Heat transfer engineering an international journal, tập 31, số 7, 2010

Heat Transfer Engineering, 31(7):527–554, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903425320 Gas Turbine Blade Tip Heat Transfer and Cooling: A Literature Survey BENGT SUNDEN and GONGNAN XIE Department of Energy Sciences, Lund University, Lund, Sweden Gas turbines are widely used for aircraft propulsion, land-base power generation, and other industrial applications like trains, marines, automobiles, etc To satisfy the fast development of advanced gas turbines, the operating temperature must be increased to improve the thermal efficiency and output work of the gas turbine engine However, the heat transferred to the turbine blade is substantially increased as the turbine inlet temperature is continuously increased Thus, it is very important to cool the turbine blades for a long durability and safe operation Cooling the blade must include cooling of the key regions being exposed to the hot gas The blade tip region is such a critical area and is indeed difficult to cool This results from the tip clearance gap where the complex tip leakage flow occurs and thereby local high heat loads prevail This paper presents a literature survey of blade tip leakage flow and heat transfer, as well as research of external and internal cooling technologies The present paper does not intend to review all published results in this field, nor review all papers from the past to now This paper is limited to a review of recently available published works by several researchers, especially from 2001 to present, concerning blade tip leakage flow associated with heat transfer, and external or/and internal tip cooling technologies INTRODUCTION A gas turbine is an engine designed to convert the energy of a fuel into some form of useful power, such as shaft power or thrust Today, gas turbines (GTs) are widely used in aircraft propulsion, land-based power generation, and other industrial applications For example, GTs are used to power commercial airplanes, marines, trains, electric power generators, automobiles, and gas pipeline compressor drivers Figure illustrates a commercial gas turbine engine The reasons that gas turbine engines are widely used for aircraft propulsion include that they are light, compact, and have a high power-to-weight ratio As shown in Figure 1, there are three main components of a gas turbine engine: compressor, combustor, and turbine The compressor is used to compress the intake air to a specific high pressure, the combustor is used to burn the input fuel and produce the high temperature gas, and the turbine extracts the energy of the gas and converts it into power work A number of components sometimes occurs in the gas turbine system to improve the The authors acknowledge financial support from the TURBO POWER consortium funded by the Swedish Energy Agency (STEM), SIEMENS Industrial Turbomachinery, and VOLVO AERO Corporation Address correspondence to Professor Bengt Sunden, Division of Heat Transfer, Department of Energy Sciences, Lund University, PO Box 118, S-22100, Lund, Sweden E-mail: Bengt.Sunden@energy.lth.se network output or thermodynamic efficiency, e.g., intercooler, recuperator, regenerator, and combustion reheater However, the balance of additional power, efficiency, cost, complexity, durability, compactness, etc must be carefully evaluated The temperature–entropy diagram for the basic cycle of a gas turbine engine with friction is shown in Figure The ideal standard cycle is assumed to be adiabatic, reversible, and frictionless The overall thermodynamic efficiency depends on the efficiencies of all components, such as compressor efficiency, turbine efficiency, and combustion efficiency Clearly, the turbine efficiency will affect the cycle efficiency at some degree Thus, improving the turbine efficiency will help to improve the overall performance of a gas turbine engine, while losses in turbine efficiency and/or output work will reduce the overall performance of the system Apart from the component efficiencies, the operating temperature of gas turbine system affects the overall performance It is well recognized that one way to increase the power output and thermodynamic efficiency of a gas turbine engine is to increase the turbine inlet temperature (TIT) From the principles of engineering thermophysics [1, 2], the reason is that at a fixed pressure ratio the net work output of a gas turbine increases with increasing turbine blade (also called rotor) inlet temperature Figure shows recent development of TIT from 1950 to 2010 Current advanced gas turbine engines are operating at TIT of about 1200–1500◦ C To pursue higher power, the inlet 527 528 B SUNDEN AND G XIE Turbine Inlet Temoperature, K 2600 New cooling concept 2400 Projected trend new material 2000 Film,Impingement