New method predicts turbulence for non-Newtonian fluids in pipelines

An excellent agreement was obtained between the values calculated from the new criterion and experimental data of Metzner and Reed and Dodge and Metzner.^{1 2}

Turbulence onset

For Newtonian fluids, the onset of turbulence is set by calculating the Reynolds number. In the case of non-Newtonian fluids, there exist two criteria: the values of the critical Reynolds number and the stability parameter. Both criteria determine the end of the laminar region and not the beginning of the turbulence.

Metzner and Reed first investigated the laminar-turbulent-transition criterion and determined the critical Reynolds number by plotting the friction factor of power-law fluids against the generalized Reynolds number under laminar flow conditions.

They concluded that the power-law fluids leave the region of stable laminar flow when the friction is ≤0.008 and when the critical Reynolds number is >2,000.¹

Dodge and Metzner, Shaver and Merrill, and Metzner and Park performed extensive experiments for power-law fluids under turbulent conditions. They plotted the friction factor against the generalized Reynolds number.

The critical Reynolds numbers determined from their charts were in close agreement with those obtained from the Metzner and Reed chart.^2-4

Inasmuch as the critical Reynolds number criterion was only limited to one type of pseudo-plastic fluids (power-law fluids), it cannot be used for the other 19 types of pseudo-plastic fluid equations.⁵

The stability parameter criterion developed by Ryan and Johnson is shown in Equation 1 in which the value of z is equal to 0 at the pipe center and at the pipe wall. It reaches the maximum value at the critical radial position r_c at which the end of the laminar region can be fixed from the laminar velocity distribution.⁶

For power-law fluids, Ryan and Johnson derived the equation shown in Equation 2.

Inasmuch as the value of z changes as the laminar velocity distribution u changes, the stability parameter is not a generalized one for all types of pseudo-plastic fluids. Hence, a generalized criterion for all types of pseudo-plastic fluids is necessary to establish realistic pipeline design calculations.

The correlation developed in this study for pseudoplastic crude oils (non-Newtonian yieldless fluids) is a new one, but the approach followed to get such a correlation is analogous to that previously ouτlined for non-Newtonian yield fluids (Bingham, yield pseudoplastic and yield dilatent).¹⁰

The correlation can be used by the design engineers who worked in pipeline transportation of non-Newtonian for predicting the beginning of turbulent flow, which cannot be obtained by previous methods.

The previous methods only predict the end of laminar flow or the beginning of transition flow, which leads to serious errors in pipeline design calculations because laminar and transition zones become one region.

Analysis

A different criterion from those previously discussed is presented here. Both the critical Reynolds number and stability parameter criteria determine the end of the laminar region for one type of pseudo-plastic fluids.

The criterion developed in this study, however, can be used to determine the onset of turbulence for all types of pseudoplastic fluids. The viscous interaction coefficient proposed by Zand and Rust and Murthy and Zandi was used to derive the new criterion, shown in Equation 3.^{7 8}

The viscous interaction coefficient λ signifies the concept that, as a result of turbulence and chaotic motion, the viscous shear stress t^t_v will be considerably amplified and yields much higher values than the laminar shear stress t¹. The value of λ is equal to unity at the laminar flow conditions.

Under turbulent pipe flow conditions, λ is still equal to unity in the laminar sublayer and should be greater than unity in the turbulence core. Conceptually, it has no upper bound. The term t¹ can be expressed (Equation 4) in terms of the apparent viscosity for all types of pseudo-plastic viscosity.^{9 10}

Metzner and Reed proposed the equation (Equation 5) for the generalized Reynolds number.¹ Substituting Equation 5 in Equation 4 yields Equation 6.

Kundson and Katz defined the viscous shear stress as shown in Equation 7.¹¹

The friction factor f can be deduced from the generalized pressure-loss equation developed by Desouky and El-Emam, as shown in Equation 8.12 Substituting Equation 8 in Equation 7 yields Equation 9.

Substituting Equations 6 and 9 in Equation 3 yields the interaction coefficient (λ) given by Equation 10. This equation can be applied to the laminar sublayer to determine the thickness of that layer (Equation 11).

The thickness of the laminar sub-layer is plotted against the generalized Reynolds number in Fig. 1. This figure shows that the thickness of the laminar sublayer increases as the value ofn’ or N_reg decreases.

Inasmuch as the values of n’ for all pseudo-plastic fluids are always greater than 0.1 , a practical maximum value for the term (1-r/R) can be set atn’ = 0.1. Equation 12 gives the relation between (1-r/R) and N_reg. Substituting Equation 12 into Equation 10 yields the proposed new equation, Equation 13.

Calculations procedure

The following steps are used to program Equation 13:

1. Transfer the basic shear data (τl, du/dr) into the flow shear data (τlw, [8v/D]) using Equation 14.⁹
2. The values of n’ and K9 are given by Equation 15. K9 = The inter_cept of the resulting straight line in the plot of log(τlw) vs. log(8V/D) at 8V/D = 1. It can easily be obtained by use of the regression analysis method.

