NATL INST. <> F STAND & TECH R.LC11105 bBMb^H
DATE DUE

Juu-.z-i l°\ld m?/ ;

' fm 11

CAVLORD PRINTED IN U-S.A.




•."wr

KH*

NBS TECHNICA

v

Comparison of Incompressible

Flow and Isothermal Compressible

Flow Formulae

356

y

«.»T OF

o\

V

10 Q

>(/*CAU 0^ *

U.S DEPARTMENT OF COMMERCE

National Bureau of Standards

iliiiKiaillllil .: WMMM^&MWM^MM^0^B::€MWM
THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards1 provides measurement and technical information services

essential to the efficiency and effectiveness of the work of the Nation's scientists and engineers. The

Bureau serves also as a focal point in the Federal Government for assuring maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. To

accomplish this mission, the Bureau is organized into three institutes covering broad program areas of research and services: THE INSTITUTE FOR BASIC STANDARDS . . . provides the central basis within the United

States for a complete and consistent system of physical measurements, coordinates that system with the measurement systems of other nations, and furnishes essential services leading to accurate and uniform

physical measurements throughout the Nation's scientific community, industry, and commerce. This

Institute comprises a series of divisions, each serving a classical subject matter area: —Applied Mathematics—Electricity—Metrology—Mechanics—Heat—Atomic Physics—Physical

Chemistry—Radiation Physics—Laboratory Astrophysics2—Radio Standards Laboratory,2 which

includes Radio Standards Physics and Radio Standards Engineering—Office of Standard Refer- ence Data. THE INSTITUTE FOR MATERIALS RESEARCH . . . conducts materials research and provides

associated materials services including mainly reference materials and data on the properties of materials. Beyond its direct interest to the Nation's scientists and engineers, this Institute yields services which are essential to the advancement of technology in industry and commerce. This Institute is or- ganized primarily by technical fields: —Analytical Chemistry—Metallurgy—Reactor Radiations—Polymers—Inorganic Materials—Cryogenics2—Office of Standard Reference Materials. THE INSTITUTE FOR APPLIED TECHNOLOGY . . . provides technical services to promote the use of available technology and to facilitate technological innovation in industry and government. The

principal elements of this Institute are: —Building Research—Electronic Instrumentation—Technical Analysis—Center for Computer Sci- ences and Technology—Textile and Apparel Technology Center—Office of Weights and Measures —Office of Engineering Standards Services—Office of Invention and Innovation—Office of Vehicle

Systems Research—Clearinghouse for Federal Scientific and Technical Information3—Materials

Evaluation Laboratory—NBS/GSA Testing Laboratory.

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.

20234.

2 Located at Boulder, Colorado, 80302.

3 Located at 5285 Port Royal Road, Springfield, Virginia 22151.
UNITED STATES DEPARTMENT OF COMMERCE

Alexander B. Trowbridge, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin. Director

NBS TECHNICAL NOTE 356

ISSUED AUGUST 17, 1967

COMPARISON OF INCOMPRESSIBLE FLOW AND

ISOTHERMAL COMPRESSIBLE FLOW FORMULAE

JESSE HORD

Cryogenics Division

Institute for Materials Research

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402

Price: 25 cents

TABLE OF CONTENTS

Page

Abstract 1

1. Introduction 1

2. Review of Theoretical Relations 4

2. 1 Incompressible Flow 5

2.2 "Modified-incompressible" Flow 5

2. 3 Isothermal Compressible Flow 7

3. Comparison of Incompressible and Isothermal

Compressible Flows 7

4. Comparison of "Modified-incompressible" and

Compressible Flows 8

5. Discussion of Figures 16

6. End Effects 19

7. Summary 20

8. Nomenclature 21

9. References 22

in
List of Figures

Page

Figure 2. 1 Notation for fully developed isothermal

flow of ideal gas in horizontal pipes .... 6

Figure 3. 1 Comparison of incompressible flow and

isothermal compressible flow at various

static pressure ratios in long pipes, i.e.,

fL/2D >>in (P

1 /P 2

). Laminar flow (3=1);

Turbulent flow (3=0. to 0.2) 9

Figure 3. 2 Comparison of incompressible flow and

isothermal compressible flow as a function

of fL/D and P2 /Pj for laminar flow (3= 1).

Mo = 1/3 is evaluated at k = 1 . 4 10

Figure 3. 3 Comparison of incompressible flow and

isothermal compressible flow as a function

of fL/D and P2

/P]_ for turbulent flow

(3=0.0 to 0. 05). M2 = 1/3 is evaluated at

k = 1.4 and 3=0. 0. 11

Figure 3.4 Comparison of incompressible flow and

isothermal compressible flow as a function

of fL/D and P2 /Pi for turbulent flow (8=0.2).

