Modelling of high speed impellers under various loading parameters and optimisation for exponentially varying disk profile S. Bhowmick

D. Misra

K.N. Saha

P. G. Student Dept. of Mech. Engg. Jadavpur University Kolkata 700032, India [email protected]

Faculty member Dept. of Mech. Engg. Jadavpur University Kolkata700032, India [email protected]

Faculty member Dept. of Mech. Engg. Jadavpur University Kolkata 700032, India [email protected]

Abstract A numerical solution procedure is developed for the analysis of rotodynamic impellers to investigate the optimum disk profile, which will maximize the rotational speed of an impeller. The impeller is modelled as an externally loaded rotating solid disk having an exponential thickness variation. The loading is in form of attached masses distributed radially at various points. These attached masses represent the impeller blades attached to the shrouds. The problem is formulated through a variational method and the solution of the governing partial differential equation is obtained by assuming a series solution and using Galerkin’s principle. The Von Mises stress distribution in the disk is obtained at various rotational speeds and corresponding to the yield criterion, the limit angular speed is computed. Effect of geometry and thickness parameters, attached mass ratio and its location on the limit angular speed is studied and the relevant results are reported in dimensionless form. It is observed that there exists a critical value of geometry parameter where maximum limit angular speed is obtained. Keywords:

Limit angular speed, Variational method.

Rotating

disk,

Nomenclature a, b {c } h h0 m n, k rm , ξ m u α β

∆ µ, E ξ

Inner and outer radii of the disk The vector of unknown coefficients General thickness of the disk at any radius Thickness at the center of disk Mass of the rotating disk Parameters controlling the thickness variation of disk Dimensional and non-dimensional radii of the attached mass Displacement field of the disk Ratio of attached mass to mass of disk Thickness ratio for uniform disk Parameter ( b - a ) Poisson’s ratio and elasticity modulus Normalized coordinate in radial direction, ( r − a ) / ∆

ρ σ0 φi ω Ω

Density of the disk material Yield stress of the disk material The set of orthogonal polynomials used as coordinate functions Angular velocity of the disk Dimensionless angular velocity,

ω b ρ /σ 0

1. Introduction Higher rotational speed enhances the aerodynamic performances of the impellers and hence their demand is ever increasing in industry. However, if a rotating disk is run at an escalating speed it bursts at a certain speed, i.e. at that particular speed the stresses generated in the disk exceeds the yield stress of the material. This particular speed is referred to as limit angular speed of the rotating disk. It has been observed that under such conditions maximum stresses occur near the innermost fibers of the disk. As a result the disks are generally made thick near the hub and the thickness is gradually reduced towards the periphery. The design results in variable thickness disks having higher bursting speeds as compared to those of constant thickness disks. Apart from that it reduces weight and rotary inertia of the disk. Historically, the analysis of rotating disks was carried out based on classical theory of elasticity. Ranta [6] proposed a method based on approximations in principal stresses and stress trajectories, using analytic law to determine stress strain relations and by assuming the ratio of tangential stress to radial stress to be constant to obtain optimum shape of rotating disk. In this context “optimum” meant disk having uniform strength throughout. The resulting shape of the optimal disk was found to be the classical exponential shape. Ray and Sinha [4] presented a numerical solution to the equations of equilibrium and compatibility and optimized the disk geometry by controlling the stress distribution through out the disk within the strength of the material. Chaudhry and Gupta [5], applying large elastic deformation theory, analyzed limit angular speed of variable thickness solid and annular disks. Analytical approach based on laws of equilibrium has been presented by Eraslan and Orcan [2] to investigate the elastic and plastic deformation in rotating disks of exponentially varying thickness. Eraslan

