Solution approach

3. Calculate the mass flux rate m^∗ then calculate equation of pressure correction to receive p′.

4. Correct velocity, pressure and mass flux rate and we receive UK^∗∗、

p^∗∗及m^∗∗。 5. Solve k equation and ε equation to receive k^∗ and ε^∗ then we calculate new

turbulent viscosity μ_t^∗.

6. Regard new value that we got as initial value, covered the step 2 again until it converges.

Chapter 4 Results and discussion for single impeller

4.1 Introduction

In this chapter, flow agitated by single disc turbine with various pitched blade angles is predicted by the Multiple Reference Frame method that is different from Shen’s model. The flow field, power number, and pumping number resulting from present work are compared with Shen’s work. The results are also compared with Ranade et al.’s experimental data when the impeller is located at the center of the vessel and the oblique angle of blades is 90 degree.

4.2 Geometry of stirred tank with single disc turbine

The geometry of stirred tank with single disc turbine is the same with Shen’s work, but the definition of Multiple Reference Frame is different. In shen’s model, the rotational frame was defined only at impeller blades swept region in radius direction and in axial direction; stationary frame was used on the other places (Fig 1.5). In present work, the rotational frame is defined encompass all the impeller and the flow surrounding it, and the other place is defined as stationary frame (Fig.1.6). The geometry of the stirred tank is shown at Fig.4.1. There are four baffles mounted vertically and equally spaced on the flat bottomed cylindrical vessel wall. The vessel is lidded and impeller rotates counterclockwise with top view. The shaft of the impeller is concentric with the axis of the vessel. The impeller has six blades on the disc; each of them equally spaced around the disc perimeter. We changed four geometrical parameters to affect the flow structure, they are oblique angle of blades α, the clearance between impeller center and vessel bottom C, the diameter of impeller D, and diameter of impeller disc d (Fig.4.2). The cylindrical coordinate system has been

used with the origin located at the center of the impeller. Under the fully developed and steady flow conditions, flow in the baffled cylindrical vessel could be divided into two similar parts. We could just simulate one of them by using the periodic boundary condition. This could decrease the computational effort required. The calculation domain and grids of straight blades are shown in Fig.4.3 and the calculation domain and grids of pitched blade turbine are shown in Fig.4.4. Periodic boundary conditions were imposed in the surface of 0 and 180.

4.2.1 Comparison with experimental data of Ranade et al.

Before comparison with Ranade et al.’s experimental data, many numbers for telling the efficiency of different stirred tanks are introduced.

(a)Power number Np: The definition of Np is:

3 5

p /

N =W ρN D

N and D are the rotational speed and diameter of the impeller respectively, W is determined to be the product of the resistance torque by the angular velocity. In experiments, the torque is directly measured. In the present numerical simulation, it is computed by summing the contributions due to (1) the pressure difference between the upstream and downstream faces of the blades, and (2) the wall shear stress on walls of shaft and impeller.

N_p =2π τ ρN / N³D⁵

τ is the torque generated by the pressure and shear force act on impeller blades when the impeller is rotating.

(b)Pumping number: the definition of pumping number is:

/ ³

Qp P

N =Q ND

Q is the flow rate of volume that fluid discharged by the impeller. N and D are as P

defined previously. In order to analyze the flow conditions that surround the impeller swept region, we calculated the volume flow rate in four directions. First is the upper part of the impeller swept region, the rest three are lower, outer (tip), and inner of the impeller swept region. The positive number of volume flow rate is defined as fluid flow out from the impeller swept region, or the negative number of volume flow rate is defined as fluid flow into the impeller swept region.

(c)Pumping efficiency: The pumping efficiency is defined by the following equation

η=N_Q /N_P

Greater pumping effectiveness means greater liquid circulation for a given power consumption. Ranade et al. have shown that the mixing time (for a given homogeneity) decrease with an increase in liquid circulation.

(d) k* and ε*: The definitions of them are * / ^{2 5}

k =

∫

v dv N D

ε =

∫

k* is the dimensionless kinetic energy of all the stirred tank, it is an index of kinetic energy of flow in stirred tank. ε* is the dimension less local energy dissipation, it is an index of energy dissipation of flow in stirred tank.

