Calculation of flow velocity in pipes

What is the material and type of pipe, and is there a pipe resistance coefficient? Is the medium a liquid or a gas? What type of medium is it, and what is its viscosity?

Calculation of pressure loss in pipes: When a viscous liquid flows, resistance exists, and energy must be expended to overcome this resistance, resulting in energy loss. In hydraulic transmission, energy loss is primarily manifested as pressure loss, which is the meaning of the hw term in Bernoulli’s equation for actual fluid flow. Pressure losses in hydraulic systems can be divided into two categories. One is the pressure loss that occurs when the oil flows through straight pipes of constant diameter, and this is known as frictional pressure loss. This type of pressure loss is caused by the internal and external friction forces during liquid flow. Another type is the local pressure loss, which occurs when the fluid flows past local obstacles such as elbows, joints, or sudden expansions or contractions in the pipe cross-section. Due to the sudden changes in the direction and velocity of the flow, vortices are formed locally, leading to collisions between fluid particles as well as between these particles and the solid wall surfaces, thereby generating intense friction and resulting in pressure losses. Excessive pressure loss means an increase in power loss within the hydraulic system, which leads to increased heating of the oil, higher leakage rates, reduced efficiency, and deteriorated performance of the hydraulic system. In hydraulic technology, the purpose of studying resistance is: ① to correctly calculate the resistance in hydraulic systems ； ②To find ways to reduce flow resistance ； ③To utilize the pressure difference p generated by resistance to control the operation of certain hydraulic components. I. Pressure loss when liquid flows in a straight pipe The pressure loss that occurs when liquid flows in a straight pipe is caused by friction associated with the flow of the liquid; this is known as frictional pressure loss. It depends primarily on factors such as the length of the pipe, its inner diameter, the flow velocity of the liquid, and its viscosity. The pressure loss along the flow path varies depending on the flow regime of the liquid. Laminar flow of fluid in circular pipes is the most common in hydraulic systems; therefore, when designing hydraulic systems, it is often desired that the fluid flow within the pipes remain in a laminar state. 1. Pressure loss in laminar flow: In hydraulic transmission, the flow state of the liquid is mostly laminar flow; under such conditions, the pressure loss as the liquid flows through a straight pipe can be calculated theoretically. ? Figure 2–21 Laminar flow in a circular pipe (1) Velocity distribution pattern of the fluid across the cross-section. As shown in Figure 2-21(a), the liquid flows laminarly in a circular tube of diameter d. The tube is placed horizontally, and a small cylinder whose axis coincides with that of the tube is taken inside it; its radius is r and its length is l. The forces acting on this small cylinder in the direction of the tube axis are: the pressure at the left end, p1, and the pressure at the right end, p2. The frictional force on the cylindrical surface is Ff. Therefore, the equation of force equilibrium is: ? (2-44)? From equation (2-6), it can be seen that: ? (2-45)? Here, μ represents the dynamic viscosity. ? Since the velocity increment du and the radius increment dr have opposite signs, a negative sign is added to the equation. Furthermore, ?Δp＝p1- p2?. By substituting ?Δp and equation (2-45) into equation (2-44), we obtain: (2-46). Integrating equation (2-46) gives: ? (2-47). When r＝R, u＝0; substituting this value into equation (2-47) yields: ?. Thus, (2-48). From equation (2-48), it can be seen that the flow velocity u within the pipe varies in a parabolic pattern along the radial direction, with the maximum velocity occurring at the axis, and its value is: ? (2-49). (1) (1) The flow rate in the pipe. The volume of the projectile shown in Figure 2-21(b) is the volume of liquid that passes through the flow cross-section per unit time, that is, the flow rate. To calculate its volume, the area of a tiny circular ring with radius r and thickness dr can be considered; the flow rate through this ring is given by equation (2-50). By integrating equation (2-50), the flow rate q can be obtained: equation (2-51). (2) Average flow velocity. Let the average flow velocity inside the pipe be υ. By comparing equation (2-52) with equation (2-49), the relationship between the average flow velocity and the maximum flow velocity can be obtained: υ = ?? (2-53). (4) Pressure loss along the pipe. In the laminar flow state, the pressure loss along a straight pipe as the liquid flows through it can be determined using equation (2-52): ? ? (2-54). It can be seen from equation (2-54) that, in laminar flow, the pressure loss associated with the flow of liquid through a straight pipe is proportional to the dynamic viscosity, pipe length, and flow velocity, and inversely proportional to the square of the pipe diameter. ? When actually calculating the pressure loss, to simplify the calculations, from equations (2-8) and (2-41), we obtain μ = υdρ/Re. Substituting μ = υdρ/Re into equation (2-54), and multiplying both the numerator and denominator by 2g, we get: ? (2-55)? Here, λ is the friction factor along the flow path. Its theoretical value is λ=64/Re, but due to various factors, for smooth metal pipes λ=75/Re is used, and for rubber pipes λ=80/Re is used. 2. Pressure loss in turbulent flow: In laminar flow, the particles move in a regular manner along the axial direction. And there is no lateral movement. One of the important characteristics of turbulence is that the various particles in the liquid no longer move in a regular axial manner; instead, they mix with each other and exhibit fluctuations during their motion. This highly irregular motion causes collisions between particles and the formation of vortices, resulting in much greater loss of turbulent energy compared to laminar flow. Due to the complexity of turbulent flow phenomena, satisfactory results have not yet been achieved through purely theoretical approaches; therefore, experimental methods are still used, supplemented by theoretical explanations. As a result, the pressure loss in liquid flow under turbulent conditions is still calculated using equation (2-55). The value of λ in this equation depends not only on the Reynolds number Re but also on the roughness of the pipe wall surface, Δ; the specific values of λ are given in Table 2-5. Table 2-5 λ values for turbulent flow in circular pipes 2. Local pressure loss: Local pressure loss is the pressure loss that occurs when a fluid flows through valve openings, bends, or areas where the cross-sectional area changes. As the fluid flows through these areas, changes in both the direction and speed of the flow result in the formation of vortices (as shown in Figure 2-22). This causes the particles in the fluid to collide with each other, leading to significant energy loss. Figure 2-22 Local losses at sudden expansions. The formula for calculating the local pressure loss can be expressed as follows: = 2 /2 (2—56). In this equation, is the local resistance coefficient; its value can be determined theoretically only when the fluid flows through a suddenly expanding cross-section, while in other cases it must be determined through experiments ； It refers to the average flow velocity of a liquid, and generally denotes the velocity downstream of the local resistance. 3. Total pressure loss and efficiency in piping systems The total pressure loss in a piping system is equal to the sum of all frictional losses along the pipeline plus all local losses, that is: = + = + (2–58) Table 2: Pressuring values for pressure resistance tests Rated pressure test pressure for flanges: JIS 10K – 2.1 MPa (21 kgf/cm2); JIS 20K – 5.0 MPa (50 kgf/cm2). Parameter selection: ※ Range of measurable flow rates (including values outside the accuracy guarantee range); ※ Flow rate range within which accuracy is guaranteed (within ±1.0%). Minimum flow rate, Maximum flow rate: 0.2 m/s, or the flow rate obtained when the Reynolds number is 5000, whichever is higher; 6 m/s. Minimum flow rate, Maximum flow rate: 0.2 m/s (0.3 m/s at 25 A), or the flow rate obtained when the Reynolds number is 2000, whichever is higher; 6 m/s. ※ Flow rate range within which accuracy is guaranteed (within ±1.0%) (applicable only to/HAC, except for 25 A). Minimum flow rate, Maximum flow rate: 0.2 m/s, or the flow rate obtained at a Reynolds number equal to the nominal diameter × 1000, whichever is higher; 6 m/s; or the flow rate obtained at a Reynolds number equal to the nominal diameter × 4000, whichever is lower. ※ The flow rate at a Reynolds number of 5000 can be determined using Figure 4; by multiplying this value by 4, the flow rate at a Reynolds number of 20,000 can be calculated. Table 3: Nominal pulse rate and K coefficient
Table 4: Operating range at 20°C – Inner diameter of flow passage (mm), nominal K coefficient (Pulse/l), nominal pulse rate (Hz/m/s, Hz/m³/h)
15A: 12.85, 4069.51, 50
25A: 23.48, 7.03, 7.42, 4.2
40A: 36.62, 22.72, 3.96, 6.31
50A: 47.51, 10.41, 18.42, 2.89
80A: 71.03, 1.11, 12.30, 0.863
100A: 93.81, 0.35, 9.32, 0.375
150A: 138.80, 0.42, 76.47, 0.119
200A: 185.60, 0.17, 94.84, 0.050

