Hydraulic resistance in pipelines
The calculation of hydraulic resistance is one of the most important issues in hydrodynamics, it is necessary to determine the pressure loss, the energy consumption for their compensation and the selection of a thrust driver.
Pressure losses in pipelines are due to resistance friction and local resistances. They enter the Bernoulli equation for real liquids.
a) Friction resistance exists during the motion of a real fluid along the entire length pipeline and depends on the mode of fluid flow.
b) local resistance occur with any change. flow velocity in magnitude and direction(pipe inlet and outlet, bends, elbows, tees, fittings, expansions, narrowings).
Friction head loss
1) laminar flow.
Under laminar flow can be calculated theoretically using the Poiseuille equation:
;
According to the Bernoulli equation for a horizontal pipeline constant cross section pressure lost to friction:
;
;
;
Substituting the value into the Poiseuille equation and substituting, we get:
;
;
;
Thus, with laminar motion in a straight line round pipe:
;
the value is called the coefficient of hydraulic friction.
Darcy-Weisbach equation:
;
This equation can be obtained in another way - with the help of the theory of similarity.
It is known that
;
For laminar flow found: .
;
;
Darcy-Weisbach equation:
;
Let's define the pressure loss: .
Darcy-Weisbach equation:
Substituting the value for the laminar mode, we get:
;
Thus, for the laminar regime:
Hagen-Poiseuille equation:
;
This equation is valid for and is especially important when studying the flow of liquid in pipes of small diameter, as well as in capillaries and pores
Therefore, for steady laminar motion:
For non-circular section: 
, where depends on the shape of the section:
;
The expression is called the drag coefficient.
Hence:
;
;
2) Turbulent mode.
For the turbulent regime, the Darcy-Weisbach equation is also valid:
;
However, the coefficient of friction cannot be theoretically determined in this case due to the complexity of the turbulent flow structure. Calculation equations for determination are obtained by generalizing experimental data by methods of similarity theory.
a) Smooth pipes.
;
;
;
Therefore, for turbulent flow in smooth pipes:
Blasius formula:
b) Rough pipes.
For rough pipes, the coefficient of friction depends not only on but also on the roughness of the walls.
The characteristic of rough pipes is relative roughness: the ratio of the average height of protrusions (bumps) on the pipe walls (absolute roughness) to the equivalent pipe diameter:
Example approximate values of absolute roughness:
New steel pipes 
;
· Steel pipes with slight corrosion;
· Glass pipes;
· Concrete pipes;
The influence of roughness on the value is determined by the ratio between the absolute roughness and the thickness of the laminar sublayer .
1. At , when the fluid smoothly flows around the protrusions, the effect of roughness can be neglected, and the pipes are considered as hydraulically smooth(conditional) - smooth friction zone.
2. As the value increases, the value decreases, and friction losses increase due to vortex formation near the roughness protrusions - mixed friction zone.
3. When large values, ceases to depend on and is determined only by the roughness of the walls, i.e. automodel mode by - self-similar zone.
It should be noted that, since , the pipe can be rough at one flow rate and hydraulically smooth at another.
For this pipe approximately:
;
For rough pipes in turbulent motion, the following equation applies:
;
For the region of smooth friction- or according to the Blasius equation, or according to the equation:
;
;
Dividing by 1.8, you can get the Filonenko formula.
Filonenko's formula:
;
For self-similar area:
;
Practically calculation is carried out according to nomograms. The dependence of the friction coefficient on the criterion and the degree of roughness - Fig. 1.5, Pavlov, Romankov.
For non-isothermal flow the viscosity of the liquid changes over the pipe section, the velocity profile changes and .
Special correction factors are introduced into the equations for determining (except for the self-similar region) (Pavlov, Romankov)
Loss of pressure on local resistances
In various local resistances, the speed measurement occurs:
a) in size =>
b) in the direction =>
c) in size and direction =>
In addition to losses associated with friction, additional pressure losses occur (the formation of eddies due to the action of inertial forces (when changing direction), the formation of eddies due to reverse fluid flows, etc. (with a sudden expansion)).
