The interaction of components on the vacuum stability milking machines

J. agric. Engng Res. (1970) 15 (4) 33 I-346

The Interaction

of Components

Stability

Milking

on the Vacuum

Machines

C. J. MULDOWNEY*; J. R. O’~ALLACHAN*

The vacuum supply of a milking machine may be represented by a closed loop pneumatic control system. Transfer functions are derived for regulators whose static and dynamic performances were studied experimentally. An analogue computer model of the pneumatic control system was developed and verified, by comparing its performance with that of real systems. The model was used to examine the performance of seven installations and to suggest improvements in the stability of them. 1.

Introduction

A milking machine withdraws milk from a cow’s teat by suction. For physiological reasons it is not permissible to apply suction continuously and the suction is interrupted by the pulsator, which allows the pressure in the teat-cup annulus to approach atmospheric pressure intermittently. The complete cycle is repeated approximately 60 timesjmin. An important requirement in a milking machine is that the suction pressure should be maintained at a constant predetermined level, which should not be influenced by disturbances due to pulsator action, pump ripple, removal of milking clusters or evacuation of recorder vessels. It is known that the stability of the vacuum system depends on such factors as pump capacity, volume of piping and vessels, diameter and condition of pipelines, type of vacuum regulator and its steady-state and transient characteristics. The interrelationships between these factors have been studied in order to obtain a quantitative understanding of their importance in the design of milking machines. 2.

The vacuum system

A vaccum pump evacuates air from the system at a volumetric rate which is dependent on the pump characteristics and the pressure difference across the inlet and outlet parts of the pump. Atmospheric air flows into the system in two ways: (a) leakage

and load flow;

(b) controlled

flow.

1

KEY TO SYMBOLS effective area discharge coefficient

volumetric flow rate

gas reaction force

volume weight

damping factor aerodynamic spring constant spring constant initial compression of spring mass

linear displacement length of pipe impedance of pneumatic bulk modulus of fluid density of fluid

pressure

frequency

I *Dept.

of Agricultural

Engineering.

University

of Newcastle

upon Tyne

331

system

332

VACUUM

STABlLlTY

MILKINCi MACHINES

In order to establish a vacuum in the first instance, the pump flow rate must be greater than the load and leakage flow rate. The controlled flow rate prevents the occurrence of excessive vacuum and is obtained by the use of a vacuum regulator. The regulator senses the pressure inside the system and compares it with a predetermined pressure. If the pressure is too low, the error signal is transmitted in the form of a force to the mechanical system of the regulator which opens a valve and admits air to the system. If the system pressure is too high the valve closes and the correct pressure is restored, provided that the load flow and leakage flow combined are less than the pump flow. The system may be represented by the block diagram shown in Fig. I. The control element is the vacuum regulator, the controlled system is the combination of piping and volumes, which Leakage flow

Steady-state operating pressure

I

I Pressure in vacuum line

Fig. 1. Block diagram

of milking machine

constitute the milking machine, and the controlled variable is the pressure inside the system. The sensing characteristics of the regulator give the feedback signal and the difference between it and the signal representing the desired system pressure actuates the control element. The load, leakage and pump flows are also fed into the controlled system. In deriving the transfer functions of the control systems three main elements were analysed: the regulator, the pneumatic system and the feedback characteristics.

The transfer rate of change derivatives

function

2.1 The regulator transfer function of the regulator may be found by analysing

of flow with pressure

az ax Zj’Z

differential

the relationship

This can be broken

between

the

down into 3 partial

aQ

and ax where x is the valve displacement

and z is the force activating the mechanical system. The relationship between sensed forces and pressure differential in all regulators is approximately, if not exactly, a constant. The regulator mechanical system can be represented (Appendix 1) by a second-order linear differential equation : d?x dx Mdt2 +f z+(km+K) In Laplace

x = PA,

form the Eqn becomes 1 2(s)

0

=

M+U+(k,+K)

.

m

which is the transfer function of the mechanical system. The presence of K in this transfer function is usually the governing feature of the mechanical system. Actual values of K for a dead weight, a spring-loaded and a compensated regulator have been measured and are shown in Fig. 2, where

C.

1.

MULDOWNEY:

J.

R.