Measurement range of flow rate or volume flow rate (m³/h), guaranteed accuracy range (±1.0%), guaranteed accuracy range (±0.5%)
15A: 0.23–2.77, 0.55–0.77 –
25A: 0.34–9.21, 1.4–9.2 –
40A: 0.76–222, 2.1–224, 4.2–16
50A: 1.3–382, 2.7–386, 6.8–6
80A: 2.9–854, 4.1–851, 17–4
100A: 5.0–1495, 5.4–1492, 27–106
150A: 11–3261, 11–3265, 9–235
200A: 20–58320, 58310, 5–419

Method for calculating volumetric flow rate:
· Qf = υ × D²/354 or Qf = 3600 × υ × s

Method for calculating flow velocity when the Reynolds number is 5000:
· υ = 5 × υ/D
· Re = (354 × 10³ × Qf) / (υ × D)
· υ = (μ × 10³) / ρf

Where:
Qf = Volumetric flow rate (m³/h)
D = Inner diameter of ULTRA YEWFLO (mm)
S = Inner area of ULTRA YEWFLO (m²)
υ = Flow velocity (m/s)
Re = Reynolds number (unitless)
ρf = Density under operating conditions (kg/m³)
μ = Viscosity under operating conditions (cP)
V = Kinematic viscosity under operating conditions (cSt)

