Compressor characteristic

Performance of a compressor is usually specified by curves of delivery pressure against mass flow rate for various fixed values of rotational speed and inlet temperature.[2][3][4] These performance characteristic curves are usually plotted with dimensionless variables. For this, we start with implicit functional relationships for all the variables as [4]

${\displaystyle F(D,N,{\dot {m}},p_{01},p_{02},RT_{01},RT_{02})=0}$ [4][5][6]

Where

${\displaystyle D}$ = characteristic linear dimension of the compressor

${\displaystyle N}$ = rotational speed

${\displaystyle {\dot {m}}}$ = mass flow rate

${\displaystyle p_{01}}$ and ${\displaystyle p_{02}}$ = stagnation pressure at compressor inlet and outlet

${\displaystyle T_{01}}$ and ${\displaystyle T_{02}}$ = stagnation temperature at compressor inlet and outlet

${\displaystyle R}$ = characteristics gas constant

By making use of π-theorem, we obtain the non-dimensional groups (π terms) as [4]

${\displaystyle {\frac {p_{02}}{p_{01}}}\ }$, ${\displaystyle {\frac {T_{01}}{T_{02}}}\ }$, ${\displaystyle {\frac {{\dot {m}}{\sqrt {RT_{01}}}}{{D^{2}}{p_{01}}}}\ }$, ${\displaystyle {\frac {{N}{D}}{\sqrt {RT_{01}}}}\ }$

When we are concerned with the performance of a machine of fixed size compressing a specified gas and D may be omitted from the groups and we can write

${\displaystyle F\left({\frac {p_{02}}{p_{01}}}\ ,{\frac {T_{01}}{T_{02}}}\ ,{\frac {{\dot {m}}{\sqrt {T_{01}}}}{p_{01}}}\ ,{\frac {N}{\sqrt {T_{01}}}}\ \right)=0}$ [4]

Figure 1. Pressure ratio vs.Non-dimensional mass flow

Though the terms ${\displaystyle {\frac {{\dot {m}}{\sqrt {T_{01}}}}{p_{01}}}\ }$ and ${\displaystyle {\frac {N}{\sqrt {T_{01}}}}\ }$ are truly not dimensionless, they are referred as 'non-dimensional mass flow' and 'non-dimensional rotational speed' for practical purpose. The stagnation pressure ratio ${\displaystyle {\frac {p_{02}}{p_{01}}}\ }$ is plotted against${\displaystyle {\frac {{\dot {m}}{\sqrt {T_{01}}}}{p_{01}}}\ }$ in the form of family of curves, each curve of a family being drawn for fixed values of ${\displaystyle {\frac {N}{\sqrt {T_{01}}}}\ }$ as shown in Figure 1.

The plot between pressure rise and flow rate is same as pressure ratio vs. mass flow rate with a small difference as in here we take pressure difference between inlet and outlet of the compressor instead of their ratio.[4]

${\displaystyle {\Delta p=f(Q)}\ }$

${\displaystyle {\Delta p=f({\dot {m}})}\ }$

First we have to know about the loading, pressure and flow coefficient which are as follow-[4]

Loading coefficient${\displaystyle ~}$ ${\displaystyle {\Psi ^{\prime }}={\frac {{g}{h}}{u^{2}}}\ }$

Pressure coefficient${\displaystyle ~}$ ${\displaystyle {\Psi }}$ = ${\displaystyle {\frac {p_{02}-p_{01}}{{\rho }{u^{2}}}}\ }$

So we can see that

${\displaystyle {\Psi }~{\propto }~{\Psi ^{\prime }}}$

Where

${\displaystyle u}$ = impeller tip speed

${\displaystyle h}$ = head

and also

Flow coefficient ${\displaystyle {\phi }~{\propto }~{\frac {Q}{{N}{D^{3}}}}\ }$

So assuming a function to relate Loading coefficient and the Flow coefficient

${\displaystyle {\Psi }=f({\phi })}$

A compressor gives its best performance while operating at its design point but it is also expected to operate away from the design point. Therefore, knowledge about its behavior at off-design operation is also necessary. The quantity ${\displaystyle (tan\beta _{2}+tan\alpha _{1})}$ can be assumed constant in a wide range of incidence. This is justified in view of small variations in the air angles at the rotor and stator exits. Therefore, writing

