Analysis of the Sustainable Cooperation between a Multi-Piped Impeller and a Concentric Casing Using Experimental Planning

Abstract

Centrifugal pumps are one of the most commonly used devices in all branches of industry. Their application area is very broad and practically related to all sectors of the economy. Such pumps are widely applied in the chemicals industry, liquid gas technology, and machine lubrication systems, among others. On the basis of available data, it can be assumed that pumps consume approximately 27% of the world’s electricity. For this reason, their efficiency has a significant impact on the energy consumption of many technological processes. This article presents a summary of the research concerning the cooperation between a multi-piped impeller and a concentric casing. A multi-piped impeller is a completely new concept of a centrifugal impeller, which is characterized by the fact that the energy of the liquid is transferred as a result of the flow of media through the internal flow channels, and also the flow of media around it. The type and design of the casing, with regard to the flow in it and around the impeller’s pipes, plays an extremely important role. Numerical analysis was used as the main research method, which was based on a rotatable experimental plan. The CFD simulation results were validated on a test stand. Finally, the mathematical model and the rules for selecting the geometric features of such a stator were proposed in order to achieve the highest possible efficiency when cooperating with the multi-piped impeller, which constitutes a scientific novelty.

1. Introduction

Centrifugal pumps are one of the most frequently used devices in all branches of industry. It is estimated that the transport of liquids in the economy consumes nearly 27% of the electricity produced in the world [1]. It is extremely difficult to design centrifugal pumps with low specific speed and at an acceptable level of efficiency due to increasing losses. For this reason, their efficiency has an important impact on energy consumption in the global economy. Sustainability of centrifugal pump components and increasing the efficiency of all pump units by only 1% would result in annual savings of EUR 27 billion (electricity consumption in the global economy is estimated at 2700 TWh per year [1]). To illustrate the scale of the phenomenon, we will use an example from the Polish energy sector. In 2017, electricity production in Poland amounted to slightly more than 165 TWh (according to the Polish Power Grid Operator), with the majority generated from coal-fired power plants: approximately 50% from hard coal and about 32% from lignite. Assuming that the electricity consumption for the self-operation of thermal power plants related to pump operation is approximately 6.2%, this translates to an energy utilization figure of around 8.1 TWh, which corresponds to CO₂ emissions at a level of 5.3 Tg (assuming 630 kg CO₂ per 1 MWh). The savings achieved by optimizing the operation of pumps in professional power plants lead to a significant reduction in the carbon footprint as well as considerable economic and environmental benefits.

Although centrifugal pumps have been well known and used for many years, there is still room to develop innovative solutions for their flow elements. An example of such an element is a multi-piped impeller, the concept and name of which were developed and patented at Wroclaw University of Science and Technology [2]. The idea of such a solution uses flow-through internal flow channels, as in a classic centrifugal impeller. Additionally, it uses the external flow around the channels, which in turn generates an additional 30% of lifting height [3,4,5].

Information regarding such impellers can be found in the publication [3], which presents the results of energy tests of such an impeller in comparison with hole impellers and also presents the principles of flow modeling in such structures. Publication [4] presents the results of comprehensive research on the influence of the geometric features of a multi-piped impeller on operating parameters.

An extremely important factor affecting the total efficiency of a pump is the cooperation between the impeller and the casing. This aspect becomes particularly important in the case of a multi-piped impeller due to the phenomena of flow around the impeller’s channels. For this reason, the casing must be selected and designed in a completely different way than for a classic centrifugal or hole impeller, as illustrated in Figure 1.

Concentric casings (Figure 2) are one of the possible types of elements for removing liquid from an impeller—mainly in single-stage pumps. They are used, for example, for the hydrotransport of liquids containing solids. The principles of constructing such elements are limited to calculating the cross-sectional area of the casing, while at the same time maintaining a constant average velocity of the liquid in the c_u2 channel (in the case of a channel without a bladeless casing). They are widely described in the literature [6,7]. Additionally, it is assumed that the ratio of the casing’s width to the impeller’s outlet width should be approximately b₃/b₂ = 1.2 ÷ 1.4, and the ratio b₃ to the casing’s height h will probably be optimal within the range b₃/h = 1.5 ÷ 2.5.