M2 = 1/3 is evaluated at k = 1 . 4 12

Figure 4. 1 Comparison of "modified-incompressible"

flow and isothermal compressible flow as a function of fL/D and P2 /Pi for laminar

flow (3=1). M2 = 1/3 is evaluated at k = 1.4. 13

Figure 4.2 Comparison of "modified-incompressible"

flow and isothermal compressible flow as a function of fL/D and P2

/P]_ for turbulent

flow (3=0.0 to 0.05). M2 = 1/3 is evaluated

at k = 1.4 and 3 = 0. 14

Figure 4. 3 Comparison of "modified-incompressible"

flow and isothermal compressible flow as a function of fL/D and P^/P^ for turbulent

flow (3= 0.2). M2= 1 /3 is evaluated at

k = 1.4. 15

iv
COMPARISON OF INCOMPRESSIBLE FLOW AND

ISOTHERMAL COMPRESSIBLE FLOW FORMULAE

by

Jesse Hord

Mass flow formulae for incompressible and "modified - incompressible" flow are compared with the isothermal compressible flow relation under the following conditions: The gas flow is

steady, isothermal, and fully developed in a horizontal pipe of

constant cross section with a prescribed static pressure drop

(P, -Po). The comparative data are limited to static pressure

ratios (P^/P,) > -J- , and subsonic isothermal flow. Laminar and

turbulent flows are treated. Under the limitations of the comparison, modified-incompressible flow and isothermal gas flow relations are identical when fL/2D >?>in (Pi /P?)* Graphical plots

indicate the degree of approximation or error involved in using

incompressible relations to solve compressible flow problems.

Pressure losses due to end effects are briefly discussed.

Key Words: Compressible flow, flow comparison, fluid flow,

incompressible flow, mass flow, pressure drop.

1 . Introduction

Engineers frequently [l] use incompressible flow theory to solve

compressible flow problems; a large class of fluid problems may be

solved within the framework of isothermal compressible flow, and many

textbook authors justifiably devote considerable attention to this subject.

While compressible flow computations are not difficult to perform -- working charts are available [2] --they are frequently replaced by the

simpler incompressible flow calculations; this is particularly true in

analyses where the flow expression is part of an integrand and closed

form solutions are impossible using the compressible formula. Examples are the computations of 1) rate of gas transfer between two vessels
and 2) the response of a pressure gage with an interconnecting tube.

Limits of applicability of the incompressible flow relations must then be

established. These limits are obviously dependent upon the desired

computational precision. The textbook author will usually advise use of

the compressible flow relations if any question of application exists.

However, intelligent use of a significant portion of the literature requires

knowledge of the limits of application of the incompressible theory. The

applicability of incompressible flow theory to evaluate compressible flow

problems has not been clearly established, and the purpose of this paper

is to provide this information.

Comparisons Ll, 3] between incompressible formulae and isentropic and adiabatic compressible relations have appeared in the literature. The approximations involved in computing pressure drop for

isothermal gas flow (at fixed inlet velocities) via incompressible flow

relations have also been treated in the literature [2,4, 5], This paper

treats the parallel problem of determining the error involved in using

incompressible flow theory to predict mass flow for fixed pressure drop

ratios and constant gas temperature in horizontal pipes.

Some examples of the limits of applicability of the incompressible

relations given in several textbooks will emphasize the need for this

information. Binder [6] states that "sometimes the rule is given that the

incompressible flow relation can be applied to isothermal compressible

flow problems if the pressure drop (P -P?

) is less than 10 percent of

the initial pressure P . " He then compares the two theories for flow in

"long, " horizontal pipes. The assumption of long pipes permits the

compressible relation to be simplified for comparison. In a later edition,

Binder [5] does not impose the "long pipe" restriction and illustrates the

difference between the two flow relations in terms of pipe inlet Mach
number. The error involved in using the incompressible relation to

compute pressure drop at a fixed inlet Mach number can be deduced from

these illustrations. King, Wisler, and Woodburn L7j also cite the 10 percent rule for pressure drop ratio, i.e., [(P -P ) / P ] < 0. 10 is the criterion used to determine the applicability of incompressible relations

to compressible flow problems. Schlichting [8j points out that gaseous

flow may be considered incompressible when the pipe line Mach number

is less than 0. 3. Rouse and Howe L9J neglect the momentum term in the

compressible flow relation (which is equivalent to assuming a long pipe)

