形状阻力公式大全(形阻公式).doc

Flow in valves and fittings Resistance coefficient K, valves and fittings head loss and flow velocity | Pipe equivalent length L/D Pressure drop or head loss is proportional to the velocity in valves or fittings. For the most engineering practices it can be assumed that pressure drop or head loss due to flow of fluids in turbulent range through valves and fittings is proportional to square of velocity. To avoid expensive testing of every valves and every fittings that are installed on pipeline, the experimental data are used. For that purpose resistance coefficient K, equivalent length L/D and flow coefficient Cv, Kv are used. These values are available from different sources like tables and diagrams from different authors and from valves manufacturers as well. Kinetic energy, which is represented as head due to velocity is generated from static head and increase or decrease in velocity directly is proportional with static head loss or gain. Velocity head is: where is: hL - head loss; v - velocity; gn - acceleration of gravity; The number of velocity heads lost due to resistance of valves and fittings is: where is: hL - head loss; K - resistance coefficient; v - velocity; gn - acceleration of gravity; The head loss due to resistance in valves and fittings are always associated with the diameter on which velocity occurs. The resistance coefficient K is considered to be constant for any defined valves or fittings in all flow conditions, as the head loss due to friction is minor compared to the head loss due to change in direction of flow, obstructions and sudden or gradual changes in cross section and shape of flow. Head loss due to friction in straight pipe is expressed by the Darcy equation: where is: hL - head loss; f - friction factor; L - length; D - internal diameter; v - velocity; gn - acceleration of gravity; It follows that: where is: K - resistance coefficient; f - friction factor; L - lengt; D - internal diameter; The ratio L/D is equivalent length in pipe diameters of straight pipe that will cause the same pressure drop or head loss as the valves or fittings under the same flow conditions. As the resistance coefficient is K is constant the equivalent length L/D will vary inversely with the change in friction factor for different flow conditions. For geometrically similar valves and fittings, the resistance coefficient would be constant. Actually there are always smaller or bigger geometrical non similarity in valves and fittings of different nominal size, so the resistance coefficient is not constant. The resistance coefficient K for a given type of valves or fittings, tends to vary with size as does friction factor for straight clean commercial steel pipe at the same flow conditions. Some resistances in piping like sudden or gradual contractions and enlargements, as well as pipe entrances or exists are geometrically similar. Therefore the resistance coefficient or equivalent length L/D is for these items independent of size. The values for resistance coefficient or equivalent length L/D are always associated with internal pipe diameter where the resistance is occurring. If the resistance coefficient or equivalent length L/D should be used for different internal pipe diameter than the diameter for which existing values can be found following relationship can be used: where is: K - resistance coefficient; D - internal diameter; where subscript a defines K and d with the reference to internal pipe diameter, and subscript b defines K and d with the reference to the internal diameter for which values of K can be found in tables or diagrams. This equation can also be used if the piping system has more than one size of valves and fittings to express the resistance coefficient or equivalent length L/D in terms of one size. Resistance coefficient K calculator for valves and fittings can be used. Resistance coefficient K for internal diameter sudden and gradual contraction and enlargement Using momentum, continuity and Bernoulli equation the resistance due to sudden enlargements may be expressed as: and the resistance factor due to sudden contraction as: where is: K1 - resistance coefficient; d1 - internal diameter (smaller); d2 - internal diameter (larger); Using as diameter ratio, both equation can be expressed as: where is: K1 - resistance coefficient; - diameter ratio d1/d2; In order to express the resistance coefficient in terms of larger pipe diameter, following relation should be used: where is: K1 - resistance coefficient based on smaller internal diameter; K2 - resistance coefficient based on larger internal diameter; - diameter ratio d1/d2; If the enlargement is not sudden but gradual, or if angle of gradual enlargement is different from 180O, Gibson coefficient Ce can be used for different angle of divergence as follows: In other words, if angle of divergence is bigger than 45O, the resistance coefficient is equal to one for sudden enlargement. For gradual contraction the resistance coefficient on the same basis based on Crane test data, contraction coefficient Cc can be used for different angles of convergence, as follows: Using above expressions for enlargement and contraction coefficient, resistance coefficient can be calculated as: For gradual enlargement: where is: Ce - coefficient of enlargement; K1 - resistance coefficient based on smaller internal