Mao Zhongyu

Zhengwei Wang

Structural Characteristic in Prototype Runner of Francis Turbine Analysis

Keywords:

Fatigue and cracks at the blade of Francis runner seriously affect the safe operation of the power stations. It is important to analyze the structural characteristic in order to avoid fatigue and cracks of runner and ensure the safe operation of power stations. Since there are always filled with water in clearance around the runner structure, the structural characteristic of runner would be changed. The whole flow passage including the spiral case, stay vane, guide vane, runner with clearance around it and draft tube was established. The modal of runner in air, flow passage and flow passage with clearance was analyzed. The static stress characteristic of runner based on one-way fluid-solid coupling was calculated. The results show that the natural frequency of runner would be reduced due to the effect of clearance. Flow field calculation with clearance has little impact on the statics stress of runner while the water pressure load at the surface of clearance would change the location of the maximum stress point and reduce the static stress of runner.

INTRODUCTION

The hydraulic stability of Francis turbines is very important for safe operation of the station. However, cracks in the runner blades of Francis turbine often threaten the safety, stability and economic profits of the power station. The combined residual stress, static stress, and dynamic stress on the runner blade are thought to be the primary causes of cracks and fatigue failure, since the runner were designed without proper consideration on its dynamic behavior [1,2] . Therefore, an accurate understanding of the structure characteristic of runner such as natural modal and static stress, especially when it is submerged in water, is of most importance.

As the runner is submerged in water in actual operation, the fluid added mass would influence the natural frequencies. Rodriguez and Egusquizaa et al. [START_REF] Rodriguez | Experimental investigation of added mass effects on a Francis turbine runner in still water[J][END_REF] found that the same mode-shapes obtained in air were obtained in water but with lower natural frequencies in water via experiment investigation. And the difference in the natural frequencies is shown to be dependent basically on the added mass effect of the water. Then both experimental test and numerical analysis has discovered that the natural frequencies in water are different from that in air the added mass effect of surrounding water and the result of numerical agree well with experiment [4,5,6,7] . Due to the deformations of runner is very small, the analysis of static stress in runner has been handled as oneway fluid-structural interaction problem. Namely the pressure load is calculated by whole passage flow analysis that is ignored the structural deformation. Many researchers have engaged in calculated static stress in Francis runner caused by hydraulic force. R. Negru et al. [START_REF] Negru | Computation of stress distribution in a Francis turbine runner induced by fluid flow[END_REF] analyzed the static stress distribution at constant head and seven variable discharge and discovered the static stress values change nonlinearly with the dimensionless discharge. XIAO Ruofu et al. [START_REF] Xiao | Dynamic stresses in a Francis turbine runner based on fluid-structure interaction analysis[END_REF] found that the maximum static stresses are in general related with the turbine power for both low and high heads. R.A. Saeed et al. [START_REF] Saeed | Modelling of flowinduced stresses in a Francis turbine runner[END_REF] found that the stresses in the trailing edge of the runner blade near the crown reach a critical state in all operating points.

Both modal and static stress analysis in Francis runner doesn't take account into the effect of clearance flow, which is between crown and runner chamber or between band and chamber. However, the clearance flow would have obviously effect on the modal and static stress of runner.

This paper thoroughly researches the dynamic behavior of prototype runner of Francis turbine. The natural frequencies and modal in air, flow passage and flow passage with clearance were analyzed in detail. Then the one-way FSI method was used to calculate the static stresses in the Francis turbine runner in different conditions. And the influence of clearance flow on the static stress was analyzed and compared.

METHODS

The model is based on a Francis turbine runner. Table 1 shows the defining parameters for the runner. In this research, the whole model of flow passage was established, including the clearance in crown band (Fig 1). The surface pressure load on fluid-solid interface was calculated by whole flow passage analysis to obtain accurate fluid field calculation. The modal and static stresses were calculated by only the runner domain including the structure and fluid. The meshes of runner for the structure domain and fluid domain were generated together to ensure the same nodes distribution at the fluid-solid interface for accurate transmission of the water pressure load. As the local stress concentration often occur at the blade root [START_REF] Saeed | Simplified model of the turbine runner blade[END_REF] , the fillet of blade with the runner crown and runner band have been accurately modeled. And this part of mash was refined to avoid stress concentration due to mesh.

RESULTS AND DISCUSSION 2.1 Modal analysis and results

The natural frequencies and modal in air, flow passage and flow passage with clearance was calculated.

Fig 3 shows the calculated results typical mode shapes of the runner which are 0ND(U),0ND(Z),1ND and 2ND. The '0ND(U)' means the mode shape with 0 nodal diameter line and the impeller twists along the tangential direction. Similarly, the '0ND(Z)' means the runner vibrates along the axial direction.

The comparison of natural frequencies in air, flow passage and flow passage with clearance are summarized in Table 3 and displayed in Fig 4 . It is observed that the reduction ratio of frequency(FRR) in flow passage with clearance is larger than that in flow passage without clearance. This is due to the added mass effect of the water and this effect will increase when the water is confined in a narrow space. As the width of the narrowest point in clearance is only 2.5mm, the water around the runner move with much larger amplitude than the runner itself. In consequence, the natural frequencies reduce more in passage flow with clearance than that in only passage flow as the In contrast, the surface pressure load of clearance flow has a significant effect on the static stress of runner, seen from the comparison between the original and the third model. In terms of the maximum stress, it concentrates on quarter of the intersection of the blade and crown while that concentrated on the link between the inlet edge and the runner crown originally. Moreover, the value of the maximum stress significantly reduces since the surface pressure load of clearance flow has been considered. Similarly, the stress on the link between the inlet edge and the runner crown as well as on the outlet edge close to the runner band also reduced apparently due to the clearance.

As can be seen from the

CONCLUSIONS

Investigations of the influence of the clearance flow on modal and static stress of the Francis runner were presented in this paper. The results show that:

(1) The reduction ratio of the natural frequency in flow passage with clearance is larger than that in flow passage without clearance.

(2) When the clearance flow is considered in flow field calculation, the influence of the clearance flow on the flow field might not be reflected in the static stress of runner.

(3) The surface pressure load of clearance flow would cause much change on the static stress of runner. The stress of whole runner would reduce and the maximum stress concentrates on quarter of the intersection of the blade and crown but not on the link between the inlet edge and the runner crown as original.

Figure 2 .

 2 Fig 2 shows the meshes of runner structure and the fillet part. Fig 3 shows the finite element model of runner submerged in passage flow and in passage flow with clearance. It is observed that the fluid-structure interfaces are different when the clearance is considered or not. To get a full understanding of the influence of passage flow and the clearance on the dynamic behavior of runner in different condition, this paper analyzed the structure characteristics in 4 typical conditions, which is shown in table 2. Structure Model for Francis Turbine Runner (a)without clearance (b)with clearance Figure 3. The Finite Element Model of Runner in Fluid

Figure 1 .

 1 Figure 1. Computational Model for the Flow Field Calculation for the Francis Turbine

FigFigure 3 .

 3 Typical Mode Shapes of the Runner

Figure 4 .Figure 6 .Figure 7 .

 467 Figure 4. The Comparison of Natual Frequencies in Air, Flow Passage and Flow Passage with Clearance 2.2 Static stress analysis and results In order to identify the effect of the clearance flow on the static stress of runner, 3 different calculation models in 4 typical operating conditions have been calculated. Since the stress distributions are similar in different conditions, Fig 5 only shows the stress distribution of three models in the rated condition. Fig 6 shows the comparison of the maximum stress of three models in different conditions. And since the stress concentration always occurred on the link between the inlet edge and the runner crown as well as on the outlet edge close to the runner band, the stress of these two positions of three models in different conditions is compared in Fig 7.• In the first model, the flow field is calculated without clearance and the static stress is calculated with the surface pressure load on the fluid-solid interface of the internal flow of runner. This model will be called 'case A' in the figures below. • In the second model, the flow field is calculated with clearance and the static stress is calculated with the surface pressure load on the fluid-solid interface of the internal flow of runner like the first. This model will be called 'case B' in the figures below. • In the third model, the flow field is calculated with clearance and the static stress is calculated with the surface pressure load on the fluid-solid interface of both the internal flow of

Fig 5 ,

 5 there are displacements upwards at the runner crown and downwards at the band originally. Since the runner is surrounded by the clearance flow, the flow on the top of runner causes downward forces and the flow under the runner causes upward forces. As a result, there is little axial displacement of the runner crown and band. Under the action of same torque and pressure distribution, the stress of runner blade would significantly reduce with smaller axial deformation.

Table 1 .

 1 Defining Runner Parameters

	Runner diameter [mm]	Number of blades [-]	Number of guide blades [-]	Rated head [m]	Rated speed [r/min]	Material density ρ [kg/m3]	Young's modulus E [GPa]	Poisson's ratio γ [-]
	2665	17	20	250	375	7.75	207	0.3

Table 2 .

 2 The Typical Operating Conditions Operation condition

			Rated head	Rated head	High head	High head
			Rated output	High output	Low output	High load
	Head(m) Discharge(m	3 /s)	250 34.9	250 39.6	293 14.6	293 35.4

Table 3 .