and derive a simple expression for estimating pressure drop in isothermal compressible flow. This expression is then mathematically compared with the incompressible relation to indicate the error involved in

pressure drop calculations for flow of gases at constant temperature in

long pipes. Shapiro [2] provides a mathematical relationship for the

comparison of incompressible and compressible formulae at "low" pressure drop ratios and inlet Mach numbers. Hall [4] presents a unique

comparison of the theories by developing a "compressibility correction

factor" in terms of the kinetic energy of the flowing fluid. He carefully

defines the Mach number limitations of the approximate comparative

formulae which he gives for computing the differences between static

and stagnation temperatures and pressures.
It is the purpose of this paper to indicate the error involved in

using incompressible and simplified compressible flow relations to compute the isothermal mass flow of gas through horizontal tubes with prescribed static pressure ratios. The results maybe used to 1) serve as

a guide to determine whether a compressible calculation is needed,

2) obtain a "correction factor" which converts the incompressible flow

to compressible flow and 3) estimate the applicable error or appropriate

"correction factor" when using incompressible formulae in integrands

where compressible relations cannot be integrated. With respect to

item 2), it is to be emphasized that there is no net advantage in using

the comparative data given here as a calculation method for compressible flow. This topic is discussed in detail in section 5. The purpose

of the paper is to assess the error involved in using the incompressible

flow relations and express the results in terms of dimensionless flow

parameters.

2. Review of Theoretical Relations

In arriving at the limits of application of the simplified flow formulae, it is necessary to make some assumptions and note pertinent

restrictions. It is assumed that the flow in a horizontal pipe is isothermal, steady, and fully developed. Binder [5] gives the critical (limiting)

pressure ratio for isothermal flow of gas in a horizontal pipe as (P/P ) = M. 4k, i. e. , isothermal sonic velocity is attained at the

1 cr 1

}

point in the pipe where this relation is satisfied. He points out that isothermal flow involves heat transfer and friction and exists only at Mach

numbers below 1 I \jk. From the pressure drop comparisons given by

Binder [5], it is apparent that static pressure ratios (P/P ) smaller

than one -half involve rapidly increasing differences between compressible and incompressible theories; static pressure ratios smaller than

one -half indicate large errors (up to 2:1) in computing the pressure
drop via incompressible formulae. Thus, it becomes clear why manyauthors limit the use of simplified flow expressions to static pressure

ratios greater than one -half.

The assumptions and restrictions which pertain to the following

comparison of compressible and incompressible theories are tabulated

below:

a) Steady, isothermal, fully developed gas flow in a horizontal

pipe of constant cross section. See figure 2. 1.

b) The limiting (critical) pressure ratio is given by (P/P ) =

1 cr

M -\/k.

c) Pipe line exit Mach number, M < lVk .

d) The comparative data will be limited to static pressure

ratios, (P

2

/P )> & .

e) The gas viscosity is a function of temperature only.

2. 1 Incompressible Flow

The mass flow of an incompressible fluid may be written as

^ic

= A

{pl gc (Pl " P2

)/(fL/2D) }* * {Z ' l)

2.2 "Modified-incompressible" Flow

The mass flow of gas given by the product of incompressible

volumetric flow and arithmetic mean density is

^mic " A {p"g

c (P r

P

2

)/(fL/2D) }^ ' (2 ' 2)
2> v2

Figure 2. 1 Notation for fully developed isothermal flow of ideal gas in

horizontal pipes.
2. 3 Isothermal Compressible Flow

The mass flow of an ideal gas is given by

mc = AJgc (P

2

1 -P2

2

) / 2RG

T [in(P

1 /P2

) + fc L/2D]}^. (2.3)

3. Comparison of Incompressible and Isothermal Compressible Flows

Dividing (2. 3) by (2. 1) and utilizing the perfect gas law we obtain

the mass flow ratio,

/m.

c = N = { T(fL/2D) (l+P2

/P

i )/2 "] / Ul,/ZD + In (P

i /P^ ]

\*.(3. 1)

o , o mc 1C

To account for laminar and turbulent flow the friction factor may

be expressed in the Blasius form

f = a/Re . (3.2)

For laminar flow a = 64, and 6 = 1; in turbulent flow typical values of a

and 8 are cc = 0. 01 to 0. 2 and g = 0. to 0. 2.

