COMPRESSIBLE FLUID FLOW HERSCHEL NATHANIEL WALLER, JR., B.S. A THESIS IN MATHEMATICS


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1 ^ '^' 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
2 ^ ACKNOWLEDGEMENTS I would like to thank Dr. Wayne Ford for allotting time to direct the writing of my thesis and for the interest 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
3 TABLE OF CONTENTS ACKN0V7LEDGMENTS LIST OF ILLUSTRATIONS ii 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. NAVIERSTOKES EQUATIONS 20 VII. BERNOULLI'S EQUATIONS 35 VIII. FLOW EQUATIONS FOR THE ORIFICE METER IX. SUMMARY AND CONCLUSIONS 51 LIST OF REFERENCES
4 LIST OF ILLUSTRATIONS Figure page 1. An incompressible fluid element 3 2. Forces on a fluid element 8 3. A fluid element in twodimensional flow Viscous fluid elements, (a) at rest, and (b) in motion A diagrammatic comparison of onedimensional (a) nonviscous flow and (b) viscous flow in a pipe Stresses on an infinitesimal volume of a viscous fluid An orifice type differential meter with Utube manometer 41 IV
5 CHAPTER I INTRODUCTION The purpose of this thesis is to show the development 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 relationships, the NavierStokes 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 discussed, 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.
6 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 (V^ + ^^ ^^x dx)dydz. X 9x Therefore the net change in volume for the xdirection is 9V ir^ dxdydz. dx
7 For the ydirection, the volume change is 9V 9y ^ dxdydz and for the zdirection is 9V j ^ dxdydz 9V (V^+y dz)dxdy V dxdz y V^dydz 9V X (V^+^3^x) dydz 9V 'y^ (V +  Z dy)dxdz y ^Y V dxdy Fig. 1. An incompressible fluid element Therefore, the total change in volume is 9V 9V 9V (^ 9x + ^rr^ 9y + y^) dxdydz
8 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 incompressible 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 continuity 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
9 (pv^)dydzdt, and the mass leaving the right yz face of the element is (pv X 9(pV ) + _Ji_ dx)dydzdt. dx Therefore, the net outflow of mass in the xdirection is 9(pV^) 7z dx dxdydzdt. Similarly, the net outflow for the ydirection is 9(pV ) ^ ^ dxdydzdt and for the zdirection is 9(PV^) ^r 9z dxdydzdt 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
10  ( ) dxdydzdt For the total mass to remain unchanged, the following equation 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 compressible fluids. If density is constant, as for incompressible fluids. Equation (3) reduces to Equation (1). Another way to express Equation (3) is 1^ + div (pv) =0. (4) d t
11 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 zaxis, respectively. The total differential force F is, therefore, ^ =  ('^i +?i + 5^lf ) ^^y^^ '^'
12 8 If the operator del, V, is defined to be V = ^ 9x ^ =" 3y ^ ^ 3z ' then F =  Vpdxdydz. (6) pdxdy + (^ dz)dxdy pdxdz pdydz pdydz+(^ dx)dydz pdxdz + (^ dy)dxdz 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
13 p = dm Therefore, Equation (6) becomes dxdydz ^ n dm,, 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.
14 10 ^ = ^ ^ + 9 V ^ + 9Vdz. 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
15 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 zdirection, respectively. Equation (14) then becomes 9V 9V 9V 3V V ^^ X +. V,, ^^ X +, V ^^ X + ^x^ _^Ji + ± 1 ^ 9p x3x y9y z9z 9t p9x  X = 0. (15) The vector form of Euler's equations with body forces is (V.V)V + ^+±Vpg=J. (16) dt P
16 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 twodimensional 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 dx # X Fig. 3. A fluid element in twodimensional flow 12
17 13 and at O' is ^ + dv. Now, V has components V, in the xdirection, and V, in the ydirection. Therefore the components of V' = V + dv are and 9V 9V V' = V + ^ dx + ^ dy X X 9x 9y ^ 9V 9V V' = V + ^ dx + ^ dy. y y 9x 9y ^ Let a = 9V^ 9V T 3V 3V _Ji b = ^ c = i( ^ + ^ ) 9x ' ^ 9y ' ^ 2^ 9x ^ 3y ^ ' and (17) Then 9V 9V e = i( 1( ^ _ ^ ). 2^ 9x 9y V' = V + adx + cdy  edy XX ^ and (18) V' = V y y + bdy + cdx + edx The components V and V in Equations (18) are, X y therefore, the components of strictly translational velocity.