[1] carried out analysis of variable thickness annular rotating disks under pressurized boundary conditions. Güven [10] also presented a similar work by taking rigid inclusion into consideration. They have used Tresca’s yield criterion, its associated flow rule and linear strain hardening to examine the effect of boundary conditions on elastic-plastic behavior. Calladine [3] applied plastic theory to determine the performance of rotating disks loaded at edges. Effect of edge loading was also considered by Sterner et al. [9] wherein the governing equilibrium equations and the constitutive relations were expressed in terms of radial stress. They presented a numerical scheme based on Taylor’s expansion combined with iterative root finding method. Since 1980’s the energy approach has been employed and has produced good results. Recently, the application of variational method, proposed by Bhowmick et al. [7,8] has yielded a generalized approach in design of high-speed impellers and profile optimization of externally loaded parabolic disks used for modeling of high speed impellers. The review work indicates that substantial amount of research have been carried out over the past few decades on the mechanical design of rotating disks, but the effect of attached mass and its location on the limit angular speed has not yet been considered by any researcher. In the present study the problem is formulated using a variational method and analysis is carried out for various exponentially varying disk geometries with additional masses attached at different locations and burst speed is obtained for each case. The effects of geometry and loading parameters on limit angular speed is studied and reported graphically. The generated results have provided a substantial insight in understanding of the modeling and analysis of high speed impellers.

2. Mathematical formulation The mathematical model is framed assuming the disk is symmetric with respect to the mid-plane, and is under plane stress condition. The material of disk is assumed to be homogeneous, isotropic and linear-elastic. The thickness of the exponentially varying disk (Fig.1) is given by h ( ξ ) = h 0 exp[ − n ( ξ ) k ] .

The variation in the geometry is controlled through parameters n and k. In each case weight of the disk is kept fixed by adjusting the value of h0, which is the disk thickness at r=0. With increase in rotational speed the stresses increase due to enhanced effect of centrifugal force, the magnitude of which is governed by the boundary conditions of the disk. The solution for the displacement field is obtained from the minimum potential energy principle δ(π) = δ(U+V) = 0, where, U and V are strain energy stored in the disk and potential energy arising out of centrifugal force, respectively. Substituting the expressions of U and V in the energy principle δ(π) = 0, the governing equilibrium equation is obtained [7] 2  2 2µu  du  ( ∆ ξ + a )  du   ∆ πE 1  u     + + δ   d ξ  hd ξ 2  1 − µ 2 0∫  ( ∆ ξ + a ) ∆  d ξ     ∆   − 2 πρ ∆ ω

2

1

∫ ( ∆ξ + a )

0

2

 uhd ξ  = 0 - - - - - - - - - - - - - - - -(1) 

The above equation is expressed in normalized space using the transformation, ξ = ( r − a)/(b − a) . Here, a and b represent the inner and outer radii of the disk and for a solid disk a=0, as in the case of this study. The displacement functions u(ξ) in equation (1), is approximated by u (ξ ) ≅ ∑ c i φ i , i=1, 2, …, n, where φ i is the set of orthogonal functions developed through GramSchmidt scheme. The necessary starting function to generate the higher order orthogonal functions is selected by satisfying the relevant boundary conditions. The effect of attached mass is taken into consideration through a factor α, which indicates the ratio of the attached mass to the mass of the total disk m. This attached mass will contribute to the centrifugal force by an additional amount ofα mω 2 rm , where rm indicates the radius of the location of the attached mass. The contribution of this term is included in the expression of equation (1) and substituting the assumed series solution of u (ξ ) the following equation is obtained.

(

)

2   d µ ∆E 1  ∑ciφi + 2  ∑c φ δ i i dξ 1 − µ2 ∫  (∆ξ + a) ∆  0  

+

(∆ξ + a)  d  ∆2 dξ

(

) (∑ciφi ) + (∆ξ 2+ a) ddξ (∑ciφi ) 

∆



2



(∑ciφi )hdξ − 2ρ∆ω2∫ {(∆ξ + a)2∑ciφi} hdξ] 1



0

 αm  = 0 − − − − − − − − − − − − − (2) +  ω2ξ ∑c φ  π  m i i ξm

Figure 1. A disk of exponentially varying thickness

The operator ‘δ’ is replaced by ∂ ∂c j , j = 1, 2, … , n, according to Galerkin’s error minimization principle and the system governing equation in matrix form is obtained as [K]{c}={R}. The unknown coefficients are obtained from {c} = [K]-1{R}. Von Mises yield criterion is considered here, in which the yield condition is given by σ 2 − σ σ + σ 2 ≥ σ 2 , where σ and σ represent r θ θ r r θ 0 radial and tangential stresses respectively.

