Unit Pressure

Unit pressure or simply called pressure is the amount of force exerted by a fluid distributed uniformly over a unit area.
 

$p = \dfrac{Force, \, F}{Area, \, A}$

 

If the unit pressure is not uniform over the unit area, it can be expressed as the sum of differential pressure.
 

$\displaystyle p = \int \dfrac{dF}{dA}$

 

001-blaise-pascal.jpg
Blaise Pascal (1623 – 1662)

Since fluid at rest cannot resist shearing stress, pressure is always at right angle to the area where it is acting. It is also worthy to note that the total hydrostatic force F = pA, which can be found by cross multiplication.
 

Pascal’s Law

The French mathematician Blaise Pascal (1623 – 1662) states that the pressure is the same in all directions at any point in a fluid at rest.
 

From the figure shown below, summation of forces in y-direction:
$\Sigma F_y = 0$

$p_2A_{ABCO} = (p_1 A_{ABED}) \cos \theta $

$p_2A_{ABCO} = p_1 (A_{ABED} \cos \theta)$

Since $A_{ABCO} = A_{ABED} \cos \theta$, $p_2 = p_1$.
 

000-pascals-law-proof.png

 

Summation of forces in z-direction:
$\Sigma F_z = 0$

$p_3A_{OCED} = (p_1 A_{ABED}) \sin \theta$

$p_3A_{OCED} = p_1 (A_{ABED} \sin \theta)$

Since $A_{OCED} = A_{ABED}\sin \theta$, $p_3 = p_1$.
 

Thus,
$p_1 = p_2 = p_3$   which can be used to conclude Pascal's Law.
 

Summation of forces in x-direction:
$\Sigma F_x = 0$

$p_4A_{AOD} = p_5A_{BCE}$

Since   $A_{AOD} = A_{BCE}$,   $p_4 = p_5$.
 

Atmospheric, Gauge, and Absolute Pressures

Atmospheric pressure is the weight of all gasses above the surface in which it comes in contact. Under normal conditions, atmospheric pressure at sea level is equal to 101.325 kPa (14.696 psi), usually rounded off to 100 kPa (14.7 psi) by engineers. With increase in altitude, atmospheric pressure decreases.
 

Gauge pressure, measured with the use of pressure gauges, is the pressure above or below atmospheric pressure. Negative gauge pressure indicates a vacuum which cannot go below –101.325 kPa. Positive gauge pressure indicates that the pressure is above atmospheric. Gauge pressure is also called relative pressure.
 

Absolute pressure is equal to gauge pressure plus atmospheric pressure. There is no such thing as negative absolute pressure. In the absence of all matter (complete vacuum), the absolute pressure is zero.

$p_{absolute} = p_{atmospheric} + p_{gauge}$

 

Pressure Gauges

Just for the purpose of completeness of this page, pressure gauges (or pressure instruments) are listed here. For more detailed discussion about pressure gauges, refer to the links in each type of pressure instrument. Some general types of pressures instruments are as follows.

  • Barometer - used to measure atmospheric pressure.1
  • Manometer - a U-tube that contains liquid of known specific gravity.2
  • Bourdon gauge - used to measure large pressure difference.3

 

Variation of Pressure with Depth in a Fluid

Consider two points 1 and 2 lie in the ends of fluid prism having a cross-sectional area dA and length L. The difference in elevation between these two points is h as shown in Figure 02 below. The fluid is at rest and its surface is free. The prism is therefore in equilibrium and all forces acting on it sums up to zero.
 

000-variations-in-pressure.gif

 

Note: FFS stands for Free Fluid Surface which refers to fluid surface subject to zero gauge pressure.
 

The volume of the prism is equal to the length times the base area of the fluid.
$V = L \, dA$
 

The weight of the fluid prism shown is equal to the product of the unit weight and volume.
$W = \gamma V$

$W = \gamma L \, dA$
 

Sum up all the forces in x-direction
$\Sigma F_x = 0$

$F_2 = F_1 + W_x$

$F_2 - F_1 =W \sin \theta$

$p_2 \, dA - p_1 \, dA = \gamma L \, dA \sin \theta$

$p_2 - p_1 = \gamma L \sin \theta$

but L sin θ = h, thus

$p_2 - p_1 = \gamma h$

Therefore, in any homogeneous fluid at rest, the difference in pressure between any two points is equal to the product of the unit weight of the fluid and the difference in elevation of the points.
 

