tion of Angular momentum and Mass (CAM Designs) [1].
International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 1380
ISSN 2229-5518
Mathcad Tool for Design of Volute of Centrifugal
Air Compressor
P V Shyam, Vithuraj T
Abstract— This paper presents a project work aimed at developing a Mathcad tool capable of designing different types of industrial centrifugal air compressor volutes. Six different volutes are designed in this project with the help of suitable algorithms, which are Rectangular Cross Sectional External and Semi-External volutes; Elliptical Cross Sectional External and Semi External volutes; Circular Cross Sectional External and Semi-External volutes. The geometry generated incorporates the manufacturing and performance requirements necessary for a practical volute design. The Mathcad tool capability is validated by comparing with the prediction of an established Commercial Tool called COMPAERO©. Thus, this project ultimately looks in introducing a volute design tool compatible to the industry.
Index Terms— Algorithms for Mathcad, CAM design, Circular Cross Sectional External and Semi-External Volutes, Elliptical Cross Sectional External and Semi External Volutes, Rectangular Cross Sectional External and Semi-External Volutes, Validation with COMPAERO©, Volute or Scroll
—————————— ——————————
HE main objective of the project is to generate the cross sectional profiles of the volute and hence arrive at the 3D structure of the volute. The tool is developed as a Mathcad script using the commercially available design software called “Mathcad-14”. Design methodology adopted is to establish the area distribution in the volute from the principles of Conserva-
tion of Angular momentum and Mass (CAM Designs) [1].
Fig. 1 Full Collection Plane of the Volute
The tool developed is capable of designing six types of com- monly used industrial volutes which are as follows:-
I. Rectangular Cross sectional External Volute
————————————————
II. Rectangular Cross sectional Semi-External Volute
III. Elliptical Cross sectional External Volute
IV. Elliptical Cross sectional Semi-External Volute
V. Circular Cross sectional External Volute
VI. Circular Cross sectional Semi-External Volute
The algorithms used for designing the Elliptical and Circular cross sectional volutes are done in such a way that 3/4th of the volute profile is an ellipse or circle and the rest is rectangle for manufacturability. The equations used for the design can be referred in the Appendix A.
a) Algorithm for Geometrical properties for the Full Col- lection Plane.
Input Values:- Volute Inlet Radius; Volute inlet Width; Sizing parameter; Aspect Ratio; Inlet Flow angle; Number of Stations for Volute generation
Equations for finding Variables like Height, Width, Area, Mean Radius and Maximum Radius for the Volute Passage cross section are derived from CAM designs and Geometrical concepts
If Width ≤ Volute inlet width then, Width = Volute inlet width and continue solving the equations by keeping this value in finding other variables
Otherwise, continue with the value obtained for Width from CAM design equations and use that to find further variables
This process is continued for the corresponding number of Stations
Display Values of Variables in a table format
b) Algorithm for 3D Co-ordinates for Full Collection
Plane
Input Values:- Height, Width, Mean Radius and
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Maximum Mean Radius of the Volute Passage, Number of Stations for Volute Generation
Equations for finding the 3D Coordinates(X, Y, Z) at 5 points of the Rectangle which are Volute inlet Point; the three Corner points of the Rectangle and the Volute inlet width point derived from mathe- matical modeling concepts
The process is continued for all the stations speci- fied
Display the coordinate values in a table format
c) Algorithm for Geometrical Properties of the Exit Cone
Input Values:- Ideal Divergence angle needed for
the Exit cone; the geometric properties of the rec-
tangle at 3600 of the Full Collection Plane; Number
of sections for Exit cone generation
Equations derived from basic geometrical concepts
of frustum to find the height, width and area of the
Exit cone Cross section at each section after giving
a fixed value to the Exit Flange Area
All the parameters of the Exit cone are calculated
and displayed in the Output
d) Algorithm for Coordinates of the Exit Cone
Input Values:- Geometrical Properties & Coordi-
nates of the Rectangle at 3600 of the Full Collection
Plane; Geometrical Properties of the Exit Cone at
each section
Equations are derived from Mathematical Model-
ing concepts for finding the 3D spacing of the Exit
Cone at each section
Results are displayed in Array format showing the
Coordinate values of the Exit cone
a) Algorithm for Geometrical Properties of Full Collec- tion Plane
Input Values:- Volute Inlet Radius; Volute inlet Width; Sizing parameter; Aspect Ratio; Inlet Flow angle; Number of Stations for Volute generation; Thickness between the annular passage and the Volute Passage
