5 HYDRAULIC CALCULATION OF PIPELINES
5.1 Simple pipeline of constant cross-section
The pipeline is called simple, if it has no branches. Simple pipelines can form connections: series, parallel or branched. Pipelines can be complex, containing both serial and parallel connections or branches.
Liquid moves through a pipeline due to the fact that its energy at the beginning of the pipeline is greater than at the end. This difference (difference) in energy levels can be created in one way or another: by the operation of the pump, due to the difference in liquid levels, or by gas pressure. In mechanical engineering, one has to deal mainly with pipelines, the movement of fluid in which is caused by the operation of a pump.
When hydraulically calculating a pipeline, it is most often determined by its required pressureH consumption - a value numerically equal to the piezometric height in the initial section of the pipeline. If the required pressure is given, then it is usually called available pressureH disp. In this case, hydraulic calculation can determine the flow rate Q liquid in the pipeline or its diameter d. The pipeline diameter value is selected from the established range in accordance with GOST 16516-80.
Let a simple pipeline with a constant flow area, arbitrarily located in space (Figure 5.1, A), has a total length l and diameter d and contains a number of local hydraulic resistances I and II.
Let us write the Bernoulli equation for the initial 1-1 and final 2-2 sections of this pipeline, assuming that the Coriolis coefficients in these sections are the same (α 1 =α 2). After reducing the velocity pressures we get
Where z 1 , z 2 - coordinates of the centers of gravity of the initial and final sections, respectively;
p 1 , p 2 - pressure in the initial and final sections of the pipeline, respectively;
Total pressure loss in the pipeline.
Hence the required pressure
, (5.1)
As can be seen from the resulting formula, the required pressure is the sum of the total geometric height Δz = z 2 – z 1 , to which the liquid rises as it moves through the pipeline, the piezometric height in the final section of the pipeline and the sum of the hydraulic pressure losses that occur when the liquid moves in it.
In hydraulics, it is customary to understand the static pressure of a pipeline as the sum 
.
Then, representing the total losses as a power function of the flow rate Q, we get
Where T - a value depending on the fluid flow regime in the pipeline;
K is the resistance of the pipeline.
Under laminar fluid flow conditions and linear local resistances (their equivalent lengths are given l eq) total losses
,
Where l calc = l + l eq - estimated pipeline length.
Therefore, in laminar mode t = 1, 
.
In turbulent fluid flow
.
Replacing the average fluid velocity through flow rate in this formula, we obtain the total pressure loss
. (5.3)
Then, under turbulent conditions 
, and the exponent m= 2. It should be remembered that in the general case the friction loss coefficient along the length is also a function of the flow rate Q.
By doing the same in each specific case, after simple algebraic transformations and calculations, you can obtain a formula that determines the analytical dependence of the required pressure for a given simple pipeline on the flow rate in it. Examples of such dependencies in graphical form are shown in Figure 5.1, b, V.
Analysis of the formulas given above shows that the solution to the problem of determining the required pressure H consumption at known consumption Q liquids in the pipeline and its diameter d is not difficult, since it is always possible to assess the fluid flow regime in the pipeline by comparing the critical value ReTop= 2300 with its actual value, which for round pipes can be calculated using the formula
After determining the flow regime, you can calculate the pressure loss, and then the required pressure using formula (5.2).
If the values Q or d are unknown, then in most cases it is difficult to assess the flow regime, and, therefore, to reasonably select formulas that determine pressure losses in the pipeline. In such a situation, it can be recommended to use either the successive approximation method, which usually requires a fairly large amount of computational work, or a graphical method, in the application of which it is necessary to construct the so-called characteristic of the required pipeline pressure.
5.2. Constructing a characteristic of the required pressure of a simple pipeline
Graphical representation in coordinates N-Q analytical dependence (5.2) obtained for a given pipeline is called in hydraulics characteristic of the required pressure. In Figure 5.1, b, c Several possible characteristics of the required pressure are given (linear - for laminar flow conditions and linear local resistances; curvilinear - for turbulent flow conditions or the presence of quadratic local resistances in the pipeline).
As can be seen in the graphs, the value of the static pressure N st can be either positive (liquid is supplied to a certain height Δ z or there is excess pressure in the final section p 2) and negative (when the liquid flows downwards or when it moves into a cavity with rarefaction).
The slope of the characteristics of the required pressure depends on the resistance of the pipeline and increases with increasing pipe length and decreasing its diameter, and also depends on the number and characteristics of local hydraulic resistance. In addition, in a laminar flow regime, the quantity under consideration is also proportional to the viscosity of the liquid. The point of intersection of the required pressure characteristic with the abscissa axis (point A in Figure 5.1, b, V) determines the fluid flow in the pipeline when moving by gravity.
Graphical dependencies of the required pressure are widely used to determine flow Q when calculating both simple and complex pipelines. Therefore, let us consider the methodology for constructing such a dependence (Figure 5.2, A). It consists of the following stages.
1st stage. Using formula (5.4) we determine the value of the critical flow Q kr, corresponding ReTop=2300, and mark it on the expense axis (x-axis). Obviously, for all expenses located to the left Q kr, there will be a laminar flow regime in the pipeline, and for flow rates located to the right Q cr, - turbulent.
2nd stage. We calculate the required pressure values H 1 And H 2 at a flow rate in the pipeline equal to Q kr, accordingly assuming that N 1 - calculation result for laminar flow regime, and N 2 - when turbulent.
3rd stage. We construct a characteristic of the required pressure for a laminar flow regime (for flow rates less than Q cr) . If local resistances installed in the pipeline have a linear dependence of losses on flow, then the characteristic of the required pressure has a linear form.
4th stage. We construct a characteristic of the required pressure for a turbulent flow regime (for flow rates large QTop). In all cases, a curvilinear characteristic is obtained, close to a parabola of the second degree.
Having a characteristic of the required pressure for a given pipeline, it is possible, based on the known value of the available pressure Hdisp. find the required flow rate Qx (see Figure 5.2, A).
If you need to find the internal diameter of the pipeline d, then, given several values d, it is necessary to construct the dependence of the required pressure Hconsumption from diameter d (Fig. 5.2, b). Next by value N disp. the nearest larger diameter from the standard range is selected d st .
In some cases, in practice, when calculating hydraulic systems, instead of the required pressure characteristic, the pipeline characteristic is used. Pipeline characteristics- this is the dependence of the total pressure losses in the pipeline on the flow rate. The analytical expression of this dependence has the form
Comparison of formulas (5.5) and (5.2) allows us to conclude that the characteristics of the pipeline differ from the characteristics of the required pressure in the absence of static pressure H st, and at H st = 0 these two dependencies coincide.
5.3 Connections of simple pipelines.
Analytical and graphical calculation methods
Let's consider methods for calculating connections of simple pipelines.
Let us have serial connection several simple pipelines ( 1 , 2 And 3 in Figure 5.3, A) different lengths, different diameters, with different sets of local resistances. Since these pipelines are connected in series, each of them has the same fluid flow Q. Total head loss for the entire connection (between points M And N) consists of pressure losses in each simple pipeline ( , , ), i.e. for a series connection the following system of equations is valid:
(5.6)
The pressure loss in each simple pipeline can be determined through the values of the corresponding flow rates:
The system of equations (5.6), supplemented by dependencies (5.7), is the basis for the analytical calculation of a hydraulic system with a series connection of pipelines.
