SENSITIVITY ANALYSIS OF VOLUTE PARAMETERS USING FDM · 2016-04-11 · identify the geometric...

International Journal of Mechanical And Production Engineering, ISSN: 2320-2092, Volume- 2, Issue-9, Sept.-2014

Sensitivity Analysis of Volute Parameters Using FDM

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SENSITIVITY ANALYSIS OF VOLUTE PARAMETERS USING FDM

1ADERSH R ALORNEKAR, 2BEENA D. BALONI, 3S. A. CHANNIWALA

1M.tech student, SVNIT, Surat; 2Asst. Prof. Dept. of Mech. Engg., SVNIT, Surat; 3Prof. Dept. of Mech. Engg., SVNIT, Surat, E-mail: [email protected], [email protected]

Abstract- Sensitivity analysis plays a critical role in numerical optimisation and reliability studies. The present paper estimates the sensitivities of the output matrices with respect to the input parameters of a volute casing geometry. A finite difference method has been applied to evaluate the impact of different volute parameters like volute width, tongue angle, throat area and radial location of the volute on the pressure distribution and pressure loss occurring inside the volute of a centrifugal blower. The geometric parameters are varied from their nominal values and numerical simulation is carried out, by validated numerical model, to evaluate the sensitivity of these input parameters on the performance output variables. Later on, two parameter sensitivities are also investigated and their results are discussed. The results of sensitivity analysis can help to identify the geometric parameters which have a major impact in deciding the performance of centrifugal turbomachines. This study provides a base for selecting the input parameters for the optimisatio of volute geometry

Keywords- Centrifugal blower volute, Sensitivity analysis, Finite difference method. I. INTRODUCTION Centrifugal fans and blowers are widely used for supplying air for the combustion chamber in industrial boilers, heating, ventilation and air conditioning systems and many domestic appliances. Such increasing demand has necessitated the study of flow phenomenon occurring inside fans and blowers to improve their performance in respective fields. The flow in a centrifugal fan is complex and three dimensional with turbulence and boundary layer separation. The geometry is complex and the geometric parameters affect the flow pattern and in turn the performance of these machines. The design of fans and blowers is well established in literature and optimisation of various impeller parameters like the blade angles, blade shape etc. has been carried out to meet the requirements in particular fields. Computational fluid dynamics (CFD) has proved to be an effective and reliable way to understand the flow behavior and the effect of various parameters on the performance of fans and blowers. A number of authors have studied the effects of different components of fans and blowers on the performance and noise generation by analyzing the flow through CFD simulations. Experimental analysis is also carried out to check the effect of volute design on the performance of a turbocharged compressor. Numerical simulation is carried out to understand the impeller-diffuser interaction in radial diffuser pumps and it was concluded that the flow rate, blade number and the radial gap significantly affect the pressure fluctuations and the associated unsteady flow. To enhance the performance of turbomachines, optimisation of various geometric parameters of impeller has been carried out and reported in literature.

Reference shows the application of multi-objective robust design exploration method (MORDE) on centrifugal fans for its optimisation. They tried to determine the design variables which have the minimum turbulent noise level and the maximum efficiency. In reference, a GMDH type neural networks is used to optimise the geometric parameters of the impeller blade. Finally, a pareto front is established and four points containing optimum head loss and head rise have been defined. As can be seen from the forgoing paragraph, much of the published worked has been aimed at understanding the flow phenomenon and interaction of the fluid with the physical elements of turbomachines. The results of such studies have been utilised to meet the needs of industrial applications. However, the quantitative effect of individual parameters of the volute casing on the performance of such machines is not investigated. Thus, the present work is aimed at analyzing the effect of individual parameters of the volute casing on the performance of a centrifugal blower. Such analysis can help identify the prominent parameters which affect the outcome to a much higher degree as compared to other parameters. In turn, the optimisation of these parameters can bring about a significant improvement in performance. The present paper analyses the sensitivity of volute parameters on the performance variables such as pressure distribution across the impeller, stagnation pressure at impeller exit and the pressure loss occurring inside the volute. The sensitivity analysis of these parameters is carried out using finite difference (FD) method, which is a gradient based method to estimate the impact of input parameters and thus identify the parameters that have maximum impact on the selected performance vaiables.

Table 1:Specifications of centrifugal blower



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Impeller specifications No. of

blades 12 Outer width 13 cm

Inner diameter 30 cm Inlet bade

angle 20˚

Outer diameter 42 cm Outlet blade

angle 48˚

Inner width 17 cm Thickness of

blade 0.5 cm

Volute specifications Width

(b) 26 cm volute

radius(r ) at: =0˚

23.38 cm

Tongue radius

(r ) 24.60 cm

=90˚ 30 cm

Tongue

angle(θ ) 18.5˚

=180˚ 38.5 cm

Throat area (A)

1647.6 cm2

=270˚ 49.4 cm

Throat length

40 cm =360˚ 63.4 cm

II. NUMERICAL ANALYSIS The numerical analysis of the single stage centrifugal blower is carried out using the commercial CFD package GAMBIT (v2.4.6) and FLUENT (v6.3.26). The geometry construction and grid generation is carried out in GAMBIT, while the flow simulation is carried out in FUENT. FLUENT solves the Navier-Stokes equation using the finite volume method (FVM), which has been applied widely in fluid mechanics and engineering applications. A. Geometric model

The geometry under consideration is a single stage backward curved centrifugal blower. It comprises of the inlet duct, the impeller and the volute. The impeller has backward inclined blades. The inlet duct has a diameter of 30 cm and length of 50 cm. The details of the impeller and the volute are presented in Table 1. The computational domain is showed in Fig. 1.

Fig. 1 The computational domain

B. Grid generation ad solution parameters The geometry of the fluid domain is created as per the data give in table 1, and the meshing is done by step by step method, i.e. creating vertexes, edges, faces and at last volumes. A structured meshing scheme is used for the geometry. The inlet and outlet face of suction duct are meshed by Quad/ Tri wedge primitive meshing scheme. The volute is meshed using a combination of map and sub-map quadratic scheme. The impeller is meshed by Tet/hybrid T-grid scheme. At the inlet boundary, a uniform axial velocity (along the Z- axis) is imposed based on the specified mass flow rate based on the design condition. The outlet boundary is specified as pressure outlet. The impeller rotates at a speed of 2900 rpm. Mixing plane model is used to replicate the stator rotor interactions. The turbulence modelling is done by using the realizable k-ε model. The SIMPLE numerical algorithm is use

Fig. 2 Meshed view of fluid domain

for pressure-velocity coupling. The pressure parameter is discretised using the standard scheme, while the rest of the parameters are discretised using second-order upwind scheme. The convergence criterion is set to 10-4 for all the residuals.This numerical simulation of the single stage centrifugal blower is already validated with experimental data III. SENSITIVITY ANALYSIS Sensitivity analysis studies the “sensitivity” of the outputs of a system to changes in the parameters, inputs or initial conditions, which are often poorly known. Sensitivity analysis is an important tool for decision making in engineering practice and a necessary step in optimisation procedures and reliability analysis. Sensitivity analysis can be divided into two categories: local and global sensitivity analysis. Local sensitivity analysis techniques evaluate thepartial derivative at one specific parameter value, also called the nominal parameter value. On the



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other hand global techniques evaluate this partial derivative in various points of the parametric domain. Various techniques for local sensitivity analysis are (a) the finite difference method, (b) the direct differential method, (c) the Green’s function method, (d) the polynomial approximation method and (e) automatic differentiation. All above techniques except for the finite difference technique have one thing in common. They all require complex manipulations of the model equations. In many studies this is not practically feasible because the models are too complicated or the model equations are not directly accessible (e.g. because they are compiled in executable commercial code). This is the basic reason why the finite difference method, although inefficient, is still used very often.

A. The finite difference method The finite difference technique is often used to obtain a sensitivity measure, most of the time without considering the nonlinearity of the model or the round-off error introduced by the output calculations. It takes a black box approach to the simulation system to estimate the gradients. The output variables can be shown as a function of input parameters.

f = (f(θ ,θ , … ,θ )) (1)

Where f is the output variable written as a function of θ , i = 1,2, . . , n the input parameters. The partial derivative of f with respect to a particular input parameter gives the sensitivity (S (θ)) of f with respect to that parameter

S (θ) = ∂f/ ∂θ (2)

The derivative can be mathematically formulated by the equation given below (central difference).