Equations for finding the Height, Width, Area, Mean Radius and Maximum Mean Radius of the Volute Passage Derived from the CAM design and geometrical concepts
If Aspect Ratio (Width/Height) > 1 then take the Equation of Height and check for all the stations and if Height ≤ 2*Volute inlet width then Height
=2* Volute inlet width and continue solving the equations by keeping this value in finding other Variables. Otherwise continue with the value ob- tained from CAM design equations
If Aspect Ratio (Width/Height) < 1 take the Equa- tion of Width and check for all the stations and if Width ≤ 2*(Volute inlet width +Thickness) then Width = 2*(Volute inlet width +Thickness) and continue solving the equations by keeping this value in finding other Variables. Otherwise con- tinue with the value obtained from CAM design
equations
This process is continued for the corresponding
number of Stations
Display the Values in a table format
b) Algorithm for 3D Co-ordinates for Full Collection
Plane
Input Values:- Height, Width, Mean Radius and Maximum Mean radius of the Volute Cross sec- tional Passage; Number of Stations for Volute gen- eration; Thickness between the annular passage
and the Volute Passage
Equations derived from the basic concepts of
Mathematical modeling to get the values for X, Y,
Z Coordinates for 6 points in the rectangle which
include Volute inlet point, Volute width + thick-
ness point, Volute inlet width point and the rest of
3 corner points of the Rectangle
The process is continued for all the stations speci-
fied
Display the coordinate values in a table format
c) Algorithm for Geometrical Properties of the Exit Cone
Input Values:- Ideal Divergence angle needed for
the Exit cone; the geometric properties of the rec-
tangle at 3600 of the Full Collection Plane; Thick-
ness between the annular and Volute passage;
Number of sections for Exit cone generation
Equations derived from basic geometrical concepts
of frustum to find the height, width and area of the
Exit cone Cross section at each section providing a
fixed value to the Exit Flange Area
All the parameters of the Exit cone are calculated and displayed in the Output
d) Algorithm for Coordinates of the Exit Cone
Input Values:- Geometrical Properties & Coordi-
nates of the Rectangle at 3600 of the Full Collection
Plane; Geometrical Properties of the Exit Cone at
each section; Thickness between the annular and
Volute passage
Equations are derived from Mathematical Model-
ing concepts for finding the 3D spacing of the Exit
Cone at each section
Results are displayed in Array format showing the
Coordinate values of the Exit cone
a) Algorithm for Geometrical properties of Full Collec- tion Plane
Input Values:- Volute Inlet Radius; Volute inlet Width; Sizing parameter; Aspect Ratio; Inlet Flow angle; Number of Stations for Volute generation
Equations for finding Variables like Total Height, Width, Area, Mean Radius and Maximum Radius for the Volute Passage are derived from CAM de- signs and Geometrical concepts
If Width ≤ Volute inlet width then Width = Volute inlet width and continue solving the equations us- ing this value to find values of other variables
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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 1382
ISSN 2229-5518
Otherwise, continue with the value obtained for Width from CAM design equations and use that to find further variables
This process is continued for the corresponding number of Stations
Display Values of Variables in a table format
b) Algorithm for Co-ordinates of Full Collection Plane
Input Values:- Height, Width, Mean Radius and
Maximum Mean radius of the Volute Cross sec-
tional Passage; Number of Stations for Volute gen-
eration; Number of points for Ellipse Generation
Equations derived from the basic concepts of
Mathematical modeling to get the values for X, Y,
Z Coordinates for ‘n+3’ points in the Cross section
which are Volute inlet point, first corner point of
1/4th rectangle portion, ’n’ points for 3/4th Ellipse
and Volute inlet width point
The process is continued for all the stations speci-
fied
Display the coordinate values in a table format
a) Algorithm for Geometrical properties of Full Collec- tion Plane
Input Values:- Volute Inlet Radius; Volute inlet Width; Sizing parameter; Aspect Ratio; Inlet Flow angle; Number of Stations for Volute generation; Thickness between the annular passage and the Volute Passage
Equations for finding the Height, Width, Area, Mean Radius and Maximum Mean Radius of the Volute Passage Derived from the CAM design and geometrical concepts
If Aspect Ratio (Width/Height) > 1 then take the Equation of Height and check for all the stations and if Height ≤ Volute inlet width then Height = Volute inlet width and continue solving the equa- tions by keeping this value in finding other Varia- bles. Otherwise continue with the value obtained from CAM design equations
