^ '-^' J

COMPRESSIBLE FLUID FLOW

THROUGH AN ORIFICE

by

HERSCHEL NATHANIEL WALLER, JR., B.S.

A THESIS

IN

MATHEMATICS

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

Chairmarr of the Committee

Accepted

Dean/of the I Graduate/School

May, 1973

• ^ ACKNOWLEDGEMENTS

I would like to thank Dr. Wayne Ford for allotting

time to direct the writing of my thesis and for the inter­

est he has shown in my work. I am also indebted to Dr. L.

R. Hunt for consenting to serve as a member of my committee

11

TABLE OF CONTENTS

ACKN0V7LEDGMENTS ii

LIST OF ILLUSTRATIONS iv

I. INTRODUCTION

II. EQUATIONS OF CONTINUITY

III. EULER'S EQUATIONS

IV. THE THREE TYPES OF FLUID MOTION 12

V. ROTATIONAL MOTION AND EULER'S EQUATIONS . . 16

VI. NAVIER-STOKES EQUATIONS 20

VII. BERNOULLI'S EQUATIONS 35

VIII. FLOW EQUATIONS FOR THE ORIFICE METER . . . . 41

IX. SUMMARY AND CONCLUSIONS 51

LIST OF REFERENCES 53

111

LIST OF ILLUSTRATIONS

Figure page

1. An incompressible fluid element 3

2. Forces on a fluid element 8

3. A fluid element in two-dimensional flow . . . . 12

4. Viscous fluid elements, (a) at rest,

and (b) in motion 21

5. A diagrammatic comparison of one-dimensional

(a) nonviscous flow and (b) viscous flow in

a pipe 22

6. Stresses on an infinitesimal volume of a

viscous fluid 23

7. An orifice type differential meter with

U-tube manometer 41

IV

CHAPTER I

INTRODUCTION

The purpose of this thesis is to show the develop­

ment of the fluid flow equation used almost everywhere in

the United States to calculate the rate of flow of natural

gas through an orifice.

This purpose is accomplished in, essentially, two

steps:

(1) Starting with the most fundamental relation­

ships, the Navier-Stokes equations for compressible fluids

are derived. These equations allow for not only the usual

hydrostatic forces but also the forces due to friction

between adjacent fluid elements and between the fluid and

its container. The various types of fluid flow are dis­

cussed, and the Euler equations are developed.

(2) Using a multitude of assumptions the Navier-

Stokes equations are reduced to the fluid flow equation

used in the natural gas industry to calculate flow rate

through an orifice. Several of the assumptions are listed

and discussed.

CHAPTER II

EQUATIONS OF CONTINUITY

Incompressible Fluids

Consider an element of incompressible fluid volume.

Let the element be a rectangular parallelepiped with sides

dx, dy, and dz parallel to, respectively, the mutually

perpendicular axes x, y, and z. Let the instantaneous

velocity of the fluid be V with magnitude |v| , and let

the scalar components of the velocity vector parallel to the

X, y, and z axes be, respectively, V , V and V .

As seen in Figure 1, the volume of fluid entering

the left yz face of the element is

V^dydz,

and the volume leaving the right yz face of the element is

^^x (V^ + ^-^ dx)dydz. X 9x

Therefore the net change in volume for the x-direction is

9V ir-^ dxdydz. dx

For the y-direction, the volume change is

9V

9y ^ dxdydz

and for the z-direction is

9V j ^ dxdydz

9V (V^+-y| dz)dxdy

V^dydz

•9V 'y^ (V + -—Z dy)dxdz y ^Y V dxdy

V dxdz y

9V X (V^+^3^x) dydz

Fig. 1.—An incompressible fluid element

Therefore, the total change in volume is

9V 9V 9V (^ + ^rr^ + y^) dxdydz 9x 9y

This, in other words, is the net outflow volume. The

volume of fluid leaving the element must equal the volume

entering the element because the fluid being considered is

incompressible. Therefore,

9V 3V 3V

Equation (1) is called the equation of continuity for incom­

pressible fluids.

The sum of the partial derivatives of the scalar

components of velocity (the left side of Equation (1)) is

called the divergence of V, abbreviated div V. Hence, for

an incompressible fluid,

div V = 0. (2)

Compressible Fluids

Now suppose the fluid is compressible; that is,

volume is a function of pressure. The equation of contin­

uity for a compressible fluid must be based not upon the

constancy of volume but upon the constancy of mass.

Consider Figure 1 again. If p is the density of

the fluid, then the mass of the fluid entering the left yz

face of the element in time dt is

(pV^)dydzdt,

and the mass leaving the right yz face of the element is

9(pV ) (pV + —_Ji_ dx)dydzdt.

X dX

Therefore, the net outflow of mass in the x-direction is

9(pV^) 7z dxdydzdt. dX

Similarly, the net outflow for the y-direction is

9(pV ) — ^ ^ dxdydzdt

and for the z-direction is

9(PV^) —^r dxdydzdt

9z

The total net outflow is, then.

