# Continuity Equation

Written by Jerry Ratzlaff on . Posted in Fluid Dynamics

Continuity equation is the moving of a quantity through a pipe in a steady flow.

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

## Continuity Equation for Area formula

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

This formula calculates the initial density of the fluid.

## Continuity Equation for Density formula

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

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

## Continuity Equation for Mass formula

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

This formula calculates the initial velocity in a pipe.

## Continuity Equation for Velocity formula

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