I. GARCIA, Contributing Author, Houston, Texas; and A. GARCIA, Kuraray America Pasadena, Pasadena, Texas
Calculating pipe diameter through long, straight pipe under laminar flow regime is simple using Poiseuielle’s equation for Re < 2,000. It is known that real piping systems contain a considerable number of valves and fittings that cause resistance to the flow of fluids—this makes calculation more difficult since the system’s energy loss is the sum of the losses of pipe and fittings.
Most textbooks and papers discussing fluid mechanics facilitate the solution assuming that the pipe length is long enough that fittings can be disregarded, or by tedious trial and error methods using classical representative values of loss coefficients evaluated in conditions of fully turbulent flow that do not accurately reflect the dependence of loss coefficients on both the Reynolds number and the fitting size. Consequently, engineers tend to use fittings correlations that lead to inaccurate results when flow regime is laminar.1,2
For solution of these problems, this article presents a set of equations based on the three constants equation (3K equation) and the Crane method, using them to reflect the Reynolds number dependence and the fitting size. The Crane method uses the constant (L/D)e and representative loss coefficients for a particular type of fittings (Ko ). These equations are an excellent resource for calculating the equivalent of straight pipe for fittings and allow engineers to accurately calculate the pipe diameter in piping systems in laminar flow conditions where fittings cannot be ignored.
In this type of problem, the following information is known:
Physical properties of fluid (viscosity and density)
Flowrate
Number and type of valves and fittings installed in the piping system
Straight length of pipe
Maximum allowable pressure drop in the piping system.
In flow problems where viscosity is high, it is necessary to check whether the flow is laminar or not. For this reason, a critical head (hcr ) for Re = 2,000 is defined as follows (Eq. 1):
hcr = 0.0022 × (μ / ρ)5 × (Ls/q3) (1)
If the total head loss in the piping system is ≤ to this critical head, the flow regime through the pipe is laminar.
Equivalent length of pipe. The equivalent length of straight pipe to account for fittings can be estimated by the following equations (Eq.2, Eq.3 and Eq.4):
Le1 = 0.0027 × ∑ K1 × (μ ______ ρ×× Lℎs ×fq)0.25 (2)
Le2 = 0.56 × ∑(L / D)e × (μ/ρ)–1.0525 × q0.9475 × (Ls /hf )–0.0525 (3)
Le3 = 29.6 × ∑Ko × (ρ/μ) × q (4)
The adjusted length of pipe is then (Eq. 5):
La = Ls + ∑ Le1-3 (5)
The pipe diameter can be calculated now using the rearranged Poiseuielle’s law (Eq.6):
D = 0.17077 × [(μ × La × q) / (ρ × hf )]0.25 (6)
A pipe size is then selected by table, the most common being Schedule 40.
Checking the results. The mean velocity of flow, the Reynolds number, the friction factor, and the equivalent length of straight pipe to account for fittings are calculated using classical equations of fluid mechanics.3,4,5 The equivalent length of pipe (Eq.8) is checked using the 3K equation for each fitting (Eq. 7):
3K = (K1 / Re) + k2 × [1 + (K3 / dn0.3 )] (7)
Le = [(∑3K + ∑Ko ) / f ] × D (8)
The pipe diameter can also be checked using the Darcy’s Equation (Eq. 9):
D = 0.4788 × q0.4 × [f × (La/hf )]0.2 (9)
An important last step is to confirm that the maximum allowable pressure drop requirement is satisfied, according to Eq. 10:
∆P = 0.001295 × [(f × La~ρ × v2 )/d) (10)
A sample problem demonstrates the usefulness of this pipe diameter calculation.
A line is being run from a vessel to a heat exchanger. The flow will be 436 gallons per min (gpm) of oil (μ = 148 cP; ρ = 56 lb/ft3). The path includes ten 90° welded elbows r/d = 1, one globe valve, two gate valves and 190 ft of pipe. A loss coefficient of Ko = 0.5 will account for the vessel to pipe transition and Ko = 2 for the velocity head at the pipe exit if the flow turns out to be laminar. With all valves wide open, what pipe size is required if 11 psi of pressure drop is allowable for the run?
Flowrate can be calculated using Eq. 11:
q = 0.002228 × 436 = 0.9714 (11)
Available head loss can be calculated using Eq.12:
hf = 144 × (11/56) = 28.2857 say 28 (12)
Critical head can be calculated using Eq.13:
hcr = 0.0022 × (148/56)5 × (190/0.97143) = 59 (13)
Since hf > hcr , the flow is laminar. A summation is shown in TABLE 1.
