There are two types of hydraulic resistance: friction resistance and local resistance. In the former case hydraulic resistance is due to momentum transfer to the solid walls. In the latter case the resistance is caused by dissipation of mechanical energy when the configuration or the direction of flow is sharply changed, by the formation of vortices and secondary flows as a result of the flow breaking away, by the centrifugal forces, etc. To categorize local resistances, we usually refer the resistances of adapters, nozzles, extension pieces, diaphragms, pipeline accessories, swivel knees, pipe entrances, etc.
In defining a total resistance (pressure loss Δp_{f}) a conditional superposition is used
The friction resistance (pressure differential along the channels length) is calculated from Darcy's empirical formula
where is the Moody friction factor (4 times the Fanning friction factor - see Friction Factor), 1 and D_{H} = 4S/P are the length and the hydraulic diameter of the channel, ρ is the fluid density, and u is the mean velocity of flow.
In order to define the local hydraulic resistance (ΔP_{1}) the Weisbach formula is used
where ζ is the coefficient of local resistance.
For flow in smooth channels the friction factor f depends on the flow conditions and is only a function of Re = ūD_{H}/ν. For a laminar flow, the value for straight pipes is determined from the Poiseuille formula:
The values of C depend on the shape of the section and are given in Table 1.
We can see from Eq. (2) that in a laminar flow the pressure differential varies with the mean velocity of motion to the first power: a linear law of resistance (area I, Figure 1). In a turbulent flow the hydraulic friction resistance increases sharply (area II). Such a rise in the resistance is due to the heavy loss of energy associated with pulsating motion of turbulent vortices in the fluid flow. The value of in a turbulent flow in a round pipe may be calculated from the Blasius formula for 5 × 10^{3} ≤ Re ≤ 10^{5}
and from the Nikuradze formula for 10^{5} ≤ Re ≤ 4 × 10^{6}
The above formulas are valid for flow in channels with smooth walls with fully developed hydraulic and thermal conditions. In the inlet zone of the channel (up to 20D_{H} long) has a higher value than that calculated by Eqs. (5) and (6). The friction factor is affected by variations of fluid physical properties caused by variations in temperature and by the action of buoyancy forces.
In rough channels the hydraulic resistance increases due to formation of vortices at the roughness elements leading to additional loss of flow specific energy. Three types of roughness can be distinguished:
Natural roughness, which is formed as a result of long operation of pipelines.
Sand roughness, characterized by a high density and various forms of nodules.
Artificial (or regular) roughness, when the elements of roughness have a particular geometrical shape and location.
Each type of roughness has its own specific character of variation of the resistance friction coefficient with Re. In the case of sand roughness the ratio of the pipe radius r_{0} to the mean protruberance height δ_{r} on the wall surface (k = r_{0}/δ_{r}) is taken as the roughness parameter. Up to a certain value of Re, the resistance of the rough pipe varies in the same manner as for a smooth one (Figure 2) (in a laminar flow it varies according to Eq. (4) (curve 1) in a turbulent flow, according to Eq. (5) (curve 2). This is because at first the thickness of the laminar sublayer near the wall δlam exceeds the average height of the roughness protruberances. (δ_{lam} > δ_{r}). As Re increases further, δ_{r} becomes greater than δ_{lam} . This brings about an increase in the friction resistance of a rough pipe as compared to a smooth one above a certain transition number Re_{tr}, whose value depends on the roughness parameter: Re_{tr} 100k. For Re > Re_{tr} (self-similarity flow) a square law of resistance is observed, when the friction resistance coefficient depends only on the value of the parameter k (curve 3 in Figure 2): . The value of for pipes with commercial roughness can be evaluated from the Colebrook-White formula
Here k_{s} is the equivalent sand roughness, which for new pipes drawn out of ferrous metals is about 0.01 mm and for new steel pipes is about 0.014 mm; after some years of operation, it increases up to about 0.2 mm. For old rusty pipes k_{s} 1 – 3 mm and for new zinc-plated pipes 0.5 mm; for new asbestos cement pipes it is 0.085 mm.
For artificial roughness, because of its diversity, there are no unique generalizing parameters for roughness. In such a case in order to determine the hydraulic resistance special calculation procedures can be used. Values of for typical fittings, etc. are given in the book by Idel'chik (1992).
In smooth bends and in coiled pipes with R/r_{0} ≥ 3 we assume that ΔP_{1} = 0, and the effect of centrifugal forces is taken into account by substituting the effective meaning of friction resistance coefficient in Eq. (2): for a laminar flow
for a turbulent flow (Re > 10^{4})
where is the friction resistance coefficient for a straight pipe; D = 1/2 Re is the Dean number, r_{0} is the pipe radius, R is the radius of curvature.
REFERENCES
Idel'chik, I, (1992) Handbook of Hydraulic Resistance (2nd edn.) Begell House, New York.
Schlichting, H. (1979) Boundary Layer Theory, McGraw Hill, New York.
Heat Exchanger, Design Handbook (1983) vol. 1 and 2, Hemisphere Publishing Corporation.
References
- Idel'chik, I, (1992) Handbook of Hydraulic Resistance (2nd edn.) Begell House, New York.
- Schlichting, H. (1979) Boundary Layer Theory, McGraw Hill, New York.
- Heat Exchanger, Design Handbook (1983) vol. 1 and 2, Hemisphere Publishing Corporation.