# The Pressure - Flow Relationship of Coiled Tubing Are Tested; Radius, Orientation, And

Abstract

Method

The pressure - flow relationship of coiled tubing are tested; radius, orientation, and solution viscosity are varied. The experimental apparatus consisted of a tank of water that is drained through cylindrical 1/8”ID tubing. A manometer is positioned to measure the change in pressure across the horizontal tube. The needle valve at the base of the water tank controls the flow rate. Low flow rates are limited by the experimenter’s inability to read very low pressure values (<2.0cm) and the height of the manometer (>99.0 cm). The coiled tubing iss wrapped around four different varying radii of curvature (12.33cm, 7.8cm, 1.51cm, 26.55cm). At constant radius, orientation is varied. (horizontal, top-bottom, bottom-top). Both water and10 % sucrose solution is tested with viscosities of 1.00 cP and 1.66 cP respectively. Flow rates are measured by collecting between 20 – 50 mL water in a graduated cylinder over a period of approximately 45 sec. For each set of experimental conditions, (radius, orientation, viscosity), three different flow rates are used. From the raw data, a friction factor (straight tube approximation) and corresponding Reynold’s number are calculated. However, this model must be replaced by one that takes into account the helical friction factor and a unitless parameter named Dean’s number.

Results

Pressure losses during internal flows through a coiled smooth tube can be characterized by dimensional analysis as a function of dimensionless parameters:

As previously seen in the analysis of Poiseuille flow through straight smooth tube, we can separate D/L from the functional form and create an expression for head loss, which itself is a function of friction factor:

Using this formula, we calculated the Darcy friction factor for each trial. We then wanted to relate the Darcy friction factor (which applies for straight smooth tubes) with a dimensionless parameter unique to coiled tubes, which is called the Dean number, which is defined:

; D=inner tube diameter, R=radius of coil

We plotted our calculated friction factor against the Dean number in order to determineif all the data lie on a single curve (Figure 1). Power regression analysis was used to observe that the data reasonably fits well (R2=0.826, N=119) with a declining exponential function given by:

Figure 1. The straight tube friction factor is an approximation that can be modeled experimentally as a function of the Dean’s number.

We then investigated whether this curve closely resembles the Darcy friction factor curve that is observed in Poiseuille flow through straight smooth tubes. We superimposed the plot of the Darcy friction factor (function of Reynolds number) on our graph in order to examine this potential correlation (Figure 2). We find that our data weakly traces the friction factor curve given by straight smooth coil approximation since although our exponent is off by 17%, our coefficient is off by 89%. This finding forces us to examine alternative models to fully characterize pressure flow through coiled tubes.

Figure 2. The experimental data points lie below the 64/Re curve fit.

C.M. White offers a potentially stronger model of pressure flow through coiled tubes and defines the friction factor through coiled tubes based on the equation:

; fw = White friction factor

We calculated the White friction factor for each of our trials, and plotted it against the Dean number. We found that there was no obvious correlation between the White friction factor (for coiled tubes) and the Dean number when aggregating all the data (see Figure 3). White proposed that the functional form of the White friction factor should be:

White experimentally found that A=0.37 and n=0.36 for Dean numbers greater than 29 but less than 1000. In order to more closely analyze our correlation, we investigated possible correlations between the White friction factor and three variables in our study: (1) coil orientation, (2) fluid viscosity, and (3) coil radius.

Figure 3.

When examining the relationship between the White friction factor and the coil orientation, we discovered that there is no statistical difference (p>0.05, N=15). When applying power regression analysis for each orientation (bottom-up, up-bottom, and horizontal), we noticed that the power fits do not agree with White’s function, since the coefficients are off by at least five orders of magnitude and the exponentsare off by more than one order of magnitude.

Figure 4. The relationship between the friction factor and Dean’s number is independent of orientation.

We then examined the effect of fluid viscosity on the White friction factor and noticed that a change in viscosity translates into a movement along the White friction factor curve (Figure 5). The Dean number is proportional to the Reynolds number, which is inversely proportional to viscosity. Increasing the viscosity of the fluid (from pure water to 10% sucrose solution) reduces the Dean number, leading to decreased values of the friction factor. The fact that these points lie along the same curve is shown when a power fit can be applied across both sets of data. We find that the power fit we obtain resembles the approximation of f(De) that White empirically determined:

Our power fit has the same coefficient as White and an exponent that is off by 52% (R2=0.5769).

Figure 5. Experimental data points for both water and 10% sucrose solution are fit to the power function y=.372*De.169.

