esdu-01016

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.

01016

Endorsed by The Institution of Mechanical Engineers

ESDUAn IHS GROUP Company

Engineering Sciences Data Unit

TM

Issued March 2002Supersedes ESDU 89040

Pressure losses in flow through asudden contraction of duct area

Associated software: ESDUpac A0116VIEWpac 0116A

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.

01016ESDUEngineering Sciences Data Unit

TM

ESDU DATA ITEMS

Data Items provide validated information in engineering design and analysis for use by, or under the supervision of,professionally qualified engineers. The data are founded on an evaluation of all the relevant information, both published andunpublished, and are invariably supported by original work of ESDU staff engineers or consultants. The whole process issubject to independent review for which crucial support is provided by industrial companies, government researchlaboratories, universities and others from around the world through the participation of some of their leading experts on ESDUTechnical Committees. This process ensures that the results of much valuable work (theoretical, experimental andoperational), which may not be widely available or in a readily usable form, can be communicated concisely and accuratelyto the engineering community.

We are constantly striving to develop new work and review data already issued. Any comments arising out of your use of ourdata, or any suggestions for new topics or information that might lead to improvements, will help us to provide a better service.

THE PREPARATION OF THIS DATA ITEM

The work on this particular Data Item, which supersedes Item No. 89040, was monitored and guided by the Internal FlowPanel. This Committee first met in 1979 and now has the following membership:

The construction and subsequent development of the Data Item was undertaken by

The person with overall responsibility for the work in this subject area is Mr S.J. Pugh, Head of Transfer, Internal Flow andPhysical Data.

ChairmanDr J.A. Eaton � National University of Ireland, Galway, IrelandVice-ChairmanMr D.A. Campbell � Independent, UKMembersDr W.R. Geddes � BNFL Consultancy Services, UKDr M.E. Gill � Royal College of Military Science, Cranfield University, UKMr J. Campbell � Ove Arup and PartnersMr A.J. Green � BHR Group Ltd, UKProf. J.L. Livesey � Independent, UKDr P.J.G. Long � Cambridge University Engineering Department, UKDr S.J. Murray � W.S. Atkins Science and Technology, UKDr J.T. Turner � University of Manchester, UK

Prof. D.H. Freeston*

* Corresponding Member

� Auckland University, New Zealand

Dr C.H.N. Yuen � EngineerMr B.C. Freeeman � Senior Engineer

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

PRESSURE LOSSES IN FLOW THROUGH A SUDDEN CONTRACTION OF DUCT AREA

CONTENTS

Page

1. NOTATION AND UNITS 1

2. INTRODUCTION 3

3. CALCULATION FOR INCOMPRESSIBLE FLOW 5

4. CALCULATION FOR COMPRESSIBLE FLOW 64.1 Flow choking for sudden contractions 7

5. NOTES ON THE EXPERIMENTAL DATA 85.1 Loss Coefficients for Incompressible Flow Through Sharp-Edged Contractions 85.2 Effect of Edge Radius 85.3 Loss Coefficients for Compressible Flow 9

6. WORKED EXAMPLES 106.1 Example 1, Incompressible Flow 106.2 Example 2, Compressible Flow 10

7. REFERENCES AND DERIVATION 127.1 References 127.2 Derivation 12

APPENDIX A COMPUTER PROGRAM FOR CALCULATION OF PRESSURE CHANGES IN FLOW THROUGH A SUDDEN CONTRACTION OF DUCT AREA

A1. INSTALLING THE PROGRAM 23A1.1 Subscribers to Internal Flow Series in Hard Copy (Printed Items) 23A1.2 Subscribers to Internal Flow Series on CD-ROM 23A1.3 Subscribers to Internal Flow Series via the Web (Internet) 23

A2. RUNNING THE PROGRAM 23

A3. PROGRAM INPUT 24

A4. OUTPUT FROM THE PROGRAM 28

A5. WORKED EXAMPLES 28A5.1 Input file for example 28A5.2 Output file for example 29

i

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

APPENDIX B OUTLINE OF COMPRESSIBLE FLOW SOLUTION

B1. METHOD 31B1.1 Derivation of Graphical Solution 32B1.2 Additional Reference 32

ii

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

PRESSURE LOSSES IN FLOW THROUGH A SUDDEN CONTRACTION OF DUCT AREA

1. NOTATION AND UNITS

SI Unit

cross-sectional area of duct m2

speed of sound in gas m/s

hydraulic diameter of duct, m

kinetic-energy profile factor defined by Equation (3.3) �

total-pressure loss coefficient for compressible flow �

total-pressure loss coefficient for incompressible flow �

Mach number, �

mass flow rate kg/s

isentropic choking mass flow rate associated with , defined by Equation (4.2)

kg/s

internal perimeter of duct m

static pressure Pa (N/m2)

static-pressure loss Pa (N/m2)

total pressure Pa (N/m2)

total-pressure loss Pa (N/m2)

specific gas constant ( 287.03 J/kg K, = 96.03 ft lbf/lb K, = 3089.56 ft lbf/slug K for air)

J/kg K

Reynolds number, �

radius of contraction edge m

total temperature of fluid K

local absolute velocity m/s

bulk or continuity-mean velocity in duct, m/s

ratio of specific heat capacities of gas ( 1.4 for air) �

A

a

D 4A/PE

JKE

Kt

Kti

M V/a

m

m* A2

PE

p

p∆

pt

pt∆

R ≈

Re ρVD/µ

r

Tt

u

V m/ρA

γ ≈

Issued March 2002

1

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

Subscripts

Superscripts

factor for effect of contraction edge radius �

dynamic viscosity of fluid N s/m2

density of fluid kg/m3

u refers to conditions in larger duct where approach flow is unaffected by presence of contraction, at least upstream of contraction plane

refers to extrapolated conditions in larger duct at contraction plane (see Sketch 2.1)

refers to extrapolated conditions in smaller duct infinitesimally downstream of contraction plane, where all irreversible processes due to the contraction are deemed to be completed (see Sketch 2.1)

d refers to conditions in smaller duct where flow can be considered to be fully developed, normally at least downstream of contraction plane

' a prime denotes value for sharp-edged contraction

* denotes a value associated with choked flow conditions.

λKtiK′ti--------

µ

ρ

4D1

1

2

4D2

2

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

2. INTRODUCTION

This Item, which supersedes ESDU 89040, gives information on the pressure changes that occur whensingle-phase, Newtonian fluids flow through a straight duct that has a sudden contraction in area; noconsideration has been given to the effects of phase change such as flashing or cavitation. The methods aresupported by data obtained from experiments on sudden contractions in straight, axi-symmetric ducts ofcircular section. The data also apply to the losses at entry to a duct from a large space, when the area ratiois approximately zero.