3. Results and discussion The numerical values of the different system parameters, considered are, E = 207 GPa, µ = 0.3, ρ = 7850 Kg/m3 and σ0 = 350 MPa. The dimensionless angular speed is defined as Ω = ω b ρ /σ 0 and the value corresponding to the onset of yielding is denoted by Ω1. Thickness control parameters k and n, loading effect of attached mass α and its location rm and thickness ratio β are the parameters considered here. Extensive numerical work is carried out to observe the effect of different system parameters on Ω1. Effect of disk geometry on limit angular speed of an exponential disk loaded with and without masses attached at specified locations is investigated for values of n ranging from 0.0 to 2.0 and k ranging from 0.0 to 1.0. The results obtained have been validated with some existing studies based on profile optimization [6] and are found to agree with their analytical conclusions.

been observed that variation of parameter n, for given values of k, affects the disk geometry, as well as limit angular speed for a fixed value of thickness parameter (Fig. 3). The figure indicates that for given loading parameters, at a given value of k and β, increase in n increases the root thickness thereby increasing the limit angular speed Ω1. The effect of parameter k on disk profile is shown in Fig.4 keeping the other system parameter values fixed as in the case of Fig. 3. Unlike in the case of parameter n, variation of parameter k controls the profile and hence the mass distribution of the disk, keeping the root thickness of the disk unchanged. Further investigations revealed that for given values of n and β, increase in k increases limit angular speed till a critical value of k is reached beyond which the magnitude of limit angular speed falls (Fig. 5). This critical value of k produces the optimum disk profile for the given value of system parameters. 0.4

*-,-/-&-

3.1 Effect of thickness ratio

0.6

*-,-/-&-

0.4

n=0.5, W1 =1.08778 n=1.2, W1 =1.39649 n=2.0, W1 =1.79073

0.2

0.1

0 0

0.2 0.4 0.6 0.8 Dimensionless Radius

1

Figure 3. Variation in disk profile for parameter n (β=0.10, k=0.72, α=0.25, ξ m =0.25).

b=0.10, W1 =1.35730 b=0.15, W1 =1.50726

0.4

b=0.20, W1 = 1.60365

*-, -/ -& -

b=0.25, W1 =1.66644

-

0.3

0.3

Disk Profile

Disk Profile

0.5

0.3

Disk Profile

Thickness ratio, β defines the ratio of thickness to the outer radius of a uniform disk and governs the mass of the disk. Increase in β increases mass of the disk. For given mass, the thickness near the hub (root) of an exponential disk is greater than that of a uniform disk. This results in larger mass distribution near the hub thereby increasing the operating speed range of the disk. Effect of thickness ratio on limit angular speed is studied and presented in (Fig. 2) for values of β ranging from 0.10 to 0.25. For a given geometry, it is observed that for given loading and geometry parameters, increase in β increases the mass distribution near the hub thereby increasing the limit angular speed Ω1.

n=0.2, W1 =0.93843

0.2 0.1

k=0.02, W1= 1.21822 k=0.50, W1= 1.38921 k=0.76, W1= 1.39655 k=1.00, W1= 1.35730

0.2

0.1

0 0

0.2

0.4 0.6 0.8 Dimensionless Radius

1

Figure 2.Variation of disk profile for different β values (n=1.2, k=1.0, α=0.25, ξ m =0.25).

3.2 Effect of disk geometry Although selection of β sets the upper limit of Ω1, it has

0 0

0.2

0.4 0.6 Dimensionless Radius

0.8

1

Figure 4. Variation in disk profile for parameter k (β=0.10, n=1.2, α=0.25, ξ m =0.25).

