Concept of Static Pressure | Additional Documents & Tests for Civil Engineering (CE) PDF Download

Concept of Static Pressure
 

• The thermodynamic or hydrostatic pressure caused by molecular collisions is known as static pressure in a fluid flow and is usually referred to as the pressure p. 

• When the fluid is at rest, this pressure p is the same in all directions and is categorically known as the hydrostatic pressure. 

• For the flow of a real and Stoksian fluid (the fluid which obeys Stoke’s law) the static or thermodynamic pressure becomes equal to the arithmetic average of the normal stresses at a point. The static pressure is a parameter to describe the state of a flowing fluid. 

• Let us consider the flow of a fluid through a closed passage as shown in Fig. 16.1a
  
  
   

• If a hole is made at the wall and is connected to any pressure measuring device, it will then sense the static pressure at the wall. This type of hole at the wall is known as a wall tap. 

• The fact that a wall tap actually senses the static pressure can be appreciated by noticing that there is no component of velocity along the axis of the hole.

• In most circumstances, for example, in case of parallel flows, the static pressure at a cross-section remains the same. The wall tap under this situation registers the static pressure at that cross-section. 

• In practice, instead of a single wall tap, a number of taps along the periphery of the wall are made and are mutually connected by flexible tubes (Fig. 16.1b) in order to register the static pressure more accurately.

Hydrostatic, Hydrodynamic, Static and Total Pressure

• Let us consider a fluid flowing through a pipe of varying cross sectional area. Considering two points A and B as shown in Figure 16.1(c), such that A and B are at a height Z_A and Z_B respectively from the datum
  
           Figure 16.1 (c)
   
• If we consider the fluid to be stationary, then,        where the subscript ‘hs’ represents the hydrostatic case.
  So,   p_Ahs - p_Bhs = ρg( Z_B–Z_A)    (16.1)
  
  where p_Ahs is the hydrostatic pressure at A and p_Bhs is the hydrostatic pressure at B.
   
• Thus, from above we can conclude that the Hydrostatic pressure at a point in a fluid is the pressure acting at the point when the fluid is at rest or pressure at the point due to weight of the fluid above it. 
   
• Now if we consider the fluid to be moving, the pressure at a point can be written as a sum of two components, Hydrodynamic and Hydrostatic.
  p_A = p_Ahs + p_Ahd    (16.2)
  
  where p_Ahs is the hydrostatic pressure at A and p_Ahd is the hydrodynamic pressure at A. 
   
• Using equation (16.2) in Bernoulli's equation between points A and B.
     (16.3)

From equation (16.1), the terms within the square bracket cancel each other.

Hence,
                (16.4)
   (16.5)

 

• Equations (16.4) and (16.5) convey the following. The pressure at a location has both hydrostatic and hydrodynamic components. The difference in kinetic energy arises due to hydrodynamic components only. 

• In a frictionless flow, the sum of flow work due to hydrodynamic pressure and the kinetic energy is conserved. Such conservation shall apply to the entire flow field if the flow is irrotational.

• The hydrodynamic component is often called static pressure and the velocity term, dynamic pressure. The sum of two, p₀ is known as total pressure. This is conserved in isentropic, irrotational flow.

Stagnation Pressure

• The stagnation pressure at a point in a fluid flow is the pressure which could result if the fluid were brought to rest isentropically. 
• The word isentropically implies the sense that the entire kinetic energy of a fluid particle is utilized to increase its pressure only. This is possible only in a reversible adiabatic process known as isentropic process.
  
  Fig 16.2   Measurement of Stagnation Pressure
  
   
• Let us consider the flow of fluid through a closed passage (Fig. 16.2). At Sec. l-l let the velocity and static pressure of the fluid be uniform. Consider a point A on that section just in front of which a right angled tube with one end facing the flow and the other end closed is placed.
   
• When equilibrium is attained, the fluid in the tube will be at rest, and the pressure at any point in the tube including the point B will be more than that at A where the flow velocity exists. 
    
• By the application of Bernoulli’s equation between the points B and A, in consideration of the flow to be inviscid and incompressible, we have,
      (16.6)
  where p and V are the pressure and velocity respectively at the point A at Sec. I-I, and p₀ is the pressure at B which, according to the definition, refers to the stagnation pressure at point A
   
• It is found from Eq. (16.6) that the stagnation pressure p₀ consists of two terms, the static pressure, p and the term ρV²/2 which is known as dynamic pressure. Therefore Eq. (16.6) can be written for a better understanding as
      (16.7)
   
• Therefore, it appears from Eq.(16.7), that from a measurement of both static and stagnation pressure in a flowing fluid, the velocity of flow can be determined. 
   
• But it is difficult to measure the stagnation pressure in practice for a real fluid due to friction. The pressure p'₀ in the stagnation tube indicated by any pressure measuring device (Fig. 16.2) will always be less than p₀, since a part of the kinetic energy will be converted into intermolecular energy due to fluid friction). This is taken care of by an empirical factor C in determining the velocity from Eq. (16.7) as
     (16.8)

Pitot Tube for Flow Measurement

Construction: The principle of flow measurement by Pitot tube was adopted first by a French Scientist Henri Pitot in 1732 for measuring velocities in the river. A right angled glass tube, large enough for capillary effects to be negligible, is used for the purpose. One end of the tube faces the flow while the other end is open to the atmosphere as shown in Fig. 16.3a.

Working:

• The liquid flows up the tube and when equilibrium is attained, the liquid reaches a height above the free surface of the water stream. 
• Since the static pressure, under this situation, is equal to the hydrostatic pressure due to its depth below the free surface, the difference in level between the liquid in the glass tube and the free surface becomes the measure of dynamic pressure. Therefore, we can write, neglecting friction
  
  
  where p₀, p and V are the stagnation pressure, static pressure and velocity respectively at point A (Fig. 16.3a).

