# Characteristic Velocity

on . Posted in Classical Mechanics

Characteristic velocity, abbreviated as U, measure the effectiveness of the combustion of a rocket engine at high temperature and pressure, seperate from nozzle performance.  It is used to compare different propellant and propulsion systems.  Characteristic velocity is used in the field of rocketry and aerospace engineering and refers to a specific velocity that is characteristic of a particular propulsion system or rocket engine.

In rocketry, the characteristic velocity is a measure of the performance of a rocket engine.  It represents the average exhaust velocity of the propellant expelled from the engine.  The exhaust velocity is the speed at which the propellant gases exit the rocket nozzle relative to the rocket itself.  It is an important parameter in rocketry as it helps determine the rocket's performance, such as its maximum achievable velocity and its ability to overcome the force of gravity and reach orbit or escape Earth's atmosphere.

It's worth noting that the characteristic velocity is not an actual velocity that the rocket achieves, but rather a figure of merit used in rocket engine design and analysis.  It provides a convenient way to compare the performance of different rocket engines or propulsion systems.

### Characteristic velocity formula

$$U = p_c \; A \;/\; \dot m_f$$     (Characteristic Velocity)

$$p_c = U \; \dot m_f \;/\; A$$

$$A = U \; \dot m_f \;/\; p_c$$

$$\dot m_f = p_c \; A \;/\; U$$

### Solve for U

 chamber pressure, pc throat area, A mass flow rate, mf

### Solve for pc

 characteristic velocity, U mass flow rate, mf chamber area, A

### Solve for A

 characteristic velocity, U mass flow rate, mf chamber pressure, pc

### Solve for mf

 chamber pressure, pc throat area, A characteristic velocity, U

Symbol English Metric
$$U$$ = characteristic velocity $$ft \;/\; sec$$ $$m \;/\; s$$
$$p_c$$ = chamber pressure $$lbf \;/\; in^2$$ $$Pa$$
$$A$$  (Greek symbol rho) = throat area $$ft^2$$ $$m^2$$
$$\dot m_f$$ = mass flow rate $$lbm \;/\; sec$$ $$kg \;/\; s$$

### Characteristic velocity formula

$$U = \sqrt{ 2 \; Ec \; c \; \Delta T }$$     (Characteristic Velocity)

$$Ec = U^2 \;/\; 2 \; c \; \Delta T$$

$$c = U^2 \;/\; 2 \; Ec \; \Delta T$$

$$\Delta T = U^2 \;/\; 2 \; Ec \; c$$

Symbol English Metric
$$U$$ = characteristic velocity $$ft \;/\; sec$$ $$m \;/\; s$$
$$Ec$$ = Eckert number $$dimensionless$$
$$c$$ = specific heat $$btu \;/\; lbm-F$$ $$kJ \;/\; kg-K$$
$$\Delta T$$ = temperature change  $$F$$ $$K$$

### Characteristic velocity formula

$$U = \sqrt{ \Delta p \;/\; Eu \; \rho }$$     (Characteristic Velocity)

$$\Delta p = U^2 \; Eu \; \rho$$

$$Eu = \Delta p \;/\; \rho \; U^2$$

$$\rho = \Delta p \; Eu \;/\; U^2$$

Symbol English Metric
$$U$$ = characteristic velocity $$ft \;/\; sec$$ $$m \;/\; s$$
$$\Delta p$$ = pressure differential $$lbf \;/\; in^2$$ $$Pa$$
$$Eu$$ = Euler number $$dimensionless$$
$$\rho$$  (Greek symbol rho) = density $$lbm \;/\; ft^3$$ $$kg \;/\; m^3$$