0 02

0

333

O’CALLAGHAN

004

O-06

0 08 Valve

Fig. 2. Gas reaction

forces

(existing

regulators)

. _-

0 IO

0 12

displacement

0 14

0 16

0 I8

(ml

weighted;

-.-

-~ -

spring loaded;

- - - - - compensated

the aerodynamic force is plotted against displacement for 3 experimental valves. It can be seen that where the compensated regulator has a force which increases with valve displacement, it has an unstabilizing efrect on the valve performance. Also it can be seen that the aerodynamic spring constant K is approximately - 1l-5 lbf/in for the weighted regulator, -2 lbf/in for the springloaded regulator and 5 lbf/in for the compensated regulator. Since the mechanical sensitivity and dynamic performance of existing regulators are interrelated, the gas reaction force determines the regulator performance. The flow gain characteristics under operating conditions for these regulators are compared in Fig. 3 and their dynamic response to a step function input is shown in Fig. 5.

I503

Vacuum

Fig. 3. Flow-gain

(in

characteristics

Hg)

(existing

regulators)

334

VACUUM

STABILITY

MILKING

MACHINES

The performance of vacuum regulators may be improved by using flow force compensation to reduce the effects of the gas reaction force or by separating the various control features of the regulator from one another. As an aid to studying the effects of flow force compensation and other design parameters, a new regulator was designed in which these factors could be examined experimentally and their effects on performance measured. A line diagram of the new regulator is shown in Fig. 4, Top. The flow gain characteristics are shown in Fig, 4, Bottom, and the dynamic

Vacuum (in&g) Fig. 4, (Top) Schematic

diagram of a regulator to separate sensing and control characteristics (Bottom) Flow-gain characteristic of designed regulator

response in Fig. 5. Both the flow gain characteristics ment over that of existing regulators. The third derivative ment.

‘g

It can be calculated

and the transient

is the one which defines the relationship from the equation

:

response between

show an improveflow and displace-

C. .I. MULDOWNEY;

335

J. R. O’CALLAGHAN

DesIgned

Fig. 5. Comparison of transient response of regulators

2.2. The parameters (i) (ii) (iii) (iv) (v)

the the the the the

involved

The pneumatic system transfer function

in the pneumatic

control

system of the milking

machine

are:

pressure difference causing flow; fluid flow in the system; resistance to flow within the system; inductive impedance of the system; capacitive impedance of the system.

Pressure difference

and air flow are related by the following P(t) = Q(t)

In the Laplace

Eqn (Appendix

R,+joPdA ,jcoVt

2)

. ..(3) I

form . .(4)

Therefore

the transfer

function

of the pneumatic

system is:

. ..(5)

2.3.

The feedback loop

In existing regulators the feedback mechanism is inseparable from the control orifice. If there is a large orifice area the sensing force will be large; however, because of the necessity to provide a reasonable preset force the practice is to keep the orifice area small. The relationship between sensed force and change in the operating pressure for existing regulators is :

336

VACUUM

where z = sensed force AP = deviation from operating A = orifice diameter.

STABILITY

MILKING

MAC‘HINkS

pressure

One effect of the orifice in the designed regulator (Fig. 4) is to introduce a time delay in the transmission of a disturbance from the vacuum line to the sensing chamber. The delay may be represented by a single exponential term.

3. The milking machine, which is represented jected to 6 different disturbances:

Disturbances by the control

system described

above may be sub-

(i) high frequency, low amplitude disturbance, such as the pump ripple; (ii) low frequency, high amplitude disturbance, known as pulsator ripple; (iii) a high amplitude step-function input, when a cluster becomes dislodged when changing clusters from one cow to another; (iv) a high amplitude impulse signal when a recorder vessel is re-evacuated; (v) zero drift of vacuum due to control inadequacies; (vi) disturbances due to the hydro-dynamic forces during milk flow.