 3 The Results of Natural Frequency and FRR

		In air	In flow passage	In flow passage with clearance
	Mode shape Frequency(Hz) Frequency(Hz) FRR(%)	Frequency(Hz)	FRR(%)
	0ND(U)	212.99	177.42	83.30	166.08	77.98
	0ND(Z)	392.17	358.57	91.43	237.48	60.56
	1ND	207.53	184.16	88.74	143.56	69.18
	2ND	279.47	236.65	84.68	206.39	73.85


C Syamsundar

Dhiman Chatterjee
email: dhiman@iitm.ac.in

M Kamaraj

Combined silt and cavitation erosion resistance of nanoparticle reinforced polyurethane coatings on 16Cr-5Ni Martensitic stainless steel substrate

Keywords:

Hydropower generation faces challenge in withdrawal of clean water from sand-laden rivers. This may lead to silt erosion and which, when compounded with cavitation-induced erosion of turbine blades, may lead to unexpected shutdowns. To address this issue, we have developed novel polyurethane (PU) coatings reinforced with boron carbide (B4C) or silicon carbide (SiC) nanoparticles on 16Cr-5Ni martensitic stainless steel substrate to improve resistance to silt and cavitation erosion. To the best of our understanding, the synergy between silt and cavitation erosion behaviors cannot be simulated with the available ASTM or non-standard test rigs. To overcome such shortcoming we developed a new test facility, where the submerged slurry jet impinges upon a stationary specimen, and focused ultrasonic transducer is used for cavitation erosion studies and this facility explores the possibility of synergetic effect of silt-cavitation erosion mechanisms. An interesting finding from the results is that there is an optimum amount of nanoparticles (20 wt.% B4C and 10 wt.% SiC) with respect to silt erosion and (10 wt.% B4C and 2 wt.% SiC) with respect to cavitation erosion, where mass removal is the minimum. Comparison of pure silt, pure cavitation and combine silt-cavitation erosion characteristics obtained using this experimental test facility, demonstrate enhanced erosion in case of combined silt-cavitation studies. This has been explained in terms of surface characteristics of coatings using scanning electron microscopy (SEM) images.

INTRODUCTION

Hydropower is one of the most promising renewable sources of energy. In most of hydropower plants withdrawal of clean water from sand-laden rivers is a major challenge, specifically during the monsoon season. Due to high cost, effective sediment settling systems remove only coarser particles (>300 µm) [START_REF] Grewal | Parametric study of slurry-erosion of hydroturbine steels with and without detonation gun spray coatings using taguchi technique[END_REF]. More than 90% of silt is quartz which causes severe silt erosion [START_REF] Kaushish | Silting problems in hydropower plants. 2 nd International Conference on Silting Problems in Hydro Power Plants[END_REF]. Mann and Arya [START_REF] Mann | Abrasive and erosive wear characteristics of plasma nitriding and HVOF coatings: their application in hydro turbines[END_REF] reported that silt erosion causes a huge loss every year due to drop in efficiency and repairs.

Another problem that often plagues hydroturbine is cavitation. One of the detrimental effects of cavitation in hydraulic machinery is cavitation erosion. The violent process of cavity collapse takes place in a very short time of about few microseconds and could result in the emission of large amplitude shock waves, as demonstrated by Avellan and Farhat [START_REF] Avellan | Shock pressure generated by cavitation vortex collapse[END_REF]. A high-speed liquid micro-jet directed towards the boundary can also occur in cavities collapsing close to a solid surface [START_REF] Philipp | Cavitation erosion by single laser-produced bubbles[END_REF]. The impingement pressure caused by the cumulative collapse of bubbles was estimated in the literature to be in the order of 900 MPa [START_REF] Franc | Fundamentals of cavitation[END_REF] which can lead to local strain hardening and an eventual fatigue failure of ductile materials, or to crack propagation in brittle materials as demonstrated by Hammit [START_REF] Hammit | Cavitation erosion: the state of the art and predicting capability[END_REF] and Arndt [START_REF] Arndt | Recent advances in cavitation research[END_REF].

There are some preliminary results showing that cavitation erosion enhances in silt environment and greatly accelerate damage mechanisms due to the following reasons [START_REF] Zhang | Assessment of resistance of nonmetallic coatings to silt abrasion and cavitation erosion in a rotating disk test rig[END_REF][START_REF] Dunstn | Cavitation enhancement of silt erosion: numerical studies[END_REF]:

 The silt particles can act as nuclei for the formation of cavitating bubbles.  The eroded surface formed by solid particles could act as preferential site for cavitation to occur.  The silt erosion of rough surface is more prominent than that of smoother surface. Thus roughened surface created by cavitation could enhance the silt erosion.

Erosion due to individual effect and synergistic effect of silt and cavitation are more complex which cause serious degradation and are unsolved problem in hydraulic machines [START_REF] Truscott | Literature survey of abrasive wear in hydraulic machinery[END_REF][START_REF] Borkent | The acceleration of solid particles subjected to cavitation nucleation[END_REF]. Reduction of material damage under combined cavitation and silt erosion is therfore an important research area and forms the focus of the present work.

In Section 1 of this paper, we discuss design and development of experimental facility and the protocol that was followed. Section 2 presents results of erosion obtained under different experimental condition. It also brings out different possible mechanisms of wear and finally Section 3

summarizes the major findings.

EXPERIMENTAL METHODOLOGY

In the present investigation 16Cr-5Ni (nominal composition of wt. 0.09% C, 15.6% Cr, 5.1% Ni, 1.00% Mn, 1.00% Si, 0.50% Mo and bal. Fe) martensitic stainless steel was used as a substrate material, because of good high mechanical properties combined with high toughness [START_REF] Syamsundar | Experimental characterization of silt erosion of 16Cr-5Ni steels and prediction using artificial neural network[END_REF].

Hard metallic coatings are found to be not advisable in the long run [START_REF] Barber | The development of subsurface damage during high energy solid particle erosion of a thermally sprayed WC-Co-Cr coating[END_REF], and so in this work we have used polyurethane (PU) coating for it high elasticity and damping characteristics. In one of an earlier works [START_REF] Syamsundar | Erosion characteristics of nanoparticle reinforced polyurethane coatings on stainless steel substrate[END_REF], we have used nanoparticle-reinforced PU coating which showed improved resistance against silt erosion. Similar coating was also studied here. Surface preparation and PU coating technique are addressed elsewhere [START_REF] Syamsundar | Erosion characteristics of nanoparticle reinforced polyurethane coatings on stainless steel substrate[END_REF] and is not discussed here. The coating thickness was maintained to be around 1 mm and Table 1 shows details of different coatings performed for current studies.

Experimental facility and methodology

Figure 1 (a and b) shows a schematic of the developed experimental facility. The test facility consists of a conical vessel of 50 litres capacity made with SS 304 steel grade, in which the erodent particles (SiO2) were mixed to a required concentration and circulated by a pump (Johnsonmake).The pump directed the erodents through a tungsten carbide nozzle with an internal diameter of 6 mm, on a test specimen placed in its path (Fig. 1(a)). Flow rate was measured by magnetic flow meter (Mag-Flow -7501E). We also used focused ultrasonic transducer (FUT) of maximum power of 250 W, with center frequency of 20 kHz,(Concord Electroceramic Industries) to produce required acoustic field (Fig. 1(b)).

All tests were conducted by using tap water. Dissolved oxygen levels were measured using a LDO101 DO probe (Hach-make) and found to be 5.6 ppm at typical water temperature around 30°C. For all experiments, temperature was found to be ~30°C.

Before and after each test, each specimen was rinsed in tap water, dried in a jet of cold air. The specimens were weighed on an analytical balance with a precision of 0.1 mg and weight loss was calculated at different time interval and bulk fluid temperature was monitored by using a thermocouple. To characterize and study the morphology of the eroded surfaces and to understand the mechanism of material removal, worn surfaces were observed using a scanning electron microscope (SEM), and erosion wear mechanisms were discussed.

Experimental procedure of silt erosion tests

Exhaustive literature review on erosion behavior reveal that impact velocity, impingement angle, average erodent size and slurry concentration mainly influence the erosion rate [START_REF] Syamsundar | Experimental characterization of silt erosion of 16Cr-5Ni steels and prediction using artificial neural network[END_REF][START_REF] Syamsundar | Erosion characteristics of nanoparticle reinforced polyurethane coatings on stainless steel substrate[END_REF]. In the present work, the experiments were carried out on PU coatings reinforced with B4C and SiC nanopowders with different wt.% and subjected the test specimen (50 mm x 50 mm x 5 mm) to erodents with impact velocity of 20 m/s, impingement angle of 45°, and average erodent particle size of 305 μm of silica sand (SiO2) of irregular particles as well as slurry concentration of 60 kg/m3. These parameters were chosen as this combination represents the worst case scenario for maximum mass removal [START_REF] Syamsundar | Experimental characterization of silt erosion of 16Cr-5Ni steels and prediction using artificial neural network[END_REF][START_REF] Syamsundar | Erosion characteristics of nanoparticle reinforced polyurethane coatings on stainless steel substrate[END_REF].The angle of impingement and 50 mm stand-off distance was maintained. Fresh slurry was prepared for each test run to ensure the same erodent conditions for each specimen.

