Equation of Continuity

The equation of continuity in Fluids stems from the conservation of mass in a closed system. It only applies to incompressible fluids. It also can be used to define volume flow and mass flow

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Definition §

Formula - 1 §

Equation of Continuity (Fluids)

$$A_{in}\cdot v_{in}=A_{out}\cdot v_{out}$$

Formula - 2 §

Volume Flow & Mass Flow

Terms §

• $A$ = Cross-sectional area (in ${\text{m}}^2$)
• $v$ = Velocity of fluid (in $\text{m/s}$)
• $Q$ = Volumetric flow rate, sometimes denoted with $\dot{V}$ or $\frac{\text{d}V}{\text{d}t}$ (in ${\text{m}}^3\text{/s}$)
• $\dot{m}$ = Mass flow rate, sometimes denoted with $\frac{\text{d}m}{\text{d}t}$ (in $\text{kg/s}$)
• $\rho$ = Density (in ${\text{kg/m}}^3$)

Derivation §

Let’s start with taking a pipe of total volume $V$ with two unequal ‘holes’, of surface areas $A_1$ and $A_2$ respectively. In order to talk about flow, we’re going to assume the pipe is already filled with an incompressible fluid, moving at a velocity, i.e. flowing

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We’ll be using fluid particles in order to understand this equation better. A fluid is composed of infinite fluid particles. For the fluid to flow, these particles must move at a velocity. Let’s call it $v_1$ Then the distance travelled by a fluid particle at a time period, $\Delta t$, is given by:

$$\Delta x_1=v_1\cdot \Delta t$$

If we take a test volume of this pipe, near $A_1$, we have $V_1$ given by (roughly):

$$V_1=A_1\cdot \Delta x_1$$

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Because this fluid is incompressible, this volume must not change as it moves to the shorter section of the pipe, i.e. towards $A_2$. I.e.:

$$V_1=V_2$$

$$A_1\Delta x_1=A_2\Delta x_2$$

Since we can represent $\Delta x=v_{in}\cdot \Delta t$ and $V=A\cdot x$

$$A_1\cdot v_1\cdot \cancel{\Delta t}=A_2\cdot v_2\cdot \cancel{\Delta t}$$

Resulting in:

$$A_1\cdot v_1=A_2\cdot v_2$$

Equation of Continuity _3.excalidraw