Storyboard

When hydraulic elements are connected in series, the flow remains constant, but each hydraulic element experiences a pressure drop. The sum of these pressure drops equals the total drop, and therefore, the total hydraulic resistance is equal to the sum of all individual hydraulic resistances. On the other hand, the inverse of the total hydraulic conductivity is equal to the sum of the inverses of the hydraulic conductivities.

>Model

ID:(1466, 0)

Iframe

>Top

ID:(15727, 0)

Concept

>Top

In the case of a sum where the elements are connected in series, the total hydraulic resistance of the system is calculated by summing the individual resistances of each element.

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:

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:

the total pressure difference ($\Delta p_t$) will be equal to the sum of the individual pressure difference in a network ($\Delta p_k$):

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:

ID:(3630, 0)

Concept

>Top

In the case of a sum where the elements are connected in series, the total hydraulic conductance of the system is calculated by adding the individual hydraulic conductances of each element.

the total hydraulic resistance in series ($R_{st}$), along with the hydraulic resistance in a network ($R_{hk}$), in

and along with the hydraulic conductance in a network ($G_{hk}$) and the equation

leads to the total Series Hydraulic Conductance ($G_{st}$) can be calculated with:

$\Delta p_k = \displaystyle\frac{J_{Vk}}{K_{hk}}$

So, the sum of the inverse of the hydraulic conductance in a network ($G_{hk}$) will be equal to the inverse of the total Series Hydraulic Conductance ($G_{st}$).

ID:(11067, 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)

Top

>Top

Parameters

$R_k$

R_k

Cylinder k radio

m

$G_{hk}$

G_hk

Hydraulic conductance in a network

m^4s/kg

$R_{hk}$

R_hk

Hydraulic resistance in a network

kg/m^4s

$\pi$

pi

Pi

rad

$R_{st}$

R_st

Total hydraulic resistance in series

kg/m^4s

$G_{st}$

G_st

Total Series Hydraulic Conductance

m^4s/kg

$\eta$

eta

Viscosity

Pa s

Variables

$\Delta p_k$

Dp_k

Pressure difference in a network

Pa

$\Delta p_t$

Dp_t

Total pressure difference

Pa

$\Delta L_k$

DL_k

Tube k length

m

$J_V$

J_V

Volume flow

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:(15732, 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

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 ($R_h$):

$R_{hk} =\displaystyle\frac{8 \eta | \Delta L_k | }{ \pi R_k ^4}$

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

Conditions

Variables

$R_h$

$R_{hk}$

Hydraulic resistance in a network

$kg/m^4s$

9887

$\pi$

Pi

3.1415927

$rad$

5057

$\Delta L$

$\Delta L_k$

Tube k length

$m$

10375

$R$

$R_k$

Cylinder k radio

$m$

10376

$\eta$

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:

$G_{hk} =\displaystyle\frac{ \pi R_k ^4}{8 \eta | \Delta L_k | }$

we can conclude that:

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

ID:(3629, 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:

$R_{hk} =\displaystyle\frac{8 \eta | \Delta L_k | }{ \pi R_k ^4}$

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 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, 1)

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, 2)

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:

$G_{hk} =\displaystyle\frac{ \pi R_k ^4}{8 \eta | \Delta L_k | }$

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

Conditions

Variables

$G_h$

$G_{hk}$

Hydraulic conductance in a network

$m^4s/kg$

10134

$\pi$

Pi

3.1415927

$rad$

5057

$\Delta L$

$\Delta L_k$

Tube k length

$m$

10375

$R$

$R_k$

Cylinder k radio

$m$

10376

$\eta$

Viscosity

$Pa s$

5422

Relation

Development

ID:(15102, 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:

$J_V = G_{st} \Delta p_t$

$J_V = G_h \Delta p$

Conditions

Variables

$G_h$

$G_{st}$

Total Series Hydraulic Conductance

$m^4s/kg$

10135

$\Delta p$

$\Delta p_t$

Total pressure difference

$Pa$

9842

$J_V$

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:

$G_{hk} =\displaystyle\frac{ \pi R_k ^4}{8 \eta | \Delta L_k | }$

to arrive at:

$J_V = G_h \Delta p$

ID:(14471, 1)

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:

$J_V = \Delta p_k G_{hk}$

$J_V = G_h \Delta p$

Conditions

Variables

$G_h$

$\Delta p_k$

Pressure difference in a network

$m^4s/kg$

10132

$\Delta p$

$G_{hk}$

Hydraulic conductance in a network

$Pa$

10134

$J_V$

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:

$G_{hk} =\displaystyle\frac{ \pi R_k ^4}{8 \eta | \Delta L_k | }$

to arrive at:

$J_V = G_h \Delta p$

ID:(14471, 2)

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

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

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)