Experimental procedure of cavitation erosion

FUT was driven by analog output from a multifunction data acquisition device (National Instruments, BNC-2120) amplified by a power amplifier (Brüel&Kjaer, 2713). The transducer was characterized in a stand-alone setup to determine driving frequency and focal length, which helped us to determine the location of the target material in actual experiments. Based on the test in stand-alone setup, cavitation induced erosion studies were conducted at a frequency of 20.4 kHz and input voltage of 95 V.

Experimental procedure of synergistic studies

Specimen exposed to combined submerged slurry suspension jet and ultrasonic field produced by FUT was used to investigate combined silt and cavitation erosion characteristics and helped to evaluate the actual performance of all these coatings.

EROSION KINETICS OF COATINGS-RESULTS AND DISCUSSION

Silt erosion studies

Silt erosion studies are carried out for different nanoparticle concentrations till 75 wt% (with respect to Part A-polymer) for both B4C and SiC reinforced PU coating. With the increasing reinforcement of nanoparticles, the wear resistance (measured as the reciprocal of the wear rate) of both the coatings rose gradually, reached the maximum (20 wt.% for the B4C and 10 wt.% for SiC nanoparticle reinforced PU coating) and then declined gradually (Fig. 2).

Wear mechanism analysis

SEM observations of eroded surfaces both 16Cr-5Ni steel and nanoparticle reinforced PU coatings give important information about the material removal mechanisms in solid particle impingement (Figs. 3456). After removal of the SiC particle, there could be a layer of PU material bonded on the SiC particle. This indicated the favorable effect of good interfacial bond strength on the solid particle erosion wear performance-which helped prolong the lifetime of the SiC particle to bear solid particle erosion and protect the coating before it is removed away. However, observations at high magnification of eroded surfaces of the coatings revealed (Fig. 6 (a)) crater formation associated with microcutting.

Less surface wear with use of nanoparticle reinforced PU coating percentage shown in these Figs. 5 and6, is achieved due to the good adhesion property of the coating with the substrate when compared with coatings having higher concentration of nanoparticles [START_REF] Syamsundar | Erosion characteristics of nanoparticle reinforced polyurethane coatings on stainless steel substrate[END_REF]. When too much B4C and SiC nanoparticles is added (>20 wt% B4C and > 10 wt% SiC nanoparticles), the interface area may have been too large and particles may touch each other directly which leads to interfacial defects and cracks could prominently emerge as explained in an earlier publication [START_REF] Syamsundar | Erosion characteristics of nanoparticle reinforced polyurethane coatings on stainless steel substrate[END_REF].

To summarize the wear mechanism, the erodent silica particle has much higher hardness than PU coatings. At high velocity impact, the sharp edges of erodent plough the PU coatings and form cluster of crater formation (Fig. 4). In case of embedded B4C/SiC particles in PU coatings, during the erosion process there is a local removal of PU material from the impacted surface, results to exposure of embedded B4C/SiC particles because of high hardness, there may be damage to erodent's or less erosion to B4C/SiC particles.

Cavitation erosion studies

To find out the kinetics of erosion caused by imploding cavitation bubbles on PU coatings, an overall examination, that is changes in the rate of mass loss as a function of testing time (which is not constant with time), and eroded surface analysis helps us to describe the erosion behavior. Erosion rate of base material (16Cr-5Ni MSS) is compared with pure PU and (B4C and SiC) nanoparticle-reinforced PU coatings at different weight percent's for baseline information as shown in Fig. 7 (a andb).

After 10 hours of testing, results showed that PU coatings has superior cavitation wear resistance compared with substrate material (16Cr-5Ni steel) and it is interesting that there is an optimum nanoparticle weight percent (10 B4C and 2 SiC) to give maximum performance which declined gradually. All the results and conclusions presented herein are limited to the steady state zone of damage (after 10 hrs of cavitation exposure), where the rate of damage does not change with time. -16.The 16Cr-5Ni MSS specimen is examined after a longer period (10 hrs) of cavitation exposure, more and more cavitation bubbles implode with different amounts of energy in already deformed surface, there are different sizes of pits with large depressions, and lip formations are observed on the surface (Fig. 8 (a) and (b)).The effect of successive impacts is to flatten and to strain further the lips, creating thin platelets of highly stressed metal that are finally knocked off the surface by succeeding impacts.

Wear mechanism analysis

For pure PU coatings (Fig. 9(a) and (b)), the appearance of the whole specimen surface shows crack interaction, pit clusters and large size grooves. In general at the beginning, the deformation corresponds to the cohesion strengthening up to a maximum. A continued loading would lead to cracks begin to form. The spreading cracks because of alternating load of jets at high velocity, extend widely over the surface as shown in Fig. 9 (a).

Surface morphology of the PU coated samples with 10 and 2 wt.% B4C and SiC nanoparticle reinforcement PU coated samples respectively after 10 hours of cavitation exposure are shown in Figs. 10 and11 (a andb). The erosion damage is associated with localized material removal from the coating surface resulting mainly due to elongated pits and extensive micro cracking in the coating surface. The observed microcracks developed in the coating. However, they did not penetrate to the coating/substrate interface, no coating delamination is observed. Brittle fracture is evident form the SEM micrographs, indicated by the formation of cracks.

Silt-cavitation synergy erosion studies

An interesting finding from Sections 2.1 and 2.2 is that there is an optimum amount of nanoparticles (20 wt.% B4C and 10 wt.% SiC) with respect to silt erosion and (10 wt.% B4C and 2 wt.% SiC) with respect to cavitation erosion, where the mass removal is the minimum. Now comparison of silt, cavitation and combined silt and cavitation erosion characteristics are demonstrated here, to show the enhanced erosion rate in case of combined silt and cavitation studies.

Figure 12 (a-d) shows and compares the results of 16Cr-5Ni MSS, pure PU, 10 and 2 wt. % B4C and SiC reinforced PU coatings, for pure silt erosion, pure cavitation erosion and combined silt-cavitation erosion, which clearly shows the existence of synergy. This accelerated damage is because of silt particles causing more roughened surfaces which make these rough surfaces more liable to cavitation. In a reverse fashion, silt erosion produced more intense damage on the pitted surface produced by cavitation, since the uneven surfaces are much more prone to silt erosion than smooth ones. Also silt particles act as nuclei for the formation of cavitation bubbles.

Wear mechanism analysis

The silt-cavitation synergy erosion mechanism is probably one of the complex mechanisms to understand and explain. The paper successfully demonstrated preparation of stable PU coatings embedded with B4C and SiC nanoparticles to increase the wear resistance properties and to protect the material which is very much useful for hydroturbine applications. 2. Extent of erosion increased several times in the case of combined silt and cavitation induced case and thereby establishes the synergistic effect. 3. Considering the maximum mass loss rate, it is found that nanoparticle-reinforced PU-coated substrate performs the best. Base material shows a mass loss rate of ~22 mg/hr, pure PU has ~9 mg/hr while nanoparticle (B4C or SiC) has ~ 1.5 mg/hr. 4. The higher wear resistance of PU coatings reinforced with B4C and SiC nanoparticles compared to pure PU is attributed to the enhanced hardness due to particle reinforcement and absorb the force of the cavitation impacts and reduce the pit formations on the surface. 5. There is an optimum level of nanoparticles (20 wt% for B4C and 10 wt% for SiC) for silt erosion and (10 wt.% B4C and 2 wt% SiC reinforced particles) for very good cavitation erosion resistance, beyond which the performance deteriorates, and these coatings with appear to be virtually unchanged and provide longterm resistance compared with other coatings.

Figure 1 .

 1 Figure 1. Schematic showing facility for combined slurry jet and cavitation induced erosion studies, (a) front view of the test chamber with flow loop of slurry jet, and, (b) side view showing ultrasonic transducer and instrumentation.For purposes of brevity, an attempt has been made to analyze the SEM micrographs taken at impact area, and at trailing edge (away from the impact area). To bring out the roles played by PU and nanoparticles, comparisons are shown with base material (16Cr-5Ni steel) also.

Figure 2 .

 2 Figure 2. Content of nanoparticles vs. silt erosion wear resistance of polyurethane coatings The SEM features of the eroded surface of 16Cr-5Ni steel are shown in Figs. 3 (a) and (b). It reveals that lips and crater formation (Fig. 3(a)), and several deep ploughing marks at the trailing edge (Fig. 3(b)) are responsible for erosion. After impingement of the first few erodent particles crater and lip are found, further elongate along the surface by the subsequent impingement of erodent, leading to a

Fig 5 (

 5 Fig 5 (a) shows good wear resistance due to 20 wt.% B4C reinforced particle. Furthermore, craters and grooves are observed on the surface and there is no appreciable material removal and the surfaces of the coatings are found to be relatively smooth even on a scale of microns (Fig 5 (b)). Similar studies are conducted for different SiC wt.% and Figs. 6(a and b) shows SEM images of 10 wt.% of SiC reinforced PU coating.

