# Compressible Fluid Notes

### Compressible Fluid Notes

Introduction

The notes below relate primarily to compressible fluids flowing in pipes.  The notes are of a basic level sufficient for a mechanical engineer to be able to estimate operating conditions in a pipeline transferring vapours or gases.

It is much more difficult to determine the operating characteristics of compressible fluids (vapours and gases) as the density is not constant under flowing conditions.

The extremes conditions encountered are adiabetic flow (PVγ = constant ) or isothermal flow (PV = constant).     Adiabetic conditions occur when no heat is transferred across system boundaries.  Isothermal conditions occur when the system changes occur at constant temperatures.  For short insulated pipes adiabetic conditions can be assumed.  For long pipes with reasonable levels of insulation isothermal conditions provide good approximations to real conditions...  Operating conditions occuring at some point between the extremes can often be related to the polytropic process (PVn = constant)

Compressible fluid flows also have a maximum velocity which is limited by the speed of propogation of a pressure wave travelling at the speed of sound for the fluid under consideration.   If the differential pressure along a pipe is such that the fluid velocity approaches sonic speed then any further increase in differential pressure will not be accompanied by an increase in fluid velocity.

For many real life pipe flow conditions it is possible to use the Darcy equations and factors as provided on webpage Pipe Flow Calcs subject to the following restrictions.

1).. If the calculated (estimated ) pressure drop (upstream (P1)  -  downstream (P2 ) is less than 10% of the inlet pressure (P1) then reasonable accuracy is achieved if the specific volume used is based on the known conditions. (Upstream or downstream).

2)..If the calculated (estimated ) pressure drop is greater than 10% but less than about 40% of the inlet pressure P1 then reasonable accuracy is achieved by using a specific volume based on the average of the downstream and upstream pressures.

3)..If the calculated (estimated) pressure drop is greater than 40% of P1 then methods as identified on this page should be used.

Symbols

 a = Acceleration (m/s2 A = Area (m2) a = Speed of sound (m/s) F = Force (N) g = acceleration due to gravity (m/s2 ) h = fluid head (m) K = Bulk modulus (MPa ) L = Pipe length (m ) m = mass (kg) m = mass flow rate (kg/s) M = mach number u /a M = Molecular weight P1 = Inlet fluid pressure (gauge) (N /m2 ) P2 = Outlet fluid pressure (gauge) (N /m2 ) P1 = Inlet fluid pressure (abs) (N /m2 ) P2 = Outlet fluid pressure (abs) (N /m2 ) P - Absolute pressure (N /m2 ) pgauge - gauge pressure (N /m2 ) patm - atmospheric pressure (N /m2 ) p s= surface pressure (N /m2 ) Q = Volume flow rate (m3 /s) q = Heat transfer /unit mass (J/kg) R = Gas Constant (J/(kg.K) Ro = Universal Gas Constant (J/(kg.mol.K) ρ = fluid density (kg /m2 ) s = specific volume (m3 /kg) u = fluid velocity (m/s) v = specific volume (m3/kg) v1 = specific volume at inlet conditions(m3/kg) x = depth of centroid (m) β = Compressibility (1/MPa) θ =slope (radians) ρ = density (kg/m3) ρ r = density (kg/m3) τ = shear stress (N /m2) μ = viscosity (Pa.s) ν kinematic viscosity (m2s-1) υ = Specific volume (m3 / kg) γ= Ratio of Specific Heats

Compressible Fluid Flow equations.

The flow of compressible fluids in long lines approximates isothermal conditions.   The flow rate in a pipe under isothermal conditions is provided by the equation below...

This equation has been developed on the basis of a number of assumptions including: Isothermal flow, steady flow, perfect gas laws apply, constant friction value, straight and horizontal pipe.   The equation below is a simplified version and assumes no acceleration along streamlines.

Limiting Flows.

The equations above do not take into account the fact that , for a particular fluid, there is a maximum speed which cannot be exceeded in the compressible fluid flowing in a pipe.

The maximum velocity of a compressible fluid is limited by the velocity of a pressure wave travelling at the speed of sound in the fluid. Reference Sonic velocity.    Clearly the maximum velocity will be at the downstream end of the pipes as the velocity will progressively rise as the pressure falls resulting in a increase in the specific volume.  Now if the pressure drop is sufficiently high such that sonic velocities is about to be exceeded the resulting pressure decrease and hence driving force will not be transmitted upstream and consequently there will be no increase in flow rate.

The sonic velocity which cannot be exceeded is expressed as

vs = (γRT ) = (γP v )

More Notes to follow -March -2007

Table of Air Flows through sched 40 Piping.

Notes : Factors for other conditions..

1) For inlet pressures  (p o) other than 7 bar gauge... [    Multiply table pressure drop value by 8,013/(p o + 1,013)     ]
2) For inlet temperatures  (t o) other that 15 deg C... [   Multiply table pressure drop values by (273 + t o ) / 288    ]
3) For Pipe sizes (d o ) other than sched 40 ( d 40 )... [   Multiply table pressure drop values by (d 40 /do ) 5    ]
4) Pressure drop is proportional to length. For pipe lengths l o other than 100m.... [   multiply table presuure drop by l o /100   ]
It is important to note that this table should only be used for crude estimates. For serious work then detailed calculations should be used.

Pipe Sizes 1/8" to 2"

 Pressure drop of air in bars per 100m of schedule 40 commercial pipe Air Flow m3/min 15 Deg C 1,013 bar abs Pipe Size (Sched. 40) 1/8" 1/4" 3/8" 1/2" 3/4" 1" 1 1/4" 1 1/2" 2" Inches 3 6 10 12 20 25 32 40 50 mm 6,8 9,2 12,5 15,8 21 26,6 35,1 40,9 52,5 ID(mm) 0,03 0,093 0,021 0,0045 - 0,06 0,337 0,072 0,016 0,0051 0,09 0,719 0,154 0,033 0,011 0,12 1,278 0,267 0,058 0,018 0,15 1,942 0,405 0,087 0,027 0,0067 0,2 3,357 0,698 0,146 0,047 0,011 0,0035 0,3 7,554 1,57 0,319 0,099 0,024 0,0073 0,4 2,71 0,548 0,17 0,041 0,012 0,5 4,1 0,842 0,257 0,062 0,018 0,6 5,9 1,19 0,37 0,088 0,026 0,0066 0,7 8,03 1,62 0,494 0,117 0,035 0,0086 0,0041 0,8 2,12 0,634 0,15 0,044 0,011 0,0053 0,9 2,64 0,803 0,187 0,055 0,014 0,0065 1,0 3,26 0,991 0,231 0,067 0,017 0,0079 1,25 4,99 1,55 0,353 0,102 0,026 0,012 1,5 7,2 2,19 0,499 0,147 0,036 0,017 0,0048 1,75 9,79 2,98 0,679 0,196 0,047 0,022 0,0064 2,0 3,82 0,871 0,257 0,062 0,029 0,0082 2,25 4,84 1,1 0,325 0,076 0,036 0,01 2,5 5,97 1,36 0,393 0,094 0,045 0,012 3,0 8,6 1,92 0,565 0,135 0,063 0,018 3,5 2,61 0,754 0,184 0,086 0,024 4,0 3,41 0,984 0,236 0,11 0,03 4,5 4,32 1,25 0,298 0,136 0,038 5,0 5,34 1,54 0,368 0,164 0,046 6 7,68 2,17 0,518 0,236 0,066 7 2,95 0,689 0,321 0,09 8 3,85 0,9 0,419 0,115 9 4,88 1,14 0,53 0,145 10 6,02 1,41 0,64 0,179 11 7,29 1,71 0,774 0,217 12 8,67 2,02 0,921 0,252 13 2,38 1,08 0,295 14 2,76 1,25 0,343 15 3,13 1,44 0,393 16 3,57 1,64 0,443 17 4,01 1,85 0,5 18 4,49 2,07 0,558 19 5,01 2,31 0,618 20 5,49 2,53 0,685 22 6,65 3,07 0,825 24 7,91 3,61 0,982 26 9,28 4,22 1,15 28 4,86 1,33 30 5,62 1,52 32 6,39 1,73 34 7,22 1,94 36 8,09 2,17 38 2,41 40 2,67 45 3,36 50 4,15 60 5,98 70 8,14 Inches 1/8" 1/4" 3/8" 1/2" 3/4" 1" 1 1/4" 1 1/2" 2" mm 3 6 10 12 20 25 32 40 50 ID(mm) 6,8 9,2 12,5 15,8 21 26,6 35,1 40,9 52,5