If a graphical calculation method is used, then there is a need to construct a summary characteristic of the connection.
In Figure 5.3, b shows a method for obtaining the summary characteristics of a serial connection. For this purpose, the characteristics of simple pipelines are used 1 , 2 And 3
To construct a point belonging to the total characteristic of a series connection, it is necessary, in accordance with (5.6), to add up the pressure losses in the original pipelines at the same flow rate. For this purpose, an arbitrary vertical line is drawn on the graph (at an arbitrary flow rate Q" ). Along this vertical, the segments (pressure loss, and) obtained from the intersection of the vertical with the initial characteristics of the pipelines are summed up. The point thus obtained A will belong to the summary characteristics of the connection. Consequently, the total characteristic of a series connection of several simple pipelines is obtained by adding the ordinates of the points of the initial characteristics at a given flow rate.
Parallel called a connection of pipelines that have two common points (a branch point and a closure point). An example of a parallel connection of three simple pipelines is shown in Figure 5.3, V. Obviously, the expense Q fluid in the hydraulic system before branching (point M) and after closure (point N) the same and equal to the amount of expenses Q 1 , Q 2 and Q 3 in parallel branches.
If we designate total pressures at points M And N through NM And H N, then for each pipeline the pressure loss is equal to the difference of these pressures:
; 
; 
,
that is, in parallel pipelines the pressure loss is always the same. This is explained by the fact that with such a connection, despite the different hydraulic resistance of each simple pipeline, the costs Q 1 , Q 2 And Q 3 distributed between them so that the losses remain equal.
Thus, the system of equations for a parallel connection has the form
(5.8)
The pressure loss in each pipeline included in the connection can be determined using formulas of the form (5.7). Thus, the system of equations (5.8), supplemented by formulas (5.7), is the basis for the analytical calculation of hydraulic systems with parallel connection of pipelines.
In Figure 5.3, G shows a method for obtaining the summary characteristics of a parallel connection. For this purpose, the characteristics of simple pipelines are used 1 , 2 And 3 , which are built according to dependencies (5.7).
To obtain a point belonging to the total characteristic of a parallel connection, it is necessary, in accordance with (5.8), to add up the flow rates in the original pipelines at the same pressure losses. For this purpose, an arbitrary horizontal line is drawn on the graph (with an arbitrary loss). Along this horizontal line, the segments (expenses) are graphically summarized Q 1 , Q 2 And Q 3), obtained from the intersection of the horizontal line with the initial characteristics of the pipelines. The point thus obtained IN belongs to the summary characteristics of the connection. Consequently, the total characteristic of a parallel connection of pipelines is obtained by adding the abscissas of the points of the original characteristics for given losses.
Using a similar method, summary characteristics are constructed for branched pipelines. Branched connection is a collection of several pipelines that have one common point (the place where the pipes branch or meet).
The series and parallel connections discussed above, strictly speaking, belong to the category of complex pipelines. However, in hydraulics under complex pipeline As a rule, they understand the connection of several simple pipelines connected in series and in parallel.
In Figure 5.3, d an example of such a complex pipeline consisting of three pipelines is given 1 , 2 And 3. Pipeline 1 connected in series with respect to the pipelines 2 And 3. Pipelines 2 And 3 can be considered parallel, since they have a common branching point (point M) and supply liquid to the same hydraulic tank.
For complex pipelines, calculations are usually carried out graphically. The following sequence is recommended:
1) a complex pipeline is divided into a number of simple pipelines;
2) for each simple pipeline its characteristics are constructed;
3) by graphical addition, the characteristics of a complex pipeline are obtained.
In Figure 5.3, e shows the sequence of graphical constructions when obtaining the summary characteristic () of a complex pipeline. First, the characteristics of pipelines are added up according to the rule for adding the characteristics of parallel pipelines, and then the characteristic of a parallel connection is added with the characteristic according to the rule for adding the characteristics of series-connected pipelines and the characteristic of the entire complex pipeline is obtained.
Having a graph constructed in this way (see Figure 5.3, e) for a complex pipeline, you can simply use a known flow rate Q 1 entering the hydraulic system, determine the required pressure H consumption = for the entire complex pipeline, costs Q 2 and Q 3 in parallel branches, as well as pressure loss, and in each simple pipeline.
5.4 Pump-feed pipeline
As already noted, the main method of supplying fluid in mechanical engineering is its forced injection by a pump. Pump called a hydraulic device that converts the mechanical energy of the drive into the energy of the flow of working fluid. In hydraulics, a pipeline in which fluid movement is ensured by a pump is called pipeline with pump supply(Figure 5.4, A).
The purpose of calculating a pump-fed pipeline is usually to determine the pressure generated by the pump (pump head). Pump head N n is the total mechanical energy transferred by the pump to a unit weight of liquid. Thus, to determine N n it is necessary to estimate the increment in the total specific energy of the liquid as it passes through the pump, i.e.
, (5.9)
Where N in,N out - specific energy of the liquid at the inlet and outlet of the pump, respectively.
Let's consider the operation of an open pipeline with pump supply (see Figure 5.4, A). The pump pumps liquid from the lower reservoir A with pressure above the liquid p 0 to another tank B, in which the pressure R 3 . Height of the pump relative to the lower liquid level H 1 is called the suction lift, and the pipeline through which the liquid enters the pump is suction pipeline, or hydraulic suction line. The height of the final section of the pipeline or the upper liquid level N 2 is called the discharge height, and the pipeline through which the liquid moves from the pump is pressure, or hydraulic injection line.
Let us write the Bernoulli equation for the fluid flow in the suction pipeline, i.e. for sections 0-0 And 1-1 :
, (5.10)
where is the pressure loss in the suction pipeline.
Equation (5.10) is the main one for calculating suction pipelines. Pressure p 0 usually limited (usually atmospheric pressure). Therefore, the purpose of calculating the suction pipeline is usually to determine the pressure in front of the pump. It must be higher than the saturated vapor pressure of the liquid. This is necessary to prevent cavitation at the pump inlet. From equation (5.10) you can find the specific energy of the liquid at the pump inlet:
. (5.11)
Let us write the Bernoulli equation for fluid flow in a pressure pipeline, i.e. for sections 2-2 And 3-3:
, (5.12)
where is the pressure loss in the pressure pipeline.
The left side of this equation represents the specific energy of the fluid leaving the pump Hout. Substituting the right-hand sides of dependencies (5.11) into (5.9) for Hinput and (5.12) for Hout, we get
As follows from equation (5.13), the pump pressure H n provides rise of liquid to a height (H 1+H 2), increasing pressure from R 0 before p 3 and is spent on overcoming resistance in the suction and pressure pipelines.
If on the right side of equation (5.13) 
designate H st and replace 
on KQm
, then we get Hn=
Hcr +
KQm.
Let's compare the last expression with formula (5.2), which determines the required pressure for the pipeline. Their complete identity is obvious:
those. the pump creates a pressure equal to the required pressure of the pipeline.
