Hydraulic

Written by Jerry Ratzlaff on . Posted in Fluid Dynamics

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Hydraulics is the force or motion applied on a confined liquid.

 

 

 

Hydraulic Conductivity

Hydraulic conductivity ( \(k\) ) is the ease with which a fluid can move through porous spaces or fractures.  See the Darcy's Law.

Hydraulic Depthhydraulic depth

Hydraulic depth ( \(d_h\) ) (also called hydraulic mean depth) is the cross section area of water in a pipe or channel divided by the wetting perimeter.

Hydraulic Depth formula

\(\large{ d_h =  \frac  {  A }  { w }   }\)         

Where:

\(\large{ d_h }\) = hydraulic depth

\(\large{ A }\) = area of section flow

\(\large{ P }\) = cross section wetting perimeter

\(\large{ w }\) = top width of fluid

Solve for:

\(\large{ A =  d_h  w  }\)

\(\large{ w = \frac { A } { d_h}  }\)

Hydraulic Diameter

Hydraulic diameter ( \(d_h\) ) is normally used when the flow is in non-circular pipe or tubes and channels.  Circular pipe has the same pressure drop of a rectangular channel but a greater average velocity.  Square or rectangular pipes have a greater weight and a greater pressure drop compared with a circular pipe with the same section.

Hydraulic Diameter FORMULA

\(\large{ d_h = \frac { 4  A } { P  }  }\)         

Where:

\(\large{ d_h }\) = hydraulic diameter

\(\large{ A }\) = cross section area

\(\large{ P }\) = cross section wetting perimeter

Hydraulic Diameter of an Ellipsehydraulic diameter of a ellipse 1

Hydraulic Diameter of an Ellipse FORMULA

\(\large{ d_h =  \frac { 4wh \left( 64 - 16 e^2 \right)  }  {  \left( w + h \right)  \left( 64 - 3 e^4 \right)   }   }\)                

\(\large{ e = \frac { w - h  }  { w + h }   }\)         

Where:

\(\large{ d_h }\) = hydraulic diameter

\(\large{ h }\) = height of ellipse

\(\large{ w }\) = width of ellipse

Hydraulic Diameter of a Duct, Pipe or tubehydraulic diameter of a pipe

Hydraulic diameter ( \(d_h\) ) of a tube is when the flow is within the tube.

Hydraulic Diameter of a Duct, Pipe or tube FORMULA

\(\large{ d_h = \frac  { 4 \pi r^2 } {  2 \pi r  }   }\)         

Where:

\(\large{ d_h }\) = hydraulic diameter

\(\large{ r }\) = pipe inside radius

 

Hydraulic Diameter of a tube with a tube on the Insidehydraulic diameter of a pipe in pipe

Hydraulic diameter ( \(d_h\) ) of tube in tube is when the flow is between the inside tube and the outside tube. 

Hydraulic Diameter of a tube with a tube on the Inside FORMULA

\(\large{ d_h =   \frac  { 4     \left( \pi r_{o}{^2}  -  \pi r_{i}{^2}  \right)  }  {  2 \pi r_o  +  2 \pi r_i  }   }\)         

\(\large{ d_h =   \frac  { 4 \pi    \left( r_{o}{^2}  -  r_{i}{^2}  \right)  }  {  2 \pi   \left( r_o  +   r_i \right)  }   }\)         

Where:

\(\large{ d_h }\) = hydraulic diameter

\(\large{ r_i }\) = pipe outside radius of the inside tube

\(\large{ r_o }\) = pipe inside radius of the outside tube

Hydraulic Diameter of an Isosceles Trianglehydraulic diameter of a rectangle

Hydraulic Diameter of an Isosceles Triangle FORMULA

\(\large{ d_h =  \frac { w\; sin \; \theta   }  {  1 + sin \frac {\theta}{2}  } }\)                

Where:

\(\large{ d_h }\) = hydraulic diameter

\(\large{ w }\) = length of side

\(\large{ \theta }\)   (Greek symbol theta) = degree

 

Hydraulic Diameter of a rectangular Tubehydraulic diameter of a rectangle

Hydraulic diameter ( \(d_h\) ) of a rectangular tube is when the flow is within the tube.