If Aspect Ratio (Width/Height) < 1 take the Equa- tion of Width and check for all the stations and if Width ≤ (Volute inlet width +Thickness) then Width = (Volute inlet width +Thickness) and con- tinue solving the equations by keeping this value in finding other Variables. Otherwise continue with the value obtained from CAM design equa- tions
This process is continued for the corresponding number of Stations
Display the Values in a table format
b) Algorithm for 3D Co-ordinates for Full Collection
Plane
Input Values:- Height, Width, Mean Radius and
Maximum Mean radius of the Volute Cross sec-
tional Passage; Number of Stations for Volute gen-
eration; Number of points for Ellipse Generation;
Thickness between the annular passage and the
Volute Passage
Equations derived from the basic concepts of
Mathematical modeling to get the values for X, Y,
Z Coordinates for ‘n+3’ points in the Cross section
which include Volute inlet point, Volute width +
thickness, Volute inlet width point, the rest of n
points for the Ellipse and an inside corner point of the 1/4th rectangle
The process is continued for all the stations speci- fied
Display the coordinate values in a table format
a) Algorithm for Geometrical Properties of Full Collec- tion Plane
Input Values:- Volute Inlet Radius; Volute inlet Width; Sizing parameter; Inlet Flow angle; Num- ber of Stations for Volute generation
Equations for finding Variables like Total Height, Width, Area, Mean Radius and Maximum Radius for the Volute Passage are derived from CAM de- signs and Geometrical concepts
If Width ≤ Volute inlet width then Width = Volute inlet width and continue solving the equations us- ing this value to find values of other variables
Otherwise, continue with the value obtained for Width from CAM design equations and use that to find further variables
This process is continued for the corresponding number of Stations
Display Values of Variables in a table format
b) Algorithm for Co-ordinates of Full Collection Plane
Input Values:- Height, Width, Mean Radius and
Maximum Mean radius of the Volute Cross sec-
tional Passage; Number of Stations for Volute gen-
eration; Number of points for Circle Generation
Equations derived from the basic concepts of
Mathematical modeling to get the values for X, Y, Z Coordinates for ‘n+3’ points in the Cross section which are Volute inlet point, first corner point of
1/4th of rectangle portion, ’n’ points for 3/4th of
Circular portion and Volute inlet width point
The process is continued for all the stations speci-
fied
Display the coordinate values in a table format
a) Algorithm for Geometrical properties of Full Collec- tion Plane
Input Values:- Volute Inlet Radius; Volute inlet Width; Sizing parameter; Aspect Ratio; Inlet Flow angle; Number of Stations for Volute generation; Thickness between the annular passage and the Volute Passage
Equations for finding the Height, Width, Area, Mean Radius and Maximum Mean Radius of the Volute Passage Derived from the CAM design and
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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 1383
ISSN 2229-5518
geometrical concepts
If Aspect Ratio (Width/Height) > 1 then take the
Equation of Height and check for all the stations
and if Height ≤ Volute inlet width then Height =
Volute inlet width and continue solving the equa-
tions by keeping this value in finding other Varia-
bles. Otherwise continue with the value obtained from CAM design equations
If Aspect Ratio (Width/Height) < 1 take the Equa- tion of Width and check for all the stations and if Width ≤ (Volute inlet width +Thickness) then Width = (Volute inlet width +Thickness) and con- tinue solving the equations by keeping this value in finding other Variables. Otherwise continue with the value obtained from CAM design equa- tions
This process is continued for the corresponding number of Stations
Display the Values in a table format
b) Algorithm for 3D Co-ordinates for Full Collection
Plane
Input Values:- Height, Width, Mean Radius and
Maximum Mean radius of the Volute Cross sec-
tional Passage; Number of Stations for Volute gen-
eration; Number of points for Circle Generation;
Thickness between the annular passage and the
Volute Passage
Equations derived from the basic concepts of
Mathematical modeling to get the values for X, Y,
Z Coordinates for ‘n+3’ points in the Cross section
which include Volute inlet point, Volute width + thickness, Volute inlet width point, the rest of n points for the Ellipse and an inside corner point of the 1/4th rectangle
The process is continued for all the stations speci- fied
Display the coordinate values in a table format
The algorithm for the generation of Exit cone for Elliptical and
circular cross sectional External as well as Semi-External vol-
utes are not mentioned because the existing algorithm used for
rectangular cross sectional External as well as Semi-External
volutes are repeated over there.