accidentally

or

The first type of disturbance is relatively unimportant as most installations have sufficient volume to damp it out. In addition, because its frequency is usually in the region of 25 Hz, the system acts as a filter and attenuates the fluctuation in pressure. Pulsator ripple is much more significant than pump ripple and it can have a considerable effect on the vacuum stability. The frequency of pulsator ripple is approximately I Hz and its amplitude can be as high as O-5 ft3/sec at 15 in Hg vacuum. A disturbance of this nature can cause a l-2 in Hg fluctuation in the vacuum. By far the most serious type of disturbance is that due to a sudden flow of air into the system. This is represented by a step input of up to 0.5 ft3/sec at 15 in Hg vacuum; it is usually not attenuated by the system characteristics and the only controlling factors are the regulator and the vacuum reserve. Such disturbance may be found in pipelines when the recorder jar is re-evacuated. Inadequate regulator characteristics cause overall zero drift from the desired vacuum level. If the regulator sensitivity is too low, any change in load, and therefore in regulator control flow, will cause a corresponding vacuum change. This change becomes greater as the regulator sensitivity decreases. A regulator which has poor dynamic characteristics will also cause a change in vacuum under operating conditions. No consideration has been given to the effects of the disturbances caused by the hydrodynamic forces which occur during milking and all measurements have been made on “dry” machines. 4.

Models of the control system

The control system may be studied by experimenting with the real system which represents all non-linear as well as linear parameters. This approach has disadvantages due to lack of versatility, slow speed of operation and the fact that for a parameter change, a considerable amount of fabrication may be necessary. Techniques such as Laplace transformations may be used to simplify a mathematical study and Nyquist and Routh criteria may be used for a graphical study. Although some non-linearities can be represented mathematically or graphically, it is difficult to use such methods under non-linear conditions. The general purpose electronic analogue computer is very suited to the study of control systems and especially to the study of dynamic or transient behaviour. Because the system transfer functions have been found already, it is a simple matter to change these into the computer format. Fig. 7 shows a computer circuit diagram of the real system block diagram of Fig. 6.

C.

J.

MULDOWNEY;

J.

R.

337

O’CALLAGHAN

Pulsator input *

;

I

13.

-.+(L+-(n-I)l/,)s

-I+T,s

step Input

W + k,l,,

_-A! -

_

= + Qm-----

I 00 4.1 +, + P” ,’- qY QP

Ad

I I+

TIs

Fig. 6. Block diagram of control s.vstem using dhtributed

parameters

utput

Fig. 7. Analogue

romputer

program

using distributed

parameters

338

VACUUM

STABlLITY

MILKING

MACHINES

It can be seen from Fig. 6 that there are a number of transfer functions to be transferred into computer terms :

Ad

-___-

l+T,S

1 ‘L+T,S

B” ’ V,S’

B” I’,-(n-l)VRS’

1

_____

K

I-+;++(;

’

“AD

m

and the non-linear characteristics of the regulator. lt is also necessary to represent pulsator, recorder vessel and cluster disturbances as voltages that can be fed into the model. When these quantities have been simulated it is only necessary to perform any magnitude and/or time scale changes before the model is complete. 5. Analogue model 5.1. Comparison of experimental and model results In order to check the computer model it was compared with real systems by taking experimental recordings of the vacuum trace in a physical installation and also in the model. Table I shows the experimental data concerning 7 actual milking machine installations. These 7 installations comprise 4 pipeline systems with recorder vessels, 2 “lowline” systems and one “round the shed” pipeline installation. The pump capacities of these installations varied from 26 to 70 ft3/min at 15 in Hg vacuum and the reserve per milking unit varied from 1.25 to 6 ft3/min of air at 15 in Hg vacuum per milking unit. Also the volume capacity of the plants varied widely over the range 3-10 ft3. The following 5 readings were made for each installation : (i) disturbance time (time to reach steady state conditions) when a cluster was dislodged. The vacuum recording was taken at the teat cup; (ii) recovery time (time to reach steady state conditions) as the cluster was returned: (iii) absolute value of the disturbance as a cluster became dislodged ; (iv) amplitude of the pulsator wave at the teat cup; (v) amplitude of the disturbance as the recorder vessel was re-evacuated. The values of the above 5 parameters, for the 7 installations studied, are shown in Table II together with the corresponding values obtained from the computer model. The experimental recordings from the milking machines and the computer model may be compared in Fig. 8. It can be seen that the correlation between the experimental and model results is good. The variation in disturbances for each installation is extremely wide indicating that the installations do not reach the standards expected for the vacuum supply. Although the “lowline” systems have a high frequency response, their transient response is poor and the amplitude of the pulsator wave at the teat cup is so large that it is almost equivalent to the disturbance caused by a cluster becoming dislodged. 5.2. System optimization Optimization studies were carried out in order to reduce the vacuum fluctuations of the seven physical systems to within 1 in Hg vacuum. Precise optimization was not possible in so far as only one parameter was varied during a series of runs to determine its optimum value. The pneumatic system of the milking machine is a closed loop control system in which there are many variables which are dependent on one another, as well as numerous independent variables which affect the system. Although a certain value of parameter x may give optimum vacuum line stability for fixed values of the remainder of the parameters (A, B, C, etc.), at a different value of A, B or C there will exist a new value of parameter x for optimum line stability. Despite the dependence of the parameters on one another the optimization procedure was made less difficult by the fact that certain parameters are found to have more effect than others on the vacuum stability. Fig. 8 shows the experimental, model and optimized recordings for the

