48967_02

7/21/2019 48967_02

http://slidepdf.com/reader/full/4896702 1/7555

2.0 Introduction

Fluid mechanics is the study of the behavior of fluids at rest, fluid statics,

and in motion, fluid dynamics, and of the properties of fluids insofar as they

affect the fluid motion. A fluid may be either a gas or a liquid. The molecules

of a gas are much farther apart than those of a liquid. Hence a gas is compressible

while a liquid is relatively incompressible.

A vapor is a gas whose temperature and pressure are such that it is very

near the liquid phase. Steam is considered a vapor because its state is

normally not far from that of water. A gas may be defined as a highly super-

heated vapor, that is, its state is far removed from the liquid phase. Air is

considered a gas because its state is normally very far from that of liquid air.

The volume of a gas or vapor is greatly affected by changes in pressure or

temperature or both. It is usually necessary to take into account the changesin volume and temperature in dealing with gases or vapors. Whenever

significant temperature or phase changes are involved in dealing with vapors

and gases, the subject is largely dependent on heat phenomena, thermodynamics.

2

Fluid Mechanics

7/21/2019 48967_02


AIR-COOLED HEAT EXCHANGERS AND COOLING TOWERS

56

The objective of this chapter is to introduce the reader to certain termi-nology, concepts, laws, and equations that are directly applicable to the

design of air-cooled heat exchangers and cooling towers.

2.1 Viscous Flow

Consider the flow of a fluid over a flat plate (Fig. 2.1.1).

Figure 2.1.1 Boundary Layer Development along a Flat Plate

Beginning at the leading edge of the plate, a region develops where the

influence of viscous forces is felt. These viscous forces are described in terms

of a shear stress, τ, between the fluid layers. If this stress is assumed to be pro-

portional to the normal velocity gradient, the defining equation for viscosi-

ty, known as Newton’s equation of viscosity, is

(2.1.1)

The constant of proportionality, µ, is called the dynamic viscosity. The

values of µ for some fluids are given in appendix A.

dv

d y τ = –µ

7/21/2019 48967_02


The velocity or hydrodynamic boundary layer is the region of flow thatdevelops from the leading edge of the plate in which the effects of viscosity

are observed. The y-position where the boundary layer ends is arbitrarily

chosen at a point where the velocity becomes 99% of the free stream value.

The boundary layer thickness, δ, is defined as the distance between this point

and the plate.

Initially, the boundary layer development is laminar. At some critical

distance from the leading edge, small disturbances in the flow begin to

become amplified and a transition process takes place until the flow becomesturbulent. This depends on the flow field and fluid properties.

The physical mechanism of viscosity is one of momentum exchange. In

the laminar portion of the boundary layer, molecules move from one lamina

to another and carry momentum corresponding to the velocity of the flow.

There is a net momentum transport from regions of high velocity to regions

of low velocity, which creates a force in the direction of flow. This force may

be expressed in terms of the viscous shear stress as given by Equation 2.1.1.

The rate at which the momentum transfer takes place is dependent onthe rate at which the molecules move across the fluid layers. In a gas, the

molecules would move about with some average speed proportional to the

square root of the absolute temperature since we identify temperature with

the mean kinetic energy of a molecule in the kinetic theory of gases. The

faster the molecules move, the more momentum they will transport. Hence

we should expect the viscosity of a gas to be approximately proportional to

the square root of temperature, and this expectation is corroborated fairly

well by experiment.

The laminar velocity profile is approximately parabolic in shape. The

transition from laminar to turbulent flow occurs typically when

where

v ∞ = free stream velocity

x = distance from the leading edge of the plate

ν = µ/ρ, the kinematic viscosity of the fluid

qv ∞x

l

v ∞x

m

= ≥3.2x105

FLUID MECHANICS

7/21/2019 48967_02



58

This particular dimensionless group or ratio of inertial force to viscousforce is called the Reynolds number after the British scientist-engineer who

first did extensive research on flow in the late 1800s.

(2.1.2)

Although the critical Reynolds number for transition on a flat plate isusually taken as 3.2 x 105 for most analytical purposes, the critical value in a

practical situation is strongly dependent on the surface roughness conditions

and the turbulence level of the free stream. The normal range for the beginning

of transition is between 3.2 x 105 and 106. With very large disturbances

present in the flow, transition may begin with Reynolds numbers as low

as 105. For flows which are very free from fluctuations, it may not start until

Rex = 2 x 106 or more. In reality, the transition process covers a range of

Reynolds numbers. Completed transition and fully developed turbulent flow

usually is observed at Reynolds numbers twice the value at which transition began.

A qualitative picture of the turbulent flow process may be obtained by

imagining macroscopic chunks of fluid transporting momentum instead of

microscopic transport on the basis of individual molecules. The turbulent

boundary layer is more complex than the laminar boundary layer because the

nature of the flow in the former changes with distance from the plate surface.

The zone adjacent to the wall is a layer of fluid, which, because of the

stabilizing effect of the wall, remains laminar even though most of the flow

in the boundary layer is turbulent. This very thin layer is called the laminarsublayer, and the velocity distribution in this layer is related to the shear

stress and viscosity using Newton’s viscosity law.

The flow zone outside the laminar sublayer is turbulent. The turbulence

alters the flow regime so much that the shear stress, as given by τ = - µ dv/dy,

is not significant. The mixing action of turbulence causes small fluid masses

to be swept back and forth in a direction transverse to the mean flow direction.

As a small mass of fluid is swept from a low-velocity zone next to the sublayer

into a relatively high-velocity zone farther out in the stream, the mass has a

retarding effect on the high-velocity stream. This mass of fluid, through an

exchange of momentum, creates the effect of a retarding shear stress applied

to a high-velocity stream. A small mass of fluid originates farther out in the

boundary layer in a high-velocity flow zone and is swept into a region of

ρv ∞x

µ Rex =

7/21/2019 48967_02


relatively low velocity. This has an effect on the low-velocity fluid much likeshear stress augmenting the flow velocity. In other words, the mass of fluid

with relatively higher momentum will tend to accelerate the lower velocity

fluid in the region into which it moves. Although the process described

previously is a momentum-exchange phenomenon, it has the same effect as

a shear stress applied to the fluid. In turbulent flow, these stresses are termed

apparent shear stresses or Reynolds stresses. The turbulent velocity profile has a

nearly linear portion in the sublayer and a relatively flat profile outside this region.

Consider the flow in a tube shown in Figure 2.1.2. A boundary layerdevelops at the entrance. Eventually the boundary layer fills the entire tube,

and the flow is said to be fully developed. If the flow is laminar, a parabolic

velocity profile is experienced as illustrated in Figure 2.1.2a. When the flow

is turbulent, a somewhat blunter profile is observed (Fig. 2.1.2b). In a tube,

the Reynolds number based on the mean fluid velocity and the tube diameter

is again used as a criterion for laminar and turbulent flow. For Red = ρvd/ µ ≤ 2300,

the flow is usually observed to be laminar, whereas for Red ≥ 10,000, it is turbulent.

Fig. 2.1.2 Velocity Profiles in a Tube: (a) Laminar Flow (b) Turbulent Flow

FLUID MECHANICS

7/21/2019 48967_02



60

Again, a range of Reynolds numbers for transition may be observeddepending on the roughness of the pipe and smoothness of the flow. The

generally accepted range for transition, also referred to as the critical region,

is 2000 < Red < 4000. Laminar flow has been maintained up to Reynolds

numbers of 25,000 in carefully controlled laboratory conditions.

The mass flow rate or continuity relationship for one-dimensional flow

in a tube is

(2.1.3)

where

m = mass rate of flow

v = mean velocity

A = cross-sectional area of the tube

The mass flux or mass velocity is defined as

(2.1.4)

so the Reynolds number may be written as

(2.1.5)

Similar flow patterns are observed in ducts that do not have a circular

cross section. In those cases, it is convenient to define the following equivalent

or hydraulic diameter for calculating the Reynolds number:

(2.1.6)

This particular grouping of terms is used because it yields the value of the

physical diameter when applied to a circular cross section.

d e =4 x cross-sectional flow area

wetted perimeter

Red = Gd /µ

G = m/ A = ρv

m = ρvA

7/21/2019 48967_02


2.2 Flow in Ducts

Real flows in ducts experience boundary stresses due to frictional effects,

which cause a pressure drop to occur between any two cross sections.

Typically, a force due to such frictional effects acts on the incompressible

fluid in the elementary control volume shown in Figure 1.4.2, i.e., dF = - τapp P edx.

Here, τapp is the apparent shear stress at the fluid-wall interface, and P e is the

wetted perimeter of the duct according to Shames. If this force is substituted

into Equation 1.4.27 for flow in a round duct or pipe of length L and fixeddiameter d (v 1 = v 2), find

or (2.2.1)

Dimensional analysis shows that, for fully developed pipe flow, the

frictional pressure drop, ∆p, between any two sections is generally related to

the pipe geometry and fluid properties in the following way:

The quantity ρv 2 / 2 is known as the dynamic pressure. The term L/d

considers the geometry of the pipe and ε/d is a measure of the roughness of

the pipe surface.

Based on this analysis, the pipe friction equation, also commonly

referred to as the Darcy-Weisbach equation for pressure drop in a circular

pipe, is obtained according to Weisbach, i.e.,

(2.2.2)∆ p = f D( L/d )(ρv 2/2)

= Function

(

), ,∆p

ρv 2 / 2

ρvd

l

ε

d

L

d

∆p = p1 – p2 = 4τapp L / d

∫ –τapp P edx / A = ∫ –τapp(πd )dx / (πd 2/4) = –4τapp L / d = p2 – p1

2

1

L

0

FLUID MECHANICS

7/21/2019 48967_02



62

The number of velocity heads, v 2 / 2, lost for a given pressure drop, ∆ p, isexpressed by the product of the Darcy friction factor f D and the geometric

factor L/d . The friction factor under consideration corresponds to fully developed

velocity profiles, both laminar and turbulent. These are encountered only

after 25 or more diameters downstream of a pipe inlet.

By equating Equations 2.2.1 and 2.2.2, find

or

Other definitions of the friction factor appear in the literature. In some

cases, the right side of this equation is divided by a factor of 4, giving a

friction factor f = f D

/4, also referred to as the Fanning friction factor. The

friction factor is a function of Re, the cross-sectional shape of the duct and,

in the turbulent flow regime, the relative roughness of the duct surface.

Equations 2.2.1 and 2.2.2 are also applicable to ducts other than circular

pipes, in which case d is replaced by d e.

Laminar flow

An extensive summary of Fanning friction factors for laminar flow in a

variety of ducts is presented by Shah. Using the Hagen-Poiseuille solution for

fully developed laminar flow in a circular duct or pipe, f = 16/ Re = f D/4

according to Shames. The friction factor is independent of the roughness of

the surface in the case of laminar flow.

The results for other duct shapes are shown in Figure 2.2.1. For the

rectangular duct, the friction curve in Figure 2.2.1 may be expressed as

(2.2.3)fRe = 24 [1–1.3553 ( ) + 1.9467 ( )

– 1.7012 ( )

b

a

b

a

b

a

b

a

b

a

2 3

4 5

+ 0.9564 (

) – 0.2537 ( ) ]

f D = 4τapp / (ρv 2/2)

4τapp L/d = f D( L/d )(ρv 2/2)

7/21/2019 48967_02


Fig. 2.2.1 Friction Factors for Fully Developed Flow

In certain applications where very viscous fluids are to be cooled, e.g., oil

coolers, a twisted tape may be inserted into a circular heat exchanger pipe or

tube. Smooth heat exchanger pipes are usually referred to as tubes. The tape

is inserted to establish swirl flow, thereby increasing the heat transfer coefficientand is twisted around the longitudinal axis shown in Figure 2.2.2.

