Mass injection flow ( Limbach Flow) refers to inviscid, adiabatic flow through a constant area duct where the effect of mass addition is considered. For this model, the duct area remains constant, the flow is assumed to be steady and one-dimensional, and mass is added within the duct. Because the flow is adiabatic, unlike in Rayleigh flow, the stagnation temperature is a constant. Compressibility effects often come into consideration, though this flow model also applies to incompressible flow.
For supersonic flow (an upstream Mach number greater than 1), deceleration occurs with mass addition to the duct and the flow can become choked. Conversely, for subsonic flow (an upstream Mach number less than 1), acceleration occurs and the flow can become choked given sufficient mass addition. Therefore, mass addition will cause both supersonic and subsonic Mach numbers to approach Mach 1, resulting in choked flow.
The 1D mass injection flow model begins with a mass-velocity relation derived for mass injection into a steady, adiabatic, frictionless, constant area flow of calorically perfect gas:
where represents a mass flux, . This expression describes how velocity will change with a change in mass flux (i.e. how a change in mass flux drives a change in velocity ). From this relation, two distinct modes of behavior are seen:
From the mass-velocity relation, an explicit mass-Mach relation may be derived:
Although Fanno flow and Rayleigh flow are covered in detail in many textbooks, mass injection flow is not. For this reason, derivations of fundamental mass flow properties are given here. In the following derivations, the constant is used to denote the specific gas constant (i.e. ).
We begin by establishing a relationship between the differential enthalpy, pressure, and density of a calorically perfect gas:
From the adiabatic energy equation () we find:
Substituting the enthalpy-pressure-density relation () into the adiabatic energy relation () yields
Next, we find a relationship between differential density, mass flux (), and velocity:
Substituting the density-mass-velocity relation () into the modified energy relation () yields
Substituting the 1D steady flow momentum conservation equation (see also the Euler equations) of the form into () yields
From the ideal gas law we find,
and from the definition of a calorically perfect gas we find,
Substituting expressions () and () into the combined equation () yields
Using the speed of sound in an ideal gas () and the definition of the Mach number () yields
This is the mass-velocity relationship for mass injection into a steady, adiabatic, frictionless, constant area flow of calorically perfect gas.
To find a relationship between differential mass and Mach number, we will find an expression for solely in terms of the Mach number, . We can then substitute this expression into the mass-velocity relation to yield a mass-Mach relation. We begin by relating differential velocity, mach number, and speed of sound:
We can now re-express in terms of :
Substituting () into () yields,
We can now re-express in terms of :
By substituting () into (), we can create an expression completely in terms of and . Performing this substitution and solving for yields,
Finally, expression () for in terms of may be substituted directly into the mass-velocity relation ():
This is the mass-Mach relationship for mass injection into a steady, adiabatic, frictionless, constant area flow of calorically perfect gas.