Compressor design procedure

With the total pressure and temperature at the inlet, 𝑃₀₁and 𝑇₀₁,which are inputs to the code, it is possible to obtain the stagnation enthalpy and the stagnation entropy through the following equations.

ℎ₀₁ = 𝑓(𝑃₀₁, 𝑇₀₁) (5-68)

𝑠01 = 𝑓′(𝑃01, 𝑇01) (5-69)

Since the entropy between static and stagnation states remains the same, it is feasible to obtain the thermophysical properties directly using the static enthalpy and the stagnation entropy at that same point. The absolute velocity at the inlet is a known variable, which is constant throughout the entire span of the blade.

88 Inverse Problem ℎ1 = ℎ01− 𝑐₁2 2 (5-70) 𝑠1 = 𝑠01 (5-71) 𝑃₁ = 𝑓(ℎ₁, 𝑠₁) (5-72) 𝑇1 = 𝑓′(ℎ1, 𝑠1) (5-73) 𝜌₁ = 𝑓′′(ℎ₁, 𝑠₁) (5-74) 𝑐_𝑃1= 𝑓′′′(ℎ₁, 𝑠₁) (5-75)

At the outlet, in order to obtain the thermophysical properties, the stagnation properties in this stage need to be addressed first. Then, the same procedure is repeated for the impeller inlet.

The pressure ratio is an input to the compressor design model, so the stagnation enthalpy at the outlet can be obtained through the equation:

ℎ03 = ℎ02−

ℎ_03𝑠− ℎ₀₂

𝜂_𝑡𝑡 (5-76)

Where ℎ03𝑠 is the enthalpy that the flow would have at the outlet if the compression process was isentropic. As the entropy at the inlet is already known and the total pressure at the outlet is determined through the pressure ratio and its inlet counterpart:

ℎ₀₃= 𝑓(𝑃₀₃, 𝑠₂) (5-77)

Therefore, the efficiencyneeds to be guessed and the model will be iterating until a convergence is achieved with the results of the performance analysis via loss models. Consequently, assuming identical total and static entropy again, all the outlet static thermophysical properties can be obtained, using the procedure described above.

The following steps are taken in order to complete the design procedure for the inducer and impeller:

Centrifugal Compressor Design Methodology 89

1. A hub to tip ratio is selected and a shroud inlet radius is guessed.

2. The velocity diagram at the inlet and the geometry are derived assuming null incidence (𝛽₂ = 𝛼₂)

3. The static conditions are checked in order to ensure supercritical flow. 4. The impeller outlet radius is derived by preliminary design.

5. It is assumed that the meridional velocity at inlet and outlet are identical. 6. The total-to-total efficiency

 

 

of the system apart from impeller [79], are guessed in order to meet the desired conditions at the impeller exit.

𝐿𝐶 =

𝑃₀₃− 𝑃₀₅ 𝑃03

(5-78)

7. The total pressure at impeller’s exit 𝑃03 is calculated as:

𝑃₀₃= 𝑃01𝑃𝑅𝑠𝑡𝑎𝑔𝑒

1 − 𝐿_𝐶 (5-79)

8. The slip factor, distortion factor and loss coefficients for impeller are guessed. 9. The velocity diagram at the impeller exit is estimated.

10. The impeller blade exit angle is estimated:

𝛽₃= 𝑎𝑡𝑎𝑛 [( 𝑉𝜃3

𝜎 − 𝑈3)

𝜆𝑉_𝑚3 ] (5-80)

11. The static conditions at the impeller exit are calculated.

12. The geometry of the impeller is derived based on the assumptions. Then the direct model presented in the previous section is used in order to estimate slip, distortion factor and the pressure loss coefficients.

13. The whole procedure is repeated until the convergence between the assumptions and the results for step 12.

Having completed the previous steps, the design of the inducer and the impeller is fixed, and the procedure continues to vaneless space and diffuser.

The significant parameters for the vaneless space design are the radius, the blade height and the inlet blade angle. For the blade height the assumption that ℎ_𝑏4 ≤ ℎ𝑏3 is used. Monge [79] presents the equation:

2 3 4 4 3 90 1 360 15 M r a r     (5-81)

90 Inverse Problem

Assuming a small diffusion of the flow, the meridional velocity at the inlet of the diffuser is guessed and for a radius ratio between 1.06 and 1.12 [78], the inlet flow angle of the diffuser

 

Finally, for the design of the vaned diffuser some significant parameters presented in the previous section have to be defined according to Aungier [78]. The design for the vaned diffuser comprised the acceptance of ranges presented in Monge [79]:

 Divergence angle lower than 11o_.  Loading parameter lower than 1/3.  A design parameter: 1.5 ≤ 𝐸 ≤ 1.7

The following diagram presents the design procedure flowchart.

Centrifugal Compressor Design Methodology 91

5.3 Numerical Method

The investigation of the flow phenomena depicting the centrifugal compressor performance, require the solution of a set of non-linear partial differential equations. The set of equations for compressible fluid are applied for the compressor flow in three dimensions and RANS equations are employed.

The differences between the flow in the heat exchanger and the compressor are firstly the rotating flow in the impeller that has to be predicted, and secondly the interface definition between the rotating and stationary compressor parts. Moreover, the flow in a centrifugal compressor is highly turbulent, presenting Reynolds numbers in the range of 105-107, leading in the development of turbulent boundary layers at the blades’ walls. Therefore, the turbulence model used in the compressor study is the shear stress transport (SST), which prevents the overprediction of the turbulence viscosity.

The CFD analysis requires a connection between the rotating and stationary domains. There is an interface connecting the two domains, so when a change occurs in one domain it would be transferred to the other. For this study, the critical interfaces are these connecting the impeller with the stationary inlet and the diffuser. There are different methods that handle this interface, such as frozen rotor and stage/mixing plane. For this analysis, the frozen rotor method is used. The frozen rotor model, which is also known as Multiple Reference Frame (MRF) model, treats the flow from one component to the next by changing the frame of reference across the interface, while keeping the relative orientation of the components fixed. The position of the mesh elements does not change between the two interfaces, but is fixed for the whole computational domain. The frozen rotor model produces a steady-state solution at each time instant, taking into consideration the two frames interaction, but the phenomena varying with time are neglected.