  

• Such a tube is known as a Pitot tube and provides one of the most accurate means of measuring the fluid velocity.
  
  Fig 16.3  Simple Pitot Tube  (a) tube for measuring the Stagnation Pressure
                                        (b) Static and Stagnation tubes together

• For an open stream of liquid with a free surface, this single tube is suffcient to determine the velocity. But for a fluid flowing through a closed duct, the Pitot tube measures only the stagnation pressure and so the static pressure must be measured separately. 
   
• Measurement of static pressure in this case is made at the boundary of the wall (Fig. 16.3b). The axis of the tube measuring the static pressure must be perpendicular to the boundary and free from burrs, so that the boundary is smooth and hence the streamlines adjacent to it are not curved. This is done to sense the static pressure only without any part of the dynamic pressure.

Pitot Static Tube

• The tubes recording static pressure and the stagnation pressure (Fig. 16.3b) are usually combined into one instrument known as Pitot static tube (Fig. 16.4).
   
        Fig 16.4  Pitot Static Tube
   
• The tube for sensing the static pressure is known as static tube which surrounds the pitot tube that measures the stagnation pressure. 
   
• Two or more holes are drilled radially through the outer wall of the static tube into annular space. The position of these static holes is important. Downstream of the nose N, the flow is accelerated somewhat with consequent reduction in static pressure. But in front of the supporting stem, there is a reduction in velocity and increase in pressure. 
   
• The static holes should therefore be at the position where the two opposing effects are counterbalanced and the reading corresponds to the undisturbed static pressure. Finally the flow velocity is given by
  
     (16.9)
  where ∆p is the difference between stagnation and static pressures
   
• The factor C takes care of the non-idealities, due to friction, in converting the dynamic head into pressure head and depends, to a large extent, on the geometry of the pitot tube. The value of C is usually determined from calibration test of the pitot tube.

Flow Through Orifices And Mouthpieces

• An orifice is a small aperture through which the fluid passes. The thickness of an orifice in the direction of flow is very small in comparison to its other dimensions.
• If a tank containing a liquid has a hole made on the side or base through which liquid flows, then such a hole may be termed as an orifice.The rate of flow of the liquid through such an orifice at a given time will depend partly on the shape, size and form of the orifice. 

• An orifice usually has a sharp edge so that there is minimum contact with the fluid and consequently minimum frictional resistance at the sides of the orifice. If a sharp edge is not provided, the flow depends on the thickness of the orifice and the roughness of its boundary surface too.

Flow from an Orifice at the Side of a Tank under a Constant Head

• Let us consider a tank containing a liquid and with an orifice at its side wall as shown in Fig. 16.5. The orifice has a sharp edge with the bevelled side facing downstream. Let the height of the free surface of liquid above the centre line of the orifice be kept fixed by some adjustable arrangements of inflow to the tank. 
   

• The liquid issues from the orifice as a free jet under the influence of gravity only. The streamlines approaching the orifice converges towards it. Since an instantaneous change of direction is not possible, the streamlines continue to converge beyond the orifice until they become parallel at the Sec. c-c (Fig. 16.5). 
   

• For an ideal fluid, streamlines will strictly be parallel at an infinite distance, but however fluid friction in practice produce parallel flow at only a short distance from the orifice. The area of the jet at the Sec. c-c is lower than the area of the orifice. The Sec. c-c is known as the vena contracta.
  
  Fig 16.5  Flow from a Sharp edged Orifice
  
   

• The contraction of the jet can be attributed to the action of a lateral force on the jet due to a change in the direction of flow velocity when the fluid approaches the orifice. Since the streamlines become parallel at vena contracta, the pressure at this section is assumed to be uniform. 

• If the pressure difference due to surface tension is neglected, the pressure in the jet at vena contracta becomes equal to that of the ambience surrounding the jet. 
   

• Considering the flow to be steady and frictional effects to be negligible, we can write by the application of Bernoulli’s equation between two points 1 and 2 on a particular stream-line with point 2 being at vena contracta (Fig 16.5).
      (16.10)

• The horizontal plane through the centre of the orifice has been taken as datum level for determining the potential head. 

• If the area of the tank is large enough as compared to that of the orifice, the velocity at point 1 becomes negligibly small and pressure p1 equals to the hydrostatic pressure p₁ equals to the hydrostatic pressure at that point as p₁=p_atm +ρg(h-z₁).
• Therefore, Eq. (16.10) becomes
    (16.11)
  
     (16.12)
  
   
• If the orifice is small in comparison to h, the velocity of the jet is constant across the vena contracta. The Eq. (16.12) states that the velocity with which a jet of liquid escapes from a small orifice is proportional to the square root of the head above the orifice, and is known as Torricelli’s formula.
   
• The velocity V₂ in Eq. (16.12) represents the ideal velocity since the frictional effects were neglected in the derivation. Therefore, a multiplying factor C_v known as coefficient of velocity is introduced to determine the actual velocity as
  
   
• Since the role of friction is to reduce the velocity, C_v is always less than unity. The rate of dischargethrough the orifice can then be written as, 
      (16.13)
  
  where a_c is the cross-sectional area of the jet at vena contracta
   
• Defining a coefficient of contraction C_c as the ratio of the area of vena contracta to the area of orifiice, Eq. (16.8) can be written as
      (16.14)
  
  where, a₀ is the cross-sectional area of the orifice. The product of C_c and C_v is written as C_d and is termed as coefficient of discharge. Therefore