Equation (3. 1) may be evaluated for a fixed pressure ratio by

assuming as a first try f = f . Since the friction factor is related to the

c mass flow through the Reynolds number, a correction in f is required

for the second and succeeding iterations. Where N > -g-, the compressible and incompressible friction factors may be evaluated from (3.2)

using identical values of a and g, i.e., a 2:1 variation in Re is permissible for a given a and g. Then, (3. 2) may be used to obtain the following relation

ill = N

5

. (3.3)
Combining (3. 1) and (3. 3) we obtain

N= {^/(c^ + N

-0 )}*, (3.4)

where a = U +' P, /P ) /2 and CT , =

~ in(P /P )1 / (fL/2D) . 1 2 1 ^ L J-^j

We may now examine the effect of assuming a long pipe, i. e. fL/2D >> in (P

f /P ). Since N

M is near unity the first term ii\ the

J. w denominator of (3.4) is negligible and (3.4) becomes

N = a

1/

(2 -

B)

. (3.5)

Figure 3. 1 illustrates the results given by (3. 5) for various values of

P /P and g . For arbitrary values of fL/2D, a trial and error solution

of (3.4) is required. A graphical solution of (3.4) is given on figures

3.2, 3.3, and 3.4. Comparison of figure 3.1 and figures 3.2 - 3.4 at

identical values of P /P and g indicates that a long pipe requires fL/D

2 3

values of 10 to 10 (the exact value of fL/D is dependent upon the computational precision desired). In laminar flow (3.4) becomes a simple

quadratic equation with the solution

(N,

R=1

1 + (1"+ 4cr1

cT2 )^/2a2

• (3.6)

4. Comparison of "Modified -incompressible"

and Compressible Flows

In a similar manner, we obtain the mass flow ratio for modified

incompressible flow; dividing (2.3) by (2.2),

m /m . = N = {1/(ct, + N

" 8

)f . (4.1)

c mic L 2

J
1.00

0.95

0.90

N

0.85

0.80

0.75

r&

% (ft

^S^S

\^XA0^

/*

\ /

h v

0.5 0.6 7 0.8

P2/P,

0.9 .0

Figure 3. 1 Comparison of incompressible flow and isothermal compressibL

flow at various static pressure ratios in long pipes, i. e. , fL/2D > >in (P

1 /P2

). Laminar flow (6=1); Turbulent flow

(3=0. to 0.2).
1.0

0.9

N

0.8

0.7

0.6

q> .Z. ~0^

/n" ///

<*v

<*>/

J-4* IS

r\

v%\ / A.,

^°/ XNK J --S&C

f \ \

\

\ <OZ \o/

£ = 1

! i 1 10 50

fL/D

100 500 1000

Figure 3.2 Comparison of incompressible flow and isothermal compressible

flow as a function of fL/D and P^/Pl for laminar flow ($=1).

M2= 1/3 is evaluated at k = 1.4.

10
1 .u

0^>^

ON*, — 91\

< ^r9>

fV ^ oS^

0.9

"> C

oo

>i ) ; / 5? G-v _r

// // •JV/Ch- 0^ ^ ^y /\

oy^

0.8

0.7

"

£ =• 0.0 to 0.05

n fi 10 50 100 500 1000

fL/D

Figure 3. 3 Comparison of incompressible flow and isothermal compressible

flow as a function of fL/D and P9/P1 f° r turbulent flow (3 =0. to

0. 05). M2 =l/3 is evaluated at k = 1.4 and 3 = 0. 0.

11
1.0

0.9

N

0.8

0.7

0.6

rJ*>

f

,1 9^/

CS ^'

A*eOX

P

3V i <^^

O

Cm. ^ ^N->.-- o^^« — / \ n^ ^

/ \v

i

/:* = 0.2

10 50 100 500 1000

fL/D

Figure 3.4 Comparison of incompressible flow and isothermal compressible

flow as a function of fL/D and P2/P1 for turbulent flow (3= 0.2).

M2 = 1/3 is evaluated at k = 1.4.

12
.0

0.9 —

N

0.8

0.7

<v o/ ft * £/

<*y\

Oy (s\

*i /

<Cy «%_ rc}y i to/

/ j&'^ c _/$£ L^v

7T7

1

""

/

/ / /

/ /

y

f /

/

/

/ /

'

( /

/ /3 = 1 _ /

/

5 10 50 100

fL/D

500 ioa

Figure 4. 1 Comparison of "modified -incompressible" flow and isotherma

compressible flow as a function of fL/D and P2/P1 for lamina

flow (3=1). M2 = 1/3 is evaluated at k = 1.4.