Sophisticated cooling system 1600 Simple cooling 1200 Allowable metal temperature Uncooled turbines 1000 Figure Gas turbine illustration From http://www.epower-propulsion.com/ epower/gallery/GasTurbines.htm 1950 1960 1970 1980 1990 2010 Year temperature should be raised increasingly to higher certain targets For example, to double the power of the aircraft, the TIT should be increased from 1500 to 2000◦ C However, the heat transferred to the blade increases with the increase of the blade inlet temperature, and the allowable melting temperature of materials increases at a slower rate This means that the turbine blade inlet temperature may exceed the material melting temperature by more than 500◦ C Thus, it is critical to cool turbine blades for a safe and long-lasting operation The blades can only survive if effective cooling methods are used Various internal and external cooling techniques are employed to decrease the blade material temperature below its melting point Figure depicts the typical cooling technology for internal and external zones The leading edge is cooled by jet impingement with film cooling, the middle portion is cooled by internal serpentine ribbed-turbulators passages, and the trailing edge is cooled by pin-fins with ejection In internal cooling, the relatively cold air, bypassed/discharged from the compressor, is directed into the hollow coolant passages inside the turbine blade In external cooling, the bypassed air is ejected through those small holes, which are located in the turbine blade discretely The commonly used cooling technique for the highpressure turbine blade is a combination of internal and film cooling Most recent developments in TIT increase have been Figure Developments of gas turbine inlet temperature over recent years Reproduction from Rolls Royce plc achieved by better cooling of the turbine blade and have improved the understanding of the heat transfer mechanisms in the turbine passages Several recent publications reviewing the gas turbine heat transfer and cooling technology investigations are available These include a relevant book [3], edited volumes [4, 5], and journal papers [6–9] Internal convective cooling Film cooling Hot gas Tip cap cooling Trailing edge ejection Rib turbulated cooling Impingement cooling Pin-fin cooling Cooling air Figure Temperature–entropy diagram for a basic gas turbine cycle heat transfer engineering Figure Typical cooling techniques for a blade vol 31 no 2010 B SUNDEN AND G XIE 529 Shroud Shroud Blade Leakage flow Clearance Gap Blade Tip Pressure side Moving rotor Suction side Rotation Figure Clearance gap and leakage flow of unshrouded turbine blade Cooling of the blade should include the cooling of all regions exposed to high-temperature gas and thermal load Among such regions, particularly for high-pressure turbines, is the blade tip area Gas turbine blades usually have a clearance gap between the blade tip and stationary casing or the shroud (as schematically shown in Figure 5) The clearance gap is necessary to allow for the blade rotation and for its mechanical and thermal expansion However, due to the pressure difference between the pressure side and suction side, the hot gas leaks through the gap This is known as the tip leakage flow The leakage flow is undesirable, because it is associated with the generation of a secondary flow resulting in reduction of the work done and hence of the overall efficiency, and results in higher heating at the high pressure tip corner from mid-chord to trailing edge The hot leakage flow increases the thermal loads on the blade tip, leading to high local temperature Thus, it is essential to cool the turbine blade tip and near the tip regions However, it is difficult to cool such regions and to seal against the hot leakage flow The blade tip operates in an environment between the rotating blade and the stationary casing, and experiences the extremes of the fluid-thermal conditions within the turbine [10–12] A more detailed discussion of the blade tip can be found in [13] Because the blade lifetime may be reduced by a factor of if the blade metal temperature prediction is off by only 30◦ C, it is very critical to predict accurately the local heat transfer and local blade temperature to prevent hot spots and thus increase the turbine blade life It is important for the gas turbine designers to know the effects of increased heat load in the area exposed to hot gas and be able to design efficient cooling schemes to protect the blade Therefore, fundamental and detailed studies of heat transfer and flow relating to the blade tip or near blade tip regions are needed to provide better understanding and prediction of the heat loads on such regions accurately Besides conventional techniques of experimental measurements with advanced apparatus, computational