3.3.2. Determination of gas load intensities
The mass air (gas) flow rate per second through an elementary ring of a rotor wheel air-gas channel of width dr, located at the radius r (see Fig. 3.2), is determined by the formula
(3.17)
where 1 is a density of gas in a rotor wheel inlet, kg/m3; C1a is an axial component of gas absolute speed in a rotor wheel inlet, m/s; dF is an area of an elementary ring, m2.
The movement of this air will cause force, which affects a rotor blade both in axial, and circumferential planes. The intensity of dynamic force, affecting the length unit of dr element in a plane of rotation qy, is equal to change of air flow momentum per second in a circumferential direction, which flows through the area dF/Z:
(3.18)
where Z is the number of rotor wheel blades; C1u, C2u are the circumferential components of gas absolute speed before and behind a rotor wheel, m/s.
The dynamic force (it depends on a difference of axial speeds), and static force (it depends on a pressure difference on a rotor wheel) affect dr element in an axial plane. The total intensity of these forces qx is equal to
(3.19)
where C2a is an axial component of gas absolute speed behind a rotor wheel, m/s; p1, p2 are static pressures of gas before and behind a rotor wheel, Pa.
Calculated by formulas (3.18) and (3.19), intensities of gas load for turbine rotor blades will be positive, and for compressor rotor blades – negative.
If an intensity of loads is a constant value, then
(3.20)
where G is a mass air (gas) flow rate through the engine, kg/s; h is a blade length (length of airfoil part), m.
In most cases the axial velocity of gas when going through a blade passage changes slightly, that is why C1a=C2a. Therefore first addend in expression (3.19) can be neglected, and formula for determination of load intensity qx(r) will be put this way:
(3.21)
At pressures p1 and p2 being constant regarding blade passage length, the load intensity will also be a constant value and will be equal to load intensity on mean radius:
Determination of the bending momenta in axial and circumferential planes
The concentrated gas forces qx(r)dr and qy(r)dr affect a blade element dr in axial and circumferential planes. These forces create the elementary bending momenta in design section, located at some radius R:
The full bending momenta in selected section can be found by integration:
(3.22)
It is necessary to put “minus” before integrals in the formulas (3.22), as values qx and qy are negative for compressor blades according to the commonly accepted rule of signs for bending momenta vectors.
We get constant loads intensities qx and qy along the length of the blade in previous calculations of the bending momenta, including course and diploma projects. They are calculated with the help of air or gas parameters on a rotor wheel mean radius. In this case we will get:
The calculation by these formulas is approximated and in practice it requires some precision with the help of the formulas (3.22).
The maximum values of the bending momenta at assumptions made will be in the root of a blade, that is at R=Rr:
