Coker A.K. Fortran Programs for Chemical Process Design, Analysis, and Simulation
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Fluid Flow 157
Table 3-2
Excess Head Loss 'K' Correlation for Fitting
Caused by a Change in Pipe Size
I
.--t ~ z --~
z ~---= _._z ~
z
' ~ z ~ z z z
r~ r~ -~ r~ r~ ro r-~
I~-... ~r It.-- ~ -,4 it.., -----!
o o ~ ~ ,._r-, ~
z
o o ~z~ ~ c:, c::,':=::'
o o o
0
I
::::w- | ~ ~ ~ f, ~ ..| - --"
.... ~. ,---, o~ ~-,= ~w, ~
,~
.c~=,., i i --r
NR e =
pVD = 50.6 Qp - 5.31 W (3-12)
g dg dg
where Q- volumetric flowrate, gals/min
W- Mass flow rate, lb/h
- fluid viscosity, cP
For laminar flow with
NRe
~ 2000
64
fo = (3-13)
NRe
The Darcy friction factor is four times the Fanning friction factor, fF,
i.e., fD = 4fF" For fully developed turbulent flow regime in smooth and
rough pipes, the Colebrook [5] equation or the Chen [6] equation can
be used.
The Colebrook equation is expressed as:
1 _ { ~ 251 }
-0.8686 In 3.7D
+
NRe~" (3-14)
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Fortran Programs for Chemical Process Design
Equation 3-14 is implicit in fD, as it cannot be rearranged to derive fD
directly and thus requires an iterative solution. Here, the Chen equation
is used for calculating fD, (i.e, fD = 4fc). The Chen equation is explicit
and easier to use than Equation 3-14). This can be expressed as:
1 __ 4log{ e 5.02 logA} (3-15)
~/fc 3.7D
NRe
where
A __
( / ~
e/D 6.7
+
3.7
and
- pipe roughness, ft
A detailed review of other explicit equations is given by Gregory and
Fogarasi [7]. Different piping materials are often used in the chemical
process industries, and at a high Reynolds number, the friction factor is
affected by the roughness of the surface. This is measured as the ratio ~/D
of projections on the surface to the diameter of the pipe. Glass and plas-
tic pipe essentially have ~ = 0. Values of ~ are shown in Table 3-3.
The Overall Pressure Drop
Designers involved in sizing process piping often apply trial and error
procedure. The designer first selects a pipe size and then calculates the
Table 3-3
Values of Absolute Pipe Roughness
Pipe Material
Riveted steel
Concrete
Wood stave
Cast iron
Galvanized iron
Asphalted cast iron
Commercial steel or wrought iron
Drawn tubing
e, ft
0.003-0.03
0.001-0.01
0.0006-0.003
0.00085
0.0005
0.0005
0.00015
0.000005
Fluid Flow 159
Reynolds number, friction factor, and coefficient of resistance. The pres-
sure drop per 100 feet of pipe is then computed. For a given volumetric
rate and physical properties of a single-phase fluid, AP~00 for laminar
and turbulent flows is:
laminar flow
APlo o -
0. 0273
d4 ,
psi / 100ft (3-16)
turbulent flow
pQ2
APlo o - 0.0216f D d5 , psi / 100ft (3-17)
Alternatively, for a given mass flow rate and physical properties of a
single-phase fluid, AP~00 for laminar and turbulent flows respectively is:
laminar flow
APlo o - 0. 0034 gW
d4p ,
psi / 100ft (3-18)
turbulent flow
W 2
9 ~, psi / 100ft (3-19)
APlo o -0 000336f D d59
Multiplying Equations 3-16, 3-17, 3-18, and 3-19 by the total length
between two points and adding the pipe elevation yields the overall
pressure drop.
LTotal 9AZ
AP - AP 9 + ~, psi (3-20)
100 144
Equation 3-20 is valid for compressible isothermal fluids of shorter lines
where pressure drops are no more than 10% of the upstream pressure.
In general, pipe size for a given flow rate is often selected on the as-
sumption that the overall pressure drop is close to or less than the avail-
able pressure difference between two points in the line.
Nomenclature
A - pipe cross-sectional area,
ft 2
d = internal pipe diameter, inch
160
Fortran Programs for Chemical Process Design
D = internal pipe diameter, ft
fc =Chen friction factor
fp - Darcy friction factor
gc = dimensional constant 32.174 (lbm/lb f ~ ft/s 2)
h L = head loss, ft
K = excess head loss for a fitting, velocity heads
K~ = K for fitting at NR~ = 1, velocity heads
K= = K for very large fitting at NRe = ~o, velocity heads
Leq- equivalent length of pipe, ft
Lst =
actual length of pipe, ft
LTota I =
total length of pipe, ft
n = number of fittings
NRe = Reynolds number
AP~00 = pressure drop per 100 ft of pipe, psi/100 ft
AP = overall pressure drop of pipe, psi
9 = fluid density, lb/ft 3
= absolute roughness of pipe wall, ft
~t = fluid viscosity, cP
COMPRESSIBLE FLUID FLOW IN PIPES
The flow of compressible fluids (e.g., gases and vapors) through pipe-
lines and other restrictions is often affected by changing conditions of
pressure, temperature, and physical properties. The densities of gases
and vapors vary with temperature and pressure. During isothermal flow,
i.e., constant temperature (PV = constant) density varies with pressure.