Now we start to simulate the flow that agitated by Ranade’s experimental model.

The Geometry of the vessel is the same with Ranade’s work. The stirred tank modeled in this study is standard configuration cylindrical vessels with four equal-spaced wall-mounted baffles of width d=T/10(30mm) as shown in Fig.4.2. T is the vessel diameter; the vessel height H=T=300mm. The impeller used is of diameter equal to one-third of the vessel diameter (100mm) with blade angle α equal to 90 degree. The width and length of the blades is w=D/5(20mm) and l=D/4(25mm)

respectively. The diameter of the disc is equal to 2D/3(67mm). The impeller is located halfway between the lid and the base, that is C=T/2(150mm). The diameter of the shaft Ds is 19mm. Rotation speed is 5 rps counter-clockwise.

Fig.4.5 shows the radius profiles of radial, tangential, and axial velocity. Fig.4.6 shows the location of various Z/R. The location of Z/R=0.933 is close to the vessel bottom, and the location of Z/R=0.2 is close to the lower part of the impeller. Fig.4.5a can be seen that present work underpredicts the radial velocity neat the bottom region of the vessel. The magnitude of maximum radial velocity near the bottom is around 0.05 times the tip speed in present work. But it is almost 0.1 times the tip speed in Ranade’s experimental data. The radial profiles of radial velocity is better when Z/R=0.2. But there are still some underprediction near the shaft and some overprediction near the vessel wall. The most striking difference between Shen’s data and present work occurs in the case of tangential velocity shown in Fig.4.5b. Shen’s work predicted negative tangential velocity near the symmetry axis especially at the vessel bottom and present work shows better agreement. But there are still some underpredictions near the vessel bottom. Predicted radial profiles of axial velocity at various axial locations are compared with Ranade et al.’s experimental data and Shen’s data in Fig.4.5c. The agreement can be said to be satisfactory except in the region of flow reversal. The difference between predictions and experimental data increases as the vessel bottom is approached.

The table below shows the comparison of Power number and Pumping number between present work, Ranade et al.’s experimental data, and Shen’s work (D=T/3, C=T/2, d=2D/3, α=90⁰).

Present Work Ranade’s Exp Data Shen’s Work

Power number 5.038 4.85 2.961

Pumping number 0.8429 0.75 0.7383

It shows that Shen’s work seriously underpredict Power number. Multiple Reference Frame is used in problems that contain regions of nearly-stationary flow. This occurs in the region midway between the blades and baffles in a mixing vessel. In Shen’s work, she defined the interface between rotational frame and stationary frame just next to the impeller blade tip that must not be a region of nearly-stationary flow. That may lead to inaccurate results. Following, we will re-do Shen’s work and compare the result with Shen’s data.

4.2.2 Grids test cases

Four sets of grids with different density are used to run the grids test. The total cells of them are 180000, 123360, 84480 and 46080 respectively. Fig.4.7 shows the radial profiles of radial, tangential, and axial velocity of the four kinds of grids. The table below shows the power number and pumping number of the four set of grids respectively.

46080 84480 123360 180000

Power number 4.931 5.028 5.038 5.045

Pumping number 0.813 0.831 0.843 0.849

It is seen that the velocity profile distributions of these grid are very close except the 46080 grids and the power number and pumping number are almost independent of the grids. In order to raise the accuracy of the result and to decrease the computational effort required, we choose grid of 123360 cells as the computational grid.

4.3 The effects of blade angle on flow pattern for different off-bottom clearance. (D=T/3, d=2D/3)

4.3.1 The effects of blade angle on flow patterns when C=T/2. (D=T/3, d=2D/3) We change the blade angle to observe the influence of the flow in the stirred tank when the impeller is located at the vertical center plane of the vessel. Fig. 4.8 shows the stream lines of various blade angles at the vertical plane of Φ= -30°. Φ is defined at Fig.1.5. We can see the flow field changes from axial flow to radial flow when the blade angle α changes from 47° to 48°. The blade angle that made the flow field changes is defined as critical angle. The critical angle in this case is 47°.