The pressure loss due to local resistances is expressed in terms of velocity pressure. The ratio of the head loss in a given local resistance to the velocity head in it is called the coefficient of local resistance:
For all local resistance pipeline:
(summed up if there are straight sections with a length of at least 5d)
The coefficients are given in tables, for example:
· the entrance to the pipe ;
Exit from the pipe
· valve to => ;
crane, =>
valve =>
valve =>
Complete loss of power
The value is expressed in meters of liquid column and does not depend on the type of liquid, and the magnitude of the pressure loss depends on the density of the liquid.
Hydraulic calculations devices, in principle, do not differ from the calculations of pipelines.
Calculation of pipeline diameter
The cost of pipelines is a significant part of capital investments and large operating costs. Regarding this great importance It has right choice pipeline diameter.
The diameter value is determined by the fluid velocity. If a high speed is selected, then the diameter of the pipeline is reduced, this provides:
Reducing metal consumption;
Reduced manufacturing, installation and repair costs.
However, at the same time, the pressure drop required to move the liquid increases. It requires high costs for fluid movement.
Optimum diameter must provide a minimum operating costs. (the sum of the cost of energy, depreciation and repair).
Annual operating costs => M (rubles/year)=A+E;
A - depreciation costs (cost / years) and repairs;
E is the cost of energy.
Based on technical and economic considerations, the following speed limits are recommended:
drip liquids:
Gravity = 0.2 - 1 m/s
When pumping = 2 - 3 m/s
gases:
With natural draft = 2 - 4 m/s
At low pressure (fan) = 4 – 15 m/s
At high pressure (compressor) = 15 - 25 m/s
Couples:
Saturated water vapor = 20 - 30 m/s
Superheated water vapor = 30 - 50 m/s.
Typically, pressure losses should be no more than 5-15% of the discharge pressure.
The optimal diameter of the pipeline must comply with GOST. GOST establishes the concept conditional diameterDy. This is the nominal inside diameter of the pipeline. According to this diameter, connecting parts are also selected - flanges, tees, plugs, etc., as well as fittings: taps, valves, gate valves, etc.
Each conditional diameter corresponds to a certain outside diameter, while the wall thickness can be different. For example (mm) (there may be deviations from this table).
Pipe material
Apply various materials, which is associated with different ambient temperatures and aggressiveness.
Most often used steel pipes:
Cast iron pipes up to 300 0 С
Others are also used metal pipes=> copper, aluminum, lead, titanium, etc. And non-metallic => polyethylene, fluoroplastic, ceramic, asbestos-cement, glass, etc.
Ways of connecting pipelines
a) One-piece - welded
b) Detachable
Flanged
Threaded
Socket (used for cast iron, concrete and ceramic pipes)
Pipe Fittings
1. steam traps.
In steam and gas communications, due to cooling, condensation of water, resin or other liquid contained in the gas in the form of vapor can always occur. The accumulation of condensate is very dangerous, because, moving through the pipes at high speed ( 
), a liquid plug with a large inertia will cause the strongest hydraulic shocks. They loosen pipelines and can cause their destruction.
Therefore, gas pipelines are mounted with a slight slope, and a condensate pipe is placed at the lowest point.
hydraulic shutter. For vacuum lines =>
through a barometric tube.
At high pressures, special designs of steam traps are used (discussed below).
2. Valves.
1 - body;
3 - valve;
4 - spindle;
5 - stuffing box.
The valve is lapped to the seat and tightly blocks the movement of the medium.
The spindle has a threaded part and is connected to the flywheel. Tightness is ensured by an oil seal.
Valves are shut-off and control valves, i.e. allow smooth flow control.
3. Cranes.
A ground conical or spherical plug rotates in the body with through hole. Cranes are mainly used as stop valves. It's hard to control flow.
4. Gate valves.
Shibernaya
There are plane-parallel and wedge gate valves. The gate is moved by means of a spindle perpendicular to the axis of the pipeline and its overlap occurs.
This valve is shut-off and control. For automation purposes, the drive can be pneumatic, electric, hydraulic, etc.
5. There is also safety and protective fittings(safety and check valves), control armature(level gauges, test taps, etc.)
All fittings are indexed:
for example: 15 cz 2br.
15=>valve; kch => malleable cast iron (case material); 2=> model number according to the catalog; br => sealing surface made of bronze.
The fittings are selected depending on the pressure in the pipeline.
Distinguish:
1) Operating pressure - the highest overpressure at which the valve operates for a long time at operating temperature environments.