diameter; - diameter ratio d1/d2; - enlargement angle; For gradual contraction: where is: Cc - coefficient of contraction; K1 - resistance coefficient based on smaller internal diameter; - diameter ratio d1/d2; - enlargement angle; For resistance coefficient based on the large pipe diameter expression: should be used, with above equations. where is: K1 - resistance coefficient based on smaller internal diameter; K2 - resistance coefficient based on larger internal diameter; - diameter ratio d1/d2; Equations for gradual enlargement and contraction can be used for resistance coefficient calculation for reduced bore straight-through valves like ball valves and gate valves. The total resistance coefficient for this type of ball and gate valves is the summation of resistance coefficient for gradual contraction and gradual enlargement. You can calculate resistance coefficient using resistance coefficient K and equivalent length l/d calculator. Flow coefficient Cv, pressure drop, control valve flow rate Selecting the correct valve size for a given application requires knowledge of process conditions that the valve will actually see in service. In the industry of control valves it is practice to use flow coefficient and flow characteristics. In the UK and in the USA coefficient Cv is used and it is defined as flow rate of water in gpm at 60OF that creates pressure drop of 1 psi across the valve. Basic equation for valve sizing for liquid service is: where is: Cv - flow coefficient gpm; q - flow rate gpm; p - pressure drop bar; S - specific gravity (relative density) - ; To aid in establishing uniform measurement of liquid flow coefficients Cv, standardized testing facility by Fluid Control Institute (FCI) are used by manufacturers. The effect of viscosity of fluids other than water should be considered when selecting the valve, as increased viscosity of fluid is reducing the valve capacity. Another coefficient Kv is used in some countries, particularly in Europe and is defined as flow rate of water in m3/h that creates pressure drop of 1kg/cm2 across the valve (1 kg/cm2 is equal to 0.980665 bar). Control valve sizing is based on the calculation of flow coefficient for given pressure drop and flow rate. Liquid flow capacity of a valve in metric units can be converted to Cv as: where is: Cv - flow coefficient gpm; qm - flow rate l/m; - density kg/m3; p - pressure drop bar; Also, liquid flow capacity of a valve can be converted to Kv as: where is: Kv - flow characteristic m3/h; qh - flow rate m3/h; S - specific gravity (relative density) - ; p - pressure drop bar; Above equations are used in flow coefficient Cv, pressure drop and control valve flow rate calculator. Flashing and cavitation, vapor pressure at valve vena contracta Flashing or cavitation inside a valve can have a significant influence on valve capacity. Flashing and cavitation can reduce the flow through valve in many liquid services. Also, damage can be made to the valve as well as to the piping system. The effect is represented by the change from liquid to vapor state of fluid, resulting in the velocity increase downstream from the valve. As liquid passes through the restriction area inside the valve flow stream is contracted. The smallest cross section area of stream is just downstream of the actual physical restriction at a point called vena contracta. At that point the velocity is at its maximum and pressure at the minimum. As the fluid exits the valve, away from vena contracta, velocity decrease and pressure increase, so the critical point for flashing and cavitation is at the point where the pressure is smallest which is in vena contracta. If pressure at vena contracta drops bellows the vapor pressure of the fluid, due to increased velocity at this point, bubbles will form in the flow stream. If pressure downstream of the vena contracta increase above the vapor pressure, bubbles will collapse or implode producing cavitation. Cavitation releases energy and produces a noise. If cavitation occurs close to solid surfaces, the energy released gradually wears the material leaving the rough surface. Cavitation can also damage the downstream pipeline, if at that place the pressure rises above the vapor pressure and bubbles collapse. Chocked flow valve pressure drop and cavitation in high pressure recovery valve Formation of bubbles in the valve resulting of flashing and cavitation effect reduces the flow rate through valve and limits the capacity. This is called chocked flow. Limiting pressure drop in valve is determined by experiment for each valve. Limiting pressure drop for chocked flow in valve can also be calculated using: where is: pallow - maximum allowable pressure drop for chocked flowpsi; Km - valve recovery coefficient from manufacturer literature - ; p1 - valve inlet absolute pressure psia; pv - vapor absolute pressure of the liquid at inlet temperature psia; rc - critical pressure ratio 0,70 - 0,95; In high recovery valve, cavitation can occur on pressure drop below that produces chocked flow. Therefore cavitation index is used to determine the chocked flow pressure drop at which cavitation damage will begin in high recovery valve: where is: Kc - cavitation index from manufacturer literature - ; p1 - valve inlet absolute pressure psia; pv - vapor absolute pressure of the liquid at inlet temperature psia; pc - pressure