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The Methodology of Analysis for Piping Network

Li Ming

Abstract

This paper gives out a methodology analysis for pipe network, and introduces method of software for piping network. It gives out Linear Equation for control network or no control (basic) network numeric model; the pipe and related elements can be defined as components of a segment, in the segment, there are pipe, pump, regulator, tee, valve and control so on. According to Kirchhoff’s laws, this network model solved by linear equation and can be used in analysis, optimization and simulation of the water distribution network model.

Key word:  network analysis, piping network.

 

Introduction

The knowledge about water network models is presented here can be a general description in graph theoretical and the fundamental of Kirchhoff’s physical laws. It is useful to distinguish between basic two terminal components with regular characteristics and complex components with control loops that potentially have irregular (non-monotonic, non-smooth) characteristics. The two terminal components are described by an equation relating element or component flow q and the head loss h_f , or . For the complex components the origin and destination heads may appear explicitly as separate variables, e.g. . Equations (8) and (17) are the set of fundamental equations for analyzing a water distribution network. The nodal model has nodal heads for unknown variables and finally the mixed model has both branch flows and nodal heads as the variables. The numerical algorithms for solving the non-linear equations are based on iterative techniques where during each iteration; a system of linear equation shall be solved.

This paper offers the rationale of analysis in piping network and forwards the methodology of piping analysis.

Component model

The basic two terminal components with regular characteristics and complex components with local control loops that potentially have irregular (non-monotonic, non-smooth) characteristics.

The basic component can be used Darcy-Weisbach formula or Hazen-Williams formula to represent it math model. In the paper^[3], it is given out the Theorem 1 and 2, that can be used to solve the pipe network component math’s model. The model vector represent as following.

From , if defines , we get  that will easy to solve the equation, because, when t iteration, we can consider the t-1 q(t-1) is a temporary fixed value, then:

                                                                           (1)

Here,

- basic two terminal component head difference,

,  is two terminal pressure head of component;

-vector of component’s resistance;

-vector of component’s conductance;

No Control Segment

In concept, the segment of pipe network can be given expression as following:

Fig.1.      The concept of k segment

Where, k – the k segment; q_k is the flow of k segment; ⊿h_k is the k segment head difference;

q_ek – the pipe element flow of the k segment; y_ek is the conductance, while r_ek is the resistance;⊿h_ek is the pipe element head difference of upstream subtract downstream;

P_k, Q_k – is the independence pressure or independence flow source of the k segment. In the system P_k maybe as online pump, Q_k maybe as flow injector, like something to put on steam prescription.

Control Segment

The segment of network with control on it can be generalized as following:

Fig.2.       The k segment with control

Like above beside the P_ck, Q_ck is the control pressure or control flow element in the k segment. P_ck maybe represent the pressure regulator which control command comes from the control room by some schedule or rule. Q_ck is the same as P_ck, but adjusting the flow instead of pressure.

No control Network model

From components to network model, the better way is to use matrix to represent the system. Let us consider following vectors:

 ; The network head vector of nodes, assume there are n number of nodes in the network.

; The network segment head difference between two ends, assume there are b number of segments in the network.

; The network segment flow rate, assume there are b number of segments in the network.

 

; The network

independence pressure vector of segment, assume there are b number of segment in the network.

 ; The network

independence flow source vector of segments, assume there are b number of segments in the network.

; The network pipe element flow rate, assume there are b number of pipes in the network.

; The network pipe

conductance, assume there are b number of pipes in the network.

; The network pipe

resistance, assume there are b number of pipes in the network.

; The network pipe head loss, assume there are b number of pipes in the network.

Element Kirchhoff’s laws

Following the flow continuity (I Kirchhoff’s law), head-loss continuity (II Kirchhoff’s law), for the no control segment, there are:

                                (2)

Network Kirchhoff’s laws

    

                                                                                                  (3)

    

                                                                                             (4)

   

                                                                                                  (5)

From the formulas (3), (5) and (4) we get:

                                     (6)

According to flow continuity (I Kirchhoff’s law), head-loss continuity (II Kirchhoff’s law), in the network, there will be:

    

                                                                                          (7)

Where, A – is n×b connection node incidence matrix, d is the demand flow vector.

