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Hydraulic element networks
Storyboard 
When comparing Darcy's law to Ohm's law in electricity, we notice an analogy where the flow of liquid resembles electric current, the pressure difference relates to the voltage difference, and hydraulic elements are compared to their hydraulic resistances, similar to electric resistors.
This analogy implies that, just as there are electrical networks, hydraulic networks can also be defined in which total hydraulic resistances are calculated based on partial hydraulic resistances.
ID:(1388, 0)
Hydraulic conductance (3)
Equation
In the context of electrical resistance, there exists its inverse, known as electrical conductance. Similarly, what would be the hydraulic conductance ($G_h$) can be defined in terms of the hydraulic resistance ($R_h$) through the expression:
 $ R_{pt} = \displaystyle\frac{1}{ G_{pt} }$ |
 $ R_h = \displaystyle\frac{1}{ G_h }$ |
ID:(15092, 3)
Hydraulic conductance (2)
Equation
In the context of electrical resistance, there exists its inverse, known as electrical conductance. Similarly, what would be the hydraulic conductance ($G_h$) can be defined in terms of the hydraulic resistance ($R_h$) through the expression:
| $ R_{st} = \displaystyle\frac{1}{ G_{st} }$ |
| $ R_h = \displaystyle\frac{1}{ G_h }$ |
ID:(15092, 2)
Hydraulic conductance (1)
Equation
In the context of electrical resistance, there exists its inverse, known as electrical conductance. Similarly, what would be the hydraulic conductance ($G_h$) can be defined in terms of the hydraulic resistance ($R_h$) through the expression:
| $ R_{hk} = \displaystyle\frac{1}{ G_{hk} }$ |
| $ R_h = \displaystyle\frac{1}{ G_h }$ |
ID:(15092, 1)
Hydrodynamic networks
Description
The hydraulic resistance ($R_h$) for an element modeled as a cylindrical tube can be calculated using the tube length ($\Delta L$), the cylinder radio ($R$), and the viscosity ($\eta$) through the following equation:
| $ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$ |
and the hydraulic conductance ($G_h$) can be calculated using:
| $ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$ |
which are related by:
| $ R_h = \displaystyle\frac{1}{ G_h }$ |
Both the hydraulic resistance ($R_h$) and the hydraulic conductance ($G_h$) allow for a relationship between the variación de la Presión ($\Delta p$) and the volume flow ($J_V$) using:
| $ \Delta p = R_h J_V $ |
or
| $ J_V = G_h \Delta p $ |
ID:(11098, 0)
Sum of hydraulic resistances in series
Description
In the case of hydraulic resistances connected in series:
the sum of the pressure drop pressure difference in a network ($\Delta p_k$) across each hydraulic resistance in a network ($R_{hk}$) corresponds to the total pressure difference ($\Delta p_t$):
| $ \Delta p_t =\displaystyle\sum_k \Delta p_k $ |
while the total hydraulic resistance in series ($R_{st}$) is described by:
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
and the total Series Hydraulic Conductance ($G_{st}$) is defined by:
| $\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$ |
ID:(15736, 0)
Process for the addition of hydraulic resistances in series
Description
First, values for the hydraulic resistance in a network ($R_{hk}$) are calculated using the variables the viscosity ($\eta$), the cylinder k radio ($R_k$), and the tube k length ($\Delta L_k$) through the following equation:
| $ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$ |
These values are then summed to obtain the total hydraulic resistance in series ($R_{st}$):
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
With this result, it is possible to calculate the volume flow ($J_V$) for the total pressure difference ($\Delta p_t$) using:
| $ \Delta p_t = R_{st} J_V $ |
Once the volume flow ($J_V$) is determined, the pressure difference in a network ($\Delta p_k$) is calculated via:
| $ \Delta p_k = R_{hk} J_V $ |
For the case of three resistances, the calculations can be visualized in the following chart:
ID:(11069, 0)
Sum of hydraulic resistances in parallel
Description
In the case of hydraulic resistances connected in series:
the sum of the pressure drop pressure difference in a network ($\Delta p_k$) across each hydraulic resistance in a network ($R_{hk}$) corresponds to the total pressure difference ($\Delta p_t$):
| $ \Delta p_t =\displaystyle\sum_k \Delta p_k $ |
while the total hydraulic resistance in series ($R_{st}$) is described by:
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
and the total Series Hydraulic Conductance ($G_{st}$) is defined by:
| $\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$ |
ID:(15737, 0)