Calculation of pressure losses for incompressible flow is considered in Section 3 and for compressible flowin Section 4. Background to the correlations is given in Section 5. A computer program is available and isdescribed in Appendix A. Appendix B outlines the solution for compressible flow.

The experimental data used to derive the graphs, equations and methods presented in this Item areinsufficient to support all the permutations of variables that are known to influence the non-dimensionalpressure-loss coefficients. These include area ratio, Reynolds number, edge radius and, in the case ofcompressible flow, a characterising Mach number. The information that is supported by experimental datais identified within the various Sections and graphs. Extrapolated results must be treated as tentative.

Sketch 2.1 Static- and total-pressure distributions near a sudden contraction

u 12 d

Total pressure

Static pressure ∆pt01

∆pt12

∆pt23

d1, 2u

∆p12

≥4D2≥4D1

D2D1

3

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

Sketch 2.1 illustrates the pressure changes for incompressible flow along a duct with a sudden contractionof area. For compressible flow, the pressure changes are similar (away from choking) except that the rateof pressure loss due to friction in the straight duct increases along the duct due to flow acceleration.

The total-pressure loss, , due to the contraction is modelled as occurring between Station 1 in thelarger duct and Station 2 in the smaller duct although, in practice, the influence of the contraction extendsboth upstream and downstream of the contraction plane. The total-pressure loss due to the contraction isdefined from measurements upstream and downstream as

, (2.1)

where and are the losses due to friction for fully-developed flow in the upstream anddownstream ducts respectively, extrapolated to the contraction plane.

The location of the measurement Stations is important in assessing pressure changes across contractions.The influence of the contraction extends both upstream and downstream of the contraction plane andmeasurements must be made in regions of fully-developed flow outside this region of influence. If thiscondition is not met, the measured pressure loss is a function of flow conditions at the measurement Stations.

Derivations 14, 15 and 16 suggest that a settling length of about 4D2 downstream of the contraction planeis sufficient to restore the pressure gradient to a fully-developed flow value. Upstream of the contractionplane, a length of 4D1 is sufficient to ensure that the measurement plane is not in the region of therecirculation zones in the corners. To ensure fully-developed flow at Station u, a duct length of typically50D1 is required upstream of Station u. However, the pressure loss due to sudden contractions is fairlyinsensitive to inlet conditions, although increasing as / increases, and a much shorter length is usuallyacceptable.

∆pt( )12

∆pt( )12 ∆pt( )ud= ∆pt( )u1� ∆pt( )2d�

∆pt( )u1 ∆pt( )2d

A2 A1

4

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

3. CALCULATION FOR INCOMPRESSIBLE FLOW

Incompressible flow conditions apply to the flow of liquids and, in the present context, are approximatelyvalid for the flow of gases when the Mach number downstream of the contraction is less than about 0.3,provided that the static-pressure change is less than about 10-15 percent of the upstream static pressure.This corresponds approximately to a Mach number upstream of the contraction of less than 0.3( ).The correlations derived from experimental data are shown in Section 5.

The total-pressure loss due to the contraction, , expressed in terms of the downstream kinetic pressure,is given by

. (3.1)

Values of for laminar and transitional flow and turbulent flow are presentedin Figures 1a and 1b respectively. A factor, , allowing for the effect of a radius on the edge of thecontraction, is given in Figure 1c for turbulent flow. No data were found for any corresponding effect inlaminar flow, but some reduction of pressure loss is to be expected. The data from which the curves ofFigure 1b are derived were restricted to incompressible flow, Reynolds numbers of the order of 105, and

. It can be deduced from the trend shown by the data that the radius correction factor varieslittle for . Extrapolated results should be treated as tentative. Values of for /are typical of loss coefficients for commercial �pipe-fitting� contractions with a small edge radius.

The static-pressure loss, , due to the contraction can be expressed as

, (3.2)

where , the kinetic-energy profile factor, allows for the effects of non-uniform velocity profiles and isgiven by

. (3.3)

As a guide in the absence of measurements, the value of may be taken as equal to 2 (the value forfully-developed laminar flow) for and equal to 1.06 (the value for fully-developed laminarflow) for . The uncertainty of these values for is very high due todifficulties in predicting the transition.

The methods are incorporated into the computer program, A0116 for calculation of pressure loss. Theprogram and its use are described in Appendix A.

A2/A1

pt12∆

pt∆ 12 Kti ½ρV22×=

Kti Re2 104≤( ) Re2 104>( )λ

0 r/D2 0.2≤ ≤r/D2 0.2> Kti r D 0.005=

p12∆

p12∆ ½ρV22 Kti JKE ,2 JKE ,1 A2 A1⁄( )2�+ =

JKE

JKE1m---- u

V--- 2

mdm∫

1A--- u

V---

3

AdA∫= =

JKERe 2000<

Re 2000≥ 1000 Re 4000< <

5

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

4. CALCULATION FOR COMPRESSIBLE FLOW

The compressible flow data are valid for and should be used for flow of gases when thedownstream Mach number exceeds about 0.3.

The analytical model used is based on an adiabatic, one-dimensional flow of a calorically-perfect (constant and ) gas and this is applied in conjunction with experimental data from Reference 7. Further details

are given in Section 5 and Appendix B.

The total-pressure loss data are presented in terms of , the total-pressure� ratio across the contraction,which is related to the total-pressure loss by

. (4.1)

The total-pressure ratio, , can be related to the total-pressure loss coefficient, , through functionsshown in Appendix B.

Values of , , , and are presented graphically as functions of the area ratio,, and the mass flow ratio, in Figures 2 to 6, respectively. These Figures apply for turbulent

flow of gases with through sharp-edge contractions. In general, rounding thecontraction edge will increase the pressure ratios and decrease and . If a measure of the effect ofedge radius is required, the computer program should be used.

Note that, although values of down to zero are included on the Figures, at small values the flowmay be laminar and the Figures are then invalid.

The isentropic choking mass flow rate, , associated with , is given by

(4.2)

for and . (4.3)

The methods are incorporated into the computer program A0116. The program and its use are described inAppendix A. The program allows for use of any appropriate values of and includes allowance for arounded contraction entry with the assumption that the edge radius affects the total-pressure loss coefficientto the same extent as it does in incompressible flow.

� Total pressure for compressible flow is given by . Further guidance on subsonic compressible flow

relationships is given in Appendix B and in Item No. 740282.

