Storyboard

When a liquid is in motion, we refer to it as flow, and its measurement is based on the volume that passes through a section in a given period of time. Assuming that the volume moves without deformation, the velocity at which the liquid passes through the section remains constant. In this case, flow can also be defined as the product of velocity and the cross-sectional area.

>Model

ID:(875, 0)

Iframe

>Top

ID:(15485, 0)

Concept

>Top

During a time elapsed ($\Delta t$), the fluid with a mean Speed of Fluid ($v$) moves a tube element ($\Delta s$). If the section ($S$) represents the amount of fluid crossing that section in the time elapsed ($\Delta t$), it is calculated as:

$\Delta V = S \Delta s = Sv \Delta t$

This equation states that the volume of fluid flowing through section the section ($S$) during a time elapsed ($\Delta t$) is equal to the product of the cross-sectional area and the distance the fluid travels during that time.

This facilitates the calculation of the volume element ($\Delta V$), which is the volume of fluid flowing through the channel in a specific period of the time elapsed ($\Delta t$), corresponding to the volume flow ($J_V$).

ID:(2212, 0)

Concept

>Top

Flow is defined as the volume the volume element ($\Delta V$) divided by time the time elapsed ($\Delta t$), which is expressed in the following equation:

and the volume equals the cross-sectional area the section Tube ($S$) multiplied by the distance traveled the tube element ($\Delta s$):

Since the distance traveled the tube element ($\Delta s$) per unit time the time elapsed ($\Delta t$) corresponds to the velocity, it is represented by:

Thus, the flow is a flux density ($j_s$), which is calculated using:

It is important to note that in this model:

The flow density acts as an average velocity across the entire flow section.

ID:(15715, 0)

Concept

>Top

Considering a tube that neither leaks nor receives additional liquid, the flow entering at point 1 The volume flow 1 ($J_{V1}$) will be equal to the flow exiting at point 2 The volume flow 2 ($J_{V2}$):

Within a channel or tube, there may be a change in cross-sectional area, either widening or narrowing.

This variation will directly affect the flow through the flux density ($j_s$), which represents the velocity, increasing (if the section narrows) or decreasing (if it widens) according to the section Tube ($S$) to keep the volume flow ($J_V$) constant, as indicated by:

The conservation of flow, along with the definition of flow density, leads to the conservation law such that the section in point 1 ($S_1$), the section in point 2 ($S_2$), the flux density 1 ($j_{s1}$), and the flux density 2 ($j_{s2}$) satisfy:

ID:(2213, 0)

Concept

>Top

The continuity equation assumes that the flow is uniform, with no reverse flows or turbulence present. Therefore, it is necessary to verify that the flow is indeed laminar and free of turbulence, especially when applying the equation to analyze fluid flows in pipes and channels.

There are various methods for detecting turbulence in the flow, such as using flow meters or visually observing the flow. It is essential to ensure that the flow is stable before applying the continuity equation, as any disturbance in the flow can affect the accuracy of the calculations and the overall efficiency of the system.

ID:(978, 0)

Top

>Top

Parameters

$\pi$

pi

Pi

rad

Variables

$j_{s1}$

j_s1

Flux density 1

m/s

$j_{s2}$

j_s2

Flux density 2

m/s

$\Delta s_1$

Ds_1

Length of element 1

m

$\Delta s_2$

Ds_2

Length of element 2

m

$r_1$

r_1

Radius of Section 1

m

$r_2$

r_2

Radius of Section 2

m

$S_1$

S_1

Section in point 1

m^2

$S_2$

S_2

Section in point 2

m^2

$\Delta t$

Dt

Time elapsed

s

$J_{V1}$

J_V1

Volume flow 1

m^3/s

$J_{V2}$

J_V2

Volume flow 2

m^3/s

$\Delta V_1$

DV_1

Volume of the element 1

m^3

$\Delta V_2$

DV_2

Volume of the element 2

m^3

Calculations

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

Equation

Number

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

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Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

ID:(15488, 0)

Equation

>Top, >Model

If we have a tube with a the section Tube ($S$) moving a distance the tube element ($\Delta s$) along its axis, having displaced the volume element ($\Delta V$), then it is equal to:

$\Delta V_1 = S_1 \Delta s_1$

$\Delta V = S \Delta s$

Conditions

Variables

$S$

$S_1$

Section in point 1

$m^2$

5257

$\Delta s$

$\Delta s_1$

Length of element 1

$m$

10294

$\Delta V$

$\Delta V_1$

Volume of the element 1

$m^3$

10292

Relation

ID:(3469, 1)

Equation

>Top, >Model

If we have a tube with a the section Tube ($S$) moving a distance the tube element ($\Delta s$) along its axis, having displaced the volume element ($\Delta V$), then it is equal to:

$\Delta V_2 = S_2 \Delta s_2$

$\Delta V = S \Delta s$

Conditions

Variables

$S$

$S_2$

Section in point 2

$m^2$

5413

$\Delta s$

$\Delta s_2$

Length of element 2

$m$

10295

$\Delta V$

$\Delta V_2$

Volume of the element 2

$m^3$

10293

Relation

ID:(3469, 2)

Equation

>Top, >Model

The volume flow ($J_V$) corresponds to the volume flowing ($\Delta V$) flowing through the channel at the time elapsed ($\Delta t$). Therefore, we have:

$J_{V1} =\displaystyle\frac{ \Delta V_1 }{ \Delta t }$

$J_V =\displaystyle\frac{ \Delta V }{ \Delta t }$

Conditions

Variables

$\Delta t$

Time elapsed

$s$

5103

$\Delta V$

$\Delta V_1$

Volume of the element 1

$m^3$

10292

$J_V$

$J_{V1}$

Volume flow 1

$m^3/s$

8478

Relation

ID:(4347, 1)

Equation

>Top, >Model

The volume flow ($J_V$) corresponds to the volume flowing ($\Delta V$) flowing through the channel at the time elapsed ($\Delta t$). Therefore, we have:

$J_{V2} =\displaystyle\frac{ \Delta V_2 }{ \Delta t }$

$J_V =\displaystyle\frac{ \Delta V }{ \Delta t }$

Conditions

Variables

$\Delta t$

Time elapsed

$s$

5103

$\Delta V$

$\Delta V_2$

Volume of the element 2

$m^3$

10293

$J_V$

$J_{V2}$

Volume flow 2

$m^3/s$

8479

Relation

ID:(4347, 2)

Equation

>Top, >Model

The flux density ($j_s$) is related to the distance traveled in a time ($\Delta s$), which is the distance that the fluid travels in the time elapsed ($\Delta t$), as follows:

$j_{s1} =\displaystyle\frac{ \Delta s_1 }{ \Delta t }$

$j_s =\displaystyle\frac{ \Delta s }{ \Delta t }$

Conditions

Variables

$j_s$

$j_{s1}$

Flux density 1

$m/s$

10288

$\Delta t$

Time elapsed

$s$

5103

$\Delta s$

$\Delta s_1$

Length of element 1

$m$

10294

Relation

ID:(4348, 1)

Equation

>Top, >Model

The flux density ($j_s$) is related to the distance traveled in a time ($\Delta s$), which is the distance that the fluid travels in the time elapsed ($\Delta t$), as follows:

$j_{s2} =\displaystyle\frac{ \Delta s_2 }{ \Delta t }$

$j_s =\displaystyle\frac{ \Delta s }{ \Delta t }$

Conditions

Variables

$j_s$

$j_{s2}$

Flux density 2

$m/s$

10289

$\Delta t$

Time elapsed

$s$

5103

$\Delta s$

$\Delta s_2$

Length of element 2

$m$

10295

Relation

ID:(4348, 2)

Equation

>Top, >Model

A flux density ($j_s$) can be expressed in terms of the volume flow ($J_V$) using the section or Area ($S$) through the following formula:

$j_{s1} = \displaystyle\frac{ J_{V1} }{ S_1 }$

$j_s = \displaystyle\frac{ J_V }{ S }$

Conditions

Variables

$j_s$

$j_{s1}$

Flux density 1

$m/s$

10288

$S$

$S_1$

Section in point 1

$m^2$

5257

$J_V$

$J_{V1}$

Volume flow 1

$m^3/s$

8478

Relation

Development

Flow is defined as the volume the volume element ($\Delta V$) divided by time the time elapsed ($\Delta t$), which is expressed in the following equation:

$J_V =\displaystyle\frac{ \Delta V }{ \Delta t }$

and the volume equals the cross-sectional area the section Tube ($S$) multiplied by the distance traveled the tube element ($\Delta s$):

$\Delta V = S \Delta s$

Since the distance traveled the tube element ($\Delta s$) per unit time the time elapsed ($\Delta t$) corresponds to the velocity, it is represented by:

$j_s =\displaystyle\frac{ \Delta s }{ \Delta t }$

Thus, the flow is a flux density ($j_s$), which is calculated using:

$j_s = \displaystyle\frac{ J_V }{ S }$

ID:(4349, 1)

Equation

>Top, >Model

A flux density ($j_s$) can be expressed in terms of the volume flow ($J_V$) using the section or Area ($S$) through the following formula:

$j_{s2} = \displaystyle\frac{ J_{V2} }{ S_2 }$

$j_s = \displaystyle\frac{ J_V }{ S }$

Conditions

Variables

$j_s$

$j_{s2}$

Flux density 2

$m/s$

10289

$S$

$S_2$

Section in point 2

$m^2$

5413

$J_V$

$J_{V2}$

Volume flow 2

$m^3/s$

8479

Relation

Development

Flow is defined as the volume the volume element ($\Delta V$) divided by time the time elapsed ($\Delta t$), which is expressed in the following equation:

$J_V =\displaystyle\frac{ \Delta V }{ \Delta t }$

and the volume equals the cross-sectional area the section Tube ($S$) multiplied by the distance traveled the tube element ($\Delta s$):

$\Delta V = S \Delta s$

Since the distance traveled the tube element ($\Delta s$) per unit time the time elapsed ($\Delta t$) corresponds to the velocity, it is represented by:

$j_s =\displaystyle\frac{ \Delta s }{ \Delta t }$

Thus, the flow is a flux density ($j_s$), which is calculated using:

$j_s = \displaystyle\frac{ J_V }{ S }$

ID:(4349, 2)

Equation

>Top, >Model

One of the most basic laws in physics is the conservation of mass, which holds true throughout our macroscopic world. Only in the microscopic world does a conversion between mass and energy exist, which we will not consider in this case. In the case of a fluid, this means that the mass entering through a pipe must be equal to the mass exiting it.

If density is constant, the same applies to volume. In such cases, when we treat the flow as an incompressible fluid, it means that a given volume entering one end of the pipe must exit the other end. This can be expressed as the equality between the flow in Position 1 ($J_1$) and the flow in Position 2 ($J_2$), with the equation:

$J_{V1} = J_{V2}$

Conditions

Variables

$J_{V1}$

Volume flow 1

$m^3/s$

8478

$J_{V2}$

Volume flow 2

$m^3/s$

8479

Relation

ID:(939, 0)

Equation

>Top, >Model

The surface of a disk ($S$) of ($$) is calculated as follows:

$S_1 = \pi r_1 ^2$

$S = \pi r ^2$

Conditions

Variables

$r$

$r_1$

Radius of Section 1

$m$

10286

$\pi$

Pi

3.1415927

$rad$

5057

$S$

$S_1$

Section in point 1

$m^2$

5257

Relation

ID:(3804, 1)

Equation

>Top, >Model

The surface of a disk ($S$) of ($$) is calculated as follows:

$S_2 = \pi r_2 ^2$

$S = \pi r ^2$

Conditions

Variables

$r$

$r_2$

Radius of Section 2

$m$

10287

$\pi$

Pi

3.1415927

$rad$

5057

$S$

$S_2$

Section in point 2

$m^2$

5413

Relation

ID:(3804, 2)

Equation

>Top, >Model

The principle of continuity dictates that the flow at the first point, which is equal to the flux density 1 ($j_{s1}$) times the section in point 1 ($S_1$), must be equal to the flow at the second point, given by the flux density 2 ($j_{s2}$) times the section in point 2 ($S_2$). From this, it follows that:

$S_1 j_{s1} = S_2 j_{s2}$

Conditions

Variables

$j_{s1}$

Flux density 1

$m/s$

10288

$j_{s2}$

Flux density 2

$m/s$

10289

$S_1$

Section in point 1

$m^2$

5257

$S_2$

Section in point 2

$m^2$

5413

Relation

ID:(4350, 0)