9(pV ) 9(pV ) 9(pV^) [ ^ ^ ^ + - ^ + - ^ ] dxdydzdt

Because of this outflow, however, the mass inside the

element is reduced by

- (||-) dxdydzdt

For the total mass to remain unchanged, the following equa­

tion must hold:

9(pV ) 9(pV ) 3(pV )

9x 9y 9z ^9t' '

or

30 (PV,,) 9(PV^) ^(9^ J HL + £L_ + 1_ + : 9t 9x 9y 9z

z = 0. (3)

Equation (3) is called the equation of continuity for com­

pressible fluids. If density is constant, as for incom­

pressible fluids. Equation (3) reduces to Equation (1).

Another way to express Equation (3) is

1^ + div (pV) = 0 . (4) d t

CHAPTER III

EULER'S EQUATIONS

Again consider an element of fluid volume as in

Figure 1. In this case, hov/ever, consider the forces acting

upon the element. If the pressure is denoted by p, as

seen in Figure 2, the differential force caused by the

pressure across the yz faces of the element is

- (1 |£ dx) dydz.

Similarly, for the xz faces, the force is

- (l 1^ dy) dxdz

and for the xy faces,

- (Jc |£ dz) dxdy. d Z

The vectors i, 3, and k are the unit vectors parallel to

the X-, y-, and z-axis, respectively. The total differen­

tial force F is, therefore,

^ = - ('^i + ? i + 5 lf ) ^-^y^^- ' '

If the operator del, V, is defined to be

8

V = ^ 9x ^ =" 3y ^ ^ 3z '

then

F = - Vpdxdydz. (6)

pdxdy + (^ dz)dxdy

pdydz

pdxdz + (- dy)dxdz

pdxdz

pdydz+(-^ dx)dydz

pdxdy

Fig. 2.—Forces on a fluid element

Also, if the mass of the element is dm, the density, p,

of the fluid is defined to be

dm p =

dxdydz

Therefore, Equation (6) becomes

•^ n d m , ,

F = - Vp -^ . (7)

Even though the fluid element may change in shape as it

moves, its mass remains constant. Hence, if external forces

(such as gravity) are ignored, Newton's second law of

motion gives

^ dV , F = ^ dm , (8)

where t is time. Substituting Equation (8) into Equation

(7),

dV . _ „^ dm _ dm - - Vp —

dV ^ P ^ + Vp = 0. (9)

Velocity, in general, depends not only upon position (x,y,z)

and time (t) but also upon initial position (x^,y^,z ) at a

reference time (t«). Assuming a fixed initial position and

time.

10

^ = ^ ^ + 9 V ^ + 9 V d z . 9V ,, v dt 9x dt " 9y dt 9z dt " It • ^^^^

Letting

V = , V = V = ^ ""x dt' V dt' ^z dt'

^X33^ ^ \ ^ ^ ^Z37 = -^- (1^)

Using Equation (11), Equation (10) becomes

i = ( •V)V . Il . (12)

Substituting Equation (12) into Equation (9) ,

9^ p[(V.V)V + 1^ ] + Vp = S . (13)

This is the vector form of Euler's equations of motion.

Equation (13) can be separated into three scalar equations

The first of these is

9V 9V 9V 9V . o(v — ^ + V — - + V — - + ^r-^ ) + -^ = 0; P^ X 3x y 9y z 3z 9t ' 9x

or

9V 9V 9V 9V , J.

X 3x y 9y z 9z 9t p 9x

11

If body forces (external forces, such as gravity)

are considered, their vector sum, B, can be denoted as

follows:

S = Xi + Y^ + zS ,

where x, Y, and Z are the scalar components of the body

forces per unit mass in the x-, y-, and z-direction, respec­

tively. Equation (14) then becomes

9V 9V 9V 3V X . ,, X , „ X + _^Ji + 1 9p V ^-^ + V ^-^ + V ^-^ + ^x^ + ± ^ - X = 0. (15) x3x y9y z9z 9t p9x

The vector form of Euler's equations with body forces is

(V.V)V + | ^ + ± V p - g = J . (16) dt P

CHAPTER IV

THE THREE TYPES OF FLUID MOTION

Fluid motion is of three basic types:

(1) Translation

(2) Rotation

(3) Deformation

To see the relationships among these types, consider

two-dimensional fluid flow. Figure 3 shows a point 0(x,y)

in the fluid and a point 0'(x+dx, y+dy) a distance

= i 0 o

dr = V(dx) + (dy) from 0. The velocity at 0 is V

dy

I "x

•#• X

dx

Fig. 3.—A fluid element in two-dimensional flow

12

13

and at O' is ^ + dV. Now, V has components V , in the

x-direction, and V , in the y-direction. Therefore the

components of V' = V + dV are

9V 9V V' = V + ^ dx + ^ dy X X 9x 9y -

and

Let

Then

9V 9V V' = V + ^ dx + ^ dy . y y 9x 9y -

a = 9V^ 9V T 3V 3V _Ji b = — ^ c = i( — ^ + — ^ ) 9x ' ^ 9y ' ^ 2^ 9x ^ 3y ^ '

and (17)

9V 9V e = i(

2^ 9x 9y

= 1( _ ) .

V' = V + adx + cdy - edy XX -^

and (18)

V' = V + bdy + cdx + edx y y

The components V and V in Equations (18) are, X y

therefore, the components of strictly translational velocity.

