Introduction

The runner is the heart of the cross flow turbine. For low head cross flow turbine, there are two important specific factors in the runner design: D/B and R₂/R₁. D is the runner outer diameter and B is the runner breadth. R₁ and R₂ are outer runner radius and inner runner radius, respectively.

To get a high efficiency, a low ratio of D/B and a high ratio of R₂/R₁ are needed. However, with a high R₂/R₁, the runner blade will suffer a lot of stress that may result in structural problems. All considered, a runner breadth of 600 mm with a ratio of R₂/R₁=0.67 is employed. As shown in Fig. 1, outer runner diameter D=2R₁=385 mm; inner runner radius R₂=129 mm.

Fig. 1 
Runner of the very low head cross flow turbine with open inlet duct

To enhance the strength of the blades in the runner, two circular side plates and a mid-plate of 5 mm thickness are used. The number of the blade is 26.

Figure 2 presents the overall view of the turbine model for CFD analysis. Open inlet duct, other than closed duct, is used to avoid blockage by small particles, e.g., wood blocks. To reduce the cost, the structure is simplified with only a nozzle to guide the water running into the runner. The schematic view of the developed cross flow turbine is shown in Fig. 3. The circumferential location starts from the runner inlet in clockwise direction. Water coming from the upstream runs into the runner by the guide of nozzle and passes through the runner twice, i.e., water flow is directed towards the centre of the wheel and then again crossing other blades before exiting. Power is transferred to the runner when the water is leaving the turbine. Moreover, particles such as wood blocks are washed away with the over flow from the outside passage.

Fig. 2 
Whole view of 2-D flow field including the turbine model for CFD analysis

Fig. 3 
Schematic view of cross flow turbine with open inlet duct

To improve the performance of the turbine, an air suction pipe is attached to the rear part of the runner in this study. The runner is divided into three parts: Stage 1, Stage 2 and Region 1. The following symbols occur in the figure:

θ:circumferential location starting from the inlet of runner;

H:water level in the draft tube;

The design parameters for the cross flow turbine are as follows: the inflow low rate is 0.25 m³/s, the head is 1.5 m and rotating speed is 132 min^-1.

Table 1 shows the studied cases for performance improvement of the cross flow turbine. Case 0 is the designed case, which has no air suction pipe included in the design. From Cases 1 to 5, the air flow rate increases step by step to check the influence of air suction. In particular, for Case 6, the air flow rate is not controlled, and only the pressure at the runner chamber determines the air flow rate. The ratio of air suction ranges from 0.21 to 1.40.

Actual efficiency can be determined by either measurement or computational simulations of the fluid passing through the turbine runner. In this study, as the velocity at the outlet is unstable, the efficiency is simply calculated based on the potential energy difference between the upstream water and downstream water. Namely

where η is the efficiency of the cross flow turbine, T is the torque at runner; ω is the rotational speed of runner rotating on its own axis; H is the turbine head.

Time serial efficiency curves of the turbine are shown in Fig. 6. To shorten the calculating time, a water level close to the bottom of the runner is set at the beginning in Case 0. The efficiency starts to increase at first, and then decreases. The final efficiency is about 0.6. Taking the final state of Case 0 as initial condition, Cases 1 to 6 are calculated to investigate the effect of air suction through the pipe behind the runner chamber.

Fig. 6 
Time serial efficiency curves versus time by different cases

According to Fig. 6, efficiencies of Cases 1 to 5 start to increase and the growth ratios get larger with the increasing of the air suction ratio, indicating that the air suction has the effect of improving turbine efficiency. Finally, the efficiency reaches the maximum point up to 0.75 in Case 4. Case 2 is also found to have the optimal air suction ratio of 0.09 with the maximum efficiency of 0.7.

However, when the opening condition is applied to the pipe inlet, which means no control of air flow rate through the air suction pipe, the efficiency curve suddenly decreases to a very low level around 0.3, which is half of that at the beginning. In other words, the excessive air suction also results in energy loss of tail water in the draft tube, which simultaneously causes the efficiency to decrease.

The distributions of water volume fraction in the fluid domain are shown in Fig. 7. A value 1 means that the covered region is only water and a value 0 means there is only air. Between 0 and 1 means is a mixture. Since variations of water volume fraction distributions with time in Cases 1 to 5 are quite similar, and Case 6 includes all the variation process of water level in the draft tube. Case 6 is chosen for description of the change of water level in the draft tube with air suction.

Fig. 7 
Distributions of water volume fraction at (a) beginning (b) maximum efficiency point and (c) final state in Case 6

From Fig. 7, it can be found that part of the water runs into the runner by the guide of nozzle, and part of the water passes over the nozzle and overflows on the water tank (in Fig. 7(a)). The air in the draft tube is rushed out with the water, forming a vacuum. Thus water level in the draft tube rises and finally the draft tube is filled with water. With constant air suction, water level gradually drops down to a certain level. When the water level drops until Stage 2, the efficiency of the cross flow turbine reaches its maximum point, as shown in Fig. 7(b). However, at the final state of Case 6, the water level in the draft tube equals to that of water tank (in Fig. 7(c)), since too much air is sucked into the draft tube, thus the efficiency decreases.