Figure 2. Velocity triangle for compressor Stator and Rotor

${\displaystyle \alpha _{1}=\alpha _{3}}$

${\displaystyle A=tan\beta _{2}+tan\alpha _{3}}$

Where

${\displaystyle U}$ = blade peripheral velocity

${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$ are absolute air velocity at the rotor inlet and outlet

${\displaystyle V_{r1}}$ and ${\displaystyle V_{r2}}$ are relative velocity at rotor inlet and outlet

${\displaystyle V_{3}}$ = absolute air velocity at the stator outlet

${\displaystyle \alpha _{1},\alpha _{2},\alpha _{3},\beta _{1}}$ and ${\displaystyle \beta _{2}}$ are the angles shown in the following Figure 2.

If the design values are identified by the superscript

${\displaystyle {\Psi }^{\ast }=1-A{\phi }^{\ast }}$

At off-design conditions [4]

${\displaystyle {\Psi }=1-A{\phi }^{\ast }}$

${\displaystyle {\Psi }=1-(1-A{\Psi }^{\ast })}$ ${\displaystyle {\frac {\phi }{{\phi }^{\ast }}}\ }$

This equation also gives the off-design characteristic of an axial-flow compressor. The figure shown below depicts theoretical characteristic curves for some values of the constant ${\displaystyle A}$. For positive values of ${\displaystyle A}$, the curves are falling, while for negative values rising characteristics are obtained.

Figure 3.Off design compressor characteristic curve

Theoretical characteristic curve

Theoretical characteristic curve of the compressor is the plot to show behavior of the fluid without considering losses due to shock and friction[7][8] as it passes through the compressor at different constant speeds. The curve as shown in Figure 3. is plotted between pressure coefficient ${\displaystyle {\Psi }}$ and flow coefficient ${\displaystyle {\phi }}$ for different values of ${\displaystyle A}$ as shown in fig. From fig. It is clear that as the flow coefficient ${\displaystyle {\phi }}$ increases the pressure gained per unit mass flow rate of the fluid decreases and due to this pressure coefficient ${\displaystyle {\Psi }}$ at higher mass flow rate will be low.

Figure 4. Actual compressor characteristic curve

Figure 5. Surging

Surging is the complete breakdown of steady flow in the compressor which occurs at low flow rate. Surging takes place when compressor is operated off the design point and it affects the whole machine and this is aerodynamically and mechanically undesirable. It can damage the rotor bearings, rotor seals, compressor driver and affect the whole cycle operation. It results in high temperature, high vibration and leads to flow reversal.[4][11][12]

Figure 6. Stalling process

Stalling is the separation of flow from the compressor blade surface as shown in the Figure 6. At low flow rates the angle of attack increases over the critical or maximum angle that the aerodynamic profile can sustain, and due to this there occurs the flow separation on the suction side of the blades which is known as positive stalling. If the flow separation occurs on the pressure side of the blade then it's known as negative stalling and this occurs due to negative attack angle. But generally positive stalling is taken into consideration.

In the compressor at high pressure stages if there occurs a deviation from design point (at which compressor is designed to operate) the angle of attack exceeds its stalling value and stall cells (which are the regions where fluid starts to whirl at a particular location and doesn't move forward) to form at hub and tip of the blade. The size of these cells increases with decreasing flow rate. If the flow rate is further reduced these cells grow larger and it affects the whole blade height and this causes significant drop in the delivery pressure and at very low flow rate, flow reversal takes place which is known as surge. It also results in drop in stage efficiency of the compressor and its delivery pressure.[15]

Following are some other characteristics of the compressor