Figure 6 (

 6 Figure 6(b) shows a hole formed after a SiC particle is removed from the surface. The inside surface of the hole indicates that one SiC particle wholly debonded from PU coating with the propagation of interfacial cracks.

Figure 3 (

 3 Figure 3 (a and b). Worn surface of substrate material 16Cr-5Ni MSS.

Figure 4 (

 4 Figure 4 (a and b). Worn surface of pure PU coatings.

Figure 5 (

 5 Figure 5 (a and b). Worn surface of PU coatings reinforce with 20 wt.% B4C nanoparticles.

Figure 6 (

 6 Figure 6 (a and b). Worn surface of PU coatings reinforce with 10 wt.% SiC nanoparticles.

  Cavitation eroded samples are examined by SEM for mode of erosion damage and erosive processes. For purposes of clear understanding, an attempt has been made to analyze the SEM micrographs taken at central heavily damaged region (shown as (a)) and a transition region (shown as (b)) in Figs. 8-11 and Figs. 13

Figure 7 .

 7 Figure 7. Rate of mass loss vs exposure time of PU coatings exposed to cavitation for 10 hrs. duration, (a) refer to reinforcement of B4C nanoparticles, and, (b) refer to reinforcement of SiC nanoparticles in PU coatings.

Figure 8 .

 8 Figure 8. 16Cr-5Ni MSS specimen after 10 hrs of cavitation load.

Figure 9 .

 9 Figure 9. Surface damage of PU coatings on 16Cr-5Ni MSS specimen after 10 hrs of cavitation load.

Figure 10 .

 10 Figure 10. Surface damage of PU coatings with 10 wt.% B4C particles after 10hrs of cavitation load.

Figure 11 .

 11 Figure 11. Surface damage of PU coatings with 2 wt.% SiC particles after 10hrs of cavitation load.

Figure 12 .

 12 Figure 12. Rate of erosion vs exposure time for pure silt erosion, pure cavitation and combine silt-cavitation erosion of (a) 16Cr-5 Ni MSS, (b) PU coating, (c) PU coating with 10 wt.% B4C nanoparticles and (d) PU coating with 2 wt.% SiC nanoparticle reinforced polyurethane coatings.

Figure 13 .Figure 14 .Figure 15 .Figure 16 .

 13141516 Figure 13. 16Cr-5Ni MSS specimen after 5 hrs of combined silt and cavitation load

Table 1 :

 1 Details of different experiments performed

	No. Coating	Description of sample
	1 PU-0	Pure PU coatings (0 wt.% nanopowder)
	2 PU-1 B4C	PU coatings with 1 wt. % nano-B4C powder
	3 PU-2 B4C	PU coatings with 2 wt. % nano-B4C powder
	4 PU-10 B4C	PU coatings with 10 wt. % nano-B4C powder
	5 PU-20 B4C	PU coatings with 20 wt. % nano-B4C powder
	6 PU-35 B4C	PU coatings with 35 wt. % nano-B4C powder
	7 PU-50 B4C	PU coatings with 50 wt. % nano-B4C powder
	8 PU-75 B4C	PU coatings with 75 wt. % nano-B4C powder
	9 PU-1 SiC	PU coatings with 1 wt. % nano-SiC powder
	10 PU-2 SiC	PU coatings with 2 wt. % nano-SiC powder
	11 PU-10 SiC	PU coatings with 10 wt. % nano-SiC powder
	12 PU-20 SiC	PU coatings with 20 wt. % nano-SiC powder
	13 PU-35 SiC	PU coatings with 35 wt. % nano-SiC powder
	14 PU-50 SiC	PU coatings with 50 wt. % nano-SiC powder
	15 PU-75 SiC	PU coatings with 75 wt. % nano-SiC powder

ACKNOWLEDGMENTS

The PU material Irathane 155 and intercoat adhesive coat system Irabond UU55/52A used here was provided by M/s IrathaneFutura-A Division of ITW Limited, UK and this help is gratefully acknowledged.


Bo Hu

Friedrich-Karl Benra

Weigang Lu

Hans Josef Dohmen

Investigation on solid-liquid flow in the rear chamber of multi-stage slurry pumps with leakage flow

Keywords:

Submersible multi-stage slurry pumps are widely used in mining industry. A sand discharge groove is designed to accomplish sand exclusion. The leakage flow through the broken balance hub is common in this kind of pumps. Whether the sand discharge groove will work when there is leakage flow is very important. The commercial RANS solver Ansys CFX 14.0 is used to predict the cavity flow in the rear chamber. The influences of leakage flow on the cavity flow are analyzed in this paper. The sand discharge capacities of the groove with and without leakage flow are also compared with each other. The simulation results show that most of the sand is discharged out of the rear chamber even if there is leakage flow. The results from the abrasion tests indicate that the simulation results are reasonable. The sand discharge groove will work even if there is leakage flow. The service life of the pump is dramatically improved.

INTRODUCTION

Flow in rotating cavity is an interesting issue.

Batchelor [START_REF] Batchelor | Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow[END_REF] noticed the presence of both the disk layer and wall layer, which are separated by a central core of fluid. Daily and Nece [START_REF] Daily | Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks[END_REF] distinguished four different flow regimes based on the circumferential Reynolds Number Re φ and the dimensionless gap width G. Djaoui et al. [START_REF] Djaoui | Mise en evidence expérimentale de la sensibilité de l'écoulement de type rotorstator aux effets de bord[END_REF] investigated the influence of peripheral conditions on the flow pattern in an open cavity without superimposed flow. Kurokawa et al. [START_REF] Kurokawa | Flow in a narrow gap along an enclosed rotating disk with through-flow[END_REF] put forward the three-layer model for flow Regime IV, which is widely used in the industry. Will et al. [START_REF] Hazra | Computation of dilute two-phase flow in a pump[END_REF]11] investigated the flow in the side chambers of a centrifugal pump. Poncet et al. [START_REF] Poncet | Turbulent Rotating Disk Flow With Inward Throughflow[END_REF] put forward a equation to calculate the values of core rotation factor, noted as β. β is the ratio of the angular velocity of fluid in the central core to that of disk. With centripetal inflow,expressed as follows:

= ⁄ -1 (1)
Bo Hu et al. [START_REF] Hu | Investigation on the solid-liquid flow in the rear chamber of a submersible multi-stage slurry pump[END_REF] investigated the solid-liquid flow in the rear chamber of a submersible multi-stage slurry pump, noted as SMSP. A sand discharge groove is designed to accomplish sand exclusion,shown in Figure 1(a). It includes two parts: an annular groove at the outer radius of stator and a spiral groove on the inner surface of the stator. The diameter and the lead of the spiral groove are 5 mm and 85 mm, respectively. The diameter of the annular groove is also 5 mm. There are four sand discharge grooves in circumferential direction. From the results of former studies [START_REF] Hu | Investigation on the solid-liquid flow in the rear chamber of a submersible multi-stage slurry pump[END_REF][START_REF] Hazra | Computation of dilute two-phase flow in a pump[END_REF] , the micro-sized sand, whose maximum length does not exceed 0. The geometries of the rear chamber with sand discharge groove are shown in the Figure 2. The sand discharge groove, however, is designed assuming that there is no leakage flow. In mining applications, however, the SMSPs start to work when there is some sand to discharge. In the rest of the time, they are switched off. Since there is large pressure difference between the rear chamber and the environment, the O-ring marked in Figure 2 is fragile. From abrasion test, when the p*, the ratio of p t to p 0, drops by 10%, the O-ring is broken. It is quite hard to find whether the O-ring in the mechanical seal is broken before a pump starts to work. Sometimes, the SMSP has to work with a broken O-ring. The leakage flow is shown in Figure 2. From Eq. ( 1), the values of β increase with the increase of leakage flow rate, noted as q. With larger tangential velocities, the micro-sized is more likely to move radial outwards. After that, the sand may enter the annular grooves and be discharged along the spiral grooves. This paper presents the flow in the rear chamber of a SMSP with (after the O-ring is broken) and without leakage flow (before the O-ring is broken) when the SMSP is started.

(a) Geometries of the Sand Discharge Groove Three-layer model (Regime IV)

NUMERICAL SIMULATION

The numerical analyses on the last stage of the SMSP are carried out with the commercial ANSYS CFX 14.0 code. From Figure 3, the flow domains include five parts: inlet, impeller, cavity with rear chamber, leakage gap, and the volute with two outlets. The height of leakage gap is measured during the abrasion test (when p* drops by 10%). The meshes are also shown in the Figure 3. The cavity with both rear chamber and spiral groove are generated with hybrid grids. The rest of domains are meshed with structured grids. The analysis type is set as transient. This study focuses on the concentration of sand in the rear chamber. If the total time is too short, the concentration of sand may not change dramatically. To shorten the simulation period, the time step should not be too small. In this study, the time duration is set as total time t=3 s. and the time step is set as 0.05 s. The transient scheme is set as second order backward Euler to improve simulation precision. Standard k-ε model is used to predict the turbulent flow. The movements of dispersed phase (sand) are predicted based on the Euler-Euler approach. All the particles in the cavities are considered to have a spherical shape. The diameters of the particles are set as 0.1 mm. The granular temperature model is set as zero equation. Schiller-Naumann drag model [13] is used to calculate the drag force. The lift force of particles is calculated with Saffman-Mei lift force model [14] .

