Some Investigation of Incompressible Inviscid Steady Flow in Engineering Systems: A Case Study of The Bernoulli’s Equation with Its Application to Leakage in Tanks

Some Investigation of Incompressible Inviscid Steady Flow in Engineering Systems: A Case Study of The Bernoulli’s Equation with Its Application to Leakage in Tanks

Mutua, N. M 1, Wanyama, D. S 2, Kiogora, P. R 3

1Mathematics and Informatics, Taita Taveta University College, Voi, 80300, Kenya

2Mathematics and Informatics, Taita Taveta University College, Voi, 80300, Kenya

3Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, 62000-00200, Kenya

Abstract

During fluid spill incidents involving damaged tanks, the amount of the product released may be uncertain. Many accidents occur under adverse conditions, so determining the volume lost by sounding the tanks may not be practical. In the first few hours, initial volume estimates often are based on visual observations of the resulting levels, a notorious unreliable approach. This study has described a computer based model by Bernoullis equation that will help in responding to leakages in tanks.

1. Introduction

   Fluid storage tanks can be damaged in the course of their life and a substantial amount of the fluid lost through leakage. There is need therefore to calculate the rate at which the tank leaks in order to adequately respond to the situation. The volume of a fluid escaping through a hole in the side of a storage tank has been calculated by the Bernoullis equation which has been adapted to a streamline from the surface to the orifice.

   1.1 Objectives

   The objectives of the study are:

   gravitational forces on the fluid. Bernoullis theorem has since been the basis for flow equation of flow meters that expresses flow rate to differential pressure between two reference points. Since the differential pressure can also be expressed in terms of height or head of liquid above a reference plane. An Italian scientist, Giovanni Venturi (1797), demonstrated that the differential pressure across an orifice plate is a square root function of the flow rate through the pipe. This is the first known use of an orifice for measuring flow rate through a pipe. Prior to Giovannis experimental demonstration, the only accepted flow measurement method was by filling a bucket of known volume and counting the number of buckets being filled. The use of orifice plates as a continuous flow rate measuring device has a history of over two hundred years. Subsequent experiments, Industry standards online in their demonstration showed that liquids flowing from a tank through an orifice close to the bottom are governed by the Bernoullis equation which can be adapted to a streamline from the surface to the orifice. They evaluated three cases whereby in a vented tank, the velocity out from the tank is equal

   to the speed of a freely body falling the distance

   h -also known as Torricellis Theorem. In a

   pressurized tank, the tank is pgrehssurized so that

   1. To determine the velocity of the fluid

      product of gravity and height

      is much lesser

      through the opening in the tank.

   2. To determine the volume flow rate through the opening.

   3. To determine the mass flow rate through the opening.

      1. Justification

         Fluid storage tanks e.g. water reservoirs & petroleum oil tanks are commonly used in our society today. This storage tanks are prone to accidents which when not adequately responded to, great losses will be incurred in terms of money and also pollution to the environment. The volume of a fluid escaping through a hole in the side of a storage tank is calculated by the Bernoullis equation calculator the results of which are useful for developing the intuitively skills of responders and planners in spill releases.

      2. Literature Review

      Castelli and Tonicelli (1600) were first to state that the velocity through a hole in a tank varies as square root of water level above the hole. They also stated that the volume flow rate through the hole is proportional to the open area. It was almost another century later that a Swiss physicist Daniel Bernoulli (1738) developed an equation that defined the relationship of forces due to the line pressure to energy of the moving fluid and earths

      than the pressure difference divided by the density. The velocity out from the tank depends mostly on the pressure difference. Due to friction the real velocity will be somewhat lower than these theoretic examples. On leaking tank models, a recent literature review reveals numerous papers describing formulas for calculating discharges of non-volatile liquids from tanks Burgreen (1960); Dodge and Bowls(1982); Elder and Sommerfeld (1974); Fthenakis and Rohatgi (1999); Hart and Sommerfeld (1993); Koehler(1984); Lee and Sommerfeld (1984); Shoaei and Sommerfeld (1984); Simecek-Beatty et al. There are also computer models available in ship salvage operations. These types of models estimate the hydrostatic, stability, and strength characteristic of a vessel using limited data. The models require the user to have an understanding of basic salvage and architectural principles, skills not typical of most spill responders or contingency planners. A simple computer model, requiring input of readily available data, would be useful for developing the intuitively skills of responders and planners in spill releases. Mathematical formulas have been developed to accurately describe releases for light and heavy oils. Dodge and Bowles (1982); Fthenakis and Rohatgi (1999); Simecek-Beatty et al. (1997). However, whether any of these models can describe the unique characteristics of fluids