2. Using the crude oil density (ρ) and pipeline characteristics (V and D), determine the generalized Reynolds number with Equation 16.
3. With the values ofn’, K9, and N_reg, the value of λ can easily be determined from Equation 13.

Verification

Hungarian Algyo pipeline¹³ data were used to assess the validity of Equations 12 and 13.

The length of the pipeline was divided into seven sections, and then the temperature, density, and the rheological properties of Algyo crude oil were measured at the midpoint of each section.

The rheological properties, plotted in Fig. 2, were used to calculate the values of k9 andn’. The value of l was calculated with Equation 13, and the type of flow in each section was then determined (Table 1).

The values of N_regc were calculated with Equation 2 and compared with the corresponding value of N_regto determine the type of flow in each section. The results appear in Table 2. This table shows that Equation 13 can determine the type of flow in each section as efficienτly as Equation 2.

Data comparison

A comparison was made between the critical Reynolds number Nregc calculated from Equation 13 and those measured by Metzner and Reed and Dodge and Metzner.^{1 2}

Because the experimental data of the critical Reynolds number were determined at the end of the laminar region, the value of λ is equal to unity at N_regc= _Nreg.

Thus Equation 13 can be rearranged as shown in Equation 17. The values of N_regc calculated from Equation 17 and those measured by Metzner and Reed and by Dodge and Metzner^{1 2} are plotted in Fig. 3.

This figure shows an excellent agreement between the calculated critical Reynolds numbers and the measured ones with average absolute relative errors of 3.2% for Metzner and Reed data and 4.7% for Dodge and Metzener data.

References

1. Metzner, A.B., and Reed, J.C., "Flow of non-Newtonian fluids-correlations of the laminar, transition and turbulent flow regions," American Institute of Chemical Engineering Journal, 1955, Vol. 1 (No. 4), pp. 434-40.
2. Dodge, D.W., and Metzner, A.B., "Turbulent flow of non-Newtonian system," American Institute of Chemical Engineering Journal, 1959, Vol. 5 (No. 2), pp.189-204.
3. Shaver, R.G., and Merrill, E.W., "Turbulent flow of pseudo-plastic polymer solutions in straight cylindrical tubes," American Institute of Chemical Engineering Journal, 1959, Vol. 5 (No. 2), pp. 181-88.
4. Metzner, A.B., and Park, M.G., "Turbulent flow characteristics of visco-plastic fluids," Journal of Fluid Mechanics, 1964, Vol. 20 (No. 2), pp. 291-303.
5. Desouky, S.E.M., and El-Emam, N.A., "Program designs for pseudo-plastic fluids," OGJ, Dec. 18, 1989, p. 48.
6. Ryan, N.W., and Johnson, M.M., "Transition from laminar to turbulent flow in pipes," American Institute of Chemical Engineering Journal, 1959, Vol. 5 (No. 4), pp. 433-35.
7. Zandi, I.M.A., and Rust, R.H., "Turbulent non-Newtonian velocity profiles in pipes," Journal of Hydrol. Div., 1965, Vol. 91 (No. 6), pp. 37-55.
8. Murthy, V.R.K., and Zandi, I.M.A., "Turbulent flow of non-Newtonian suspension in pipes," Journal of Engineering Mech. Div., 1965, Vol. 95 (EMI), pp. 271-88.
9. Govier, G.W., and Aziz, K., The Flow of Complex Mixtures in Pipes. New York: Van Nostrand Reinhold, 1977.
10. Desouky, S.E.M., and Al-Awad, M., "A New laminar-Turbulent Transition Criterion for Yield Fluids," Journal of Petroleum Science and Engineering, 1997, Vol. 19 (Nos. 3-4), pp. 171-76.
11. Kundson, J.G., and Katz, D.L., Fluid Dynamics and Heat Transfer. New York: McGraw-Hill, 1958.
12. Desouky, S.E.M., and El-Emam, N.A., "Turbulent non-Newtonian fluids in pipes," Journal of Can. Pet. Tech., 1990, Vol. 29 (No. 5), pp. 48-54.

13. El-Emam, N.A., "The Rheological characteristics of non-Newtonian thixotropic crude oil under pressure," PhD thesis, Technical University, Heavy Industries, Miskolc, Hungary, 1979.

The author

Saad El-Din M. Desouky ([email protected]) is assistant professor in the Petroleum Department, College of Engineering, King Saud University, Riyadh, Saudi Arabia. Previously, he was assistant professor in the Egyptian Petroleum Research Institute, Cairo, conducting core analysis and studies for oil companies in Egypt. His main interests are reservoir engineering and pipeline transportation and storage of Newtonian and non-Newtonian fluids. He holds BSc, MSc, and PhD degrees in petroleum engineering from Al-Azhar University (Cairo) and from Cairo University.