Re2 104>

cp γ

pt2/pt1

pt p 1 γ 1�2

---------- M2 +

γγ 1�( )

------------------

=

pt12∆ pt1 1 pt2/pt1�( )=

pt2/pt1 Kt

pt2/pt1 p2/p1 p2/pt1 M2 KtA2/A1 m /m*

Re2 104>( ) γ 1.4=M2 Kt

m /m*

m* A2

m∗ A2pt1γ

RTt-------- 2

γ 1+-----------

γ 1+γ 1�-----------

½

=

0.04042A2pt1

Tt

--------------= γ 1.4= R 287.03=

γ

6

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

4.1 Flow choking for sudden contractions

For the contraction alone, the idealised flow model adopted assumes that the flow chokes when reachesunity in the smaller duct just downstream of the contraction plane. However, several aspects of the realflow act to invalidate this limit in practice.

Firstly, in many real systems the flow will choke at the end of the overall duct system due to pressure lossesdownstream of a contraction, thus limiting the Mach number at the contraction exit to less than unity.

Related to this is the requirement that the real duct configuration must include a minimum downstreamlength in which the flow recovers to a fully-developed flow condition. Friction losses in that duct lengthreduce the maximum Mach number and mass-flow ratio from the values that can be achieved for theidealised model flow.

In addition, in the real flow, the flow would actually become choked when the Mach number at the venacontracta reached unity�. Since the vena contracta area is always smaller than , the correspondingMach number based on area will be less than unity and the achievable mass flow ratio will be reduced.

The data available do not allow these effects to be quantified precisely but tentative boundaries have beenimposed on Figures 2 to 6 and are included within the computer program. In practice, Mach numbers, ,above 0.7 to 0.8 are unlikely to be achieved and the maximum / * is likely to be restricted to around85% to 90% of the choking values predicted by the flow model.

� If the Mach number at the vena contracta reaches unity, the flow may become supersonic initially downstream, reducing to subsonicthrough a normal shock and the increasing again towards choking in the downstream duct.

M2

A2A2

M2m m

7

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

5. NOTES ON THE EXPERIMENTAL DATA

5.1 Loss Coefficients for Incompressible Flow Through Sharp-Edged Contractions

The laminar flow data presented in Figure 1a were derived using the theoretical equations of Derivation 12with the coefficients fitted to the experimental data� of Derivation 9. The resulting expression for isgiven by

. (5.1)

The limit of applicability of Equation (5.1) is given by

. (5.2)

Few data were found for the transitional region, which is defined here as the region between the limitingvalues of given by Equation (5.2) and of 2000. The curves in this region are based on the data ofDerivation 4 for / , extrapolated to other / using the theory of Derivation 4. The flowcondition is uncertain for , and is presented as a shaded area in Figure 1a.

Data from Derivations 4, 7, 10, 11, 13, 14, 16 and 18 were used to construct Figure 1b. The upper curve ofFigure 1b for sharp-edged contractions is given by

, (5.3)

and was derived from data from Derivations 7, 10, 14�, 16 and 18. This equation fits the data to withinpercent.

5.2 Effect of Edge Radius

The values of given in Figure 1c were obtained for incompressible flows. Values of were availablefrom Derivations 13, 14, 16 and 17, for , 0.2 and 0.53 respectively. The data were correlated by

. (5.4)

The data are restricted to . Also, if exceeds

, (5.5)

the radius cannot be tangential to both the downstream duct and the contraction face, introducing a furtheruncertainty.

� Data from Derivation 8 are considered unreliable because the upstream pressure tapping is too close to the plane of the contraction.� The data from Derivation 14 for sharp-edged contractions are consistently low. These data relate to measurements taken with a test rig

with steady, uniform upstream flow. There is some evidence (Derivation 17) that increased pressure losses would result if the upstreamflow conditions were disturbed.

Kti

Kti 0.32 159/Re2+[ ] 1 A2/A1( )2�[ ]=

Re 165 7.515 A2 A1⁄( )1.8[ ]exp+=

Re Re2A2 A1 0.28= A2 A1

2000 Re 4000≤ ≤

K ′t i 0.54 0.004 A2/A1( ) 1.285 A2/A1( )2 0.741 A2/A1( )3+�+=

10±

λ λA2/A1 0=

λ 0.455 e 17.5 r/D2( )� e 19.5 r /D2( )� 0.198+ +{ }=

r/D2 0.2≤ r/D2

r /D2 ½ 1A2 A1⁄---------------- 1�=

8

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

5.3 Loss Coefficients for Compressible Flow

An empirical relationship for the total-pressure loss coefficient for compression flow, , was derived fromthe data of Derivation 7 as

. (5.6)

The equation also meets the constraints that for and for . The data forDerivation 7 cover a range of from about 0.3 to 0.7 and apply for sharp-edged contractions only.

The application of Equation (5.6) is outlined in Appendix B.

No data were found for the effect of edge radius in compressible flow. For the computer program it hasbeen assumed that he values of given by Equation (5.4) can be applied to compressible flow within thevalue of in Equation (5.6).

Kt

Kt Kti 0.2M21.85 1.23 M2

6.3� 10.6 Kti

1.1�

0.6-------------------------

2

�½

+=

Kt 0= A2/A1 Kt Kti= M2 0=M2

λKti

9

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

6. WORKED EXAMPLES

6.1 Example 1, Incompressible Flow

Find the total-pressure loss due to a reduction in area from a 20 mm bore pipe to a 13 mm bore pipe forwhich the reduction fitting is a sudden contraction with sharp edges. The fluid is water, flowing at a rateof 0.5 kg/s, and its density and dynamic viscosity may be taken as 1000 kg/m3 and 0.001 N s/m2 respectively.

The flow velocity, , is given by , i.e.

m/s.

The downstream Reynolds number, is , i.e.

.

The contraction ratio is given by so

.

For water flow, the assumption of incompressible flow is valid and Figure 1b is appropriate because. From Figure 1b, and hence the total-pressure loss across the contraction from

Equation (3.1) is

Pa.

6.2 Example 2�, Compressible Flow

Find the downstream static pressure and downstream Mach number in a planar intake with an intakediameter of 0.2 m for two mass flow rates of (a) 1.0 and (b) 5.0 kg/s. The other data are

Pa, K,

J/kg K, and N s/m2.

The intake may be approximated as a contraction of with and . The methodof Section 4, dealing with compressible flow, will be used because the downstream Mach number may begreater than 0.3. For this method to be valid, the downstream Reynolds number must be greater than .To check the validity, is calculated as follows:

(a) for the mass flow rate of 1.0 kg/s

.� This example is evaluated in Appendix A, using the computer program for both a sharp edge and with an edge radius of 0.008m as an

illustration of the effect of edge radius.