      accurately is not known. Observational data of these types of releases are needed to test existing models, and probably modify them in the case of a given fluid. J. Irrig and Drain. Engrg (August 2010) carried out experiments under different orifice diameters and water heads. The dependence of the discharge coefficient on the orifice diameter and water head was analysed, and then an empirical relation was developed by using a dimensional analysis and a regression analysis. The results showed that the larger orifice diameter or higher water head have a smaller discharge coefficient and the orifice diameter plays more significant influence on the discharge coefficient than the water head does. The discharge coefficient of water flow through a bottom orifice is larger than that through a sidewall orifice under the similar conditions of the water head, orifice diameter, and hopper size.

      In this research work a simple computer model, based on Bernoullis equation requiring input of readily available data, is used to investigate leakage from vessels or tanks: the case of two pipeline companies (Oil refineries and Kenya pipeline co.) in the Coast province of Kenya. The results are of vital importance in responding to leakages to minimize losses.

      This template, created in MS Word 2003 and saved as Word 2003 doc for the PC, provides authors with most of the formatting specifications needed for preparing electronic versions of their papers. All standard paper components have been specified for three reasons: 1) ease of use when formatting individual papers, 2) automatic compliance to electronic requirements that facilitate the concurrent or later production of electronic products, and 3) Margins, column widths, line spacing, and type styles are built-in; examples of the type styles are provided throughout this document. Some components, such as multi- levelled equations, graphics, and tables are not prescribed, although the various table text styles are provided. The formatter will need to crate these components, incorporating the applicable criteria that follow. Use the styles, fonts and point sizes as defined in this template, but do not change or redefine them in any way as this will lead to unpredictable results. You will not need to remember shortcut keys. Just a mouse-click at one of the menu options will give you the style that you want.

2. Governing Equations

2. 1 Overview

The equations governing the flow of an incompressible inviscid steady fluid through an

orifice in a tank are presented in this chapter. This chapter first considers the assumptions and approximations made in this particular flow problem and the consequences arising due to these assumptions. The conservation equations of mass, momentum, energy to be considered in the study are stated, Bernoullis equation derived followed by a model description of the fluid flow under consideration.

Finally a simple computer model, requiring input of readily available data is built which will be used in investigating flow rate in chapter three.

1. Assumptions and approximations

   The following assumptions have been made.

   1. Fluid is incompressible i.e. density is a

      constant and t 0

   2. Fluid is inviscid i.e. the fluid under investigation is not viscous.

   3. Flow is steady in that is if F is a flow

      variable defined as F(x, y,t) then

      F 0

      t

   4. Flow is along a streamline.

   5. The fluid speed is sufficiently subsonic

      V mach 0.3

   6. Discharge coefficient C value is typically between 0.90 and 0.98.

2. Consequences as a result of the assumptions

   The fact that as the water issues out of the orifice from the tank, the level in it changes and the flow is, strictly speaking, not steady. However,

   if A1 A2 , this effect (as measured by the ratio of the rate of change in the level and velocity V2 ), is so small that the error made is insignificant.

   The assumption of incompressibity would not hold at varying temperatures for a fluid like water which has differing densities at different temperatures. However this is taken care of by our model which allows the user to enter determined values of densities at various temperatures.

3. The Governing Equations

   1. Equation of conservation of mass

      Consider the fluid element of volume v fixed in space surrounded by a smooth surface S . If the density of the fluid is and dv is the volume element of v at a point p in v , then the total mass of the fluid in v is given by

      dv

      1

      f

      f

      dv

      0 7

      v v

      t

      We let s be an element of the surface S .

      Since v is an arbitrary volume, then for equation

      f

      f

      7 to hold, we have

      Then if the fluid flows outwards through s , and after time t this fluid is displaced a distance

      0 8

      t

      equal to x . Then the rate at which mass is leaving

      the volume element is given by

      And this is the general equation of mass conservation.