Both turbulent dispersion force and wall lubrication force between particles and wall are ignored during the simulation. Instead of resolving boundary layers, scalable wall functions are used to model the viscous effects in the near wall regions. The basic idea behind the scalable wall-function approach is to limit the value of y*, which is used in the logarithmic formulation by the value of y + =max (y*, 11.06). 11.06 is the intersection between the logarithmic and the linear near wall profile. All the mesh points are outside the viscous sub-layer and all fine mesh inconsistencies are avoided. The temperature of slurry is set as 20 o C. Since the volute has two outlets, the flow rates through the two outlets are considered as an equal value. Based on the design requirements, the values of mass flow rate at inlet, pressure at outlet and solid volume fraction are set as 5.4 kg/s, 2.6 MPa and 10%, respectively. The leakage flow rate is set as according to the results from abrasion test, obtained by a turbine flow meter (about 0.139 kg/s). The surface roughness value of all the domains is 12.5 μm. The speed of rotation interface between rotatable parts and liquid is 2900/min. The locations of monitor points are shown in Figure 4. Their positions are given in Table 1. Monitor point 1, 2 and 3 are close to the disk and the rest of the monitor points are near the wall. The dimensionless axial coordinate z and the dimensionless radial coordinate r are defined as: . In former studies [START_REF] Hu | Investigation on the solid-liquid flow in the rear chamber of a submersible multi-stage slurry pump[END_REF] , the mesh independence studies for impeller and volute have been accomplished. Hence, only the leakage gap is meshed with different grid numbers to perform mesh sensitivity analyses. The grid numbers of the rest of domains do not change. The total grid number of the leakage gap is noted as N lg . p r1 is the ratio of value of pressure at monitor point 1 to that when N lg =0.29 million. According to the results in Figure 5, the fine grid number of the leakage gap is 0.2 million.

, 𝑟 𝑏 (2)

Figure 5: Mesh Sensitivity Analysis

The history of scaled residuals of simulation with leakage flow is given in Figure 6. The residuals without leakage flow are similar. The analyses are considered to be converged. During the simulation, the values of pressure, both radial and tangential velocity at the monitor points are monitored.

RESULTS AND DISCUSION

In this section, experimental data and simulation results are presented in order to analyze the sand discharge capacity. All the velocities are made dimensionless by dividing Ω•b.

Movements of particles

In this study, only the movements of sand and contours on the cut plane, pointed out in Figure 1(a), are shown. The annular groove and the spiral groove are connected on this surface. If the sand is predicted in the groove on that surface, it is more likely to be discharged along the spiral groove. When t=3 s, the movements of particles on the meridian plane (the cut plane pointed out in Figure 1(a)) are predicted to analyze the influence of leakage flow on the sand discharge capacity, shown in Figure 7. The directions of arrows show the movements of particles. When the sand enters the leakage gap, it may rotate and remain inside when there is no leakage flow. A lot of sand enters the groove in the rear chamber with or without leakage flow. With large tangential velocity, the sand may be discharged along the spiral groove when enters the sand discharge groove. The leakage flow does not have large influence on the cavity flow. It seems that the sand discharge groove will work even if there is leakage flow.

Distribution of v r

The variations of v r at the first three seconds are shown in Figure 8. The positive and negative values represent radial outward flow and radial inward flow, respectively. The values of v r vary relative more widely near the disk (point 1 to point 3) than near the wall. The values of v r near the disk are slightly smaller in general when leakage flow occurs, as indicated in Figure 7

t (s) t (s) -0.1 t (s) t (s) v r v r 0 v r v r

Leakage gap

Leakage gap

Distribution of v θ

The simulation results of dimensionless tangential velocity, noted as v θ , at the first three seconds, are shown in Figure 9. The values of v θ vary more widely near the wall than the near the disk. The values of v θ in the cavity with leakage flow are slightly higher than those without leakage flow in general. The reason is that the radial inward superimposed flow will push the fluid, which has larger values of tangential velocity, radial inwards. With large tangential velocity, the micro-sized sand in the disk layer and the central core is more likely to move radial outwards and then enter the sand discharge groove. The results indicate that the sand discharge capacity may be better when leakage flow occurs.

Prediction of the sand discharge capacity

The The variations of φ s in the leakage flow are shown in the Figure 12. When leakage flow occurs, the value of φ s in the leakage flow is higher than the value of φ s (10%) when t=0 s. The values of φ s plunge in three second from around 20% to around 2%. This indicates that most of the sand is discharged out of the rear chamber. The sand discharge capacity of groove is better when there is leakage flow. Since the concentration of sand in the leakage flow drops, the service life of mechanical seal may be improved.

Axial thrust analysis

The variations of the axial thrust of the last stage, noted as F a , are shown in the Figure 13. Since the axial thrust of the 6-stages pump is large, the axial thrust should be carefully analyzed to make sure that the bearing system is safe when leakage flow occurs. The values of axial thrust of the last stage vary greatly from t=0 s to t=3 s. When leakage flow occurs, the values of axial thrust are much smaller, which indicate that the load on the bearing will decrease and the service life of the bearing may be longer.

TEST RESULTS

The results from abrasion test are provided in this part in order to verify the correctness of simulation results. The test rig meets the requirements of international B-grade precision (ISO9906-1999) [START_REF] Hu | Investigation on the solid-liquid flow in the rear chamber of a submersible multi-stage slurry pump[END_REF] . A pressure transmitter with the uncertainty of 0.1% measures the pressure at pump outlet to calculate the pressure head, noted as H. The flow of the pump is measured by a turbine flow meter with the uncertainty of 0.2%. The density of the slurry is 1160 kg/m 3 . The temperature of the slurry is 20℃. This test rig has circulation system at the bottom to reduce the settlement of sand. The size of the sand is selected with a screen mesh. During the test, the concentration of sand is measured every 48 hours. Appropriate sand is added to the slurry when the sand concentration decreases by more than 20%. The values of outlet pressure are recorded during the test. When the O-ring is broken, large pressure difference contributes to large leakage flow from the rear chamber (approximate 2 MPa) to the environment. It results in the plunge of pressure head, shown in Figure 14. One of the SMSP is produced with sand discharge groove. The other is used for comparison. When p* drops by 10%, the mechanical seal is considered to fail because the O-ring is broken. Two pumps are switched off and the geometries of the mechanical seal are measured. When the pumps are restarted, there are leakage flows in both of the pumps. The decrease rate of the p* in the SMSP with sand discharge groove is less than that without sand discharge grove. The results show that the sand discharge groove will work even if there is leakage flow.

CONCLUSION

The sand discharge groove, designed based on Batchelor type flow, will work even if there is leakage flow. From Poncet's equation, the leakage flow will results in greater value of β. Associated with the simulation results, the leakage flow contributes to larger tangential velocity in the central core and near the disk. The sand is more likely to move radial outward and enter the groove. The sand discharge capacity of the groove may be better when there is leakage flow. The abrasion test shows that the simulation results are reasonable and the service life of SMSP is dramatically improved.

  1 mm, moves with the conveying fluid. The flow regime in the cavity is still Regime IV, even if the flow contains the micro-sized sand. A central core separates the disk layer and the wall layer. Due to mass conservation, an axial convection of fluid from the outer radius of the disk to that of stator takes place. Although the cavity is open and the disk is not infinite, effective sand exclusion can be accomplished based on the axial convection and the three-layer model, shown in Figure 1(b).

Figure 1 .Figure 2 .

 12 Figure 1. Sand Dischage Groove and Flow Regime in the Rear Chamber

Figure 3 .

 3 Figure 3. Domains and Meshes of Simulation Model

Figure 4 .

 4 Figure 4. Arrangement of Monitor Points in the Rear Chamber

Figure 6 .

 6 Figure 6. Convergence Behavior of Simulation with Leakage Flow

Figure 7 .Figure 8 .

 78 Figure 8. Variation of the Radial Velocity with Leakage Flow

Figure 9 .Figure 10 .

 910 Figure 9. Variation of the Tangential Velocity with Leakage Flow

Figure 11 .

 11 Figure 11. Distributions of Solid Volume Fraction with Leakage Flow

Figure 12 .

 12 Figure 12. Variation of Solid Volume Fraction in the Leakage Flow

Figure 13 .Figure 14 .

 1314 Figure 13. Variations of the Axial Thrust of the Last Stage

Table 1 .

 1 Position of Monitor Points

			Cavity with sand discharge groove with sand discharge groove with sand discharge groove	Volute with two outlets
			Impeller
						Leakage gap
	Inlet
		Point 1	Point 2	Point 3	Point 4	Point 5	Point 6
	z	0.1	0.1	0.1	0.9	0.9	0.9
	r	0.88	0.92	0.96	0.88	0.92	0.96

ACKNOWLEDGEMENTS

This study is funded by CSC (China Scholarship Council).