Figure 8 shows the approximate path (dash line) of jet through cross flow turbine. The water enters into the runner at point A at an angle of α, with the velocity tangent to the periphery. The absolute velocity of the water is V. u is the peripheral velocity of the runner at point A, which is known as R_1ω. w is the relative velocity. β is the angle in the forward direction of the w and u, and α is the angle between V and u.

Fig. 8 
Velocity triangle at the entrance of a water path

To calculate the torque transmitted by a blade channel, the following equation can be used [13]:

where T_c is the theory output torque, and Q_mc is the mass flow per blade channel, R₁ and R₂ are outer runner radius and inner runner radius, respectively, and ω is the angular velocity. Therefore, to get a large torque, a small α is needed. Moreover, for maximum efficiency, β should be equal to the angle of the blade. Figure 9 presents the averaged α and β at the entrance of Stage 1. It is observed that as air suction flow rate increases, α shows an increasing trend by almost 2 degrees. Case 1 shows to have the minimum α but the distribution fluctuates by circumferential angle as shown in Fig. 10. The 0 of circumferential angle ratio means the starting point at Stage 1, and the 1of that means the ending point at the Stage 1. With air suction, the averaged β at the entrance of Stage 1 increases. However, because of excess air suction, the difference between β and blade angle become larger, and energy loss in the blade channel increases.

Fig. 9 
Averaged α and β at the entrance of Stage 1 versus air suction

Fig. 10 
Distribution of α at the entrance of Stage 1 in Case 1

Figure 11 shows the distribution of β at the entrance of Stage 1 in Case 2. It is observed that the efficiency becomes maximum from t=32s onwards as presented in Fig. 6, because it has the closest angle, β, to the blade angle as observed in Fig. 11. Even though the flow angle varies by the circumferential angle, the averaged flow angle agrees well with the blade inlet angle and thus the efficiency of Case 2 is higher than the other cases.

Fig. 11 
Distribution of β at the entrance of Stage 1 in Case 2

Figure 6 shows that in Cases 1 to 5, the efficiency firstly increases to a maximum point and then decreases. Detailed examination is carried out and it is found that the distributions of water volume fraction in the runner are quite similar. As shown in Fig. 12, the runner passage is filled with water from circumferential position of 60 degree. By the end of Stage 2, however, the water volume fraction transforms from 1 to 0. And the runner passage at Region 1 is filled with air. Considering the efficiency curve in Fig. 6, the effect of air layer on the turbine efficiency can be deduced. That is, when the air layer area is small, the increasing of air suction can improve turbine efficiency and when Region 1 is filled with air, it reaches maximum efficiency. When air layer area is larger than that of best efficiency point, further increasing air suction leads to the decrease in turbine efficiency.

Fig. 12 
Water distributions in the runner passage versus circumferential location at maximum efficiency point

In order to identify the influence of air suction, the interrelationships among water level, pressure and torque distributions are compared as shown in Fig. 13. Here the change in pressure is demonstrated as

Fig. 13 
Water level, pressure and torque distributions by air suction

where ΔH is the head of the potential energy generated from the draft tube, p₀ is the atmospheric pressure and p is the local pressure of the low pressure region in the runner.

Figure 13 shows that the pressure is proportional to the water level as they present similar falling tendency when the air suction increases from Cases 1 to 5. However, the torque at the runner first increases and then decreases with different air suctions. In Case 2, where air suction ratio is 0.09, water level is close to h_BEP, and the torque reaches its maximum point. In other words, when air suction increases, the power obtained from the potential energy in the draft tube decreases. However, the power generated from the runner remains larger while the water level in the draft tube is higher Fig. 12 Water distributions in the runner passage versus circumferential location at maximum efficiency point than h_BEP. In this situation, torque at Stages 1 and 2 increases evidently as shown in Fig. 14 and the total power generated from the water increases. It is worth mentioning that Stage 1 is more sensitive to the air suction. In Case 2, β is closest to the angle of blade inlet and its efficiency is the highest. However, when the water level is below hBEP, the situation is reversed. The increase of air suction leads to the increase of β, which deviates the flow from the pressure side of runner blade. As a result, the flow in the runner passage is no longer uniform, so the torque at the runner decreases. On the other hand, increase of air suction leads to the potential energy loss at the draft tube. These two factors cause the decrease in turbine efficiency.

Fig. 14 
Torque distributions at three parts of runner

The torque distribution curve of the runner in the circumferential direction is presented in Fig. 15. At Stages 1 and 2, the magnitude of the blade torque is positive to produce output power but at Region 1 it is negative. The curve reaches the crest near the border between Stages 1 and 2. Similarly, this curve presents to have another crest near the border between Stage 2 and Region 1.

Fig. 15 
Local blade torque versus circumferential position

At Region 1 in Case 0, two torque wave troughs occur, which means there is some power loss. With air suction, torque increases at Stages 1 and 2 from Cases 1 to 3 while Case 2 presents the best performance, but it decreases from Cases 4 to 6. At Stage 2, the torque distribution curve of Case 6 presents the trend that goes to zero from the tangential angle of 180°. From Region 1, the torque curve tends to remain at zero.