Fig.4.9 shows the angular profiles of Power number and Pumping number. The power number increases by the increases of the blade angle α. When the blade angle increases from 47° to 48° the Pumping number increases rapidly, but pumping number decreases rapidly, similar to the change of Power number and Pumping number at critical angle reporter by Chou [18]. This is the different from Shen’s result.

In Shan’s work, the Power number and Pumping number increase smoothly in the same condition. Then we separate the Pumping number into four pars, they are flow rate out from the outer, inner, lower, and upper side of the impeller swept region. At the lower side of the impeller swept region, the flow rate increases by the increase of blade angle α until 47°. When the blade angle changes to 48°, the flow rate of lower side decreases rapidly, and continue decreasing when the oblique angle goes bigger.

The flow rate of outer side increases rapidly when the blade angle α changes from 47°

to 48° and continue increasing by the increasing of blade angle α. That the flow rate of outer side of impeller increase rapidly and the flow rate of lower side of impeller decrease rapidly is an apparent characteristic when the flow field changes from axial flow to radial flow. We can also find the critical angle by the angular profiles of flow rate except from the vertical plane of stream line.

Fig. 4.10 shows the angular profile of power number and pumping number

compare with Shen’s work. The Power number and Pumping number of Shen’s work are generally lower then the present work, especially at high blade angle. Fig. 4.11 shows the pressure contour of α=47° and α=48° at the vertical plane of Φ=-30°. The pressure gradient below the impeller of α=47° is bigger then that of α=48° and there is a high pressure in the right down position of the stirred tank.

4.3.2 The effects of blade angle on flow patterns when C=T/3. (D=T/3, d=2D/3) Now we posit the impeller at one third height of the stirred vessel and observe the flow field. Fig.4.12 is the Stream lines of various blade angles at vertical plane of φ=

-30°. Critical angle in this case is found at 72° by the observation of Fig.4.12. The flow field changes from axial flow to radial flow when the impeller blade oblique angle changes from 72° to 73°.

Then we observe the angular profiles of power number and pumping number in Fig.4.13. The power number increases as the oblique angle increases, and it increases rapidly when the oblique angle increase from 72° to 73°. In the figure of Pumping number profile, the flow rate at lower side of the impeller increases by the increasing of oblique and reaches a maximum at about 50° of the oblique angle, then it decreases slowly by the increases of oblique angle and decreases rapidly when the oblique angle reaches critical angle. The flow rate at outer side of the impeller becomes from flowing into the impeller swept region to flowing out the impeller swept region by the increasing of oblique angle, and it increases rapidly when the oblique angle reaches the critical angle, and then keep slow increasing by the increasing of oblique angle. The flow rate through inner side of the impeller does not change a lot; it decreases slowly as the oblique angle increases. The flow rate through upper side of the impeller increases in the beginning and reaches a maximum value at about α=50°

and then decreases with the oblique angle decreases. Fig.4.14 is the angular profiles

of Power number and Pumping number compare with Shen’s work. The critical angle of Shen’s work is 63° in this condition. Power number in her case decreases rapidly when the oblique angle reaches critical angle. The quantity of Power number and Pumping number at Shen’s work is lower then present.

Fig. 4.15 shows the pressure contour at the vertical plane of α= 72° and α= 73°.

There is an area of higher pressure at the right down corner of the vessel when the oblique angle is 72°. High pressure only exist at the upper side of the impeller disc when oblique angle is 72°, but high pressure exist both side of the impeller disc when the oblique angle is 73°. There is also a place where the pressure gradient is apparent that is about one third the stirred tank height near the wall with oblique angle equal to 73°. It is also the separation point of upper circulation and lower circulation.

4.3.3 The effects of blade angle on flow patterns when C=T/4. (D=T/3, d=2D/3) By the observation of stream lines shown in Fig.4.16, we find the critical angel is 88° when the clearance between impeller and stirred tank bottom is one fourth the vessel height. Fig.4.17 shows the Power number and Pumping number of various impeller blade angles. The power number increases as the blade angle increases, and it decreases rapidly when the oblique angle increase from 88° to 89°, it is similar to the change of Power number and Pumping number at critical angle reporter by Shen [19]. In the figure of Pumping number profile shown in Fig.4.17, the flow rate at lower side of the impeller increases by the increasing of blade angle and reaches a maximum at about 55° of the impeller blade angle, then it decreases slowly by the increases of blade angle and decreases rapidly when the blade angle reaches critical angle. The flow rate at outer side of the impeller becomes from flowing in to flowing out the impeller swept region by the increasing of blade angle, and it increases rapidly when the blade angle reaches the critical angle, and then keeps slow increasing by

the increasing of blade angle.