Total volume of pipelines (ft3) Total vol. of pneumatic system (ft”) Reserve/Unit System vol./res. pump cap. (ft3/ft3 set)

-

25.6

I .22 9.38 2.75

-

-

064 1 I.12 1 201 96 ft ,- 4 in 4 in 48 ft

Vol of moisture traps (ft”) No. of interceptor jars Vol. of interceptor jars (ft3) Pipeline diam. vat/milk (in) Pipeline length (ft) Other pipelines

20.0

0.83 4.67 4.67

140 45 ft $ in

1

0.64 1 0.8

gal

I9

0.96 1 0.96 1 ICOO 29ft z l&in 42 ft : in 42 ft # in 5.69 7.61 3.43

I /I gal

1.i5

0.785 None None

0307 8 6.4 1

Valve orifice area (in”) No. of recorder vessels Vol. of recorder vessels (fty) No. of moisture traps 0442 3 2.4 1

24 18 Dead weight Conical

15 Round the shed pipeline

14 8 Dead weight Conical

Type

22 13 Dead weight Spherical

15 Tandem

3

60 stall cocks I 60

14

2 _______~

6 3 26

(in Hg)

I

-

No. stalls No. units Pump capacity fF/min Reserve capacity (operating condition fF/min) Pulsator load ft3/min Regulator type Valve type

vacuum

_

-

12 8 70

15 Tandem

4 ~_____

60

0.64 1 0.8 1 377 48 ft I 3 in 64 ft ’ i in 96 ft *’ 2. /4in 9.98 1.25

0442 8 6.4 1

10 15 Dead weight Spherical

_

of 7 on-farm installations

Herringbone Pipeline Recorder Jar 16 8 60

Operating

Installation

-

Specification

4ft 50ft

6.95

1.45 3.37 2.42

l&in

x l&in

47

0.8 1 I.12

29 21 Dead weight Conical flat ended 0.307 None None 1

13 Herringbone Lowline Pipeline 12 12 65

5 ~_____ --

6

129

6.45 6

I

0.64 1 0.8 1 137 20ft . l*in 65 ft ,: &in 50 ft d $ in

0442 5 4 1

30 5 Dead weight Conical

5 5 41

15 Tandem

___-

-

7

13.4

44 28ft ‘: 14 in 17 ft * 1 in 30 ft / * in 1.34 2.9 1.3

13 35 Dead weight Conical flat ended 0.307 None None 1 J 5 gal I f 2 gal 1.12 1 044

12 Herringbone Lowline Pipeline 10 10 65

_____-

T

340

VACUUM

f=O

Installation 1

Cluster

T 2 in Hg L

I

t=o

STABILITY

l----l k-20

MILKING

MACHINES

1=0

set--I

Experimental “p”“+tiyvacuum line +Ty------

Recorder vessel

Fig. 8. Comparison of exaerimental,

model and optimized vacuum readings

C.

J.

MULDOWNEY;

3.

R.

341

O’CALLAGHAN

TABLE II

Comparison of readings from full scale installations and analogue model

-

-

Dimrbance time (sets)

lnstallution

Recovery

__ Full .si,ale

I

Full scale

23 20 28 I.7 5

‘7

Model

22

.Ampl. of disturbance as recorder vessel i.s re-evacuated CHK)

Full stole

Full scale

Full scale

20 9.7 25 14 5 6

11 28 16 9 9 12

IO

I3 4

Ampl. of pulsator wave of teat cup (H.)