14

linear velocity in the x- and y-direction, respectively. If

V and V were the only components in Equations (18) , ^ y

( a = b = c = e = 0 ) , then the rectangular fluid element in

Figure 3 would remain rectangular at every point in the

field of flow; that is, the flow would be ideal parallel

flow. The terms adx and bdy are expressions of the

change of velocity in the x- and y-direction, respectively;

they represent the "stretching rate" of the edge of the

element in each direction. From Figure 3,

3V 9V

^1 = 93E ^^^ ^2 = 97^ '

or

3V 9V

Equation (19) represents the change of the angle between

the two edges of the rectangle at point 0. The terms a, b,

and c, then, represent the deformation of the fluid element

between point O and point 0'.

Now assume a = b = c = 0. Then

3V 9V __J1 = ^ 9x 9y

so that

15

Yi = - Y 2 '

or

- Yi = Y2

Therefore,

e = I (Yi - (-Yi))

e = Y-L .

Thus,

^ 9V 9V

Equation (20) is an expression of the angular velocity with

which the rectangular element moves about an axis through

point O and normal to the plane of flow; that is, e

represents the rotational velocity of the element.

The terms V , V , a, b, c, and e, then,

describe the three types of motion in a fluid: translation,

rotation, and deformation. Equations (18) completely

express the relationship among these types of flow for the

two-dimensional case.

CHAPTER V

ROTATIONAL MOTION AND EULER'S EQUATIONS

As shown in Chapter IV, angular (rotational) velo­

city about an axis normal to the plane of flow can be

expressed as

T 9V 3V 1( _jz: X 2^ 3x 3y

for two-dimensional flow. If the axis normal to the plane

of flow in Figure 3 is thought of as the z-axis, then the

above expression represents rotational flow about the

z-axis.

For three-dimensional flow rotational velocity is

represented by three terms; one is the above expression.

The other two are

T 3V 3V 1( _Ji 2 . 2^ 9z 9x ^

for rotational velocity about the y-axis, and

9V 9V 1( _ ^ y ) 2^ 9y 9z ^

16

17

for rotational velocity about the x-axis.

Tne vector, V , formed from the three expressions.

1 -J. 9V 9V 3V 9V T 9V 9V

is called the vorticity vector.

Now, since

^ ^ 9V 3V 3V^ 3V^ _ 9V 3V V X V = i( ^ ^) + tf ^ 1\ + t( Z ^\ f o i \

^9y 9z ' ^ ^9z 93r^ + ^alT " 9^^ ' ^^1)

the following relationship is established:

V* = (V X V) . (22)

Equation (21) is an expression of the curl of the velocity

vector; that i s .

c u r l V = V X V . (23)

From the theory developed in Chapter III, if the flow is

irrotational,

V* = |(curl V) = ;

then

18

V X V = ? .

Euler's equations of motion, as developed in Chapter

III, consist of three scalar equations or one vector equa­

tion. The scalar form can be written as follows:

9V 9V 9V 9V T . ^ v ^ + V , + V ^ ^ + ^ = X - i ^ (24a) X dx y oy z 3z 9t p 9x

9V 9V 9V 9V T . V ^ ^ + V ^ + V ^ + ^ = Y - 1 | P (24b) x 9x y 9y z 9z 9t P 9y

^^z ^^z ^ \ ^^z 1 9P V TT- + V -r—^ + V ,7-^ + —-^ = z - - 4^ (24c) X 3x y 3y z 9z 3t p 3z ^ '

Consider the left side of Equation (2 4a) , the Euler equa­

tion for the x-direction:

3V 3V 9V 9V V ^ + V — ^ + V — ^ + — ^ X 9x y 9y ^ ^z 9z ^ 9t

(25) 9V 9V 9V 9V 9V

Since

1 9V 9V^ 1( _JL ^ 2^ 9x 9y

represents rotational velocity about the z-axis, and

19

T 9V 9V 1^/ X z X 2^ 9z ~ 9x ^

represents rotational velocity about the y-axis, the middle

two terms on the right side of Equation (25) have coeffi­

cients that are merely twice the rotational velocities

about the axes perpendicular to the direction of flow. The

left sides of Equations (24b) and (24c) can be written

similarly.

Therefore, for irrotational flow. Equation (24a)

reduces to

3V 1 9 , 2 2 2, X „ 1 9p ±. _^ (V + V + V ) + TTT^ = X - - ^ . 2 9x ^ X y z' 9t p 9x

Similarly, Equations (24b) and (24c) become, respectively.

3V 1 3 f„2 , v2 2 y - Y - ^ -P 2 9? ^ x "*• ^y ^ ^z^ 9t ^ P 9y

and

3V

2 9z X y z' 9t P 9z

CHAPTER VI

NAVIER-STOKES EQUATIONS

In the preceding derivations friction forces have

been ignored. Friction between one fluid element and

another and between the fluid and its container must be

considered if a truly general fluid flow equation is to be

developed.

That property of a real fluid which causes shearing

(friction) forces is called viscosity. A fluid whose flow

is affected by viscosity is called a viscous fluid.

Incompressible Fluids

Consider, first, simple parallel flow of a viscous

incompressible fluid, illustrated in Figure 4.