      Lim s x s Lim x

      If is a constant, then equation 8 becomes

      t 0 t

      t 0 t

      2

      f 0

      9

      x

      t is taken to be speed.

      If n is the unit normal vector to s , then

      This is the equation of continuity of an incompressible fluid flow.

      On the other hand, if the flow is steady i.e. the

      f

      f

      Lim x n

      ( f denotes

      flow variables dont depend on time, and then

      f

      f

      equation 8 reduces to

      t 0 t

      velocity).

      Thus

      0

      10

      Lim s x

      t0 t

      s. fn

      3

      This is the equation of continuity of a steady fluid flow.

   2. Equation of conservation of energy

      This gives the rate of mass flux flowing out of v .

      From 3, the total mass flux flowing out of v

      through S is given by

      Consider the fluid element of shown below.

      v dxdydz as

      fns

      s

      From Gauss theorem, equation 4

      fns dv

      fns dv

      written as

      f

      4

      can be

      5

      s v

      From equation 1 the rate at which the mass is changing in volume v is

      Figure 1: Fluid Element

      v

      v

      t dv

      6

      By Fouriers law heat transferred through the face perpendicular to the x- axis by conduction is

      T

      Applying the law of conservation of mass, we have,

      given by K x dydz

      (11)

      f

      dv

      t

      dv

      t dv

      The element of heat which leaves the volume element along the x-axis is given by

      v v v

      K T

      K T dxdydz

      (12)

      f

      f

      dv

      dv

      0

      0

      x

      x

      x

      dv

      dv

      Or v v t

      Amount added to x axis =

      V through conduction along

      K T K T

      x x

      K T dx dzdy

      x x

      where is known as the dissipation function given by

      (13)

      u 2

      v 2 w 2

      u

      v 2

      K T dxdzdy

      x x

      x

      y z

      2

      y

      x

      Similarly the amount added to

      v along the y-

      2 u w

      (20)

      axis and z- axis through conduction is

      z

      x

      K

      T

      dxdydz and

      T

      K dxdydz .

      v

      w 2

      2 u

      v w 2

      y

      x

      z

      x

      z y

1. x y z

Total heat added to

v by conduction

Q T T T

(19) is the general equation of energy of the fluid

t x K x y K y z K z dxdydz (14) (Q is purely due to heat conduction)

If we let e be the internal energy per unit mass,

then total internal energy of the fluid

flow whose velocity components are u, v and w if

is as defined by (20)

It is noted that if the fluid is incompressible K = a constant and the internal energy E is a function of t alone, then equation (19) reduces to

element v i.e. v.e edxdydz

(15)

2T

2T

2T

df T

From this we have

K x2

y 2

z 2 dt

de v de de

dxdydz

(16)

(T=temperature)

dt dt dt

2 df (T )

So K T

(21)

T T T e dt

x K

x y K y z K

z dxdydz

If is a function of T alone then

e CV T f T CV T

(22)

de dxdydz dw

substituting this in (21) we have

V

V

dt dt

or

K2T C

T

t

(23)

T T T

de 1 dw

(23) is the equation of energy of a non- viscous

x K x y K y z K z dt v dt

incompressible fluid with constant coefficient of

(17)

conductivity.

2.43 Equation of motion for compressible fluids

If the fluid is incompressible then

dw 0

for

incompressible fluids equation (17) reduces to

x x y y z z dt

x x y y z z dt

T T T de

K K K

Consider a control volume ABCD (of unit depth) in a 2-D flow field with its centre located at (x, y). If the state of stress at (x, y) in this 2-D flow field is

represented by , , and , then the

xx yy xy yx

(18)

Adding heat generation by frictional force to equation (18) we have,

T T T de

surface forces on the four faces of the CV can be written in terms of these stresses and their derivatives using Taylor series. A few of these are shown in the figure below.

x K x y K y z K z dt

(19)

