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.

>Model

ID:(1388, 0)

Equation

>Top, >Model

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:

$R_{pt} = \displaystyle\frac{1}{ G_{pt} }$

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

Conditions

Variables

$G_h$

$G_{pt}$

Parallel total hydraulic conductance

$m^4s/kg$

10136

$R_h$

$R_{pt}$

Total hydraulic resistance in parallel

$kg/m^4s$

5429

Relation

ID:(15092, 3)

Equation

>Top, >Model

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:

$R_{st} = \displaystyle\frac{1}{ G_{st} }$

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

Conditions

Variables

$G_h$

$G_{st}$

Total Series Hydraulic Conductance

$m^4s/kg$

10135

$R_h$

$R_{st}$

Total hydraulic resistance in series

$kg/m^4s$

5428

Relation

ID:(15092, 2)

Equation

>Top, >Model

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:

$R_{hk} = \displaystyle\frac{1}{ G_{hk} }$

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

Conditions

Variables

$G_h$

$G_{hk}$

Hydraulic conductance in a network

$m^4s/kg$

10134

$R_h$

$R_{hk}$

Hydraulic resistance in a network

$kg/m^4s$

9887

Relation

ID:(15092, 1)

Iframe

>Top

ID:(15729, 0)

Description

>Top

The hydraulic resistance ($R_h$) for an element modeled as a cylindrical tube can be calculated using the tube length ($\Delta L$), the tube radius ($R$), and the viscosity ($\eta$) through the following equation:

and the hydraulic conductance ($G_h$) can be calculated using:

which are related by:

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:

or

ID:(11098, 0)

Description

>Top

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 ($\Delta p_t$):

while the total hydraulic resistance in series ($R_{st}$) is described by:

and the total Series Hydraulic Conductance ($G_{st}$) is defined by:

ID:(15736, 0)

Description

>Top

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:

These values are then summed to obtain the total hydraulic resistance in series ($R_{st}$):

With this result, it is possible to calculate the volume flow ($J_V$) for the total pressure difference ($\Delta p_t$) using:

Once the volume flow ($J_V$) is determined, the pressure difference in a network ($\Delta p_k$) is calculated via:

For the case of three resistances, the calculations can be visualized in the following chart:

ID:(11069, 0)

Description

>Top

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 ($\Delta p_t$):

while the total hydraulic resistance in series ($R_{st}$) is described by:

and the total Series Hydraulic Conductance ($G_{st}$) is defined by:

ID:(15737, 0)

Description

>Top

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:

These values are then summed to obtain the total hydraulic resistance in series ($R_{st}$):

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:

Once the variación de la Presión ($\Delta p$) is determined, the volume flow in a network ($J_{Vk}$) is calculated via:

For the case of three resistances, the calculations can be visualized in the following chart:

ID:(11070, 0)

Top

>Top

Parameters

$G_h$

G_h

Hydraulic conductance

m^4s/kg

$G_{hk}$

G_hk

Hydraulic conductance in a network

m^4s/kg

$R_h$

R_h

Hydraulic resistance

kg/m^4s

$R_{hk}$

R_hk

Hydraulic resistance in a network

kg/m^4s

$G_{pt}$

G_pt

Parallel total hydraulic conductance

m^4s/kg

$\pi$

pi

Pi

rad

$R_{pt}$

R_pt

Total hydraulic resistance in parallel

kg/m^4s

$R_{st}$

R_st

Total hydraulic resistance in series

kg/m^4s

$G_{st}$

G_st

Total Series Hydraulic Conductance

m^4s/kg

$R$

R

Tube radius

m

$\Delta p$

Dp

Variación de la Presión

Pa

$\eta$

eta

Viscosity

Pa s

Variables

$J_{Vt}$

J_Vt

Flujo de Volumen Total

m^3/s

$\Delta p_k$

Dp_k

Pressure difference in a network

Pa

$\Delta p_t$

Dp_t

Total pressure difference

Pa

$\Delta L$

DL

Tube length

m

$J_V$

J_V

Volume flow

m^3/s

$J_{Vk}$

J_Vk

Volume flow in a network

m^3/s

Calculations

• Alternative method
• Traditional method
• Strategy
• 2D
• 3D

Equation

Number

First, select the equation: to , then, select the variable: to

---

Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

ID:(15734, 0)

Equation

>Top, >Model

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 ($R_h$):

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

Conditions

Variables

$R_h$

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

Relation

Development

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:

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

ID:(3629, 0)

Equation

>Top, >Model

With the tube radius ($R$), the viscosity ($\eta$) and the tube length ($\Delta L$) we have that a hydraulic conductance ($G_h$) is:

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

Conditions

Variables

$G_h$

Hydraulic conductance

$m^4s/kg$

10124

$\pi$

Pi

3.1415927

$rad$

5057

$\Delta L$

Tube length

$m$

5430

$R$

Tube radius

$m$

5417

$\eta$

Viscosity

$Pa s$

5422

Relation

Development

ID:(15102, 0)