A concentric casing is less hydraulically advantageous than a collecting spiral due to the fact that it mixes liquid streams with different velocities. Paper [8] states that the efficiency of extremely low-specific speed centrifugal pumps with a concentric casing is sometimes higher than the efficiency of a unit with a spiral volute.

The disadvantage of this type of stator is the fact that liquid streams with different velocities mix with each other, causing an increase in hydraulic losses. Preliminary research by the authors of [8] also shows that a concentric casing can be a highly efficient structure that competes with a spiral liquid drainage element. Papers [9,10] present the results of experimental research concerning the geometry of a concentric casing that cooperates with semi-open and open blade impellers. The optimized geometry of the stator was characterized by having higher efficiency than spiral channels that were designed for the same discriminants of specific speed.

There is a lack of data and knowledge of how a concentric casing will cooperate with a multi-piped impeller, as well as what flow phenomena will occur (and what their intensity will be), and also how to optimally determine the dimensions of such a stator in order to ensure the highest possible efficiency of the entire pump.

The need to know and understand the working process that takes place in a concentric casing, as well as the answer to the question of how to optimally design the flow geometry of this element, constitute an important problem of pump technology. In this paper, numerical calculations were used as the main research method, and they were based on a rotatable experimental plan. The CFD simulation results were validated on a test stand. Statistical analysis of the results allowed a mathematical model that describes the energy equation of the pump in question to be proposed.

2. Research Object and Test Rig

The object of the numerical and experimental research is a model pump (operating parameters: capacity Q = 4.8 m³/h, lifting height H_u = 24.43 m, and kinematic-specific speed factor n_q = 9.54), which consists of a multi-piped impeller (Figure 3a) and a casing with a radial diffuser (Figure 3b). The liquid drainage element was designed in accordance with guidelines in the literature [6,7,8]. Experimental tests were performed in accordance with the PN—EN ISO 9906:2012 standard (Grade 1) [11]. The measurement is made from the pump’s maximum capacity to its minimum capacity, and then from its minimum capacity to its maximum capacity. Each measured quantity is sampled 10 times in order to eliminate random errors. The experimental tests were carried out on a fully automated test stand, which is shown schematically in Figure 3c.

The geometry of the impeller and its dimensions were unchanged during all the performed tests. The geometric parameters of the multi-piped impeller and the model concentric casing, which were tested on the measurement station, are presented in

Table 1. Summary of the main geometric parameters of the multi-piped impeller.

No.	Name	Symbol	Value	Unit
1	Hub diameter	dp	20	mm
2	Inlet diameter	d1	40	mm
3	Outer diameter	d2	150	mm
4	Pipe diameter	dk = b2	6	mm
5	Inlet angle	β1	50	°
6	Outlet angle	β2	80	°
7	Number of pipes	z	5	–

and

Table 2. Summary of the main geometric parameters of the concentric casing.

No.	Name	Symbol	Value	Unit
1	Channel width	b3	22.5	mm
2	Inlet diameter	d3	155	mm
3	Outer diameter	d4	180	mm
4	Inlet diameter (into the diffuser)	dt	22	mm
5	Diffuser opening angle	δmax	2	°

.

3. Numerical Modeling

In order to understand the flow phenomena that occur in a pump with a multi-piped impeller and a concentric casing, and to obtain a reliable research tool for further research work, numerical simulations of fluid dynamics were used. They were performed as non-stationary calculations using the commercial software ANSYS FLUENT 16.2 [12]. The domain moving at a given rotational speed of n = 2870 rpm was a multi-piped impeller. The Sliding Mesh Model (SMM) [12,13] was used while taking into account the following settings:

• Pressure–velocity model according to the SIMPLE scheme;
• Isothermal calculations;
• Discretization of all equations using the second-order upwind scheme;
• The criterion of convergence of continuity equations was set at 1∙10⁻⁵ (for RMS values);
• K-ω SST turbulence model (the boundary conditions of the model for clear water);
• The model made one rotation of the impeller, which was divided into 120 time steps, in turn giving 3° of rotation per time step—its value amounted to t_1s = 1.74·10⁻⁴ s, with one rotation of the impeller lasting t_1obr = 2.09·10⁻² s;
• A maximum number of 1000 iterations was assumed for one time step;
• Fluid: pure water with a density of ρ = 998.2 kg/m³, dynamic viscosity of μ = 1003 ·10⁻³ Pa·s, and at a temperature of t = 20 °C;
• The initial parameters of the numerical model were determined based on the stationary flow solution.