Fig. 2.2.2 A Tube with a Twisted Tape Insert

FLUID MECHANICS

7/21/2019 48967_02



64

Using Hong and Bergles, the friction factor for fully developed flow in atube with such an insert is given by

where

Re = m/[µ(πd/ 4-t t )]

m = mass flow rate

t t = tape thickness

A more recent equation for the friction factor was derived numerically by

Du Plessis and Kröger:

(2.2.4)

where

a1 = a2/[ Re(15.767 – 0.14706 t t /d )]

a2 = Ats d 2/(a3a42)

a3 = 2 P 2 (a6 – 1) /p – dt t

a4 = 4a3/a5

a5 = 2d – 2t t + πd /a6

a6 = [1 + (πd/ 2 P )2]0.5

Ats = πd 2/4

This equation is valid for 50 ≤ Re ≤ 2000 and for P/d ≥ 2.

In the case of hydrodynamically developing flow in a duct from an initial

uniform velocity distribution, an apparent Fanning friction factor is defined.

The factor takes into account both the skin friction and the change in

momentum rate caused by a change in the shape of the velocity profile in the

hydrodynamic entrance region. In a long duct , the apparent friction factor

may be expressed in terms of an incremental pressure drop number K ∞ as

f = a1 [1 + { Re/(70( P /d )1.3)}1.5]0.333

f = 45.9 / Re

7/21/2019 48967_02


(2.2.5)

For a circular duct or pipe,

(2.2.6)

Du Plessis proposes the following general correlation which can beapplied to developing laminar flow in ducts of various cross sections:

(2.2.7)

where

fRe = the value for fully developed flow

n = an exponent dependent on the duct geometry

This correlation agrees well with similar ones by Shah.

For concentric annular ducts having inner and outer radii of r i and r o, values

of n are listed in Table 2.2.1.

Table 2.2.1 Values of Exponent for Annular Ducts

The case r i /r o = 0 corresponds to a pipe while r i /r o = 1 can be used for

parallel plates.

f app Re=[ (fRe)n+{3.44/( L/(d e Re))0.5}n

]1/n

f app Re=16+0.313dRe/ L for L/(dRe)≥0.06

f app Re = f Re + K ∞d e Re/4 L

FLUID MECHANICS

7/21/2019 48967_02



66

For rectangular ducts, fRe is determined using Equation 2.2.3. The valuesfor n are listed in Table 2.2.2.

Table 2.2.2 Values of Exponent for Rectangular Ducts

For isosceles triangular ducts having apex angles of 2θ degrees, the val-

ues of n are listed in Table 2.2.3.

Table 2.2.3 Values of Exponent for Isosceles Triangular Ducts

In general, the hydraulic entry length Lhy = x/(d e Re) is the dimensionless

length required for the centerline velocity to attain 99% of its fully developed

value. Values for Lhy and K ∞ for different duct sections are listed in Table 3.2.1.

When heat is transferred to or from the fluid, all physical properties are

evaluated at the mean fluid temperature. The latter is also referred to as the

bulk or mixing cup temperature according to Holman. For those problems

involving large temperature differences between the fluid and the duct wall,

Shah introduced corrections to provide for the temperature dependence of the fluid properties.

In the case of gases, the friction factor evaluated at the bulk mean

temperature is multiplied by one of the following factors:

7/21/2019 48967_02


(T w /T) for 1 < (T w /T ) < 3 (heating) (2.2.8a)

(T w /T)0.81 for 0.5 < (T w /T) < 1 (cooling) (2.2.8b)

For liquids, the friction factor evaluated at the bulk mean temperature is

multiplied by one of the following factors to obtain the correct value.

(µw /µ)0.58 for µw /µ < 1 (heating) (2.2.8c)

(µw /µ)0.54 for µw /µ > 1 (cooling) (2.2.8d)

The subscript w refers to the mean value of the duct wall temperature and

the temperatures are in degrees Kelvin. These relationships are also applicable to

developing flow.

Example 2.2.1

Air at a pressure of p = 101,025 N/m2 and a temperature of T = 16.87 °C flows

uniformly into a rectangular duct with

a = 50 mm

b = 3.5 mm

at a rate of m = 6.403 x 10-4 kg/s

the duct length is L = 200 mm

Determine the pressure differential between the inlet and the outlet of the duct.

Solution

Using the perfect gas law given by Equation 1.4.13, the density of the air at

the specified conditions can be expressed as

where the gas constant for air is R = 287.08 J/kgK.

ρ = = = 1.2134 kg/m3 p

RT

101025

287.08x(273.15+16.87))

FLUID MECHANICS

7/21/2019 48967_02



68

Using Equation A.1.3, the dynamic viscosity of the air at 16.87 °C or (273.15+ 16.87) = 290.02 K is

The mean air speed in the duct follows from Equation 2.1.3, i.e.,

v = m/( ρab) = 6.403 x 10-4/(1.2134 x 0.05 x 0.0035) = 3.015 m/s

Using Equation 2.1.6, the hydraulic diameter of the duct is

The Reynolds number for the air flowing in the duct is, using Equation 2.1.5,

Re = ρv d e / µ = 1.2134 x 3.015 x 0.006542/(1.8007 x 10-5) = 1329.1

It follows from Equation 2.2.3 that for duct flow

For b/a = 0.0035/0.05 = 0.07, find n ≈ 2.3 from Table 2.2.2.

Substitute the values for fRe and n into Equation 2.2.7 to find

or

f app = 30.16/1329.1 = 0.02269

f app Re=[(21.938)2.3+{3.44/(0.2/0.006542x1329.1)0.5}2.3]1/2.3=30.16

fRe=24[1–1.3553(0.0035/0.05)+1.9467(0.0035/0.05) 2–1.7012(0.0035/0.05)3

+0.9564(0.0035/0.05)4–0.2537(0.0035/0.05)5]=21.938

d e= = = 0.006542 m4ab

2(a+b)

4x0.05x0.0035

2(0.05+0.0035)

µ=2.287973x10–6 +6.259793x10–8T –3.131956x10–11 T 2+8.15038x10–1 5T 3

=2.287973x10–6+6.259793x10–8 x 290.02–3.131956x10–1 1x290.022

+8.15038x10–15 x290.023=1.8007x10–5kg/sm

7/21/2019 48967_02


Using Equation 2.2.2, the pressure drop between the inlet and the outlet of the duct is

Turbulent flow

With fully developed turbulent flow in ducts, the friction loss depends

on flow conditions as characterized by the Reynolds number and on the

nature of the duct wall surface. The quantity, ε, having the dimension of

length is introduced as a measure of the surface roughness. From dimensional

analysis, it follows that the friction factor is a function of the Reynolds

number and the relative roughness ε /d . The graphical representation of thisrelationship is known as the Moody diagram and is presented in Figure 2.2.3

according to Moody.

Fig. 2.2.3 Friction Factors for Pipe Flow

∆ p=4f app( ) =4x0.02269( )( ) =15.3 N /m2 L

d e

ρv 2

2

0.2

0.006542

1.2134x3.0152

2

FLUID MECHANICS

7/21/2019 48967_02



70

As shown, the laminar friction factor for pipe flow is a single straight line andis not influenced by the relative roughness. Reynolds numbers in the range from

2000 to 4000 lie in a critical region where flow can be either laminar or turbulent.

For Reynolds numbers larger than those in the critical region, turbulent

flow exists. The two regions into which the turbulent zone is divided, transition

and complete turbulence, categorize the state of the viscous sublayer as

influenced by roughness.

Based on Nikuradse’s data, the following implicit relationship is applicable

to turbulent flow in smooth pipes

(2.2.9)

or according to Filonenko,

(2.2.10)

Inspection indicates, for high Reynolds numbers and relative roughness,

the friction factor becomes independent of the Reynolds number in the

region of complete turbulence. Then

(2.2.11)

Transition between this region and the smooth wall friction factor is

represented by an empirical implicit transition function developed by Colebrook.

(2.2.12)

For purposes of computation, the following explicit relationship from

Benedict is of value:

(2.2.13)

Haaland recommends an equation that yields results comparable to the

implicit Colebrook equation:

0.9 Re5.74 +

3.7/d log0.25 = f 10

-2

Dε

f Re

2.51 +

3.7

/d log =

f

10.5

D

-2

100.5 D

ε

]/d)(n0.86-[1.14=f -2

D ε

)1.64- Relog = (1.82f -2

10 D

0.8-)f ( Ren= 0.86f /1 0.5 D

0.5 D

7/21/2019 48967_02


For ε/d > 10-4

(2.2.14)

For situations where ε /d is very small, Haaland proposes

(2.2.15)

The curves in the turbulent and transitional zones in Figure 2.2.3 were

drawn employing Equation 2.2.11 and the Haaland relations, respectively.

An approximate indication of the relative roughness of typical pipe surfaces

encountered in practice is shown in Figure 2.2.4 according to Kirschmer.

Fig. 2.2.4 Surface Roughness in Pipes

3.75

/d +

Re

7.7 log2.7778=

f

3.333

10

-2

D

ε

1.11

3.7

/d +

Re

6.9log0.3086 =f 10

-2

D

ε

FLUID MECHANICS

7/21/2019 48967_02



72

Dimensional analysis does not relate the performance of ducts havingcircular and noncircular cross sections. The fully turbulent friction factor for

noncircular cross sections (annular spaces, rectangular and triangular ducts,

etc.) may be evaluated from the data for circular pipes. This applies if the pipe

diameter is replaced by an equivalent diameter, also referred to as the

hydraulic diameter , defined by Equation 2.1.6.

The equivalent or hydraulic diameter for an annulus of inner and outer

diameter d i and d o is

(2.2.16)

For a rectangular duct having sides a and b, it is

(2.2.17)

Launder and Ying show, for a rectangular duct, the secondary velocity

distribution gives rise to an increase in the friction factor of about 10%. Even so,

their full theory slightly underestimates the measurements of Hartnett et al.

According to White, the friction factor for turbulent flow between parallel

plates as given by the following equation is higher than the value that would

be obtained if the equivalent diameter were substituted into the equation for

pipe flow.

(2.2.18)

where Re is based on the hydraulic diameter which is equal to twice the

distance between the plates.

For developing turbulent flow near the entrance of a duct, the friction factor

is considerably higher than for fully developed flow according to Deissler.

When heat transfer occurs in turbulent duct flow, changes in thermo-

physical properties should be considered. This effect is taken into consideration

by multiplying the friction factor, evaluated at the bulk temperature of the

fluid, by one of the following appropriate correction factors from Petukhov:

( ) 1.19- Ref log2=f /1 0.5D10

0.5 D

b)+/(aab2=d e

d -d )d +d (

)d -d 4)(/4(d io

io

2i

2o

e ππ

= =

7/21/2019 48967_02


(7 – µ/µw )/6 for (µw /µ) < 1 (heating) (2.2.19a)

(µw /µ)0.24 for (µw /µ) > 1 (cooling) (2.2.19b)

for 1.3 < Pr < 10 and where µw is evaluated at the duct wall temperature.

For air and hydrogen—temperature in degrees Kelvin

(2.2.20a)

(2.2.20b)

Example 2.2.2

Calculate the approximate mean Darcy friction factor when air

• at a pressure of pa = 1.013 x 105 N/m2

• a bulk temperature of T = 93.33 °C

• flows at a speed of 6.096 m/s through a smooth pipe having an inside

diameter of 25.4 mm

• T w = 426.67 °C is the inside pipe wall temperature• µaw = 3.355 x 10-5 kg/ms is the dynamic viscosity of air at 426.67 °C

Solution

Evaluate the Reynolds number of the airstream at bulk temperature.

Using Equation A.1.1, the density of the air at the bulk temperature of

(273.15 + 93.33) = 366.48 K is

m3kg/0.9628=93.33)+(273.15x287.08

x1.013 =

T 287.08

p =

105a

aρ

[ ](cooling) )/T T ( )/ Re0.79(+-0.6

w

-0.11w w ρρ

[ ](heating) )/T T ( )/ Re(5.6+-0.6

w

-0.38w w ρρ

FLUID MECHANICS

7/21/2019 48967_02



74

The dynamic viscosity of the air at this temperature follows from Equation A.1.3.