Darrell Robinette
email: darrell.robinette@gm.com

Jason Blough

Transient Heat Transfer and Energy Balance Model for Hydrodynamic Torque Converters While Operating at Extreme Speed Ratio

Keywords:

This investigation details the development of a simple transient model to predict the bulk toroidal flow fluid temperature of the automotive torque converter based upon an energy balance and the derivation of an overall convective heat transfer correlation for the entire turbomachine. Data collected on a torque converter specific dynamometer setup is used to derive a Nusselt correlation for overall convective heat transfer from the shell using measured parameters. Dimensional analysis applied to three torque converters of nearly exact geometric similitude was used to show the applicability of the correlation to prediction of the overall external convective heat transfer mechanism. Discussion is focused on operation at low speed ratios for varying lengths of time. This particular condition represents the worst-case scenario for torque converter operating efficiency and heat rejection. It was found that the external heat rejection from the torque converter shell surface is a function of the rotational Reynolds number and proportional to the characteristic dimensionless performance number unit input speed. The model is shown to trend with temperature measurements made at the external shell of the torque converter over a variety of zero speed ratio transient maneuvers. Overall, the first order, linear differential equation model and Nusselt correlation for overall heat transfer are a first step in developing an adequate tool for use in future torque converter designs predicated on scalability of the turbomachines performance quantified through dimensionless numbers.

INTRODUCTION

The torque converter is a particular class of turbomachine used in a wide variety of power transfer applications to couple the prime mover to the gear system of an automatic transmission. Torque converters can be found in on or off highway powertrain applications and is most commonly utilized in automatic transmissions to enhance the initial acceleration performance of the vehicle. It is also a means of improving system durability since the torque converter has superior cooling characteristics and does not experience wear like that of a friction launch clutch. The modern threeelement torque converter consists of a closed loop torus containing a mixed flow pump, a mixed flow turbine and an axial flow stator. The device is required to operate over a range of speed ratios from zero to values greater than one. Speed ratio is defined as the rotation of turbine speed divided by pump speed. At zero speed ratio, maximum torque multiplication is achieved, while maximum efficiency occurs at speed ratios between 0.85 and 0.95 depending on the details of the design. As a result, the flow field is threedimensional within the torus, producing secondary flow structures and the potential for two-phase mixture, e.g. cavitation, depending on the severity of the specific operating condition. A detailed description of the torque converter and previous experimental or numerical studies can be found in [START_REF] Jandasek | Design of Single-Stage, Three Element Torque Converter[END_REF][START_REF] Abe | Development of Super Flat Torque Converter with Multi Plate Lock-up Clutch[END_REF][START_REF] Usui | Development of a Compact Ultra-Flat Torque Converter Equipped with a High-Performance Damper[END_REF] for example.

Selection of a torque converter design is performed numerically to match the torque and speed characteristics of the prime mover to a particular powertrain application to balance acceleration and efficiency for the range of operation conditions expected. The geometry of the torus and individual elements is developed through iterative CFD and other numerical methods, and then validated through physical hardware testing on a dynamometer to achieve specific performance targets, see [START_REF] Bahr | Laser Velocimeter Measurements in the Stator of a Torque Converter[END_REF][START_REF] By | Measurement and Analysis of Static Pressure Field in a Torque Converter Turbine[END_REF][START_REF] Schweitzer | Computational Fluid Dynamics in Torque Converters: Validation and Application[END_REF][START_REF] Robinette | Torque Converter Clutch Optimization: Improving Fuel Economy and Reducing Noise and Vibration[END_REF][START_REF] Robinette | Predicting the Onset of Cavitation in Automotive Torque Converters -Part II: A Generalized Model[END_REF]. To meet physical packaging constraints or to minimize the occurrence of performance inhibiting toroidal flow structures such as stalled or reversed flow and large volume cavitation, further refinement of the torus dimensions, pump and turbine blade angles and stator blade geometry may be undertaken. Additionally, the working conditions of the hydraulic fluid imposed upon the torque converter pressure vessel can be calibrated to suppress cavitation or create sufficient through flow cooling to maintain adequate toroidal flow operating temperatures.

In automotive powertrain applications, the mega trend is towards increasingly power dense prime movers with greater torque availability at lower operating speeds due to turbocharging. Simultaneously, the dimensions of the torque converter torus are being significantly reduced to accommodate increased gear and clutch content of the automatic transmission to achieve 8, 9, 10 or more forward fixed gear ratios. Additionally, more torus space is being allocated for the lockup clutch damper to bypass the hydrodynamic circuit to further reduce fuel consumption once the vehicle has launched, see Figure 1. The combination of these two design trends results in greater demands on the turbomachine, particularly for extended operation at extreme speed ratios and elevated levels of torque transfer. The purpose of this investigation is to develop a first principal based model validated with experimental data that can predict the internal toroidal flow temperature of the torque converter. The method presented is based upon that reported by [START_REF] Williams | Transmission and Torque Converter Cooling Control[END_REF] for powertrain cooling and protection strategy based upon a predicted temperature internal to the torque converter. Predictions of these temperatures can be critical in determining required cooling capacity, establishing operating limitations for component durability, preventing cavitation and preservation of oil quality from excessive thermal cycling.

TORQUE CONVERTER FUNDAMENTALS

The principals of torque converter operation and details of flow theory are contained in a number of sources found in the literature, [1, 4-8 and 10-14], for example. A brief discussion is provided here for reference and continuity of the material contained in this paper.

Performance Characteristics

Torque converter performance is typically defined by dimensionless or semi-dimensionless quantities to match the turbomachine to specific applications are summarized in Equations 1 through 3. The quantities of torque ratio, TR, output (turbine) to input (engine or pump) torques and Kfactor, K, input speed to square root of input torque are plotted versus speed ratio, SR, output to input speeds and forms the mainstay of matching torque converter -prime mover, see Figure 2. Unit input speed is the inverse, square root of the more common dimensionless number referred to as the torque coefficient. The designation for a given torque converter are normally specified by the diameter of the pump and the Kfactor value realized at zero speed ratio, known as stall.

One Dimensional Flow Theory

A significant amount detailed research, both analytical and experimental, has been performed on torque converters, [4-8 and 10-14]. The fundamental laws of fluid mechanics and thermodynamics can be applied to create a one-dimensional flow model capable of useful predictions. The principles of conservation of angular momentum and energy for a control volume, CV, encompassing the fluid and bladed surfaces of the turbomachine elements are used to the define the model as given by Equations 5 and 6, though a thorough description is outside of scope for this paper and is readily available in the literature, see [START_REF] Ishihara | Torque Converter as a Vibration Damper and Its Transient Characteristics[END_REF][START_REF] Tsangarides | Dynamic Behavior of a Torque Converter with Centrifugal Bypass Clutch[END_REF][START_REF] Xia | A Dynamic Model for Automotive Torque Converters[END_REF][START_REF] Hrovat | Bond Graph Modeling and Computer Simulation of Automotive Torque Converters[END_REF][START_REF] Fujita | Transient Characteristics of Torque Converter-Its Effect on Acceleration Performance of Auto-Trans. Equipped Vehicles[END_REF].

    A d v v r dV v r t CV                  (5) W Q dt dE     (6)
The key output of the one-dimensional model are torus fluid operating parameters that are not easily measured experimentally, namely the volumetric or mass flow rate of the toroidal flow. Having this information is important in the prediction of heat transfer from the toroidal flow to the external surface of the torque converter shell. This topic will be discussed in detail later in the paper. One-dimensional models for the torque converters tested for this investigation and example data showing correlation with dynamometer data is provided in Figure 2. Excellent correlation of the model is noted at low speed ratios, which are the focus of the discussion. Figure 3 contains toroidal mass flow rate for the stall, zero speed ratio, condition at a range of pump torques and mass flow rate across the speed ratio range at a pump torque of 150 Nm. At stall, mass flow rate is a linear function with pump torque, corresponding to an increasing pressure head, which will be later shown to adversely affect cooling through at elevated pump speeds. Toroidal mass flow rate follows an inverse power law with respect to speed ratio.

Energy Balance

An energy balance for the torque converter operating with or without the lockup clutch applied is shown in Figure 4. Shaft work input from the prime mover at the cover and pump (green) is converted to useful shaft work through the torus fluid and extracted in the turbine and lockup clutch damper (blue) assembly and then transmitted to the automatic transmission via the turbine shaft. The remainder of the energy not converted to useful shaft work is rejected as heat through two principal pathways. Cooling through flow, c Q  , is supplied to the torque converter from the automatic transmissions high pressure oil supply which enters through the turbine shaft and mixes with the torus flow and exhausts back to the sump via passages between stator and pump shafts. Energy is also dissipated as heat from the torus to the external shell via forced convection from torus to shell, torus Q  , and shell to air, shell Q  , as well as conduction through the shell. The overall heat transfer mechanism from the external surface can be difficult to measure or quantify using traditional methods and will be a focus of discussion in the paper. Based upon the energy balance in Figure 4, a linear, first order, differential equation can be written for the temperature of the toroidal fluid temperature, torus T , as seen in Equation 7.

It should be noted that for this model, it is assumed that torus temperature is equivalent to the exhaust or outlet temperature from the torque converter.

    W T T C m T T UA dt dT C m torus init c oil p oil torus air overall torus oil p oil         , , , (7)
T T UA Q   ,  (8)
.

The solution of the differential equation of Equation 7for torus temperature becomes,

  t t init torus e e T T          1 (9),
where  and  are equal to the following, T , , which is assumed to be time invariant. A simplification can be made to Equation ( 9) if the amount of heat energy rejected through the external surface is assumed a known quantity,

shell overall Q ,  ,   W T T C m Q dt dT C m toruc init c oil p oil shell overall torus oil p oil         , , , , (12)
, which results in the following solution for torus temperature with the same initial condition,

                       
. Equation 13will be the foundation for the remainder of the discussion in this paper for estimating the toroidal flow temperature of the torque converter during transient operation. Detail will be provided in the results section as to the mechanism enabling the simplification through the development of a proportional relationship between overall external surface heat rejection and input power based upon the specific design details of the torque converter.

EXPERIMENTAL SETUP

A torque converter specific dynamometer test cell, Figure 5, was used to test three torque converters of nearly exact geometric scaling, see Figure 6, at stall. The testing was originally performed to determine the appropriateness of applying the dimensional analysis to developing a prediction tool for incipient cavitation in torque converters using a nearfield acoustical technique, see [START_REF] Robinette | Predicting the Onset of Cavitation in Automotive Torque Converters -Part II: A Generalized Model[END_REF]. The dynamometer data collected during this testing can be further utilized to develop a model to predict torus temperatures during transient, high input power maneuvers when operating at low overall efficiency. The dynamometer was setup for the stall condition, zero output speed, only and all three torque converters were painted flat black to create the same radiation surface emissivity and to enable a more suitable infrared surface temperature measurement. The details of this measurement setup can be seen in Figure 5, along with nearfield microphones to detect cavitation enclosed in a nitrile cover to prevent oil contamination. The torque converters tested were 245, 258 and 300 mm diameters as measured at the pump outside diameter and had a stall unit input speed rating of 140, corresponding to K-factors of 163, 146 and 101, respectively. Figure 7 contains performance data, K-factor and torque ratio as a function of speed ratio, from dynamometer testing for the three torque converters tested in this investigation. The near perfect geometric scaling of all three torque converters results in the value of the dimensionless unit input speed to be equivalent across the range of speed ratios tested as noted in Figure 8.

RESULTS

Stall Speed Sweeps

A common set of stall speed sweep tests were performed for this investigation, ramping pump speed of the torque converter from 100 r/min to maximum speed ranging between 2500 to 2900 r/min depending on the design being tested. Pump speed was swept at four ramp rates of 40, 100, 200 and 500 r/min/s. These ramp rates were selected to span those realized during vehicle operation. The worst case condition is 40 r/min/s for internal heat buildup in the torus fluid to a ramp rate of 500 r/min/s more typical of operation in a vehicle powertrain. Once the peak pump speed was achieved, pump speed was ramped down to a 100 r/min at a ramp rate of -300 r/min/s. Once pump speed reached 100 r/min, an extended cool down period was allowed for torus fluid temperature to stabilize to that of the oil supply. The stall speed sweep test procedure and measurements of cooling through flow rate and temperatures at the inlet, outlet and surface of the torque converter shell is summarized in Figure 9 for a sweep rate of 40 r/min/s. Figure 10 As expected, the rate of temperatures rise and absolute value achieved is proportional to the sweep rate, however, cooling flow rate through the torque converter drops equivalently and independently of the sweep rate. The drop in cooling flow rate is a direct result of the pressure head developed in the torus as pump speed increases and rises above the charging pressure, impeding flow. Cooling through flow increases once the charging pressure is higher than the pressure developed in the toroidal flow. Thus, a time lag between the rise in outlet temperature of the fluid exiting the torque converter torus and the peak pump speed during the test can be noted in Figure 9. As sweep rate increases the rise in temperatures decreases sharply as the amount of energy input into the torque converter is reduced significantly. The most severe operating scenario occurs with the slow sweep rates combined extended operation at elevated power input.

The effect of torque converter size on output and shell surface temperatures are found in Figure 11. As the physical size of the torque converter increases, the rate of change of temperatures decreases, though the time required to return to a steady state oil temperature equal to the inlet oil temperature remains roughly constant for all three diameters.

Cooling Flow Heat Transfer

The cooling through flow to the torque converter torus acts as a mixing flow heat exchanger. Cooler automatic transmission oil enters the torus through flow passages in the turbine shaft and mixes with warmer torus fluid, causing displacement of warmer fluid to exit through passages in the stator shaft. This process is roughly depicted in Figure 4 and can be approximated as a heat exchanger with the amount of heat transferred determined from the following,

  torus init c oil p oil c T T C m Q   , ,   (14).
It was assumed that the outlet or exhaust temperature from the torque converter was equal to the internal torus temperature. Any heat transfer occurring in the passages within the shafting or test fixture were ignored. The heat rejection from cooling can directly be calculated since the mass flow rate of cooling oil, inlet and exhaust temperatures are measured during the test. Heat rejected during a 40 r/min/s stall speed sweep and following cooling cycle is plotted in Figure 12 along with mass flow rate of cooling for the three diameter torque converters tested. As indicated in Figure 12, heat rejection is minimal during the ramp up in pump speed as an increasing temperature differential between inlet and outlet is offset by a decreasing flow rate of cooling oil. All significant removal of heat from the through flow cooling circuit occurs after the stall event, ranging between 36 and 45 kW. During the stall event, the dominant mode of heat removal is forced convection at the outer shell.

Overall External Heat Transfer Coefficient -Conventional Approach

This section will detail calculations of heat transfer coefficients and rates related to determining the amount of heat rejected through the external surface of the torque converter. The rotational Reynolds number, Equation 15, was used in this investigation to characterize the turbomachines operating flow regime for both internal and external flows. For internal flow, the critical Reynolds number for transition to turbulent flow is 2500 while for external flow the transition point was taken to be 250000.

  R

Internal Convection

The heat transfer coefficient for forced convection from the torus flow was determined using the Nusselt correlations in Equations 16 and 17 for laminar and turbulent flow, respectively, approximating the individual blade passages within the torque converter elements as rectangular passages. Given the torus geometry, operating pump speeds and resultant toroidal flow velocities, the flow regime is turbulent with heat transfer coefficients in the range of 0.3 to 10 kW/m2-K. Although numerous Nusselt correlations exist with an applicable set of assumptions, nearly all produced forced convection coefficients within the same range and influenced results negligibly.

. 4 

Nu

External Convection

External heat transfer coefficients were sought out from multiple sources, namely flow over a flat plate, [START_REF] Incorpera | Fundamentals of Heat and Mass Transfer[END_REF], and rotating disks or cylinders in air, [16, 17 and 18]. Both approaches yielded similar heat transfer coefficients for the given geometry and boundary conditions. However, those derived specifically for rotating disks in air were selected due to the closer approximation to the rotating torque converter in the test fixture. The Nusselt correlations utilized are from testing conducted by [START_REF] Cardone | Heat Transfer Measurements on a Rotating Disk[END_REF] and are as follows, For the external surface geometry, rotation speeds between 100 and 3000 r/min and the fluid properties of air result in heat transfer coefficients in the range of 0.01 to 0.25 kW/m2-K. These values are an order of magnitude less than those computed for internal heat transfer and will be a limiting factor on overall heat rejection from the external surface. The net result will be prediction of torus fluid temperatures being unrealistically high for the stall speed sweep tests as will be presented next.

Overall Heat Transfer Coefficient

The internal and external heat transfer coefficients found using available Nusselt relationships were combined with the conduction coefficient through the torque converter shell to define an overall heat transfer coefficient as described in many heat transfer textbooks, [START_REF] Bergman | Fundamentals of Heat and Mass Transfer, 7th Edition[END_REF] for example. The limiting factor for the overall heat transfer coefficient becomes the external forced convection with a heat transfer coefficient an order of magnitude less than that of internal convection for the torus to the shell. The result is minimal heat rejection through the external shell to the air, which during a slow stall speed sweep in which power input is relatively high throughout the majority of the stall maneuver will produce extreme torus temperatures. An example prediction of torus flow temperature using Equation 9and an overall heat transfer coefficient from Equations 16 to 19 for a 258 mm torque converter is shown in Figure 13. Also included in Figure 13 are measured surface and outlet temperatures during the 40 r/min/s stall speed sweep. As expected with the minimal overall heat transfer coefficient for external surface heat rejection, torus temperature is predicted to peak at an unreasonably value of 900 ˚C. It was concluded that the available Nusselt relationships cannot be applied to this particular situation with good confidence and that an alternative Nusselt relationship must be derived.

Overall External Heat Transfer Coefficient -Derived Nusselt Correlation

A Nusselt correlation unique to this particular heat transfer aspect for torque converter applications is derived to predict the external heat transfer mechanism more appropriately. Similar approaches to predict heat transfer have been defined by [20 and 21] as applied to automotive turbochargers and torque converter lockup clutches, respectively. The heat energy rejected through the torque converter external surface can be reduced to the relationship of Equation 20 assuming the heat energy rejected is proportional to the input power by the scaling factor,  .

p surf W Q     (20)
Such an assumption is appropriate given the energy balance of Equation 12, knowing that the balance of input power must be absorbed in the torus fluid to be removed by the cooling through flow or rejected from the external shell surface. As shown earlier, minimal amount of energy is removed by cooling flow during the high power input stall speed sweep, thus a significant amount of the input power is transferred to the torus fluid to be rejected from the external shell surface. Utilizing traditional Nusselt correlations and combining internal and external convection with conduction through the shell into an overall heat transfer coefficient, an insufficient amount of heat rejection was determined to be computed by this method. The result was an unrealistic torus fluid temperatures approaching 900 ˚C for the slowest sweep rates.

Owing to similitude and previous Nusselt correlation work for rotating disks and assemblies by [16, 17 and 18], it is known that a similar Nusselt relationship should result for this particular turbomachine. The scaling factor,  , should be a function of the torque converter design, size and performance attributes, such as unit input speed and operating fluid properties. For this investigation, it was found that scaling the input power by Equation 21 resulted in a universal Nusselt correlation, see Equation 22, for the three diameter torque converters tested. as shown in Figure 14. The rotational Reynolds number for the testing performed extend to a value of 300,000, corresponding to the transition region reported by [START_REF] Cardone | Heat Transfer Measurements on a Rotating Disk[END_REF] and thus can be assumed to reside mainly in the laminar region for external forced convection from a rotating assembly in quiescent air. Interestingly, the values of the power law curve for Nusselt verse Reynolds number are similar in form and value to those found by [START_REF] Cardone | Heat Transfer Measurements on a Rotating Disk[END_REF] for the transition region for rotating disk heat transfer. The Nusselt correlation in Equation 22 is applied in the prediction of torque converter torus fluid temperature expressed by the first order, linear differential equation expressed by Equation 13. Equation 22 is valid for the stall operating condition only and for torque converters with dimensions scaled from the designs tested in this investigation that yield a stall unit input speed of 140. From a torque converter design standpoint, the scaling factor found in Equation 21shows that torque converters of increasingly higher unit input speed and K-factor will result in higher heat rejection through the external shell. However, further dynamometer testing of such designs is required to substantiate the observation and establish design guidelines. The same general observations are made for the case of the 245 mm torque converter performing a stall speed sweep at 200 r/min/s followed by a subsequent cooling cycle. Predicted torus temperature leads measured surface temperature with good association followed with a lagging outlet temperature during the cooling phase, continuing until convergence after adequate mixing of cooling and torus flows at approximately 80 seconds in Figure 16.

Model Correlation and Torus Fluid Prediction

Future Developments

The transient model presented in this investigation was derived focusing on the stall operating condition, which is an extreme condition, but most severe for thermal loading of the torus fluid. Future testing should focus on the development and validation of the model for transient or steady state speed ration operation between 0 and 1. This is will require potential further development of the derived Nusselt relationship which may take the following functional relationship,

  SR U fnc Nu , Re,  (23)
including the additional terms of speed ratio, SR, and unit input speed,  , in the case of different torque converter designs. Expansion of the Nusselt relationship and scaling factor on external heat rejection as a function of input power would increase the predictive capability of the model and methodology presented to future torque converter designs across the entirety of their operating range.

CONCLUSIONS

A first order, linear differential equation model based upon an energy balance of the torque converter was developed to predict internal torus fluid temperatures at the extreme stall operating condition, zero speed ratio. A Nusselt correlation was derived specific to the overall turbomachine for external heat transfer as it was found that conventional relationships were inadequate in accurate prediction. The heat rejected from the torque converter external surface was found to be proportionally scaled by the two times the square root of the torque converters diameter to the power of 5 and operating fluid density and a semi-dimensionless attribute, K-factor, raised to the power of 1.5. This relationship was shown to collapse the Nusselt relationship to a single function of the rotation Reynolds number for three torque converters of geometric similitude. Prediction of torus temperatures trended with measured temperature data acquired at the shell surface and outlet flow during dynamometer testing. The next steps in this research would be to develop fully the Nusselt relationship to include speed ratios other than stall to enhance its prediction capability for all operating conditions the torque experiences during typical powertrain maneuvers in vehicle applications.

NOMENCLATURE Variable Description Units

Figure 1 .

 1 Figure 1. Cross-sections of typical three element, automotive torque converters showing typical design trend over past 20 years, from left to right, oldest to newest.

  is transformed into the dimensionless quantity unit input speed,  , with the addition of diameter, D, to the 5/2 power and square root of the density, ρ, of the operating fluid,

Figure 2 .

 2 Figure 2. Correlation of one-dimensional flow theory model with measured dynamometer data.

Figure 3 .

 3 Figure 3. Toroidal mass flow rate as a function of pump torque at stall and speed ratio for a fixed pump torque.

Figure 4 .

 4 Figure 4. Energy balance across a three torque converter with lockup clutch and cooling through flow circuit.

  The individual heat transfer mechanisms of torus Q  and shell Q  in additional to conduction through the shell can be combined into an overall heat transfer term,

  an initial condition for the torus temperature, init T , to be equal to the cooling through flow inlet temperature, init c

Figure 5 .

 5 Figure 5. Torque converter specific dynamometer test cell and test fixture instrumentation.

Figure 6 .

 6 Figure 6. Geometrically scaled torque converters tested with U = 140, from left to right, 300 mm, 258 mm and 245 mm pump diameters.

Figure 7 .

 7 Figure 7. K-factor and torque ratio vs. speed ratio for the three torque converters tested for this investigation.

Figure 8 .

 8 Figure 8. Unit input speed and torque ratio vs. speed ratio for the three torque converters tested for this investigation.

  contains the stall speed sweep temperature data for all sweep rates for the 258 mm torque converter showing only the ramp up in speed portion of the test.

Figure 9 .

 9 Figure 9. 258 mm torque converter, cooling mass flow rate and operating temperatures during a 40 r/min/s stall speed sweep and subsequent cool down period.

Figure 10 .

 10 Figure 10. 258 mm torque converter surface and outlet temperatures during various stall speed sweep tests.

Figure 11 .

 11 Figure 11. Effect of torque converter size on surface and outlet temperatures at a 40 r/min/s stall speed sweep rate and cooling cycle.

Figure 12 .

 12 Figure 12. Heat rejection from cooling through flow for 245 mm, 258 mm and 300 mm torque converters during 40 r/min/s stall speed sweep and cool down cycle.

Figure 13 .

 13 Figure 13. Torus fluid temperature prediction using available Nusselt relationships and Equation 9 for 258 mm, 140 unit input speed torque converter during 40 r/min/s stall speed sweep and cooling cycle.

  fitting a power function to the data reveals the Nusselt correlation of the form

Figure 14 .

 14 Figure 14. Derived Nusselt correlation for overall heat transfer coefficient from torque converters of constant unit input speed and geometric scaling.

  The derived Nusselt correlation of Equation 22 was used as an input for the solution of the differential equation describing the energy balance of the torque converter, Equations 12 and 13, at stall to predict the torus fluid temperature. The models prediction capability is shown for two examples, the results of which are contained in Figures15 and 16. In the first example, the 258 mm torque converter is performing a 40 r/min/s stall speed sweep and cool down cycle. The predicted torus temperature leads the surface temperature measurement in very close association, while outlet temperature lags predicted torus temperature as cooling flow drops during high stall speeds then surges once pump speed drops. It can be noted that torus temperature drops rapidly once cooling mass flow rate increases, and sufficient cooling flow mixes with the hot torus fluid, being fully replaced within a period of approximately 15 seconds.

Figure 15 .

 15 Figure 15. Torus fluid temperature prediction using derived Nusselt relationship and Equation 13 for 258 mm torque converter during a 40 r/min/s stall speed sweep and cool down period overlaid with surface and outlet temperatures.

Figure 16 .

 16 Figure 16. Torus fluid temperature prediction using derived Nusselt relationship and Equation 13 for 245 mm torque converter during a 200 r/min/s stall speed sweep and cool down period overlaid with surface and outlet temperatures. Overall, the linear first order, differential equation describing the torus fluid temperature based on the energy balance of the torque converter combined with the derived Nusselt relationship for overall heat transfer from the external surface provides sufficient capability for adequate prediction of toroidal flow temperatures during transient stall maneuvers. The predicted values show that extended operation at extreme speed ratios can be detrimental to the fluid, breaking down viscosity modifiers and compromising long term durability of internal transmission components as temperatures can reach beyond the typical recommended operating temperature of 120 ˚C for most automatic transmission fluids. Additionally, the predicted torus fluid temperatures during the extended stall operation at high input powers can promote cavitation.