29

DVy

Dt

fy

• yy

  y

• xy

  x

  DVx

  Dt and DVy Dt

  may be expressed in

  terms of the field derivatives using the Euler acceleration formula which gives

  Vx V

  Vx V

  Vx f

  xx yx

  30

  t x x y y x x y

  With similar equations obtained from 30, we can now use the expression for

  • Vy Vx

    Figure 2: Control Volume

    The net surfaces forces acting on this element in

    yx

    xy

    xy

    x

• y

  and

  the x- and y-directions are easily seen to be

  • Vx

  xx

  yx

  xx p

  2 x

  Fsurface,x

  x

• y

  .x.y.1

  24

  (From Fluid Mechanics and its Applications by Vijay Gupta and Santosh K Gupta page 150), we obtain,

  And

  V V

  V

  p 2V

  2V

  x y x

  x y x

  x V x V x f

  x x

  yy

  xy

  t

  x y

  x x2

  y 2

  Fsurface, y

  y

• x

  .x.y.1

  31

  And

  25

  V V

  V

  p 2V

  2V

  If f x and f y are the components of the body

  y V

  y V y

  f

  y y

  force per unit mass, then

  t

  x x y y

  y y x2 y 2

  Fbody, x

  And

  fx .x.y.1

  26

  32

  For a general 3-D flow these components can be

  Fbody, y

  fy .x.y.1 27

  written in the vector form

  By Newtons law of motion

  V

  t

  V .V f

  p 2V

  33

  x.y.1 DVx

  Dt

  fxx.y.1

  Known as Navier-Stokes equation. This contains three equations forVx ,Vy ,Vz .The LHS of

  xx

  yx

  equation 33 represents the fluid acceleration

  So that

  x

• y

  x.y.1

  (local and convective) and is referred to as the inertial term. The term on the right represents the body force, the pressure and the viscous forces respectively.

  In many flows of engineering interest, the

  DVx

  Dt

  fx

• xx

  x

• yx

  y

  viscous terms are much smaller than the inertial terms and thus can be neglected, giving

  28

  And similarly

  V

  t

  V .V f

  • p

  This is known as the Euler equation which is applicable to non-viscous flows and plays an

  h p p

  36

  important role in the study of fluid motion.

  1. Bernoullis Equation

     Bernoullis equation states that the total

     Since (1) and (2)s heights from a common

     reference are related as (36), and the equation of continuity can be expressed as (37), it is possible to transform (35) to (38).

     A2

     mechanical energy (consisting of kinetic, potential

     V1 A .V2

     37

     1

     and flow energy) is constant along a streamline.

     It is a statement of the conservation of energy in

     2 P1 P2

     a form useful for solving problems involving fluids.

     V2

     A 2

• g.h

  38

  For a non-viscous, incompressible fluid in steady flow, the sum of pressure, potential and kinetic

  1

  2

  A 2

  energies per unit volume is constant at any point.

  The form of Bernoulli's Equation we have used arises from the fact that in steady flow the particles of fluid move along fixed streamlines, as on rails, and are accelerated and decelerated by the forces acting tangent to the streamlines.

  Bernoullis equation has been obtained by

  1

  A special case of interest for equation (38) is when the orifice area is much lesser than the surface area and when the pressure inside and outside the tank is the same- when the tank has an open surface or vented to the atmosphere. At this situation (38) can be transformed to (39)

  directly integrating the equation of motion of an

  V2

  2.g.h

  (39a)

  inviscid fluid.

  Thus equation (34) is the Bernoullis equation

  Due to friction, the real velocity will be somewhat lower than the theoretic velocity in (39).

  V 2

  p V 2 p

  The actual velocity profile at A2 will depend on the

  2

  34

  • gz

  i

  gz

  2 0

  nature of the orifice. A sharp-edged orifice will result in much 2-D effects than will a smooth one. In engineering practice, these non-ideal effects are usually taken care of by introducing an

  experimentally obtained correction factor termed

  1. Illustration of the fluid flow

     Fluids flow from a tank or container through a hole/orifice close to the bottom. The Bernoulli equation derived can be adapted to a streamline from the surface of the fluid to the orifice.

     the discharge coefficient Cd . Thus (39a) can be expressed as (39b). The coefficient of discharge can be determined experimentally. For a sharp edged opening it may be as low as low as 0.6. For

     smooth orifices it may be between 0.95 and 1.