The resulting equation (5.14) allows you to analytically determine the pump pressure. However, in most cases, the analytical method is quite complex, so the graphical method for calculating a pipeline with pump supply has become widespread.
This method consists of jointly plotting on a graph the characteristics of the required pipeline pressure (or pipeline characteristics) and pump characteristics. The pump characteristic refers to the dependence of the pressure generated by the pump on the flow rate. The point of intersection of these dependencies is called operating point hydraulic system and is the result of a graphical solution of equation (5.14).
In Figure 5.4, b An example of such a graphical solution is given. Here is point A and there is the desired operating point of the hydraulic system. Its coordinates determine the pressure H n created by the pump and flow rate Qn fluid flowing from the pump into the hydraulic system.
If for some reason the position of the operating point on the graph does not suit the designer, then this position can be changed by adjusting any parameters of the pipeline or pump.
7.5. Water hammer in the pipeline
Water hammer is an oscillatory process that occurs in a pipeline when there is a sudden change in the speed of the liquid, for example when the flow stops due to the rapid closing of a valve (faucet).
This process is very fast and is characterized by alternating sharp increases and decreases in pressure, which can lead to destruction of the hydraulic system. This is due to the fact that the kinetic energy of a moving flow, when stopped, is converted into work on stretching the walls of the pipes and compressing the liquid. The greatest danger is the initial pressure surge.
Let us trace the stages of hydraulic shock that occurs in the pipeline when the flow is quickly blocked (Figure 7.5).
Let at the end of the pipe through which the liquid moves at a speed vq, The tap is instantly closed A. Then (see Figure 7.5, A) the speed of liquid particles colliding with the tap will be extinguished, and their kinetic energy will be transferred into the work of deformation of the walls of the pipe and liquid. In this case, the walls of the pipe are stretched and the liquid is compressed. The pressure in the stopped liquid increases by Δ p beat Other particles run into the inhibited particles of liquid at the tap and also lose speed, resulting in a cross-section p-p moves to the right with speed c, called shock wave speed, the transition region itself (section p-p), in which the pressure changes by an amount Δ p oud is called shock wave.
When the shock wave reaches the reservoir, the liquid will be stopped and compressed throughout the pipe, and the walls of the pipe will be stretched. Shock pressure rise Δ p the shock will spread throughout the entire pipe (see Fig. 7.5, b).
But this state is not equilibrium. Under the influence of increased pressure ( R 0 + Δ p beat) liquid particles will rush from the pipe into the tank, and this movement will begin from the section directly adjacent to the tank. Now the section p-p moves along the pipeline in the opposite direction - to the tap - at the same speed With, leaving behind pressure in the liquid p 0 (see Figure 7.5, V).
The liquid and pipe walls return to the initial state corresponding to the pressure p 0 . The work of deformation is completely converted into kinetic energy, and the liquid in the pipe acquires its original speed , but directed in the opposite direction.
At this speed, the “liquid column” (see Figure 7.5, G) tends to break away from the tap, resulting in a negative shock wave (the pressure in the liquid decreases by the same value Δ p ud). The boundary between two states of a liquid is directed from tap to tank at speed With, leaving behind compressed pipe walls and expanded liquid (see Figure 7.5, d). The kinetic energy of the liquid again transforms into work of deformation, but with the opposite sign.
The state of the liquid in the pipe at the moment the negative shock wave arrives at the tank is shown in Figure 7.5, e. Same as for the case shown in Figure 7.5, b, it is not in equilibrium, since the liquid in the pipe is under pressure ( R 0 + Δ p beat), less than in the tank. In Figure 7.5, and shows the process of equalizing pressure in a pipe and a tank, accompanied by the occurrence of fluid movement at a speed .
It is obvious that as soon as the shock wave reflected from the tank reaches the tap, a situation will arise that already occurred when the tap was closed. The entire water hammer cycle will repeat.
Theoretical and experimental studies of hydraulic shock in pipes were first carried out by N.E. Zhukovsky. In his experiments, up to 12 complete cycles were recorded with a gradual decrease in Δ p beat As a result of the research, N.E. Zhukovsky obtained analytical dependencies that made it possible to estimate the shock pressure Δ p beat One of these formulas, named after N.E. Zhukovsky, has the form
where is the shock wave propagation speed With determined by the formula
,
Where TO - volumetric modulus of elasticity of the liquid; E - modulus of elasticity of the pipeline wall material; d and δ are the internal diameter and wall thickness of the pipeline, respectively.
Formula (7.14) is valid for direct water hammer, when the flow shutoff time t closed is less than the water hammer phase t 0:
Where l- pipe length.
Water hammer phase t 0 is the time during which the shock wave moves from the tap to the tank and returns back. At t closed > t 0 the shock pressure is less, and such a water hammer is called indirect.
If necessary, you can use known methods of “mitigating” water hammer. The most effective of these is to increase the response time of taps or other devices that shut off the flow of liquid. A similar effect is achieved by installing hydraulic accumulators or safety valves in front of devices that block the flow of fluid. Reducing the speed of fluid movement in the pipeline by increasing the internal diameter of the pipes at a given flow rate and reducing the length of the pipelines (reducing the hydraulic shock phase) also help reduce the shock pressure.
The movement of liquid in a pipeline is determined by the difference between two pressures: the pressure before entering the pipeline and the pressure at the exit from it. However, if the reference plane is combined with the free surface of the liquid in a piezometer connected to the output section, then the specific potential energy of the output section with respect to the comparison plane will be equal to zero. In most practical problems, the kinetic energy in the exit section is either very small or is not of interest for calculation. Thus, the main quantity that determines the movement of liquid in a pipeline is the pressure in the initial section relative to the liquid level in the piezometer connected to the outlet section. This pressure is called design pressure pipeline.
The magnitude of the design pressure can be estimated as follows. In general, the difference between the energies of the input and output cross sections
Typically, liquid enters a pipeline from a tank or reservoir of such large dimensions that the velocity before entry can be considered negligible. The kinetic energy at the output, as already noted, can also be neglected. In addition, if both sections communicate with the atmosphere (as usually happens), then 
. Then
that is, in this simple case, the design pressure is the difference in the geometric heights of the centers of gravity of the inlet and outlet sections of the pipeline.
Let's first consider the calculation scheme simple pipeline, that is, a pipeline that does not have branches. Such a pipeline can supply water from one pressure tank to another or from a canal (reservoir) to a point where water from the water supply flows directly into the atmosphere.
Pipe length l and diameter d can be horizontal or inclined, flow flows through it Q(Fig. 6.1).
Let's create Bernoulli's equation for two sections: one of them 1 –1 coincides with the free surface of the water in the tank, other 2 –2 passed through the outlet of the pipeline. We draw the 0–0 comparison plane through the center of the outlet section of the pipe. Bernoulli's equation will be written as
.
The comparison plane is drawn through the center of the outlet section, that is z 1 = H, z 2 = 0. The pressure in both sections is equal to atmospheric: . The liquid level in the tank remains constant, therefore.
For long pipelines, the kinetic energy of the liquid in the outlet section is always very small compared to the amount of losses; it can be neglected in the same way as local losses. Taking all this into account, from the Bernoulli equation we obtain
| . | (6.1) |
This ratio means that almost all the available pressure is spent on overcoming friction resistance along the length of the pipeline. To find out the required pressure value, you should calculate the energy loss along the length of the pipeline. The calculation of long pipelines is based on this position.