Conversely, in adiabatic flow, i.e., no heat loss (PV k = constant), a decrease
in temperature occurs when pressure decreases, resulting in a density
increase. At high pressures and temperatures, the compressibility factor
can be less than unity, which results in an increase in the fluid density.
The condition of high AP in compressible flow frequently occurs in
venting systems, vacuum distillation equipment, and long pipelines [8].
Some design situations involve vapor flows at very high velocities
resulting in AP > 10% of the upstream pressure. Such cases are vapor
expanding through a valve, high speed vapor flows in narrow pipes, and
vapors flowing in process lines under vacuum conditions. In many cases,
AP is critical and requires accurate analysis and design. For instance,
the inlet pipe AP of a safety relief valve should not exceed 3% of the
relief valve set pressure (guage) at its relieving capacity for stable
operation. This limit is to prevent the rapid opening and closing of the
Fluid Flow 161
valve, a phenomenom known as chattering, resulting in lowered fluid
capacity and subsequent damage of the value seating surfaces. Con-
versely, the tailpipe or ventline of a relief valve should be designed in
such a way that AP < 10% of the relieving pressure, i.e., set pressure +
over pressure in gauge. Figure 3-3 shows a typical tail pipe and relief
valve connected to a heat exchanger.
Maximum Flow and Pressure Drop
Determining the maximum fluid flow rate or pressure drop for pro-
cess design often has the dominant influence on density. As pressure
decreases due to piping and component resistance, the gas expands and
its velocity increases. A limit is reached when the gas or velocity cannot
exceed the sonic or critical velocity. Even if the downstream pressure is
lower than the pressure required to reach sonic velocity, the flow rate
will still not increase above that evaluated at the critical velocity. There-
fore, for a given AP, the mass discharge rate through a pipeline is greater
for an adiabatic condition (i.e., insulated pipes, where heat transfer is
TAIL
P IPE
RELIEF VALVE
II !1
! '
1
P2
P2
~ ......
9
liH
i"
....
J-L
tube fluid in
J-L
23 -L_F
Ishell fluid in ~tube fluid
out
Figure
3-3. Fluid flow through a heat exchanger, relief valve, and tail pipe.
162
Fortran Programs for Chemical Process Design
nil) than the rate for an isothermal condition by as much as 20%. There
is, however, no difference if the pipeline is more than 1,000 pipe diameters
long [ 10]. In practice, the actual flows are between the two conditions,
and the inflow rates are often below 20% even for lines less than 1,000
pipe diameters.
Critical or Sonic Flow and the Mach Number
The flow rate of a compressible fluid in a pipe with a given upstream
pressure will approach a certain maximum rate that it cannot exceed
even with reduced downstream pressure. The maximum velocity is lim-
ited by the velocity of propagation of a pressure wave that travels at the
speed of sound in the fluid. The maximum velocity that a compressible
fluid can attain in a pipe is known as the sonic velocity, V~, and can be
expressed as:
)0.5
V~ - 223 kT ,
Mw
ft/s
/0.5
kP~
=68.1 -~1
ft/s (3-21)
With a high velocity vapor flow, the possibility of attaining critical or
sonic flow conditions in a process pipe should be investigated. These
occur whenever the resulting pressure drop approaches the following
values of AP as a percentage of the upstream pressure [ 11 ]:
1. saturated steam, AP = 42%
2. diatomic gases, (e.g.,
H2, N 2,
02) , AP =
47%
3. triatomic and higher molecular weight gases including hydrocar-
bon vapors and superheated steam, AP = 45%. Vapor flow at or
near this maximum velocity should be avoided because a critical
pressure, Pc, is attained at the sonic velocity and any AP beyond Pc
will translate into shock waves and critical turbulence instead of
being converted into useful kinetic energy.
In the case of a high pressure header, the flow may be sonic at the
exit. Therefore, it is often necessary to check that the outlet pressure of
each pipe segment is not critical. If Pc is less than terminal
P2,
the flow
is subcritical. If however, Pc is greater than
P2,
then the flow is critical.
Fluid Flow 163
Although, it may be impractical to keep the flow in high pressure
subheaders below sonic, Mak [12] suggests that the main flare header
should not be sized for critical flow at the outlet of the flare stack. This
would obviate the undesirable noise and vibration resulting from sonic
flow. Crocker's [13] equation for critical pressure can be expressed as:
( /
p = G RT , psia (3-22)
c l1400d 2 k(k+l)
where R- (1544/29oSpGr), molar gas constant
SpGr - molecular weight of the gas + molecular weight of air
The upstream fluid velocity is"
0.0509G
V = ft/s
d291 (3-23)
A recommended compressible fluid velocity for trouble-free operation
is V < 0.6V~. The design criteria for compressible fluid process lines
(e.g., carbon steel) are shown in Table 3-4.
