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4.7 Pipe resistance, compressibility and inertia
94
Q [[PV
Q [PV (11.2litres/min)
95
The flow regime around Re = 2000 is actually quite complex and the transition zone from laminar to turbulent flow is not quite matched at Re = 2000. However this fluid mechanics detail is not too important for servodrive applications where the flow is usually well defined as either laminar for low-power applications or turbulent for high-power applications. For turbulent flow conditions a linear resistance is not applicable but the use of a non-linear resistance is easy to implement if computer simulation is used. For large flow, large diameter and long pipe applications, such as forging presses, the use of a laminar flow resistance rather than a turbulent flow resistance can give serious errors in system damping
Example 4.4
Consider a forging press as shown below and similar in concept to two industrial systems analysed by the author for the manufacturer.
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I joined MITAS because
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�e Graduate Programme for Engineers and Geoscientists
Month 16 I was a construction supervisor in the North Sea
advising and helping foremen solve problems I was a
he s
Real work International opportunities
�ree work placements al
Internationa or
�ree wo
I wanted real responsibili�
I joined MITAS because
Maersk.com/Mitas
�e Graduate Programme for Engineers and Geoscientists
Month 16 I was a construction supervisor in the North Sea
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he s
Real work International opportunities
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96
One half of the press is shown where cylinders 3 and 4 (plus two others) are used for lowering the press into position and cylinders 1 and 2 (plus two others) are used for pressing. The servovalves shown are used to bleed flow for rotation control when the press is operating off-centre. Determine the line pressure drop when forging and the total power dissipated by friction when forging. Data for the lines are as follows:
Pressing cylinder net cross-sectional-area = P Line length Ɛ =150m Line diameter d = 80mm Full flow rate when pressing 4000litres/min
Fluid viscosity ȝ 1VPFluid density ȡ NJP
Flow rate when pressing [ P V
4
Pressing speed PPV
S [
8
Pipe cross-sectional-area [ P
[
D ʌ
Pipe mean velocity PV
[ 8PHDQ [
Reynolds Number
ȝ G 8 H ȡ
5 PHDQ
Hence the flow is highly turbulent and therefore the friction factor is given by;
5H
I
97
The pressure drop is then given by ¸
¹
¨ ·
©
¸§
¹
¨ ·
©
§ ȡ8 G I
ǻS PHDQ "
[ 1P
ǻS ¸
¹
¨ ·
©
¸§
¹
¨ ·
©
§ EDU
Power dissipated down one line only when pressing:
ǻS4 [[ [:N:
The total power loss is extremely large and explains why large-force forging presses require a substantial electrical power supply to drive the array of pumps to overcome friction losses and provide the high flow rate at the high pressure required during operation.
Fluid compressibility for a pipe has already been dealt with since it is no different to that defined earlier for a general fluid volume. However the problem is how the compressibility is handled if it is significant for a long line and this will be considered after fluid inertia has been discussed. Fluid inductance for a pipe is defined as follows:
Pipe fluid inductance
D ȡ D
/S P "
Pipe total capacitance
ȕ
&S 9
Therefore for laminar flow the pressure drop down a pipe has resistive and inductive components. The compressibility flow and pressure drop are given symbolically by:
GW
& G3 ǻ4 S
GW / G4 4 5
ǻS S S
Now having defined fluid resistance, capacitance and inductance the important question is how are these fluid properties distributed along a line? The true answer is that a distributed parameter analysis applies and there are well-established ways of modelling a line using this approach. For the present study, given the context of this book, just one of several approximations is used, sometimes called a lumped approximation, and actually can be a good representation of a line as a first-estimate. A common lumped approximation is a π network as shown in figure 4.5.
(4.26) (4.27)
(4.28) (4.29)
98
4L4L4R
5/
3L&S&S3R
Figure 4.5 A π network approximation for a line
In this approximating network half the line capacitance is lumped at each end with the total resistance and total inductance being lumped in series in the middle. Two networks may be connected with each R, L, C elements halved for each network. This aspect of line modelling will be pursued later when a servodrive undamped natural frequency is considered.