Question

What is the pressure energy per unit mass stored in liquid of density ρ at pressure P?

Solution

Consider a tube of varying cross-section through which an ideal liquid is in streamline flow.
Let,
P₁ = Pressure applied on the liquid at A,
P₂ = Pressure at end B, against which the liquid is to move out
a₁, a₂ = Area of cross section at the tube at A and B respectively.
h₁, h₂ = mean height of section A and B from ground or a reference level.
v₁, v₂ = normal velocity of the liquid flow at section A and B,
$straight rho$ = density of ideal liquid flowing though the tube.
Liquid flows from A to B.
That is,
P₁ > P₂
Mass m of the liquid crossing per second through any section of the tube is in accordance with the equation of continuity, given by
a₁v₁p = a₂v₂ p = m
$rightwards double arrow space straight a subscript 1 space straight v subscript 1 space equals space straight a subscript 2 space straight v subscript 2 space equals space straight m over straight rho space equals space straight V space As comma space straight a subscript 1 space greater than thin space straight a subscript 2 therefore space straight v subscript 2 space greater than space straight v subscript 1 space Force space on space liquid space at space section space straight A space equals space straight P subscript 1 straight a subscript 1 Force space on space liquid space at space section space straight B space equals space straight P subscript 2 straight a subscript 2 space Work space done space per space second space on space the space liquid space at space section space straight A space equals straight P subscript 1 straight a subscript 1 space cross times space straight v subscript 1 space equals space straight P subscript 1 straight V space Work space done divided by second space by space the space liquid space at space section space straight B space equals space straight P subscript 2 straight a subscript 2 space cross times space straight v subscript 2 space equals space straight P subscript 2 straight V Net space work space done divided by second space on space the space liquid space by space the space pressure space energy space in space moving space the space liquid space from space section space straight A space to space section space straight B space is comma space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight P subscript 1 straight V space space minus space straight P subscript 2 straight V space When space the space mass space straight m space of space the space liquid space flows space in space 1 space second space from space straight A space to space straight B comma space it apostrophe straight s space height space increases space from space straight h subscript 1 space to space straight h subscript 2. Therefore comma space Increase space in space straight P. straight E space divided by sec space of space the space liquid space from space straight A space to space straight B space is comma space mgh subscript 2 subscript space end subscript space minus space mgh subscript 1 space Increase space in space straight K. straight E space divided by sec space of space the space liquid space from space straight A space to space straight B space is comma space space space space space space space space space space space space space space space space space space space space space space 1 half mv subscript 2 squared space minus space 1 half mv subscript 1 squared Now comma space ussing space work space energy space principle comma space Work space done divided by sec space by space the space pressure space energy space equals increase space in space straight P. straight E divided by sec space space plus space increase space in space straight K. straight E divided by sec space straight P subscript 1 straight V space minus space straight P subscript 2 straight V space equals space left parenthesis mgh subscript 2 subscript space end subscript space minus space mgh subscript 1 space right parenthesis space plus space open parentheses space space space space space space 1 half mv subscript 2 squared space minus space 1 half mv subscript 1 squared close parentheses That space is comma space straight P subscript 1 straight V space plus space mgh subscript 1 space plus space 1 half mv subscript 1 squared space equals space space straight P subscript 2 straight V space plus mgh subscript 2 subscript space end subscript plus 1 half mv subscript 2 squared Dividing space throughout space by space straight m comma space we space get fraction numerator straight P subscript 1 straight V over denominator straight m end fraction space plus space gh subscript 1 space plus 1 half straight v subscript 1 squared space equals fraction numerator straight P subscript 2 straight V space over denominator straight m end fraction space plus gh subscript 2 subscript space end subscript space plus space 1 half straight v subscript 2 squared Therefore comma space straight P over straight rho plus space gh space plus space 1 half straight v squared space equals space straight a space constant$

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Mechanical Properties Of Fluids

What is the pressure energy per unit mass stored in liquid of density ρ at pressure P?

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