The discrete model of the base pump consisted of five water solids (Figure 4a). Unstructured, tetragonal meshes were used for discretization, with a densification of elements near the walls (the mapping of the boundary layer)—Figure 4b,c. The criterion of y+ < 1 was used as the criterion for evaluating the computational mesh, which is consistent with the assumptions for the k-ω SST turbulence model [13,14] for non-stationary calculations. The thickness of the first layer was Δy₁ = 0.02 mm. Detailed information regarding modeling has been included in the publications [3,4].

In order to determine the optimal mesh size in terms of accuracy and speed of calculations, GIT (Grid Independence Test) [15,16] was used, the results of which are presented in

Table 3. Main parameters of the numerical meshes and the pump parameters for the GIT test.

No.	Number of Mesh Elements	Average Skewness	Average Quality of Elements	Average Aspect Ratio	Q	Mwir	Hu
1	12,956,566	0.26	0.72	2.85	4.8	2.86	20.82
2	16,843,536	0.24	0.77	2.45	4.8	2.78	23.53
3	22,065,033	0.23	0.78	2.22	4.8	2.73	25.34
4	28,684,543	0.20	0.80	2.08	4.8	2.63	25.78
5	37,003,060	0.20	0.81	2.04	4.8	2.61	25.90
6	47,363,917	0.19	0.82	2.02	4.8	2.59	25.84

. On its basis, it can be concluded that mesh no. 4 (marked in the table in bold) is the smallest possible discrete mesh size, the number of elements of which does not affect the solution. The difference in values for the effective lifting height of the pump H_u between mesh variants no. 3 and no. 4 is δH_{u_3/4} = 1.70%, and for mesh no. 4 and mesh no. 5—δH_{u_4/5} = 0.46%. In the case of torque M_wir, the difference in values is δM_{wir_3/4} = 3.63% and δM_{wir_4/5} = 0.65%.

To obtain a reliable research tool in the form of a numerical model of a base pump with a multi-piped impeller, the discretization error was analyzed using the GCI (Gird Convergence Index) test [17,18]. The obtained values of GCI_fine²¹ = 1.39% and GCI_fine³² = 1.42% for the considered model parameters M_wir and H_u were at a very low level.

The pump’s operating parameters, obtained from the numerical calculations, were determined on the basis of the following relationships:

• Pump lifting height:

$$H_u={\displaystyle \frac{p_{out}-p_{in}}{\rho g}}$$

(1)

• where p_out is the total pressure at the outlet of the model (Figure 4—OUTLET) and p_in is the total pressure at the inlet to the model (Figure 4—INLET);
• Power on the pump’s shaft:

$$P_w=M_{wir}\mathsf{\omega }$$

(2)

• where M_wir is the total torque on the moving outer and inner walls of the multi-piped impeller;
• Computational (CFD) hydraulic efficiency:

$${\eta }_h={\displaystyle \frac{\rho gQ{H}_u}{M_{wir}\mathsf{\omega }}}$$

(3)

• Total unit efficiency:

$${\eta }_c={\eta }_h{\eta }_v{\eta }_m\left(1-{\zeta }_t\right)$$

(4)

• where η_v is the volumetric efficiency of the pump (η_v = 0.92), η_m is the mechanical efficiency of the unit (η_m = 0.96), and ζ_t is the friction loss coefficient of the rotating discs assumed based on the diagram ζ_t = f(Re) [7] (ζ_t = 0.0047).

In the presented considerations, volumetric and mechanical losses were not modeled. The assumed efficiency values of η_v and η_m in Equation (4) result from the authors’ previously conducted research [2,3,4]. They had a constant value over the entire efficiency range, which may contribute to the errors in the numerical modeling, especially in the case of the extreme values of the pump’s efficiency.

In order to determine the accuracy of the obtained numerical simulation results for the model pump with regard to the experimental test results, the numerical characteristics were compared with the experimental characteristics, as shown in Figure 5.

