A graphical version of this slide is available. In the text only version presented here, * denotes multiplication, / denotes division, ^ denotes exponentiation, ^2 means quantity squared, sqrt means square root. r is the density, p is the pressure, T is the temperature and g is the gas constant. v is the velocity, A is the area, M is the Mach number, gam is the ratio of specific heats, and mfr is mass flow rate. pt, and Tt are total values of the variable.

Equation 1. Mass flow rate: mfr = r * v * A

Equation 2. For an ideal compressible gas: mfr = A * pt / sqrt(Tt) * sqrt(gam / g) * M * (1 + (gam -1) / 2 * M ^2) ^ -((gam + 1) / (2 * (gam -1)))

Equation 3. At the throat of a nozzle, the area is a minimum and the Mach number is equal to 1. For these conditions, with air, the mass flow rate is: mfr = .532 * A * pt / sqrt (Tt) in lbs/sec for A given in sq feet, p expressed in psi, and T in degrees Rankine.

The conservation of mass is a fundamental
concept of physics. Within some problem domain, the amount of mass
remains constant --mass is neither created or destroyed. The
**mass** of any object is simply the volume that the object
occupies times the density of the object.
For a **fluid** (a liquid or a gas) the
density, volume, and shape of the object can all change within the
domain with time. And mass can move through the domain.

__Mass Flow Rate__

The **conservation of mass** (continuity) tells us that the
mass flow rate through a tube is a constant
and equal to the product of the density, velocity, and flow area.
Considering the mass flow rate equation, it appears that for a given
area, we can increase the mass flow rate to any desired point by
increasing the velocity. However, in real fluids, compressibility
effects limit the speed at which a flow can be forced through a
given area.

__Mass Flow Rate for an Ideal Compressible Gas__

For a compressible, ideal gas, the mass flow rate is a unique function of the flow area, total pressure, temperature of the flow, properties of the gas, and the Mach number. This relationship is shown in the red box on this slide. To see the mathematical derivation of this equation, click here.

What does this equation tell us and how can we use this equation?
If we had a flow through a tube at a **fixed Mach number** (fixed
velocity), we could increase the mass flow rate through the tube by
either __increasing__ the area, __increasing__ the total
pressure, or __decreasing__ the total temperature. Similarly, if
we had a tube with a known area, we could increase the mass flow rate
by __increasing__ the total pressure or __decreasing__ the
total temperature. The effect of Mach number is a little hard to
determine because the form of the Mach number term is fairly complex.
Using calculus, we can determine that the maximum mass flow rate
occurs when the Mach number is equal to one. To see the mathematical
derivation of this equation, click
here.

__Mass Flow Rate at Nozzle Throat__

If we have a tube with changing area, like the nozzle
shown on the slide, the maximum mass flow rate through the system
occurs when the Mach number is one at the smallest area. This
location is called the **throat** of the nozzle, and the limit on
the amount of mass flow rate that can be passed through the nozzle is
called **choking**. The equation at the bottom of the slide gives
some actual values for the maximum mass flow rate that can move
through a nozzle throat. Going back to the mass flow rate equation in
the box, if we know the throat area of a nozzle and know the mass
flow rate, we can solve this equation for the **Mach number**
(velocity) at any place in the nozzle, since the total pressure and
temperature remain constant and only the area changes from place to
place.

So with a nozzle, we can set the mass flow rate by setting the area of the throat. And we can set the exit Mach number by setting the area ratio from the throat to the exit. Using the isentropic relations, we can determine the pressure and temperature at the exit of the nozzle. And from the Mach number and temperature we can determine the exit velocity. If we consider the rocket thrust equation, we have now determined all the values necessary to determine the thrust of the rocket. You can explore the operation of a nozzle with our interactive nozzle simulator and design your own rockets!

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- Beginner's Guide Home Page

*byTom
Benson
Please send suggestions/corrections to: benson@grc.nasa.gov *