2) Nominal pressure- the highest pressure (ex.) created by the medium at 20 0 С.
There are a number of conditional pressures, according to which fittings are made:
P y \u003d 1;2.5;4;6;10;16;25;40;64;100;160;200;250;320;400 ... atm.
The choice of P y is carried out according to the tables depending on the steel grade, the highest medium temperature and operating pressure.
Example: Steel Х12H10T
t environment \u003d 400 0 С P slave \u003d 20 atm: P y \u003d 25 atm
P slave \u003d 80 atm: P y \u003d 100 atm
t environment \u003d 660 0 С P slave \u003d 20 atm: P y \u003d 64 atm
P slave \u003d 80 atm: P y \u003d 250 atm
Local hydraulic resistances are sections of pipelines (channels) on which the fluid flow undergoes deformation due to a change in the size or shape of the section, or in the direction of movement. The simplest local resistances can be roughly divided into expansions, contractions, which can be smooth and sudden, and turns, which can also be smooth and sudden.
But most local resistances are combinations of these cases, since turning the flow can lead to a change in its cross section, and expansion (narrowing) of the flow can lead to a deviation from the rectilinear motion of the fluid (see Figure 3.21, b). In addition, various hydraulic fittings (faucets, valves, valves, etc.) are almost always a combination of the simplest local resistances. Local resistances also include sections of pipelines with separation or merging of fluid flows.
It must be borne in mind that local hydraulic resistances have a significant impact on the operation of hydraulic systems with turbulent fluid flows. In hydraulic systems with laminar flows, in most cases, these head losses are small compared to friction losses in pipes. In this section, local hydraulic resistances will be considered in the turbulent flow regime.
Head loss in local hydraulic resistance is called local losses.
Despite the variety of local resistances, in most of them, pressure losses are due to the following reasons:
Curvature of streamlines;
A change in the magnitude of the speed due to a decrease or increase in living sections;
Separation of transit jets from the surface, vortex formation.
Despite the variety of local resistances, in most of them, a change in the speed of movement leads to the emergence of vortices, which use the energy of the fluid flow for their rotation (see Figure 3.21, b). Thus, the main reason hydraulic losses head in most local resistances is vortex formation. Practice shows that these losses are proportional to the square of the fluid velocity, and the Weisbach formula is used to determine them.
When calculating the head loss using the Weisbach formula greatest difficulty is the definition of the dimensionless coefficient of local resistance. Due to the complexity of the processes occurring in local hydraulic resistances, it is theoretically possible to find only in individual cases, so most of the values of this coefficient are obtained as a result of experimental studies. Let us consider methods for determining the coefficient for the most common local resistances in a turbulent flow regime.
For a sudden expansion of the flow (see Figure 3.21, b) there is a theoretically obtained Borda formula for the coefficient , which is uniquely determined by the ratio of areas before expansion (S1) and after it (S2):
It should be noted special case when the liquid flows from the pipe into the tank, i.e. when the cross-sectional area of the flow in the pipe S1 much less than that in the tank S2. Then from formula (3.35) it follows that for the pipe exit to the tank = 1. To estimate the head loss coefficient during a sudden narrowing, the empirical formula proposed by I.E. Idelchik, which also takes into account the area ratio before expansion (S1) and after it (S2):
. (3.36)
For a sudden narrowing of the flow, it is also necessary to note the particular case when the liquid flows out of the tank through the pipe, i.e. when the cross-sectional area of the flow in the pipe S2 much less than that in tank S 1 . Then from (3.36) it follows that for the pipe entry into the tank = 0.5.
AT hydraulic systems quite often there is a smooth expansion of the flow (Figure 3.21, in) and smooth narrowing of the flow (Figure 3.21, G). An expanding channel in hydraulics is commonly called a diffuser, and a narrowing channel is called a confuser. In this case, if the confuser is made with smooth transitions in sections 1 "-1 "and 2 "-2 ", then it is called a nozzle. These local hydraulic resistances can have (especially at small angles α) a fairly large length l. Therefore, in addition to losses due to vortex formation caused by a change in the geometry of the flow, these local resistances take into account the pressure loss due to friction along the length.
The values of the coefficients for smooth expansion and smooth narrowing are found by introducing correction factors into formulas (3.35) and (3.36): and .