After analyzing the data presented in Figure 5, the following conclusions can be drawn:

• In the pump’s efficiency range of 2.2 m³/h < Q_n < 5.2 m³/h, the maximum discrepancy between the numerical and experimental results does not exceed 1.4%;
• In the case of lifting height, the difference between the numerical and experimental values at the operating point is equal to δH_{u_BEP} = 0.36%;
• For the total efficiency, the difference between the results at the operating point is equal to δη_{c._BEP} = 0.64%—assuming a volumetric and mechanical efficiency at the level of η_v = 0.92 and η_m = 0.96;
• The accuracy of matching the results of the numerical simulations of the curve of the power on the pump’s shaft to the real results, for the efficiency of Q = Q_n, is at the level of δP_{w_BEP} = 0.71%.

4. Testing the Cooperation between the Multi-Piped Impeller and the Concentric Casing

CFD numerical tests were planned in order to analyze the flow phenomena that occur during cooperation between the multi-piped impeller and the concentric casing, as well as to determine the impact of the geometric features of the stator on the pump’s operating parameters. In order to limit the number of the performed numerical simulations, experiment planning techniques were used. From the dimensional analysis [19,20,21], the relationship between the unit energy and geometric parameters of the concentric casing was obtained. To limit the number of parameters that influence the energy conversion process in a centrifugal pump with a multi-piped impeller, the following assumptions were made:

• A constant flow geometry of the multi-piped impeller;
• A constant value of the opening angle of the outlet diffuser (δ_max = 8°);
• The liquid discharge element is symmetrical in relation to the axis of the impeller’s flow channels (pipes);
• A rectangular shape of the cross-section of the stator [8].

Based on the theory of dimensional analysis [19,20,21], and by taking into account the above assumptions, a function expressing the influence of the geometric dimensions of the concentric casing on the amount of unit energy related to the outer diameter of the impeller d₂ was obtained and written in the form:

$${\displaystyle \frac{g{H}_u}{n^2{d}_2^2}}=f\left({\pi }_8,{\pi }_9\right)=f\left({\displaystyle \frac{d_4}{d_2}},{\displaystyle \frac{b_3}{d_2}}\right)=>{d_4=Z}_8, {b_3=Z}_9$$

(5)

Taking into account that the concentric casing is an extremely simple element, and that only two parameters determine its geometry—width and diameter—(Figure 6), variability in these parameters was assumed within the following limits: width b₃ = 18 ÷ 22 mm; external diameter of the casing d₄ = 160 ÷ 170 mm. In order to design the research, a rotatable experimental plan was used, which is widely applied in hydraulic machines [22,23]. Nine stator geometries were prepared and then numerically tested. The values of the geometric parameters, which were determined according to the experimental plan, are presented in

Table 4. Values of the geometric parameters according to the numerical simulation plan (abbreviations: kk1–kk9 denote subsequent geometries of the concentric casing).

N	Z8	π8	Z9	π9
	d4 (mm)	–	b3 (mm)	–
kk1	160	1.067	18	0.120
kk2	170	1.133	18	0.120
kk3	160	1.067	22	0.147
kk4	170	1.133	22	0.147
kk5	158	1.053	20	0.133
kk6	172	1.147	20	0.133
kk7	165	1.100	17.20	0.115
kk8	165	1.100	22.90	0.153
kk9	165	1.100	20	0.133

.

4.1. Results of Numerical Calculations

Figure 7 shows the results of the numerical calculations for the nine tested pump geometries, which are presented with regard to the lifting height in the function of the pump’s efficiency.

The highest value of the lifting height, which is a measure of the energy transported to the liquid, was obtained by the kk7 casing geometry with dimensions b₃ = 17.2 mm and d₄ = 165 mm (model pump b₃ = 22.5 mm and d₄ = 180 mm). As a result of reducing the flow cross-section of the concentric casing by more than 30%, the effective lifting height of the pump increased by almost 17%.

4.2. Analysis of the Results of the Experiment Plan

In order to check the influence of the pump capacity Q on the change in the unit energy Y, a statistical analysis [24,25] of the obtained results of the numerical simulations for the pump efficiency Q (shown in Figure 7) was carried out. Based on the statistical analysis of the regression function, a mathematical model of the pump with the multi-piped impeller and concentric casing was determined, and written as

$$\begin{array}{ll}Y=& g{H}_u={\displaystyle \frac{Q^2}{d_2^4}}\left(-173.64{\zeta }_{d4}-994.66{\zeta }_{b3}+274.74\right)\\ & +{\displaystyle \frac{nQ}{d_2}}\left(27.87{\zeta }_{d4}+126.31{\zeta }_{b3}-178.847\right)\\ & +n^2{d}_2^2\left(-5.199{\zeta }_{d4}-7.962{\zeta }_{b3}+13.202\right)\end{array}$$

(6)

where

ζ_d4—discriminant of the external diameter of the stator d₄/d₂ = π₈

ζ_b3—discriminant of the cross-sectional width of the stator b₃/d₂ = π₉.