Network equation

From (7) and (6) get:

                                                                  (8)

Let,

Then, we get the equation of network as following:

  

                                                                                                        (9)

This is the no control network Nodal EQUATION. It is the shape as the Linear Equation of math, so it can be solved by the way of linear solution.

After we get the result of H we can calculate the other as:

                                                                  (10)

Control network model

Beside the network as above the control network has the control parameter:

 ; The network

control pressure vector of segment, assume there are b number of segments in the network.

 ; The network

control flow source vector of segments, assume there are b number of segments in the network.

Element Kirchhoff’s laws

Following the flow continuity (I Kirchhoff’s law), head-loss continuity (II Kirchhoff’s law), for the control segment, see Fig.2, there are:

  

                    (11)

Control components with control loops are control elements such as pressure reducing valves (PRVs), pressure sustaining valves (PSVs), and pressure control pumps (PCP).

Water networks often have controls in order to achieve the desired behavior of the network. The control loops can be local, e.g. around pressure reducing valves (PRVs) or global, e.g. a pump controlled by a reservoir level. In the first case the local control loop can be masked and the component together with the control loop can be represented as a control segment. The global control pressure is defined by a function P_ck(x,t), Q_ck(x,t) relating a control parameter P_ck , Q_ck(x,t) of a hydraulic component to a state variable in the network, such as x which is a flow or head, and t which is time. These control loops can not be included within a component characteristic and have to be considered as a separate part of the model. The control loop characteristic can be continuous or discontinuous.

Network Kirchhoff’s laws

       

                                                                                             (12)

                                                                                           (13)

                                                                                               (14)

From the formulas (3), (5) and (4) we get:

        (15)

According to flow continuity (I Kirchhoff’s law), head-loss continuity (II Kirchhoff’s law), in the network, there will be:

 

                                                                                             (16)

Where, A – is n×b connection node incidence matrix, d is the demand flow vector.

Network equation

From (16) and (15) get:

  

                                             (17)

Let’s define following:

Then, we get the equation of network as following:

 

                                                                                                         (18)

This is the control network Nodal EQUATION. It is same shape of the Linear Equation of math, so it also can be solved by the way of LU linear solution.

After we get the result of H we can calculate the other as:

                                                        (19)

 

Conclusion

From Fig. 1 and 2, the pipe network of distribution is defined as segment, includes pipe element, independence pressure or flow source and control pressure or flow regulator components, according to KL-I, II, the paper gives out Linear Equation for analysis of network.

Equation (8) and (17) is the basic equation for the network hydraulic analysis. According to that the pipe network simulation or optimization can be approached. Through segment concept, fitting together with components we can analyse the real-time system and dynamic analysis of them.

Reference

1.      Mays, L. W. editor.  1999.  Hydraulic design handbook.  McGraw-Hill Book Co.

2.      Munson, B.R., D. F. Young, and T. H. Okiishi.  1998.  Fundamentals of Fluid Mechanics.  John Wiley and Sons, Inc.  3ed.

3.      Li Ming, The Basic Theorem and Math Model of Piping Network, J. of Water Saving Irrigation, 2001(1); January /2001.

4.      Li Ming, THE METHOD OF AUTOMATIC SUITABLE SIMULATION FOR SPRAY SYSTEM, Proc. Vol.1, (2B‑2) Fourth Int. Micro‑irrigation Congress, Albury, Australia.

5.      Haestad Methods, Thomas M. Walski, Advanced Water Distribution Modeling and Management [M], Haestad Press, 2003.

 

The Basic Theorem and Math Model of Piping Network Li Ming

The Basic Theorem and Math Model of Piping Network

Li Ming

Abstract

This paper gives out a hydraulic model and its basic concept about piping net element, and introduces theorem of its joined elements for in piping network. In the paper, author gives the basic shape of hydraulic model as a power function in mathematics, and points out two important coefficients of the function is coefficient of proportionality and coefficient of exponent. According to the concept of pipe element and its combination model, author presents the application of the theorem and its network analysis method.

Key word: pipe joined, hydraulics, model, piping network.