µa = 2.287973 x 10-6 + 6.259793 x 10-8 x 366.48 - 3.131956 x 10-11 x 366.482

+ 8.15038 x 10-15 x 366.483 = 2.14236 x 10-5 kg/ms

Thus,

The flow is turbulent, and the friction factor for the smooth tube may be

determined at the bulk temperature using Equation 2.2.10, i.e.,

This factor must be corrected by multiplying it by Equation 2.2.20a, which

includes the Reynolds number of the air evaluated at the wall temperature

T w = 426.67 °C

or

(273.15 + 426.67) = 699.82 K

The air density at this temperature is

With this density and the specified dynamic viscosity of the air evaluated at

the pipe wall temperature, find the Reynolds number

The corrected friction factor is thus

0.03019=)366.48/(699.820.03489=

)/T T (f =f

])0.9628/0.5042x(2326.965.6+[-0.6

])/ Re(5.6+[-0.6w D Dc

0.38-

-0.38aaw w ρρ

2326.96= x3.355

0.0254x6.096x0.5042 = Re 510

-w

kg/0.5042=699.82x287.08

x1.013 = m

3105

aw ρ

0.03489=)1.64-6958.61log(1.82=f -2

10 D

6958.61= x2.14236

0.0254x6.096x0.9628 =

vd = Re

10-5

a

aµ

ρ

7/21/2019 48967_02


Transition laminar-turbulent flow In the critical zone where transition from laminar to turbulent flow takes

place, the friction factor is uncertain, and there is an uncertainty in pressure

drop estimates if the Reynolds number falls in this range, i.e., 2000 ≤ Re ≤ 4000.

A single correlating equation that covers the entire range from the laminar

through the critical region to turbulent flow in smooth pipes or tubes is

proposed by Churchill:

(2.2.21)

A more comprehensive equation including the effect of surface roughness

is also presented by Churchill.

(2.2.22)

where

and(37530) 16

a2 = [ Re ]

2.3 Losses in Duct Systems

As a result of the frictional resistance experienced during flow in ahorizontal duct, the mechanical energy (the p/ ρ + αev 2/2 terms in Equation

1.4.4) between any two sections of the duct is reduced, i.e., converted to

thermal energy. Similar reductions or losses in mechanical energy may

occur at inlets, outlets, abrupt changes in duct cross-sectional area, valves,

bends, and other appurtenances.

/d 0.27+0.9/ Re)(7

1n 2.457 = a

16

1 ε

1.5)a+a(

1 +

12

Re

88 = f D

21

0.0833

10

7

Ren2.21+

20

36500

Re +

10

Re

8

18f D =

0.5

-0.2

7/21/2019 48967_02



76

A dimensionless loss coefficient, in general, can be defined between twocross sections in a horizontal duct as

(2.3.1)

where v is usually based on conditions at either section 1 or section 2.

Since most loss coefficients are determined experimentally, it is important tospecify the velocity on which a loss coefficient for a particular duct element

is based, i.e., inlet, outlet, or some mean condition.

If the flow is incompressible and the velocity distribution at sections 1

and 2 is uniform, as is approximately the case in turbulent flow, the kinetic

energy coefficient is αe ≈ 1, and Equation 2.3.1 can be written as

(2.3.2)

or

where

pt 1 = total pressure at section 1

pt 2 = total pressure at section 2

K is also referred to as the total pressure loss coefficient

A static loss coefficient is sometimes defined as

(2.3.3)

This loss coefficient is equivalent to K if there is no change in velocity

between sections 1 and 2.

( ) ( )2/v / p- p= K 2

21s ρ

( ) ( )2/v /t 2 p-t 1 p= K 2ρ

( ) ( )2/v

2/v + p-2/v + p

= K 2

222

211

ρρρ

( ) ( )2/v

2/v +/ p-2/v +/ p = K

2

222e22

211e11 αραρ

7/21/2019 48967_02


For frictional, fully developed, incompressible flow in a pipe of constantdiameter, the pressure drop is given by the empirical Darcy-Weisbach

Equation 2.2.2. A loss coefficient, using Equation 2.3.2, may be expressed as

K f = ∆ p/(ρv 2/2) = f D L/d (2.3.4)

The region of influence for a component can be determined experimentally.

Attach straight pipes to the exit and entry of the component to attain fullydeveloped conditions upstream and downstream.

Consider the pressure distribution along a pipeline containing a bend for

incompressible turbulent flow shown in Figure 2.3.1. The variations in static

pressure, present across a section within the bend, extend for a diameter or

two into the straight pipes upstream or downstream. The constant pressure

gradient associated with fully developed flow in a straight pipe is not re-

established until fifty or more diameters downstream from the bend.

Fig. 2.3.1 Pressure Distribution in Horizontal Pipeline Containing a Bend

For most other pipework components, the variations in static pressure at

a cross section are far less marked than for bends, and the fully developed

velocity profile is recovered more quickly. For example, the flow recovers after

about five diameters downstream of a sudden enlargement according to Hall.

FLUID MECHANICS

7/21/2019 48967_02



78

When the flow upstream and downstream of a component is fullydeveloped, the component is said to be free of interference effects, since

its performance is independent of the flow beyond the regions of fully

developed flow. In the calculation of system pressure losses, it may be

necessary to approximate the real situation by using data obtained from

tests in interference-free flow. This approach should not lead to large

errors, provided that a spacer length of at least 5 diameters or, in the case

of bends, 10 diameters separates one component from another.

In many practical systems, interference exists between the componentsof the pipe system. This occurs when the regions of influence of two

components overlap. According to ESDI, the pressure loss through combinations

may be higher or lower than the sum of the losses of the components in

interference-free flow.

There are so many possible combinations of components that, at

present, the prediction of the performance of a system with interferences

cannot be made with any accuracy except in special circumstances. If the

interference in the system is confined to the interactions between a few

of the components for which data are available, the principles outlined

for interference-free flow can be applied by considering these components

as single entities. In some circumstances, it may be possible to deduce,

by broad physical arguments, that the interference effects will not be

important. Consequently, the performance of the system can be estimated

quite accurately using data obtained under interference-free conditions.

Alternatively, there may be one or two particularly large sources of pressure

loss in the system, and a much lower order of accuracy is acceptable for the

other components—if their magnitudes are known with reasonable accuracy.

However, these situations must be regarded as special cases. With the

present state of knowledge, it is necessary to resort to model tests or

computer analysis if an accurate assessment is required of the performance

of a duct flow system in which interferences occur.

Extensive data for loss coefficients of different components in pipe

and duct systems is presented in the literature in works by Miller, Holms,

Crane, Idelchik, and Fried.

7/21/2019 48967_02


Abrupt contractions and expansionsHead or mechanical energy losses occur at abrupt changes in duct flow

cross-sectional areas. Consider incompressible flow in the duct shown in

Figure 2.3.2, which includes a contraction at the inlet and a sudden enlargement

at the outlet.

Fig. 2.3.2 Static Pressure Distribution in a Duct

From Equation 2.3.2 for uniform velocity distributions, the total inlet

pressure drop due to a reduction in the flow area resulting in an acceleration

of the flow and a loss due to separation of the boundary layer can be

expressed as

=- 2/v K 22c ρ( )2/v + p 2

22 ρ( )2/v + p 211 ρ

FLUID MECHANICS

7/21/2019 48967_02



80

The static pressure drop is

(2.3.5)

where

σ21 = A2/ A1

K c = the contraction coefficient containing the irreversible loss

At the outlet of the duct, there will be a rise in static pressure owing to

the increase in flow area, whereas a loss will occur due to boundary layer

separation and momentum changes following the abrupt expansion.

The resultant change in pressure is

(2.3.6)

where

K e = expansion coefficient

σ34 = A3/ A4

The loss coefficients K c and K e refer to the kinetic energy of the flow in the

smaller cross-sectional area. For highly turbulent flow, Kays expressed thesecoefficients using the following two equations:

(2.3.7)

and

Ke

= (1 – σ34

)2 (2.3.8)

( )σσσ c 22

c c c /1-1=/1+/2-1= K

[ ])-(1- K 2)/v ( p- 2

34e2343 σρ=

[ ] K +)-(12) / v (= p- c 221

2221

σρ

7/21/2019 48967_02


Equations 2.3.7 and 2.3.8 apply to square-edged abrupt changes in crosssection for single tubes or for tube bundles.

The contraction ratio σc = Ac / A2 , shown in Figure 2.3.3 for two-dimen-

sional and three-dimensional circular contractions, is usually determined

experimentally according to Weisbach. The minimum area of the jet between

sections 1 and 2, Ac , is referred to as the vena contracta.

Fig. 2.3.3 Contraction Ratio for Round Tubes and Parallel Plates

The curves shown in Figure 2.3.3 are approximated by the following

empirical relations for round tubes

(2.3.9)

and for parallel plates according to Rouse.

(2.3.10)

σσσ6

21

5

21

4

21 3.558944+5.963169-2.672041+

σσσσ 321

22121c 0.4082743+0.336651-0.04566493+0.6144517=

σσσσ 32122121c 0.51146+0.26095-0.13318+0.61375=

FLUID MECHANICS

7/21/2019 48967_02



82

It is possible to reduce the contraction loss coefficient for a tube significantlyby rounding off the inlet edge. This is illustrated in Fig. 2.3.4 and Table 2.3.1,

according to Fried.

Fig. 2.3.4, Table 2.3.1 Contraction Loss Coefficient for Rounded Inlet

For tubes penetrating into a manifold, Fried shows a higher loss coefficient

is applicable (Fig. 2.3.5 and Table 2.3.2).

Fig. 2.3.5, Table 2.3.2 Contraction Loss Coefficient for Penetrating Tube

7/21/2019 48967_02


Example 2.3.1

A heat exchanger bundle consists of tubes having an inside diameter of 22.09 mm

and a total length of 15,024 mm. The tubes are arranged in a staggered pattern

and are welded into tube sheets (Fig. 2.3.6).

Water at 52.5 °C flows through each tube at a rate of 0.4015 kg/s. The tube

inlet has a square edge. Determine the difference in static pressure between theheaders for a smooth tube and for the case where ε/d i = 10-3.

Fig. 2.3.6 Tube Dimensions and Layout

Solution

The thermophysical properties of water are listed in appendix A. Evaluate

properties at 52.5 °C (325.65 K).

Density of water from Equation A.4.1:

ρw = (1.49343 x 10–3 – 3.7164 x 10–6 x 325.65 + 7.09782 x 10–9 x 325.652

– 1.90321 x 10–20 x 325.656)–1 = 986.9767 kg/m3

FLUID MECHANICS

7/21/2019 48967_02



84

Dynamic viscosity of water from Equation A.4.3:

µw = 2.414 x 10–5 x 10247.8/(325.65 – 140) = 5.21804 x 10–4 kg/ms

The Reynolds number for the water flowing in the tube is

The flow in the tube is turbulent, e.g., transition zone. The mean velocity of

the water in the tube is determined from

The frictional pressure drop may be determined using Equation 2.2.2.

For a smooth tube, the friction factor follows from Equation 2.2.10:

f D = (1.82 log10 44349.9 – 1.64)–2 = 0.021516

The frictional pressure drop is thus

For the rough pipe, it follows from Equation 2.2.14 that

f D = 0.3086/[log10 {6.9/44349.9 + (10-3/3.7)1.11 }]2 = 0.0241234

The resultant frictional pressure drop is

m N/9122.122

061441.x986.9767

22.09

15024 0.0241234 p

2

2

f

∆ ==

m N/8136.152

061441.x986.9767

22.09

15024 0.021516 p

22

f

∆ ==

2v d

L f = p

2

w

i

Df

ρ

∆

m/s1.06144 = )(0.02209x986.9767

0.4015x4 =

d

m4 = v

22iw πp q

44,349.95.21804x 22.09x

10x0.4015x4

d

m4d v Re

7

w iw

iw

πµπµρ

= = = =

7/21/2019 48967_02


For the particular tube layout, the area ratio for the entering water stream is

σ = π x 22.092/(4 x 58 x 50.22) = 0.13158

whereas the jet contraction ratio is, using Equation 2.3.9,

σc = 0.61375 + 0.13318 x 0.13158 – 0.26095 x 0.131582 + 0.51146 x

0.131583 = 0.628

For turbulent flow, the inlet contraction loss coefficient may be approximated

by Equation 2.3.7, i.e.,

K c = 1 – 2/0.628 + 1/0.6282 = 0.351

The static pressure drop at the inlet to the tube follows from Equation 2.3.5:

∆ pi = 0.5 x 986.9767 x 1.061442 [(1 – 0.131582) + 0.351] = 741.5 N/m2

The outlet expansion loss coefficient is approximated by Equation 2.3.8:

K e = (1 – 0.13158)2 = 0.7542

The static pressure drop at the outlet of the tube follows from Equation 2.3.6:

∆ pe = 0.5 x 986.9767 x 1.061442 [0.7542 – (1 – 0.131582)] = –127.0 N/m2

For a smooth tube, the total static pressure differential between the headers is

∆ p = 8136.15 + 741.5 – 127.0 = 8751 N/m2

For the rough tube, the pressure drop is

∆ p = 9122.12 + 741.5 – 127.0 = 9737 N/m2

FLUID MECHANICS

7/21/2019 48967_02



86

Reducers and diffusersWhen the duct flow area is reduced gradually (Figs. 2.3.7a and b), the

number of velocity heads lost is very small. Based on the smaller flow area, a

loss coefficient of K red = 0.04 or less is commonly quoted.