S (θ) ≈ lim∆ → {[f(θ + ∆θ ) − f(θ − ∆θ )/2∆θ ]}

(3)

It should be noted that only one parameter is perturbed at a time while all others are kept at their nominal value. The choice of this perturbation will determine the quality of the sensitivity function. . If the perturbation is taken too small it will result in numerical inaccuracies. On the other hand, ∆θ should not become too large because then the nonlinearity of the model will start to play an important role in the sensitivity calculations. If there are p number of input parameters, the central difference formula requires 2p solutions. Similarly the two parameter sensitivity can be calculated using (4)

S (θ) = (f ′ − f ′ − f ′ + f ′ )/ 4∆θ ∆θ (4) Where;

f ′ = f θ − ∆θ ,θ − ∆θ , f ′

= f θ − ∆θ ,θ + ∆θ , f ′ = f θ + ∆θ ,θ − ∆θ , f ′

= f θ + ∆θ ,θ + ∆θ

Authors in [6] have studied the effect of volute tongue on centrifugal fan performance and have shown that modifying the volute design led to an improvement in performance. Research carried out in [9] has showed that the width of the volute has a major roll on determining the construction of flow in the volute and the building of the pressure gradient and the loss level. Thus in the present study, the width of the volute, tongue angle, throat area and the radiallocation of volute are considered as the input parameters for sensitivity analysis. The output variables include the total pressure at impeller outlet, percentage variation of static pressure at impeller outlet and the total pressure loss occurring inside the volute. The perturbation is set different for each input parameter. The details of the input parameters are presented in Table 2. IV. RESULTS AND DISCUSSION The sensitivity analysis is carried out with the help of validated numerical results of the fluid domain. Total four input parameters with their perturbation selected. Thus, total eight simulations are performed to investigate the single parameter effects on the output variables using the central difference formula. The eight cases are numerically simulated by adding and subtracting the perturbations of the input parameters from their nominal value (i.e.(θ ∓ ∆θ ) ).

Table 2:Perturbation for Input parameters Factor: Parameter ∆θ

A:Width of volute 0.075b

B:Tongue angle 0.135θ

C:Throat area 0.05A D:Radial location of volute 0.05r

Fig. 3 represents the results of the numerical simulation for the eight cases considered. A, B, C and D represents the input parameters like volute width, tongue angle, throat area and radial location of volute cross-section respectively. A bold line, parallel to X- axis in figures, represents the values of the output variable at nominal values of the input parameter. Fig. 3(a) reveals that giving a positive perturbation to the input parameters, the total pressure at the impeller exit decreases in all cases and vice versa. For factor A, the total pressure at the impeller exit shows a larger deviation from the nominal value amongst all whereas, the tongue angle shows the least deviation. The large



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deviation, in case of volute width, may be due to the sudden enlargement/decrement after the impeller. According to Fig. 3(b), the percentage pressure variations increase in both positive and negative perturbation, for all input parameters except positive perturbation in case of A and D. This suggests that any change from their nominal values is not recommended for factors B and C as it increases the static pressure variation. This increase in turn can lead to higher bearing loads. However, positive perturbation in factor A and D suggest favorable change. Fig. 3(c) shows the percentage total pressure loss occurring inside the volute. The figure reveals a greater deviation in the pressure loss with factor B i.e. tongue angle as compared to the other input parameters. This is due to the fact that the tongueangle greatly determines the flow behavior and the accompanied losses near the volute entry. This in turn determines the flow pattern in the remaining portion of the volute. For minimizing the pressure loss inside the volute, increasing the width and the tongue angle is advisable. However, increasing the throat area and the radial location of volute is not recommended for the same. To check the effects of input parameters on the output variables, the sensitivity analysis is carried out. The individual parameter sensitivities are calculated as per (3).It was observed that, the width of the volute showed a high sensitivity to all the considered output variables except pressure loss inside the volute. The tongue angle showed the highest influence on the pressure loss inside the volute of the centrifugal blower. The throat area showed the least sensitivity to all the output variables. To get a better understanding of the input parameters on the performance of the centrifugal blower, two parameter sensitivity analyses was also carried out. For two parameter sensitivity analysis, the throat area has been neglected as it showed the least effect on the output variables amongst all the considered input parameters. The results of both, the single parameter sensitivity and the two parameter sensitivity analysis, are representing in Table 3. As per single parameter analysis, the tongue angle shows a higher impact on the pressure loss inside the volute as compared to others. However, the combined effect of two parameters negates the influence of tongue angle. The results reveal that width of the volute and the radial location

Fig. 3. Effect of input parameters: a) Total pressure at impeller

outlet b) Static pressure variation at impeller outlet c) Total pressure loss inside volute

of volute, when considered together; have a prominent effect on all output performance parameters. CONCLUSION The preset paper reveals the sensitivity analysis of volute input parameters like its width, tongue angle, throat area and the radial location of volute on the performance output parameters. The outcomes of study suggest the level of sensitivity of individual and combined input parameters. One can considered suggested higher sensitive parameters of volute as control factors for the optimisation of a turbomachines for significant improvement in performance.

Table 3: Results of sensitivity analysis

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