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## Influence of specific heat capacity ratio on the optimum design of the leading edge of centrifugal compressor impeller considering the presence of IGV | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Gas Processing Journal | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

دوره 9، شماره 2، مهر 2021، صفحه 103-112 اصل مقاله (1.16 M) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

نوع مقاله: Research Article | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

شناسه دیجیتال (DOI): 10.22108/gpj.2022.131687.1110 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

نویسنده | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Jafar Nejadali^{*}
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^{}Department of Mechanical Engineering, Faculty of Engineering and Technology, University of Mazandaran, Mazandaran, Iran | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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In this paper, a theoretical analysis was carried out for optimum designing of the leading edge of centrifugal compressor blades. The effect of change in specific heat capacity ratio on the optimal design of impeller blades' leading edge was investigated theoretically considering the inlet pre-whirl. It was found that with the growth in heat capacity ratio, the maximum achievable mass flow function was reduced, while the optimum blade angle at the leading edge was increased. Results showed that the maximum achievable mass flow function for γ = 1.13, was about 0.77 and occurred at a pre-whirl angle (α) of 60.3° and blade angle (β) of 48.2°. For γ = 1.4, the maximum achievable mass flow function was about 0.64 and occurred at α=59°, β=51°. For the case of γ = 1.67, the maximum mass flow function was obtained at about 0.55 and took place at α=57.9°, β=52.7°. It was found that there is a limitation for the hub to shroud radius ratio in impeller designing. The interval between hub to shroud radius is reduced by increasing the angle of inlet guide vanes. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Centrifugal compressor؛ Heat capacity ratio؛ Optimum inlet؛ Inlet guide vanes؛ Mach number | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