18 14 linear velocity in the x and ydirection, 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 ydirection, respectively; they represent the "stretching rate" of the edge of the element in each direction. From Figure 3, or 3V 9V ^1 = 93E^ ^^^ ^2 = 97^ ' 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 J1 = 9x 9V ^ 9y so that
19 15 Yi =  Y 2 ' or  Yi = Y2 Therefore, e = I (Yi  (Yi)) e = YL. 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 twodimensional case.
20 CHAPTER V ROTATIONAL MOTION AND EULER'S EQUATIONS As shown in Chapter IV, angular (rotational) velocity about an axis normal to the plane of flow can be expressed as T 9V 3V 1( _jz: X 2^ 3x 3y for twodimensional flow. If the axis normal to the plane of flow in Figure 3 is thought of as the zaxis, then the above expression represents rotational flow about the zaxis. For threedimensional 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 yaxis, and 9V 9V 1( _^ y ) 2^ 9y 9z ^ 16
21 17 for rotational velocity about the xaxis. 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 ^\ foi\ ^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 is. curl V = V X V. (23) From the theory developed in Chapter III, if the flow is irrotational, V* = (curl V) = ^ ; then
22 18 V X V =?. Euler's equations of motion, as developed in Chapter III, consist of three scalar equations or one vector equation. 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 ^ + ^ = Y1 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 equation for the xdirection: 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 zaxis, and
23 19 T 9V 9V 1^/ X z X 2^ 9z ~ 9x ^ represents rotational velocity about the yaxis, the middle two terms on the right side of Equation (25) have coefficients 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 13 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
24 CHAPTER VI NAVIERSTOKES 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
25 21 where y is a proportionality factor called the dynamic viscosity of the fluid. y y (a) X ^ v+dv E, ^ T V Cb) X 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.
26 22 (a) (b) Fig. 5. A diagrammatic comparison of onedimensional (a) nonviscous flow and (b) viscous flow in a pipe. 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.
27 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. Z A dx Q(x,y,z) ^x / 9p^ 9x z / V ^T T  2SZd^^y^x xy 9x / zy zy 3z 9T ' zx / 3z dz / / xz, T  T: dx XZ 9x T / yz / / / T yx / / / 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
28 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 = Vx' ^xz = ^zx' ^"'^ ^yz = ^zy" The differential force in the positive xdirection is formulated as follows: ^x= 9p [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 zdirection, and 9p 9T 9T F = (1Z +  ^ + ^) dxdydz (27b) y 9y 9z 9x
29 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 coordinate 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 proportionality is 2y, where the 2 is required for agreement between Equation (26) and Equations (18). Therefore, for the threedimensional 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
30 26 9V Px = P ^ Px = P ^ 2y^, (28a) and 9V Py = P + Py = P + 2y^, 9V P^ = P + p; = P + 2y3f. (28b) (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 Equation (27a), 9V. 9V 9V 3 ^^z ^^X
31 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 xdirection, r, 9 V^ 9 V 9 V F' =  ^ + u( ) X dx ^^ ^ 2 ^. 2 ^ ^ 2 ^ ^ 9,^^x dx ^ ^^ 3y ^ ^^z 3z, (30a) Corresponding substitutions give the following expressions for force per unit volume in the y and zdirection: V 9 V 3 V y ^y 3x2 9y2 3z2 (30b) ^ ^ 9?^~93F ^ 3? ^ "3^ ^ ' and V 3 V 3 V^ =" ^^ 3x2 3^2 3^^ (30c)