For the calculated regression coefficients of the polynomial (6), the obtained coefficients of determination R² ∈ (0.802; 0.941) prove that the model fits very well. Moreover, it was shown, based on the F-Snedecor test, F ∈ (5,33, 39,43) > F_kr = 5.14), and t-student test, t ∈ (17.79; 37.01) > t_kr = 2.45), that all the variables have a significant impact on the polynomial. In engineering practice, the analytical model from Equation (6) can facilitate the design process of the concentric casings of pumps with multi-piped impellers.

4.3. Verification of the Test Results

In order to verify the results of the tests of the derived mathematical model and the results of the numerical calculations, the flow geometry of the optimal solution of the concentric casing was designed. This stator was then printed (PET-G) using FDM technology (Figure 8a,b) and analyzed on a modernized test stand (Figure 8c,d).

The results of the conducted research include the energy characteristics of the pump, which are shown in Figure 9. These characteristics were then compared with the initial characteristics of the base model.

As can be seen in Figure 9, the increase in the operating parameters of the optimized geometry is identical to the results obtained within the experimental design. The experimental studies confirm that the numerical modeling that was used to conduct the research as part of the experimental design can be treated as a reliable research tool for assessing the impact of the geometric parameters of the casing of a pump on the energy characteristics of a pump with a multi-piped impeller. Validation of the discrete model showed that the maximum error between the results of the numerical simulations and the results of the experimental measurements did not exceed 3% (in BEP the discrepancy in the results was below 1%). The total efficiency of the unit increased by over 16 percentage points (for Q = 4.8 m³/h). Power demand dropped by nearly 24%.

The accuracy of the mathematical model of the centrifugal pump with the multi-piped impeller and the optimized concentric casing in relation to the numerical and experimental calculations is shown in Figure 10.

The analysis of the lifting height characteristics (Figure 10) allows for the conclusion that the mathematical model and the performed numerical calculations allow (with satisfactory accuracy) the actual operating parameters of the pump to be approximated and the following observations to be formulated:

• The maximum discrepancy (in the case of lifting height) between the experimental results of the pump with the modernized kk7 concentric casing and the numerically modeled pump with the same casing geometry amounts to ∆H ≈ 4.5 m for Q = 6.2 m³/h. In the optimal point Q_{n_opt} = 4.8 m³/h, this deviation amounted to ∆H ≈ 0.9 m (approximately 3.2% more than for the pump from the experimental stand);
• When comparing the mathematical model of the pump with the kk7 stator and the results of the numerical CFD analysis for the same geometry of the stator, ∆H = 0.76 m (2.91% when compared to the experimental tests) for Q = 6.2 m³/h. In the optimal point Q_{n_opt} = 4.8 m³/h, this deviation is ∆H = 0.132 m (approximately 0.46% more than for the discrete pump with the kk7 stator). Within the range of 2.2 m³/h < Q_n < 5.2 m³/h, the maximum error between the results obtained from the mathematical model of the pump and the results of the pump with the kk7 concentric casing (numerically simulated) does not exceed 0.82%;
• For Q_{n_opt} = 4.8 m³/h, the difference in the lifting height between the base pump and the unit with the optimized flow geometry of the kk7 stator is ∆H ≈ 3.3 m;
• The obtained results prove that the mathematical model fits well with the results of the numerical simulations of the pump with the stator kk7, as well as the results of real tests of the modernized model pump.

From the analysis of the partial characteristics of the influence of the geometric features of the stator on the increase in unit energy Y, the following dimensional relationships can be determined:

• In the case of diameter d₄:

$$d_4=(1.053÷1.1)d_2$$

(7)

• In the case of width b₃:

$$b_3=(0.115÷0.133)d_2$$

(8)

The obtained Equations (7) and (8) can be used to correctly design stators that cooperate with the multi-piped impeller in slow-running centrifugal pumps.