13
0.9-

N

0.8

0.7

- Ok

1^

o^i

x

vr^

- P'ov

y/f <c•/

/<D/<Ci/

/o>

s?/

~ £ =: 0.0 to 0.05

10 50 100

fL/D

500 1000

Figure 4.2 Comparison of "modified -incompressible" flow and isothermal

compressible flow as a function of fL/D and P^/Pi for turbulent

flow (3= 0. to 0. 05). M2 = 1/3 is evaluated at k = 1 . 4 and

fl= 0.0.

14
1.0

0.9

N

0.8

0.7

<>>

1<? N /

\ > ^f

<b/

c\7^

/_ —»

"^ of >/—

Coy ^

/ v

/c r»> / ji

xj

£ = 0.2

10

fL/D

50 100 500 1000

Figure 4. 3 Comparison of "modified -incompressible" flow and isothermal

compressible flow as a function of fL/D and P2 /Pi for turbulent

flow (3 = 0.2). M2 = 1/3 is evaluated at k = 1.4.

15
When the pipe is long, fL/2D > >in (P /P ), N = 1, and identical results

are obtained from the compressible and modified incompressible theories.

A trial and error solution of (4. 1) is required for arbitrary values of

fL/2D, and a simple quadratic solution arises in laminar flow (8 = 1),

(N) =[-l+(l+4a

2 )*]/2a2

• (4.2)

The comparative results given by (4. 1) and (4.2) are shown on figures

4. 1, 4. 2, and 4. 3.

5. Discussion of Figures

Figure 3. 1 depicts the mass flow ratio, N, as a function of the

pressure ratio P /P and 3 for long pipes. For the range of turbulent

Ct J. flow, 8 = 0.0 to 0. 2, there is little variation in the flow ratio for a given

pressure ratio; however, there is an appreciable difference between

laminar (8 = 1) and turbulent flows. A large portion of isothermal flow

problems fall into the long pipe category. By comparing figure 3. 1 and figures 3. 2 - 3.4 at identical values of 8 and P /P > the implications

of a long pipe become clear. Figures 3.2 - 3.4 are not restricted to

long pipes and the various pressure ratio curves asymptotically approach

the flow ratio given by figure 3. 1. Depending upon the degree of agree

2 3 ment required the long pipe requires fL/D of 10 to 10 for the range of

pressure ratios plotted.

The counterpart of figure 3. 1 for modified -incompressible flow

is not given since the mass flow ratio, N, is unity for long pipes.

Accordingly, all of the curves shown in figures 4. 1 - 4. 3 approach N= 1 as ymptotically.

A few words concerning the construction of figures 3.2 - 3.4 and

figures 4.1 - 4. 3 will clarify their use. Figures 3 and 4 are plotted for

16
values of g which are representative of the entire flow range: i. e. ,

g = 0. to 0. 05 is appropriate for constant friction at large values of Re

and for turbulent flow in very rough pipes at lower Re; 6=0-2 applies to

4 5

turbulent flow in smooth pipes at lower values of Re (10 to 10 ), and

g = 1 is applicable to laminar flow. Comparison of figure 3. 3 with

figure 3.4 and figure 4.2 with figure 4. 3 indicates very small differences

in flow ratio occur for a particular P /P due to change in g. Hence we

have essentially two sets of figures; one is for laminar flow (figures 3.2

and 4. 1) and the other is for turbulent flow (figures 3. 3 and 4. 2 or fig3

ures 3. 4 and 4. 3 may be used where Re >2 x 10 ). The pressure ratio

curves are limited to values of fL/D greater than those at which sonic

flow would occur at the pipe outlet, i. e. , M < 1 A/Tc . To indicate the

implications of the rule that M < 0. 3 permits neglect of compressibility,

the locus of M =1/3 has been plotted on figures 3.2 -3.4 and figures

4. 1 - 4. 3 This locus was constructed from the following relation which

is derived by combining (2. 3) and (3. 3);

(fL/D) = N

8 [(l/kM2

2

) {(P/P^2

-l} -2in(P

1 /P

2 )] f (5. 1)

where k is evaluated at 1.4 and M = 1/3. N was used in (5. 1) for figures 4. 1 - 4. 3. In these figures, M < 1/3 exists only for values of

fL/D larger than those intersected by the dotted line (M = 1/3) for a prescribed pressure ratio. For M = 1/3, the error is observed to

increase with decreasing pressure ratios for incompressible flow (figures 3.2 - 3.4), and to decrease with decreasing pressure ratios for

modified -incompressible flow (figures 4. 1 - 4. 3). In both cases, N or

N increases with decreasing values of M at a fixed pressure ratio.