     V2 Cd

     2.g.h

     39b

     If the tank is pressurized so that the product of gravity and height (g.h) is much lesser than the pressure difference divided by the density, (38) can be transformed to (40). Thus the velocity out from the tank depends mostly on the pressure difference. However in our study, we have thus concentrated on the vented tank. The pressurized tank will be

     dealt with in our subsequent studies.

     V2

     2.P1 P2

     40

     In the case of the vented tank, the discharge rate is given by

     C A

     2.g.h

     41

     Q d 2

     P2

     P2

     Figure 3: Flow from a Tank

     The source of error in the equation is the fact that as the water issues out of the orifice from the tank,

     P1

     P1

     g.p

     Where

• V1

  2

  2

  2

  g.p

• V2

2

2

2

35

the level in it changes and the flow is, strictly speaking, not steady. However, if A1 A2 , this effect (as measured by the ratio of the rate of

change in the level and velocityV2 ), is so small that the error made is insignificant.

1. Water flowing (discharging) from a vented tank, pond, reservoir containing water or other liquid

   Figure 4: Water discharging from a vented tank

   The Borda type is also known s a re-entrant since it juts into the tank.C values were obtained from Dally et al. (1993) for circular orifices.

   Water (or other liquid) draining out of a tank, reservoir, or pond is a common situation. Our calculation allows you to compute for final liquid depth given the time the tank has discharged and the initial liquid depth of the fluid. This will be convenient for a tank that we know the initial liquid depth of the fluid but we dont know the final liquid depth. The final liquid depth will give us the spout depth. (This works for high tanks that one cant climb to measure the fluid depth from the top of the tank).

   Alternatively, the user can compute the time needed to lower the water from one depth to a lower depth or to empty the tank.

   The tank (or pond or reservoir) is open to the atmosphere and it can be cylindrical or other cross- section but must have the same cross-section for its entire height. The orifice can be circular or non- circular, but for our study, we have modelled the circular orifice. Hi, Hf, and h are measured vertically from the centerline of the orifice.

   Hydrostatic pressure will impart a velocity to an exiting fluid jet. The velocity and flow rate of the jet depend on the depth of the fluid.

   Given the time the tank has discharged, the initial liquid depth of the fluid, tank diameter/side dimensions (which will give us tank area), orifice diameter (will give us orifice area) and discharge coefficient we will compute for final liquid depth which will give us the spout depth.

   To calculate the jet velocity and flow rate (both volume and mass), enter the parameters specified on the calculator application. (Any interaction of the fluid jet with air is ignored.)

   2.9 Equations used in the Calculation

   V2 Cd

   2.g.h

   Figure 5: Free discharge view

   Q AjetVjet CAspouVt

   jet

   Or Q Cd A2 2.g.h

   2.8 Design of Calculator Application

   In this study, the free discharge orifice has been modelled. We have studied on circular orifice geometry where B is orifice diameter. A pull-down menu allows you to select an orifice type. Discharge coefficients for the four orifice types are built into the calculation.

   Built-in values for orifice discharge coefficients are:

   Rounded Sharp-edged Short-tube Borda 0.98 0.6076 0.8 0.5096

   Note: The above equations are valid if both the tank and orifice are at the same pressure, even if the pressure is not atmospheric

   For a tank with a constant cross-sectional geometry A in the plan view (i.e. as you look down on it), substitute:

   Q A dh

   dt

   Integrate h from Hi to Hf and integrate t from 0 to t, then solve for time t, which is the time required for the liquid to fall from Hi to Hf:

   t A

   aC

   Hi

   2

   H f g

   1. Effect of Discharge coefficient on Exit

      If the tank is circular in plan view (i.e. looking down on it):

      D2

      velocity

      Table 2: Effect of Discharge coefficient on Exit velocity

      Discharge coefficient	0.98	0.8	0.6076	0.5096
      Exit velocity	62.45577	38.43185	18.99865	11.58102

      Discharge coefficient	0.98	0.8	0.6076	0.5096
      Exit velocity	62.45577	38.43185	18.99865	11.58102

      A

      4

      If the orifice is circular:

      d 2

      a

      4

      Our calculation allows you to solve for any of the variables: a, A, Hf, or t. The orifice and tank can be either circular or non-circular. If non- circular, then the diameter dimension is not used in the calculation.