Fig. 2.3.7 Reducers (a) (b) (c)

For the conical reducer shown in Figure 2.3.7(c), the loss coefficient

based on the smaller area may be obtained from the following equation

according to Fried:

(2.3.11)

where

σ21 = A2/ A1

θ is in radians

The loss coefficient is based on the velocity at 2.

( )θθπθ 20-4-8x 23

( )0.00745-0.00444+0.00723-0.0224+0.0125-= K 21221

321

421red σσσσ

a b c

7/21/2019 48967_02


Whenever it is necessary to increase the flow area of a pipe gradually, theconical diffuser shown in Figure 2.3.8 may be used.

Fig. 2.3.8 Conical Diffuser

Ideally, in the absence of losses, the total pressure remains constant, i.e.,

and the pressure recovery is

(2.3.12)

where

σ12 = A1 /A2

id = ideal conditions

In practical diffusers, only a part of this pressure recovery is possible, and

a diffuser efficiency is defined as

(2.3.13)( ) 2/-1v

p- p

p- p

p- p2

1221

12

1id 2

12dif

σρη = =

( ) ( ) 2/-1v =2/v -v = p-2

1221

22

211id 2 σρρ

2/v + p=2/v +211

22id 2 ρρ

FLUID MECHANICS

7/21/2019 48967_02



88

For relatively small expansion angles, the diffuser efficiencies have been

determined by Patterson, whose results are shown in Figure 2.3.9.

The loss coefficient of a diffuser with uniform inlet and outlet flow is

(2.3.14)

Fig. 2.3.9 Conical Diffuser Efficiencies

Substitute Equation 2.3.13 into Equation 2.3.14 and find

(2.3.15)

For practical applications, Daly suggests it may be convenient to employ

Figure 2.3.10.

( )( )ση 212dif dif -1-1= K

( ) ( )2/v

2/v + p-2/v + p =

2/v

p- p = K 2

1

222

211

21

2t 1t dif

ρ

ρρ

ρ

7/21/2019 48967_02


Fig. 2.3.10 Losses in Duct Diffusers

The losses for open-outlet diffusers are shown in Figure 2.3.11. A uniform

approach velocity, such as a venturi nozzle flow meter, allows more rapid

expansion and lower loss, shown by the broken lines. Extensive data on flat

and conical diffusers is presented by Runstadler et al.

Fig. 2.3.11 Losses in Open Outlet Diffusers

FLUID MECHANICS

7/21/2019 48967_02



90

Three-leg junctionsPressure loss data for dividing flows through planar three-leg junctions

are cited in various references such as Engineering Sciences Data Unit (ESDU),

Verein Deutscher Ingenieure-Wärmeatlas, the General Electric Fluid Flow

Data Book, and Miller.

The total pressure differences across pairs of inlet and outlet legs of a

junction (Fig. 2.3.12) are calculated from

pt 3 – pt 1 = K 31ρv 32/2 + f D3 L3ρv 3

2/2d 3 + f D1 L1ρv 12/2d 1 (2.3.16)

and

pt 3 – pt 2 = K 32ρv 32/2 + f D3 L3ρv 3

2/2d 3 + f D2 L2ρv 22/2d 2 (2.3.17)

Fig. 2.3.12 Variation of Total Pressure in the Vicinity of a Junction

The last two terms on the right side of Equations 2.3.16 and 2.3.17 are

the straight-pipe friction losses over lengths L3, L1 , and L2.

The loss coefficient for a 90° junction, K j90 = K 31, with Re3 ≥ 2 x 105

between leg 3 and leg 1 is given in Figure 2.3.13 for square corners (r 31 = r 12 = 0)

as a function of A1/ A3 and V 1/V 3.

7/21/2019 48967_02


Fig. 2.3.13 Loss Coefficient for a 90° Junction with Square Corners

With rounded corners, the loss coefficient may be reduced, i.e.,

(2.3.18)

for r 12/d 1 < 0.15 and r 31/d 1 < 0.15, and

(2.3.19)

for r 12/d 1 > 0.15 and r 31/d 1 > 0.15.

The information strictly applies to junctions where the inlet flow is fullydeveloped and where there is a long downstream duct length. However, in practice

it can be applied without significant error when there are 15 or more equivalent

diameters upstream and at least 4 diameters downstream of the junction. It is

possible to reduce the loss coefficient by installing suitable guide vanes.

A / A

V / V 0.45- K = K = K

31

31

2

90 jr 90 jr 31

d

r

A / A

V / V 0.26 –

d

r

A / A

V / V 0.9- K = K = K

1

12

0.5

31

31

2

1

31

0.5

31

31

2

90 jr 90 jr 31

FLUID MECHANICS

7/21/2019 48967_02



92

The loss coefficient between leg 3 and 2 can be assumed to be unaffect-ed by the geometry of leg 1 and is given in Figure 2.3.14 as a function of the

flow ratio only. There is no significant change in the loss coefficient, K 32, due

to rounding of the junction corners.

Fig. 2.3.14 Loss Coefficient K32 for a 90° Junction

In the case of a square-edged T-junction, ( K T = K 31, the loss coefficient is

shown in Figure 2.3.15.

Fig. 2.3.15 Loss Coefficient for a T-Junction with Square Corners

7/21/2019 48967_02


Curved ducts or bendsA few common types of curved ducts or bends are shown in Figure 2.3.16.

Pressure loss data for flow through such bends are available from ESDU.

Fig. 2.3.16 Bends (a) Circular-Arc Bend (b) Single Miter Bend (c) Composite Miter Bend

The pressure loss due to a square bend and, in particular, a miter bend

may be reduced by fitting guide vanes. It is common practice to use a num-

ber of guide vanes in a cascade in a miter bend. The vane geometry is not

fixed by the bend geometry, and a number of designs exist. One example of

such a miter bend is shown in Figure 2.3.17.

Fig. 2.3.17 Miter Bend with Cascade and Circular-Arc Guide Vanes

FLUID MECHANICS

a b c

7/21/2019 48967_02



94

Insufficient systematic data exist to provide detailed information onpressure losses in miter bends with cascades shown in Figure 2.3.18. The

probable range of the loss coefficient, K bm, is between 0.15 and 0.4, the lower

values requiring very careful construction for their achievement. These values

compare with a loss coefficient equal to approximately 1.1 for a similar bend

without a cascade.

Fig. 2.3.18 Miter Bends Kbm = 0.28, Kbm = 0.25, Kbm = 0.4 (Jorgensen)

2.4 Manifolds

The design of manifolds for the distribution or division of a fluid

stream into several branching streams is of importance in the design of

different types of heat exchangers. The same is true for the formation of

a single main stream by the collection or confluence of several smaller

streams. A manifold basically consists of a main channel, a header, towhich several smaller conduits, tubes, or laterals are attached at right

angles. Manifolds commonly used in flow distribution systems can be

classified in the following five categories (Fig. 2.4.1).

• simple distributing or dividing

• collecting or combining

• cocurrent flow or parallel flow

• countercurrent flow or reverse flow

• mixed flow or combined flow

7/21/2019 48967_02


The cocurrent or countercurrent flow configurations are also referred toas Z or U type heat exchangers.

Fig. 2.4.1 Types of Manifolds

FLUID MECHANICS

7/21/2019 48967_02



96

Often, the objective of the design is to provide equal flow rates throughthe branches or laterals. This can be achieved if the cross-sectional area of the

header is designed such that the fluid velocity and the pressure in the header

remain constant.

The variations in fluid pressure in the header are due to frictional effects

and changes in momentum. The frictional loss is always in the direction of

the flow. The momentum changes cause an increase in pressure in the direction

of flow in the distributing header and a decrease in pressure in the direction

of flow for the collecting header.

By applying the energy equation to the flow in the header, Miller

and Hudson quantified the flow and pressure distribution in the header.

Enger and Levy, Keller, Acrivos et al., Markland and Bassiouny, and

Martin based their continuous mathematical model on local momentum

balance considerations in the header. Discrete mathematical models

based on local momentum balances are derived by Kubo and Euda,

Majumdar and Datta, and Majumdar. Bajura and Bajura and Jones

derived continuous mathematical models based on integral momentum

balances of the fluid in the header. They integrated the momentum

equation in vector form over a control volume to quantify the flow and

pressure distribution. A discrete mathematical model based on an integral

momentum balance is due to Nujens.

To find the approximate flow distribution and pressure change in a

manifold, consider a section of a header of uniform cross section, Ah, and

wetted perimeter, P eh, fitted with closely spaced laterals shown in Figure

2.4.2.

Fig. 2.4.2 Header Control Volume (a) Dividing Header (b) Combining Header

7/21/2019 48967_02


By applying a mass balance to an incompressible fluid in an elementarycontrol volume, Ah∆x, in a header, find

ρ Ah(dv h/dx)∆x + ρv A

= 0 (2.4.1)

where the subscripts h and

refer to the header and the laterals, for a

header of length Lh and n

laterals, ∆x = Lh/n.

The approximate momentum balance applicable to the control volumemay be written as

(2.4.2)

where the factor c = 1 for v h > 0 and c = -1 for v h < 0. The momentumcorrection factors, αmh and αm

provide for nonuniformities in local header

and lateral inlet velocity distributions.

Using Equations 2.2.1 and 2.2.2, the header shear stress can be expressed

in terms of the Darcy friction factor, i.e.,

(2.4.3)

Substitute Equations 2.4.1 and 2.4.3 into Equation 2.4.2, and simplify to find

(2.4.4)

where the overall momentum correction factor αmh = 2αmh - αmand

d e = 4 Ah/ P eh. Bajura et al. graphically present some values for overall momentum

correction factors.

0=d 2v f c

+dxdv v +

dx

dp

e

2h Dh

hmhh ρα

8/v f =2h Dh ρτ

( )[ ] Av -v Av + Ax/dxdv +v = h2hmhhmhhh

2mh ραραα δρ

( )[ ] x P c - Axdx/dp+ p- A ehhhhhhh ∆∆ τ

FLUID MECHANICS

7/21/2019 48967_02



98

If a lateral is located between a dividing and a combining header, the staticpressure difference between these headers can be expressed as

(2.4.5)

In this equation, K i and K o are the lateral inlet and outlet loss coefficients

while αed

and αec

are the header energy correction factors. In a conventional

single phase heat exchanger, the lateral velocity v i = v

o; however, in a

condenser, the inlet velocity is greater than the outlet velocity.

Bassiouny and Martin, after making some simplifying assumptions,

present solutions for the manifold momentum equations for single phase

flow in U or Z type configurations.

The more complicated problem of flow distribution in an air-cooled

steam condenser was analyzed by Zipfel and Kröger. While the lateral inlet

loss in a single-phase heat exchanger is usually negligible, this is not the casein a condenser. They show, under certain circumstances, backflow of vapor

can occur in some of the laterals.

