Coker A.K. Fortran Programs for Chemical Process Design, Analysis, and Simulation

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Fluid Flow 177

Figure

3-7. Lockhart-Martinelli pressure drop correlation. Source: Lockhart and

Martinelli [19].

where YG is the two-phase flow modulus. YG is a function of the

Lockhart-Martinelli two-phase flow modulus X, which is defined as:

X (APL) ~

- (3-61)

APG

YG = f(x)

The value of YG for the different flow regimes is determined as follows:

For bubble or froth flow

yG [14 2x07512

(WL/A) ~

(3-62)

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Fortran Programs for Chemical Process Design

For plug flow

yG _ [27.315X~ 12

(WL/A) ~

(3-63)

For stratified flow

E ] 2

YG - 15,400 X

(3-64)

For wave flow, the Huntington friction factor [20] is used to determine

the two-phase pressure loss:

(3-65)

In FH - 0.2111 In (H x ) - 3.993

(3-66)

APT 0.000336(FH)(WG)2

lOOft dSPG

(3-67)

For slug

1190X ~

(3-68)

For annular flow

- (aXb) 2

(3-69)

a - 4.8- 0.3125d

b - 0.343 - 0.021d

d - pipe inside diameter, inch

d - l0 for 12 inch and larger sizes

For dispersed or spray flow

YO - {exp[~0 + ~l ~,n • + ~ ~,n • + ~ ~ln

• 1}~

(3-70)

Fluid Flow 179

C o - 1. 4659

C 1 - 0. 49138

C 2 - 0.04887

C 3 - -0.000349

Once the two-phase flow modulus (YG) for the particular flow regime

has been calculated, Equation 3-60 can be used to calculate APT/100ft,

the two-phase, straight-pipe pressure drop. The pressure drop of liquid

or gas flowing alone in a straight pipe can be expressed as"

APT 0. 000336 f D W 2x

lOOft

dSPG

psi/100ft (3-71)

Equation 3-71 can be modified for calculating the overall pressure drop

of the two-phase flow for the total length of pipe plus fittings, L from

Equation 3-5, based on the gas-phase pressure drop.

0.000336

9

WG2 9 YG 9 L

APToverall

"- ,

psi (3-72)

100dSPG

The velocity of the two-phase fluid is

F 7

V - 0.0509 | WG WL /

- ~ + ~ (3-73)

PG PL

Erosion-Corrosion

Depending on the flow regime, the liquid in a two-phase flow system

can be accelerated to velocities approaching or exceeding the vapor

velocity. In some cases, these velocities are higher than what would be

desirable for process piping. Such high velocities lead to a phenomenon

known as erosion-corrosion, where the corrosion rate of a material is

accelerated by an erosive material or force (in this case, the high-veloc-

ity liquid).

An index [21 ] based on velocity head can indicate whether erosion-

corrosion may become significant at a particular velocity. It can be used

to determine the range of mixture densities and velocities below which

erosion-corrosion should not occur.

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Fortran Programs for Chemical Process Design

This index is:

IOM U2

10,000

where the mixture density is:

WL + WG

and the mixture velocity is:

UM -- UG --[-U L

(3-74)

(3-75)

WG WL /

- +

3600pGA 3600pLA

(3-76)

Nomenclature

A = inside cross-sectional area of pipe,

ft 2

B x, By - Baker parameters for determining for regime

d = pipe inside diameter, inch

D = pipe inside diameter, ft

fc =Chen friction factor

fD = Darcy friction factor

fF = Fanning friction factor

FH = Huntington friction factor

H~ = Huntington flow factor

ID = internal diameter of pipe, inch

K = excess head loss for a fitting, velocity heads

K 1 = K for fitting at NRe = 1 velocity heads

K~ = K for large fittings at NRe = ~ velocity heads

L = total fitting equivalent pipe length plus straight pipe

length, ft

Leq- equivalent length of pipe due to fittings, ft

L~t = straight pipe length, ft

LTota 1 =

total length of pipe, ft

MW = molecular weight

NRe = Reynolds number

NRe L = liquid Reynolds number

NRe G = gas Reynolds number

APL/100ft = pressure drop of liquid if flowing alone in the pipe (psi/100 ft)

Fluid Flow 181

APG/100ft =pressure drop of gas if flowing alone in the pipe (psi/100 fi)

APT/100ft = pressure drop of the two-phase mixture in straight pipe

(psi/100 ft)

APToveral 1 = overall pressure drop of the two-phase mixture for the total

length plus fittings, psi

UL = liquid velocity, ft/s

U G = gas velocity, ft/s

UM = velocity of the two-phase mixture, ft/s

V = velocity of fluid in a pipe, ft/s

WL = liquid flow rate, lb/h

WG = gas flow rate, lb/h

W~ = flow rate of either liquid or gas, lb/h

X = Lockhart-Martinelli two-phase flow modulus (APL/APG) ~

YG = two-phase flow modulus

= absolute roughness of pipe wall, ft (0.00015 ft for carbon

steel)