1. Results AND Discussion

1. Overview

   This section details the results of our work in form of graphs together with discussion of the results obtained.

2. Effect of Spout Depth on Exit velocity

Table 1: Effect of Spout Depth on Exit velocity

Figure 3a: Effect of Spout Depth on Exit velocity

Spout exit velocity decreases with decrease in the depth of the spout. This is due to hydrostatic pressure which imparts a velocity to an exiting fluid jet but this pressure decreases when the fluid level drops. Thus the velocity of the jet depends on the depth of the fluid.

Figure 3b: Effect of Discharge coefficient on Exit velocity

Orifices with higher orifice coefficients give higher exit velocities. This implies that the type of orifice will affect the orifice exit velocity. In our case rounded orifice type will give the highest exit velocity followed by short-tube, sharp-edged then finally borda in decreasing velocities.

1. Effect of spout depth on discharge rate

   Table 3: Effect of spout depth on discharge rate

   Spout Depth	20	18.89763	17.82652	16.78664	15.77802
   Discharge	0.024394	0.023712	0.02303	0.022349	0.021667

   Figure 3c: Effect of spout depth on discharge rate

   The highest depth of spout gives the highest discharge. Decrease in the depth of the spout reduces the rate of discharge. Hydrostatic pressure reduces on reduction of fluid level imparts a velocity to an exiting fluid jet. Thus the velocity and flow rate of the jet depend on the depth of the fluid.

2. Effects of Orifice area on Discharge

   Table 4: Effects of Orifice area on Discharge

   Orifice area	0.001257	0.0028278	0.005027	0.007855	0.01131 1
   Discha rge	0.024394	0.0548863	0.097576	0.152462	0.21954 5

   Figure 3d: Effects of Orifice area on Discharge

   The larger the orifice area the larger the rate of discharge and vice versa. However, increasing orifice size beyond the carrying capacity of the system has no effect on the flow rate. (From the paper Flow Rate as a Function of Orifice Diameter

   by Ginger Hepler, Jeff Simon & David Bednarczyk)

3. Effects of Discharge coefficient on discharge/flow rate

   Table 5: Effects of Discharge coefficient on discharge/flow rate

   Discharge coefficient	0.98	0.8	0.6076	0.5096
   Flow rate	0.024394	0.0199134	0.015124	0.012685

   Figure 3e: Effects of Discharge coefficient on discharge/flow rate

   Orifices with higher orifice coefficients give

   higher discharge rates. This implies that the type of orifice will affect the discharge rate. In our case rounded rifice type will give the highest discharge rate followed by short-tube, sharp-edged then finally borda in decreasing discharge rate.

4. Effects of density on mass flow rate

Table 6: Effects of density on mass flow rate

Figure 3f: Effects of density on mass flow rate

Higher fluid densities cause higher mass flow rates. Thus the rate of mass flow of a fluid from the tank is directly proportional to the fluid density

4. Conclusions

Many phenomena regarding the flow of liquids and gases can be analyzed by simply using the Bernoulli equation. However, due to its simplicity, the Bernoulli equation may not provide an accurate enough answer for many situations, but it is a good place to start. It can certainly provide a first estimate of parameter values.

We analyzed the effects of spout depth, orifice area, friction coefficient on the fluid flow rate in a vented tank.

The Bernoullis equation that modelled the fluid flow was obtained by directly integrating the equation of motion of an inviscid fluid. The effect of a certain parameter value on another was investigated by varying the parameter and observing the behavior of the other.

In this study the pressure is constant that is both the tank and orifice are at the same pressure, (even if the pressure is not atmospheric), in that the exiting fluid jet experiences the same pressure as the free surface.

The major analysis is made on orifice geometry more than on tank geometry. A change in orifice geometry in terms of spout depth and type affected the flow rate. Knowing the duration the tank has leaked enables one to determine the current fluid depth in cases where tanks are too large and the depth cant be determined physically. This way will know whether the tank is empty already or not.

In conclusion, the exit velocity of the fluid is affected by the significant changes in spout depth, orifice area discharge coefficient. The mass flow rate is only affected by change in the fluid density.