In Figure 4(a) the fluid is at rest, fluid element

E^ lies atop fluid element E^, and viscosity has no

effect. In Figure 4(b), however, the fluid is in motion.

As shown, E, has scalar velocity v, and E2 has scalar

velocity v + dv. The friction force (or shear stress) per

unit area, T, is defined as follows:

dv I ^c\

T = y , (26)

20

21

where y is a proportionality factor called the dynamic

viscosity of the fluid.

y

X

(a)

y

•^ v+dv

- T

E, V

X

Cb)

Fig. 4.—Viscous fluid elements,

(a) at rest, and (b) in motion.

The quantity ^ is the angular velocity of deformation

of the element, originally a rectangle. The difference

between ideal (nonviscous) fluid flow and viscous fluid

flow in a pipe is illustrated in Figure 5.

(a)

(b)

Fig. 5.—A diagrammatic comparison

of one-dimensional (a) nonviscous flow

and (b) viscous flow in a pipe.

22

Consider Figure 6, which illustrates the three

dimensional case of viscous incompressible fluid flow. The

figure shows both normal, or direct, stresses (i. e.,

stresses due to pressure) and shear stresses (i. e.,

stresses due to friction) that affect a parallelepiped of

infinitesimal volume dxdydz. Note that, in viscous fluids,

even the normal stresses are dependent upon the orientation

of the axes, as shown by the subscripts x, y, and z on p.

23

The stresses are those acting at a point Q(x,y,z) in the

fluid, and they are shown for only five sides of the

parallelepiped.

9p^ ^x 9x

z

/

Z A

dx Q(x,y,z)

• ^ T

T - --2SZd^--^y--^x--xy 9x / zy

zy 3z

zx / 3z dz /

/

9T ' xz, T - —T: dx XZ 9x T„„ / yz

/ /

/ •• T

/ yx

/ /

/

V- T • ^ x 9P.

P^" z 3z kiz

Fig. 6.—Stresses on an

infinitesimal volume of a viscous fluid,

The stresses can be arranged in the form of a matri x

24

7P T T / x xy xz

T P T yx ^y yz

W_.. T P zx zy ^z

However, to avoid rotation of the infinitesimal element.

^xy = V x ' ^xz = ^zx' "' ^yz = zy"

The differential force in the positive x-direction is form­

ulated as follows:

9p ^x= [Px- (P, - 33^dx)]dydz

9T

+ [T - (T ^ dy)]dxdz yx yx 9y -

9T

+ [T - (T ^ dz)]dxdy zx zx 9z

9p 9T 9T

F = (_ii + - ^ + _2X)(jxdydz (27a) ^x 9x 9y 9z ^

Similarly, for the y- and z-direction,

9p 9T 9T F = (-1Z + - ^ + - ^ ) dxdydz (27b) y 9y 9z 9x

and

25

9p 9T 9T

z = ^JT* - ^ " - - l ^ )i^Ay^z. (27c)

The relationships between shear and normal stresses will

now be developed.

As already mentioned, in a viscous fluid the normal

and shear stresses depend upon the orientation of the coor­

dinate axes. The stress system can be divided into the

hydrostatic pressure p and any additional normal and

tangential stresses that cause only deformation of the fluid

by the action of viscosity.

For plane flow, three terms with coefficients a, b,

and c, as defined in Equations (17), characterize the rate

of deformation of a fluid element. The coefficient c

represents half the angular rate of deformation betv/een the

two edges of the plane rectangular fluid element. These

rates of deformation must be proportional to the extra

normal and tangential stresses; the constant of propor­

tionality is 2y, where the 2 is required for agreement

between Equation (26) and Equations (18) . Therefore, for

the three-dimensional case, the additional normal stresses

caused by the action of viscosity are

9V 9V , 9V^ Px = ^^-^' Py = 2^-^' ^^^ P- ^ '""^ '

so that the total normal stresses are

and

26

9V Px = -P ^ Px = -P - 2y-^ , (28a)

• 9V Py = -P + Py = -P + 2y-^ , (28b)

9V P^ = -P + p; = -P + 2y-3f . (28c)

The hydrostatic pressure term p has a negative sign

because p , p , and p were assumed to be positive X y z ^

outward.

The shear stresses are related to the velocity of

angular deformation (cf. Equations (17)) as follows:

3V 3V

xy yx ^ 3x 9y

^^z ^^x T = T = y ( ^ + -^) ; (29b) xz zx ^ 3x 9z

3V 3V T = T = y (- + -5^) . (29c) yz zy 9z dy

Substituting Equations (28a), (29a), and (29b) into Equa­

tion (27a),

9V . 9V 9V

3 ^^z ^^X

27

rearranging.

3 s \ d\ d\

9x 9y 9z

. 9V 3V 3V

If F^ is defined to be the force per unit volume in the

x-direction,

2 2 2 r, 9 V^ 9 V 9 V

F' = - + u( - + - + - ) X dx ^^ ^ 2 ^ . 2 ^ ^ 2 ^

dx 3y 3z (30a)

^ 9 , x ^ ^ ^ ^^z ,

Corresponding substitutions give the following expressions

for force per unit volume in the y- and z-direction:

2 2 2 3 V 9 V 3 V

y ^y 3x2 9y2 3z2

^ ^ 9?^~93F ^ - 3 ? ^ " 3 ^ ^ '

(30b)

and

2 2 2 3 V 3 V 3 V^

=" ^^ 3x2 3^2 3^^ (30c)

28

However, for incompressible fluids, the equation of contin­

uity states that

9V 3V 3V ii + - J : 4- — ^ = 0.