2.5 Drag

Drag is defined as the force component, parallel to the relative approach

velocity exerted on the body by the moving fluid. Mathematically it can be

expressed as

F D = C D A ρ v 2/2 (2.5.1)

where

A = the characteristic projected area normal to the flow

ρv 2/2 = the dynamic pressure of the main stream

C D = the drag coefficient

[ ] ( ) /v v /v + K +2 2/v K +v /v - K = p-2o

2o

2hc ec o

2ii

2hd ed ihc hd ραρα

7/21/2019 48967_02


The drag coefficient is found by dimensional analysis to be a function of thegeometrical configuration of the immersed body, the Reynolds number, the

turbulence characteristics of the incoming free stream, and the surface roughness

of the body. Experimental data on drag coefficients versus Reynolds numbers for

several different two-dimensional bodies are plotted in Figure 2.5.1.

Fig. 2.5.1 Coefficient of Drag for Two-Dimensional Bodies

A summary of results on forces associated with flow across circular

cylinders is reported by the Engineering Sciences Data Unit (ESDU). As

shown in Figure 2.5.1, the drag coefficient remains almost constant at a

value of 1.2 for 104 < Re < 2 x 105 in the case of infinitely long cylinders.

The drag coefficient is not significantly affected by surface roughness or

free-stream turbulence for Re < 3 x 104. Above this value, these effects do

become important according to ESDU.

FLUID MECHANICS

7/21/2019 48967_02



100

If the cylinder axis is rotated through an angle θ D with respect to thenormal approach flow direction, the drag coefficient may be calculated

approximately from

C Dθ = C D (cos θ D)3 (2.5.2)

for 0° < θ D < 45°.

Empirical equations based on experimental data have been developed byHoerner for predicting the drag coefficient about elliptical sections for different

Reynolds numbers, i.e.,

C D = 2.656 (1 + a/d ) Re–0.5 + 1.1 (d /a) (2.5.3)

for 103 < Re < 106 and where a and d are the dimensions of the major and

the minor axes. This relationship is plotted in Figure 2.5.1.

If the elliptical section is inclined relative to the flow, the drag coefficient

is corrected in the same way as the circular cylinder, i.e. Equation 2.5.2.

The drag coefficient for an infinitely long square section is C D = 2 for

Re > 104 (Fig. 2.5.1). An extensive study on square and rectangular sections

is presented by the ESDU.

The drag coefficients for other two-dimensional structural shapes are listed

in Table 2.5.1 according to Simiu.

Table 2.5.1 Two-Dimensional Drag Coefficients for Structural Shapes

7/21/2019 48967_02


The drag coefficient for all objects with sharp corners is independent of the Reynolds number because the separation points are fixed by the geometry

of the object.

A more detailed list of drag coefficients is presented by Sachs. Data for

estimating the mean fluid forces acting on lattice frameworks are presented

in ESDU Item Number 75011.

According to Turton and Levenspiel, the drag coefficient for a sphere is

given by

C D = 24(1 + 0.173 Re0.657)/ Re + 0.413/(1 + 16300 Re-1.09) (2.5.4)

for Re ≤ 200000.

2.6 Flow through Screens or GauzesA screen may be defined as a regular assemblage of elements forming a

pervious sheet, which is relatively thin, in the direction of flow through the

screen. Screens of various types may be installed in systems to:

• remove foreign objects from the fluid stream

• protect equipment (e.g., against hailstones)• reduce fouling or clogging in heat exchangers

• smooth flow

• produce turbulence

In such cases, the prediction of the total pressure loss caused by the

screen is of interest.

An expression for the loss coefficient across a plane screen has been

deduced by Cornell for a compressible fluid. Since the losses across screens

FLUID MECHANICS

7/21/2019 48967_02



102

employed in air-cooled heat exchangers are normally small, the loss coefficientcan be expressed approximately in terms of the pressure difference across the

screen and mean, free, stream velocity, i.e.,

K s = 2( pt 1 – pt 2)/ρv 2 = 2( p1 - p2)/ρv 2 (2.6.1)

A round-wire screen or gauze of square mesh shown in Figure 2.6.1 is

usually specified by the mesh, defined by the number of openings per unit

length, 1 /P s, and by the diameter, d s, of the wires.

Fig. 2.6.1 Geometry of a Square-Woven Screen

An example of standard mesh data is listed in Table 2.6.1.

7/21/2019 48967_02


Table 2.6.1 Standard Mesh Data

The porosity of the screen is defined as

βs = area of holes/total area = (1 – d s / P s)2 (2.6.2)

According to Simmons, the following equation holds for a screen placed atright angles to an airstream at velocities above 10 m/s under ambient conditions:

K s = (1 – βs)/βs2 (2.6.3)

This equation can be recommended for application in the case of most

screens of practical interest where screen Reynolds numbers exceed 300.

FLUID MECHANICS

7/21/2019 48967_02



104

The screen Reynolds number is defined as

Res = ρvd s /(βs µ) (2.6.4)

According to Wieghardt, the loss coefficient may be expressed as

K s = 6(1 – βs)βs –2 Res

–0.333 (2.6.5)

for 60 < Res < 1000.

In Figure 2.6.2, Equations 2.6.3 and 2.6.5 are compared with the

experimental results obtained by various investigators. Since the screen

geometry can have a significant influence on the loss coefficient, specific

tests should be performed when these losses are of importance.

Fig. 2.6.2 Screen Loss Coefficient

7/21/2019 48967_02


For cases where ds/Ps << 1, the effective drag force per screen area may beexpressed as follows. This corresponds to one screen opening and uses

Equation 2.5.1

F D/ A = F D/[d s (2 P s – d s)] ≈ C D ρ v 2/2 (2.6.6)

Using Equation 2.6.1, the loss coefficient is

K s = 2F D/ ( P s2 ρ v 2) (2.6.7)

Substitute Equation 2.6.6 into Equation 2.6.7, and find

K s = C D (2 d s / P s – d s2/ P s2) = C D (1 – βs) (2.6.8)

Equation 2.6.8 can also be written as

K s βs2/ (1 – βs) = C D βs

2 ≈ C D (2.6.9)

for d s / P s << 1.

With the drag coefficient, C D, known for an infinitely long cylinder, the

approximate relationship (Eq. 2.6.9) can be applied over a wide range of

Reynolds numbers (Fig. 2.6.2).

Fan guards consisting of expanded metal or wire woven mesh are

required in most air-cooled heat exchangers. According to the American

Petroleum Institute, the recommended openings in the petrochemical industry

for woven mesh are usually not more than 2600 mm2 with a wire diameterof not less than 2.8 mm or 12 Birmingham wire gage (BWG).

FLUID MECHANICS

7/21/2019 48967_02



106

2.7 Two-Phase Flow

Two-phase flows are found in many industrial processes. Gas-liquid flow,

which occurs in condensation and boiling operations, is of particular interest

in the design of heat exchangers. Generally, these flows are more complicated

physically than single-phase flow. In addition to the usual inertia, viscous,

and pressure forces present in single-phase flow, two-phase flows are also

affected by interfacial tension forces, the wetting characteristics of the liquid

on the tube wall, and the exchange of momentum between the liquid andthe vapor phases in flow.

Two-phase flow patterns

In gas-liquid or vapor-liquid flow, the two phases can adopt various

geometric configurations known as flow patterns or flow regimes. Many flow

patterns are present during flow in tubes, including those observed by bothCollier and Hewitt, and a few of the more important ones will be described.

The total mass flow rate through a tube is equal to the sum of the mass

flow rates of gas, m g , and liquid, m.

m = m g + m

(2.7.1)

The ratio of the mass flow rate of gas to the total mass flow rate is known

as the dryness factor or quality, i.e.,

(2.7.2)

where G is the mass flux based on the total cross-sectional area of the

tube—also referred to as the superficial mass velocity. The subscript g refers to

the gas.

The void fraction, α is defined as the ratio of the gas flow cross-section-al area, A g , to the total cross-sectional area, Ats.

(2.7.3) A/ A= ts g α

/GG=/mm=x g g

7/21/2019 48967_02


When both phases are flowing upwards in a tube, cocurrent flow,possible flow patterns obtained are shown in Figure 2.7.1 for increasing

gas flow or quality.

Fig. 2.7.1 Flow Patterns in Vertical Upflow in Tube

• Bubble flow. At low gas or vapor flow rate and low to moderate liquid

flow rates, the gas phase tends to become distributed throughout theliquid continuum as bubbles.

• Slug flow (also called plug flow). If the gas flow rate of a dispersed

bubble flow is increased, the bubbles coalesce to form large bullet

shaped bubbles, which approach the tube diameter in size. The

bubbles are separated by slugs of liquid containing small entrained

gas bubbles.

• Churn flow. With a further increase in gas flow rate, the liquid slugs

break down and give rise to this unstable flow of an oscillatory

nature according to Jayanti, Mao, and Hewitt.• Annular flow. At high gas velocities and low liquid flow rates, the

liquid travels partly as wavy annular film on the wall of the tube and

partly as small drops distributed in the gas, which flows in the

center of the tube.

FLUID MECHANICS

7/21/2019 48967_02



108

• Mist flow. At a gas velocity much higher than that of annular flow,there is an increase in the number of liquid droplets being sheared

away from the film flow until nearly all liquid is entrained in the gas

core flow.

More details of vertical cocurrent upward flow are given in the literature

from sources such as Oschinowo, Taitel, Hewitt, McQuillan, and Cindric.

Vertical cocurrent downward flow patterns have been described by

Oschinowo and Charles, Barnea et al., Crawford et al., and Mukherjee andBrill. A few of these patterns are shown in Figure 2.7.2 for increasing gas flow rate.

Fig. 2.7.2 Flow Patterns in Vertical Downflow in Tube

• Bubble flow. At very low gas flow rates and high liquid flow rates, the

gas phase flows as discrete bubbles in a downward flowing continu-

um. Shear stresses decrease from the tube wall to a minimum at thetube center, therefore the resultant imbalance in shear force acting

on each bubble causes it to migrate towards the center of the tube.

7/21/2019 48967_02


• Slug flow. At higher gas flow rates, the dispersed bubbles coalesce toform larger bubbles separated by liquid slugs.

• Falling film flow . At low liquid and gas flow rates, the liquid flows in

the form of a thin film. The gas core flow entrains very few or no liq-

uid droplets. There is a tendency for dry spots to develop on the tube

wall. This flow pattern is sometimes classified as an annular flow

although its nature is different.

• Churn flow. If the gas flow rate in a slug flow pattern is increased, a

point is reached where the high gas concentration in the liquid slug

causes the slug to collapse. As this happens, the liquid slug falls then

regroups with other liquid drops until it is able to bridge the tube

again. The mixture is turbulent but much less agitated than for ver-

tical upward churn flow.

• Annular flow. At high gas flow rates and moderate liquid flow rates,

the liquid flows as an annular film while the gas forms a continuous

core flow at the center of the tube. The high velocity gas core flow

may entrain a portion of the wavy liquid film.

• Mist flow. At very high gas flow rates, all or most of the liquid film isentrained into the gas core flow in the form of small droplets.

The results of many studies of countercurrent two-phase flows in vertical

tubes have been reported by sources such as Wallis and Taitel. Flow patterns

include bubble, slug, and annular flow or falling film.

Horizontal two-phase flows in tubes generate patterns which are more

complex than vertical flow patterns because the gravitational influence is no

longer in the axial direction of flow. The horizontal flow pattern descriptionshown in Figure 2.7.3 are combinations of the categorization of a number of

researchers such as Taitel, Hewitt, Mukherjee, and Cindric.

• Stratified flow. This flow pattern occurs at low gas and liquid volume

flow rates. The liquid phase flows along the bottom of the tube, and

its surface is relatively smooth—also referred to as stratified smooth

flow. Increasing gas flow causes the liquid surface to become wavy—

also referred to as stratified wavy flow.

• Bubble flow. Bubble flow occurs at low gas flow rates and moderate

liquid flow rates. Bubbles tend to concentrate near the top of the

tube due to buoyancy forces.

FLUID MECHANICS

7/21/2019 48967_02



110

• Slug flow. At higher gas flow rates than for dispersed bubble flow, coa-lescence of bubbles occurs forming large bubbles separated by slugs

of liquid.

