燃气涡轮工程手册 Gas Turbine Engineering Handbook 1(47)

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1
Thus， the total energy transfer can be written:
m的 E -gc (U1V， 1 -U2V， 2 )(3-18)
where U1 and U2 are the linear velocity of the rotor at the respective radii. The previous relation per unit mass flow can be written:
1H -gc (U1V， 1 -U2V， 2 )(3-19)
where H is the energy transfer per unit mass flow ft-lbjllbm or fluid pressure. Equation (3-19) is known as the Euler turbine equation.
The equationof motionas given interms ofangular momentum can be trans-formed into other forms that are more convenient to understanding someof the basic design components. To understand the flow in a turbomachine， the concepts of aboslute and relative velocity must be grasped. Absolute velocity (V ) is gas velocity with respect to a stationary coordinate system. Relative velocity (W) is the velocity relative to the rotor. In turbomachinery， the air entering the rotor will have a relative velocity component parallel tothe rotorblade， and an absolute velocity component parallel to the station-ary blades. Mathematically， this relationship is written:
  
V-W一卡U(3-2O)
where the absolute velocity (V ) is the algebraic addition of the relative velocity (W ) and the linear rotor velocity (U ). The absolute velocity canbe resolved into its components， the radial or meridional velocity ( Vm) and the tangential component V，. From Figure3-3， the following relationships are obtained:
2
V1 -V， 21 +Vm21

V22 -V， 22 +Vm22
W12 -(U1 -V，1 )2 +Vm21
2W22 -(U2 -V，2 )+Vm22 (3-21)

By placing these relationships into the Euler turbine equation， the follow-ing relationship is obtained:
[(2 的2 (22 的(22的]
H -V1 -V2 + U1 -U2 + W2 -W1 (3-22)
2gc
The Energy Equation
The energy equation is the mathematical formulation of the law of con-servation of energy. It states that the rate at which energy enters the volume of a moving fluid is equal to the rate at which work is done on the surroundings by the fluid within the volume and the rate at which energy increases withinthe moving fluid. The energy in a moving fluid is composed of internal，flow，kinetic， and potential energy
句1 + P1 + V12 +Z1 +1Q2 -句2 + P2 + V22 +Z2 +1(Work)2 (3-23)ρ12gc ρ22gc
For isentropicflow， the energy equation can be written as follows， noting that the addition of internal and flow energies can be written as the enthalpy
(h) of the fluid:
V12 V22 1(Work)-(h1 -h2)+ -+(Z1 -Z2)(3-24)
22gc 2gc
Combining the energy and momentum equations provides the following relationships:
(h1 -h2)+ V12 -V22 +(Z1 -Z2)-1 [U1V，1 -U2V，2 川(3-25)2gc 2gc gc
Assuming that there is no change in potential energies， the equation can be written:
--[川(3-26)
h1 + V2g1 c 2 h2 + V2g2 c 2 -h1t -h2tg1 cU1V，1 -U2V，2
Assuming that the gas is thermally and calorifically perfect， the equation can be written:
T1t -T2t -Cp1 gc [U1V，2 -U2V，2 川(3-27)
For isentropicflow，
， -1
，

T2tP2t
- (3-28)
T1tP1t
By combining Equations (3-27) and(3-28)，
，-1
，
T1t 1 -PP12tt -Cp1 gc [U1V，1 -U2V，2 川 (3-29)
Efficiencies Adiabatic Efficiency
The work in a compressor or turbine under ideal conditions occurs atconstant entropy as shown in Figures 3-4 and3-5， respectively. The actual work done is indicated by the dotted line. The isentropic efficiency of the compressor can be written in terms of the total changes in enthalpy

Isentropic work (h2t -h1t)id
Tadc --(3-3O)
Actual work (h2't -h1t)act
This equation can be rewritten for a thermally and calorifically perfect gas in terms of total pressure and temperature as follows:
Tadc -「，-1-1「，匹-1 (3-31)
，
P2tT2t
P1tT1t
The process between 1 and 2' can be defined by the following equation of state:
P
-const (3-32)
　
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