The improvement in computational precision offered by the modified -incompressible flow relation is elucidated by comparing figure 3.2

17
with figure 4. 1, figure 3. 3 with 4. 2, etc. It should be mentioned that

equation (2.2) represents the familiar Poiseuille flow relation when

laminar flow (g = 1) is considered. Consequently, figure 4. 1 illustrates

the agreement between Poiseuille and compressible flows. Poiseuille

flow is frequently utilized by vacuum textbook authors in analyzing the

flow of gas in systems at low pressures.

Use of the various figures is straightforward and some applications were listed in the introduction to this paper. For the class of

problems of interest, P /P and the pipe dimensions are known. The

J. Lt flow may be classified as laminar or turbulent by conventional means

using the Darcy-Weisbach friction formula, and the Hagen -Poiseuille

relation for laminar friction factor. For laminar flow the friction factor is computed and then the appropriate chart (figure 3. 2 or 4. 1) is

entered with known values of fL/D and P /P and the mass flow ratio

read directly. When turbulent flow is indicated the friction factor can

usually be estimated within a factor of about two. Again the appropriate

figure is used --e.g. , figure 4. 2 --and the mass flow ratio read directly.

If the degree of approximation of the incompressible calculation is inadequate, as indicated by the mass flow ratio obtained, a compressible calculation is required. If fL/D >>in(P /P_) exact results are obtained by

using equation (2.2), or figure 3. 1 may be used to assess the error

involved in a true incompressible flow calculation. Although there is no

advantage in using the various figures to compute the compressible mass

flow, they may be used this way. The incompressible flow is computed

in the usual manner, i. e. , by assuming a value for the friction factor

and performing an iterative calculation to obtain f and m , or by using

1C

one of the direct solution nomographs [lO] available in the literature. A

mass flow ratio is obtained from the pertinent figure and used as a "correction factor" to obtain the compressible mass flow. Where

18
incompressible relations are substituted for compressible formulae in

integrands, the applicable error or appropriate "correction factor" maybe estimated as follows: the "correction factor" associated with the

initial and final pressure differences in the gas transfer process are

obtained as outlined above; an average "correction factor" may then be

obtained and applied over the appropriate time interval.

6. End Effects

In most practical applications involving isothermal gas flow the

effect of end losses is not significant [l 1J ; however, in some short pipe

installations, end losses may warrant consideration. Entrance and exit

losses are dependent upon the area expansion or contraction ratio and

the geometry (i. e. , abrupt, contoured, etc. ) of the transition. The pressure loss, due to abrupt [ll, 12] enlargements and contractions, has been

shown to be a function of dimensionless flow parameters, e. g. , Reynolds Number, pressure ratios and area ratios.

It has also been shown[ll-13] that incompressible loss coefficients may be used for compressible fluids at low Mach numbers (<0. 15

according to Kays [ll]). At higher Mach numbers the compressible and

incompressible loss coefficients differ [l2, 13] appreciably. The loss

coefficients also depend on whether the flow is laminar or turbulent [l 1J.

In addition to the pressure losses due to sudden changes in flow

cross section there are additional friction losses associated with the

establishment of "fully developed" flow in the entrance region of a tube.

The establishment of a fully developed boundary layer and velocity profile in a pipe requires a certain entrance or "entry" length. The "entry

length" which is sometimes referred to as the critical length is given

[l4] for laminar flow by L /D = 0. 0575 Re. The entry length for turbulent flow is given by Schlichting [8] (attributed to Nikuradse) as 25 to 40

19
pipe diameters. Latzko[l5] has developed an expression for turbulent

0. 25

flow in a bellmouthed tube, L /D = 0. 693 Re . Foust, Wenzel, Clump,

e Maus, and Anderson [l 6] state that "the pressure drop through the entrylength is considerably greater than the pressure drop after fully developed

flow is established." They suggest the pressure drop through the entry

length be taken as two to three times the value for fully developed flow

in an equivalent length downstream of the entrance region. Most analy- ses treating entrance region losses include the momentum and friction

losses incurred, while a simple contraction loss coefficient neglects the

additional friction loss required to establish flow. Lundgren, Sparrow,

and Starr [l7] have shown that the friction component accounts for 25 to

40 percent of the total loss in the entrance region of ducts of arbitrary

cross section in laminar incompressible flow. The bulk of the literature

concerning entrance region losses has treated laminar incompressible

flow[l7-27]. A summary of the literature is given in reference [27]

Few [28] articles concerning turbulent entry regions have appeared.

From the foregoing discussion it is apparent that correction of

the mass flow calculations for "end losses" will vary with each individual

problem; therefore, it is not possible to account for end effects in the

comparison figures presented herein. Further discussion of end effects

is beyond the scope and objective of this paper and the reader is referred

to the literature cited in this section for detailed information.

7. Summary

1. Incompressible and isothermal compressible mass flow relations

have been graphically compared for prescribed static pressure ratios.

The comparative data are dependent upon the level of turbulence. "Modified -incompressible" flow is shown to agree more closely with the compressible case than true incompressible flow; the compressible and

20
modified -incompressible formulae are identical when fL/D >>in(P /P ?

). The results of the study are given in figures 3. 1 through 4. 3 and some

applications have been outlined.

2. The error involved in neglecting compressibility when M < 1/3, or (P -P )/P < 0. 10, is indicated on figures 3.2 - 3.4 and figures 4. 1

X Li X

- 4. 3. It has been demonstrated that the assumption of a long pipe

2

requires fL/D > 10 .

8. Nomenclature

A = cross sectional area of pipe [ =TTD /4]

D = inside diameter of pipe

f = friction factor (Darcy-Weisbach) in incompressible

flow [ =a/Re p

], dimensionless

g

= conversion factor in Newton's law of motion [given in

engineering units as g =32.2 (ft) (pounds mass)/

2 n ° (sec ) (pounds force)J

k = ratio of specific heats [ =C /C ], dimensionless

p v

L. = length of pipe

L = entry length [defined in text]

M = Mach Number [=gas velocity/adiabatic acoustic

velocity], dimensionless

o m = mass flow rate

N = ratio of isothermal gaseous mass flow to incompresto . O -I = m /m. J, dimensionless

c 1C

N = ratio of isothermal gaseous mass flow to "modified

incompressible" mass flow [ =m /m . ], dimensionc mic

less

P = absolute pressure of gas in pipe

P = arithmetic mean pressure in the pipe [=(P +P )/2]

X Ct

21
Re = Reynolds Number (based on pipe diameter), dimensionless

R = gas constant in mechanical units, equals P/pT, equals G

universal gas constant divided by the molecular mass

[typical engineering units are (ft)(pounds force)/(pounds

mass((degree Rankine)]

T = absolute temperature of gas in pipe

V = bulk velocity of gas in pipe

Greek

a = constant in friction factor formula

3 = constant in friction factor formula

p = gas mass density

p" = arithmetic mean gas mass density [ =(p + p )/2J

a

= variable in comparative mass flow ratio [=(1 + P /P )/2j

1 Lm i. a - variable in comparative mass flow ratio [ = {in(P /P9

)}/

Cd \. Ld (fL/2D)]

Subscripts

c = denotes isothermal compressible flow

ic = denotes isothermal incompressible flow

mic = denotes "modified -incompressible" flow

1 = denotes pipe inlet condition

2 = denotes pipe exit condition

9. References

1. Benedict, R. P., "Some Comparisons Between Compressible and

Incompressible Treatments of Compressible Fluids," Journ.

of Basic Eng. , ASME Trans. Vol. 86, p. 527-537 (Sept. 1964).

2. Shapiro, A. H. , Compressible Fluid Flow, Vol. 1, Chap. 6,

p. 178-183 (The Ronald Press Co., New York, N. Y. , 1953).

22
3. Benedict, R. P., and N. A. Carlucci, "Flow With Losses,"

Journ. of Eng. for Power, ASME Trans., Vol. 87, p. 37-49,

(Jan. 1965).

4. Hall, N. A., Thermodynamics of Fluid Flow, Chap. 6, p. 77-80,

(Prentice Hall, Inc., Englewood Cliffs, N. J., 1956).

5. Binder, R. C. , Fluid Mechanics, Chap. 14, p. 265-268, fourth

edition (Prentice Hall, Inc., Englewood Cliffs, N.J. 1962).

6. Binder, R. C, Fluid Mechanics, Chap. 13, p. 186-188 (Prentice

Hall, Inc., New York, N. Y. 1945).

7. King, H. W. , C. O. Wisler, and J. G. Woodburn, Hydraulics , Chap. 7, p. 180, fifth edition (John Wiley & Sons, Inc. , New

York, N. Y. 1955).

8. Schlichting, H. , Boundary Layer Theory, Chap. 1, p. 10

(McGraw-Hill Book Co., Inc., New York, N. Y. 1955).

9. Rouse, H. , and J. W. Howe, Basic Mechanics of Fluids , Chap. 10, p. 219-221, (John Wiley & Sons, Inc., New York,

N. Y. , 1956).

10. Ramalho, R. S. , F. M. Tiller, and V. J. Berry, Jr., "New

Friction -Factor Chart For Pipe Flow, " Chem. Eng. , Vol. 71,

No. 15, p. 153-154 (July 1964).