Correction factors kp and kc have numerical values less than one, depend on the angles α, as well as on the smoothness of transitions in the sections and 1 "-1 " and 2 "-2 ". Their meanings are given in reference books.
Reversals of streams are also very common local resistances. They can be with a sudden turn of the pipe (Figure 3.21, d) or with a smooth turn (Figure 3.21, e).
A sudden turn of the pipe (or elbow) causes significant vortex formation and therefore leads to significant head losses. The coefficient of resistance of the knee is determined primarily by the angle of rotation δ and can be selected from the handbook.
A smooth turn of the pipe (or branch) significantly reduces the formation of vortexes and, consequently, pressure losses. The coefficient for a given resistance depends not only on the turning angle δ, but also on the relative turning radius R/d. To determine the coefficient, there are various empirical dependencies, for example, , (3.37) or are in the reference literature.
The loss factors of other local resistances encountered in hydraulic systems can also be determined from a handbook.
It should be borne in mind that two or more hydraulic resistances installed in one pipe may have mutual influence if the distance between them is less than 40d(d- pipe diameter).
Determination of local hydraulic resistance
Head loss in local resistances is determined by the Weisbach formula: , (39)
· where x - dimensionless coefficient, depends on the type and design of local resistance, the state of the inner surface and Re.
· J - the speed of fluid movement in the pipeline, where local resistance is installed.
If between sections 1-1 and 2-2 flow, there are many local resistances and the distance between them is greater than the length of their mutual influence (»6 d ), then local losses pressures are summed up. In most cases, this is what is assumed when solving problems.
.
· In our task local pressure losses are equal to:
å h m= h ext.narrow . + h in + 2h pov . + h out = (x inner narrow . + x in + 2x pov . + x out )× Q 2 /(w 2 × 2g);
å h m= å x× Q 2 /(w 2 × 2g); where å x \u003d x ext. . + x in + 2x pov . + x out
· In our task the total head loss is equal to:
h 1-2 =(l×l/d+åx) × Q 2 /(w 2 × 2g.
· With developed turbulent motion in local resistance ( Re > 10 4) there is a turbulent self-similarity - the head loss is proportional to the velocity to the second power, and the drag coefficient does not depend on the number Re( quadratic zone for local resistances). Wherein x sq = const and is determined by reference data (Appendix 6).
· In most practical problems, turbulent self-similarity takes place and the coefficient of local resistance is a constant value.
in laminar mode x = x sq × j, where j- number function Re (Appendix 7).
With a sudden expansion of the pipeline, the coefficient of sudden expansion is determined as follows:
x ext. ext = (1-w 1 /w 2 ) 2 =(1-d 1 2 /d 2 2) 2 (40)
· When w 2 >>w 1 , which corresponds to the exit of liquid from the pipeline into the tank, . x out.=1.
In the event of a sudden narrowing of the pipeline, the coefficient of sudden narrowing
x ext. narrow equals:
, (41)
where w 1 is the area of the wide (inlet) section, and w 2 - the area of the narrow (outlet) section.
· When w 1 >>w 2 , which corresponds to the inlet of liquid from the tank into the pipeline, x input=0.5 (with a sharp leading edge).
valve resistance coefficient x in depends on the degree of valve opening (Appendix 6).
.
In our problem, the law of conservation of energy looks like:
This is the calculation equation for determining the value R - forces on the piston rod.
4. Calculate the quantities included in equation (42). We substitute the initial data in the SI system.
sectional area 1-1 w 1 = p×d 1 2 /4 \u003d 3.14 × 0.065 2 /4 \u003d 3.32 × 10 -3 m 2.
sectional area of the pipeline w = p×d 2 /4 \u003d 3.14 × 0.03 2 /4 \u003d 0.71 × 10 -3 m 2.
sum of coefficients of local resistances
å x \u003d x ext. . + x in + 2x pov . + x out = 0.39+5.5 + 2×1.32+1=9.53.
sudden contraction factor
90° sharp turn ratio x pov.= 1.32 (Appendix 6);
coefficient of resistance at the outlet of the pipe x out.= 1 (formula 40);
coefficient of friction l
Since the Reynolds number Re >Re cr (2.65×10 5 >2300), the friction coefficient was calculated using formula (38).