V2 m / ρA2( )

V2 0.5/ 1000 π4--- 0.0132××

3.77= =

Re2 ρV2D2/µ

Re2 1000 3.77 0.013/0.001 4.9 104×=××=

A2/A1 D2/D1( )2=

A2/A1 0.013/ 0.02( )( )2 0.42= =

Re2 104> Kti 0.37=

pt12∆ Kti½ρV2= 2 0.37 ½ 1000 3.772 2629=×××=

pt1 105= Tt 288=

R 287.03= γ 1.4= µ 18 6�×10=

A2/A1 0= M1 0= pt1 p1=

104

Re2

Re2 ρ2V2D2/µ 4m / D2πµ( ) 4 5/ 0.2 π 18 10 6�×××( )×===

3.54 5×10=

10

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

(b) for the mass flow rate of 5.0 kg/s

.

Since , the method of Section 4 is valid for both cases.

To evaluate the downstream static pressure and Mach number, the ratios and must first becalculated.

From Equation (4.2)

where m2. Thus, using Equation (4.3),

kg/s

and

(a) m / m* = 1.0 /7.38 = 0.135,

(b) m / m* = 5.0 /7.38 = 0.68.

Hence, from Figure 3,

(a) so that

Pa,

(b) so that

Pa.

From Figure 5,

(a) M2 = 0.08,

(b) M2 = 0.49.

Re2 1.77 106×=

Re2 104>

m /m* r/D2

m∗ A2pt1γ

RTt--------- 2

γ 1+------------- γ 1+( )/ γ 1�( )

½

=

A2 π 0.2( )2/4 0.031==

m∗ 0.04042 0.031 105××γ288

--------------------------------------------------------- 7.38= =

p2/p1 0.994=

p2 0.994 105×=

p2/p1 0.78=

p2 0.78 105×=

11

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

7. REFERENCES AND DERIVATION

7.1 References

The references given are recommended sources of information supplementary to that in this Item.

7.2 Derivation

The derivation lists selected sources that have assisted in the preparation of this Item.

1. ESDU Friction losses for fully-developed flow in straight pipes. Data Item No.66027, ESDU International, London, 1966.

2. ESDU One-dimensional compressible gas flow in ducts. Data Item No. 74028,ESDU International, London, 1974.

3. ESDU Friction losses for fully-developed flow in straight pipes of constantcross section � subsonic compressible flow of gases. Data Item No.74029, ESDU International, London, 1974.

4. KAYS, W.M. Loss coefficients for abrupt changes in flow cross section with lowReynolds number flow in single and multiple-tube systems. Trans. am.Soc. mech. Engrs, Vol.72, pp.1067-1074, November 1950.

5. IDEL'CHIK, I.E. Handbook of hydraulic resistance. AEC-tr-6630. US Atomic EnergyComm., 1966. (Available from US Dept Commerce, Springfield, Va.Transl. from Spravochnik po gidravlicheskim soprotivleniyam. Gos.Energ. Izd., Moscow, 1960.)

6. BONNINGTON, S.T. Measurements of the pressure losses in copper fittings. BHRA Rep.RR719, Brit. Hydromechanics Res. Assoc., Cranfield, UK, 1962.

7. BENEDICT, R.P.CARLUCCI, N.A.SWETZ, S.D.

Flow losses in abrupt enlargements and contractions. J. Engng Power,Vol.88, No.1, pp.73-81, 1966.

8. ASTARITA, G.GRECO, G.

Excess pressure drop in laminar flow through sudden contraction. Ind.Engng Chem. Fundam., Vol.7, No.1, pp.27-31, February 1968.

9. KAYE, S.E.ROSEN, S.L.

The dependence of laminar entrance loss coefficients on contractionratio for Newtonian fluids. Am. Inst. chem. Engrs J., Vol.17, No.5,pp.1269-1270, September 1970.

10. LEVIN, L.CLERMONT, F.

Étude des pertes de charge singulières dans les convergents coniques.Le Génie Civil, T. 147, No. 10, pp. 463-470, 1970.

11. MILLER, D.S. Internal flow � a guide to losses in pipe and duct systems. Brit.Hydromechanics Res. Assoc., Cranfield, UK, 1971.

12. CHRISTIANSEN, E.B.KELSEY, S.J.CARTER, T.R.

Laminar tube flow through an abrupt contraction. Am. Inst. chem. EngrsJ., Vol.18, No.2, pp.372-380, March 1972.

12

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

13. IDEL'CHIK, I.E. Some refinements of the contraction and resistance coefficients in thecase of a sudden contraction of the flow. Hydrotech Construction,Vol.20, No.10, pp.591-594, October 1986.

14. BULLEN, P.R.CHEESEMAN, D.J.HUSSAIN, L.A.RUFFELL, A.E.

The determination of pipe contraction pressure loss coefficients forincompressible turbulent flow. Int. J. Heat Fluid Flow, Vol.8, No.2,pp.111-118, June 1987.

15. DURST, F.SCHIERHOLZ, W.F.WUNDERLICH, A.M.

Experimental and numerical investigations of plane duct flow withsudden contraction. Trans. am. Soc. mech. Engrs, Vol.109, pp.376-383,December 1987.

16. BULLEN, P.R.CHEESEMAN, D.J.HUSSAIN, L.A.

The effects of inlet sharpness on the pipe contraction pressure losscoefficient. Technical note, Int. J. Heat Fluid Flow, Vol. 9, No. 4, pp.431-433, December 1988.

17. CHEESEMAN, D.J.BULLEN, P.R.

Private communication, February 1989.

18. PAL, R.HWANG, C.Y.J.

Loss coefficients for flow of surfactant-stabilised emulsions throughpipe components. Trans. Instn. Chem. E, Vol. 77, Part A, pp. 685-691,November 1999.

13

ESDU product issue: 2003-03. For current status, contact ESDU. Observe Copyright.

01016E

SD

UE

ngineeringS

ciencesD

ataU

nit TM

D TRANSITIONAL FLOW

1042 3 4 5 6 8

||||||||||

||||||||||

||||||||

||

A2 / A1

0

0.6

0.8

0.9

0.95

A10.6 0.8 1.0

] [ 1 � (A2 /A1)2 ] (5.1)

applicability of Equation (5.1)

ertain

10

40

100

104

14

FIGURE 1a TOTAL-PRESSURE LOSS COEFFICIENT FOR INCOMPRESSIBLE LAMINAR AN

Re2

100 101 102 103

K'ti + 0.1

2

3

4

5

6

8

2

3

4

5

6

8

2

3

4

5

6

8

A2 / A1

0

0.6

0.8

0.9

0.95

21 2 3 4 5 6 8 2 3 4 5 6 8

||||||

||||||

||||||

||||||

||||||

||||||

||||||

||||||||

||||||||

||||||||

|||

||||||||

||||||||

||||||||

||

||||||||||

||||||||||

|||||||

||||||||||

|||||||

0.1

1

10

102

3

WARNINGThe ordinate of this Figureis K'ti + 0.1

8654

A2/0.0 0.2 0.4

K'ti + 0.1

0.1

1

10

100

K'ti = [ 0.32 + 159Re2

Limit of||||||||||||||||||||||||| |||||||||||||||||||||||||||

Data unc

Re2

103Limit on main figure


15

01016E

SD

UE

ngineeringS

ciencesD

ataU

nit TM

T FLOW,

0.8 0.9 1.0

}198.0)/(5.19� 2 +Dr

3

1

2

2

1

2 741.0285.