Losses distributed along the length of the pipeline can be calculated using formula (5.2) – the Weisbach–Darcy formula:
.
The speed of fluid movement through a pipeline in a fully developed turbulent flow regime, that is, in the case of quadratic resistance, is determined by formula (4.7) - Chezy’s formula:
Then the fluid flow will be determined as
The complex expresses the amount of fluid flow that the pipe in question can pass with a hydraulic slope equal to one. This quantity is called flow module pipes. Recalling the Expression for Hydraulic Slope i at steady flow
and using the designation of the flow module, we can obtain a formula relating energy losses and fluid flow:
| . | (6.2) |
The pipe flow modulus is related to its diameter and degree of roughness. Using Manning's formula (4.9) for the coefficient C, and taking into account the value of the hydraulic radius for round pipes, we can write
.
For industrially produced pipes of standard diameters (range), the values of the flow modulus K calculated and compiled into hydraulic reference books.
Thus, the basic formulas for all three types of problems arising when calculating a simple pipeline can be obtained from formula (6.2) taking into account formula (6.1), that is, using the value of energy losses as the design pressure:
| , | (6.3) | |
| , | (6.4) | |
| . | (6.5) |
The procedure for calculations for problems of the first type (determining the required pressure) is as follows.
1. Using a known pipe diameter, the cross-sectional area and average flow speed are calculated
2. Reynolds number is calculated
3. In accordance with the material and condition (new or used) of the pipeline, its roughness is determined using hydraulic tables.
4. Based on the calculated Re number and roughness from Nikuradze’s graphs, it is determined which case of resistance along the length occurs. This will allow you to select the type of formula for calculating the coefficient C.
5. The value of the flow module is calculated or determined from hydraulic tables K.
6. With known Q, l And K formula (6.3) is used to find the pressure value. Often the value found in this way H increase slightly (by 2–5%) to provide a margin for unaccounted local losses.
In problems of the second type (determining flow), it is initially impossible to calculate velocities, calculate the Reynolds number and determine the law of resistance along the length of the pipe. In problems of the third type (calculation of required diameters), the initial roughness characteristics of the pipeline are also unknown. Such problems are solved by successive approximations, in which preliminary calculations are carried out by specifying some initial values of unknown parameters. After obtaining the result, the initial assumptions are corrected and the calculations are repeated. When using the capabilities of modern computer technology, these methods do not cause fundamental difficulties.
If pipelines with a known high flow velocity and significant roughness are considered, then this allows us to confidently assume the presence of a quadratic resistance law. Then, using the Chezy, Pavlovsky or Manning formulas, you can solve such problems without selection.
5 HYDRAULIC CALCULATION OF PIPELINES
5.1 Simple pipeline of constant cross-section
The pipeline is called simple, if it has no branches. Simple pipelines can form connections: series, parallel or branched. Pipelines can be complex, containing both serial and parallel connections or branches.
Liquid moves through a pipeline due to the fact that its energy at the beginning of the pipeline is greater than at the end. This difference (difference) in energy levels can be created in one way or another: by the operation of the pump, due to the difference in liquid levels, or by gas pressure. In mechanical engineering, one has to deal mainly with pipelines, the movement of fluid in which is caused by the operation of a pump.
When hydraulically calculating a pipeline, it is most often determined by its required pressureH consumption - a value numerically equal to the piezometric height in the initial section of the pipeline. If the required pressure is given, then it is usually called available pressureH disp. In this case, hydraulic calculation can determine the flow rate Q liquid in the pipeline or its diameter d. The pipeline diameter value is selected from the established range in accordance with GOST 16516-80.
Let a simple pipeline with a constant flow area, arbitrarily located in space (Figure 5.1, A), has a total length l and diameter d and contains a number of local hydraulic resistances I and II.
Let us write the Bernoulli equation for the initial 1-1 and final 2-2 sections of this pipeline, assuming that the Coriolis coefficients in these sections are the same (α 1 =α 2). After reducing the velocity pressures we get
Where z 1 , z 2 - coordinates of the centers of gravity of the initial and final sections, respectively;
p 1 , p 2 - pressure in the initial and final sections of the pipeline, respectively;
Total pressure loss in the pipeline.
Hence the required pressure
, (5.1)
As can be seen from the resulting formula, the required pressure is the sum of the total geometric height Δz = z 2 – z 1 , to which the liquid rises as it moves through the pipeline, the piezometric height in the final section of the pipeline and the sum of the hydraulic pressure losses that occur when the liquid moves in it.
In hydraulics, it is customary to understand the static pressure of a pipeline as the sum 
.
Then, representing the total losses as a power function of the flow rate Q, we get
Where T - a value depending on the fluid flow regime in the pipeline;
K is the resistance of the pipeline.
Under laminar fluid flow conditions and linear local resistances (their equivalent lengths are given l eq) total losses
,
Where l calc = l + l eq - estimated pipeline length.
Therefore, in laminar mode t = 1, 
.
In turbulent fluid flow
.
Replacing the average fluid velocity through flow rate in this formula, we obtain the total pressure loss
. (5.3)
Then, under turbulent conditions 
, and the exponent m= 2. It should be remembered that in the general case the friction loss coefficient along the length is also a function of the flow rate Q.
By doing the same in each specific case, after simple algebraic transformations and calculations, you can obtain a formula that determines the analytical dependence of the required pressure for a given simple pipeline on the flow rate in it. Examples of such dependencies in graphical form are shown in Figure 5.1, b, V.
Analysis of the formulas given above shows that the solution to the problem of determining the required pressure H consumption at known consumption Q liquids in the pipeline and its diameter d is not difficult, since it is always possible to assess the fluid flow regime in the pipeline by comparing the critical value ReTop= 2300 with its actual value, which for round pipes can be calculated using the formula
After determining the flow regime, you can calculate the pressure loss, and then the required pressure using formula (5.2).
If the values Q or d are unknown, then in most cases it is difficult to assess the flow regime, and, therefore, to reasonably select formulas that determine pressure losses in the pipeline. In such a situation, it can be recommended to use either the successive approximation method, which usually requires a fairly large amount of computational work, or a graphical method, in the application of which it is necessary to construct the so-called characteristic of the required pipeline pressure.
5.2. Constructing a characteristic of the required pressure of a simple pipeline
Graphical representation in coordinates N-Q analytical dependence (5.2) obtained for a given pipeline is called in hydraulics characteristic of the required pressure. In Figure 5.1, b, c Several possible characteristics of the required pressure are given (linear - for laminar flow conditions and linear local resistances; curvilinear - for turbulent flow conditions or the presence of quadratic local resistances in the pipeline).
As can be seen in the graphs, the value of the static pressure N st can be either positive (liquid is supplied to a certain height Δ z or there is excess pressure in the final section p 2) and negative (when the liquid flows downwards or when it moves into a cavity with rarefaction).
