j compressible fluid flow through an orifice …
TRANSCRIPT
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^ '-^' 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
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• ^ 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
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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
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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
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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.
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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
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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
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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
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(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
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- (||-) 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
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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^^- ' '
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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
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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.
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^ = ^ ^ + 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
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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
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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
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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.
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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
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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.
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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 ^
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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
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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
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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
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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)
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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.
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(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.
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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
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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
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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
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and
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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
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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)
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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
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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)
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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)
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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
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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:
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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 ' '
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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
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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)
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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 '
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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
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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.
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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
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g = acceleration of gravity and
z = elevation above some datum.
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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
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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
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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
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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)
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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
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^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
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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.
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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)
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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
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(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,
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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
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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).
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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.
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