+

AA

AA

(5.3)

(5.4)

den 05

Re2 104≥≥≥≥
FIGURE 1b TOTAL-PRESSURE LOSS COEFFICIENT FOR INCOMPRESSIBLE TURBULEN
A2 / A1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Kti

0.0

0.1

0.2

0.3

0.4

0.5

0.6

r / D20.0

0.02

0.04

0.06

0.08

0.10

0.15

$0.20

0.0

K'ti{455.0 )/(5.17� 2 += Dr eeλ

Kti = λK'ti

KNti 1

2 1004.054.0 −

+=

AA

0.005

See Section 5.2Typical commercial sudcontractions r/D2 = 0.0


TM

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.

FIGURE 1c EDGE RADIUS FACTOR λ FOR TURBULENT FLOW

r/D2

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

λ

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

extrapolation

λ = 0.455 {e�17.5(r/D2) + e�19.5(r/D2) + 0.198} (5.4)

0.9

16


01016E

SD

UE

ngineeringS

ciencesD

ataU

nit TM

ND = 0

1.00 0

Tentative limit for choked exit

*

0.10.2

0.3

0.4

0.5

6

rD2-------

17

FIGURE 2 TOTAL PRESSURE RATIO, , FOR COMPRESSIBLE FLOW, γ = 1.4 A

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

pt2/pt1

0.95

m/m

0.95

0.9

0.8

0.

0.7A2/A1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.20.1 0

pt2pt1-------


18

01016E

SD

UE

ngineeringS

ciencesD

ataU

nit TM

ND = 0


00.1

0.20.3

0.4

0.5

rD2-------
FIGURE 3 STATIC PRESSURE RATIO, , FOR COMPRESSIBLE FLOW, γ = 1.4 A
0.5

0.6

0.7

0.8

0.9

1.0

p2/p1

0.95

A2/A1

m/m*

0.6

0.950.9

0.8

0.7

0.9

0.8

0.7

0.60.5 0.4 0.3 0.2 0.1 0

p2p1-----


19

01016E

SD

UE

ngineeringS

ciencesD

ataU

nit TM

= 0


m/m*

0.4

0.30.2

0.10

rD2-------
FIGURE 4 PRESSURE RATIO, FOR COMPRESSIBLE FLOW, γ = 1.4 AND
0.5

0.6

0.7

0.8

0.9

1.0

p2/pt1

0.95

m/m*

0.40.5

0.6

0.7

0.8

0.9

0.95

A2/A1

00.10.20.30.40.50.60.70.80.9

1.0

p2pt1-------


20

01016E

SD

UE

ngineeringS

ciencesD

ataU

nit TM

.4 AND = 0

0.95

0.70.8

0.9

1.0

m/m*

0.95

0.9

0.8

0.7

0.6

0.5

rD2-------
FIGURE 5 DOWNSTREAM MACH NUMBER, , FOR COMPRESSIBLE FLOW, γ = 1
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M2


A2/A1

0 0.1 0.2 0.30.4

0.50.6

0.4

0.3

0.2

0.1

0

M2


01016E

SD

UE

ngineeringS

ciencesD

ataU

nit TM

= 1.4 AND = 0

A2/A1

0.5 0.4 0.3 0.2 0.1 0

rD2-------

21

FIGURE 6 TOTAL-PRESSURE LOSS COEFFICIENT, , FOR COMPRESSIBLE FLOW, γ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Kt


m/m*

m/m*

0.7

0.95

0.9

0.8

0.6

0.9

1.00.95

0.95

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.20.1 0

Kt

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

Blank Page.

22

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

APPENDIX A COMPUTER PROGRAM FOR CALCULATION OF PRESSURE CHANGES IN FLOW THROUGH ASUDDEN CONTRACTION OF DUCT AREA

The program is provided in two versions, as an interactive program, 0116AVxx, running through ESDUviewand as FORTRAN source code, A0116Vxx, where Vxx indicates the version number The program includescalculation procedures for both compressible and incompressible flow.

A1. INSTALLING THE PROGRAM

A1.1 Subscribers to Internal Flow Series in Hard Copy (Printed Items)

Both versions are supplied on CD-ROM or 3½� disk in the Internal Flow Software Volume. Installationinstructions are provided as appropriate. The default installation is to a directory/folderC:\ESDU\Viewpacs\0116A for the interactive version and C:\ESDU\ESDUpacs\A0116 for thesource code version.

A1.2 Subscribers to Internal Flow Series on CD-ROM

Both versions are automatically installed with the Series installation.

The default installation is to a directory/folder C:\ESDU\Viewpacs\0116A for the interactive versionand C:\ESDU\ESDUpacs\A0116 for the source code version.

A1.3 Subscribers to Internal Flow Series via the Web (Internet)

Subscribers who have opted for Web delivery can access both versions on the ESDU web site(www.esdu.com), by selecting Fluid Mechanics Internal Flow... Volume 4b... Data Item 01016... Abstract.

A2. RUNNING THE PROGRAM

The interactive program is run within ESDUview, a user-friendly environment for running ESDUpacson personal computers. ESDUview manages all operations such as setting up input files, running the Fortrancode and viewing the output files. On selecting ESDUpac A0116 from the menu of available programs, theuser is prompted for all of the input data. Guidance is provided on the features of ESDUview and on theprogram variables in the context-sensitive HELP facility. At various points throughout operation a numberof checks are carried out on the entered data. Error messages alert the user to incorrect numerical valuesand guide in their correction; other potential problems are highlighted by warning messages. TheESDUview version of the program is recommended for its ease of operation.

The ESDUview version can be operated without reference to this Data Item but the user may find usefulthe background information on the data and model. The features of ESDUview are described fully in ESDU00009.

