Compressor characteristic

Compressor characteristic is a mathematical curve that shows the behaviour of a fluid going through a dynamic compressor. It shows changes in fluid pressure, temperature, entropy, flow rate etc.) with the compressor operating at different speeds. 
A compressor increases the pressure of a fluid passing through it, so that the exit pressure is higher than the inlet pressure. Due to this property, compressors are used in a wide range of machines, such as refrigerators, cars, jet engines and industrial processes.
Compressor characteristic curves are plotted between various parameters and some are as follows
Pressure ratio vs. non-dimensional flow rate.
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. These performance characteristic curves are usually plotted with dimensionless variables. For this, we start with implicit functional relationships for all the variables as 
formula_1 
Where
formula_2 = characteristic linear dimension of the compressor
formula_3 = rotational speed
formula_4 = mass flow rate
formula_5 and formula_6 = stagnation pressure at compressor inlet and outlet
formula_7 and formula_8 = stagnation temperature at compressor inlet and outlet
formula_9 = characteristics gas constant
By making use of π-theorem, we obtain the non-dimensional groups (π terms) as 
formula_10, formula_11, formula_12, formula_13
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
formula_14 
Though the terms formula_15 and formula_16 are truly not dimensionless, they are referred as 'non-dimensional mass flow' and 'non-dimensional rotational speed' for practical purpose. The stagnation pressure ratio formula_10 is plotted againstformula_15 in the form of family of curves, each curve of a family being drawn for fixed values of formula_16 as shown in Figure 1.
Pressure rise vs. flow rate.
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.
formula_20
formula_21
Loading coefficient/pressure coefficient vs. flow coefficient.
First we have to know about the loading, pressure and flow coefficient which are as follow-
Loading coefficientformula_22 formula_23
Pressure coefficientformula_22 formula_25 = formula_26
So we can see that
formula_27
Where
formula_28 = impeller tip speed
formula_29 = head
and also
Flow coefficient formula_30
So assuming a function to relate Loading coefficient and the Flow coefficient
formula_31
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 formula_32 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
formula_33
formula_34
Where
formula_35 = blade peripheral velocity
formula_36 and formula_37 are absolute air velocity at the rotor inlet and outlet
formula_38 and formula_39 are relative velocity at rotor inlet and outlet
formula_40 = absolute air velocity at the stator outlet
formula_41 and formula_42 are the angles shown in the following Figure 2.
If the design values are identified by the superscript
formula_43
At off-design conditions 
formula_44
formula_45 formula_46
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 formula_47. For positive values of formula_47, the curves are falling, while for negative values rising characteristics are obtained.
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 as it passes through the compressor at different constant speeds. The curve as shown in Figure 3. is plotted between pressure coefficient formula_25 and flow coefficient formula_50
for different values of formula_47 as shown in fig. From fig. It is clear that as the flow coefficient formula_50 increases the pressure gained per unit mass flow rate of the fluid decreases and due to this pressure coefficient formula_25 at higher mass flow rate will be low.
Actual characteristic curve.
Actual characteristic curve of the compressor is the modified version of the theoretical characteristic curve. In this we take care of both shock and friction losses as shown in the Figure 4. As seen in the figure the steepness of the curve increases at higher mass flow rates because since there will be higher frictional losses at high flow coefficient (Darcy's friction law).
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.
Surge process.
Assuming the compressor operates at point A (formula_54) on the characteristic curve (let at constant speed formula_55) as shown in Figure 5. Now if the flow rate is reduced to formula_56 by closing a control valve on the delivery pipe, the static pressure upstream of the valve is increased. This increased pressure (formula_57) is then matched by the increased delivery pressure (at B) which is developed by the compressor. Now further reducing the flow (to formula_58 and formula_59), the increased pressures in the delivery pipe are again matched by the compressor delivery pressures at C and S on the characteristic curve.
On the characteristic curve at the flow rates below ṁS provides lower pressure as seen in the fig. at D and E. But now the pipe pressures due to further reduction of flow by valve (let at point D) will be higher than the pressure at D and E. This unbalance between the pipe pressure and the compressor delivery pressure only exist for a very short time. This is because there is higher pressure in the pipe than the gas pressure produced by the compressor and due to this reversing of the flow takes place and it leads to a complete break-down of the normal steady flow from the compressor to the pipe.
Surge cycle.
Due to flow reversal, pressure in the pipe falls and the compressor regains its normal stable operation (let at point B) delivering the gas at higher flow rate (formula_56). But the control valve still corresponds to the flow rate formula_61. Due to this compressor's operating conditions will again return to D through points C and S. And due to lower compressor pressure, the pressure falls further to formula_62 and the entire phenomenon from point E to D repeats again and again and this cycle EBCSDE known as the surge cycle.
Surge point.
Surge points are the peak points on the characteristic curves (as in Figure 5.) left of which the pressure generated by the compressor is less than the pipe pressure and these points initiates the surge cycle. These points on the curves are shown in the fig. by point S.
Surge line.
Surge line is the line which connects the surge points (S) on each characteristic curve corresponding to different constant speeds. The stable range of operation for the compressor is on the right hand side of the surge line.
Surge control line.
Surge control line is the line which works as the indicating line for the surge control mechanism so that surge can be prevented in the system and proper steps can be taken. The line can vary for different surge control systems as it is up to the system to decide the margin between operating point and the surge point.
<ref name="http://www.gmrc.org/documents"></ref>
Surge margin.
Surge margin is the indicator to show that how close is the system to the surge. It is different for the different surge control system as they employ different surge control lines. The basic surge margin can be defined as
Surge margin = formula_63
Where
formula_64 = mass flow rate at surge control line at a particular constant speed
formula_65 = mass flow rate at surge line at the same constant speed.
Stalling.
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.
Other compressor phenomena.
Following are some other characteristics of the compressor
Choke/Stonewall point.
Choking is the condition which occurs in the compressor in which it operates at very high mass flow rate and flow through the compressor can't be further increased as mach number at some part of the compressor reach to unity i.e. to sonic velocity and the flow is said to be choked. In compressor maximum volume flow rate is limited by cross-section at the inlet. This condition can be seen in the right side of the Figure 5. in which constant speed lines descends steeply. The point on constant speed line at which choking occurs is known as choke point or stonewall point.
Choke line.
Choke line is the line joining the choke points on different constant speed lines in the Figure 5. In simple words, the operation on right side of choke line is very inefficient, but is possible if the exit static pressure is low enough and blade instabilities such as choke flutter are avoided.
Constant efficiency lines.
Constant efficiency lines are the elliptical shaped curves in the compressor characteristic curve along which the efficiency of the compressor remains constant.
Maximum allowable speed.
This is the maximum speed at which compressor can work properly(shown as formula_55 in Figure 5.) and beyond this speed limit stresses and vibrations in the compressor crosses the prescribed limit and this can damage the compressor and to control this, compressor's speed is lowered.
Minimum required speed.
This is the minimum speed at which compressor should operate (shown as formula_67 in Figure 5.) and below this limit compressor can't increase the pressure of inlet fluid and goes into idle condition.