The estimated of equivalent length of pipe can be calculated using Eqs. 14–17:
Le1 = 0.0027 × 10100 × [(148 × 190 × 0.9714) / (56 × 28)]0.25 = 56 (14)
Le2 = 0.56 × 556 × (148 / 56)–1.0525 × 0.97140.9475 × (190 / 28)–0.0525 = 99 (15)
Le3 = 29.6 × 2.5 × (56/148) × 0.9714 = 27 (16)
∑(56 + 99 + 27) = 182 (17)
The adjusted length of pipe can be calculated using Eq. 18:
La = 190 + 182 = 372 (18)
The pipe diameter can be calculated using Eq.19:
D = 0.17077 × [(148 × 372 × 0.9714) / (56 × 28)]0.25 = 0.4127 (19)
A 5-in. Schedule 40 pipe is selected. Pipe data includes: d = 5.047; D = 0.4206; and fT = 0.016. The results can be checked using Eqs. 20, 21 and 22:
v = 0.4085 × (436 / 5.0472 ) = 7 (20)
Re = 124 × [(5.047 × 7 × 56) / 148] = 1657.6 Re < 2,000, the flow is laminar) (21)
f = 64 / 1657.6 = 0.03861 (22)
TABLE 2 shows the results of applying the 3K equation for fittings.
An equivalent length of pipe (Le) and actual length of pipe (La) are calculated using Eqs. 23 and 24:
Le = [(14.975 + 2.5) / 0.03861] × 0.42058 = 190 (23)
La = 190 + 190 = 380 (24)
Pipe diameter is calculated using Eq. 25:
D = 0.4788 × 0.97140.4 × [0.03861 × (380/28)]0.2 = 0.416 (25)
Pressure drop is calculated using Eq. 26:
∆P = 0.001295 × [(0.03861 × 380 × 56 × 72) / 5.047] = 10 (26)
Takeaways. The proposed given equations in this paper allow the accurate determination of the required pipe diameter under a laminar flow regime where fittings cannot be disregarded. These equations are based on the Ron Darby correlation (3K equation) that accurately reflects the dependence of loss coefficients on both the Reynolds number of the flow and the fitting size. Pipe sizing in this article is also based on process conditions in terms of the maximum allowable pressure drop required for the piping system. HP
NOMENCLATURE
d = Internal diameter of pipe, in.
D = Internal diameter of pipe, ft
dn = Nominal diameter of pipe, in.
f = Friction factor
fT = Friction factor in zone of complete turbulence
hf = Total head loss, ft
ℎcr = Critical head, ft
Ko = Fitting loss coefficient for a specific type of fitting
K1 = First constant in the 3K equation
K2 = Second constant in the 3K equation
K3 = Third constant in the 3K equation
3K = Three constants equation
Le = Equivalent length of straight pipe for fittings, ft
Ls = Straight length of pipe, ft
La = Adjusted length of pipe, ft
n = Number of fittings installed in the piping system
P = Pressure, psi
q = Flowrate, ft3/sec
Q = Flowrate, gpm
v = Velocity of flow, ft/sec
(L⁄D)e = Equivalent length of pipe in number of pipe diameters
μ = Dynamic viscosity in cP
ρ = Density of fluid, lbm/ft3
∆ = Differential
LITERATURE CITED
Churchill, S. W., “Friction factor equations span all fluid flow problems,” Chemical Engineering, November 1977.
Crane Valve Group (CVG), “Flow of fluid through valves, fittings and pipe,” Crane Technical Paper No. 410 (TP-410), 1988.
Darby, R., “Correlate pressure drop through fittings,” Chemical Engineering, April 2001.
Mott, R. L., Applied Fluid Mechanics, 4th Ed. Prentice Hall, PTR, 1996.
Verma, C. P., “Solve pipe flow problems directly,” Hydrocarbon Processing, August 1979.
Israel Garcia graduated from the University of Cienfuegos, Cuba, with an MS degree in mechanical engineering. Rodriguez has been attached to the mechanical engineering faculty of that university since 1985 as a professor in fluid mechanics, heat transfer and science materials. Garcia has more than 30 yr of industrial experience in chemical plants and power stations and has presented several papers on the design of heat exchangers, pressure vessels and piping systems. He now works as a Consulting Engineer in Houston, Texas. The author can be reached at isgaro47@gmail.com.
Alejandro Garcia is a mechanical engineer who graduated from the University of Cienfuegos, Cuba. He received his MS degree in mechanical engineering with a specialty in materials from the Autonomous University of Nuevo Leon, Mexico. He gained several years of experience in chemicals and power plants as a static and dynamic equipment specialist. He now works in the Kuraray America Pasadena, Texas Plant as Project Engineer II. The author can be reached alessandromilan88@gmail.com.