We finally relate the White friction factor to the coil radius and notice that different radii produce different power fits (Figure 6). Increasing the radius lowers the Dean number but it isn’t expected that the friction factor curve shifts entirely. Nonetheless, we obtain different power fits for each radius using power regression analysis and find that our best fit is observed at the largest radius (R=26.5 cm). At this radius, we obtain the White friction factor function to be:

Although our coefficient deviates by 93% and our exponent deviates by 134%, this power fit is at least within one order of magnitude of the empirical approximation obtained by White.

Figure 6.

White’s model for pressure flow through coiled smooth tubes did not constitute an upgrade over the approximation we obtained using the Darcy friction factor for straight smooth tube flow. We therefore decided to apply a new model that incorporates the occurrences of vortex flow, which we may be able to attribute our error to. Drawing from a recent study that investigated pressure flow through coiled tubes, we notice that the helical friction factor can be related to the Darcy friction factor in the following way, considering the effect of vortex flow and provided that De < 11.6 and Re < Re,crit:

When we applied this fit to obtain theoretical values for the helical friction factor, we plotted our experimentally calculated helical friction factors with these theoretical values and procured a 1:1 correlation (slope=1.025) with an R2=0.9311 (Figure 7).

Figure 7. The friction factor for helical flow is fit to a model equation and compared to the experimental friction factor.

Discussion

Pressure losses through straight smooth cylindrical tubes occur as a result of friction, manifested as the loss of mechanical energy from some point 1 and some point 2 along the tube. Laminar flow through coiled smooth tubes expectedly encounters greater resistance. Given a paraboloidal velocity distribution, rapidly flowing axial parts of the flow swerve outward due to centrifugal motion, while slower sections that glide against the wall are forced inward towards to center. The effective friction factor in the coiled tubes was first approximated using the Darcy friction factor (applies to straight smooth tubes) which relates to Reynolds number: f(s) = 64/Re. In plotting the Darcy friction factor against the Dean number, we sought to find a similar correlation and obtained a power fit that deviated from 64/Re by less than an order of magnitude (<89% error). The Dean number does not differ from the Reynolds number except by a factor of, which implies that the deviation from the curve 64/Re must be attributed by this factor which depends on the ratio of the inner tube radius to the coil radius.

In the hope of better characterizing pressure flow through coiled smooth tubes, we invokes White’s model which is reflected in the following relationships:

; fw = White friction factor

The White friction factor was defined to be a helical friction factor that depends on the Dean number. By examining the effect on friction factor due to (1) coil orientation, (2) fluid viscosity, and (3) coil radius, we concluded that our data largely did not adequately coincide with White’s model. White approximated that the White friction factor to be. When we fitted our data to this functional form, we found that our coefficient and exponent often differed by several orders of magnitude. We investigated how coil orientation affects that friction factor and discovered that the difference was not statistically significant (p>0.05). When we fitted these curves to a power fit, we obtained coefficients that we at least five orders of magnitude less and exponents that were exceeded White’s estimate eight-fold. We investigated how the coil radius affects friction factor and noticed that curves corresponding to each radius varied from White’s approximation by several orders of magnitude. We suspect that for very small coil radii, pressure flow may be effectively impeded or occluded as the radius of the tube becomes comparable to the radius of curvature. At these radii, the fluid was forced into the coiled tubing. Only our power fit from the largest radius varied within one magnitude. We also noticed that we obtained our best fit when examining the effect of fluid viscosity, obtaining the power fit:

The coefficient only varied by 93% while the exponent varied by 134%. Nonetheless, it remained particularly clear that White’s model did not best suit our data.

Recent studies performed on pressure flow through helical tubes show that vortex flow may indeed be responsible for deviations of our data from idealized models like that of White. Vortex flow (swirling of fluid in non-straight tube) is generally caused by the effect of centrifugal force, which makes flow transitions from laminar to turbulent flow no longer constant but dependent on the Dean number. At high Dean numbers, the effect of centrifugal force becomes greater as tube diameter approaches coil diameter, forcing fluids to traverse sharper angle bends. Adjustments due to vortex flow have been proposed, as we use the empirical correction suggested by Huaming et al:

When we compared calculated helical friction factor values with theoretical values predicted by this empirical correction, we confirmed a 1:1 correlation with an R2=0.9311. Because applying this empirical correction enabled us to reasonably explain our data, we can say that vortex flow was the major contributor to errors we were observing when using (1) straight tube friction factor approximation and (2) White’s model.