The algorithms which are made above are converted into pro- grams in Mathcad software so that a new tool could be created for designing the Volutes. The programs were created such that each algorithm produced an individual program for each volute. So, a total of 24 programs were created for the design of all the six volutes and all programs were clubbed together in another fresh Mathcad file. An additional program called Choice was added to it so that a user could choose between any of the six volutes and model the selected one. The Output of the combined programs consists of 12 parts which are:-
i. Full Collection Plane Geometrical Properties ii. Coordinates for the Full Collection Plane
iii. Exit Cone Geometrical Properties iv. Length of the Exit Cone
v. Coordinates for the Exit Cone
vi. Area Variation of Full Collection Plane
vii. Circumferential Area Increase of Full Collection Plane
viii. Radial Orientation of Full Collecton Plane
ix. Variation of Full Collection plane around Diffuser
x. 3D Contour Plot of the Volute
xi. Variation of the Exit Cone
xii. Radial Orientation of the Exit cone
In this section, the comparison results are reported for all the External volutes which were previously generated by Mathcad software. The procedure is repeated for Semi-External Volutes so, it’s immaterial to report those results here. In COMPAER- O© sofware the properties generated for internal volutes are basically the properties for Semi-External volutes. Thus, the results obtained in Mathcad for Semi-External volutes are di- rectly compared with the internal volutes. The process used for comparison of geometrical properties is COMPAERO© results were subtracted from the Mathcad outputs. For match- ing the cross sectional shapes a separate mathcad file was cre- ated for each volute. In this program 5 angles were taken on a symmetric basis and the cross sectional shapes of the Volute were compared at each of these angles. In this section the cross sectional shape at the exit of the full collection plane of the Volute is shown.
These values are procured by subtracting the COMPAERO© result values from Mathcad result values. A small difference is seen while comparing the results because COMPAERO© soft- ware has reduced the area by filleting the corner part of the rectangle with the radius having the magnitude of volute inlet width as shown in fig.2. The properties taken here for validat- ing are the Area of the Full Collection Plane, Mean Radius Variation of the Volute Passage (R), Maximum Mean Radius Variation of the Volute Passage (Rmax), Maximum Width of the Cross Section of the Volute (Wmax).
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ISSN 2229-5518
At 3600 Polar Angle
Mathcad.
At 3600 Polar Angle
Fig. 2 Comparison of Rectangular Cross sectional shapes gen- erated in Mathcad and COMPAERO©
Fig. 4 Comparison of Circular Cross sectional shapes generated in Mathcad and COMPAERO©
Avg% Area Error | Avg%R Error | Avg% Rmax Error | Avg% Wmax Error |
0.0 | 0.0 | 0.0 | 0.0 |
Elliptical cross sectional Volutes showed a great similarity between the results produced in both the soft wares. This tells about the authenticity of the programming algorithm used by the Mathcad software to design the Volutes.
At 3600 Polar Angle
Fig. 3 Comparison of Elliptical Cross sectional shapes generated in Mathcad and COMPAERO©
Avg% Area Error | Avg%R Error | Avg% Rmax Error | Avg% Wmax Error |
0.0 | 0.0 | 0.0 | 0.0 |
Circular Cross Sectional Volutes also gave a great similarity between the results obtained from COMPAERO© and from
Equations involved in Design of each volute are presented in this section. These equations are directly derived from the CAM de- sign concepts of centrifugal air compressor.
For all the below volutes the equation for Area calculation is
TABLE 1
NOMENCLATURE
r Volute Radius Measured from the center of the com- pressor
R Radius of the Mean Passage flow
Rmax Volute Outer Radius Measured from the Centre of
the Compressor
b Volute Inlet Width or Minimum width of the volute
t Thickness between the Annular passage and the
Passage flow inside the volute
H Height of the Cross section of the volute
W Width of the Cross section of the volute
A Area of the Cross section of the volute
ρ Density of the air flow in the volute
ɵ Polar angle representing the sections of volute pas-
sage
Vf Meridional velocity of the air
Vw Swirl component of the velocity of the air
V Absolute Velocity of the air
α Flow angle with respect to the volute cross section
β Divergence angle required for the Exit Cone
L Length of the Exit Cone
SP Sizing Parameter for performance
4 Volute Inlet Position
5 Full Collection Plane’s maximum radial position
6 Tongue Position
7 Position of the Exit Flange Area
C Mean Passage Flow position of the volute
E Intermediate Exit cone position
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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 1385
ISSN 2229-5518
taken from the CAM design concept [1] as,
(rV)c = (rVw )4 (1)
(ρV)cA = θ(ρrbVf )4 (2)
A = SP ∗ θ ∗ rc ∗ b4 ∗ tanα4 (3)
For Rectangular Cross sectional External Volute,
This Project has been done during the Practice School II course conducted by Birla Institute of Technology and Science, PILANI. Hence, special acknowledgement is given to the Prac- tice school division for the background support given in com- pletion of the project successfully.
The authors are thankful to Mr. Pavan Kumar Reddy and Dr.