• Annular flow. At high gas flow rates, shear forces at the vapor-liquid

interface may be large compared to gravitational forces. The interaction

between these forces and the waves on the liquid surface causes the

liquid to flow up the tube wall until the liquid film flows as an

eccentric annulus according to James and Lin. The effect of gravity

causes the liquid film to be thicker at the bottom than at the top of

the tube.

• Mist flow. At very high gas flow rates, all the liquid film tends to be

entrained into the gas core as small droplets or mist.

Fig. 2.7.3 Flow Patterns in Horizontal Flow in Tube

7/21/2019 48967_02


Flow patterns for inclined tubes are a superposition of the flow patternsfor horizontal and vertical flows. However, a quantitative prediction of flow

pattern transition for inclined tubes is more difficult according to Hewitt.

More detailed studies of two-phase flows in inclined tubes have appeared

in the literature from authors such as Spedding, Crawford, Mukherjee,

Stanislav, and Barnea.

Numerous attempts have been made to present different flow patterns

on a two-dimensional graph having different areas that correspond to the

different flow patterns. Such flow pattern maps may use the same axesfor all flow patterns and transitions, or they may employ different axes

for different transitions.

Baker was the first to develop a horizontal tube-side flow pattern map

that could be used for any fluid. His map was subsequently modified by Scott

(Fig. 2.7.4) and further evaluated by Bell et al. The map is plotted in terms of

G g /λ and Gψ where G g = m g / Ats and G

= m

/Ats are the superficial mass

velocities of the gas and liquid, and the factors λ and ψ are given by

(2.7.4)

(2.7.5)

where

σ = surface tension

a = physical properties of air

w = physical properties of water

= properties of the liquid flowing in the tube

g = properties of the gas flowing in the tube

ρw

= 1000 kg/m3

ρa = 1.23 kg/m3

( ) ( )( )[ ]///= w

2

w

0.333

w ρρµµσσψ

( )( )[ ]//= w a g

0.5ρρρρλ

FLUID MECHANICS

7/21/2019 48967_02



112

Fig. 2.7.4 Modified Baker Flow Pattern Map for Horizontal Flow in a Tube

The dynamic viscosity of water is taken as µw = 10-3 kg/ms, and the

surface tension is σw = 0.072 N/m. The Baker map works reasonably well for

water/air and oil/gas mixtures in small diameter (< 0.05 m) tubes.

One of the disadvantages of the Baker map is the parameters are

dimensional and empirical, so it is not possible to relate the map boundaries

theoretically to any known physical characteristics of the flow. Taitel and

Dukler approach the flow regime transitions theoretically and present a more

complex type of flow pattern map. Other graphical presentations have

been proposed by Breber, with some based on studies conducted during

condensation of vapors in horizontal tubes.

The Hewitt and Roberts map shown in Figure 2.7.5 is a widely used chart

for vertical upflow in a tube, while a map proposed by Oschinowo andCharles is applicable to downflow in a vertical tube. The Hewitt and Roberts

map works reasonably well for water/air and water/steam systems, again in

7/21/2019 48967_02


small diameter tubes. Note that wispy annular flow is a subcategory of annularflow, which occurs at high mass flux when the entrained drops are said to

appear as wisps or elongated droplets.

Fig. 2.7.5 Hewitt and Roberts Map for Vertical Upflow in a Tube

Maps for patterns in inclined tubes have been presented by various

authors such as Spedding, Crawford, Mukherjee, Stanislav, and Barnea.

A systematic and practical approach for determining changes in pressure

in two-phase flows is presented by Carey.

FloodingThe flooding process is illustrated in Figure 2.7.6, and a description

found in Hewitt and Bankoff is summarized in the following.

FLUID MECHANICS

7/21/2019 48967_02



114

Fig. 2.7.6 Flow Regime Transitions during the Flooding Process inside a Vertical Tube (a-h)

Liquid enters the top of the tube through a porous section and isremoved through a porous section at the bottom of the tube. At low gas flow

rates, a stable counterflow exists (Fig. 2.7.6a). As the gas flow rate is increased,

the interface becomes wavy, liquid is entrained, and the film starts creeping

up past the liquid inlet (Fig. 2.7.6b). This flow transition is defined as flooding

and is also sometimes referred to as the onset of flooding or limiting condition

for countercurrent flow . Eventually, liquid flows up past the liquid inlet

(Fig. 2.7.6c), and a state of partial liquid delivery exists. With a further

increase in gas flow, the liquid flow below the inlet porous section changes

to a climbing film flow (Fig. 2.7.6d), and a state of cocurrent annular or churnupward flow above the liquid inlet porous section is reached (Fig. 2.7.6e).

When the gas flow rate is reduced, the liquid begins to creep below the

liquid feed (Fig. 2.7.6f). This point is known as flow reversal. A further

decrease in gas flow rate results in a state of simultaneous climbing and

falling film flow (Fig. 2.7.6g). Finally, the initial state of countercurrent flow

is obtained (Fig. 2.7.6h). The last transition has been termed the deflooding

point by Clift et al.

There are various mechanisms by which flooding is said to occur. They

fall into two fundamental categories, namely: film flow theory assuming a

smooth gas-liquid interface and flow instability/wave growth theory.

7/21/2019 48967_02


McQuillan and Whalley studied air-water flow experiments in vertical tubeswith porous wall liquid injection and removal. They concluded the following

concerning flooding:

• The gas-liquid interface is wavy when the flooding point is

approached.

• These waves grow in amplitude as they travel downwards.

• The velocity of the falling waves decreases as they fall. At the floodinggas flow rate, the wave reaching the lower porous wall becomes

stationary and grows rapidly in size. The result is a large disturbance

wave, which moves upwards and causes flooding.

• Prior to the formation of the large disturbance wave, there is very

little entrainment of liquid droplets into the gas stream.

• For an airflow rate just below the flooding rate, an artificially injected

wave grows to form a disturbance wave, which is indistinguishable

from the flooding disturbance.

It is known that the geometry of liquid and gas entry can affect the gas

velocity at which flooding occurs. According to Hewitt, a square edged gas

inlet introduces more turbulence than a rounded inlet, promotes wave

growth on the liquid film, and reduces the gas velocity where flooding

occurs. If the turbulence level is low, the tube length has an effect on the gas

velocity necessary to initiate flooding. In longer tubes, Whalley found the

liquid waves have more time to build up, so the flooding occurs at lower values

of the gas velocity.

Reviews of the flooding literature are presented by numerous researchers

such as Tien, McQuillan, Bankoff, and Stephan. There tends to be a consid-

erable amount of scatter in the available data. In part this can be ascribed to

the fact that different definitions for flooding exist.

Many correlations have been proposed for predicting the onset of flooding .

The method based on the semiempirical correlation of Wallis is widely used

to predict flooding, i.e.,

(2.7.6)a= Fr a+ Fr 20.25 D1

0.25 Dg

FLUID MECHANICS

7/21/2019 48967_02



116

where the superficial densimetric Froude numbers for the gas and theliquid are defined as

(2.7.7)

(2.7.8)

The superficial velocities are defined as follows:

(2.7.9)

(2.7.10)

where

Ats = total cross-sectional area of the tube

The constants a1 and a2 depend upon the liquid inlet and outlet flow

condition and geometric characteristics. For turbulent air-water flow, the constant

a1 is close to unity while the approximate value chosen for the other constant

generally is as follows:

• a2 ≈ 1.00 for very smooth liquid inlet and outlet, e.g., porous, withminimal flow disturbance

• a2 ≈ 0.88 for very smooth liquid inlet and outlet, e.g., porous, with

high flow disturbance

• a2 ≈ 0.88 for rounded or tapered inlet and outlet flanges

• a2 ≈ 0.725 for sharp or square-edged inlet and outlet flanges

A significant increase in pressure drop is measured when flooding occurs.

Zapke and Kröger conducted adiabatic counterflow experiments to investigatethe effect of the duct geometry, duct inclination, and the liquid and gas physical

properties on flooding. Typical flooding data generated during the course of

the experiment for air-water flow in a flattened tube having a square-edged

(90°) inlet is shown in Figure 2.7.7 as a function of the duct inclination.

( ) A/m=/G=v tss ρρ

( ) A/m=/G=v ts g g g g gs ρρ

2 Fr D = ρ v s / [(ρ - ρ g )gd ]

2 Fr Dg = ρ g v gs / [(ρ - ρ g )gd ]

7/21/2019 48967_02


Fig. 2.7.7 Flooding Data for Air-Water Flow in an Inclined Duct

As a result of the different flow patterns encountered during counterflow

in inclined ducts as opposed to vertical ducts, a significant decrease in the

flooding gas velocity was observed as the duct inclination changed from just

off the vertical to the vertical. Separate correlations were developed by Zapke

for flooding in inclined and vertical ducts.

For inclined round and flattened tubes with a square-edged gas inlet,

where a3 and a4 are functions of the duct inclination ϕ, i.e.,

(2.7.11)

where ϕ is in degrees and

ϕϕϕ 3-42-24 10x5.3227-10x6.7058+1.9471-18.149=a

ϕϕϕ 3-62-4-3-23 10x1.9852-10x1.5183+10x4.9705+10x7.9143=a

( )Oh Fr a-exp a= Fr 0.20.6

D43 Dg

FLUID MECHANICS

7/21/2019 48967_02



118

The Ohnesorge number is a dimensionless parameter that accounts forthe effect of the liquid properties on flooding and in this case is defined as:

where

d e = the hydraulic diameter of the duct

In the case of flattened or elliptical tubes, Fr Dg is based on the inside

height, while Fr Dis based on the hydraulic or equivalent diameter.

Equation 2.7.11 is based on tests conducted within these ranges:

10 mm ≤ W (inside duct width) ≤ 20 mm (flattened tubes)

50 mm ≤ H (inside duct height) ≤ 150 mm (flattened tubes)

d = 30 mm (tubes)

2° ≤ ϕ ≤ 80° (duct inclination to the horizontal)

For vertical round and flattened tubes with a square-edged gas inlet,

(2.7.12)

within the ranges:

10 mm ≤ W (inside duct width) ≤ 20 mm (flattened tubes)

50 mm ≤ H (inside duct height) ≤ 100 mm (flattened tubes)

d i = 30 mm (tubes)

ϕ = 90°

( )Oh Fr 0.0055= Fr

0.30.2

D

-1

Dg

0.04Oh Fr 0.20.6

D ≤

( )σρ l d /=Oh e0.5

7/21/2019 48967_02


Limited extrapolation outside the geometric ranges specified is possible.

Pressure drop in two-phase flow

During two-phase flow in inclined ducts, the total pressure gradient,

dptp/dz, can be considered to be composed of three components arising from:

• friction losses, dpf /dz

• acceleration of the fluid, dpm /dz

• change of pressure or static head due to gravitational forces, dps /dz

A momentum balance on an element of the duct leads to the followingexpression for the various terms:

(2.7.13)

The normal procedure for using Equation 2.7.13 is to make some arbitrary

assumption about the nature of the flow. One such assumption would be to

treat the flow as a homogeneous mixture of the phases with constant phasecontent and constant and equal velocities for the two phases across the

whole duct. This is called the homogeneous model for two-phase flow. Usually

this model is not very satisfactory.

An alternative to the homogeneous model is to consider the flow to be

completely separated into two zones within the duct, one occupied by the

first phase and the other by the second phase.

In the separated flow models, a knowledge of phase content is essential

to calculate the accelerational and gravitational terms. The most common

way of correlating the frictional term is to express it in terms of a friction

dz

dp +

dz

dp +

dz

dp =

dz

dpsmf tp

0.02Oh Fr 0.110.30.2

D ≥≥

FLUID MECHANICS

7/21/2019 48967_02



120

multiplier . This is defined as the ratio of the frictional pressure gradient tothat for one or the other phase flow alone or for a flow at the same total mass

flux having gas or liquid properties.