drop that creates cavitation in high recovery valves psi; This equation can be used anytime outlet pressure is greater than the vapor pressure of the liquid. Flow and discharge through Venturi, nozzle and orifice | Discharge coefficient, pressure and diameter ratio The rate of flow of any fluid through an orifice or nozzle, may be calculated using following equation: where is: q - flow rate; Cd - coefficient of discharge; A - cross section area; - diameter ratio d1/d2; gn - acceleration of gravity; hL - head loss; Instead of coefficient of discharge Cd, more convenient is the use of flow coefficient C which is represented by: where is: C - flow coefficient; Cd - coefficient of discharge; - diameter ratio d1/d2; Flow rate through nozzles and orifices are than calculated as: where is: q - flow rate; C - flow coefficient; A - cross section area; p - pressure drop; - density; gn - acceleration of gravity; hL - head loss; The values of hL and p are measured differential static head or pressure before and after the nozzle or orifice. Values for coefficient of discharge or flow coefficient (C or Cd) can be calculated based on applicable standards like ISO 5167 or similar ASME standards. Coefficient of discharge for orifice flow can be calculated using Reader-Harris/Gallagher (1998) equation (ISO 5167): where is: - diameter ratio d1/d2; ReD - Reynolds number based on bigger diameter; d1 - internal diameter (smaller); d2 - internal diameter (larger); L1 and L2 are functions on tap type and it is: L1=L2=0 - for corner tapsL1=1; L2=0.47 - for D and D/2 tapsL1=L2=0.0254/d1 - for d1m for 1 taps Coefficient of discharge for Venturi tubes can be obtained based on the type of Venturi tube. There are three types of Venturi tubes and each type has different range of diameters and Reynolds number for which coefficient of discharge is defined as follows: Venturi tubes with as cast convergent section Cd=0.984; Range for which coefficient of discharge is defined:100 mm D 800 mm0.3 0.752x10e5 ReD 2x10e6 Venturi tubes with a machined convergent section Cd=0.995; Range for which coefficient of discharge is defined:50 mm D 250 mm0.4 0.752x10e5 ReD 2x10e6 Venturi tubes with a rough-welded sheet-iron convergent section Cd=0.985; Range for which coefficient of discharge is defined: 200 mm D 1200 mm0.4 0.72x10e5 ReD 0.75. For compressible flow through orifices expansion factor is (ISO 5167): where is: Y - expansion factor; - specific heat ratio; - diameter ratio d1/d2; p1 - inlet pressure; p2 - pressure in Venturi throat or after the orifice or nozzle; Above equations are used in Venturi tube flow rate meter and Venturi effect calculator and in orifice plate sizing and flow rate calculator. This equation can be used for gas flow though the orifice and discharging to the atmosphere. For that purpose the pressure difference equals to the upstream gauge pressure. This applies only if absolute atmospheric pressure divided by absolute upstream pressure is bigger than critical pressure ratio for sonic flow conditions. When the smoothly convergent nozzle is used compressible fluid can reach the speed of sound at minimum cross section or throat, if upstream pressure is high enough. When the velocity of compressible fluid reaches the speed of sound, maximum flow has been reached, and increase of upstream pressure or decrease of downstream pressure will not increase the flow any more. For short tubes where relation L/D is not bigger than 2.5 the flow of discharge to the atmosphere can be calculated using above equations, with flow coefficient C somewhere between the values for orifice and nozzle. If the entrance to the short tube is well rounded then the flow coefficient C for nozzles can be used and if the pipe entrance is square shaped and sharp then flow coefficient C for orifice is more appropriate. Flow discharge through valves, fittings and pipe, resistance coefficient K For discharge of liquids through valves, fittings and pipes Darcy formula can be expressed as: where is: q - discharge flow rate l/min; d - internal pipe diameter mm; hL - head loss m; K - resistance coefficient - ; This equation can be used for discharge calculation from pipes, fittings and valves when resistance coefficient K, static head difference hL and internal pipe diameter d is known. The resistance coefficient is the sum of all resistances in the piping system. For discharge of compressible flow of fluid from a pipe to a larger area or larger cross section like in the case of discharge to the atmosphere, a modified Darcy formula can be used: where is: w - discharge mass flow rate kg/s; Y - expansion factor - ; d - internal pipe diameter mm; p - pressure drop Pa; - density kg/m3; K - resistance coefficient - ; Gas pressure regulator capacity - flow coefficient Cg, Kg | Critical flow rate For gas pressure regulators that are operating in the range of inlet pressure up to 100 bar following equations for sub-critical and critical flow behavior are used (EN 334): Sub-critical flow: Critical flow: where is: Q - gas flow rate m3/h; S - gas relative density (for air=1) - ; tu - gas upstream temperature OC; Cg - gas flow coefficient - ; pu - gas upstream pressure bar; pd - gas downstream pressure bar; pb - ambient atmospheric pressure bar; K1 - valve body shape factor - ; Simplified calculation can be used if K1 0,1 (pu+pb): Sub-critical flow rat