-Model

22 21 2.5 I.5

2 3 4 5 6 7

time

kc)

Absolute value of disturbance (HP)

7

I5

Model

2 5 3.4 I.5 1.8

2 5 3.1 1.3

1.6 3.1

I.6 3.9

0.27 0.25 0.26 0.8 0.25 0.14

I4

-

Model

0.73

0.23 0.15 0.20 1.4 0.1 0.4 2.0

I.1

Model 0.7

I.5 1.2 0.74 0.7 Not applicable 0.7 j 0.7 Not applicable Not applicable

-

L

TABLE III

Computer optimization

Installation

1

2

3

4

30

20

5

6

7

24

20

30

_____.__-

Pump capacity, increase (ft”/min)

30

26

_l__Regulator, increase in sensing area System volume, increase in pulsator volume (ft3)

Double 3.2

Double 3.2

0 3.2

Double

3.2

Double* -~ ____3.2

Double -__.__-_ 3.2

_

Double* _~..~_ 3.2

*Use delay orifice (I” = 0.25 set) Reduce damping (/‘= 0.55 Ibf/ft.sec) Remove dead-band

5 test readings in 7 different physical systems. Table I11 shows the modifications proposed in order to confine the disturbances to within 1 in Hg vacuum at all times and to attenuate the pulsator wave to within 0.2 in Hg. The increase in volume of 3.2 ft3 is equivalent to 4 five-gallon vessels. The stability of the vacuum in a milking machine is dependent on a number of factors, the most important of which are: the the the and

reserve capacity, total system volume, volume of the recorder vessel and connecting pipes the steady state and transient characteristics of the vacuum

regulator.

The effect of the reserve capacity and the total system volume on the vacuum stability is shown in Fig. 9. For low values of reserve capacity the recovery time from the effect of a step input disturbance is much greater than the disturbance time. As the reserve capacity rises these 2 values approach one another.

342

VACUUM

0

4

8

12

16

,

I ___

f

MILKING

24

28

32

,

,

/

MACHINLS

(sex)

Tme

300

20

STABILITY

I

I

IO -c

12.5

Time

(set)

Fig. 9. Effect of step input of 0.4 fP/sec on the vacuum stability of the system Disturbance time (time to lose 2 in Hg) _- Recovery time

Where a recorder vessel is used, it plays an important role in the attentuation of pulsator and other medium frequency disturbances. The volume of the recorder vessel and connecting piping has a pneumatic time constant which, if large, has considerable attenuating properties as shown in Fig. 10. There is little value in a regulator with good dynamic characteristics in a system where the disturbance and recovery times are long. Where the installation has a high frequency response, selection of regulator damping and/or delay orifice, and of the sensing area of the regulator, can have considerable effects on the stability of the system. The effects of damping and delay orifice on the recovery

time of a vacuum

system are shown in Fig. 11, Top. The smaller

more critical are these factors on the recovery time. The relationship the regulator and the recovery time is shown in Fig. 11, Bottom.

between the sensing area of

C.

J.

MULDOWNEY;

J.

R.

343

O’CALLAGHAN

Fix. 10. Effect of the time ronstant

6.

of the recorder

vessel on pulsator

wave attenuation

Conclusions

If the volume capacity of a milking machine pneumatic system is small there is only a small reserve of vacuum that can be used to smooth out fluctuations, and if the volume is large the fluctuations will be damped out. If the reserve pump capacity is large it will absorb changes in load or leakage flows. The effectiveness of the reserve pump capacity is dependent on a satisfactory vacuum regulator. For a given vacuum regulator, the 2 main parameters affecting the vacuum stability are the volume capacity (V,) and the reserve pump capacity (Q R). The bigger the vacuum pump reserve and the smaller the volume capacity the higher will be the frequency response of the system. Therefore

the quotient 5

response: the smaller the value of&

QIZ

gives a measure of the system frequency

the higher the frequency response of the system. An attempt

QR

has been made to rationalize this quotient and use it as an indication of the system frequency response. Many milking machine installations are sluggish in their operation. This is as a result of the volume capacity being excessively high and/or the reserve capacity being too low. If an installation has a high frequency response and a low value of 5

QR

the dynamic characteristics of the regulator

are critical in relation to the vacuum stability. Very often the advantages of having a low5 ratio QR V, are lost by an ineffective regulator. Low values ofratio give rise to regulator chatter and may QR

exaggerate regulator faults such as backlash and low frequency response. Where damping is introduced to curb ‘chattering” there is an optimum value of damping. Some existing regulators have damping constants as high as 20 lbf/ft/sec which is too high and renders their performance sluggish. In some cases the use of a delay orifice has been found to be advantageous to the dynamic response of the regulators. Increasing regulator sensitivity without increasing inertia is desirable. Attenuation of pulsator disturbances is achieved by increasing the effective volume seen by the pulsator. By reducing the diameter of the pipe connecting the recorder vessel to the vacuum line the time constant of the disturbance obtained as the vessel is re-evacuated is increased, thereby enabling it to approach the overall time constant of the system.