PL = liquid density, lb/ft 3

PG = gas density, lb/ft 3

PM = density of liquid-gas mixture, lb/ft 3

la x = viscosity of liquid or gas, cP

gL = liquid viscosity, cP

gG = gas viscosity, cP

(YL = surface tension of liquid (dyne/cm)

VAPOR-LIQUID TWO-PHASE VERTICAL DOWNFLOW

An understanding of two-phase flow is necessary for sound piping

design because almost all chemical process plants encounter two-phase

flow conditions. Two-phase vertical downflow presents its own problems

in horizontal flow. In a vertical flow, large vapor bubbles, known as slug

flow, are formed in the liquid stream. This flow regime is associated with

pipe vibration and pressure pulsation. With bubbles greater than 1 inch in

diameter and the liquid viscosity less than 100 cR slug flow region can be

represented by dimensionless numbers for liquid and vapor phases respec-

tively (Froude numbers (NFr) L and

(NFr)C).

These are related by the ratio of

inertial to gravitational force and are expressed as:

/0.5

_ VL PL (3-77)

(NFr)L

(gD) ~

PL--PG

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Fortran Programs for Chemical Process Design

/0.5

_ VG 9L (3-78)

(NFr)G --

(gD)~ 9L - PG

The velocities VG and VL are superficial velocities based on the total

pipe cross section. These Froude numbers exhibit several features in the

range 0 < (NFr) L and (NFr) G < 2.

Simpson [3] illustrates the values of (NFr) L and

(NFr)C

with water flow-

ing at an increased rate from the top of an empty vertical pipe. As the

flow rate further increases to the value (NFr) L = 2, the pipe floods and

the total cross section is filled with water. If the pipe outlet is further

submerged in water and the procedure is repeated, long bubbles will be

trapped in the pipe below

(NFr)L =

0.31. However, above

(NFr)L =

0.31,

the bubbles will be swept downward and out of the pipe.

If large long bubbles are trapped in a pipe (d >1 inch) in vertically

down flowing liquid having a viscosity less than 100 cP and the Froude

number for liquid phase, (NFr)L < 0.3, the bubbles will rise. At higher

Froude numbers, the bubbles will be swept downward and out of the

pipe. A continuous supply of vapor causes the Froude number in the

range 0.31 < (NFr) L < 1 tO produce pressure pulsations and vibration.

These anomalies are detrimental to the pipe and must be avoided. If the

Froude number is greater than 1.0, the frictional force offsets the effect

of gravity, and thus requires no pressure gradient in the vertical downflow

liquid. This latter condition depends on the Reynolds number and

pipe roughness.

The Equations

The following equations will calculate Froude numbers for both the

liquid and gas phases. The computer program will print out a message

indicating whether the vertical pipe is self-venting, whether pulsating

flow occurs, or whether no pressure gradient is required.

D = ~, ft (3-79)

gD 2

Area = --, ft (3-80)

VL = , ft/s (3-81)

3600 ,, 9L " Area

Fluid Flow 183

VG = , ft/s (3-82)

3600 9 9G 9 Area

10.5

FRNL - VL PL (3-83)

(gD) ~ PL -- 9G

10.5

FRNG - VG PG (3-84)

(gD) ~ PL - PG

The Algorithm

If FRNL < 0.31, vertical pipe is SELF-VENTING

else

0.3 < FRNL < 1.0, PULSE FLOW, and may result in pipe vibration

FRNL > 1.0, NO PRESSURE GRADIENT

Nomenclature

Area = inside cross-sectional area of pipe,

ft 2

d = inside diameter of pipe, inch

D = inside diameter of pipe, ft

FRNL,

(NFr)L =

Froude number of liquid phase, dimensionless

FRNG,

(NFr)C -

Froude number of vapor phase, dimensionless

g = gravitational constant, 32.2 ft/s 2

VL- liquid velocity, ft/s

VG = vapor velocity, ft/s

WL = liquid flowrate, lb/h

WG = vapor flowrate, lb/h

9L = liquid density, lb/ft 3

9G = vapor densit, lb/ft 3

LINE SIZES FOR FLASHING STEAM CONDENSATE

When a liquid is flowing near its saturation point (i.e., the equilibrium

or boiling point) in a pipe line, decreased pressure will cause vaporization.

The higher the pressure difference, the greater the vaporization result-

ing in flashing of the liquid. Steam condensate lines cause a two-phase

flow condition, with hot condensate flowing to a lower pressure through

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Fortran Programs for Chemical Process Design

short and long lines. For small lengths with low pressure drops, and the

outlet end being a few pounds per square inch of the inlet, the flash will be

assumed as a small percentage. Consequently, the line can be sized as an all

liquid line. However, caution must be exercised because 5% flashing can

develop an important impact on the pressure drop of the system [22].