1. Recommendations

   For one to respond to any leak scenario, one has to first establish that a leakage really exists. This is done by various leak detecting techniques e.g. the leak-measurement computer LTC 602 offers virtually unlimited possibilities for all kinds of leak measurement tests. The model built requires that the user has prior knowledge of the time the tank has leaked. Thus this model will be of use practically when used together with a leak detecting model which establishes the time the leak commenced. Some of the areas that need further research include:

   1. A pressurized tank, where the tank is pressurized so that the velocity out from

      the tank depends mostly on the pressure difference.

   2. An application model that establishes that a leakage really exists and then calculate for the discharge rate.

   3. Fluid flow for unsteady, compressible fluids

ACKNOWLEDGEMENT

The authors wish to thank NCST for the financial support to carry out this research.

REFERENCES

1. American Society for Nondestructive Testing, Columbus, OH – McMaster, R.C. editor, 1980. Leak Testing Volume of ASNT Handbook.

2. American Society of Mechanical Engineers (ASME). 2001. Measurement of fluid flow using small bore precision orifice meters. ASME MFC- 14M-2001.

3. Avallone, E.A., Baumeister, T. (eds.) (1997), Marks' Standard Handbook for Mechanical Engineers, 11th ed., McGraw-Hill, Inc. (New York).Avallone, E.A., Baumeister, T. (eds.)

4. Binder, R.C. (1973), Fluid Mechanics, Prentice- Hall, Inc. (Englewood Cliffs, NJ).

5. Burgreen, D.1960. Development of flow in tank draining.Journal of Hydraulics Division, Proceedings of the American Society of Civil Engineers. HY 3, pp. 12-28.

6. Colebrook, C.F. (1938), "Turbulent Flow in Pipes", Journal of the Inst. Civil Eng.

7. Fluid Mechanics, 4th Edition by Frank M. White, McGraw-Hill, 1999.

8. Gerhart, P.M, R.J. Gross, and J.I. Hochstein. 1992. Fundamentals of Fluid Mechanics. Addison-Wesley Publishing Co. 2ed.

9. H. Lamb, Hydrodynamics, 6th ed. (Cambridge: Cambridge Univ. Press, 1953)

10. King, Cecil Dr. – American Gas & Chemical Co., Ltd. Bulletin #1005 "Leak Testing Large Pressure Vessels". "Bubble Testing Process Specification".

11. Munson, B.R., D.F. Young, and T.H. Okiishi(1998). Fundamentals of Fluid Mechanics.John Wiley and Sons, Inc. 3ed.

12. Potter, M.C. and D.C. Wiggert.1991.Mechanics of Fluids. Prentice-Hall, Inc.

13. Roberson, J.A. and C.T. Crowe.1990.Engineering Fluid Mechanics. Houghton Mifflin Co.

14. White, F.M.1979. Fluid Mechanics. McGraw-Hill, Inc.

15. 2002 LMNO Engineering, Research, and Software, Ltd. Online.

APPENDIX

Program code for calculator application in visual basic

Public Class frmCalc

Private Sub ComboBox6_SelectedIndexChanged(ByV al sender As System.Object, ByVal e As System.EventArgs) Handles cmbOrificeType.SelectedIndexChange d

txtunitEVel.BackColor =

Color.AntiqueWhite

txtOArea.ReadOnly = True txtOArea.Text = "Will be

calculated"

txtOArea.BackColor =

Color.AntiqueWhite

txtspout.ReadOnly = True txtspout.Text = "Will be

calculated"

txtspout.BackColor = Color.AntiqueWhite

If cmbOrificeType.SelectedItem = "Rounded" Then

txtcoeff.Text = "0.98"

True

txtMassFlow.ReadOnly =

txtMassFlow.Text = "Will

ElseIf cmbOrificeType.SelectedItem = "Sharp-edged" Then

txtcoeff.Text =

be calculated"

txtMassFlow.BackColor = Color.AntiqueWhite

"0.6076"

ElseIf cmbOrificeType.SelectedItem = "Short-tube" Then

txtcoeff.Text = "0.8"

ElseIf cmbOrificeType.SelectedItem = "Borda" Then

txtcoeff.Text =

"0.5096"