32 28 However, for incompressible fluids, the equation of continuity 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 ^ ^ 3x^ 3y^ 2 2? dz^ :^r. 9 V^ 9 V 3^V ^y =  f ^^( i^~^ i)' (31b) <%^^\\^\^ 3x 3y 3z (31C) A rearrangement of Equation (15), the scalar Euler equation for the xdirection, gives 3V 3V 3V 3V ^XSl + \af + ^ z ^ + ^ = ^  f If (32a) The Euler equations for the y and zdirection arranged in the above form are, respectively, then, 3V 3V 3V 3V T ^^ V_Z+v^+V^+^=Yl P X 3x y 3y z 3z 3t P 9y (32b) and
33 29 9V 3V 3V 3V T ^^ V  ^ + V  ^ + V  ^ +  ^ = Z  ^ ^. X 3x y 9y z 3z 3t p 3z (32c) 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 n c. 9 V 9 V^ 9 V^ P ^^ P 3x2 3^2 3^2 (33a) and 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 3x.^2 3y.2 3z 2 3V 3V 3V 3V ^x^ ^ ^ ^ ^ ^z^ ^ Tt n. 9'^V 9'^V^ 9''V^ P 9z p 3^2 3^2 g^zi (33b) (33c) Equations (33) are the scalar NavierStokes equations for incompressible fluids. In vector notation. Equations (33) become (^.V)^ + ll = 6  ivp + ^V. (34)
34 30 Since il = (^v)^. Il, Equation (34) can also be stated as follows V = g _ 1 y^2^ dt p ^ p (35) Compressible Fluids To obtain the NavierStokes equations for compressible 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 = P2y^. e (  3. ^. ^ ). (36c)
35 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 compressible 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
36 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, Sir. 3 V 3 V 3''v ^x 3 x ^ ^. 2 ^ dx, 2 ^ 3y ^ 2 ^ 3z, ^ 9V 9V 9V F' =  le + u(  + ^ + ^ ) 3 ^ 9x ^9x 9y 9z ^ * (41a) Corresponding substitutions give the following expressions for force per unit volume in the y and zdirection:
37 V 9 V 9'^V F' =  P + u( ^+ Z+ Z ) y ^y dx^ 9y2 3z2 T. 3V 3V 3V 3 *^ 3y ^ 3x 3y 3z ^ (41b) and 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 ^ ' (41c) Substituting F', F", and F' into Euler's equations ^ X y z (Equations (32)) for  ^, g^, and  ^, respectively, gives 3V 3V 3V 3V \  ^ ^ \ ^ * ^ z ^^ ^ '^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 n. 9 V 9X ^ ^v P ^y P 3x2 3^2 3^2 3V 3V 3V^ _^ 1 y 3 / X, _y_. E ) "^ I "^ 3y ^~3^ 9y 9z ' '
38 34 and 9V 3V 3V 3V V ^ + V  + V +  X dx y 3y z 3z 3t :^r. n 3 V^ 3^V^ 3^V ^ 3x 3y 3z 3 p 3z ^ 3x 3y 3z ^ * Equations (42) are the scalar NavierStokes equations for compressible fluids. In vector notation. Equations (42) become (vv)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  Equation (43) can also be stated as follows: dv ^ g _ 1 ^ + y v^v + i ^ V(div V). (44) dt p ^ p 3 p
39 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 equation, irrotational flow will be assumed; that is, V""x ^ = "5, (45) as discussed in Chapter V. Moreover, the identity V(vV) = 2V.VV + 2V X (V X V) (46) will be utilized (see [3], p. 313). Now, the relationship / V.d? = 0, (47) 35
40 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 Equation (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 , is that j Vdr is independent of the path taken from point P^. to point Independence of the path of integration implies that V'dr can be written as the differential of a scalar function (^ ; that is, Vdr = d<t>. (48) Another v/ay of expressing this is as follows P 1 ^... / Vdr = <^{V )  (})(PQ) P ^0 (49) Since ^^ = ^ 9^ ^ ^ 97 ^ ^ yl '
41 37 V,.dr = i dx. i dy. i d. or V(j)dr = d(j). (50) Subtracting Equation (50) from Equation (48), Vdr  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
42 38 (46) becomes V(^V) = 2VVV. (52) But, the identity V = 1^1 = (^.^)^/2 further reduces Equation (52) to ^ V^ = V(^ V^). (53) Now Equation (53) can be substituted into the vector form of Euler's equations. Equation (13), to obtain pv(i V^) + p ^ + Vp = ^, or V(l v2)+ i + ^=ti. (54) 2 d t p Substituting Equation (51) into the above relationship gives V( v2) ^^(V*) +^=i5 ; that is.