The source code is written in ANSI Standard FORTRAN 77. It is supplied in 15 files. The following filesshould be present: A0116V10.FOR (the Main Program file) and A0116A10.FOR, A0116B10.FOR,A0116C10.FOR, A0116D10.FOR, A0116E10.FOR, A0116F10.FOR, A0116G10.FOR,A0116H10.FOR, A0116I10.FOR, A0116J10.FOR, A0116K10.FOR, A0116L10.FOR,A0116M10.FOR and A0116N10.FOR.

Vxx Vxx

VxxVxx

23

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

In order to run this version, the source code must be compiled� for the operating system on which it is tobe run. Some guidance on compilation of the program is given in the �Introduction to ESDUpacs� in theFluid Mechanics, Internal Flow Software Volume.

To run the executable version of the program, the input data file must be specified on the command lineusing the operating system redirection symbols, e.g. for MS-DOS© running under Windows.

A0116V10 < inputfile

where inputfile is the name of the input data file to be used. In this case, the input file must be assembledseparately using a file editor that can save the file as an unformatted ASCII script. The output file name isread from the input file and should not be specified on the command line. Section A3 describes theconstruction of the input file.

Output from the program is to a formatted file, named by the user, summarising the input data and listingcalculated results. Warnings and error messages are also output to this file.

Guidance on the input and output is given in Sections A3 and A4 respectively. Section A5 provides a workedexample that illustrates the use of the program and may be used as a test case.

A3. Program Input

The program A0116 requires a prewritten input file containing the necessary input data. Table A3.1 providesa guide to construction of an appropriate file. Each entry� in the input file must be preceded by theappropriate entry label in quotes e.g. the first entry under general data input might be 'G01', 'INPUT1.DAT'.

If the program is run within ESDUview, either directly or from ESDUscope, an interface prompts the userfor all the required entries and assembles the input file. The entry label is then automatically inserted bythe ESDU Interface program.

The input data required by the program are divided into four groups of related entries: general data, ductgeometry data, flow conditions data and fluid property data.

General data includes input and output file names, a run title and flags to define choices of incompressibleor compressible flow calculation and units of input and output data. Four choices of units are available, SI(kg, N, m, s) and 3 alternative "British" systems using different combinations of force, mass and lengthunits. Examples of the appropriate units for SI and one of the British systems are provided in Table A3.1.The units chosen for input and output can be different.

The flow conditions data input requires specification as to which two of the three parameters mass flowrate, upstream total pressure and upstream static pressure will be input. Multiple values can be entered foreach of the chosen input parameters up to a limit at which the product of the numbers of entries does notexceed 25. Total temperature is required only for compressible flow calculations.

Under fluid property data, the dynamic viscosity is required together with, for incompressible flow, thefluid density and, for compressible flow, the ratio of specific heat capacities and specific gas constant. Thefluid density can be entered as 0, in which case only Kti will be calculated. If 0 is entered for specific heat

� A compiled version of the program, in a format suitable for running in a DOS window under Microsoft Windows© may be found in theESDUpacs, Epacs16 or Epacs32 subdirectory within the appropriate Viewpacs directory.

� In the case of multiple entries such as several mass flow rates, the entry label should appear once only, preceding the set of entries. SeeSection A5 for an example input file.

24

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

capacity ratio, , or specific gas constant, , default values of 1.4 and 287.03 (typical values for air) areused.

The program reads the input file sequentially and uses the entry label to allocate entries to the appropriateparameter. If entries are missing or invalid, default values are substituted where possible. Note that,especially in the case of missing or invalid entries for file names or titles, the actual error in the input filemay sometimes occur on an entry earlier than that for which an error message is generated.

γ R

25

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

TABLE A3.1 Construction of Program Input File

Entry Label Program data entry

UnitsEntry

SI British

'G01' Enter file name for input file, up to 256 characters (including path, if required�) between quotes.

- -

'G02' Enter file name for output file, up to 256 characters (including path, if required�) between quotes.

- -

'G03''G04''G05'

Enter data set name or run title. Three lines, each of up to 75 characters between quotes. All lines must be present but can be blank (enter ' ').'Line 1 of title.''Line 2 of title.''Line 3 of title.'

---

---

'G06' Units for data output, enter:- 1 for SI units, 2 for British units, (slug, lbf/ft2, ft), 3 for British units, (lbm lbf/ft2, ft), 4 for British units, (lbm lbf/in2 (psi), in).

- -

'G07' Units of input data, enter:- 1 for SI units, 2 for British units (slug, lbf/ft2, ft), 3 for British units (lbm, lbf/ft2, ft), 4 for British units (lbm, lbf/in2 (psi), in).

Entry G08 is not used in this program.

'G09' Enter 1 for incompressible flow calculation or2 for compressible flow calculation.

- -

'S01' Enter upstream duct equivalent diameter�. m ft

'S02' Enter downstream duct equivalent diameter. m ft

'S03' Enter choice for contraction entry edge condition: 0 if sharp, 1 if radiused.

- -

'S04' If entry S03 is 0, no entry required. - -

If entry S03 is 1, enter contraction entry edge radius. m ft

'P01' Choice of flow properties input, enter:-1 for mass flow rate and upstream total pressure,2 for mass flow rate and upstream static pressure,3 for upstream total pressure and upstream static pressure,4 for mass flow rate only (only part calculation possible).

- -

'P02' If Entry P01 is 3, no entry required. - -

If Entry P01 is 1, 2 or 4, enter number of mass flow rates to be input. - -� To define a duct inlet from a large space, enter a large number here, e.g. 106.� The path will be assumed relative to the current working directory unless a path starting from the root directory is given explicitly. For ESDUview, the default installation working directory is C:\ESDU\WORKING\A0116 , where denotes the program version number.

(Table continues overleaf)

Vxx Vxx

26

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

'P03' If Entry P01 is 3, no entry required. - -

If Entry P01 is 1, 2 or 4, enter specified number of values of mass flow rate.

kg/s lbm/s

'P04' If Entry P01 is 2 or 4, no entry required. - -

If Entry P01 is 1 or 3, enter number of upstream total pressures to be input.

- -

'P05' If Entry P01 is 2 or 4, no entry required; - -

If Entry P01 is 1 or 3, enter specified number of values of upstream total pressure.

N/m2 lbf/ft2


If Entry P01 is 2 or 3, enter number of upstream static pressures to be input.

- -


If Entry P01 is 2 or 3, enter specified number of values of upstream static pressure.

N/m2 lbf/ft2

'P08' If entry G09 is 1, no entry required. - -

If entry G09 is 2, enter total temperature of flow or enter 0 to use a default value of 288 K.