End If

txtTArea.ReadOnly = True txtTArea.Text = "Will be

calculated"

txtTArea.BackColor = Color.AntiqueWhite

txtEVel.ReadOnly = True txtEVel.Text = "Will be

calculated"

txtEVel.BackColor =

Color.AntiqueWhite

txtVolFlow.ReadOnly = True txtVolFlow.Text = "Will be

calculated"

txtVolFlow.BackColor = Color.AntiqueWhite

End Sub

Private Sub btncalc_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles btncalc.Click

txtTArea.ReadOnly = False txtTArea.Text = ""

txtOArea.ReadOnly = False txtOArea.Text = ""

tankdiam = Val(txtTdiam.Text)

TCArea = 3.142 * tankdiam

* tankdiam / 4

TNCArea = (Val(txtTlength.Text) * Val(txtTwidth.Text))

If cmbTankType.SelectedItem = "Circular" Then

txtTArea.Text = TCArea Else

txtspout.ReadOnly = False txtspout.Text = ""

TNCArea

txtTArea.Text =

End If

False

False

txtMassFlow.ReadOnly =

txtMassFlow.Text = ""

txtEVel.ReadOnly = False txtEVel.Text = ""

txtVolFlow.ReadOnly =

txtVolFlow.Text = ""

Dim Orarea, subt,

coeff = Val(txtcoeff.Text)

Theight = (Val(txtInLevel.Text)) ^ 0.5

time = Val(txtTimeTank.Text)

gr = ((2 / 9.8066) ^ 0.5)

gr1 = time * Orarea * coeff / ((Val(txtTArea.Text)) * gr)

subt = Theight – gr1

finalheight = subt * subt txtspout.Text =

finalheight

finalheight, VolFlowrate, coeff, gr, EVel, gr1, TCArea, TNCArea, Theight, mass, time, tankdiam, ordiam As Double

ordiam = Val(txtODiam.Text)

Orarea = 3.142 *ordiam * ordiam / 4

txtOArea.Text = Orarea

EVel = coeff * ((2 * 9.8066 * finalheight) ^ 0.5)

txtEVel.Text = EVel

VolFlowrate = Orarea * coeff * ((2 * 9.8066 *

finalheight) ^ 0.5)

txtVolFlow.Text =

VolFlowrate

mass = VolFlowrate * Val(txtfluid.Text)

txtMassFlow.Text = mass

txtunitEVel.Text = "m/s"

txtunitMass.Text = "Kg/s" txtunitOArea.Text =

"Square Metres"

False

Used"

False

False

True

txtTdiam.ReadOnly =

txtTdiam.Text = ""

Else

txtTdiam.Text = "Not

txtTwidth.Text = "" txtTlength.Text = "" txtTlength.ReadOnly =

txtTwidth.ReadOnly =

txtTdiam.ReadOnly =

txtunitSpout.Text = "Metres"

txtunitTArea.Text = "square Metres"

txtunitVolRate.Text =

End If

End Sub

"M^3/s"

End Sub

Private Sub ComboBox1_SelectedIndexChanged(ByV al sender As System.Object, ByVal e As System.EventArgs) Handles cmbTankType.SelectedIndexChanged

If cmbTankType.SelectedItem = "Circular" Then

txtTlength.ReadOnly =

Private Sub btnrefresh_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles btnrefresh.Click

txtcoeff.Text = "" txtEVel.Text = "" txtfluid.Text = "" txtInLevel.Text = ""

txtMassFlow.Text = ""

True

True

Used"

Used"

txtTwidth.ReadOnly =

txtTwidth.Text = "Not

txtTlength.Text = "Not

txtOArea.Text = "" txtODiam.Text = "" txtspout.Text = "" txtTArea.Text = "" txtTdiam.Text = ""

txtTimeTank.Text = "" txtTlength.Text = "" txtTwidth.Text = "" txtunitEVel.Text = "" txtunitMass.Text = "" txtunitOArea.Text = "" txtunitSpout.Text = "" txtunitTArea.Text = "" txtunitVolRate.Text = "" txtVolFlow.Text = ""

cmbOrificeType.SelectedItem = ""

cmbTankType.SelectedItem =

""

End Sub

Private Sub frmCalc_Load(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles MyBase.Load

End Sub End Class