The slope of the characteristics of the required pressure depends on the resistance of the pipeline and increases with increasing pipe length and decreasing its diameter, and also depends on the number and characteristics of local hydraulic resistance. In addition, in a laminar flow regime, the quantity under consideration is also proportional to the viscosity of the liquid. The point of intersection of the required pressure characteristic with the abscissa axis (point A in Figure 5.1, b, V) determines the fluid flow in the pipeline when moving by gravity.
Graphical dependencies of the required pressure are widely used to determine flow Q when calculating both simple and complex pipelines. Therefore, let us consider the methodology for constructing such a dependence (Figure 5.2, A). It consists of the following stages.
1st stage. Using formula (5.4) we determine the value of the critical flow Q kr, corresponding ReTop=2300, and mark it on the expense axis (x-axis). Obviously, for all expenses located to the left Q kr, there will be a laminar flow regime in the pipeline, and for flow rates located to the right Q cr, - turbulent.
2nd stage. We calculate the required pressure values H 1 And H 2 at a flow rate in the pipeline equal to Q kr, accordingly assuming that N 1 - calculation result for laminar flow regime, and N 2 - when turbulent.
3rd stage. We construct a characteristic of the required pressure for a laminar flow regime (for flow rates less than Q cr) . If local resistances installed in the pipeline have a linear dependence of losses on flow, then the characteristic of the required pressure has a linear form.
4th stage. We construct a characteristic of the required pressure for a turbulent flow regime (for flow rates large QTop). In all cases, a curvilinear characteristic is obtained, close to a parabola of the second degree.
Having a characteristic of the required pressure for a given pipeline, it is possible, based on the known value of the available pressure Hdisp. find the required flow rate Qx (see Figure 5.2, A).
If you need to find the internal diameter of the pipeline d, then, given several values d, it is necessary to construct the dependence of the required pressure Hconsumption from diameter d (Fig. 5.2, b). Next by value N disp. the nearest larger diameter from the standard range is selected d st .
In some cases, in practice, when calculating hydraulic systems, instead of the required pressure characteristic, the pipeline characteristic is used. Pipeline characteristics- this is the dependence of the total pressure losses in the pipeline on the flow rate. The analytical expression of this dependence has the form
Comparison of formulas (5.5) and (5.2) allows us to conclude that the characteristics of the pipeline differ from the characteristics of the required pressure in the absence of static pressure H st, and at H st = 0 these two dependencies coincide.
5.3 Connections of simple pipelines.
Analytical and graphical calculation methods
Let's consider methods for calculating connections of simple pipelines.
Let us have serial connection several simple pipelines ( 1 , 2 And 3 in Figure 5.3, A) different lengths, different diameters, with different sets of local resistances. Since these pipelines are connected in series, each of them has the same fluid flow Q. Total head loss for the entire connection (between points M And N) consists of pressure losses in each simple pipeline ( , , ), i.e. for a series connection the following system of equations is valid:
(5.6)
The pressure loss in each simple pipeline can be determined through the values of the corresponding flow rates:
The system of equations (5.6), supplemented by dependencies (5.7), is the basis for the analytical calculation of a hydraulic system with a series connection of pipelines.
If a graphical calculation method is used, then there is a need to construct a summary characteristic of the connection.
In Figure 5.3, b shows a method for obtaining the summary characteristics of a serial connection. For this purpose, the characteristics of simple pipelines are used 1 , 2 And 3
To construct a point belonging to the total characteristic of a series connection, it is necessary, in accordance with (5.6), to add up the pressure losses in the original pipelines at the same flow rate. For this purpose, an arbitrary vertical line is drawn on the graph (at an arbitrary flow rate Q" ). Along this vertical, the segments (pressure loss, and) obtained from the intersection of the vertical with the initial characteristics of the pipelines are summed up. The point thus obtained A will belong to the summary characteristics of the connection. Consequently, the total characteristic of a series connection of several simple pipelines is obtained by adding the ordinates of the points of the initial characteristics at a given flow rate.
Parallel called a connection of pipelines that have two common points (a branch point and a closure point). An example of a parallel connection of three simple pipelines is shown in Figure 5.3, V. Obviously, the expense Q fluid in the hydraulic system before branching (point M) and after closure (point N) the same and equal to the amount of expenses Q 1 , Q 2 and Q 3 in parallel branches.
If we designate total pressures at points M And N through NM And H N, then for each pipeline the pressure loss is equal to the difference of these pressures:
; 
; 
,
that is, in parallel pipelines the pressure loss is always the same. This is explained by the fact that with such a connection, despite the different hydraulic resistance of each simple pipeline, the costs Q 1 , Q 2 And Q 3 distributed between them so that the losses remain equal.
Thus, the system of equations for a parallel connection has the form
(5.8)
The pressure loss in each pipeline included in the connection can be determined using formulas of the form (5.7). Thus, the system of equations (5.8), supplemented by formulas (5.7), is the basis for the analytical calculation of hydraulic systems with parallel connection of pipelines.
In Figure 5.3, G shows a method for obtaining the summary characteristics of a parallel connection. For this purpose, the characteristics of simple pipelines are used 1 , 2 And 3 , which are built according to dependencies (5.7).
To obtain a point belonging to the total characteristic of a parallel connection, it is necessary, in accordance with (5.8), to add up the flow rates in the original pipelines at the same pressure losses. For this purpose, an arbitrary horizontal line is drawn on the graph (with an arbitrary loss). Along this horizontal line, the segments (expenses) are graphically summarized Q 1 , Q 2 And Q 3), obtained from the intersection of the horizontal line with the initial characteristics of the pipelines. The point thus obtained IN belongs to the summary characteristics of the connection. Consequently, the total characteristic of a parallel connection of pipelines is obtained by adding the abscissas of the points of the original characteristics for given losses.
Using a similar method, summary characteristics are constructed for branched pipelines. Branched connection is a collection of several pipelines that have one common point (the place where the pipes branch or meet).
The series and parallel connections discussed above, strictly speaking, belong to the category of complex pipelines. However, in hydraulics under complex pipeline As a rule, they understand the connection of several simple pipelines connected in series and in parallel.
In Figure 5.3, d an example of such a complex pipeline consisting of three pipelines is given 1 , 2 And 3. Pipeline 1 connected in series with respect to the pipelines 2 And 3. Pipelines 2 And 3 can be considered parallel, since they have a common branching point (point M) and supply liquid to the same hydraulic tank.
For complex pipelines, calculations are usually carried out graphically. The following sequence is recommended:
1) a complex pipeline is divided into a number of simple pipelines;
2) for each simple pipeline its characteristics are constructed;
3) by graphical addition, the characteristics of a complex pipeline are obtained.
In Figure 5.3, e shows the sequence of graphical constructions when obtaining the summary characteristic () of a complex pipeline. First, the characteristics of pipelines are added up according to the rule for adding the characteristics of parallel pipelines, and then the characteristic of a parallel connection is added with the characteristic according to the rule for adding the characteristics of series-connected pipelines and the characteristic of the entire complex pipeline is obtained.
Having a graph constructed in this way (see Figure 5.3, e) for a complex pipeline, you can simply use a known flow rate Q 1 entering the hydraulic system, determine the required pressure H consumption = for the entire complex pipeline, costs Q 2 and Q 3 in parallel branches, as well as pressure loss, and in each simple pipeline.