𝑆∗𝜃
𝑆∗𝜃 2
Swaminathan Gopalakrishnan at ELGI EQUIPMENTS LIM-
𝑊 = 1 �
2 2
+ �� �
2
+ 4 ∗ 𝐴𝐴 ∗ 𝑟4 ∗ 𝑆𝑆� (4)
ITED for their valuable support during the course of the pro-
Where S = SP ∗ b4 ∗ tanα4
For Aspect Ratio Greater than 1 (AR > 1)
ject. A special gratitude is shown to the Learning and Devel- opment Department of ELGI EQUIPMENTS LIMITED for providing the opportunity to do the project.
H = 1
�− t + �t2 + 4 ∗ (S ∗ θ + b2 ∗ AR)� (5)
2∗AR 2 4 4
For Aspect Ratio Less than 1 (AR < 1)
W = 1 �b4 −t + �(b4 −t)2 + 4 ∗ θ ∗ S� (6)
2 2 4
Where S = SP ∗ AR ∗ r4 ∗ b4 ∗ tanα4
For Elliptical Cross sectional External Volute,
2
The primary project goal of development of a volute design tool using the Mathcad was successfully completed. The new tool is capable of designing centrifugal compressor volute pro- files of six different types which were mentioned all along the project. The main aim was the consideration of ease of manu- facturability and performance which are of prime concern for industry which was attained with the help of this tool. The main reason for this is that the output of the tool is in a format which could be easily imported to any CAD software for
W = �S ∗ θ
+ ��S ∗ θ
� + 4 ∗ AR ∗ r
∗ S ∗ θ
� (7)
modeling the same and for further manufacturing process.
1+3∗π
4
Where S = SP ∗ b4 ∗ tanα4
1+3∗π
4
4 1+3∗π
4
The Mathcad results obtained are authenticated and validated with commercially approved software called COMPAERO©, which is used in most of the major centrifugal compressor
For Elliptical Cross sectional Semi-External Volute, For Aspect Ratio Greater than 1 (AR > 1)
production industries. All these comparison results are report- ed and it has been observed that the results produced were greatly matching with the results of COMPAERO© software. Performance validation module could be taken up as a future
H = 1
�−t + �t2 + 4 ∗ �1 + 3 π� ∗ (S ∗ θ + b2 ∗ AR)� (8)
work. Thus, it can be concluded that the Mathcad tool gener-
�1+3π�∗AR
4
4 4
ated volute designs could be used in the manufacturing indus-
For Aspect Ratio Less than 1 (AR < 1)
tries for the Centrifugal compressors.
W = 1
�b4 −t + �(b4 −t)2 3
�1+3π� 1
4
+ 4 ∗ θ ∗ S ∗ �1 + π�� (9)
1 4
Where S = SP ∗ AR ∗ r4 ∗ b4 ∗ tanα4
For Circular Cross sectional External Volute,
2
[1] R..H. Aungier Volute design. In Centrifugal compressors: a strategy for aerody- namic design and analysis (pp. 195-200). New york: ASME.
[2] Cohen, H., & Rogers, G. (1996). Centrifugal Compressors. In Gas Turbine
Theory (pp. 141-151). England: Longman Group Limited.
[3] Meherwan. (2003). Centrifugal Compressors: a basic guide. In Boyce, Compressor
W = �S ∗ θ
+ ��S ∗ θ
� + 4 ∗ r
∗ S ∗ θ
� (10)
1+3∗π
4
Where S = SP ∗ b4 ∗ tanα4
1+3∗π
4
4 1+3∗π
4
Selections (pp. 1-30). Oklahoma: PennWell Corporation.
[4] PMFM_Lecture_Slides. (2011, February). Centrifugal compressor. Hyderabad,
Andhra, India: BITS-Pilani.
[5] Reunanen, A. (2001). “Experimental and Numerical Analysis of different
For Circular Cross sectional Semi-External Volute,
volutes in a centrifugal compressor”. High Speed Technology Research Program,
152.
W = 1
�b4 −t + �(b4 −t)2
3 [6] Ronald. (n.d.). Turbomacinery Aerodynamic Technology. Retrieved 2013,
�1+3π� 1
4
+ 4 ∗ θ ∗ S ∗ �1 + π�� (11)
1 4
from Design and Analysis software and Consulting services: http:// www.turbo-
Where S = SP ∗ r4 ∗ b4 ∗ tanα4
For the calculation of Exit Cone cross sectional parameters the
equation used is as follows,
WE =2*W360*tan(β/2) (12)
Where, W360 is the Width of the Cross section of the full collec-
tion plane.
aero.com/documents/CompAero.aspx
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