Calculate flowrate through valves, fittings and pipe under a laminar flow regime

I. Garcia, University of Cienfuegos, Cienfuegos, Cuba; and A. GARCIA, Kuraray America Inc. EVAL Plant, Pasadena, Texas

Calculating flowrate through long, straight pipe under laminar flow regime is easy using standard equations of fluid mechanics. Real piping systems, as it is known, usually contain a considerable number of valves and fittings that cause resistance to the flow of fluids; if the flow regime is laminar, the loss coefficient of fittings is highly dependent on both the Reynolds number of the flow and the fitting size. Therefore, the system’s energy loss is the sum of the losses of pipes and fittings.

This article presents a combined equation based on the 3K equation and the Crane method, which allow the accurate calculation of the equivalent length of pipe to account for valves and fittings. This equation, therefore, becomes a very useful tool to accurately determine the flowrate in piping systems where fittings cannot be ignored. The first part of this equation properly accounts for the Reynolds number dependance, while the other reflects the effect of fittings size. In this type of problem, the following information is known:

1. Physical properties of the fluid (density and viscosity)
2. Total pressure drop or total frictional losses in the piping system
3. Nominal pipe size and its schedule
4. Straight length of pipe
5. Quantity and type of valves and fittings installed in the piping system
6. Elevation of pipe above reference level (if any).

In flow problems where viscosity is high, it is necessary to  determine whether the flow is laminar or not. For this reason, a critical head for Re = 2,000 is defined by the following expression (Eq. 1):

If the total frictional losses in the piping system are less than or equal to this critical head, the flow regime through the pipe is laminar. Eq. 1 predicts laminar flow in pipe systems where frictional losses can reach up to 50% of total losses.

Equivalent length of pipe. The equivalent of straight pipe—considering fittings, based in the 3K equation and the Crane method—can be calculated by the following combined equation (Eq. 2):

where (Eq. 3):

As an alternative, ∑K_t  can also be calculated for all fittings using the values of constants K₂ and K₃ in the 3K equation, as follows (Eq. 4):

The adjusted length of pipe is then (Eq. 5):

L_a = L_s + L_e            (Eq. 5)

Flowrate. The flowrate for laminar flow can be obtained using the rearranged Poiseuille’s equation (Eq. 6) or (Eq.7):

Checking the result. The mean velocity of flow, the Reynolds number and the friction factor can be calculated by classic equations of fluid mechanics, and the equivalent length of pipe (Eq. 9) is checked using the 3K equation for each fitting (Eq. 8):

Finally, the flowrate can now be checked by using the rearranged Darcy’s equation, which applies to any flow regime (Eq. 10) or (Eq. 11):

Sample problem. An open tank contains 23 ft of oil head (ρ = 55 lb_m ⁄ ft³, μ = 98 Cp). The tank drains through a piping system containing ten 90° threaded standard elbows (r/d = 1), one conventional globe valve, two gate valves and 131 ft of 3-in. SCH 40 steel pipe. The top surface of the tank and the discharge are both at atmospheric pressure. An entrance loss factor (inward projecting) of K₀ = 0.78 will account for the tank to pipe transition and K₀ = 2 for the velocity head considered in the Bernoulli equation at the pipe exit if flow regime turns out be laminar. All common 3-in. nominal diameter will be used for all fittings to calculate the resulting flowrate in gpm (gallon per minute) and ft³/sec with all valves wide open.

The total frictional losses applying the Bernoulli equation (Eq. 12):

h_f = Z = 23 ft          (Eq. 12)

Where, the pipe data is:

Dn = 3 in., d = 3.068 in., D = 0.2557 ft and f_t = 0.018.

The critical head is (Eq. 13):

Since h_f < h_cr, the flow is laminar. TABLE 1 shows data for calculating ∑K_t.

The equivalent length of pipe and adjusted length of pipe are calculated in Eqs. 14 and 15, respectively:

Using Eq. 4, the equivalent length of pipe calculated by Eq. 2 is shown below.

∑K_t = 11.81 + 2.78 = 14.59

L_e = 106.5 ft          L_a = 237.5 ft

TABLE 2 summarizes the constants K₂ and K₃.

The flowrate is calculated using Eqs. 16 and 17:

The results are checked using Eqs. 18 and 19:

For an equivalent length of pipe applying the 3K equation (Eqs. 20–23):

TABLE 3 shows the results of the 3K equation.

The difference in flow is small enough to forego any correction.

Takeaways. The combined 3K equation avoids tedious trial and error methods, reduces execution time and finally allows the accurate calculation of flowrate through valves, fittings and pipe under laminar flow regime for any type of fluid. The combined 3K equation appears to be complicated, but it leads to a quadratic expression with a solution of (Eq. 24):

The values a, b and c are obtained from data known in the problem.^1–3 The values of constants (L⁄D)_e , f_t and K_o for valves and fittings can be found in the extensive tabulation of the Crane Technical Manual,⁴ whereas the values of the three constants K₁, K₂ and K₃, are in articles and papers by Darby,⁵ the author of the excellent 3K equation. HP

NOMENCLATURE

LITERATURE CITED

1. Churchill, S. W., "Friction factor equation spans all fluid flow regime," Chemical Engineering, November 1977.
2. Mott, R. L., Applied fluid mechanics, 6th Ed., Pearson College Div., 1996.
3. Verma, C. P., "Solve pipe flow problems directly," Hydrocarbon Processing, August 1979.
4. Crane Co., “Flow of fluids through valves, fittings and pipe,” New York, 1988.
5. Darby, R., “Correlate pressure drop through fittings,” Chemical Engineering, April 2001.

ISRAEL GARCIA is a mechanical engineer who graduated from the University of Cienfuegos, Cuba, and has been attached to the Mechanical Engineering faculty of that University since 1985 as a Professor in fluid mechanics, heat transfer and science materials. He has more than 30 yr of experience in chemical plants and power stations and has presented several papers that deal with the design of heat exchangers, pressure vessels and piping systems.

ALEJANDRO GARCIA is a mechanical engineer who graduated from the University of Cienfuegos, Cuba, and received his MS degree in mechanical engineering with a specialty in materials at the Autonomous University of Nuevo Leon, Mexico. Garcia gained several years of experience in power plants and the automotive industry as a static and dynamic equipment specialist. He works at Kuraray America Inc. EVAL Plant in Pasadena, Texas as a Project Engineer.