(2.7.14)

(2.7.15)

where

ϕ, ϕ g , ϕ

o, and ϕ go = two-phase frictional multipliers

The literature contains a large number of two-phase flow correlations for

predicting the frictional pressure drop during cocurrent flow such as those

from Lockhart, Baroczy, Chisholm, and Friedel. Whalley tentatively recommends

the use of the following correlations:

• For µ/µ g < 1000, the Friedel correlation should be used.

• For µ/u g > 1000 and G > 100 kg/m2s, the Chisholm correlation

should be employed.

• For µ/µ g > 1000 and G < 100 kg/m2s, the Martinelli correlationshould be used.

The pressure gradient due to acceleration or deceleration effects may be

expressed as

(2.7.16)

ραρα g

222m x

+

-(1

)

)

x-(1

dz

d G=

dz

dp

/dzdp=/dzdp=/dzdp go

2

goo

2

of ϕϕ

/dzdp=/dzdp=/dzdp g 2

g 2

f ϕϕ

7/21/2019 48967_02


where the term in brackets is the local effective specific volume and x isthe vapor mass fraction. The void fraction, α, can be determined using

Premoli’s correlation.

The static or geodetic pressure gradient is given by

(2.7.17)

where ϕ is the angle of the tube with respect to the horizontal plane with

downward flow.

The homogeneous and separated flow models tend to give an inadequate

representation of many real two-phase flows, and the detailed physics of the

flow, including the flow pattern, are important. By employing flow pattern

related frictional pressure drop correlations, it is possible to predict, stepwise,the pressure gradient along the tube according to Olujic. A systematic and

practical approach for determining changes in pressure in two-phase flows is

presented by Carey.

Correlations on counterflow pressure drops are presented by a few

researchers such as Feind, Dukler, Bharathan, and Stephan.

The pressure drop for nonadiabatic flow is discussed in more detail in

chapter 3.

( )[ ] ϕραρα sin g -1+=dz

dp g

s

FLUID MECHANICS

7/21/2019 48967_02



122

References

Acrivos, A., B. D. Babcock, and R. L. Pigford, “Flow Distribution in Manifolds,”

Chemical Engineering Science, 10:112–124, 1959.

American Petroleum Institute (API) Standard 661, Air-Cooled Heat Exchangers for General

Refinery Services, API, Washington, 1978.

Andrews, J., A. Akbarzadeh, and I. Sauciuc, ed., Heat Pipe Technology , 156–163,Pergamon Press, 1997.

Bajura, R. A., “A Model for Flow Distribution Manifolds,” Transactions of the American

Society of Mechanical Engineers, Journal of Engineering for Power , 70-Pwr-3, 7–14, 1971.

Bajura, R. A., and E. H. Jones, “Flow Distribution Manifolds,” Transactions of the

American Society of Mechanical Engineers, Journal of Fluid Engineering , 98:656–666, 1976.

Bajura, R. A., V. F. Le Rose, and L. E. Williams, “Fluid Distribution in Combining,

Dividing and Reverse Flow Manifolds,” American Society of Mechanical Engineers,

73-PWR-1, 1973.

Baker, O., “Simultaneous Flow of Oil and Gas,” Oil and Gas Journal, 53:185–195, 1954.

Bankoff, S. G., and S. C. Lee, “A Critical Review of the Flooding Literature,” Multiphase

Science and Technology , 2:95–180, 1986.

Barnea, D., “A Unified Model for Predicting Flow Pattern Transitions for the Whole

Range of Pipe Inclinations,” International Journal of Multiphase Flow , 13-1:1–12, 1987.

Barnea, D., and N. Brauner, “Holdup of the Liquid Slug in Two-Phase Intermittent

Flow,” International Journal of Multiphase Flow , 11-1:43–49, 1985.

Baroczy, C. J., “A Systematic Correlation for Two-Phase Pressure Drop,” Chemical

Engineering Progress Symposium Series 62, 232–249, 1965.

Bassiouny, M. K., and H. Martin, “Flow Distribution and Pressure Drop in Plate Heat

Exchangers-I,” Chemical Engineering Science, 39-4:693–700, 1984.

Bassiouny, M. K., and H. Martin, “Flow Distribution and Pressure Drop in Plate Heat

Exchangers-II,” Chemical Engineering Science, 39-4:701–704, 1984.

Bell, K. J., J. Taborek, and F. Fenoglio, “Interpretation of Horizontal Intube

Condensation Heat Transfer Correlation with a Two-Phase Flow Regime Map,”

Chemical Engineering Progress Symposium Series, 66-102:159–165, 1970.

Benedict, R. P., Fundamentals of Pipe Flow , John Wiley, New York, 1980.

Bharathan, D. and Wallis, G. B., “Air-Water Countercurrent Annular Flow,”

International Journal of Multiphase Flow , 9-4:349–366, 1983.

Breber, G., Intube Condensation, Heat Transfer Equipment Design, ed. R. K. Shah, , E. C.

Subbarao, and R. A. Mashelkar, Hemisphere Publishing Co., New York, 1988.

7/21/2019 48967_02


Carey, V. P., Liquid-Vapor Phase-Change Phenomena, Hemisphere Publishing Corp.,Washington, 1992.

Chisholm, D., “Pressure Gradients Due to Friction During the Flow of Evaporating

Two-Phase Mixtures in Smooth Tubes and Channels,” International Journal of Heat

and Mass Transfer , 16:347–438, 1973.

Churchill, S. W., “Comprehensive Correlating Equations for Heat Mass and

Momentum Transfer in Fully Developed Flow in Smooth Tubes,” Industrial and

Engineering Chemistry, Fundamentals, vol. 16, no. 1, 1977.

Churchill, S. W., “Friction Equation Spans All Fluid Flow Regimes,” Chemical

Engineering , 84-24:91–92, 1977.

Cindric, D. T., S. L. Gandi, and R. A. Williams, “Designing Piping Systems for Two-

Phase Flow,” Chemical Engineering Progress, 51–55, March, 1987.

Clift, R., C. Pritchard, and R. M. Nedderman, “The Effect of Viscosity on the Flooding

Conditions in Wetted Wall Columns,” Chemical Engineering Science, 21:87–95, 1966.

Colebrook, C. F., “Turbulent Flow in Pipes, with Particular References to the Transition

Region Between the Smooth and Rough Pipe Laws,” Journal of the Institute of Civil

Engineering of London, 11:133–156, 1938–1939.

Collier, J. G., Convective Boiling and Condensation, McGraw-Hill Book Co., New York, 1981.Cornell, W. G., “Losses in Flow Normal to Plane Screens,” Transactions of the American

Society of Mechanical Engineers, 80-4:791–799, May 1958.

Crane Company, Flow of Fluids Through Valves, Fittings and Pipes, Technical Paper No.

410, 5th ed., Crane Co., New York, 1976.

Crawford, T. J., C. B. Weinberger, and J. Weisman, “Two-Phase Flow Patterns and Void

Fraction in Downward Flow, Part 1: Steady-State Flow Patterns,” International

Journal of Multiphase Flow , 11-6:761–782, 1985.

Daly, B. B., Woods Practical Guide to Fan Engineering , Essex Telegraph Press Ltd.,

Colchester, 1978.

Datta, A. B., and A. K. Majumdar, “A Calculation Procedure for Two-Phase Flow

Distribution in Manifolds with and without Heat Transfer,” International Journal of

Heat and Mass Transfer , 26-9:1321–1328, 1983.

Datta, A. B., and A. K. Majumdar, “Flow Distribution in Parallel and Reverse Flow

Manifolds,” International Journal of Heat and Fluid Flow , 2-4:253–262, 1980.

Deissler, R. G., “Turbulent Heat Transfer and Friction in the Entrance Regions of

Smooth Passages,” Transactions of the American Society of Mechanical Engineers,

1221–1233, November 1955.

Derbunovich, G. Q., A. S. Zemskaya, Ye. U. Repik, and Yu. P. Sosedko, “Optimum Wire

Screens for Control of Turbulence in Wind Tunnels,” Fluid Mechanics-Soviet

Research, 10-5:136–147, September–October 1981.

FLUID MECHANICS

7/21/2019 48967_02



124

Du Plessis, J. P., and M. R. Collins, “A New Definition for Laminar Flow EntranceLengths of Straight Ducts,” South African Institution of Mechanical Engineers R + D

Journal, 11–16, September 1992.

Du Plessis, J. P., and D. G. Kröger, “Friction Factor Prediction for Fully Developed

Laminar Twisted-Tape Flow,” International Journal of Heat and Mass Transfer , 27-

11:2095–2100, 1984.

Dukler, A. E., and L. Smith, Two-Phase Interaction in Countercurrent Flow: Studies on the

Flooding Mechanism, Nuclear Regulatory Commission Report, NUREG/CRO.617, 1979.

Enger, M. L., and M. I. Levy, “Pressures in Manifold Pipes,” Journal of American Water

Works Association, May 1929.

Engineering Sciences Data Unit, Fluid Forces Acting on Circular Cylinders for Application

in General Engineering, Item No. 70013, Part 1: Long Cylinders in Two-Dimensional

Flow , London 1970.

Engineering Sciences Data Unit, Fluid Forces Acting on Circular Cylinders for Application

in General Engineering, Item No. 70014, Part 11: Finite-Length Cylinders, London, 1970.

Engineering Sciences Data Unit, Fluid Forces on Lattice Structures, Item No. 75011,

London, 1975.

Engineering Sciences Data Unit, Fluid Forces, Pressures and Moments on Rectangular Blocks, Item No. 71016, London, 1978.

Engineering Sciences Data Unit, Pressure Losses in Curved Ducts: Interaction Factors for

Two Bends in Series, London, 1977.

Engineering Sciences Data Unit, Pressure Losses in Three-Leg Pipe Junctions: Dividing Flow ,

London, 1973.

Feind, R., “Falling Liquid Films with Countercurrent Air Flow in Vertical Tubes,” Verein

Deutscher Ingenieure Forschungsheft , vol. 481, 1960.

Filonenko, G. K., Teploenergetika no. 4, 1954.

Fried, E., and I. E. Idelchik, Flow Resistance: A Design Guide for Engineers, Hemisphere

Publishing Co., New York, 1989.

Friedel, L., “Improved Friction Pressure Drop Correlation for Horizontal and Vertical

Two-Phase Pipe Flow,” European Two-Phase Flow Group Meeting, Paper E2, Ispra, 1979.

General Electric Corp., General Electric Fluid Flow Data Book, Genium Publishing

Corporation, Schenectady, N.Y., 1987.

Haaland, S. E., “Simple and Explicit Formulas for the Friction Factor in Turbulent Pipe

Flow,” Transactions of the American Society of Mechanical Engineers, Journal of Fluids

Engineering , 105-3:89–90, March 1983.Hall, W. B., and E. M. Orme, “Flow of Compressible Fluid Through a Sudden

Enlargement in a Pipe,” Proceedings of the Institution of Mechanical Engineers,

169:1007, 1955.

7/21/2019 48967_02


Hartnett, H. P., J. C. Y. Koh, and S. T. McComas, “A Comparison of Predicted andMeasured Friction Factors for Turbulent Flow through Rectangular Ducts,”

Transactions of the American Society of Mechanical Engineers, Series C, 84-1:82–88, 1962.

Hewitt, G. F., Flow Regimes, Handbook of Multiphase Systems, ed. G. Hetsroni, 1982.

Hewitt, G. F., and S. Jayanti, “To Churn or Not to Churn,” International Journal of

Multiphase Flow , 19-3:527–529, 1993.

Hewitt, G. F., and D. N. Roberts, Studies of Two-Phase Flow Patterns by Simultaneous Flash

and X-Ray Photography , AERE-M2159, Association of Environmental and Resource

Economists, 1969.

Hoerner, S. F., Fluid Dynamic Drag , Hoerner, New Jersey, 1965.

Holman, J. P., Heat Transfer , McGraw-Hill Book Co., New York, 1986.

Holms, E., Handbook of Industrial Pipework Engineering , McGraw-Hill Book Co., U.K., 1973.

Hong, S. W., and A. E. Bergles, “Augmentation of Laminar Flow Heat Transfer in Tubes

by Means of Twisted-Tape Inserts,” American Society of Mechanical Engineers,

75-HT-44, 1975.

