Pumps, Valves and Actuators

Storyboard

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Bomba centrifuga

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Bombas de rotor y centrifugas

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Los dos principales mecanismos sobre los que se basan las bombas son de rotor (desplazan liquido) y las centrifugas que aceleran el liquido radialmente para generar el movimiento.

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Comparación entre bombas

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Las bombas centrifugas logran un menor flujo pero parejo sobre un mayor rango de diferencia de presiones:

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Bernoulli equation, variations

Equation

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The variación de la Presión ($\Delta p$) can be calculated from the average speed ($\bar{v}$) and the speed difference between surfaces ($\Delta v$) with the density ($\rho$) using

$ \Delta p = - \rho \bar{v} \Delta v $

$\bar{v}$
Average speed
$m/s$
10298
$\rho$
Density
$kg/m^3$
5342
$\Delta v$
Speed difference between surfaces
$m/s$
5556
$\Delta p$
Variación de la Presión
$Pa$
6673

In the case where there is no hystrostatic pressure, Bernoulli's law for the density ($\rho$), the pressure in column 1 ($p_1$), the pressure in column 2 ($p_2$), the mean Speed of Fluid in Point 1 ($v_1$) and the mean Speed of Fluid in Point 2 ($v_2$)

$\displaystyle\frac{1}{2} \rho v_1 ^2 + p_1 =\displaystyle\frac{1}{2} \rho v_2 ^2 + p_2 $



can be rewritten with the variación de la Presión ($\Delta p$)

$ \Delta p = p_2 - p_1 $



and keeping in mind that

$v_2^2 - v_1^2 = \displaystyle\frac{1}{2}(v_2-v_1)(v_1+v_2)$



with

$ \bar{v} = \displaystyle\frac{ v_1 + v_2 }{2}$



and

$ \Delta v = v_2 - v_1 $



you have to

$ \Delta p = - \rho \bar{v} \Delta v $

which allows us to see the effect of the average speed of a body and the difference between its surfaces, as observed in an airplane or bird wing.

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Valvulas

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Darcy's law and hydraulic resistance

Equation

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Darcy rewrites the Hagen Poiseuille equation so that the pressure difference ($\Delta p$) is equal to the hydraulic resistance ($R_h$) times the volume flow ($J_V$):

$ \Delta p = R_h J_V $

$R_h$
Hydraulic resistance
$kg/m^4s$
5424
$\Delta p$
Variación de la Presión
$Pa$
6673
$J_V$
Volume flow
$m^3/s$
5448

The volume flow ($J_V$) can be calculated from the hydraulic conductance ($G_h$) and the pressure difference ($\Delta p$) using the following equation:

$ J_V = G_h \Delta p $



Furthermore, using the relationship for the hydraulic resistance ($R_h$):

$ R_h = \displaystyle\frac{1}{ G_h }$



results in:

$ \Delta p = R_h J_V $

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Hydraulic resistance of a tube

Equation

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Since the hydraulic resistance ($R_h$) is equal to the inverse of the hydraulic conductance ($G_h$), it can be calculated from the expression of the latter. In this way, we can identify parameters related to geometry (the tube length ($\Delta L$) and the tube radius ($R$)) and the type of liquid (the viscosity ($\eta$)), which can be collectively referred to as a hydraulic resistance ($R_h$):

$ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$

$R_h$
Hydraulic resistance
$kg/m^4s$
5424
$\pi$
Pi
3.1415927
$rad$
5057
$\Delta L$
Tube length
$m$
5430
$R$
Tube radius
$m$
5417
$\eta$
Viscosity
$Pa s$
5422

Since the hydraulic resistance ($R_h$) is equal to the hydraulic conductance ($G_h$) as per the following equation:

$ R_h = \displaystyle\frac{1}{ G_h }$



and since the hydraulic conductance ($G_h$) is expressed in terms of the viscosity ($\eta$), the tube radius ($R$), and the tube length ($\Delta L$) as follows:

$ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$



we can conclude that:

$ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$

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Parallel hydraulic conductivity

Concept

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If we have three hydraulic resistances $R_{h1}$, $R_{h2}$, and $R_{h3}$, the series sum of the resistances will be:

$ K_{pt} = \displaystyle\sum_k K_{hk}$

$R_{h1}$
Hydraulic Resistance 1
$kg/m^4s$
5425
$R_{h2}$
Hydraulic Resistance 2
$kg/m^4s$
5426
$R_{h3}$
Hydraulic Resistance 3
$kg/m^4s$
5427
$R_{st}$
Total hydraulic resistance in series
$kg/m^4s$
5428

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