In Fig.5, the variation of angular speed with parameter k is shown for different values of β and the results indicate that increase in β for given n and loading parameters results in higher values of limit angular speed and the critical value of k increases as well. To obtain more insight on the effect of geometry parameters, the location of the yielding is studied and some sample results are shown in Fig.6. At lower values of k there is a sudden reduction in disk thickness in close vicinity of the hub (Fig. 4). In such cases the radial location of yielding is observed to exist in the region of reduced thickness. This unfavorable stress distribution lowers the limit angular speed of the disk. With increase in k, the radial location of yielding is found to shift outwards, thus increasing limit angular speed. This phenomenon occurs till the critical value of k is reached. Further increase in k shifts the location of yielding towards the root once again due to 380

*- /-

,- &-

b =0.10, -

b =0.15

b =0.20,

b =0.25

3.3 Effect of attached mass ratio Variation of limit angular speed with attached mass ratio was studied for values of α between 0.25 and 1.00. For a given profile, it was found that increase in attached mass at any specified location decreases the limit angular speed. For attached mass locations near the axis, yielding occurs at the same point for different values of α and the nature of stress distribution also remains almost identical (Fig. 7). Variation of Von Mises stress for different α at location away from the axis is shown in Fig. 8. The nature of stress distribution changes appreciably but they follow a similar trend among themselves. Here also an increase in attached mass ratio decreases the limit angular speed. 4E+008

*-a=0.25, W1=1.30981 - ,- a =0.50, W 1=1.09399 - /- a =0.75, W 1=0.95849 - &- a =1.00, W 1=0.86325 -

Von Mises Stress (Pa)

345

w1(rad/sec)

fall in root thickness thus lowering the limit angular speed (Fig. 6).

310

275

3E+008

2E+008

1E+008

0E+000

240 0

0.2

0.4

0.6

0.8

0

1

k

Figure 5. Effect of parameter k on limit angular speed for different β values (n=1.2, α=0.25, ξ m =0.25).

Von Mises Stress (Pa)

8

3x10

5E+008

k=0.02, W1=1.21822

*- a=0.25, W1=1.21554 -,- a=0.50, W1=0.97812 -/- a=0.75, W1=0.83604 -&- a=1.00, W1=0.74199 -

k=0.50, W1=1.38291 k=0.76, W1=1.39655

Von Mises Stress (Pa)

*- ,- /- &-

1

Figure 7. Radial distribution of Von Mises stress for different values of α at ξ m =0.25 (β=0.10, n=1.2, k=0.2).

8

4x10

0.2 0.4 0.6 0.8 Dimensionless Radius

k=1.00, W1=1.35730

8

2x10

4E+008

3E+008

2E+008

8

1x10

0

0.2 0.4 0.6 0.8 Dimensionless Radius

1

Figure 6. Radial distribution of Von Mises stress for different k values (β=0.10, n=1.2, α=0.25, ξ m =0.25).

1E+008 0

Figure 8.

0.2 0.4 0.6 0.8 Dimensionless Radius

Radial distribution of Von Mises stress for different values of α at ξ m =0.75 (β=0.10, n=1.2, k=0.2).

1

3.4 Effect of attached mass location Effect of location of attached mass ξ m , on limit angular speed is studied separately for four values of ξ m (=rm/b) ranging form 0.25 to 1.0 and the results are shown in Fig.9. As ξ m increases, for given geometry and thickness ratio, magnitude of limit angular speed falls and the location of yielding also shifts as shown in Fig. 9. It is interesting to note that the location of yielding tries to catch up the attached mass location but attains a slightly lower value. Over a range of attached mass location, the variation of limit angular speed and parameter k is studied and results are plotted in Fig. 10. It is quite evident that for each ξ m there exists an optimum value of k yielding maximum limit angular speed. The figure also indicates that for given n, β and α, increase in ξ m lowers maximum limit angular speed and corresponding optimum value of k. 5E+008

*-

xm =0.25, W1=1.39649

,- /-

xm =0.50, W1=1.25986 xm =0.75, W1=1.16850

&-

xm =1.00, W1=1.12860

-

Von Mises Stress (Pa)

-

4E+008

-

3E+008

Figure 10. Contour and 3D plot of limit angular speed with attached mass location and k ( n=1.2, β=0.10, α=0.25)

2E+008

1E+008 0.0

Figure 9.