Hudson, H. E., R. B. Uhler, and R. W. Bailey, “Dividing-Flow Manifolds with Square

Edged Laterals,” Proceedings of American Society of Mechanical Engineers, Journal of

Environmental Engineering , vol. 105, 1979.

Idelchik, I. E., Handbook of Hydraulic Resistance, Hemisphere Publishing Co.,

Washington, 1986.

James, P. W. et al., “Developments in the Modelling of Horizontal Annular Two-Phase

Flow,” International Journal of Multiphase Flow , 13-2:173–198, 1987.

Jayanti, S., G. F. Hewitt, D. E. F. Low, and E. Hervieu, “Observation of Flooding in the

Taylor Bubble of Co-Current Upwards Slug Flow,” International Journal of

Multiphase Flow , 19-3:531–534, 1993.

Jorgensen, R., Fan Engineering , Buffalo Forge Co., Buffalo, N.Y., 1961.Kays, W. M., “Loss Coefficients for Abrupt Changes in Flow Cross Section with Low

Reynolds Number Flow in Single and Multiple-Tube Systems,” Transactions of the

American Society of Mechanical Engineers, 72-8:1067–1074, 1950.

Keller, J. D., “The Manifold Problem,” Transactions of the American Society of Mechanical

Engineers, 71:77–85, 1949.

Kirschmer, O., “Kritische Betrachtungen zur Frage der Rohrreibung,” Zeit, Verein

Deutscher Ingenieure, 94:785, 1952.

Kubo, T., and T. Ueda, “On the Characteristics of Dividing Flow and Confluent Flow in

Headers,” Bulletin of Japan Society of Mechanical Engineers, 12-52:802–809, 1969.

Launder, B. E., and W. M. Ying, “Prediction of Flow and Heat Transfer in Ducts of

Square Cross Sections,” Proceedings of the Institute of Mechanical Engineers,

187:455–461, 1973.

FLUID MECHANICS

7/21/2019 48967_02



126

Launder, B. E., and W. M. Ying, “Secondary Flow in Ducts of Square Cross Section,” Journal of Fluid Mechanics, 54-2:289–295, 1972

Lin, P. Y., and T. J. Hanratty, “Effect of Pipe Diameter on Flow Patterns for Air-Water

Flow in Horizontal Pipes,” International Journal of Multiphase Flow , 13-4:549–563, 1987.

Lockhart, R. W., and R. C. Martinelli, “Proposed Correlation of Data for Isothermal

Two-Phase Two-Component Flow in Pipes,” Chemical Engineering Progress,

45-1:39–48, 1949.

Majumdar, A. K., “Mathematical Modelling of Flow in Dividing and Combining Flow

Manifolds,” Applied Mathematical Modelling , 4:424–431, 1980.

Mao, Z. S., and A. E. Dukler, “The Myth of Churn Flow,” International Journal of

Multiphase Flow , 19-2:377–383, 1993.

Markland, E., “Analysis of Flow from Pipe Manifolds,” Engineering , 189:150–151, 1959.

Martinelli, R. C., and D. B. Nelson, “Prediction of Pressure Drop During Forced

Circulation Boiling of Water,” Transactions of the American Society of Mechanical

Engineers, 70:695–702, 1948.

McQuillan, K. W., and P. B. Whalley, “A Comparison between Flooding Correlations

and Experimental Flooding Data for Gas-Liquid Flow in Vertical Tubes,” Chemical

Engineering Science, 40-8:1425–1441, 1985.McQuillan, K. W., and P. B. Whalley, “Flow Patterns in Vertical Two-Phase Flow,”

International Journal of Multiphase Flow , 11-2:1425–1440, 1985.

Miller, D. S., Internal Flow: A Guide to Losses in Pipe and Duct Systems, British Hydraulic

Research Association, Cranfield, 1971.

Miller, D. S., Internal Flow Systems, 2d ed., BHRA Fluid Engineering Series, British

Hydraulic Research Association, Cranfield, 1990.

Moody, L. F., “Friction Factors for Pipe Flow,” Transactions of the American Society of

Mechanical Engineers, 66:671, 1944.

Mukherjee, H., and J. P. Brill, “Empirical Equations to Predict Flow Patterns in Two-

Phase Inclined Flow,” International Journal of Multiphase Flow , 11-3:299–315, 1985.

Nikuradse, J., “Gesetzmässigkeiten der turbulenten Strömung in glatten Rohren,”

Verein Deutscher Ingenieure Forschungsh, vol. 356, 1932.

Nujens, P. G. J. M., A Discrete Model for Flow Distribution Manifolds, CSIR Report ,

Chemical Engineering Research Group, Pretoria, 1983.

Olujic, Z., “A Method for Predicting Friction Pressure Drop of Two-Phase Gas-Liquid

Flow in Horizontal Pipes,” American Institute of Chemical Engineers 75th Annual

Meeting, Washington, 1983.Oschinowo, T., and M. E. Charles, “Vertical Two-Phase Flow,” Canadian Journal of

Chemical Engineering , 52:25–35, 1974.

Patterson, G. N., “Modern Diffuser Design,” Aircraft Engineering , 10:267, 1938.

7/21/2019 48967_02


Petukhov, B. S., Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties, Advances in Heat Transfer , vol. 6, ed. J. P. Hartnett and T. F. Irvine,

Academic Press, New York, 503–564, 1970.

Premoli, A., D. Di Francesco, and A. Prina, “An Empirical Correlation for Evaluating

Two-Phase Mixture Density Under Adiabatic Conditions,” European Two-Phase

Flow Group Meeting, Milan, 1970.

Rouse, H., Elementary Mechanics of Fluids, John Wiley & Sons, London, 1946.

Runstadler, P. W., F. X. Dolan, and R. C. Dean, Diffuser Data Book, Creare Inc., Hanover,

New Hampshire, 1975.

Sachs, F., Wind Forces in Engineering , Pergamon, New York, 1972.

Scott, D. S., Properties of Co-Current Gas-Liquid Flow, Advances in Chemical Engineering ,

4:199–277, Academic Press, New York, 1963.

Shah, R. K., Fully Developed Laminar Flow Forced Convection in Channels, Low Reynolds

Number Flow Heat Exchangers, ed. S. Kakac, R. K. Shah, and A. E. Bergles,

Hemisphere Publishing Corp., Washington, 1983.

Shah, R. K., and A. L. London, Laminar Flow Forced Convection in Ducts, Academic Press,

New York, 1978.

Shames, I. H., Mechanics of Fluids, McGraw-Hill Book Co., New York, 1962.

Simiu, E., and R. H. Scanlan, Wind Effects on Structures: An Introduction to Wind

Engineering , J. Wiley, Washington, 1977.

Simmons, L. F. G., Measurements of the Aerodynamic Forces Acting on Porous Screens, R & M,

no. 2276, 1945.

Spedding, P. L., and V. T. Nguyen, “Regime Maps for Air-Water Two-Phase Flow,”

Chemical Engineering Science, 35:779–793, 1980.

Spedding, P. L. et al., “Two-Phase Co-Current Flow in Inclined Pipe,” International

Journal of Heat and Mass Transfer , 41:4205–4228, 1998.Stanislav, J. F., S. Kokal, and M. K. Nicholson, “Intermittent Gas-Liquid Flow in Upward

Inclined Pipes,” International Journal of Heat and Mass Transfer , 12-3:325–335, 1986.

Stephan, M., “Untersuchungen zur gegenstrombegrenzung in vertikalen Gas-

Flüssigkeits-Strömungen,” Doctoral thesis, Thermodynamik Technische

Universität, München, August 1990.

Stephan, M., and F. Mayinger, “Experimental and Analytical Study of Countercurrent

Flow Limitations in Vertical Gas/Liquid Flow,” Chemical Engineering and

Technology , 15:51–62, 1992.

Taitel, Y., and B. Barnea, “Counter-Current Gas-Liquid Vertical Flow, Model for FlowPattern and Pressure Drop,” International Journal of Multiphase Flow , 9-6:637– 647, 1983.

FLUID MECHANICS

7/21/2019 48967_02



128

Taitel, Y., and A. E. Dukler, “A Model for Predicting Flow Regime Transition inHorizontal and Near Horizontal Gas-Liquid Flow,” American Institute of Chemical

Engineers Journal, 22-1:47–55, 1976.

Taitel, Y., D. Barnea, and A. E. Dukler, “Modelling Flow Pattern Transition for Steady

Upward Gas-Liquid Flow in Vertical Tubes,” American Institute of Chemical

Engineers Journal, 26-3:345–354, 1980.

Tien, C. L., and C. P. Liu, Survey on Vertical Two-Phase Counter-Current Flooding, Report

NP-984, Electric Power Research Institute, February 1987.

Turton, R., and O. Levenspiel, “A Short Note on the Drag Correlation for Spheres,”

Power Technology , 47:83–86, 1986.

Verein Deutscher Ingenieure-Wärmeatlas, Verein Deutscher Ingenieure, Düsseldorf, 1977.

Wallis, G. B., Flooding Velocities for Air and Water in Vertical Tubes, AEEW-R123, 1961.

Wallis, G. B., D. C. de Sieyes, R. J. Rosselli, and J. Lacombe, Countercurrent Annular Flow

Regimes for Steam and Subcooled Water in a Vertical Tube, Research Project 443-2, EPRI

NP-1336, Electric Power Research Institute, California, January 1980.

Weisbach, J., Die Experimental-Hydraulik, J. S. Engelhardt, Freiberg, 1855.

Whalley, P. B., Boiling, Condensation, and Gas-Liquid Flow , Clarendon Press, Oxford, 1987.

Whalley, P. B., and K. W. McQuillan, “Flooding in Two-Phase Flow: The Effect of Tube

Length and Artificial Wave Injection,” Physicochemical Hydrodynamics, 6-1:3–21, 1985.

Whalley, P. B., and G. F. Hewitt, Multiphase Flow and Pressure Drop, Heat Exchanger

Design Handbook, 2:2.3.2–2.3.11, Hemisphere, Washington D.C. 1983, 1980.

White, F. M., Viscous Fluid Flow , McGraw-Hill Book Co., Singapore, 1991.

Wieghardt, K. E. G., “On the Resistance of Screens,” The Aeronautical Quarterly ,

4:186–192, February 1953.

Zapke, A., and D. G. Kröger, “Countercurrent Gas-Liquid Flow in Inclined and Vertical

Ducts. Part 1: Flow Patterns, Pressure Drop Characteristics and Flooding,” International Journal of Multiphase Flow , 26:1439–1455, 2000.

Zapke, A., and D. G. Kröger, “Countercurrent Gas-Liquid Flow in Inclined and Vertical

Ducts. Part 2: The Validity of the Froude-Ohnesorge Number Correlation for

Flooding,” International Journal of Multiphase Flow , 26:1457–1468, 2000.

Zapke, A., and D. G. Kröger, “The Effect of Fluid Properties on Flooding in Vertical and

Inclined Rectangular Ducts and Tubes,” Proceedings, American Society of

Mechanical Engineers Fluids Engineering Division Summer Meeting, FED-

239:527–532, San Diego, July 1996.

Zapke, A., and D. G. Kröger, “The Influence of Fluid Properties and Inlet Geometry onFlooding in Vertical and Inclined Tubes,” International Journal of Multiphase Flow ,

22-3:461–472, 1996.

7/21/2019 48967_02


Zapke, A., and D. G. Kröger, “Pressure Drop during Gas-Liquid Countercurrent Flow inInclined Rectangular Ducts,” Proceedings, 5th International Heat Pipe Symposium,

Melbourne, November 1996.

Zipfel, T., and D. G. Kröger, “A Design Method for Single-Finned-Tube-Row Air-Cooled

Steam Condenser to Avoid Back Flow of Steam into Any Finned Tube,” South

African Institution of Mechanical Engineering R + D Journal, 13:68–75, 1997.

Zipfel, T., and D. G. Kröger, “Flow Loss Coefficients at Air-Cooled Condenser Finned

Tube Inlets,” South African Institution of Mechanical Engineering R + D Journal ,

13:62–67, 1997.

FLUID MECHANICS

48967_02

Documents