اصل مقاله | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

**Introduction**
Centrifugal compressors; also known as turbo-compressors belong to the dynamic type of compressors, meaning that compression is accomplished through the conversion of kinetic energy to static energy Achieving high efficiency and wide operating range in centrifugal compressors requires a considerable design effort. The complexity of the flow, especially at the inlet of the centrifugal compressor impeller imposes a challenge on the designer to reach good performance. Flow non-uniformity at the impeller inlet, which is generally known as ‘‘inlet distortion’’, has been investigated with experiments and numerical simulations [3, 4]. A zero-dimensional model was proposed by song et al. for the mass flow rate of a compressor with a distorted inlet flow field and the results were compared with 3D CFD data and experiments [5]. All centrifugal compressor designers want to achieve the highest efficiency as well as a wide operating range. To reach this goal, the inlet guide vane (IGV) is a convenient and economic option for various applications (Fig. 1) [16]. Variable inlet guide vanes are used to generate inlet swirl in the flow upstream of impellers to reestablish optimum impeller incidence for variable mass flow rate at the constant rotational speed [17].
Fig. 1: IGV in centrifugal compressor [18] IGVs for centrifugal compressors have been thoroughly studied both experimentally and numerically [16, 17, 18, 19, 20]. Since the centrifugal compressor performance is strictly linked to the inlet gas conditions, the design of the inlet of impeller requires careful consideration. In this paper, (a) the theoretical analysis was carried out for optimum designing of the leading edge of centrifugal compressor blades, and, (b) the effect of change in gas conditions on the optimal design of impeller blades leading edge considering the inlet pre-whirl is studied. Hence, considering the ideal gases, three different heat capacity ratio (γ) was applied and their effect
**Limitations of relative velocity at the centrifugal compressor inlet**
The inlet eye is a critical region in compressors and requires careful consideration at the design stage. In high-speed centrifugal compressors, it is necessary to limit the maximum relative Mach number (Eq. (1)) at the inlet region to avoid choking and obtain optimum inlet flow conditions. Thus, suitable design of the leading edge of the impeller blades is a crucial factor in centrifugal compressor design.
Since the greatest radius at the inlet is in the shroud region, the maximum relative velocity will occur at the shroud region of the impeller
For validation, the right-hand side of Eq. (2) is plotted with and for different relative Mach numbers in Fig. 2. The results were in complete agreement with the reference [21]. It is obvious that with the enhancement of the relative Mach number, the maximum mass flow will increase. With the limitation considered for inlet flow ( ), the maximum mass flow will occur at , which is the optimum relative angle at the shroud.
Fig. 2: Mass flow function versus relative flow angle at different relative Mach numbers ( , and ), (in full accordance with reference [21])
**Optimum angle for the leading edge of Impeller blades**
To achieve the optimum design for the leading edge of blades, it is necessary to consider the optimum angle for the shroud region. Conducting analysis for an arbitrary radius ( ), as indicated in Fig. 3 and from the velocity triangle, it can be written as (Eq. (3) and Eq. (4)):
Fig. 3: a) Velocity triangle at inlet considering the pre-whirl. b) meridional view of a centrifugal compressor
By definition of , and applying the equation of continuity (Eq. (5)),
The absolute and relative Mach number are related as follows (Eq. (6)),
Eventually, using the relations of perfect gases, Eq. (5) is written as Eq. (7).
For a perfect gas, , and , thus Eq. (7) is reworked to give Eq. 8:
Equation 8 demonstrates the mass flow function, based on the analysis done for an arbitrary radius, r
As mentioned before, to have an optimum design for the leading edge of impeller blades, the relative Mach number should be limited. Since the maximum relative velocity takes place near the shroud, this region should be considered the critical one. Indeed, other points of the leading edge must be designed based on the optimum design of shroud radius. Therefore, it is wise to apply for the critical relative Mach number ( ) for the shroud radius (r Since , thus, . By the definition of absolute Mach number and based on the velocity triangle at the inlet region of blades,
Substituting and in to Eq. (10), we get,
Thus, the mass flow function becomes,
Eq. (11) looks rather cumbersome and complicated. But indeed, this equation is quite useful. The effect of pre-whirl on the mass flow function can be determined by specifying the value of
**Results and discussion**
As was mentioned, the inlet configuration and inlet flow structure are known significantly to affect the centrifugal compressor performance. Centrifugal compressors are required to increase their operating range and efficiency, which are limited at low mass flow rates by the rotating stall and surge [22, 23]. In order to suppress the tendency to critical working conditions and extend the operating range at low flow rates, inlet swirl is often considered through the application of inlet guide vanes [21]. Limiting the relative Mach number to 0.9 at the inlet region to avoid choking, and considering γ =1.4, the right-hand side of Eq. (11) is plotted at different pre-whirl in Fig. 4. It can be seen that with growth in pre-whirl angle (α), the maximum mass flow function increases at first and then declines. Conversely, the optimum blade angle (β) decreases at first and then increases. Fig. 5 (a) shows that the maximum mass flow will occur at a pre-whirl angle of 58.75°. Considering the mass flow function as an objective function, the sensitivity analysis for the optimum angle of IGV is shown in Fig. 5(b). As shown, the angle of IGV is sensitive to mass flow function
Fig. 4: Change in Mass flow function versus relative angle at shroud for different angle of IGV
Fig. 5: (a)Maximum mass flow function versus angle of IGV, and (b) Sensitivity analysis for IGV angle
Examination of the specific heat capacity ratio of different gases at different temperature conditions revealed that the values are approximately in the range of 1.13 to 1.67 [24]. Therefore, the smallest and largest values with an intermediate value are considered to study the effect of specific heat capacity ratio on the optimal design of impeller blades' leading edge. Hence, three different specific heat capacity ratios (γ = 1.13, 1.4, 1.67) were applied. Gases were considered ideal and the relative Mach number was limited to 0.9. The contour plot in Fig. 6 illustrates the variation of mass flow function in terms of relative flow angle (β) and angle of IGV (α) for different specific heat capacity ratios. Results showed that the maximum achievable mass flow function for γ = 1.13, was about 0.77 and occurred at , . For γ = 1.4, the maximum achievable mass flow function was about 0.64 and occurred at , . For the case of γ = 1.67, the maximum mass flow function was obtained at about 0.55 and took place at , .
It is obvious that with the growth in heat capacity ratio, the maximum achievable mass flow function was reduced. Applying Eq. (9), the right-hand side of Eq. (11) is plotted in Fig. 7 with α = 15°, 30°, 45°, and 60°, for different specific heat capacity ratios. As mentioned, the maximum relative velocity takes place near the shroud. Thus, by specifying the optimum angle of the blade in the shroud radius (the maximum value of the Eq. (2) at ), the optimum blade angle at different radiuses can be obtained from Fig. 7.
Results showed that with growth in specific heat capacity ratio, the optimum blade angle was increased. But, the changes are not significant. Indeed, the difference was less than 8 percent.
**Conclusions**
In this paper, a theoretical investigation was applied to study the effect of change in gas conditions on the optimal design of centrifugal compressor impeller blades leading edge in the presence of IGV. Thus, considering the ideal gases, three different heat capacity ratios (γ = 1.13, 1.4, 1.67) were applied. Knowing that the maximum relative velocity will occur at the shroud region, the relative Mach number was limited to 0.9 to avoid choking. Results showed that with growth in pre-whirl angle (α), the maximum mass flow function increased at first and then declined. Conversely, the optimum blade angle (β) decreased at first and then increased. The maximum mass flow occurred at a pre-whirl angle of 59° for γ = 1.4. With the growth in heat capacity ratio, the maximum achievable mass flow function was reduced. Results showed that with growth in specific heat capacity ratio, the optimum blade angle was increased. But, the changes were not significant. The maximum achievable mass flow function for γ = 1.13, was about 0.77 and occurred at α=60.3°, β=48.2°. For γ = 1.4, the maximum achievable mass flow function was about 0.64 and occurred at α=59°, β=51°. For the case of γ = 1.67, the maximum mass flow function was obtained at about 0.55 and took place at α=57.9°, β=52.7°. It was found that there is a limitation for the hub to shroud radius ratio in impeller design. For the case of α = 30°, the hub to shroud radius should not be less than 0.33 for a specific heat capacity ratio of 1.4. This limitation was about 0.35 for γ = 1.13 and 0.32 for γ = 1.667. The interval between hub to shroud radius is reduced by increasing the pre-whirl angle.
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