By condition, the kinematic viscosity coefficient is given in centistokes (cSt). 1cSt \u003d 10 -6 m 2 / s.
Coriolis coefficient a 1 in section 1-1
Since the mode of motion in the section 1-1 turbulent, then a 1 =1.
Power on rod
4.6.2. Fluid flow determination
Attention!
Since all the necessary explanations and theoretical basis applications of the Bernoulli equation were made in detail when solving problem 1, the law of conservation of energy for this problem is derived without detailed explanations.
1. Choose two sections 1-1 and 2-2 , as well as the comparison plane 0-0 and write in general view Bernoulli equation:
.
Here p 1 and p 2 are the absolute pressures in the centers of gravity of the sections; J1 and J2 are average velocities in sections; z1 and z2 - heights of the centers of gravity of the sections relative to the reference plane 0-0; h 1-2 – loss of pressure during the movement of fluid from the first to the second section.
2. Determine the terms of the Bernoulli equation in this problem.
The heights of the centers of gravity of the sections: z 1 = H ; z2 =0.
Average speeds in sections: J2 = Q/w 2 =4× Q/p/d 2 ;
J1 = Q/w 1 . As w 1 >>/w2 , then J1 <<J2 and can be accepted J1 =0.
· The Coriolis coefficients a 1 and a 2 depend on the mode of fluid motion. In laminar mode a=2, and in turbulent mode a=1.
Absolute pressure in the first section p 1 \u003d p m + p at, p m - excess (gauge) pressure in the first section, it is known.
Absolute pressure in section 2-2 is equal to atmospheric p at , as the liquid flows into the atmosphere.
Head loss h 1-2 are the sum of the pressure losses due to friction along the length of the flow h dl and losses due to local hydraulic resistance å h m .
h 1-2 = h dl +å h m.
Losses along the length are equal
.
Local pressure losses are equal
å h m=å x× J 2 /( 2g) = å x× Q 2 /(w 2 × 2g); where e x given by condition
The total head loss is equal to
h 1-2 =(l×l/d+åx) × Q 2 /(w 2 × 2g);
3. So, we substitute the quantities defined above into the Bernoulli equation .
In our problem, the law of conservation of energy has the form:
We cancel the terms with atmospheric pressure, remove the zeros and give like terms. As a result, we get:
. (43)
This is a calculation equation for determining the flow rate of a liquid. It represents the law of conservation of energy for a given problem. The flow enters the right side of the equation directly, as well as the coefficient of friction l through number Re (Re = 4Q/(p×d×n) !
Without knowing the flow rate, it is impossible to determine the mode of fluid movement and choose a formula for l. In addition, in the turbulent regime, the friction coefficient depends on the flow rate in a complex way (see formula (38)). If expression (38) is substituted into formula (43), then the resulting equation cannot be solved by algebraic methods, that is, it is transcendental. Such equations are solved graphically or numerically using a computer (most often by iteration).
Hydraulic resistance or hydraulic losses are the total losses during the movement of fluid through water-carrying channels. They can be conditionally divided into two categories:
Friction losses - occur when fluid moves in pipes, channels or the flow path of the pump.
Losses due to vortex formation - occur when a fluid flows around various elements. For example, a sudden expansion of a pipe, a sudden narrowing of a pipe, a turn, a valve, etc. Such losses are usually called local hydraulic resistances.
Hydraulic resistance coefficient
Hydraulic losses are expressed either in head losses Δh in linear units of the medium column, or in pressure units ΔP:
where ρ is the density of the medium, g is the free fall acceleration.
In industrial practice, the movement of fluid in flows is associated with the need to overcome the hydraulic resistance of the pipe along the length of the flow, as well as various local resistances:
turns
Aperture
valve
valves
Cranes
Various branches and the like
To overcome local resistances, a certain part of the flow energy is expended, which is often called the loss of pressure on local resistances. Typically, these losses are expressed as fractions of velocity head corresponding to the average fluid velocity in the pipeline before or after local resistance.
Analytically, the head loss due to local hydraulic resistance is expressed as
h r = ξ υ 2 / (2g)
where ξ is the coefficient of local resistance (usually determined empirically).
Data on the value of the coefficients of various local resistances are given in the relevant reference books, textbooks and various manuals on hydraulics in the form of individual values of the coefficient of hydraulic resistance, tables, empirical formulas, diagrams, etc.
