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Therefore，
a 2 -，P/ρ  (3-8)
Substituting the general equation of state and the definition of the acous-tic velocity， the following equation is obtained:
a 2 -，gcRTs  (3-9)
where Ts (static temperature) is the temperature of the moving gas stream.
Since the static temperature cannot bemeasured， the value of statictemperature must be computed using the measurements of static pressure， and total pressure and temperature. The relationship between static temperature and total temperature is given by the following relationship:
V2Tt
-1 + (3-1O)
Ts 2gccpTs
where the specific heat cp at constant volume can be written: ，Rcp -， -1 (3-11) and where， is the ratio of the specific heats
， -cp
cU

Combining Equations (3-1O) and (3-11) gives the following relationship:
Tt -1 +， -1 M2 (3-12)Ts 2
The relationship between the total and static conditions is isentropic;therefore，
， -1
，

Tt Pt
- (3-13)
Ts Ps
and the relationship between total pressure and static pressure can be written:
Pt -1 +， -1 M2，-1，(3-14)Ps 2
By measuring the total and static pressure and using Equation (3-14)， theMach number can be calculated. Using Equation (3-12)， the static tempera-ture can becomputed， since the total temperature can be measured.Finally，using the definition of Machnumber， the velocity of the gas stream can be calculated.
The Aerothermal Equations
The gas stream can be defined by the three basic aerothermal equations:
(1)continuity， (2) momentum， and (3) energy.
The Continuity Equation
The continuity equation is a mathematical formulation of the law of conservation of mass of a gas that is a continuum. The law of conservation of mass states that the mass of a volume moving with the fluid remains unchanged
的m -ρAV
where:
m的 -mass flow rate
ρ -fluid density
A -cross-sectional area
V -gas velocity
The previous equation can be written in the differential form
dAdV dρ
+ +-O (3-15)
AV ρ
The Momentum Equation
The momentum equation is a mathematical formulation of the law of conservation of momentum. It states that the rate of change in linear momentum of a volume moving with a fluid is equal to the surface forces and body forces acting on a fluid. Figure 3-2 shows the velocity components in a generalized turbomachine. The velocity vectors as shown are resolved into three mutually perpendicular components: the axial component (Va)， the tangential component (V，)， and the radial component ( Vm).
By examining each of these velocities， the following characteristics can be noted: the change in the magnitude of the axial velocity gives rise to an axialforce which is taken up by a thrust bearing， the change in radial velocity gives rise toaradial force whichistaken up bythejournal bearing. Thetangential component is the only component that causes a force that corresponds to a change in angular momentum; the other two velocity components haveno effect on thisforce一except for what bearing friction may arise.

By applying the conservation of momentumprinciple， the change in angular momentum obtained by the change in the tangential velocity is equal to the summation of all the forces applied on the rotor. This summa-tion is the net torque of the rotor. A certain mass of fluid enters the turbomachine with an initial velocity V1， at a radius r1， and leaves with a tangential velocity V， 2 at a radius r2. A，ssuming that the mass flow ratethrough the turbomachine remains unchanged， the torque exerted by the change in angular velocity can be written:
U -m的 (r1V-r2V)(3-16)gc ， 1 ， 2
The rate of change of energy transfer (ft-lbj lsec) is the product of the torque and the angular velocity (ω )
m的 Uω -gc (r1ω V， -r2ω V， 2)(3-17)
　
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