerically studied the heat transfer in helical coils in case of uniform heat flux using an orthogonal toroidal coordinate system. They have found that for Prandtl Numbers greater than 0.7, it was shown that the local Nusselt Number in the area of the inner wall was always less than that of a straight tube, and increasing less as the Dean Number is increased till it reached a limiting value. The local Nusselt Numbers on the outer wall continued to increase with increasing Dean Number. Fully developed laminar flow and heat transfer was studied numerically by Zapryanov et al. [10] using a method of fractional steps for a wide range of Dean (10 to 7000) and Prandtl (0.005 to 2000) numbers. Their work focused on the case of constant wall temperature and showed that the Nusselt number increased with increasing Prandtl numbers, even for cases at the same Dean number.

Spiral coils have received little attention compared to helical coils, though the reported results of spiral coils show better performance than helical ones. Figueiredo and Raimundo [12] experimentally investigated the thermal response of a hot-water store and the thermal discharge characteristics from heat exchanger coils placed inside. The classical cylindrical coil and the flat spiral coil were investigated. The results indicated that the efficiency of flat spiral coil was higher than that of a cylindrical one. The results from comparison between the model and experiments were in good agreement. Naphon and Suwagrai [13] studied the Effect of curvature ratios on the heat transfer in the horizontal spirally coiled tubes both experimentally and numerically, they have found that due to the centrifugal force, the Nusselt number and pressure drop obtained from the spirally coiled tube are 1.49, 1.50 times higher than those from the straight tube, respectively.

Helical cone coils have even received lower attention than spiral coils, only very few researchers have investigated the capabilities of these coils due to the complexity of the structure, it was hard to investigate it both numerically and experimentally. Yan Ke et al. [14] have investigated the helical cone tube bundles both numerically and still some foregoing experiments, the authors found that the cone angle has a significant effect on enhancing the heat transfer coefficient, also they've found that the pitch has nearly no effect on the heat transfer.

The aim of this paper is to further numerically investigate the effect of the taper angle on Nusselt Number and the heat transfer coefficient for helical cone coils, also to further investigate the effect of the pitch on the heat transfer for these coils. Finally try to optimize the helical cone coils and provide principle formulation for it.

## Numerical Simulation

Helical Cone Coil Geometry

The Geometry of the helical cone tube is shown in Fig.1; both the curvature and torsion are variable along the tube. The bottom radius of curvature is donated (R), the pipe diameter (a), the helical pitch as (P), the straight height (H) and finally the inclined height (I). For a straight helical coil the height (H) will be equal to (I) but when changing the inclination angle (θ), the height of the coil (I) will change in accordance to that angle, while keeping (H) constant.

## Figure : Helical Coil Geometry

Three bottom radii of curvatures (R) were used 50, 75 and 100 mm, also two pipe diameters (a) were used 10 and 12.5 mm. So as to keep the height of the coil (H) constant, the height (I) which changes with respect to the taper angle (θ) was proposed. It should be noted that the helical spiral coil is mainly optimized to be used as a condenser (dehumidifier) for a solar HDH desalination unit.

Simulation Model

The laminar flow in the helical spiral coil is simulated using the commercial CFD software Fluent. In the simulation of the laminar fluid flow, the flow and pressure equations were solved with SIMPLEC algorithm, which is one of the three widely, used velocity pressure coupling algorithm in Fluent. The Second Order Upwind algorithm was employed in the discretization of the equations because of its accuracy and iterating efficiency. The parameters of laminar fluid flow model were in accordance with the default values of the CFD software:

Purf = 0.3 Murf = 0.7 (3)

Where, the Purf and Murf respectively denote the Under Relaxation Factor of pressure and momentum of the fluid flow inside the tube during the iterating of the calculation.

The commercial software 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 - Stocks equation:

u + v + w = - + μ ( + + ) (4)

u + v + w = - + μ ( + + ) (5)

u + v + w = - + μ ( + + ) (6)

The continuity equation:

+ + = 0 (7)

The energy equation:

ρ cp (u + v + w) = K ( + + ) (8)

The second step was to make mathematical model verification, and as stated previously, very few experiments and mathematical simulations have been conducted on helical cone tubes. In order to verify the accuracy of the mathematical model we are investigating, the finite element model for the circular cross sectional area made by Yan Ke et al [14] has been used in the verification. Unstructured, non-uniform grid systems are used to discretize the main governing equations. The sweep grids were used to discretize the whole volume of the spiral coil. The constant temperature and non-slip boundary conditions were applied. The results of the mathematical model were found in agreement with the results of Yan Ke et al. [14], in the case of circular cross section.

## Results and Discussion

Taper Angle

Twenty two models were used to study the effect of the taper angle on the heat transfer coefficient and Nusselt Number. Table 1 shows the details of these models. To have a better understanding for the results, each bottom radius of curvature (R) and pipe diameter (a) will be discussed separately, then a comparison between them will be made to have a complete understanding for the effect of the taper angle on each case, also to know how to optimize each case.