Fig.4.18 shows the angular profile of Power number and Pumping number compare with Shen’s work. The Power numbers and pumping numbers of both of them increase by the increasing of blade angle before the critical angle. The Power numbers of both present and Shen’s work decrease rapidly and the Pumping numbers of them increase rapidly when the blade angle reaches critical angle. Fig.4.19 shows the pressure contour at the vertical plane of α= 88° and α= 89°. There is a place where the pressure gradient is apparent that is about one fourth the stirred tank height near the wall with oblique angle equal to 89°. It is also the separation point of upper circulation and lower circulation.

4.3.4 The effects of pitched blade disc turbine position on flow patterns.

The impeller that of diameter equal to D/3 is posited in three different distances from the vessel bottom, they are T/2, T/3, and T/4 respectively. The critical angles of these three conditions are 47°, 72°, and 88° respectively. The less the clearance from impeller to vessel bottom is, the bigger the critical angle is. Fig.4.20 shows the flow field changes from axial flow to vertical flow. The flow field is easier to become axial flow when the clearance decreased. The flow fields of blade angle equal to 70° when the impeller is posited at three different locations are presented in Fig. 4.21. We can observe that when the blade angle is constant, the change of the clearance from vessel bottom to impeller will influence the flow field. The less the clearance is, the stronger the axial flow will be. This might because when the impeller is closer to the bottom, the fluid volume below the impeller is smaller. At the same impeller rotational speed, the fluid velocity of smaller volume will be faster and small blade oblique angle can produce large axial velocity component, this may be the reason why the flow field is easy to become axial flow when the clearance is small.

The power number increase rapidly when the oblique angle reaches the critical angle in the condition of C=T/2. This was happened in the flow field prediction of pitched blade impellers presented by Chou [18]. In the other hand, the power number decrease rapidly when the blade angle reaches the critical angle when C=T/4. This might because the effect of the impeller disc. When the flow field is axial flow, the flow below or above of the disc is baffled by the disc. The disc makes the fluid flow toward radial direction but the impeller blade force them flow down to form the axial flow. This may be the reason why the power number decreases rapidly when the flow field changed from axial flow to radial flow, because the impeller blades don’t have to change flow direction in radial flow condition. When the impeller is located at C=T/2 with disc=2D/3, the effort of disc is not large enough, the power number increase rapidly just like the flow field that agitated by pitched blade turbine changes from axial flow to radial flow.

4.4 The effects of blade angle on flow pattern for different off-bottom clearance. (D=T/3, d=3D/4)

4.4.1 The effects of blade angle on flow patterns when C=T/2. (D=T/3, d=3D/4) The stream lines of various blade angles at the vertical plane of Φ= -30 is reported in Fig.4.22. By the change of stream lines, we can tell the critical angle is 37°.

The power number, pumping number and pumping efficiency of radial profile are shown in Fig.4.23. The power number increases smoothly when the impeller blade angle reaches critical angle, so does the pumping number.

Predictions of Power number and Pumping number of this case are compared with Shen’s result in Fig.4.24. Overall Pumping numbers of present work are higher then Shen’s, especially at high impeller blade angle. The Pumping numbers reported

in present work are also higher then Shen’s.

Fig. 4.25 shows the pressure contour of α=37° and α=38° at the vertical plane of Φ=-30. High pressure only exist at the upper side of the impeller disc when blade angle is 37°, this may cause the higher pressure gradient on impeller blades and make the power number larger then that of oblique angle equal to 38°

4.4.2 The effects of blade angle on flow patterns when C=T/3. (D=T/3, d=3D/4)

4.4.2 The effects of blade angle on flow patterns when C=T/3. (D=T/3, d=3D/4)