Introduction

Generally, we use some formulas, known as Darcy-Weisbach formula or Hazen-Williams, to analysis the piping network in hydraulics. Many hydraulics books have a description of this, for Darcy-Weisbach formula as . For the Reynolds number, they use the Moody diagram to determine the "f” value needed for the Darcy-Weisbach equation. The same way is used when using the Hazen-Williams formula. Actually, the two formulas are the same, only that, the shapes in expression are different. Above-mentioned formula is applied as base of hydraulics analysis. This computational method only gets out static parameter of pipe section, in case of operative and dynamic state of water pump, when pump is hand-in-hand working with the pipes, the changing of whole systematic state is unable to accurately present with this approach.

The writer gave forth, at 1988, THE METHOD OF AUTOMATIC SUITABLE SIMULATION FOR SPRAY SYSTEM, which presents dynamic state simulating method of tree like network in analysis of piping net in spray system.

Today, for the reason of saving energy and water, people need more and more precision application of water. Then analytical theory is needed to apply an exact method to analyze the piping network. This paper offers the rationale of analysis in piping net and forwards the methodology of piping analysis.

Darcy-Weisbach and Hazen-Williams formula

Darcy-Weisbach formula

Hazen-Williams formula

Two formulas can be the same; only the expression of shapes is different. If diameter, length and material are assumed, that is, the layout of piping has been done; they may be going to same written standardized shape as follows

(1)

where,

a, b are coefficients to be determine as real, but not equal to zero.

h_f(Q) is, in its defining range, as one of monotonic increasing power function.

Fig.1

Definition 1:

Equation (1) expresses a function of the pipe element; the function’s curve is defined as characteristic curve of flow and pressure of element.

The pipe, equipment, valve, elbow, emitter, or tee etc., acts like the resistance of flow and consuming the energy. They may present as the same shape as equation (1).

Definition 2:

In the piping network, if element acts as the resistance of flow and consuming the energy, then it is defined as piping element of piping network. Simply, as element.

Element gets math model as equation (1), it is the element that gets the shape of power function.

As it will be readily seen, that the equation (1) also can represent the part, where having minor head lost in the piping system, in the other word, equation (1) can used to represent the valve, elbow, tee or emitter math model in the piping network analysis.

Definition 3:

The element, which can give out of water and hold pressure, is defined as element of water source. Simply, as source.

Generic water source is reservoir, higher tower or pump, etc. For math model of pump, it can normally be used as following model:

(2)

where A, B, C is a constant.

For ideal reservoir, its model is given expression as:

(3)

here, C as a constant.

Math model of general water source is expressible for:

(4)

where,

a is exponent factor.

The join model of the elements

The theorem of join elements

Theorem 1:

Any join element has the shape of its pressure head is expressed as the power function of flow, for its math model. This approach, no matter what and how they are connecting, it still gets the same shape of the math model. However, its coefficient of proportionality and coefficient of exponent, somewhat, is varied.

Here, let’s use epagoge method to proof it as follows:

(1)When n=1,

Here, Ho is the outlet section water head of position, since this head is relative, just let Ho=0, then

In other words, the Theorem 1, when n=1, came into existence.

From now on, if no specification,

H represents the head of element at pipe transect,

Q represents the flow within the element.

(2)Assuming, when n=k, upper Theorem 1 would be come into existence.

We get

then

First, for no flows exchanging element

, then

For the inlet of the element

commonly, let us assuming ,

Because the power function is monotonic increase function at first quadrant in defining range, there is a

that let

then

Let

then

that is

The Theorem 1 is verification.

For the element, which has interior exchange of flow, we get follow equation:

here, use q to signify the difference between the inlet flow and the outlet flow.

For all,

but both have interconvertible property, yet may let

in a similar way we can get

That is

Theorem 1 come into existence is proofed.

Theorem 2:

At pipe transect of element, its pressure head expresses as power function of its flow, this equation shape can be reversed for each other.

From

get

only let

that is easily got the result of theorem 2.

a, b is called as the coefficient of flow,

is called as the coefficient of pressure head.

The model of inlet and outlet

First, if no exchange influent of flow occurs in the element, and we know the inlet side power model, we can determine the model’s expression of outlet side.

That is if we know, see Fig. 2

since,

Fig. 2

that is

the right side of equation is known, let’s consider

let two sides take logarithm, then

at first quadrant in defining range of function, to take two special point Q=1, of Q=e then we can get

(5)

(6)

For the element that has interior exchange of flow, we get following, see Fig. 3

Fig. 3

If we get the inlet side

just let

to take logarithm for both sides

If we get tow points at this curve, for example, assumed as these

from the linear equations as following:

(7)

we can get the coefficient of flow a₂, b₂.