3x 3y 3z

Therefore, Equations (30) become

and

2 2 : ? cs 3 V 3 V 3 V

^x 33F " ^ — 2 ^ — T + — ^ ^' (31a) 9x 3y 3z 2 2 P

a^ 3''V 3 V 3 V y = -|f ^ ( — i ^ ~ ^ - — i ) ' (31b) ^ ^ 3x^ 3y^ dz^

2 2 ? :r. 9 V^ 9 V 3 V

< - - % - ^ ^ - \ - - \ ^ - \ ^ - (31C)

3x 3y 3z

A rearrangement of Equation (15), the scalar Euler equation

for the x-direction, gives

3V 3V 3V 3V

^X-Sl + \-af + z ^ + = - f If • (32a)

The Euler equations for the y- and z-direction arranged in

the above form are, respectively, then,

3V 3V 3V 3V T ^ V _ Z + v - ^ + V - ^ + - ^ = Y - l | P (32b) X 3x y 3y z 3z 3t P 9y

and

29

9V 3V 3V 3V T ^ V - ^ + V - ^ + V - ^ + - ^ = Z - ^ | ^ . (32c) X 3x y 9y z 3z 3t p 3z

Substituting F', F', and F' into Euler's equations for

- -^f - ^ / and - , respectively, gives

3V 3V 3V 3V V -—ii + V -rrii + V -;rii + ^ X 3x y 3y z 3z 3t

2 2 2 n c. 9 V 9 V^ 9 V^ P ^^ P 3x2 3 2 3 2

3V 3V 3V 3V V — - + V — r ^ + V —T^ + —r^ X dx y 3y z 3z 3t

2 2 2 , , 3 V 3 V 3 V P 9y p .^2 . 2 2

(33a)

(33b)

3x 3y 3z

and

3V 3V 3V 3V

^ x ^ ^ ^ ^ z ^ -Tt (33c)

9 2 2 n . 9'V 9' V 9''V P 9z p 3 2 3 2 g zi

Equations (33) are the scalar Navier-Stokes equations for

incompressible fluids. In vector notation. Equations (33)

become

(^.V)^ + ll = 6 - ivp + ^V . (34)

30

Since

i l = (^-v)^ . I l ,

Equation (34) can also be stated as follows

|V = g _ 1 y^2^ dt p ^ p

(35)

Compressible Fluids

To obtain the Navier-Stokes equations for compres­

sible fluids. Equations (33) must be modified slightly. A

term proportional to

3V 3V 3V

3x 3y 3z

must be added to Equations (2 8) .

Let e be the constant of proportionality. Then

Equations (2 8) become

3V 3V 3V 3V p. = -P+ 2y ^ + e ( - ^ + ^ + ^ ) , (36a) X 3x "" 3x 3y 3z

3V 3V 3V 3V Py = - P ^ 2 y ^ . e ( - ^ + - 3 ^ + ^ ) , (36b)

and 9V 3V^ 3V 3V

Pz = - P - 2 y - ^ . e ( - 3 | . ^ . ^ ) . (36c)

31

Summing Equations (36) ,

3V 3V 3V Px ^ Py ^ Pz = -^P ^ (2y f 30) (^ + ^ + ^ ) . (37)

Now, if the fluid were incompressible. Equation (37) would

be

Px + Py + Pz = -3P (38)

by Equation (1) . Assuming Equation (3 8) holds for compres­

sible fluids. Equation (37) becomes

3V 3V 3V •3p = -3p + (2y + 39) (• • " ' 3x 3y 3z

Solving for G,

e = - I y . (39)

Substituting this result into Equations (36) ,

3V ^ ... 3V 3V„ 3V^

3V o 9V 9V 9V^ Py = -P + 2p 33^ - I y ( ^ + a/ + g/ ) , (40b)

and

32

9V 3V 3V^ 3V

Since Equations (29) are unaffected by compressibility,

substituting Equations (40a), (29a), and (29b) into Equation

(27a) gives

a 9V 3V^ 3V 3V

g 8V 3V 3V 8V

^87(^(3^+ 5 / " •^3^(v''33r+ a/))]dxdydz;

or, the force per unit volume, F', is

; 3V^ -^ 3V 3V 3V

. 9V, 3V^ . 3V 3V

^^('^(a^^-a^r" ^ 3!'^ <air 3r^>'

Simplifying,

2 2 2 Sir. 3 V 3 V 3''v

F' = - lE + u( - + ^ + ^ ) ^x 3 x ^ ^ . 2 ^ , 2 ^ ^ 2 ^ dx 3y 3z

, ^ 9V 9V 9V

3 ^ 9x 9x 9y 9z ^ *

(41a)

Corresponding substitutions give the following expressions

for force per unit volume in the y- and z-direction:

33

and

2 2 2 9 V 9 V 9' V

F ' = - | P + u ( ^ + Z + Z ) y ^y dx^ 9y2 3z2

T . 3V 3V 3V

3 * 3y ^ 3x 3y 3z ^

2 2 2 . 3 V 3 V 9 V

Tn' o p , , Z , Z , Z »

F_ = - ^ + y ( 2 " 2 •*• 5" ^ = ^^ 3x'^ 3y'^ 3z^

T ' 9V 9V 9V^ + 1 u - ^ (—i^ + — Z + ^ ) ^ 3 ^ 9z ^ 9x 9y ^ 9z ^ '

(41b)

(41c)

S u b s t i t u t i n g F ' , F " , and F ' i n t o E u l e r ' s e q u a t i o n s ^ X y z

(Equations (32)) for -|^, -g^, and - •^, respectively,

gives

3V 3V 3V 3V

\ - ^ ^ \ ^ * ^ z ^ - ^

2 2 2 -. . 9'V 9' V 9 V

P ^^ P 3x^ 3y^ 3z^

^ 3V 3V 3V^ 1 y _3_ ,_jc + __Z + — ^ ) ,

• 3" 9x ^ 9x 9y 9z ^ '

3V 3V 3V 3V

X dx y 9y z 9z 9t

9 2 2 n . 9 V 9 X ^ v P ^y P 3x2 3 2 3 2

3V 3V 3V^ _ 1 y 3 / X , _y_ . E ) " I " 3y ^~3^ 9y 9z ' '

34

and

9V 3V 3V 3V V ^ + V — - + V + -

X dx y 3y z 3z 3t 2 2 2

1 :r. n 3 V^ 3^V^ 3 V

^ 3x 3y 3z

3 p 3z ^ 3x 3y 3z ^ *

Equations (42) are the scalar Navier-Stokes equations for

compressible fluids. In vector notation. Equations (42)

become

(v-v)v . Il = g - i vp . v^^ . i v ( ^ . % . )

But, using the definition of the divergence of V,

(t^.V)^ + |V g _ 1 p E v2^ + I H v(div V) . (43)

Again, since

i - ' •v)v Il -

E q u a t i o n (43) can a l s o be s t a t e d as f o l l o w s :

dV ^ g _ 1 ^ + y v^V + i ^ V(div V) . (44) d t p ^ p 3 p

CHAPTER VII

BERNOULLI'S EQUATION

The preceding chapters have shown the development of

increasingly more general equations describing fluid flow.

This and subsequent chapters will show how a multitude of

assumptions are used to reduce the general equations to an

equation frequently used in industry to calculate the rate

of flow of natural gas.

To obtain the first relationship, Bernoulli's equa­

tion, irrotational flow will be assumed; that is,

V""x = "5, (45)

as discussed in Chapter V. Moreover, the identity

V(v-V) = 2V.VV + 2V X (V X V) (46)

will be utilized (see [3], p. 313).

Now, the relationship

/ V.d? = 0, (47)

35

36

where the symbol }> denotes the line integral around any c

closed curve C and dr is the infinitesimal vector

(dx) 1 + (dy)] + (dz)k, can be shown to hold whenever Equa­

tion (45) does. This is a result of Stokes's theorem (see

[3] , p. 295) . A trivial consequence of Equation (47) , as

proved in [6], pp. 264-265, is that

j V-dr

i s i n d e p e n d e n t of t h e pa th t aken from p o i n t P . t o p o i n t

Independence of t h e p a t h of i n t e g r a t i o n imp l i e s t h a t

V ' d r can be w r i t t e n as t h e d i f f e r e n t i a l of a s c a l a r

f u n c t i o n (^ ; t h a t i s ,

V-dr = d<t> . (48)

Another v/ay of expressing this is as follows

P 1 ^ ...

/ V-dr = <^{V ) - (})(PQ) P ^0

(49)

Since

^ = ^ 9^ ^ ^ 97 ^ ^ y l '

37

V,.d-r = |i dx . |i dy . |i d.

or

V(j)-dr = d(J) . (50)

Subtracting Equation (50) from Equation (48),

V-dr - V(t).dr = 0 ;

therefore

(V - V(|)) .dr = 0 .

This implies that the vector in parentheses is orthogonal

to the vector dr. But, since dr is arbitrary,

V - V(t) = "5 ,

or

V = V(j) . (51)

Equation (46) can now be greatly simplified using the

assumptions and Equation (51). By Equation (45), Equation

38

(46) becomes

V(^-V) = 2V-VV . (52)

But, the identity

V = 1 1 = ( . ) /2

further reduces Equation (52) to

^•V^ = V(^ V^) . (53)

Now Equation (53) can be s u b s t i t u t e d i n to the vector form

of E u l e r ' s e q u a t i o n s . Equation (13) , to obtain

pV(i V^) + p | ^ + Vp = ^ ,

or

V(l v 2 ) + | i + ^ = t i . (54) 2 d t p

Substituting Equation (51) into the above relationship gives

V(|v2) ^^(V*) + ^ = i 5 ;

that is.