Hydraulic Diameter of a rectangular Tube FORMULA

\(\large{ d_h = \frac  { 4wh } {  2 \left( w + h  \right)  }   }\)          

\(\large{ d_h = \frac  { 2wh } { w + h }   }\)               

Where:

\(\large{ d_h }\) = hydraulic diameter

\(\large{ h }\) = height of tube

\(\large{ w }\) = width of tube

Hydraulic Diameter of a Right Trianglehydraulic diameter of a square

Hydraulic Diameter of a Right Triangle FORMULA

\(\large{ d_h = \frac  { 2wh } { w + h  +  \left(  w^2 + h^2  \right) ^ \frac{1}{2}   }   }\)                       

Where:

\(\large{ d_h }\) = hydraulic diameter

\(\large{ h }\) = height of tube

\(\large{ w }\) = width of tube

 

Hydraulic Diameter of a Square Tubehydraulic diameter of a square

Hydraulic diameter ( \(d_h\) ) of a square tube is when the flow is within the tube.

Hydraulic Diameter of a Square Tube FORMULA

\(\large{ d_h = w  }\)        

Where:

\(\large{ d_h }\) = hydraulic diameter

\(\large{ w }\) = width of tube

 

Hydraulic Gradienthydraulic gradient

Hydraulic gradient ( \(i\) ) (dimensionless number) is the change in height (pressure) to length.

Hydraulic Gradient FORMULA

\(\large{ i = \frac { h_1 \;-\; h_2} { l}  }\)         

Where:

\(\large{ i }\) = hydraulic gradient

\(\large{ h_1 }\) = pressure head at point 1

\(\large{ h_2 }\) = pressure head at point 2

\(\large{ l }\) = length of column

Solve for:

\(\large{ h_1 = i   l \;+\; h_2  }\)

\(\large{ h_2 = h_1 \;-\; i  l   }\)

\(\large{ l =  \frac { h_1 \;-\; h_2} { i}    }\)

Hydraulic Headhydraulic gradient

Hydraulic head ( \(h\) ) is the measurement mechanical energy due to pressure of a fluid from a higher elevation to a lower elevation.

Hydraulic Head Formula

\(\large{ h = \frac {p}{g \rho} }\)         

\(\large{ h =  h_1 \;-\; h_2  }\)         

Where:

\(\large{ h }\)= head

\(\large{ g }\) = gravitational acceleration

\(\large{ h_1 }\) = pressure head at point 1

\(\large{ h_2 }\) = pressure head at point 2

\(\large{ p }\) = pressure

\(\large{ \rho }\)   (Greek symbol rho) = density

Hydraulic Radius

Hydraulic radius ( \(r_h\) ) is the cross section area of water in a pipe or channel divided by the wetting perimeter.

Hydraulic Radius formula

\(\large{ r_h =  \frac  {  A }  { P }   }\)         

Where:

\(\large{ r_h }\) = hydraulic radius

\(\large{ A }\) = cross section flow area

\(\large{ P }\) = wetted perimeter

Solve for:

\(\large{ A =  r_h  P  }\)

\(\large{ P = \frac { A } { r_h}  }\)

Hydraulic Radius of a Pipehydraulic radius of a pipe 2

Hydraulic radius ( \(r_h\) ) is the cross section area of water in a pipe or channel divided by the wetting perimeter.

Hydraulic Radius of a Pipe formula

\(\large{ r_h =  \frac  {  A }  { P }   }\)         

\(\large{ r_h =  \frac  {  \frac { d^2  } {  4  \left( \theta  - sin  \;  \left( 2 \theta \right)  \right)  } }  { \theta d }   }\)

\(\large{ r_h =  \frac {d} {4}  \; \frac { 1 - sin \; \left(  2 \theta \right)  }    {  2  \theta  }   }\)

Where:

\(\large{ r_h }\) = hydraulic radius

\(\large{ d }\) = diameter ( \(2r\) )

\(\large{  A }\) = cross section flow area

\(\large{ P }\) = wetted perimeter

\(\large{ \theta }\)   (Greek symbol theta) = degree

Solve for:

\(\large{ \theta = cos^{-1}  \; \left(  1 -  \frac {h} {r}  \right)   }\)

\(\large{ A =   \frac { d^2  } {  4  \left( \theta  - sin  \;  \left( 2 \theta \right)  \right)  }  }\)

\(\large{ P =  \theta d  }\)

Hydraulic Radius of a Rectangular Channelhydraulic radius of a rectangle

Hydraulic radius ( \(r_h\) ) is the cross section area of water in a pipe or channel divided by the wetting perimeter.