Figure 4: Relationship between minimum flow velocity and kinematic viscosity (when Reynolds number is 5000)

Pressure loss: When the flow velocity is 6 m/s, the pressure is 3.9 kPa (0.4 kgf/cm²). The pressure loss can be calculated using the following formulas:
△P = 108 × 10⁻⁵ × ρf × V²……………①
(△P = 1.1 × 10⁻⁵ × ρf × V²)
△P = 135 × ρf × Qf²/D⁴………………②
(△P = 1.38 × ρf × Qf²/D⁴)

Where:
△P = Pressure loss [kPa (kg·f/cm²)]
ρf = Density of the fluid under operating conditions, kg/m³
V = Flow velocity (m/s)
Qf = Volumetric flow rate under operating conditions (m³/h)
D = Inner diameter of the flow meter (mm)

Figure 5: Relationship between pressure loss and flow rate. When the adjacent pipe is of Sch80 grade, the calculated pressure loss is about 10% lower. Example: Calculation of pressure loss: For a nominal diameter of 50A, with warm water at 80°C flowing at a rate of 20 m3/h. ① The density of warm water at 80°C is 972 kg/m3; using Equation 2, we get △P = 135 × 972 × 20² / 47.52 = 10.3 kPa (0.105 kg/cm²). ② Using Equation 1: V = 354 × Qf / D² = 354 × (20 / 47.52) = 3.14 m/s; △P = 108 × 10⁻⁵ × 972 × 3.14² = 10.3 kPa. ③ Using Figure 5: C = 10.8 (as read from Figure 5); △P = 98.1 × C × ρf × 10⁻⁵ = 98.1 × 10.8 × 972 × 10⁻⁵ = 10.3 kPa (0.105 kg/cm²). ※ Cavitation (minimum back pressure in the pipeline): When measuring fluids, if the pipeline pressure is low and the flow velocity is high, cavitation can occur, which prevents accurate measurement of the flow rate. To prevent this phenomenon from occurring, the minimum back pressure in the pipeline can be calculated using the following formula: P=3.8×△P+1.3×Po…………③ Where P represents the pipeline pressure at a distance of 2–7D downstream of the flow meter. 　　 ［kPa abs(kgf/cm2abs)］ ΔP: Pressure loss; Po: Insulation and vapor pressure of the fluid under operating conditions. ［kPa abs(kgf/cm2abs)］ Example: Verification of the presence or absence of cavitation. 　　In the example above, the pipeline pressure is 150 kPa abs, and the flow rate ranges from 0 to 2 m3/h; therefore, by determining the value at the maximum flow rate, we can determine whether cavitation occurs or not. The saturated vapor pressure of water at 80°C is 47.4 kPa, as indicated by the saturated vapor tables. Using equation 3, we get: P = 3.8△P + 1.3Po = 3.8×10.3 + 1.3×47.4 = 101 kPa. Since the pipeline pressure (150 kPa) is higher than the minimum back pressure required, cavitation will not occur. Figure 5 Relationship between pressure loss and flow rate △P=98.1×C×ρf×10-5 △P: Pressure loss (kPa) ρf: Density (kg/m3) Precautions for installation: ※Installation direction The pipeline should always be filled with liquid. When installed vertically, the flow direction of the liquid should be from bottom to top. The gas-liquid two-phase condition must be avoided, otherwise it will cause instability in the zero point. ※Adjacent pipes: It is recommended to use pipes of grade Sch80 or lower. ※Position of the valve and straight pipe length: The valve should be installed on the downstream side of the flow meter. For the required straight pipe length, refer to the following criteria; at least 5D of straight pipe length should be ensured on the downstream side. 　If the valve must be installed upstream of the flow meter, the straight pipe length on the upstream side should be 20D or more (30D or more when /HAC applies), and the straight pipe section on the downstream side should be 5D or more. ※Length of straight sections before and after the reducing/expanding pipe: For reducing pipes, there should be a straight section of at least 5D on the upstream side of the flow meter (10D or more when using /HAC), and a straight section of at least 5D on the downstream side. For expanding pipes, there should be a straight section of at least 10D on the upstream side of the flow meter (20D or more when using /HAC), with a straight section of at least 5D on the downstream side. ※ Length of straight sections at bends: There should be a straight section of at least 10D on the upstream side of the flow meter, and at least 5D on the downstream side. ※Piping with pulsating pressure ●Due to the influence of the pulsating pressure from the pump, the valve is installed on the upstream side of the flow meter. ●Due to the effect of the pulsating pressure from the T-shaped piping, the valve should be installed on the upstream side of the flow meter (V1'). ※If it leads directly into an open container on the downstream side, please refer to the piping diagram below. (Raise the piping on the downstream side.) ※Since scale tends to accumulate inside the flow meter housing, the inner walls should be cleaned regularly. ※To ensure the flow accuracy of the flow meter, it should be avoided to have gaskets protruding inside the pipeline. Even for the clamping type, a gasket with bolt holes as shown in the figure below should be used. ※When insulating the piping for high-temperature fluids, do not wrap insulation material around the converter support.

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