4.4. Analysis of the Graphical Numerical Results

The performed non-stationary numerical calculations allow for a quantitative and qualitative assessment of the energy conversion process that takes place in the pump with the multi-piped impeller and the concentric casing. The assessment of liquid flow is best done by analyzing the vorticity distributions (Figure 11), thanks to which the mechanism of energy transfer can be assessed. Vorticity is a measure of a fluid’s tendency to rotate. Together with the Shear Strain Ratio, it can be used to assess losses that occur in the flow [10]. The vorticity distributions are shown in the cross-section of the casing on two planes: P4 passing through the axis of the tube and P5 in the plane halfway between the impeller’s tubes (Figure 1a).

Figure 12 shows a comparison of the total pressure distributions in the pump with the multi-piped impeller and the concentric casing (on three control planes).

The reduction in the flow geometry of the concentric casing results in a reduction in the value and area of vorticity in the space between the impeller’s pipes (Figure 11c,d) when compared to the base pump model. In the case of the basic variant, the liquid flowing around the tubes generates an increase in vorticity, which in turn may result in an increase in losses. In the optimal variant, the intensity of vorticity around the edges of the pipes is reduced. The change in vorticity translates into the pump’s efficiency. The reduction of the cross-sectional area of the concentric casing—in particular, its width b₃—causes a decrease in the intensity of liquid turbulence. This is especially the case in the area between the impeller’s pipes, which in turn results in an increase in the hydraulic efficiency of the pump.

Similar conclusions can be drawn from the analysis of the total pressure distribution shown in Figure 12. The reduction in the cross-section of the concentric casing influenced the equalization of pressure within the impeller’s pipes (Figure 12e). As the rotor radius increases, an increase in pressure in the concentric casing space around the tubes can be observed (Figure 12b,e). In the gaps between the outer walls of the impeller’s flow channels, and the walls of the kk7 pump’s casing, the pressure fields within the impeller were equalized in the case of the basic geometry of the concentric stator (Figure 12d,f).

5. Conclusions

A single-stage centrifugal pump with a multi-piped impeller that works with a properly selected and designed concentric casing is an interesting alternative to classic centrifugal pumps, both in terms of its operating parameters and implementation costs. Stators should be designed in order to minimize hydraulic losses, and at the same time to maximize the effect of liquid circulation. This phenomenon can also be used to process the intensification of the simultaneous mixing and pumping process of mixtures or substances.

The developed calculation formulas allow the geometric features of the concentric casing to be linked with the diameter d₂ and the achieved lifting height H_u, thanks to which they can be used in the design procedures of slow-running pumps with multi-piped impellers. The obtained mathematical model may complement the design algorithm for centrifugal pumps with multi-piped impellers. It also provides a foundation for further research for the development of hybrid centrifugal-side channel pumps.

According to the results of the numerical and experimental studies, modification of the flow geometry of the casing reduces liquid recirculation in the cross-section of the pump and significantly improves its operating parameters.

The performed numerical simulations, laboratory measurements, and statistical analyses allow the following conclusions to be formulated:

• It was confirmed that numerical modeling can be treated as a reliable research tool for assessing the impact of the geometric parameters of a casing on the energy characteristics of a pump with a multi-piped impeller. Validation of the discrete model showed that the maximum error between the results of the numerical simulations and the results of the experimental measurements did not exceed 3% (in BEP the discrepancy in the results was below 1%);
• The highest value of the lifting height of the pump with the concentric casing, with an optimal efficiency of Q_n = 4.8 m³/h, was obtained for the kk7 casing. At the nominal point, H_{u_exp_opt} = 27.65 m in relation to H_{u_exp_base} = 24.43 m for the basic concentric casing;
• The highest increase in efficiency Δη_c was also obtained for the kk7 casing; Δη_c = 37.63%—this translates into over 17 percentage points of efficiency increase;
• The ranges of the geometric features of the concentric casing in relation to the outlet diameter of the multi-piped impeller d₂, for which the best operating parameters are achieved, were determined;
• Taking into account the production advantages, a rational element for draining the liquid of a single-stage centrifugal pump with a multi-piped impeller (which operates in the range of low and extremely low values of the kinematic-specific speed factor) seems to be a concentric casing with the relative dimensions d₄ = 1.1d₂ and b₃ = 0.115d₂.