43 39 V( v E) = ^. (55) Since the quantity in parentheses is evidently of position, it must be a function of time only. independent Therefore, 1 V^ + II + / ^ = f(t). (56) ^ <3t ' p Equation (56) is a very general form of Bernoulli's equation. 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
44 40 g = acceleration of gravity and z = elevation above some datum.
45 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 differential meter used in the natural gas industry. Figure 7 is a cutaway schematic drawing of an orifice 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 Utube manometer 41
46 42 measured, flowing from left to right, is partially obstructed by a metal plate. A, in which a concentricallylocated 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 locations, the particular pressure taps in Figure 7 are called flange taps. The upstream and downstream pressures are relayed to a Utube 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 , and [1], pp 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
47 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 experimental 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. (14 Equation (62) then becomes D Pi  P2 1/2 V2 = K(2g(i^^))^/^ Now, since the quantity rate of flow, Q, through the
48 44 orifice is the product of the velocity of the fluid and the crosssectional area. A, of the orifice. Pi " P2 1/2 Q = KA(2g(i ^))^/^. (63) Equation (63) is a form of the socalled "hydraulic" equation. Units will now be assigned to the quantities in Equation (63). Let 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 ' 1 ' K = second; and (14 D measurement. =y^, corresponding to the condition of 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)
49 45 where each quantity has the units designated above. Equation (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 = ft/sec^; (66a) h Y h = ^^, 12Y (66b) where
50 46 ^w ~ lb/ft 60 F., and = specific weight of water at 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 = ^ i ^ G, (66d) "f where = 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 Substituting Equations (66a) through (66d) into Equation (64),,2 h (62.37)(14.7)T. Q = K( "^^ ) (2(32.17) { ^ ^))^^''. (67) 4(144) 12( )P^(49 2)G
51 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^ = 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^ = d^k ph (_^), or T,, 1/2./2 Q^= d^k^ (^) (h^p,)^/^ (69) Eauation (69) is in the form of Equation (65), where C = d^k ^ (T^)^^^. (70) ^b ^f If the supercompressibility of the gas is considered.
52 48 2 ^b 1 1/2 ^ 1/2 C = 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 = 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 = are assumed. Equation (72a) then becomes F^ = d^kq. (72b) (2) Reynolds number factor, F^ F = 1 + B, (72c) r,^ ^,1/2 ' (Vf)
53 49 where b is calculated from a set of empirical equations The purpose of F is to allow for the difference between Kj, 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
54 50 (8) Supercompressibility factor, F PV 1 1/2 Using the symbolism of the eight factors above. Equation (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 = Vr^V^tb^tfVpv'Vf)'^" (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,
55 CHAPTER IX SUMMARY AND CONCLUSIONS Starting with the most fundamental relationships, the NavierStokes 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 , 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 Equation (57). (4) Gravity is the only body force. (5) Compressible fluid flow across an orifice is incompressible; that is, the specific weight of the fluid does not 51
56 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 crosssectional 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 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).
57 LIST OF REFERENCES 1. American Gas Association. Gas Measurement Committee Report No. 3. Orifice Metering of Natural Gas. New York: American Gas Association, Aris, Rutherford. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Englewood Cliffs, New Jersey: PrenticeHall, Inc., Hildebrand, Francis B. Advanced Calculus for Applications. Englewood Cliffs, New Jersey: Prentice Hall, Inc., Kaufmann, Walther. Fluid Mechanics. New York: McGraw Hill Book Company, Inc., Spink, L. K. Principles and Practice of Flow Meter Engineering, 8th Ed. Norwood, Massachusetts: The Foxboro Company, Wrede, Robert C. Vector and Tensor Analysis. New York: John Wiley & Sons,
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