Pipe Sizes 2 1/2" to 12"

 Pressure drop of air in bars per 100m of schedule 40 commercial pipe Air Flow m3/min 15 Deg C 1,013 bar abs Pipe Size (Sched. 40) 2 1/2" 3" 3 1/2" 4" 5" 6" 8" 10" 12" Inches 3 80 90 100 125 150 200 250 300 mm 62,7 77,9 90,1 102,3 128,2 154,1 202,7 254,5 303,3 ID (mm) 2,25 0,0042 2,5 0,0051 3,0 0,0073 3,5 0,0097 4,0 0,012 4,5 0,016 0,0051 5,0 0,019 0,0063 6 0,027 0,009 7 0,036 0,012 0,0059 8 0,047 0,015 0,0075 9 0,058 0,019 0,0094 10 0,072 0,023 0,011 11 0,085 0,028 0,014 0,0073 12 0,101 0,033 0,016 0,0085 13 0,119 0,039 0,019 0,0098 14 0,138 0,045 0,022 0,011 15 0,158 0,051 0,025 0,013 16 0,178 0,058 0,028 0,015 17 0,2 0,065 0,031 0,016 18 0,223 0,072 0,035 0,018 19 0,247 0,081 0,039 0,02 20 0,266 0,089 0,043 0,022 0,0072 22 0,328 0,107 0,052 0,027 0,0086 24 0,388 0,126 0,061 0,032 0,01 26 0,455 0,148 0,071 0,037 0,012 28 0,525 0,171 0,082 0,043 0,014 0,0054 30 0,603 0,197 0,094 0,049 0,016 0,0061 32 0,682 0,222 0,106 0,055 0,018 0,0069 34 0,77 0,251 0,119 0,062 0,02 0,0078 36 0,863 0,28 0,134 0,07 0,022 0,0087 38 0,957 0,312 0,148 0,077 0,024 0,0096 40 1,05 0,346 0,164 0,086 0,027 0,011 45 1,33 0,435 0,207 0,107 0,034 0,013 50 1,65 0,534 0,254 0,132 0,042 0,016 60 2,37 0,765 0,363 0,188 0,059 0,023 0,0058 70 3,23 1,03 0,495 0,254 0,08 0,031 0,0077 80 4,22 1,35 0,639 0,332 0,104 0,04 0,01 90 5,34 1,7 0,808 0,418 0,13 0,051 0,013 0,0041 100 6,59 2,1 0,992 0,513 0,16 0,062 0,015 0,005 110 7,97 2,54 1,19 0,621 0,192 0,075 0,019 0,006 120 9,49 3,02 1,42 0,739 0,228 0,089 0,022 0,0071 130 3,55 1,67 0,862 0,267 0,103 0,026 0,0082 140 4,12 1,93 1 0,308 0,12 0,029 0,0095 150 4,73 2,22 1,15 0,353 0,138 0,034 0,011 0,0045 200 8,4 3,94 2,03 0,628 0,243 0,059 0,019 0,0078 250 6,16 3,17 0,975 0,378 0,09 0,029 0,012 300 8,88 4,56 1,4 0,54 0,129 0,041 0,017 350 6,21 1,9 0,735 0,174 0,056 0,023 400 8,11 2,48 0,96 0,227 0,072 0,03 450 3,14 1215 0,286 0,091 0,037 500 3,88 1,5 0,352 0,112 0,046 550 4,69 1,82 0,424 0,134 0,055 600 5,58 2,16 0,504 0,16 0,066 650 6,55 2,54 0,592 0,188 0,076 700 7,6 2,94 0,686 0,218 0,089 750 8,72 3,38 0,788 0,248 0,101 800 3,84 0,896 0,282 0,115 850 4,34 1,01 0,319 0,13 Inches 2 1/2" 3" 3 1/2" 4" 5" 6" 8" 10" 12" mm 3 80 90 100 125 150 200 250 300 ID (mm) 62,7 77,9 90,1 102,3 128,2 154,1 202,7 254,5 303,3