Sizing of flashing steam condensate return lines requires techniques that

calculate pressure drop of two-phase flow correlations. Many correlations

have been presented in the literature [ 15,16,19,23]. Most flow patterns for

steam condensate headers fall within the annular or dispersed region on the

Baker map. Sometimes, they can fall within the slug flow region; however,

the flashed steam in steam condensate lines is less than 30% by weight.

Ruskin [24] developed a method for calculating pressure drop of flash-

ing condensate. Ruskin's method gave pressure drops comparable to

those computed by the two-phase flow with good agreement with

experimental data. The method employed here is based on a similar

technique given by Ruskin. The pressure drop for flashing steam uses

the average density of the resulting liquid-vapor mixture after flashing.

In addition, the friction factor used is valid for complete turbulent flows

in both commercial steel and wrought iron pipe. The pressure drop

assumes that the vapor-liquid mixture throughout the condensate line is

represented by mixture conditions near the end of the line. This assump-

tion is valid because most condensate lines are sized for low pressure

drop, with flashing occurring at the steam trap or valve close to the pipe

entrance. If the condensate line is sized for a higher pressure drop, an

iterative method must be used. For this case, the computations start at

the end of the pipeline and proceed to the steam trap.

The method employed determines the following:

1. the amount of condensate flashed for any given condensate header

from 15 to 140 psia. Initial steam pressure may vary from 40 to

165 psia.

2. the return condensate header temperature.

3. the pressure drop (psi/100 ft) of the steam condensate mixture in

the return header.

4. the velocity of the steam condensate mixture and gives a warning

message if the velocity is greater than 5000 ft/min, because this

may present problems to the piping system.

The Equations

The following equations are used to determine the pressure drop for

flashed condensate mixture [20].

Fluid Flow 185

WFRFL - B (In Pc )2

where

A - 0. 00671 (In

Ph )2.27

B -

e x

9 10 -4

+0.0088

and

X-6" 122 - (16"919/In

W G - WFRFL 9 W

WL - W- WG

TFL - 115.68(P h

)0.226

OG --

0029P~

PL --

60.827 - 0.078Ph

-I-

0.00048p2 h - 0.0000013p3 h

WG + WL

/ + WL/ :

For turbulent flow

f m

0.25

[- l~ 000486)12 d

where d- pipe diameter, inch

Pressure drop

APT -

0. 000336f

9 W 2

d5 9 PM

3.054[WG +~WL 1

d2 PG PL

(3-85)

(3-86)

(3-87)

(3-88)

(3-89)

(3-90)

(3-91)

(3-92)

(3-93)

(3-94)

(3-95)

(3-96)

(3-97)

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Fortran Programs for Chemical Process Design

If V > 5000 ft/min, print a warning message as condensate may cause

deterioration of the process pipeline.

Nomenclature

d = internal pipe diameter, inch

f = friction factor, dimensionless

Pc = steam condensate pressure before flashing, psia

Ph = flashed condensate header pressure, psia

V = velocity of flashed condensate mixture, ft/min

W = total flow of mixture in condensate header, lb/h

WG = flashed steam flow rate, lb/h

WL = flashed condensate liquid flow rate, lb/h

WFRFL = weight fraction of condensate flashed to vapor

TFL = temperature of flashed condensate, ~

APT = pressure drop of flashed condensate mixture, psi/100 ft

PG = flashed steam density, lb/ft 3

PL = flashed condensate liquid density, lb/ft 3

9M = density of mixture (flashed condensate/steam), lb/ft 3

FLOW THROUGH PACKED BEDS

Flow of fluids through packed beds of granular particles occurs fre-

quently in chemical processes. Examples are flow through a fixed-bed

catalytic reactor, flow through a filter cake, and flow through an absorp-

tion or adsorption column. An understanding of flow through packed

beds is also important in the study of sedimentation and fluidization.

An essential factor that influences the design and operation of a

dynamic catalytic or adsorption system is the energy loss (pressure drop).

Factors determining the energy loss are many and investigators have

made simplifying assumptions or analogies so that they could use some

of the general equations. These equations represent the forces exerted

by the fluids in motion (molecular, viscous, kinetic, static, etc.) to arrive

at a useful expression correlating these factor.

Ergun [25] developed a useful pressure drop equation caused by

simultaneous kinetic and viscous energy losses and applicable to all

types of flow. Ergun's equation relates the pressure drop per unit of bed

depth to dryer or reactor system characteristics, such as, velocity, fluid

gravity, viscosity, particle size, shape, surface of the granular solids and

void fraction. The original Ergun equation is: