Experimental research on similarity deviation between prototype and model pumping systems

To investigate the reasons for the flowrate deviation of the prototype pumping systems between the measured value onsite and the calculated one by the similarity law, the energy characteristic relationship of the prototype and model pumping systems under different working conditions was compared and analyzed. It was pointed out that the hydraulic loss in the outlet passages caused by the residual velocity circulation is not similar. Using the method of vessel-mounted ADCP and water level gauges to measure the flowrates and heads of pumping stations, the energy characteristics of three typical pumping stations were obtained, and the comparisons with the characteristic curves of the pumping systems predicted by similarity law are made. It was found that there was a deviation between the two in most cases, and the similarity deviation of the elbow inlet and siphon outlet passages was greater than that of the straight pipe inlet and outlet passages. Meanwhile that the larger the blade angle (or the higher the rotational speed of pump), the greater the similarity deviation. Therefore, in the design of the pumping system, the energy characteristics of the model pumping system cannot be simply converted by the similarity law to predict the characteristics of the prototype pumping system, but the influence of different factors on the similarity deviation between the prototype and model pumping system should be considered carefully.


Introduction
With the continuous development of national water diversion and city flood control projects, the construction speed of large and medium scale low-head pumping stations has been accelerated.Due to the large size of the pumping system in large and medium scale pumping stations, before the construction of the project, it is almost impossible to get the energy characteristics of the prototype pumping system by the actual measurement process.Usually, the characteristics of the prototype pump system are basically obtained by similar conversion of the characteristics measured by the model pump system test using the similarity law.For a long time, there are few studies on the similarity deviation between the prototype and model pumping system by use of the actual measurement method.By the completion of the project, the flow measurement results show that the flowrate of the prototype pumping system is compared with the conversion value of the similarity law, and there are different degrees of deviation of each pump station.The pumping stations of different types have different degrees of deviation under different working conditions.And it has certain regularity.In this paper, the ADCP flow measurement method is used to measure the characteristics of the prototype pumping systems in a pumping station group.Three typical pumping system types are selected to compare and analyze the similarity deviation rules of different prototype and model pumping systems.
2. Theoretical analysis of similarity deviation between prototype and model pumping system 2.1.Theoretical basis of model pumping system experiment The prototype and model pump segment, which meet the conditions of motion similarity, geometric similarity and dynamic similarity, the relationship between flowrate and head of the pump meet the following similarity laws when conveying the same liquid: When the size and rotational speed of the prototype pump segment and the model pump segment are not much different, the volume efficiency and hydraulic efficiency of the two similar pumps can be approximately considered to be equal when operating under similar conditions.The equation(3) and equation( 4) are established: For the same pump, it is approximately considered that the pump is still operating under similar conditions at different speeds.Equation(3) and equation( 4) can be expressed as : Equation( 5) and equation (6), also known as the proportional rate, are approximately established when there is not much change in speed.The subscript 1 represents the parameter when the pump speed is  , and the subscript 2 represents the parameter when the pump speed is  [1].

Similarity theory deviation of pumping system
The head of the pump segment generally refers to the level difference between the appropriate position in front of the water pump impeller and the appropriate position behind the guide vane.The pumping system has additional inlet and outlet flow channels comparison with the pump segment.Usually, the model pumping system is equipped with water tanks at the inlet and outlet of the flow channels, and the pressure measuring points are set in the water tanks.Obviously, the head of the model pumping system is the difference between the head of the pump segment and the hydraulic loss of the inlet and outlet channels.Therefore, at the same flowrate, the head of the model pumping system is slightly lower than that of the model pump segment.The derivation process of similarity theory is based on a series of parameter changes before and after the fluid passes through the pump impeller, which maybe not fully applicable to the system characteristics between the prototype and model pumping system.
The characteristic curves of the prototype and model pumping system are shown in Figure1.M1 and P1 are the corresponding similar working points on the characteristic curves of the model and prototype pump segments, respectively.M2 and P2 are the corresponding working points on the characteristic curves of the model and prototype pumping system, respectively.Generally, when model pumping system test, the nD values of the prototype and model pumping system are set to be equal (or approximately equal), i.e., = 1, = λ.Assuming that the volume efficiency of the prototype and model pumping system is equal, it can be known from the corresponding point relationship between the pumping system and the pump segment curve:  =  =  and  =  =  : As known as in Figure 1,  =  + ∆ , ∆H is the head loss of the inlet and outlet channels, and the hydraulic loss of the inlet channel is related to the geometry, shape and roughness of the channel.
During the normal operation of the pump, the influence of the pre-swirl of the impeller can be generally ignored [2], and the hydraulic loss of the inlet channel can be directly calculated by the loss coefficient method.Under different working conditions, the velocity distribution at the outlet of the guide vane has a great impact on the hydraulic performance of the outlet passage.The hydraulic loss of the outlet passage is not unique quadratic relationship with the flowrate [3].In summary, the formula for calculating the hydraulic loss of the inlet and outlet passage is as follows: Where,  is the sum of the local resistance coefficient and the resistance coefficient along the inlet and outlet channels,  =  +  =  +  ,, and its value is related to the geometry of the inlet and outlet channels and the roughness of the wall surface. is the cross-section areas of the inlet channel outlet.The first term in equation( 10) is the sum of the friction loss and the local losses of the inlet and outlet channels, which is proportional to the square of the flow.The second term ℎ is the loss caused by the residual circulation, which is related to the operating conditions of the pump and the dimension and shape of the outlet passage.
The formula of the prototype pumping system is the same as that of the model, and it can be derived as follows: The meaning of the left part of the equation ( −  ) is the difference between the head of the prototype pumping system calculated by the similarity law and the head of the actual prototype pumping system under similar working conditions.The first term ( −  ) on the right side is the deviation caused by the similarity of the channels of the prototype and model pumping system.It is generally believed that the local losses of the channels of the prototype and model pumping system are the same, and the friction loss of the model is slightly larger than that of the prototype due to the scale effect, i.e., ( −  ) < 0 ; and, the second term (ℎ − ℎ ) on the right side is the loss caused by the residual circulation, which is greatly related to the operating conditions and the type of the outlet passage.That is to say, there will be a certain deviation when the similarity law is used to derive the characteristic curve of the prototype pumping system from that of the model pumping system.The deviation value is related to both the hydraulic loss coefficient of the outlet passage of the pumping system and the flowrate of the operating condition, and the loss caused by the water circulation of the pumping system as well.
3. Hydraulic performance experiment of prototype pumping system onsite

Project background
There are three typical pumping stations (pumping station A, B and C ) in a large low-head pumping station group.The specific parameters of the three pumping stations are as follows: Pumping station A is equipped with 4 sets of vertical mixed flow pumps with guide vanes.The diameter of impeller is 2600mm, and the rated rotational speed is 150r/min.The blade is adjustable, and the angle adjustment range is from -6° to +2°.The pumping system has an elbow inlet channel and a siphon outlet passage, and a vacuum breaking valve is used to cut off the flow.
Pumping station B is equipped with 4 sets of shaft tubular pumps with guide vanes.The diameter of impeller is 3300 mm, and the rated rotational speed is 105.6 r/min.The blade is adjustable, and the angle adjustment range is from -6° to +2°.The shaft is front-positioned as inlet channel and the outlet passage is a straight and gradual diffusion, the quick-stop gate is used to cut off the flow.
Pumping station C is equipped with 5 sets of bulb tubular pumps.The diameter of impeller is 3050 mm.The blade angle is fixed at −2°, and the rotational speed is adjustable, and its rated speed is 107.1 r/min in power frequency.The inlet channel is straight and the bulb is back-positioned in the outlet passage, and the quick-stop gate is used to cut off the flow also.

Experimental method
The head of the pumping station is the water level difference between the upstream and downstream.The water level measurement is mainly monitored by the ultrasonic water level gauge, supplemented by manual observation.Before the experiment, the zero elevation of the water gauge is measured to ensure the accuracy and tolerance requirements, and the ultrasonic water level gauges are calibrated.The water level gauges of the pumping station are basically set at about 15m from the inlet and outlet of the pumping station to the upstream and downstream wing walls of the pumping station, as shown in Figure 2.During the measurement process, ensure that there is no dirt in front of the trash rack and safety barrier of the pumping station.The method of vessel-mounted ADCP is used for flow experiment.Before the experiment, the measured flow of the ADCP is compared with the measured by the current meters, and the result deviation is controlled within ±5%.The each downstream of the three pumping stations is all equipped with a bridge for trash-eliminating machine.In order to ensure the smooth flow of the flow measuring segment, the flow measuring segment is set at 50m downstream of the bridge, as shown in Figure2.When the experiment, the ship crossing speed is close to or slightly less than the water flow speed.By each experiment, the measured flow of the ADCP and the current meters are compared twice to ensure the accuracy of the flow measurement.
In the process of each experiment, a single flow experiment is carried out by that the condition adjustment of the pump and the smooth flow state is achieved.Three times of measurement are carried out under each working condition.The deviation between the flow value of each measurement and the average value is within ±5%.It is considered that the measurement result is effective, and the average value is taken as the flowrate under this working condition.

Analyses of measurement results
For the convenience of comparison, the flow deviation (∆) and the relative flow deviation ( ) are defined as: ∆ =  −  (12)  = ∆  ⁄ (13) Where,  represents the measured flow value of a working condition test, and  represents the theoretical flow value of the corresponding head working point.
The average value of the flow deviation and the average relative deviation of the measured operating conditions of each blade angle (or speed) of the three pumping stations are counted, as shown in Table .1.In most of the working conditions of the three pumping stations, ∆ > 0, that is, the measured flow value is smaller than the theoretical value.From Table .1, it can be discovered that the flow deviation and relative deviation of pumping station A and B with the increase of the blade angle.With the increase of rotational speed, the flow deviation of pump station C with rotational speed regulation increases, and the relative deviation is almost unchanged.
Table 1.The measured flow rate deviation and relative deviation of the three pumping stations at different blade angle (or rotational speed).In order to facilitate the comparison of relative deviation between different pumping systems, the flow measurement data of pump station A and pump station B with -2° blade angle and rated speed of pump station C are compared.
is used as abscissa ( is the theoretical maximum efficiency point flow),  is ordinate, and the relative deviation of the flow of the three pumping system is compared as shown in Figure 3.
Figure 3 Relative deviation of flowrate of three pumping stations.It can be found from Figure 3 that the relative deviation of the flow rate of the three pumping stations is smaller near the high-efficiency zone, and the relative deviation of the flowrate increases at the operating point deviating from the high-efficiency zone.In comparison, the relative deviation of flowrate of pumping station B is the smallest, followed by pumping station C, and the relative deviation of flowrate of pumping station A is the largest.
It has been revealed from the analysis in Section 2.2 that the loss caused by the residual circulation has a great relationship with the operating conditions and the type of the outlet passage.According to the analyses of the structural type of the outlet passage of the pumping system, the internal flow pattern of the outlet passage of the three types needs to gradually evolve from the inlet circular segment to the rectangular segment along the flow direction.The water flow needs to be diffused horizontally and longitudinally inside the outlet passage, while the siphon passage has three additional curved segments.There are three changes in the flow direction of the water flow inside the siphon passage, of which two changes in the flow direction are large, so the internal flow pattern of the siphon passage is complex.The outlet passage of the shaft tubular pumping system is relatively simple, with only a gradually diffusion and a circularly changing segment.In addition to the gradual expansion and the gradual change of the segment circle, there is also a bulb in the outlet passage of the bulb tubular pumping system.The outlet passage of this segment including the bulb is a gradual change from the annular segment to the circular segment, and the outlet passage of the man access hole segment under the bulb is gradually changed as well.The internal flow pattern of the bulb tubular pumping system is simpler than the siphon outlet passage and more complicated than the shaft tubular pumping system.The streamline diagrams of the outlet passages of three pumping system [4][5][6]    The internal flow pattern of the siphon outlet passage of pump station A is complex, and the relative deviation of the corresponding measured flow is also the largest, while the internal flow pattern of the outlet passage of pump station B is the simplest, and the relative deviation of the measured flow is also the smallest.That is to say, the water circulation of the pump and the structural type of the outlet passage directly affect the flow pattern inside the outlet passage, resulting in different hydraulic losses inside the outlet passage, and also affecting the similarity deviation between the prototype and model pumping system.While the specific impact of different pump systems on relative similarity deviation requires more in-depth research based on measured data.

Summary
The energy characteristic deviation of the prototype and model pumping systems is analyzed by theoretical analysis, and the measured values of the characteristics of different pumping system are compared with the characteristic curves converted by use of the similarity law according to the model pumping system tests.The following conclusions are drawn: There will be a certain deviation when the similarity law is used to derive the characteristic curve of the prototype pumping system from the characteristic curve of the model pumping system.The value is related to the hydraulic loss coefficient of the flow channel of the pumping system, the flow rate of the operating condition, and the loss caused by the residual circulation.
With the increase of blade angle, the flow deviation and relative deviation of the two pump stations with blade adjustment increase.The relative deviation of flow in pump station A is from 3.1 % to -7.9 %, and the relative deviation of flow in pump station B is from 0.4 % to -3.8 %.The relative deviation of pump station A increases rapidly with the increase of blade angle.With the increase of rotational speed, the flow deviation of pump station C with rotational speed regulation increases, and the relative deviation changes little, which is from 6.4 % to 5.4 %.
The segment and internal flow pattern of the outlet passage have a great relationship with the loss caused by the residual circulation, and affect the similarity deviation between the prototype and model pumping system.The more complex the internal flow pattern of the outlet passage is, the greater the similarity deviation is.In the design of the pumping system, the external characteristics of the model pumping system cannot be simply converted by the similarity law to predict the characteristics of the prototype pumping system.

Figure 2
Figure 2The sketch map of the water gauge and the flow measurement segment.
Figure 3 Relative deviation of flowrate of three pumping stations.It can be found from Figure3that the relative deviation of the flow rate of the three pumping stations is smaller near the high-efficiency zone, and the relative deviation of the flowrate increases at the operating point deviating from the high-efficiency zone.In comparison, the relative deviation of flowrate of pumping station B is the smallest, followed by pumping station C, and the relative deviation of flowrate of pumping station A is the largest.It has been revealed from the analysis in Section 2.2 that the loss caused by the residual circulation has a great relationship with the operating conditions and the type of the outlet passage.According to the analyses of the structural type of the outlet passage of the pumping system, the internal flow pattern of the outlet passage of the three types needs to gradually evolve from the inlet circular segment to the rectangular segment along the flow direction.The water flow needs to be diffused horizontally and longitudinally inside the outlet passage, while the siphon passage has three additional curved segments.There are three changes in the flow direction of the water flow inside the siphon passage, of which two changes in the flow direction are large, so the internal flow pattern of the siphon passage is complex.The outlet passage of the shaft tubular pumping system is relatively simple, with only a gradually diffusion and a circularly changing segment.In addition to the gradual expansion and the gradual change of the segment circle, there is also a bulb in the outlet passage of the bulb tubular pumping system.The outlet passage of this segment including the bulb is a gradual change from the annular segment to the circular segment, and the outlet passage of the man access hole segment under the bulb is gradually changed as well.The internal flow pattern of the bulb tubular pumping system is simpler than the siphon outlet passage and more complicated than the shaft tubular pumping system.The streamline diagrams of the outlet passages of three pumping system[4][5][6] are shown in Figure4.The comparison of three types of the outlet passages is shown in Table.2.

Figure 4
Figure 3 Relative deviation of flowrate of three pumping stations.It can be found from Figure3that the relative deviation of the flow rate of the three pumping stations is smaller near the high-efficiency zone, and the relative deviation of the flowrate increases at the operating point deviating from the high-efficiency zone.In comparison, the relative deviation of flowrate of pumping station B is the smallest, followed by pumping station C, and the relative deviation of flowrate of pumping station A is the largest.It has been revealed from the analysis in Section 2.2 that the loss caused by the residual circulation has a great relationship with the operating conditions and the type of the outlet passage.According to the analyses of the structural type of the outlet passage of the pumping system, the internal flow pattern of the outlet passage of the three types needs to gradually evolve from the inlet circular segment to the rectangular segment along the flow direction.The water flow needs to be diffused horizontally and longitudinally inside the outlet passage, while the siphon passage has three additional curved segments.There are three changes in the flow direction of the water flow inside the siphon passage, of which two changes in the flow direction are large, so the internal flow pattern of the siphon passage is complex.The outlet passage of the shaft tubular pumping system is relatively simple, with only a gradually diffusion and a circularly changing segment.In addition to the gradual expansion and the gradual change of the segment circle, there is also a bulb in the outlet passage of the bulb tubular pumping system.The outlet passage of this segment including the bulb is a gradual change from the annular segment to the circular segment, and the outlet passage of the man access hole segment under the bulb is gradually changed as well.The internal flow pattern of the bulb tubular pumping system is simpler than the siphon outlet passage and more complicated than the shaft tubular pumping system.The streamline diagrams of the outlet passages of three pumping system[4][5][6] are shown in Figure4.The comparison of three types of the outlet passages is shown in Table.2.

Table 2 .
Comparison of three types of the outlet passages.