Electricity + Control February 2016

ENERGY + ENVIROFICIENCY: FOCUS ON VALVES + ACTUATORS

Air density is calculated using the standard equation of state for a perfect gas [3]:

at certain sections throughout the day since agitation operations continue without interruption. The different pressure and flow re- quirements for the various consumers presents an opportunity to implement a control strategy that will allow compressed air to be supplied more efficiently. Pressure losses resulting from line friction Air flowing over a long distance will be subject to relatively large pres- sure losses as a result of line friction. To calculate this pressure loss a few fundamental principles need to be taken into account. Owing to the relatively large air flow speed, pipe wall roughness and long pipe lengths, only turbulent flow conditions are assumed to be present. The airflow in a pipe experiences resistance as a result of viscous sheer forces. The wall roughness plays an important role in determin- ing the friction factor, ƒ . Experimental methods are mostly used to determine ƒ for a specific situation. The friction factor is a function of the Reynolds number ( N R ) and relative pipe roughness factor ( ) [2]. Figure 4 illustrates what is meant by the relative roughness of a pipe. ε D

p RT

ρ =

Where the variables are: o Density

- ρ - T - p - R

[kg/m 3 ]

o Temperature

[K]

[N/m 2 ]

o Absolute pressure

o Gas constant

[N · m/kg · K]

For air, the gas constant is 287 N · m/kg · K at 298 · K, and 101 kPa [3]. The pressure loss, ∆ p = (p 1 - p 2 ) over a length of pipe can be calculated using the following equation [4]:

l pv 2

∆ p = 4 ƒ

d 2

Where the variables are defined as: o Loss of pressure

- ∆ p [Pa]

- ρ - d - l - v - ƒ

[kg/m 3 ]

o Density

o Internal pipe diameter

[m] [m]

o Pipe length

[m/s]

o Average velocity o Friction factor

[dimensionless]

Figure 4: Relative roughness of a pipe [2].

From this formula it is clear that the ∆ p over a length of pipe will increase if: • The pipe length increases • The rate of air flowing through the pipe increases • The air density increases Air flow losses through air leaks The compressed air lost through air leaks in a compressed air system is a function of the pressure of the air in the pipe as well as the size of the leak. Most of the formulas used to calculate the compressed air lost through air leaks are derived empirically. The following formula can be used to calculate the volumetric flow rate of free air through a hole with a given size [5]:

When calculating the roughness of pipes that have been in use for some time, the absolute roughness of new pipes can only be used as estimates since the effects of rust and corrosion in the pipes will result in larger relative roughness values of the pipes. Using assumed relative roughness and Reynolds number val- ues, the friction factor ƒ can be read off the Moody diagram shown in Figure 5 .

0.1 0.09 0.08

ε D

0.05 0.04 0.03

0.07

0.06

0.02 0.015 0.01 0.008 0.006 0.004 0.002

0.05

0.04

NL × (T i

+ 273) × P 1

× C

× C

× C

× π D 2 /4

/P

0.03

i

1

2

d

V

=

f

× T

+ 273

C

3

1

0.02 Coefficientof friction f

0.001 0.0008 0.0006 0.0004

Where the variables are defined as: o Volumetric flow rate - V f

0.0002

[m 3 /h]

0.0001 0.00005

0.01

o Number of air leaks

- NL

[dimensionless]

0.009

[ o C] [ o C]

o Atmospheric air temperature - T i

0.00001

0.008

3 45 10 4

2 345 10 5 2 345 10 6 2 345 10 7 2 345 10 8

ReynoldsnumberR

o Line air temperature

- T - P

l

Figure 5: Moody diagram.

[kPa]

o Line pressure

l

February ‘16 Electricity+Control

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