Equation

>Top, >Model

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:

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

Conditions

Variables

$G_h$

Hydraulic conductance

$m^4s/kg$

10124

$R_h$

Hydraulic resistance

$kg/m^4s$

5424

Relation

ID:(15092, 0)

Equation

>Top, >Model

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 ($J_V$):

$\Delta p = R_h J_V$

Conditions

Variables

$R_h$

Hydraulic resistance

$kg/m^4s$

5424

$\Delta p$

Variación de la Presión

$Pa$

6673

$J_V$

Volume flow

$m^3/s$

5448

Relation

Development

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)

Equation

>Top, >Model

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:

$J_V = G_h \Delta p$

Conditions

Variables

$G_h$

Hydraulic conductance

$m^4s/kg$

10124

$\Delta p$

Variación de la Presión

$Pa$

6673

$J_V$

Volume flow

$m^3/s$

5448

Relation

Development

If we examine the Hagen-Poiseuille law, which allows us to calculate the volume flow ($J_V$) from the tube radius ($R$), 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 tube radius ($R$), 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)

Equation

>Top, >Model

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:

$\Delta p_t =\displaystyle\sum_k \Delta p_k$

Conditions

Variables

$\Delta p_k$

Pressure difference in a network

$Pa$

10132

$\Delta p_t$

Total pressure difference

$Pa$

9842

Relation

ID:(4377, 0)

Equation

>Top, >Model

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:

$R_{st} =\displaystyle\sum_k R_{hk}$

Conditions

Variables

$R_{hk}$

Hydraulic resistance in a network

$kg/m^4s$

9887

$R_{st}$

Total hydraulic resistance in series

$kg/m^4s$

5428

Relation

Development

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)

Equation

>Top, >Model

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 ($G_{hk}$):

$\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$

Conditions

Variables

$G_{hk}$

Hydraulic conductance in a network

$m^4s/kg$

10134

$G_{st}$

Total Series Hydraulic Conductance

$m^4s/kg$

10135

Relation

Development

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)

Equation

>Top, >Model

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 ($J_{Vk}$):

$J_{Vt} =\displaystyle\sum_k J_{Vk}$

Conditions

Variables

$J_{Vt}$

Flujo de Volumen Total

$m^3/s$

6611

$J_{Vk}$

Volume flow in a network

$m^3/s$

10133

Relation

.

ID:(4376, 0)

Equation

>Top, >Model

The total hydraulic resistance in parallel ($R_{pt}$) can be calculated as the inverse of the sum of the hydraulic resistance in a network ($R_{hk}$):

$\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$

Conditions

Variables

$R_{hk}$

Hydraulic resistance in a network

$kg/m^4s$

9887

$R_{pt}$

Total hydraulic resistance in parallel

$kg/m^4s$

5429

Relation

Development

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)

Equation

>Top, >Model

The parallel total hydraulic conductance ($G_{pt}$) is calculated with the sum of the hydraulic conductance in a network ($G_{hk}$):

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

Conditions

Variables

$G_{hk}$

Hydraulic conductance in a network

$m^4s/kg$

10134

$G_{pt}$

Parallel total hydraulic conductance

$m^4s/kg$

10136

Relation

Development

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)

Equation

>Top, >Model

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 ($J_V$):

$\Delta p_t = R_{st} J_V$

$\Delta p = R_h J_V$

Conditions

Variables

$R_h$

$R_{st}$

Total hydraulic resistance in series

$kg/m^4s$

5428

$\Delta p$

$\Delta p_t$

Total pressure difference

$Pa$

9842

$J_V$

Volume flow

$m^3/s$

5448

Relation

Development

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)

Equation

>Top, >Model

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 ($J_V$):

$\Delta p_k = R_{hk} J_V$

$\Delta p = R_h J_V$

Conditions

Variables

$R_h$

$R_{hk}$

Hydraulic resistance in a network

$kg/m^4s$

9887

$\Delta p$

$\Delta p_k$

Pressure difference in a network

$Pa$

10132

$J_V$

Volume flow

$m^3/s$

5448

Relation

Development

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)

Equation

>Top, >Model

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 ($J_V$):

$\Delta p = R_{pt} J_{Vt}$

$\Delta p = R_h J_V$

Conditions

Variables

$R_h$

$R_{pt}$

Total hydraulic resistance in parallel

$kg/m^4s$

5429

$\Delta p$

Variación de la Presión

$Pa$

6673

$J_V$

$J_{Vt}$

Flujo de Volumen Total

$m^3/s$

6611

Relation

Development

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)

Equation

>Top, >Model

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 ($J_V$):

$\Delta p = R_{hk} J_{Vk}$

$\Delta p = R_h J_V$

Conditions

Variables

$R_h$

$R_{hk}$

Hydraulic resistance in a network

$kg/m^4s$

9887

$\Delta p$

Variación de la Presión

$Pa$

6673

$J_V$

$J_{Vk}$

Volume flow in a network

$m^3/s$

10133

Relation

Development

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)