The Mach Number
The Mach number, M, is the velocity of the gas divided by the velocity
of the sound in the gas and can be expressed as"
V
M - (3-24)
Vs
The exit Mach number for compressible isothermal fluid has been shown
to be M ~: 1, but 1/~fk-, where k is the ratio of the fluid specific heat
capacities at constant pressure to that at constant volume. Table 3-5
shows the k values for some common gases. The following cases are such:
1. for M < 1/~vr-k, the flow is subsonic
2. for M- 1/~/k, the flow is sonic
3. for M > 1/~]-k, the flow is supersonic
Case 3 is produced under certain operating conditions in the throttling
process (e.g., a reduction in the flow cross-sectional area). Kirkpatrick
[14] indicates that there is a maximum length for which continuous
flow is applied for an isothermal condition, and this corresponds to
Table 3-4
Recommended Fluid Velocity and Maximum
AP for Carbon Steel Vapor Lines
Type of Service
Recommended
Velocity ft/sec.
Maximum AP
psi/1 O0 ft.
1. General Recommendation
Fluid pressure psig
Subatmospheric
0<P<50
50 < P < 150
150 < P < 200
200 < P < 500
P> 500
2. Tower Overhead
Pressure (P > 50 psia)
Atmospheric
Vacuum (P < 10 psia)
3. Compressor Piping Suction
4. Compressor Piping Discharge
Gas lines with battery limits.
Refrigerant Suction lines
Refrigerant Discharge lines
Steam lines
1. General Recommendation
Maximum: Saturated
Superheated
Steam pressure in psig.
0-50
50-150
150-300
>300
2. High pressure Steam Lines
short (L < 600 ft)
Long (L > 600 ft)
3. Exhaust Steam lines
(P > atmosphere)
Leads to Exhaust Header
4. Relief valve discharge
Relief valve, Entry point at silencer
40-50
60-100
75-200
100-250
15-35
35-60
200
250
167
117
0.5V s
Vs
0.18
0.15
0.30
0.35
1.0
2.0
0.2-0.5
0.05-0.1
0.5
1.0
0.5
0.25
0.40
1.0
1.5
1.0
0.5
0.5
0.5
1.5
164
Fluid Flow 165
Table 3-5
Approximate k values for some common gases (68~ 14.7 psia)
Chemical Approximate
Formula Molecular
Gas or Symbol Weight k(Cp/Cv)
Acetylene (Ethyne) C2H 2 26.0 1.30
Air 29.0 1.40
Ammonia NH 3 17.0 1.32
Argon A 39.9 1.67
Butane C4HI0 58.1 1.11
Carbon Dioxide CO 2 44.0 1.30
Carobon Monoxide CO 28.0 1.40
Chlorine
C12
70.9 1.33
Ethane CzH 6 30.0 1.22
Ethylene C2H 4 28.0 1.22
Helium He 4.0 1.66
Hydrogen Chloride HC 1 36.5 1.41
Hydrogen H 2 2.0 1.41
Methane CH 4 16.0 1.32
Methyl Chloride CH3C 1 50.5 1.20
Natural Gas 19.5 1.27
Nitric Oxide NO 30.0 1.40
Nitrogen N 2 28.0 1.41
Nitrous Oxide N20 44.0 1.31
Oxygen O 2 32.0 1.40
Propane C3H 8 44.1 1.15
Propylene (Propene) C3H 6 42.1 1.14
Sulfur Dioxide SO2 64.1 1.26
M = 1/~-k. The limitation for isothermal flow, however, is the heat
transfer required to maintain a constant temperature. Therefore, when
M < 1/~v/k, heat must be added to the stream to maintain constant tem-
perature. For M < ~v/-k, heat must be rejected from the stream. Depend-
ing on the ratio of specific heats, either condition could occur with
subsonic flow. Therefore, to maintain isothermal flow during heat trans-
fer, high temperatures require high Mach numbers and low tempera-
tures require low Mach numbers.
166
Fortran Programs for Chemical Process Design
Mathematical Model of Compressible Isothermal Flow
The derivations of the maximum flow rate and pressure drop of com-
pressible isothermal flow are based on the following assumptions:
1. isothermal compressible fluid
2. no mechanical work done on or by the system
3. the gas obeys the perfect gas laws
4. a constant friction factor along the pipe
5. steady state flow or discharge unchanged with time
Figure 3-4 illustrates the distribution of fluid energy with work done by the
pump and heat added to the system. Table 3-6 gives friction factors for
clean commercial steel pipes with flow in zones of complete turbulence.
Flow Rate Through Pipeline
Bemoulli's equation for the steady flow of a fluid can be expressed as:
~12 dP V 2
+~+---g
AZ+h L+6W -0 (3-25)
p 2gcCZ gc
POINT I
~q (heat added to fluid)
I
I POINT 2
0
P2
V 2
Z 2
PUMP
/
ZI -w (work done by pump on fluid)
DATUM
Figure 3-4. Energy aspects of a single-stream piping system.