The study of energy losses (losses of pump head) due to various local resistances has been conducted for more than a hundred years. As a result of experimental studies carried out in Russia and abroad at various times, a huge amount of data has been obtained relating to a variety of local resistances for specific tasks. As far as theoretical studies are concerned, only certain local resistances are amenable to them so far.
This article will look at some typical local resistances that are often encountered in practice.
Local hydraulic resistance
As already mentioned above, pressure losses in many cases are determined empirically. In this case, any local resistance is similar to the resistance during a sudden expansion of the jet. There are sufficient grounds for this, given that the behavior of the flow at the moment it overcomes any local resistance is associated with an expansion or contraction of the cross section.
Hydraulic losses due to sudden narrowing of the pipe
The resistance in the case of a sudden narrowing of the pipe is accompanied by the formation of a whirlpool area in the place of narrowing and a decrease in the jet to sizes smaller than the cross section of a small pipe. Having passed the narrowing section, the jet expands to the dimensions of the internal section of the pipeline. The value of the coefficient of local resistance in case of a sudden narrowing of the pipe can be determined by the formula.
ξ int. narrower \u003d 0.5 (1- (F 2 / F 1))
The value of the coefficient ξ ext. judging by the value of the ratio (F 2 /F 1)) can be found in the appropriate hydraulics manual.
Hydraulic losses when changing the direction of the pipeline at a certain angle
In this case, the jet first compresses and then expands due to the fact that at the point of rotation, the flow, as it were, is squeezed out of the pipeline walls by inertia. The coefficient of local resistance in this case is determined by reference tables or by the formula
ξ turn = 0.946sin(α/2) + 2.047sin(α/2) 2
where α is the angle of rotation of the pipeline.
Local hydraulic resistance at the entrance to the pipe
In a particular case, the pipe inlet may have a sharp or rounded inlet edge. The pipe into which the liquid enters can be located at some angle α to the horizontal. Finally, in the inlet section there may be a diaphragm narrowing the section. But all these cases are characterized by the initial compression of the jet, and then its expansion. Thus, the local resistance at the entrance to the pipe can be reduced to a sudden expansion of the jet.
If the liquid enters a cylindrical pipe with a sharp inlet edge and the pipe is inclined to the horizon at an angle α, then the value of the local resistance coefficient can be determined by the Weisbach formula:
ξ in \u003d 0.505 + 0.303 sin α + 0.223 sin α 2
Local hydraulic resistance of the valve
In practice, one often encounters the problem of calculating local resistances created by shutoff valves, for example, gate valves, valves, throttles, taps, valves, etc. In these cases, the flow part, formed by different locking devices, can have completely different geometric shapes, but the hydraulic essence of the flow when overcoming these resistances is the same.
The hydraulic resistance of a fully open shutoff valve is equal to
valve ξ = 2.9 to 4.5
The values of the coefficients of local hydraulic resistance for each type of valves can be determined from reference books.
Hydraulic diaphragm loss
The processes occurring in locking devices are in many respects similar to the processes when liquid flows through diaphragms installed in a pipe. In this case, the jet also narrows and then expands. The degree of narrowing and expansion of the jet depends on a number of conditions:
fluid motion mode
the ratio of the diameters of the orifice of the diaphragm and the pipe
design features of the diaphragm.
For a diaphragm with sharp edges:
ξ aperture = d 0 2 / D 0 2
Local hydraulic resistance when the jet enters under the liquid level
Overcoming local resistance when the jet enters under the liquid level into a sufficiently large reservoir or into a medium not filled with liquid is associated with the loss of kinetic energy. Therefore, the drag coefficient in this case is equal to unity.
input ξ = 1
Video about hydraulic resistance
To overcome hydraulic losses, the work of various devices (pumps and hydraulic machines) is expended.
To reduce the influence of hydraulic losses, it is recommended to avoid the use of nodes in the design of the route that contribute to abrupt changes in the direction of the flow and try to use a streamlined body in the design.
Even when using absolutely smooth pipes, one has to deal with losses: in the laminar flow regime (according to Reynolds), the roughness of the walls does not have a big effect, but when switching to a turbulent flow regime, the hydraulic resistance of the pipe usually also increases.