39

V ( | v2 + | 1 + E) = ^ . (55)

S i n c e t h e q u a n t i t y i n p a r e n t h e s e s i s e v i d e n t l y i n d e p e n d e n t

of p o s i t i o n , i t mus t b e a f u n c t i o n of t i m e o n l y . T h e r e f o r e ,

1 V + II- + / ^ = f ( t) . (56) ^ <3t •' p

Equation (56) is a very general form of Bernoulli's equa­

tion. For steady (time independent) flow, the relationship

collapses to

1 v2 + / ^ = c , (57) 2 ^ P

where c is a constant.

If the fluid is incompressible, density is constant;

then Equation (57) becomes

1 v2 + E = c . 2 P

Moreover, if gravity is considered.

1 v2 + gz + H = c, (58)

where

40

g = acceleration of gravity and

z = elevation above some datum.

CHAPTER VIII

FLOW EQUATIONS FOR THE ORIFICE METER

Further understanding is best served at this time by

describing the setup for the standard orifice type differ­

ential meter used in the natural gas industry.

Figure 7 is a cut-away schematic drawing of an ori­

fice type differential meter in which a manometric liquid

is used to measure differential pressure. The fluid to be

Direction

of flow

TD

Fig. 7.—An orifice type

differential meter with U-tube manometer

41

42

measured, flowing from left to right, is partially obstructed

by a metal plate. A, in which a concentrically-located hole

has been bored. The purpose of this metal plate, called an

orifice plate, is to produce a pressure drop. The greater

pressure is sensed at location 1, called the upstream

pressure tap; the lower pressure is sensed at location 2,

called the downstream pressure tap. Because of their loca­

tions, the particular pressure taps in Figure 7 are called

flange taps. The upstream and downstream pressures are

relayed to a U-tube manometer, B, filled with mercury or

some other suitable manometric liquid.

The method of transformation of Equation (58) into

a form that utilizes data from the orifice meter to obtain

a flow rate will now be outlined. For details, see [5],

pp. 51-52, and [1], pp. 78-79.

Assume that density, p, is constant. Then, for

pressure tap locations 1 and 2,

2 2 ^1 Pi ^2 P2 ^ + z , + - i = ^ + z ^ + — , (59) 2g 1 Y 2g 2 y

where y = P^ ^^ specific weight. For horizontal pipe,

rearrangement of Equation (59) gives

.2 2 . Pi " P2 v; - Vt = 2g(--^ ^) . (60) 2 1 ^ Y

43

Using the assumption

V^= (^V,)^,

where

D = inside diameter of the pipe and

d = diameter of the orifice.

V2 = ^ 1/2 (2g( ^ ^ ) ) ^ ^ ^ (61) j

CI - 4 ; D

Because the development has been oversimplified, the experi­

mental constant C, called the coefficient of discharge, is

inserted, giving

V2 = 5-^^72 (2g(^L_^)) V2^ ( 2)

(1 - ^ ) D

and the resulting quotient j-^ ^^ renamed K.

(1 - 4 D

Equation (62) then becomes

Now

Pi - P2 1/2 V2 = K(2g(-i^-^))^/^

, since the quantity rate of flow, Q, through the

44

orifice is the product of the velocity of the fluid and the

cross-sectional area. A, of the orifice.

Pi " P2 1/2 Q = KA(2g(-i— ^))^/^. (63)

Equation (63) is a form of the so-called "hydraulic"

equation. Units will now be assigned to the quantities in

Equation (63).

Let ' 1

Q = fluid flow rate at the average specific weight, '

Y, in cubic feet per second;

A = orifice area in square feet;

g = acceleration of gravity in feet per second per

second; and

K = =-y , corresponding to the condition of

(1 - 4 D

measurement.

Pi ~ P9 The quotient — =• is the differential head, h, of

the flowing fluid in feet at the average specific weight

at the orifice. Equation (63) therefore becomes

Q = KA(2gh) 1/2^ (64)

45

where each quantity has the units designated above. Equa­

tion (64) is very unhandy to use practically; therefore, it

will be changed to the form that is used almost everywhere

in the United States to calculate natural gas flow across

an orifice plate:

Qh = ^"(Vf '^^^' <">

where \ I

Q, = hourly fluid flow rate at stated base conditions I

of temperature and pressure, I

h^ = differential pressure across the orifice in »

inches of water column, i i

P_ = absolute static pressure in pounds per square

inch (psia) at a designated tap location, and

C' = orifice flow constant.

To change Equation (6 4) into the practical form.

Equation (65), several substitutions must be made:

g = 32.17 ft/sec^; (66a)

h Y h = - ^ ^ , (66b)

12Y

where

46

^w ~ 62.37 lb/ft = specific weight of water at

60° F., and

Y = actual specific weight of the natural gas in

pounds per cubic foot at flowing conditions;

A = - ^ , (66c) 4(144)

where d = diameter of the orifice in inches; and

P Y = 0.08073 — ^ i|^ G , (66d)

14.7 -"f

where

0.08073 = specific weight of dry air at 14.7 psia

and 32°F.,

T^ = flowing temperature of the natural gas in

degrees Rankine (°R.), and

G = specific gravity of the flowing gas, where the

specific gravity of dry air is taken to be 1.000

Substituting Equations (66a) through (66d) into Equation

(64) ,

,2 h (62.37)(14.7)T . Q = K( " ^ ) (2(32.17) {—^ ^))^^''. (67)