K K

'D01' Enter dynamic viscosity of fluid. N s/m2 lbf s/ft2

'D02' If entry G09 is 1 (incompressible flow), enter fluid density(if entered as 0, only Kti values can be calculated).

kg/m3 lbm/ft3

If entry G09 is 2, no entry required. - -

'D03' If entry G09 is 1, no entry required. - -

If entry G09 is 2 (compressible flow), enter ratio of specific heat capacities for fluid, γ, or enter 0 to use a default value for air (1.4).

- -

'D04' If entry G09 is 1, no entry required. - -

If entry G09 is 2 (compressible flow), enter specific gas constant for fluid or enter 0 to use a default value for air (287.03 J/kg K).

J/kg K CHU/lbm K

THE INPUT DATA FILE IS NOW COMPLETE.

TABLE A3.1 Construction of Program Input File (Continued)

Entry Label Program data entry

UnitsEntry

SI British

27

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

A4. OUTPUT FROM THE PROGRAM

The program output is written to a file named by the user in the input file. If the file name (or path) providedis invalid or the entry is missing, the output is written to a file named RSA0116.OUT in the current directory.The program output is in two sections.

The first section lists the input data read from the input file together with error or warning messagesassociated with the input data.

The second section lists default data values assigned, the calculated duct geometry and a tabulation ofcalculated values of flow properties together with notes on the flow conditions and errors or warnings whereappropriate. The tabulation includes the mass flow rate and upstream and downstream total and staticpressures. For incompressible flow, the upstream and downstream Reynolds numbers, the pressure losscoefficient and total- and static- pressure losses are also given. For compressible flow, the mass flow ratio,upstream and downstream Mach numbers and three pressure ratios are also given together with the pressureloss coefficient and upstream and downstream mass-flow functions.

A5. WORKED EXAMPLES

Example 2 in Section 6.2 is solved in order to illustrate the use of the computer program. The required datafrom Example 2 are entered in the input file

In order to achieve the desired area ratio of zero (sudden contraction from an infinite volume) the inletdiameter, , is set to a large number, i.e. . The input to the program and the corresponding outputfile appear in Sections A5.1 and A5.2 respectively.

A5.1 Input file for example

'G01', 'EXAMPLE2.DAT''G02', 'EXAMPLE2.OUT''G03', 'Example 2 of ESDU 01016.''G04', 'Rounded duct entry, compressible air flow.''G05', 'See Section 6.2 and A5 of ESDU 01016.''G06', 1'G07', 1'G09', 2

'S01', 1.0e4'S02', 0.200'S03', 1'S04', 0.008

'P01', 1'P02' 2'P03', 1.0,5.0'P04', 1'P05', 1.0e5'P08' 288

'D01', 18e-6'D03', 1.4'D04', 287.03

D1 1 104×

28

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

A5.2 Output file for example

***************************************************************************** ESDU International plc. PROGRAM A0116V10 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ESDUpac Version: 1.0 May 2002. ESDUpac Title: PRESSURE LOSSES IN FLOW THROUGH A SUDDEN CONTRACTION OF DUCT AREA. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Data Item Number: 01016. Data Item Title: PRESSURE LOSSES IN FLOW THROUGH A SUDDEN CONTRACTION OF DUCT AREA ***************************************************************************** INPUT DATA ***************************************************************************** Data Set Title. Example 2 of ESDU 01016. Rounded duct entry, compressible air flow. See Sections 6.2 and A5 of ESDU 01016. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Input File Name: EXAMPLE2.DAT Output File Name: EXAMPLE2.OUT ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Units. SI units chosen for output. SI units chosen for input. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Calculation type 2, compressible flow. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Duct Geometry. Upstream duct equivalent diameter, D1, = .1000E+05 m Downstream duct equivalent diameter, D2, = .2000 m Rounded contraction entry edge. Contraction edge radius, r = .8000E-02 m ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Flow Conditions. Input Type 1, mass flow rate and total pressure. 2 mass flow rates entere, values in kg/s 1.0000 5.0000 Upstream total pressure, pt1, = .1000E+06 N/m^2 Upstream static pressure not specified. Total temperature, Tt, = 288.0 K ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Fluid Properties. Dynamic viscosity, mu, = .18000E-04 N s/m^2 Ratio of specific heat capacities, gamma, = 1.4000 Specific gas constant, Rg, = 287.03 J/kg K ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ No errors or warnings were generated from these input data. ***************************************************************************** CALCULATED RESULTS ***************************************************************************** General data. Compressible flow calculation. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Duct geometry. Rounded-edge duct inlet. area ratio nominally zero ( .4000E-09) edge radius ratio, r/D2, = .4000E-01

29

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Fluid properties. Dynamic viscosity, mu, = .18000E-04 N s/m^2 [input value]. Specific heat capacity ratio, = 1.4000 [input value]. Specific gas constant, Rg, = 287.03 J/kg K [input value]. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Flow properties. Input values of mass flow rate and upstream total pressure were used. Total temperature, Tt, = 288.0 K [input value].

Isentropic choking mass flow rate, m*, = 7.482 kg/s

Case mass flow pt1 p1 pt2 p2 Note rate kg/s N/m^2 N/m^2 N/m^2 N/m^2 1 1.000 .1000E+06 .1000E+06 .9988E+05 .9946E+05 D 2 5.000 .1000E+06 .1000E+06 .9608E+05 .8339E+05 D ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~

Case m/m* M1 M2 pt2/pt1 p2/p1 p2/pt1 Note 1 .1337 .3094E-10 .7772E-01 .9988 .9946 .9988 D 2 .6683 .1547E-09 .4545 .9608 .8339 .8339 D ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ Case Kt Qtt1 Qts1 Qtt2 Qts2 Note 1 .2849 .3661E-10 .3661E-10 .9163E-01 .9202E-01 D 2 .3250 .1830E-09 .1830E-09 .4763 .5487 D

~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ Notes. D. This case is in the turbulent flow regime for contraction losses. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ***************************************************************************** END OF OUTPUT FILE EXAMPLE2.OUT *****************************************************************************

These results may be compared with the following table of results for a sharp-edged entry for illustrationof the effect of using the simplified graphical calculation method in Section 6.2.