VACUUM

344 Tune constant 0,02 I

6o.ol

0 03 I

0 04 I

of delay orlflce

005 I

006 I

STABILITY

MILKING

MACHINES

(secl

0.07 I

0 08 I

0 09

,

1

“t-

3

2

1

0

Mechanical

0

2

4

6

4 dompinq

8

IO

7

6

5 ratio

8

(c = f/f,,

12

14

16

Time (set) Il.

(Top) Eflect ofdelay

orificeand mechanical damping on the recovery (Borrom) Effect of regulator sensing area on recovery time

time

Possible ways of reducing the disturbances found in milking machine installations are 6) correct selection of the reserve capacity which prevents the loss of vacuum exceeding 1 in Hg when one cluster becomes dislodged; (ii) selection of a value of 5 ratio as small as possible; QR (iii) careful selection of regulator damping or delay orifice when !!?- is less than 10; QR (iv) increasing the sensing area of a regulator improves its sensitivity; (v) increasing the effective volume as seen by the pulsator attenuates the disturbances transmitted inside the teat-cup liner; (vi) reducing the bore of the tube leading from the recorder vessel to the vacuum line attenuates both pulsator and recorder vessel disturbances. Acknowledgement

We wish to thank the Irish Institute of Industrial Research and Standards for their support by a Postgraduate Research Fellowship.

C.

J.

MULDOWNEY;

I.

R.

345

O’CALLAGHAN

Appendix 1 Equation of motion of mechanical regulator At zero displacement there is a leakage flow through the regulator valve. The reaction at the valve seat can be neglected and the valve is considered to be in equilibrium under the system of forces shown in Fig. 12; dp, is the pressure differential which balances the valve at zero displacement . ..(i) dp,A, == WSk,l.F”

W+k,l, Fig. 12. Static free body diagram

Assume

change p from the value P,:.

that the valve moves under a pressure

From the free-body

of regulator

of Fig. 13, the Eqn of motion

diagram

A42 = Ap,A,+pA,-tFSubstituting

is W--f2

-k,(l,-tx)

. ..(ii)

from (i) in (ii) M%+fi+k,x

= pA, tF--F,,

. ..(iii)

For small values of X, A, = A,. In regulators

of both the dead-weight

type (Fig. 2)

and spring-loaded F=

-Kx-IF,

:.M%+fi+(k,+K)x

= PA,,

. ..(iv)

A~,A,+PA, / yy+

, A,

_

HAx P Oh I

9

F

W

rr MLY+x)

Fig. 13. Dynamic free body diagram

of regulator

Appendix 2 Pressure-flow relationship for pneumatic system Consider a pipeline of length y and cross-section A entering a volume V, in which the air pressure is pD and the density is p”. Let the column of air in the pipeline be given a harmonic motion x = x,eJWt. Instantaneous

pressure

rise, p, in the pipe is given by p.4 = rn$ = -mmco2.x,e~*’

. ..(i)

346

VACUUM

STABILITY

MlLKlNG

MACHINES

m = pyA

But :.

Inductive impedance of the fluid zi =

p =

. ..(ii)

-pyu2x,ejmt

Excess pressure -pyw%e~oiz, = jU~ A Volume velocity = Ajwx,ejw’ ’

.(iii)

The capacitive impedance z, may be calculated by considering the conservation of mass in the system : p,,V, = constant Vdp”+p,dV,

= 0

dp, z--p,

Ff

.(iv)

For an adiabatic process, the accoustic velocity c= or the instantaneous pressure rise

p = c2dp,

- c2p, dVt Vt But

. ..(v)

dV, = :. p =

2,

. .(vi)

=

C2P” jo V,

But

c2

&

PD YP? :. z, = jw V,

P = -where .M V, The resistive impedance

z, =

fi = bulk modulus of the fluid

pressure loss in pipe volume velocity

R

0

. ..(vii)

. . .(viii)

When the elements are combined in series, the total impedance z = z,+zi+zC .(ix)

,.(x1