 
 
 
 
 
 
 
  
Before introducing the fluid section types for gases, a couple of fundamental
aerodynamic equations are introduced. For details, the reader is referred to
[56]. 
The thermodynamic state of a gas is usually determined by the static pressure
 , the static temperature
, the static temperature  and the density
 and the density  . For an ideal gas (the
case considered here), they are related by
. For an ideal gas (the
case considered here), they are related by 
 (the
ideal gas equation), where
 (the
ideal gas equation), where
 is the specific gas constant.
 is the specific gas constant.  only depends on the material, it does
not depend on the temperature.
 only depends on the material, it does
not depend on the temperature.
The energy conservation law runs like [18]:
where D denotes the total derivative. By use of the mass conservation:
and the conservation of momentum
this equation can also be written as
| ![$\displaystyle \frac{D[\rho(\varepsilon+v_kv_k/2)]}{Dt}=(v_kt_{kl})_{,l}-(pv_k)_{,k}+\rho v_k f_k-q_{k,k}+\rho h^{\theta},$](img294.png) | (21) | 
or
where 
 is the entalpy. For an ideal gas one can write
 is the entalpy. For an ideal gas one can write
 ,
,  is the heat capacity at constant pressure.
 is the heat capacity at constant pressure. 
The total temperature  is now defined as the temperature which is obtained by
slowing down the fluid to zero velocity in an adiabatic way. Using the energy
equation (22), dropping the first term on the right hand side
because of ideal gas conditions (no viscosity), the second term because of
stationarity, the third term because of the absence of volumetric forces and
the last two terms because of adiabatic conditions one obtains the relationship:
 is now defined as the temperature which is obtained by
slowing down the fluid to zero velocity in an adiabatic way. Using the energy
equation (22), dropping the first term on the right hand side
because of ideal gas conditions (no viscosity), the second term because of
stationarity, the third term because of the absence of volumetric forces and
the last two terms because of adiabatic conditions one obtains the relationship:
| ![$\displaystyle \frac{D[\rho(c_p T +v_kv_k/2)]}{Dt}=0,$](img299.png) | (23) | 
along a stream line, or
|  | (24) | 
where v is the magnitude of the velocity. The Mach number is defined by
|  | (25) | 
where  is the specific heat ratio and the denominator is the speed of sound. Therefore, the
total temperature satisfies:
 is the specific heat ratio and the denominator is the speed of sound. Therefore, the
total temperature satisfies:
|  | (26) | 
The total pressure is defined as the pressure which is attained by slowing
down the fluid flow in an isentropic way, i.e. a reversible adiabatic way. An
ideal gas is isentropic if 
 is constant, which leads
to the relationship
 is constant, which leads
to the relationship
|  | (27) | 
and consequently to
|  | (28) | 
Substituting the definition of mass flow 
 , where
, where  is the
cross section of the fluid channel, in the definition of the Mach number (and
using the ideal gas equation to substitute
 is the
cross section of the fluid channel, in the definition of the Mach number (and
using the ideal gas equation to substitute  ) leads
to
) leads
to
|  | (29) | 
Expressing the pressure and temperature as a function of the total pressure and total temperature, respectively, finally leads to
This is the general gas equation, which applies to all types of flow for ideal
gases. The right hand side exhibits a maximum for  , i.e. sonic
conditions.
, i.e. sonic
conditions. 
It is further possible to derive general statements for isentropic flow
through network elements. Isentropic flow is reversible adiabatic by
definition. Due to the adiabatic conditions the total enthalpy 
 is constant or
is constant or
|  | (31) | 
The first law of thermodynamics (conservation of energy) specifies that
|  | (32) | 
or, because of the adiabatic and reversible conditions
|  | (33) | 
Since the enthalpy 
 , one further obtains
, one further obtains
|  | (34) | 
Substituting this in the equation we started from leads to:
|  | (35) | 
The continuity equation through a network element with cross section  ,
,
 constant can be written in the following differential form:
 constant can be written in the following differential form:
|  | (36) | 
or, with the equation above
|  | (37) | 
which leads to
|  | (38) | 
Since 
 is the speed of sound, one arives at:
 is the speed of sound, one arives at:
|  | (39) | 
Therefore, for subsonic network flow an increasing cross section leads to a decreasing velocity and an increasing pressure, whereas a decreasing cross section leads to an increasing velocity and a decreasing pressure. This is similar to what happens for incompressible flow in a tube.
For supersonic flow an increasing cross section leads to an increasing velocity and a decreasing pressure whereas a decreasing cross section leads to a decreasing velocity and an increasing pressure.
Sonic conditions can only occur if  , in reality this corresponds to a
minimum of the cross section. Therefore, if we assume that the network
elements are characterized by a uniformly increasing or decreasing cross
section sonic conditions can only occur at the end nodes. This is important
information for the derivation of the specific network element equations.
, in reality this corresponds to a
minimum of the cross section. Therefore, if we assume that the network
elements are characterized by a uniformly increasing or decreasing cross
section sonic conditions can only occur at the end nodes. This is important
information for the derivation of the specific network element equations.
Using the definition of entropy per unit mass s satisfying 
 and the definition
of enthalpy the first law of thermodynamics for reversible processes runs like
 and the definition
of enthalpy the first law of thermodynamics for reversible processes runs like
|  | (40) | 
Therefore
|  | (41) | 
For an ideal gas 
 and
 and  and consequently
 and consequently
|  | (42) | 
or
|  | (43) | 
Since all variables in the above equation are state variables, it also applies to irreversible processes. If the specific heat is temperature independent one obtains
linking the entropy difference between two states to the temperature and pressure difference.
A network element is characterized by a type of fluid section. It has to be specified on the *FLUID SECTION card unless the analysis is a pure thermomechanical calculation. For gases, several types are available. At the start of the description of each type the main properties are summarized: ``adiabatic'' means that no heat is exchanged within the element; ``isentropic'' refers to constant entropy, i.e. flow without losses; ``symmetric'' means that the same relations apply for reversed flow; ``directional'' means that the flow is not allowed to be reversed.
 
 
 
 
 
 