Process for the addition of hydraulic resistances in parallel
Description
First, values for the hydraulic resistance in a network ($R_{hk}$) are calculated using the variables the viscosity ($\eta$), the cylinder k radio ($R_k$), and the tube k length ($\Delta L_k$) through the following equation:
| $ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$ |
These values are then summed to obtain the total hydraulic resistance in series ($R_{st}$):
| $\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$ |
With this result, it is possible to calculate the variación de la Presión ($\Delta p$) for the total hydraulic resistance in parallel ($R_{pt}$) using:
| $ \Delta p = R_{pt} J_{Vt} $ |
Once the variación de la Presión ($\Delta p$) is determined, the volume flow in a network ($J_{Vk}$) is calculated via:
| $ \Delta p = R_{hk} J_{Vk} $ |
For the case of three resistances, the calculations can be visualized in the following chart:
ID:(11070, 0)
Model
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Calculations
Variables
Parameters
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Calculations
Calculations
Variable 
Given Calculate 
Target : Equation To be used Equation
$\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$
1/ G_st = @SUM( 1/ G_hk, k )
$\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$
1/ R_pt =@SUM( 1/ R_hk , k )
$ \Delta p_t = R_{st} J_V $
Dp = R_h * J_V
$ \Delta p_k = R_{hk} J_V $
Dp = R_h * J_V
$ \Delta p = R_{pt} J_{Vt} $
Dp = R_h * J_V
$ \Delta p = R_{hk} J_{Vk} $
Dp = R_h * J_V
$ \Delta p = R_h J_V $
Dp = R_h * J_V
$ \Delta p_t =\displaystyle\sum_k \Delta p_k $
Dp_t =sum_k Dp_k
$ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$
G_h = pi * R ^4/(8* eta * abs( DL ))
$ G_{pt} =\displaystyle\sum_k G_{hk} $
G_pt = @SUM( G_hk , k )
$ J_V = G_h \Delta p $
J_V = G_h * Dp
$ J_{Vt} =\displaystyle\sum_k J_{Vk} $
J_Vt =sum_k J_Vk
$ R_h = \displaystyle\frac{1}{ G_h }$
R_h = 1/ G_h
$ R_{hk} = \displaystyle\frac{1}{ G_{hk} }$
R_h = 1/ G_h
$ R_{st} = \displaystyle\frac{1}{ G_{st} }$
R_h = 1/ G_h
$ R_{pt} = \displaystyle\frac{1}{ G_{pt} }$
R_h = 1/ G_h
$ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$
R_h =8* eta * abs( DL )/( pi * R ^4)
$ R_{st} =\displaystyle\sum_k R_{hk} $
R_st =@SUM( R_hk , k )
ID:(15734, 0)
Hydraulic resistance of a tube
Equation
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 cylinder radio ($R$)) 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}$ |
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 cylinder radio ($R$), 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}$ |
ID:(3629, 0)
Hydraulic Conductance of a Pipe
Equation
With the cylinder radio ($R$), the viscosity ($\eta$) and the tube length ($\Delta L$) we have that a hydraulic conductance ($G_h$) is:
| $ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$ |
ID:(15102, 0)
Hydraulic conductance
Equation
In the context of electrical resistance, there exists its inverse, known as electrical conductance. Similarly, what would be the hydraulic conductance ($G_h$) can be defined in terms of the hydraulic resistance ($R_h$) through the expression:
| $ R_h = \displaystyle\frac{1}{ G_h }$ |
ID:(15092, 0)
Darcy's law and hydraulic resistance
Equation
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 $ |
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 $ |
ID:(3179, 0)
Darcy's law and hydraulic conductance
Equation
With the introduction of the hydraulic conductance ($G_h$), we can rewrite the Hagen-Poiseuille equation with the pressure difference ($\Delta p$) and the volume flow ($J_V$) using the following equation:
| $ J_V = G_h \Delta p $ |
If we examine the Hagen-Poiseuille law, which allows us to calculate the volume flow ($J_V$) from the cylinder radio ($R$), the viscosity ($\eta$), the tube length ($\Delta L$), and the pressure difference ($\Delta p$):
| $ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
we can introduce the hydraulic conductance ($G_h$), defined in terms of the tube length ($\Delta L$), the cylinder radio ($R$), and the viscosity ($\eta$), as follows:
| $ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$ |
to arrive at:
| $ J_V = G_h \Delta p $ |
ID:(14471, 0)
Sum of series pressures
Equation
The total pressure difference ($\Delta p_t$) in relation to the various pressure difference in a network ($\Delta p_k$), leading us to the following conclusion:
| $ \Delta p_t =\displaystyle\sum_k \Delta p_k $ |
ID:(4377, 0)
Hydraulic resistance of elements in series
Equation
When there are multiple hydraulic resistances connected in series, we can calculate the total hydraulic resistance in series ($R_{st}$) by summing the hydraulic resistance in a network ($R_{hk}$), as expressed in the following formula:
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
One way to model a tube with varying cross-section is to divide it into sections with constant radius and then sum the hydraulic resistances in series. Suppose we have a series of the hydraulic resistance in a network ($R_{hk}$), which depends on the viscosity ($\eta$), the cylinder k radio ($R_k$), and the tube k length ($\Delta L_k$) via the following equation:
| $ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$ |
In each segment, there will be a pressure difference in a network ($\Delta p_k$) with the hydraulic resistance in a network ($R_{hk}$) and the volume flow ($J_V$) to which Darcy's Law is applied:
| $ \Delta p_k = R_{hk} J_V $ |
the total pressure difference ($\Delta p_t$) will be equal to the sum of the individual pressure difference in a network ($\Delta p_k$):
| $ \Delta p_t =\displaystyle\sum_k \Delta p_k $ |
therefore,
$\Delta p_t=\displaystyle\sum_k \Delta p_k=\displaystyle\sum_k (R_{hk}J_V)=\left(\displaystyle\sum_k R_{hk}\right)J_V\equiv R_{st}J_V$
Thus, the system can be modeled as a single conduit with the hydraulic resistance calculated as the sum of the individual components:
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
ID:(3180, 0)
Hydraulic conductancia of elements in series
Equation
In the case of hydraulic resistances in series, the inverse of the total Series Hydraulic Conductance ($G_{st}$) is calculated by summing the inverses of each the hydraulic conductance in a network ($G_{hk}$):
| $\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$ |
The total hydraulic resistance in series ($R_{st}$), along with the hydraulic resistance in a network ($R_{hk}$), in
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
and along with the hydraulic conductance in a network ($G_{hk}$) and the equation
| $ R_{st} = \displaystyle\frac{1}{ G_{st} }$ |
leads to the total Series Hydraulic Conductance ($G_{st}$) can be calculated with:
| $\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$ |
ID:(3633, 0)
Sum of parallel flows
Equation
The sum of soil layers in parallel, denoted as the total flow ($J_{Vt}$), is equal to the sum of the volume flow in a network ($J_{Vk}$):
| $ J_{Vt} =\displaystyle\sum_k J_{Vk} $ |
.
ID:(4376, 0)
Hydraulic resistance of parallel elements
Equation
The total hydraulic resistance in parallel ($R_{pt}$) can be calculated as the inverse of the sum of the hydraulic resistance in a network ($R_{hk}$):
| $\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$ |
The parallel total hydraulic conductance ($G_{pt}$) combined with the hydraulic conductance in a network ($G_{hk}$) in
| $ G_{pt} =\displaystyle\sum_k G_{hk} $ |
and along with the hydraulic resistance in a network ($R_{hk}$) and the equation
| $ R_{st} = \displaystyle\frac{1}{ G_{st} }$ |
leads to the total hydraulic resistance in parallel ($R_{pt}$) via
| $\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$ |
ID:(3181, 0)
Hydraulic conductance of elements in parallel
Equation
The parallel total hydraulic conductance ($G_{pt}$) is calculated with the sum of the hydraulic conductance in a network ($G_{hk}$):
| $ G_{pt} =\displaystyle\sum_k G_{hk} $ |
With the total flow ($J_{Vt}$) being equal to the volume flow in a network ($J_{Vk}$):
| $ J_{Vt} =\displaystyle\sum_k J_{Vk} $ |
and with the pressure difference ($\Delta p$) and the hydraulic conductance in a network ($G_{hk}$), along with the equation
for each element, it leads us to the conclusion that with the parallel total hydraulic conductance ($G_{pt}$),
$J_{Vt}=\displaystyle\sum_k J_{Vk} = \displaystyle\sum_k G_{hk}\Delta p = G_{pt}\Delta p$
we have
| $ G_{pt} =\displaystyle\sum_k G_{hk} $ |
.
ID:(3634, 0)
Darcy's law and hydraulic resistance (1)
Equation
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_t = R_{st} J_V $ |
| $ \Delta p = R_h J_V $ |
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 $ |
ID:(3179, 1)
Darcy's law and hydraulic resistance (2)
Equation
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_k = R_{hk} J_V $ |
| $ \Delta p = R_h J_V $ |
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 $ |
ID:(3179, 2)
Darcy's law and hydraulic resistance (3)
Equation
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_{pt} J_{Vt} $ |
| $ \Delta p = R_h J_V $ |
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 $ |
ID:(3179, 3)
Darcy's law and hydraulic resistance (4)
Equation
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_{hk} J_{Vk} $ |
| $ \Delta p = R_h J_V $ |
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 $ |
ID:(3179, 4)