# Continuity Equation

on . Posted in Fluid Dynamics

The continuity equation, also known as the conservation of mass equation, is a fundamental principle in fluid dynamics that describes the conservation of mass within a fluid flow.  It states that the rate of change of mass within a control volume is equal to the net rate of mass flow into or out of the control volume.  The continuity equation can also be stated that the change in density over time in a given region is equal to the negative divergence of the mass flux density.  This equation implies that mass is conserved, meaning that the amount of fluid entering or leaving a control volume must be balanced by the change in mass within that volume.

The continuity equation is widely applied in various fields of fluid dynamics, including hydrodynamics, aerodynamics, and fluid flow analysis.  It is a fundamental equation used in conjunction with other equations, such as the Navier-Stokes equations, to analyze and solve fluid flow problems.

## Continuity Equation for Area formula

This formula calculates the initial cross-section area of the pipe.

$$\large{ A_1 = \frac{ \rho_2 \; A_2 \; v_2 }{ v_1 \; \rho_1 } }$$     (Continuity Equation for Area Formula)

$$\large{ \rho_2 = \frac{ A_1 \; v_1 \; \rho_1 }{ A_2 \; v_2 } }$$

$$\large{ A_2 = \frac{ A_1 \; v_1 \; \rho_1 }{ \rho_2 \; v_2 } }$$

$$\large{ v_2 = \frac{ A_1 \; v_1 \; \rho_1 }{ \rho_2 \; A_2 } }$$

$$\large{ v_1 = \frac{ \rho_2 \; A_2 \; v_2 }{ A_1 \; \rho_1 } }$$

$$\large{ \rho_1 = \frac{ \rho_2 \; A_2 \; v_2 }{ A_1 \; v_1 } }$$

Symbol English Metric
$$\large{ A_1 }$$ = initial area cross-section $$\large{in^2}$$ $$\large{mm^2}$$
$$\large{ \rho_2 }$$  (Greek symbol rho) =  final cross-section density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ A_2 }$$ = final area cross-section $$\large{in^2}$$ $$\large{mm^2}$$
$$\large{ v_2 }$$ = final cross-section velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$
$$\large{ v_1 }$$ = initial cross-section velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$
$$\large{ \rho_1 }$$  (Greek symbol rho) =  initial cross-section density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$ ## Continuity Equation for Density formula

This formula calculates the initial density of the fluid.

$$\large{ \rho_1 = \frac{ \rho_2 \; A_2 \; v_2 }{ A_1 \; v_1 } }$$     (Continuity Equation for Density Formula)

$$\large{ \rho_2 = \frac{ \rho_1 \; A_1 \; v_1 }{ A_2 \; v_2 } }$$

$$\large{ A_2 = \frac{ \rho_1 \; A_1 \; v_1 }{ \rho_2 \; v_2 } }$$

$$\large{ v_2 = \frac{ \rho_1 \; A_1 \; v_1 }{ \rho_2 \; A_2 } }$$

$$\large{ A_1 = \frac{ \rho_2 \; A_2 \; v_2 }{ \rho_1 \; v_1 } }$$

$$\large{ v_1 = \frac{ \rho_2 \; A_2 \; v_2 }{ \rho_1 \; A_1 } }$$

Symbol English Metric
$$\large{ \rho_1 }$$  (Greek symbol rho) =  initial cross-section density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ \rho_2 }$$  (Greek symbol rho) =  final cross-section density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ A_2 }$$ = final area cross-section $$\large{in^2}$$ $$\large{mm^2}$$
$$\large{ v_2 }$$ = final cross-section velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$
$$\large{ A_1 }$$ = initial area cross-section $$\large{in^2}$$ $$\large{mm^2}$$
$$\large{ v_1 }$$ = initial cross-section velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$ ## Continuity Equation for Mass formula

This formula states that the mass entering a system is equal to the mass leaves the system both at the same rate.

$$\large{ A_1 \; v_1 = A_2 \; v_2 }$$     (Continuity Equation for Mass Formula)

$$\large{ A_1 = \frac{ A_2 \; v_2 }{ v_1 } }$$

$$\large{ v_1 = \frac{ A_2 \; v_2 }{ A_1 } }$$

$$\large{ A_2 = \frac{ A_1 \; v_1 }{ v_2 } }$$

$$\large{ v_2 = \frac{ A_1 \; v_1 }{ A_2 } }$$

Symbol English Metric
$$\large{ A_1 }$$ = initial area cross-section $$\large{in^2}$$ $$\large{mm^2}$$
$$\large{ v_1 }$$ = initial cross-section velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$
$$\large{ A_2 }$$ = final area cross-section $$\large{in^2}$$ $$\large{mm^2}$$
$$\large{ v_2 }$$ = final cross-section velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$

## Continuity Equation for Velocity formula

This formula calculates the initial velocity in a pipe.

$$\large{ v_1 = \frac{ \rho_2 \; A_2 \; v_2 }{ A_1 \; \rho_1 } }$$     (Continuity Equation for Velocity Formula)

$$\large{ \rho_2 = \frac{ v_1 \; A_1 \; \rho_1 }{ A_2 \; v_2 } }$$

$$\large{ A_2 = \frac{ v_1 \; A_1 \; \rho_1 }{ \rho_2 \; v_2 } }$$

$$\large{ v_2 = \frac{ v_1 \; A_1 \; \rho_1 }{ \rho_2 \; A_2 } }$$

$$\large{ A_1 = \frac{ \rho_2 \; A_2 \; v_2 }{ v_1 \; \rho_1 } }$$

$$\large{ \rho_1 = \frac{ \rho_2 \; A_2 \; v_2 }{ v_1 \; A_1 } }$$

Symbol English Metric
$$\large{ v_1 }$$ = initial cross-section velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$
$$\large{ \rho_2 }$$  (Greek symbol rho) =  final cross-section density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ A_2 }$$ = final area cross-section $$\large{in^2}$$ $$\large{mm^2}$$
$$\large{ v_2 }$$ = final cross-section velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$
$$\large{ A_1 }$$ = initial area cross-section $$\large{in^2}$$ $$\large{mm^2}$$
$$\large{ \rho_1 }$$  (Greek symbol rho) =  initial cross-section density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$ 