5.4 Pump-feed pipeline
As already noted, the main method of supplying fluid in mechanical engineering is its forced injection by a pump. Pump called a hydraulic device that converts the mechanical energy of the drive into the energy of the flow of working fluid. In hydraulics, a pipeline in which fluid movement is ensured by a pump is called pipeline with pump supply(Figure 5.4, A).
The purpose of calculating a pump-fed pipeline is usually to determine the pressure generated by the pump (pump head). Pump head N n is the total mechanical energy transferred by the pump to a unit weight of liquid. Thus, to determine N n it is necessary to estimate the increment in the total specific energy of the liquid as it passes through the pump, i.e.
, (5.9)
Where N in,N out - specific energy of the liquid at the inlet and outlet of the pump, respectively.
Let's consider the operation of an open pipeline with pump supply (see Figure 5.4, A). The pump pumps liquid from the lower reservoir A with pressure above the liquid p 0 to another tank B, in which the pressure R 3 . Height of the pump relative to the lower liquid level H 1 is called the suction lift, and the pipeline through which the liquid enters the pump is suction pipeline, or hydraulic suction line. The height of the final section of the pipeline or the upper liquid level N 2 is called the discharge height, and the pipeline through which the liquid moves from the pump is pressure, or hydraulic injection line.
Let us write the Bernoulli equation for the fluid flow in the suction pipeline, i.e. for sections 0-0 And 1-1 :
, (5.10)
where is the pressure loss in the suction pipeline.
Equation (5.10) is the main one for calculating suction pipelines. Pressure p 0 usually limited (usually atmospheric pressure). Therefore, the purpose of calculating the suction pipeline is usually to determine the pressure in front of the pump. It must be higher than the saturated vapor pressure of the liquid. This is necessary to prevent cavitation at the pump inlet. From equation (5.10) you can find the specific energy of the liquid at the pump inlet:
. (5.11)
Let us write the Bernoulli equation for fluid flow in a pressure pipeline, i.e. for sections 2-2 And 3-3:
, (5.12)
where is the pressure loss in the pressure pipeline.
The left side of this equation represents the specific energy of the fluid leaving the pump Hout. Substituting the right-hand sides of dependencies (5.11) into (5.9) for Hinput and (5.12) for Hout, we get
As follows from equation (5.13), the pump pressure H n provides rise of liquid to a height (H 1+H 2), increasing pressure from R 0 before p 3 and is spent on overcoming resistance in the suction and pressure pipelines.
If on the right side of equation (5.13) 
designate H st and replace 
on KQm
, then we get Hn=
Hcr +
KQm.
Let's compare the last expression with formula (5.2), which determines the required pressure for the pipeline. Their complete identity is obvious:
those. the pump creates a pressure equal to the required pressure of the pipeline.
The resulting equation (5.14) allows you to analytically determine the pump pressure. However, in most cases, the analytical method is quite complex, so the graphical method for calculating a pipeline with pump supply has become widespread.
This method consists of jointly plotting on a graph the characteristics of the required pipeline pressure (or pipeline characteristics) and pump characteristics. The pump characteristic refers to the dependence of the pressure generated by the pump on the flow rate. The point of intersection of these dependencies is called operating point hydraulic system and is the result of a graphical solution of equation (5.14).
In Figure 5.4, b An example of such a graphical solution is given. Here is point A and there is the desired operating point of the hydraulic system. Its coordinates determine the pressure H n created by the pump and flow rate Qn fluid flowing from the pump into the hydraulic system.
If for some reason the position of the operating point on the graph does not suit the designer, then this position can be changed by adjusting any parameters of the pipeline or pump.
7.5. Water hammer in the pipeline
Water hammer is an oscillatory process that occurs in a pipeline when there is a sudden change in the speed of the liquid, for example when the flow stops due to the rapid closing of a valve (faucet).
This process is very fast and is characterized by alternating sharp increases and decreases in pressure, which can lead to destruction of the hydraulic system. This is due to the fact that the kinetic energy of a moving flow, when stopped, is converted into work on stretching the walls of the pipes and compressing the liquid. The greatest danger is the initial pressure surge.
Let us trace the stages of hydraulic shock that occurs in the pipeline when the flow is quickly blocked (Figure 7.5).
Let at the end of the pipe through which the liquid moves at a speed vq, The tap is instantly closed A. Then (see Figure 7.5, A) the speed of liquid particles colliding with the tap will be extinguished, and their kinetic energy will be transferred into the work of deformation of the walls of the pipe and liquid. In this case, the walls of the pipe are stretched and the liquid is compressed. The pressure in the stopped liquid increases by Δ p beat Other particles run into the inhibited particles of liquid at the tap and also lose speed, resulting in a cross-section p-p moves to the right with speed c, called shock wave speed, the transition region itself (section p-p), in which the pressure changes by an amount Δ p oud is called shock wave.
When the shock wave reaches the reservoir, the liquid will be stopped and compressed throughout the pipe, and the walls of the pipe will be stretched. Shock pressure rise Δ p the shock will spread throughout the entire pipe (see Fig. 7.5, b).
But this state is not equilibrium. Under the influence of increased pressure ( R 0 + Δ p beat) liquid particles will rush from the pipe into the tank, and this movement will begin from the section directly adjacent to the tank. Now the section p-p moves along the pipeline in the opposite direction - to the tap - at the same speed With, leaving behind pressure in the liquid p 0 (see Figure 7.5, V).
The liquid and pipe walls return to the initial state corresponding to the pressure p 0 . The work of deformation is completely converted into kinetic energy, and the liquid in the pipe acquires its original speed , but directed in the opposite direction.
At this speed, the “liquid column” (see Figure 7.5, G) tends to break away from the tap, resulting in a negative shock wave (the pressure in the liquid decreases by the same value Δ p ud). The boundary between two states of a liquid is directed from tap to tank at speed With, leaving behind compressed pipe walls and expanded liquid (see Figure 7.5, d). The kinetic energy of the liquid again transforms into work of deformation, but with the opposite sign.
The state of the liquid in the pipe at the moment the negative shock wave arrives at the tank is shown in Figure 7.5, e. Same as for the case shown in Figure 7.5, b, it is not in equilibrium, since the liquid in the pipe is under pressure ( R 0 + Δ p beat), less than in the tank. In Figure 7.5, and shows the process of equalizing pressure in a pipe and a tank, accompanied by the occurrence of fluid movement at a speed .
It is obvious that as soon as the shock wave reflected from the tank reaches the tap, a situation will arise that already occurred when the tap was closed. The entire water hammer cycle will repeat.
Theoretical and experimental studies of hydraulic shock in pipes were first carried out by N.E. Zhukovsky. In his experiments, up to 12 complete cycles were recorded with a gradual decrease in Δ p beat As a result of the research, N.E. Zhukovsky obtained analytical dependencies that made it possible to estimate the shock pressure Δ p beat One of these formulas, named after N.E. Zhukovsky, has the form
where is the shock wave propagation speed With determined by the formula
,
Where TO - volumetric modulus of elasticity of the liquid; E - modulus of elasticity of the pipeline wall material; d and δ are the internal diameter and wall thickness of the pipeline, respectively.