Simple connection of the elements

1. 
   

   Series connection

For the connection as showed in the Fig. 4, we abstract it out as series connection.

Fig. 4

The series connection is coupling up element s one after one; there is not exchange of flow within the circuitry. Here, h_f is called the resistance of the element. And

If the connection of the element s is setup with the same material, and its flow state is at turbulence, then that is said, b_i =b, here

then the equifinality of parametric coefficient is

(8)

2. 
   

   Parallel connection

The parallel connection is fitting up element s, using end of element connecting all together, and the other end coupling at other side. Under the water source pressure, two-terminal hydraulic pressure of them are all the same. See Fig. 5

Fig. 5

That is

If the connection is setup with element s of the same material, and flow state is at turbulent range, we get

then

That is

3. 
   

   Series-Parallel connection

With the knowledge of series and parallel connection, we can determine the coefficients of series-parallel connection, step by step, without any difficulty.

The adaptive simulating method of piping network

From above study, we know when we get the element of water originating node and its coefficients of power function, a, b, we can use

for the element model. And for the source model, here to be assumed as water pump which might as,

from two models we can get the operating point of piping system. See Fig. 6

Fig. 6

By the data of operating point, we can get out whole and systematic state data of parameters very easily.

Model identification of piping system

For the whole piping system, if we use above- mentioned procedures, to calculate one by one of the elements, it will need a lot memory and time of machine requirement. Actually, according to the Theorem 1 we can use the model identification procedure to get the coefficients of a and b. We can use the recursive algorithm starts at the end of lateral pipe where the input is the distal elbow upstream operating pressure head, it is end on the inlet of main pipe, where we get the requirement of the pipe's system Q and H, series of input data will be result as series of Q and H, From these data we can do statistic regressive job to get the coefficients at water originating node of element. Modern Control Theory call this procedure as procedure of model identification. See Fig. 7

Fig. 7

To use this procedure we can determine the unknown model or non-determinate piping system, for example, using measure equipment, measure various number of output data, and do statistic of regressive procedure, numerically, then it comes into possession of power function in math model of the system element.

Same procedure can come into possession of systematic math model of mathematics combinatorial model of elements.

Above, we give out the method of solving the hydraulics network analysis of lateral piping, for the loop system, we can use the minor spanning tree to represent the main piping system, after adding the join or linking part of elements we would get all system parameters. The method of analyzing loop piping network is omitted to discussion as limited space here.

Conclusion

From Fig. 6, it is look like some diagram method, but it is totally in difference from the diagram method. With suggested procedure of this paper, we can come into possession of Q, H, which is not only for one working point, but the model of element, which is, imply to be, from one point to a series of points from the model. This meaning that this grow of changing is very much, also in respect, significant.

Theorem 1 point out that the element has the shape of pressure head expresses as power function of its flow in its cross-section, its coefficient of proportionality and coefficient of exponent is a and b. They are identified from a regression procedure. This approach, no matter what and how they are connecting, it still gets the same shape of math model. Through element concept, fitting together element s and power function model identification, we can analyse the real-time system and dynamic analysis of them.

Reference

1. 
   

   Mays, L. W. editor.  1999.  Hydraulic design handbook.  McGraw-Hill Book Co.

   
   
2. 
   

   Munson, B.R., D. F. Young, and T. H. Okiishi.  1998.  Fundamentals of Fluid Mechanics.  John Wiley and Sons, Inc.  3ed.

   
   
3. 
   

   Streeter, V. L., E. B. Wylie, and K. W. Bedford. 1998. WCB/McGraw-Hill. 8ed.

   
   
4. 
   

   Li Ming, THE METHOD OF AUTOMATIC SUITABLE SIMULATION FOR SPRAY SYSTEM, Proc. Vol.1, (2B 2) Fourth Int. Micro irrigation Congress, Albury, Australia.

   
   
5. 
   

   Liou C.P. (1999), Limitations and Proper Use of the Hazen Williams Equation, J. of Hydraulic Engineering, ASCE 124(9)951-170.

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