If h = 0 so that points 1 and 2 are on the same horizontal plane, p2 - p1 = 0 or

$p_1 = p_2$

Therefore, in any homogeneous fluid at rest, the pressures at all points along the same horizontal plane are equal.
 

If point 1 lie on the FFS, the gauge pressure p1 = 0, making p2 - 0 = γh or simply

$p = \gamma h$

This means that the pressure at any depth h below a continuous free fluid surface at rest is equal to the product of the unit weight of fluid and the depth h.
 

Transmission of Pressure

We can write the equation p2 - p1 = γh into the form

$p_2 = p_1 + \gamma h$

which means that any change in the pressure at point 1 would cause an equal change of pressure at point 2. In other words, a pressure applied at any point in a liquid at rest is transmitted equally and undiminished to every other point in the liquid.
 

Pressure Head

The equation p = γh may be written into the form

$h = \dfrac{p}{\gamma}$

where h or its equivalent p/γ is in hydraulics called the pressure head. Pressure head is the height of column of homogeneous fluid of unit weight γ that will produce an intensity of pressure p.
 

To convert pressure head of liquid A to equivalent pressure head of liquid B

$h_B = \dfrac{s_A}{s_B}h_A = \dfrac{\rho_A}{\rho_B}h_A = \dfrac{\gamma_A}{\gamma_B}h_A$

 

To convert pressure head of any liquid to equivalent pressure head of water4

$h_{water} = s_{liquid} \times h_{liquid}$

where,
s = specific gravity
γ = unit weight
ρ = density
 

Manometers

Manometer is a simple and inexpensive device of measuring pressure and pressure difference. It is usually bent to form a U-tube and filled with liquid of known specific gravity. The surface of the liquid will move in proportion to changes of pressure.
 

Types of Manometer

Piezometer
Piezometer is the simplest form of manometer which is tapped into the wall of pressure conduit for the purpose of measuring pressure. Though effective in many purposes, piezometer is not practical to use in lighter liquids with large pressure and cannot be used to measure gas pressure.
 

001-piezometers.gif

 

From the figure above, three piezometers A, B, and C are attached to a pressure conduit at bottom, top, and side, respectively. The column of liquid at A, B, and C will rise at the same level above M indicating a positive pressure at M. Also, the piezometer D measures the negative pressure at N.
 

Open Manometer
Open manometer is a tube bent into a U-shape to contain one or more fluids of different specific gravities. It is used to measure pressure. Example of open manometer is shown below.

 

001-open-manometer.gif

 

Differential Manometer
Differential manometer cannot measure pressure but can measure pressure difference. Frequently in hydraulic problems, difference in pressure is more useful information than the pressure itself.

 

001-differential-manometer.gif

 

Steps in Solving Manometer Problems

Ordinarily, it is easier to work in units of pressure head rather than pressure for solving any manometer problem.

  1. Draw a sketch of the manometer approximately to scale.
  2. Decide on the fluid of which head are to be expressed. Water is more desirable. In most cases, we suggest to use head in water even if there is no water in the system.
  3. Starting at a point of know pressure head, number in order the levels of contact of fluids of different specific gravities.
  4. Proceed from level to level, add pressure head in going down and subtract pressure head in going up with due regard to the specific gravity of the fluids.
  • 1. Wikipedia article about barometer
  • 2. Wikipedia article about manometer
  • 3. Schematic diagram of Bourdon gauge
  • 4. For water
    specific gravity, s = 1.0
    unit weight, γ = 9.81 kN/m3 (or 62.4 lb/ft3 in English system)
    density ρ = 1000 kg/m3 (or 1.94 slugs/ft3 in English system)
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