Formula (7.14) is valid for direct water hammer, when the flow shutoff time t closed is less than the water hammer phase t 0:
Where l- pipe length.
Water hammer phase t 0 is the time during which the shock wave moves from the tap to the tank and returns back. At t closed > t 0 the shock pressure is less, and such a water hammer is called indirect.
If necessary, you can use known methods of “mitigating” water hammer. The most effective of these is to increase the response time of taps or other devices that shut off the flow of liquid. A similar effect is achieved by installing hydraulic accumulators or safety valves in front of devices that block the flow of fluid. Reducing the speed of fluid movement in the pipeline by increasing the internal diameter of the pipes at a given flow rate and reducing the length of the pipelines (reducing the hydraulic shock phase) also help reduce the shock pressure.
Pipes connecting various apparatuses of chemical plants. With their help, substances are transferred between individual devices. Typically, several individual pipes are connected to create a single piping system.
A pipeline is a system of pipes connected together using connecting elements, used for transporting chemicals and other materials. In chemical plants, closed pipelines are usually used to move substances. If we are talking about closed and isolated parts of the installation, then they also refer to the piping system or network.
A closed pipeline system may include:
- Pipes.
- Pipe connecting elements.
- Sealing seals connecting two detachable sections of the pipeline.
All of the above elements are manufactured separately and then connected into a single pipeline system. In addition, pipelines can be equipped with heating and the necessary insulation made from various materials.
The choice of pipe size and materials for manufacturing is carried out on the basis of technological and design requirements in each specific case. But to standardize the sizes of pipes, their classification and unification was carried out. The main criterion was the permissible pressure at which the pipe can be operated.
Nominal size DN
Conditional diameter DN (nominal diameter) is a parameter that is used in pipeline systems as a characterizing feature with the help of which pipeline parts such as pipes, fittings, fittings and others are adjusted.
The nominal diameter is a dimensionless value, but is numerically approximately equal to the internal diameter of the pipe. Example of nominal diameter designation: DN 125.
Also, the nominal diameter is not indicated on the drawings and does not replace the actual diameters of the pipes. It approximately corresponds to the clear diameter of certain parts of the pipeline (Fig. 1.1). If we talk about the numerical values of conditional transitions, they are selected in such a way that the throughput of the pipeline increases in the range from 60 to 100% when moving from one conditional passage to the next.
Common nominal diameters:
3, 4, 5, 6, 8, 10, 15, 20, 25, 32, 40, 50, 65, 80, 100, 125, 150, 200, 250, 300, 350, 400, 450, 500, 600, 700, 800, 900, 1000, 1200, 1400, 1600, 1800, 2000, 2200, 2600, 2800, 3000, 3200, 3400, 3600, 3800, 4000.
The dimensions of these nominal passages are set with the expectation that there will be no problems with fitting the parts to each other. Determining the nominal diameter is based on the value of the internal diameter of the pipeline; the value of the nominal diameter that is closest to the clear diameter of the pipe is selected.
Nominal pressure PN
Nominal pressure PN is a value corresponding to the maximum pressure of the pumped medium at 20 °C, at which long-term operation of a pipeline of specified dimensions is possible.
Nominal pressure is a dimensionless value.
Like the nominal diameter, the nominal pressure was calibrated based on operational experience and accumulated experience (Table 1.1).
The nominal pressure for a particular pipeline is selected based on the pressure actually created in it, by selecting the nearest higher value. In this case, fittings and fittings in this pipeline must also correspond to the same pressure level. The thickness of the pipe walls is calculated based on the nominal pressure and must ensure the operability of the pipe at a pressure value equal to the nominal pressure (Table 1.1).
Permissible excess operating pressure p e,zul
The nominal pressure is only used for an operating temperature of 20°C. As the temperature increases, the load capacity of the pipe decreases. At the same time, the permissible excess pressure is correspondingly reduced. The p e,zul value shows the maximum excess pressure that can be in the pipeline system when the operating temperature increases (Fig. 1.2).
Pipeline materials
When choosing materials that will be used for the manufacture of pipelines, indicators such as the characteristics of the medium that will be transported through the pipeline and the operating pressure expected in this system are taken into account. It is also worth considering the possibility of corrosive effects from the pumped medium on the material of the pipe walls.
Almost all piping systems and chemical plants are made from steel. For general use in the absence of high mechanical loads and corrosive effects, gray cast iron or unalloyed structural steels are used for the manufacture of pipelines.
In the case of higher operating pressures and the absence of corrosive loads, a pipeline made of tempered steel or using cast steel is used.
If the corrosive effect of the environment is great or high demands are placed on the purity of the product, then the pipeline is made of stainless steel.
If the pipeline must be resistant to seawater, then copper-nickel alloys are used for its manufacture. Aluminum alloys and metals such as tantalum or zirconium can also be used.
Various types of plastics are becoming increasingly common as pipeline materials, due to their high corrosion resistance, low weight and ease of processing. This material is suitable for wastewater pipelines.
Pipeline fittings
Pipelines made of plastic materials suitable for welding are assembled at the installation site. Such materials include steel, aluminum, thermoplastics, copper, etc. To connect straight sections of pipes, specially manufactured shaped elements are used, for example, elbows, bends, valves and diameter reductions (Fig. 1.3). These fittings can be part of any pipeline.
Pipe connections
Special connections are used to install individual parts of the pipeline and fittings. They are also used to connect the necessary fittings and devices to the pipeline.
Connections are selected (Fig. 1.4) depending on:
- materials used for the manufacture of pipes and fittings. The main selection criterion is the possibility of welding.
- operating conditions: low or high pressure, as well as low or high temperature.
- production requirements that apply to the pipeline system.
- the presence of detachable or permanent connections in the pipeline system.
Linear expansion of pipes and its equipment
The geometric shape of objects can be changed both by force on them and by changing their temperature. These physical phenomena lead to the fact that the pipeline, which is installed in an unloaded state and without temperature exposure, undergoes some linear expansion or contraction during operation under pressure or exposure to temperature, which negatively affects its performance.
When it is not possible to compensate for expansion, deformation of the pipeline system occurs. In this case, damage to the flange seals and those places where the pipes connect to each other may occur.
Thermal linear expansion
When laying out pipelines, it is important to take into account the possible change in length as a result of increasing temperature or the so-called thermal linear expansion, denoted ΔL. This value depends on the length of the pipe, which is designated L o and the temperature difference Δϑ =ϑ2-ϑ1 (Fig. 1.5).
In the above formula, a is the coefficient of thermal linear expansion of a given material. This indicator is equal to the linear expansion of a 1 m long pipe with an increase in temperature of 1°C.
Pipe expansion compensation elements
Pipe bends
Thanks to special bends that are welded into the pipeline, it is possible to compensate for the natural linear expansion of the pipes. For this purpose, compensating U-shaped, Z-shaped and corner bends, as well as lyre compensators are used (Fig. 1.6).
They perceive the linear expansion of pipes due to their own deformation. However, this method is only possible with certain restrictions. High pressure pipelines use elbows at different angles to accommodate expansion. Due to the pressure that acts in such bends, increased corrosion is possible.