## Table : Helical Coil Details

R (mm)

a (mm)

Re

u (m/s)

θ

I (mm)

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 mm and a = 8 mm, it can be clearly seen from Fig. (2), Fig. (4) and Fig. (6),That the heat transfer coefficient has increased with increasing the taper angle (θ), Fig. (3), Fig. (5) and Fig. (7) Show that Nusselt Number increases with increasing the taper angle too, while the surface area of heat transfer decreases with increasing the taper angle, leading to a decrease in the space required for installation and material used in manufacturing. It should be noted that the increase in the Nusselt Number and heat transfer coefficient is logic, as the Nusselt Number varies directly with the Dean Number (De) which varies directly with the curvature ratio (a / R). So, as (R) decreases when increasing the taper angle, the curvature ration increases. Finally, Fig. (8, a), (8, b), (8, c) represents the angles (0, 20, 40) respectively for R = 70, these figures show the change in the velocity due to the change in the taper angle, and as it can be seen that the velocity increases which means that Reynolds Number is increasing and thus the Dean Number. Also, it could be noted that the center of the main flow is shifted towards the outwards of the pipe (Dean Vortex). A polynomial curve fitting is made to know the governing equation for Nusselt Number and the heat transfer coefficient with the taper angle (θ).

For R = 70, 80, 90 mm and a = 12 mm, it can also be clearly seen from Fig. (9), Fig (11) and Fig. (13) that the heat transfer coefficient increases 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, c) represents the angles (0, 25, 40) respectively for R = 70, these figures show that the center of the main flow is shifted towards the outwards of the pipe (Dean Vortex). A polynomial curve fitting is made to know the governing equation for Nusselt Number and the heat transfer coefficient with the taper angle (θ).

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

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

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

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

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

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

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

## Figure : The effect of the taper angle on the velocity profile

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

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

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

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

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

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

## Figure : The effect of the taper angle on the velocity profile

Pitch

The effect of the pitch variation on the heat transfer coefficient and Nusselt Number will be studied in this section using eight mathematical models. As it has been discussed in the previous section, that as the taper angle increases the heat transfer coefficient and the Nusselt Number increase, so the two coils that will be used in the simulation will have taper angle equals to sixty. The construction parameters of the two coils could be found in table 2. The height was kept constant so the number of turns was changing when changing the pitch.

It can be clearly seen in Fig. (16) and Fig. (17) That both the heat transfer coefficient and the Nusselt Number increase when increasing the pitch. That also seems to be confirming to the fact that as the pitch increases while keeping the height constant for helical cone coils the curvature ratio (a / R) increases, which leads to an increases in Dean Number and so Nusselt Number. A polynomial curve fitting is made to know the governing equations.

## Table : Construction Parameters for the Two Coils

R (mm)

a (mm)

H (mm)

u (m/s)

Re

P (mm)

θ

100

10

11

0.1

997

50,60,70,80

60

12.5

1246

## Figure : The effect of pitch variation on Nusselt Number and heat transfer coefficient (a = 10)

## Figure : The effect of pitch variation on Nusselt Number and heat transfer coefficient (a = 12.5)

Results Comparison with the MATLAB Code

A comparison between the MATLAB code, which was built based on the experimental equation of Manlapaz and Churchill for ordinary helical coils subjected to constant wall temperature and the CFD results will be discussed in this section, the reason for this comparison is to see whether these equations could be used for the helical cone coils or not. To apply the equations on the helical cone coil, the equation will be calculated for every coil turn then an average value for the Nusselt Number will be evaluated. The comparison will be on one of the previous results only, the model will be the R = 5 and a = 10. From Fig. (18) it could be seen that the equation could be used till taper angle equals to forty but after that error increases significantly.

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

## Conclusion

The heat transfer coefficient and Nusselt Number was found to increase when increasing the taper angle of the helical cone coil, the helical cone coil was found to have lower area and that leads to a better space utilization in industrial applications and better material utilization in its manufacturing. The helical coil pitch was found to be effective when changing the taper angle, for both the Nusselt number and the heat transfer coefficient.

Future experiments will be carried out to verify these mathematical results, and to study the effect of both the taper angle and the pitch on both the heat transfer coefficient and the Nusselt Number.

## Nomenclature

a: Pipe radius (mm).

H: Helical coil height (mm).

h: Heat transfer coefficient (w/m2 K).

I: Inclined height (mm).

Murf: Relaxation factor of momentum.

P: Helical Pitch (mm).

Purf: Relaxation factor of pressure.

R: Coil radius of curvature (mm).

T: Temperature.

u: Inlet velocity (m/s).

Nu: Nusselt Number.

De: Dean Number.

Greek Symbol

Θ: Taper angle.

Ρ: Density.