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of each fluidparticle， we are studying Lagrangian motion; in studying the spatial system we are studying Eulerian motion. This book will examine the Eulerian motion of the flow. The flow will be considered fully described ifthe magnitude，direction， and thermodynamic properties of the gas at every point in space are determined.
To understand the flow in turbomachines， an understanding of the basicrelationships of pressure， temperature， and type of flow must be acquired. Ideal flow in turbomachines exists when there is no transfer of heat betweenthe gas and its surroundings， and the entropy of the gas remains unchanged. This type of flow is characterized as a reversible adiabaticjlow. To describethisflow， the total and static conditions ofpressure， temperature， and the concept of an ideal gas must be understood.
Ideal Gas
Ideal gas obeys the equation of state PV -MRT or P/ρ -MRT， where Pdenotes the pressure， Vthe volume， ρthe density， Mthe mass， T thetemperature of thegas， and R the gas constant per unit mass independent of pressure and temperature. In most cases the ideal gas laws are sufficient to describe the flow within 5% of actual conditions. When the perfect gas lawsdo not apply， the gas compressibility factor Z can be introduced:
PV
Z(P T)-(3-1)
RT
Figure 3-1 shows the relationship between the compressibility factor andpressure and temperature， couched in terms of reduced pressure and tem-perature:
PT
Pr -Tr -(3-2)
Pc Tc
Pc and Tc are the pressure and temperature of the gas at the critical point.
Static pressure is the pressure of the moving fluid. The static pressure of a gas is the same in all directions and is a scalar point function. It can be measured by drilling a hole in the pipe and keeping a probe flush with the pipe wall.
Total pressure is the pressure of the gas brought to rest in a reversible adiabatic manner. It can be measured by a pitot tube placed in the flow
FPO
Figure 3-1. Generalized compressibility factor for simple fluid. (Adapted with permission from Journal of the Amerícan Chemícal Socíety， @1955， American Chemical Society.)
stream. The gas is brought to rest at the probe tip. The relationship between total and static pressure is given in the following relationship:
ρV2
Pt -Ps +(3-3)
2gc
where ρV2/2gc is the dynamic pressure head that denotes the velocity of the moving gas.
Static temperature is the temperature of the flowing gas. This temperature rises because of the random motion of the fluid molecules. The static temperature can only be measured by a measurement at rest relative to themoving gas. The measurement of the static temperature is a difficult， if notimpossible， task.
Total temperature is the temperature rise in the gas if its velocity is brought to rest in a reversible adiabatic manner. Total temperature can be measuredby the insertion of a thermocouple， RTD or thermometer in the fluid stream. The relationship between the total temperature and static temperature can be given:
V2 Tt -Ts +(3-4)2cpgc
Compressibility Effect
The effect of compressibility is important in high mach number machines. Mach number is the ratio of velocity to the acoustic speed of a gas at a given temperature M ι V/a. Acoustic speed is defined as the ratio change in pressure of the gas with respect to its density if the entropy is held constant:
a 2主P(3-5)ι主ρ S -C
With incompressibilefluids， the value of the acoustic speed tends towardinfinity. For isentropicflow， the equation of state for a perfect gas can be written:
P/ρ， -const
Therefore，
lnP -，lnρ -const (3-6)
Differentiating the previousequation， the following relationship is obtained:
dP dρ
P -，ρ -O (3-7)
For an isentropicflow， the acoustic speed can be written:
2
a -dP/dρ
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燃气涡轮工程手册 Gas Turbine Engineering Handbook 1(45)