Corrugated pipe expansion joints
This device consists of a thin-walled metal corrugated pipe, which is called a bellows and stretches in the direction of the pipeline (Fig. 1.7).
These devices are installed in the pipeline. The preload is used as a special expansion compensator.
If we talk about axial expansion joints, they are capable of compensating only those linear expansions that occur along the pipe axis. To avoid lateral movement and internal contamination, an internal guide ring is used. In order to protect the pipeline from external damage, as a rule, a special lining is used. Expansion joints that do not contain an internal guide ring absorb lateral movement as well as vibration that may come from pumps.
Pipe insulation
If a high-temperature medium moves through the pipeline, it must be insulated to avoid heat loss. When a medium with a low temperature moves through a pipeline, insulation is used to prevent it from being heated by the external environment. Insulation in such cases is carried out using special insulating materials that are placed around the pipes.
The following materials are usually used:
- At low temperatures up to 100°C, rigid foams such as polystyrene or polyurethane are used.
- At average temperatures of around 600°C, shaped casings or mineral fibers such as stone wool or glass felt are used.
- At high temperatures around 1200°C - ceramic fiber, for example, alumina.
Pipes with a nominal diameter below DN 80 and an insulation layer thickness of less than 50 mm are usually insulated using insulating fittings. To do this, two shells are placed around the pipe and fastened with metal tape, and then covered with a tin casing (Fig. 1.8).
Pipelines that have a nominal diameter greater than DN 80 must be equipped with thermal insulation with a lower frame (Fig. 1.9). This frame consists of clamping rings, spacers, and a metal cladding made of galvanized mild steel or stainless steel sheet. The space between the pipeline and the metal casing is filled with insulating material.
The thickness of the insulation is calculated by determining the costs of its manufacture, as well as losses that arise due to heat loss, and ranges from 50 to 250 mm.
Thermal insulation must be applied along the entire length of the pipeline system, including areas of bends and elbows. It is very important to ensure that there are no unprotected areas that could cause heat loss. Flange connections and fittings must be equipped with shaped insulating elements (Fig. 1.10). This provides unobstructed access to the connection point without the need to remove insulating material from the entire piping system in the event of a leak.
If the insulation of the pipeline system is chosen correctly, many problems are solved, such as:
- Avoiding a strong drop in temperature in the flowing medium and, as a result, saving energy.
- Preventing temperatures in gas pipeline systems from falling below the dew point. Thus, it is possible to eliminate the formation of condensation, which can lead to significant corrosion damage.
- Avoiding condensation in steam lines.
Pipelines are divided into short and long. If the total losses in local resistances are less than 5% of the total losses, such a pipeline is considered long.(∑h< 5%). Если суммарные потери в местных сопротивлениях больше 5% от суммарных потерь – короткий трубопровод. По способам гидравлического расчета трубопроводы делятся на простые и сложные. Простым называется трубопровод, состоящий из одной линии труб постоянного или переменного сечения без ответвлений. Отличительной особенностью простого трубопровода является постоянство расхода в любом сечении по всей длине. Сложными называются трубопроводы, содержащие какие-либо ответвления (параллельное соединение труб или разветвление). Всякий сложный трубопровод можно рассматривать как совокупность нескольких простых трубопроводов, соединенных между собой параллельно или последовательно. Поэтому в основе расчета любого трубопровода лежит задача о расчете простого трубопровода.
The movement of liquid in pressure pipelines occurs due to the fact that its energy (pressure) at the beginning of the pipeline is greater than at the end. This difference in energy levels is created in various ways: by the operation of the pump, due to the difference in liquid levels, gas pressure, etc.
Simple pipeline of constant cross-section
The main calculation relationships for a simple pipeline are: the Bernoulli equation, the flow equation Q = const and formulas for calculating friction pressure losses along the length of the pipe and in local resistances.
When applying Bernoulli's equation to a specific calculation, you can consider the following recommendations. First, you should define two design sections and a comparison plane in the figure. It is recommended to take as sections:
the free surface of the liquid in the tank, where the velocity is zero, i.e. V = 0;
the flow exits into the atmosphere, where the pressure in the cross section of the jet is equal to the ambient pressure, i.e. p a6c = p atm or p from6 = 0;
the section in which the pressure is set (or needs to be determined) (readings of a pressure gauge or vacuum gauge);
section under the piston where the excess pressure is determined by the external load.
It is convenient to draw the comparison plane through the center of gravity of one of the design sections, usually located below (then the geometric heights of the sections are 0).
Let a simple pipeline of constant cross-section be located arbitrarily in space (Fig. 1), have a total length l and diameter d and contain a number of local resistances. In the initial section (1-1) the geometric height is z 1 and the excess pressure is p 1, and in the final section (2-2) z 2 and p 2, respectively. Due to the constancy of the pipe diameter, the flow velocity in these sections is the same and equal to v.
Bernoulli equation for sections 1-1 and 2-2 taking into account 
,
will look like:
sum of local resistance coefficients.
For convenience of calculations, we introduce the concept of design pressure
.
,
٭
٭٭
Hydraulic calculation of a simple composite pipeline
,
,
Calculations of simple pipelines come down to three typical tasks: determining the pressure (or pressure), flow rate and diameter of the pipeline. Next, we consider a technique for solving these problems for a simple pipeline of constant cross-section.
Problem 1. Given: pipeline dimensions 
And 
the roughness of its walls 
, properties of liquid 
, fluid flow Q.
Determine the required pressure H (one of the quantities that make up the pressure).
Solution. The Bernoulli equation is compiled for the flow of a given hydraulic system. Control sections are assigned. Selecting a reference plane Z(0.0)
, the initial conditions are analyzed. The Bernoulli equation is compiled taking into account the initial conditions. From the Bernoulli equation we obtain a calculation formula of type ٭. The equation is solved with respect to H. The Reynolds number Re is determined and the mode of motion is established. The value is found 
depending on the driving mode. H and the desired value are calculated.
Task 2. Given: pipeline dimensions 
And 
, the roughness of its walls 
, properties of liquid 
, pressure N. Determine flow rate Q.
Solution. The Bernoulli equation is compiled taking into account the previously given recommendations. The equation is solved relative to the desired value Q. The resulting formula contains an unknown coefficient 
, depending on Re. Direct location 
under the conditions of this problem it is difficult, since when Q is unknown, Re cannot be established in advance. Therefore, further solution of the problem is performed by the method of successive approximations.
approximation: R e → ∞
, define 
2nd approximation:
, we find λ
II (R e II ,
Δ
uh )
and determine 
The relative error is found. If 
, then the solution ends (for training problems 
). Otherwise, the solution is carried out in the third approximation.
Task 3. Given: dimensions of pipelines (except diameter d), roughness of its walls 
, properties of liquid 
, pressure H, flow Q. Determine the diameter of the pipeline.
Solution. When solving this problem, difficulties arise with directly determining the value 
, similar to the problem of the second type. Therefore, it is advisable to carry out the solution using the graphic-analytical method. Multiple diameter values are specified 
.For each 
the corresponding pressure value H is found for a given flow rate Q (the problem of the first type is solved n times). Based on the calculation results, a graph is constructed 
. From the graph, the required diameter d is determined, corresponding to the specified pressure value H.