CALCULATED RESULTS FOR SHARP-EDGED CONTRACTION~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Case mass flow pt1 p1 pt2 p2 rate kg/s N/m^2 N/m^2 N/m^2 N/m^2 1 1.000 .1000E+06 .1000E+06 .9977E+05 .9935E+05 2 5.000 .1000E+06 .1000E+06 .9245E+05 .7905E+05 ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~

Case m/m* M1 M2 pt2/pt1 p2/p1 p2/pt1 1 .1337 .3094E-10 .7781E-01 .9977 .9935 .9935 2 .6683 .1547E-09 .4784 .9245 .7905 .7905 ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~

Case Kt Qtt1 Qts1 Qtt2 Qts2 1 .5419 .3661E-10 .3661E-10 .9173E-01 .9212E-01 2 .5958 .1830E-09 .1830E-09 .4949 .5789 ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~~ ~~~~~

30

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

APPENDIX B OUTLINE OF COMPRESSIBLE FLOW SOLUTION

B1. METHOD

The total-pressure loss coefficient is available from a correlation of experimental data (Equation (5.6)) as

. (B1.1)

For the adiabatic, one-dimensional flow of a calorically-perfect (constant ) gas through a component,the total-pressure loss coefficient can be expressed in terms of the upstream and downstream Mach numbersasB1

, (B1.2)

or in terms of a mass-flow ratio, , as

, (B1.3)

where is the mass flow rate for isentropic flow through the component given by Equation (4.2). Allcoefficients here are based on the kinetic pressure, , at the downstream reference station, Station2. The parameters and can be related by

. (B1.4)

With given values of and or , a solution for can be found by equating Equations (B1.1)and (B1.2) or (B1.3). An iterative solution is necessary. With and known, the total- andstatic-pressure ratios can be obtained from

, (B1.5)

and

. (B1.6)

Kt Kti 0.2M21.85 1.23 M2

6.3� 10.6 Kti

1.1�

0.6-------------------------

2

�½

+=

cp

Kt

1 γ 1�2

------------ M22 +

γ

γ 1�------------

½γM22

----------------------------------------------------A2A1------

M2M1-------

1 γ 1�2

------------M22+

1 γ 1�2

------------M12+

---------------------------------

1 γ+

2 1 γ�( )---------------------

1�=

m/m*

Kt

1 γ 1�2

------------ M22 +

γ

γ 1�------------

½γM22

----------------------------------------------------M2

m /m*--------------

1 γ 1�2

------------ M22+

γ 1+2

----------------------------------------------

1 γ+

2 1 γ�( )---------------------

1�=

m*½ρ2V2

2

M1 m/m*

1m /m*--------------

A2A1------ 1

M1-------

γ 1+2

-------------

1 γ 1�2

------------ M12+

---------------------------------

1 γ+

2 1 γ�( )---------------------

=

A2/A1 M1 m/m* M2M1 M2

pt2pt1-------

A1A2------

M1M2-------

1 γ 1�2

------------ M12+

1 γ 1�2

------------ M22+

---------------------------------

1 γ+

2 1 γ�( )---------------------

=

p2p1-----

A1A2------

M1M2-------

1 γ 1�2

------------M12+

1 γ 1�2

------------M22+

---------------------------------

½

=

31

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

Other conditions at the upstream and downstream stations can be obtained from the relationships

, (B1.7)

and , (B1.8)

noting that, for adiabatic flow, is constant.

The ratio of downstream static- to upstream total-pressure is given by

. (B1.9)

B1.1 Derivation of Graphical Solution

A graphical solution for the problem can be derived by fixing certain parameters to give families of curvesfrom which a solution can be read for many cases. The derivation of these curves is outlined below.

B1.2 Additional Reference

Choose

Choose from Equations (5.3) and (5.4)

Choose

, and from Equation (B1.4)

Choose a starting value for

, and from� Equation (B1.2)Compare and iterate Figures 5, 6

and from Equation (B1.1)

, , and from Equation (B1.5) Figures 2

, , and from Equation (B1.6) Figures 3



and from Equation (B1.9) Figures 4

�Or use , and with Equation (B1.3)

B1. ESDU Analytical pressure-loss models for one-dimensional flow ofconstant-density fluids and ideal gases in ducts and components. Item tobe issued. ESDU International, London, 2002.

ptp---- 1 γ 1�

2------------M2+

γγ 1�------------

=

TtT----- 1 γ 1�

2------------M2+

=

Tt

p2pt1-------

p2pt2-------

pt2pt1-------×=

γ

A2/A1 ⇒ Kti

m/m

m/m* γ A2/A1 ⇒ M1

M2

M1 M2 γ ⇒ Kt

M2⇒M2 Kti ⇒ Kt

M1 M2 A2/A1 γ ⇒ pt2/pt1

M1 M2 A2/A1 γ ⇒ p2/p1

M1 γ ⇒ p1/pt1

M2 γ ⇒ p2/pt2

pt2/pt1 p2/pt1 ⇒ p2/pt1

m/m* M2 γ

32

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.


TM

KEEPING UP TO DATE

Whenever Items are revised, subscribers to the service automatically receive the material required to update the appropriateVolumes. If you are in any doubt as to whether or not your ESDU holding is up to date, please contact us.

Please address all technical engineering enquiries and suggestions to:

For users in the USA, please address all Customer Service and Support enquiries and suggestions to:

ESDU International plc Tel: 020 7490 5151 (from the UK)+44 20 7490 5151 (from outside the UK)

Fax: 020 7490 2701 (from the UK)+44 20 7490 2701 (from outside the UK)

E-Mail: [email protected]: www.esdu.com

IHS Engineering Products Tel: 1 800 525 7052 (toll free number)and Fax: 1 303 397 2599Global Engineering Documents Website: www.ihs.com

www.global.ihs.com

ESD

U p

rodu

ct is

sue:

20

03

-03

. Fo

r cu

rren

t st

atus

, con

tact

ES

DU

. O

bser

ve C

opyr

ight

.

ESDUAn IHS GROUP Company

All rights are reserved. No part of any Data Item may be reprinted, reproduced, or transmittedin any form or by any means, optical, electronic or mechanical including photocopying,recording or by any information storage and retrieval system without permission from ESDUInternational plc in writing. Save for such permission all copyright and other intellectualproperty rights belong to ESDU International plc.

© ESDU International plc, 2002

Engineering Sciences Data Unit

TM

01016Pressure losses in flow through a sudden contraction of duct areaESDU 01016

ISBN 1 86246 174 0, ISSN 0141-4011

Available as part of the ESDU Series on Fluid Mechanics, Internal Flow. Forinformation on all ESDU validated engineering data contact ESDU Internationalplc, 27 Corsham Street, London N1 6UA.

This Item, which supersedes ESDU 89040, provides information on pressurechanges in flow of Newtonian fluids through a sudden contraction of duct area.Both incompressible and compressible flows are covered and data are presentedfor sharp- and rounded-edge contractions. The data may be applied also to asimple duct entry from a large space.

The methods are also implemented in computer programs ESDUpac A0116 andVIEWpac 0116A.

esdu-01016

Documents

experimental data

review data

data itemthe work

internal flow andphysical

particular data item

incompressible flow

new work

cranfield university