4(144) 12(0.08073)P^(49 2)G

47

However, flow rate in cubic feet per hour, Q^, would be

calculated as follows:

Qf = 3600 Q ,

where the subscript "f" denotes that the flow rate is based

upon flowing conditions. Equation (67) therefore becomes

^ h T. 1/2 Q^ = 218.44 d^K(^^) . (68)

Using the ideal gas law, the combined laws of Charles and

Boyle, the hourly flow rate, Q^, at base pressure P^ and

base temperature T, is obtained as follows:

^ T, h P. 1/2 Q^ = 218.44 d^K pH (_^) ,

or

T, , 1/2 ./2 Q^= 218.44 d ^ K ^ (^) (h^P,)^/^ (69)

Eauation (69) is in the form of Equation (65), where

C = 218.44 d^K ^ (T^)^^^. (70) ^b ^f

If the supercompressibility of the gas is considered.

48

2 ^b 1 1/2 ^ 1/2 C = 218.44 d' K p^ (^) (|) , (71)

b f

where Z = compressibility factor at T^ and P-.

To make computations easier. Equation (71) is, in

practice, subdivided into factors, as detailed in [1]. The

principal factors are listed below:

(1) Basic orifice factor, F, .

2 ' b 1 ^/2 b = 218.44 A \ ^ ( ) , (72a)

D f

where K^ is found from a set of empirical equations. The

values

T, = 520*'R. , b '

T^ = 520°R.,

P^ = 14.7 psia, and D

G = 1.000

are assumed. Equation (72a) then becomes

F^ = 338.17 d^KQ. (72b)

(2) Reynolds number factor, F^

F = 1 + B—— , (72c) r , ,1/2 '

(Vf)

49

where b is calculated from a set of empirical equations

The purpose of F is to allow for the difference between

K-j, used to calculate F, in Equation (72b), and K,

used in Equation (71) .

(3) Expansion factor, Y. This factor allows for the change

in specific weight of the gas across the orifice plate.

(4) Pressure base factor, F , .

F ^ = M ^ , (72d) Pb PK

where P^ is the desired pressure base.

(5) Temperature base factor, F j .

F = -A- (72e) ^tb 520 '

where T, is the desired temperature base.

(6) Flowing temperature factor, F^^

F = (i20//\ (72f) *tf T^'

(7) Specific gravity factor, F g

F = () . (72g) g G

50

(8) Supercompressibility factor, F PV

1 1/2

Using the symbolism of the eight factors above. Equa­

tion (71) becomes

C- = F^F^YFp^F^j^F^^F^Fp^ . (73)

Then the flow rate in cubic feet per hour, Q^, at T, and h b

Pj_ , is calculated using the follov/ing equation:

Qh = V r ^ V ^ t b ^ t f V p v ' V f ) ' ^ " - (74)

Three additional factors, not in universal use in the

natural gas industry, are also developed in [1]. These

factors are (1) the manometer factor, F ; (2) the location

factor, F.; and (3) the orifice thermal expansion factor,

CHAPTER IX

SUMMARY AND CONCLUSIONS

Starting with the most fundamental relationships, the

Navier-Stokes equations, which allow for friction, were

developed; several assumptions were then made to reduce

these very complicated partial differential equations to a

form used in the calculation of gas flow across an orifice

plate. The assumptions, several of which are cited and

discussed in [5], pp. 52-55, are listed below:

(1) The gas flow is irrotational. See Equation (45) .

(2) Friction does not affect fluid flow. That is, the

velocity of the fluid is the same at all points across the

diameter of the pipe, and no energy is lost as the gas

passes through the orifice. This assumption results from

the use of Euler's equations in the derivation of the

"hydraulic" equation.

(3) Fluid flow velocity is not time dependent. See Equa­

tion (57) .

(4) Gravity is the only body force.

(5) Compressible fluid flow across an orifice is incompres­

sible; that is, the specific weight of the fluid does not

51

52

change as it passes through the orifice. This assumption

is required to obtain Bernoulli's equation.

(6) The velocity at the upstream pressure tap is related to

the velocity at the downstream pressure tap as the orifice

area is related to the cross-sectional area of the pipe.

This assumption is made to derive the "hydraulic" equation.

(7) Suction or impact effects at the pressure taps are nil.

(8) The acceleration of gravity is 32.17 feet per second

per second.

All of these assumptions are at least partially

incorrect. However, as discussed in [5] , the effects of

the assumptions are either negligible or are corrected by

the construction of the piping upstream and downstream of

the orifice plate or by factors in Equation (74).

LIST OF REFERENCES

1. American Gas Association. Gas Measurement Committee Report No. 3. Orifice Metering of Natural Gas. New York: American Gas Association, 1969.

2. Aris, Rutherford. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1962.

3. Hildebrand, Francis B. Advanced Calculus for Appli­cations . Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1962.

4. Kaufmann, Walther. Fluid Mechanics. New York: McGraw-Hill Book Company, Inc., 1963.

5. Spink, L. K. Principles and Practice of Flow Meter Engineering, 8th Ed. Norwood, Massachusetts: The Foxboro Company, 19 58.

6. Wrede, Robert C. Vector and Tensor Analysis. New York: John Wiley & Sons, 19 63.

53