0.2 0.4 0.6 0.8 Dimensionless Radius

1.0

Radial distribution of Von Mises stress for different ξ m values (n=1.2, k=0.72, β=0.10 and α=0.25)

Fig. 11 represents a 3-D plot of limit angular speed with α and k for given values of β, ξ m and n providing an insight regarding variation of maximum limit angular speed and corresponding optimum value of k. The effect of variation in α on limit angular speed over a given range of k is also studied and presence of optimum k value resulting in maximum limit angular speed was observed for each value of α.

4. Conclusion The present study considers the modeling of high speed impellers as externally loaded rotating disks having exponential profile and analyzes the effect of various geometry parameters on the limit angular speed with the purpose of profile optimization. Analysis is carried out to

Figure 11. Contour and 3D plot of limit angular speed with attached mass ratio and k (n=1.2, β=0.10, ξ m =0.25)

explore the effects of attached masses and their respective locations on the behavior of rotating disks and to obtain optimum disk profile for various possible combinations of geometry and loading parameters. The static design analysis is carried out using variational method formulation and assuming a series solution of unknown displacement field.

[4] G.S. Ray and B.K. Sinha, “Profile optimization of variable thickness rotating disk”, Computers and Structures, 1992, Vol. 42(5), pp. 809-813.

The method of formulation gives a kernel for subsequent dynamic analysis and study of many other complicating effects. The method developed has application potential in various other problems e.g., shrink fitted disks, disks made of anisotropic material, etc. The results are presented graphically thus providing a better insight into the study.

[6] M.A. Ranta, “On the optimum shape of a rotating disk of any isotropic material”, International Journal of Solids and Structures, 1969, Vol. 5, pp. 1247-1257.

5. References

[8] ] S. Bhowmick, D. Misra and K.N. Saha, “On optimum disk profile for high speed impellers having parabolic thickness variation”, Proc. of 31st National Conference on Fluid Mechanics and Fluid Power, Jadavpur University, Kolkata, India, 2004, pp. 417-424.

[1] A.N. Eraslan, “Elastic–plastic deformations of rotating variable thickness annular disks with free, pressurized and radially constrained boundary conditions”, International Journal of Mechanical Sciences, 2003, Vol. 45 (4), pp. 643-667.

[5] H.R. Chaudhry and U.S. Gupta, “Rotation of hyperelastic annular and solid disks of variable thickness”, International Journal of Non-Linear Mechanics, 1992, Vol. 27(3), pp. 341346.

[7] S. Bhowmick, G. Pohit, D. Misra and K.N. Saha, “Design of high speed impellers”, Proceedings of International Conference on Hydraulic Engineering and Research Practice -2004, Department of Civil Engineering, IIT, Roorkee, India, 2004, pp. 229-241.

[2] A.N. Eraslan and Y. Orcan, “Elastic-plastic analysis of a rotating solid disk of exponentially varying thickness”, Mechanics of Materials, 2002, Vol. 34 (7), pp. 423-432.

[9] S.C. Sterner, S. Saigal, W. Kistler and D.E. Dietrich, “A unified numerical approach for analysis of rotating disks including turbine rotors”, International Journal of Solids and Structures, 1994, 31(2), 269-277.

[3] C.R. Calladine, “Engineering Plasticity”, Pergamon Press, Oxford, 1969.

[10] U. Güven, “Elastic – plastic stress distribution in a rotating hyperbolic disk with rigid inclusion”, International Journal of Mechanical Sciences, 1998, Vol. 40 (1), pp. 97–109.