Hydraulic Radius of a Rectangular Channel formula

\(\large{ r_h =  \frac  {  A }  { P }   }\)         

\(\large{ r_h =  \frac  {  b h }  { b + 2h }   }\)         

Where:

\(\large{ r_h }\) = hydraulic radius

\(\large{ A }\) = cross section flow area

\(\large{ b }\) = bottom width of fluid

\(\large{ h }\) = depth of fluid

\(\large{ P }\) = wetted perimeter

\(\large{ w }\) = top width of fluid

Solve for:

\(\large{ A =  b h  }\)

\(\large{ P = b + 2h  }\)

Hydraulic Radius of a Trapezoidal Channel (Equal Side Slopes)hydraulic radius of a trapezoid ES

Hydraulic radius ( \(r_h\) ) is the cross section area of water in a pipe or channel divided by the wetting perimeter.

Hydraulic Radius of a Trapezoidal Channel formula

\(\large{ r_h =  \frac  {  A }  { P }   }\)         

\(\large{ r_h =  \frac  {  bh  +  \left( z h \right) 2 }  { b + 2h  \left( bh  +  z h 2  \right)  \frac {1}{2} }   }\)

\(\large{ r_h =  \frac  {  h  \left( b + z h \right) }  { b + 2h  \sqrt { 1 + z^2 }  }   }\)

Where:

\(\large{ r_h }\) = hydraulic radius

\(\large{ A }\) = cross section flow area

\(\large{ b }\) = bottom width of fluid

\(\large{ h }\) = depth of fluid

\(\large{ P }\) = wetted perimeter

\(\large{ w }\) = top width of fluid

\(\large{ z }\) = width of channel slope

Solve for:

\(\large{ A =  bh  +  \left( z h \right) 2  }\)

\(\large{ P = b + 2h  \left( bh  +  z h 2  \right)  \frac {1}{2}  }\)

\(\large{ A =  h  \left( b + z h \right)  }\)

\(\large{ P = b + 2h  \sqrt { 1 + z^2 }  }\)

Hydraulic Radius of a Trapezoidal Channel (Unequal Side Slopes)hydraulic radius of a trapezoid US

Hydraulic radius ( \(r_h\) ) is the cross section area of water in a pipe or channel divided by the wetting perimeter.

Hydraulic Radius of a Trapezoidal Channel formula

\(\large{ r_h =  \frac  {  A }  { P }   }\)         

\(\large{ r_h =  \frac  {  \frac {h^2} {2}  \left( z_1 + x_2 \right) + hg }  { b + h  \left(  \sqrt {  1 + z_{1}{^2}  }  +  \sqrt {  1 + z_{2}{^2}  }   \right) }   }\)

Where:

\(\large{ r_h }\) = hydraulic radius

\(\large{ A }\) = cross section flow area

\(\large{ b }\) = bottom width of fluid

\(\large{ h }\) = depth of fluid

\(\large{ P }\) = wetted perimeter

\(\large{ w }\) = top width of fluid

\(\large{ z }\) = width of channel slope

Solve for:

\(\large{ A =  \frac {h^2} {2}  \left( z_1 + x_2 \right) + hg  }\)

\(\large{ P =  b + h  \left(  \sqrt {  1 + z_{1}{^2}  }  +  \sqrt {  1 + z_{2}{^2}  }   \right)  }\)

Hydraulic Radius of a Triangular Channelhydraulic radius of a triangle 

Hydraulic radius ( \(r_h\) ) is the cross section area of water in a pipe or channel divided by the wetting perimeter.

Hydraulic Radius of a Triangular Channel formula

\(\large{ r_h =  \frac  {  A }  { P }   }\)         

\(\large{ r_h =  \frac  {  zh^2 }  { 2h  \sqrt  {1  +  z^2 } }   }\)

Where:

\(\large{ r_h }\) = hydraulic radius

\(\large{ A }\) = cross section flow area

\(\large{ b }\) = bottom width of fluid

\(\large{ h }\) = depth of fluid

\(\large{ P }\) = wetted perimeter

\(\large{ w }\) = top width of fluid

\(\large{ z }\) = width of channel slope

Solve for:

\(\large{ A =  zh^2  }\)

\(\large{ P = 2h  \sqrt  {1  +  z^2 }  }\)

Flow Area

Flow area refers to the cross section area of the flow within the channel.

Wetted Perimeterhydraulic wetted

Wetting perimeter is the portion of the channel that is in contact with the fluid flowing.

 

 

 

 

 

 

 

Tags: Equations for Hydraulic