11. Kays, W. M. , "Loss Coefficients for Abrupt Changes in Flow

Cross Section With Low Reynolds Number Flow in Single and

Multiple -Tube Systems, " ASME Trans. , Vol. 72, 1067-1074

(1950).

12. Benedict, R. P., N. A. Carlucci, and S. D. Swetz, "Flow Losses

in Abrupt Enlargements and Contractions," Journ. of Eng. for

Power, ASME Trans., Vol. 88, p. 73-81 (Jan. 1966).

23
13. Benedict, R. P. , "On the Determination and Combination of Loss

Coefficients for Compressible Fluid Flows, " Journ. of Eng.

for Power, ASME Trans. , Vol. 88, p. 67-72 (Jan. 1966).

14. Langhaar, H. L. , "Steady Flow in the Transition Length of a Straight Tube," Journ. of Appl. Mech. , Vol. 9, Trans. ASME,

Vol. 64, p. A-55 (1942).

15. Latzko, H. , "Heat Transfer in a Turbulent Liquid or Gas Stream,"

Z. angew Math. u. Mech., Vol. 1, p. 268-290 (1921); English

Translation, NACA Tech. Memo. 1068(1944).

16. Foust, A. S. , L. A. Wenzel, C. W. Clump, L. Maus, L. B.

Anderson, Principles of Unit Operations , Chap. 13, p. 162-

163, (John Wiley & Sons, Inc., New York, N. Y. , (1964).

17. Lundgren, T. S. , E. M. Sparrow and J. B. Starr, "Pressure

Drop Due to the Entrance Region in Ducts of Arbitrary Cross

Section," Journ. of Basic Eng., ASME Trans., Vol. 86,

p. 620-626 (Sept. 1964).

18. McComas, S. T. , and E. R. G. Eckert, "Laminar Pressure

Drop Associated with the Continuum Entrance Region and for

Slip Flow in a Circular Tube, " Journ. of Appl. Mech. , ASME

Trans., Vol. 32, p. 765-770 (Dec. 1965).

19. Campbell, W. D. and J. C. Slattery, "Flow in the Entrance of a Tube," Journ. of Basic Eng., ASME Trans., Vol. 85,

p. 41-46 ('Mar. 1963).

20. Sparrow, E. M. , and S. H. Lin, "The Developing Laminar Flow

and Pressure Drop in the Entrance Region of Annular Ducts, "

Journ. of Basic Eng. ASME Trans., Vol. 86, p. 827-834

(Dec. 1964).

24
21. McComas, S. T. , "Hydrodynamic Entrance Lengths for Ducts of

Arbitrary Cross Section," Journ. of Basic Eng. , ASME Trans.,

Paper No. 67-FE-4.

22. Kline, S. J. and A. H. Shapiro, "Experimental Investigation of

the Effects of Cooling on Friction and on Boundary -Layer

Transition for Low -Speed Gas Flow at the Entry of a Tube, " NACA Tech. Note 3048 (Nov. 1953).

23. Gillis, J. and M. Shimshoni, "Initial Flow in the Entrance of a Straight Circular Tube, " Journ. Roy. Aero. Soc. , Vol. 70,

p. 368-369 (Feb. 1966).

24. Hornbeck, R. W. , "Laminar Flow in the Entrance Region of a Pipe," Appl. Sci. Res., Vol. 13, Sec. A, p. 224-232 (1964).

25. Christiansen, E. B. , and H. E. Lemmon, "Entrance Region

Flow," AICHE Journ., Vol. 11, p. 995-999 (Nov. 1965).

26. Dealy, J. M. , "A Neglected Effect in Entrance Flow Analyses, " AICHE Journ., Vol. 11, p. 745 (July 1965) Discussion by

J. D. Heliums, Vol. 12; p. 197-198 (Jan. 1966).

27. Sparrow, E. M. , S. H. Lin and T. S. Lundgren, "Flow Development in the Hydrodynamic Entrance Region of Tubes and Ducts, " The Physics of Fluids, Vol. 7, No. 3, p. 338-347 (Mar. 1964).

28. Barbin, A. R. , and J. B. Jones, "Turbulent Flow in the Inlet

Region of a Smooth Pipe, " Journ. of Basic Eng. , ASME Trans. , Vol. 85, p. 29-34 (Mar. 1963).

25

GPO 837-855


I.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COM MERC

OFFICIAL BUSINESS

^BBBU