fluid dynamics (CFD) plays an increasingly important role in design and research studies of gas turbines During the past two decades, CFD has been developed so rapidly that many advanced heat transfer engineering computational codes and commercial softwares have continuously appeared for solving the heat transfer and flow field of complex geometries like gas turbine passages By validating the codes with experimental data, many computational results based on CFD are accurate and reliable This will contribute to the prediction and design of turbomachinery components, without doubt, including the turbine blade and its tip The highly accurate computational results can contribute to the design and manufacture of gas turbine blades and improve the durability and safe operation This paper does not and cannot review all the interesting and important progress related to gas turbine heat transfer and cooling (some may be found in [1–5]), but tries to summarize the recently published results in the concerned field of blade tip heat transfer and development of cooling technology The first studies on blade tip heat transfer were reviewed earlier [11–13] In this paper, published literature from 1995 to 2008 and on, especially the recent years 2001–2008, are reviewed This paper is organized as follows Gas turbine heat transfer and the need for cooling techniques are introduced first Then the blade tip leakage complicated flow associated with heat transfer on tips or near the tip regions is reviewed Next the development of external tip cooling methodology is reviewed, while the last section reviews the development of internal tip cooling methodology A summary is presented in the final section BLADE TIP LEAKAGE FLOW AND HEAT TRANSFER Generic Flow Pattern Associated With Tip Leakage Flow The flow field in a turbine is very complex It is strongly three-dimensional, unsteady, and viscous, with several types of secondary flows, endwall flows, and vortices (passage vortex, counter vortex, horseshoe vortex, leakage flow vortex, etc.) Transition flow and high turbulence intensity result in additional complexities Figure depicts the complex flow phenomena vol 31 no 2010 530 B SUNDEN AND G XIE Figure [14] Complex flow and heat transfer phenomena in the turbine gas path in a turbine blade gas path [14] The understanding of such complex flow fields and heat transfer characteristics is necessary to improve the blade design and prediction in terms of efficiency as well as the evaluation of mechanical and thermal fatigue Tip leakage flow is a dominant source of unsteadiness and threedimensionality of the flow in turbomachineries As depicted in Figure 7, the tip leakage flow passes through the tip clearance driven by the pressure gradient between the pressure side and suction side Also, the leakage flow tends to roll up into a vortex and interacts with the secondary flow Thus, the leakage flow and its interaction with other flow features show very complex phenomena A perfect blade tip will not allow any leakage flow, and no secondary flows to reduce stage efficiency will be generated, nor losses for downstream stages created, and cooling is not required Thereby no thermodynamic losses occur [11] Thus, the two main objectives of blade design are to reduce the leakage Figure Different kinds of blade tips [11] Figure Schematics of blade tip leakage flow characteristics [11] heat transfer engineering flow as much as possible and to cool the blade tips using small quantities of extracted cooling air However, all the blade tips in modern gas turbines allow some leakage flow and secondary flows are generated Today, there are several major types of blade tips: (a) flat tip, (b) recessed tip with peripheral squealer sealing rims, and (c) attached tip shrouds [11], as shown in Figure Each blade tip has its advantages and disadvantages Although it is easy to design a flat tip and its cooling scheme, very few turbines use flat tips High leakages lead to bad tip vol 31 no 2010 B SUNDEN AND G XIE 531 Figure Streamlines of blade tip flow pattern [15] aerodynamics and results in higher heat loads on the tip A recessed tip with sealing rims is the most common design in practice today for high-pressure turbine blades A recessed tip with rim reduces the risk of blade damage if the tip rubs against the shroud; however, the design of a recessed tip is more complex because of the cooling of the rim and the need to prevent losses by oxidation and erosion Blades with attached tip shrouds are mostly used in low-pressure turbine blades This tip has the lowest aerodynamic loss when properly installed, but it requires greater attention to stresses because of the heavier weight and requires a more complex cooling system Ameri et al [15] performed calculations on flow and heat transfer of a GE-E3 rotor tip considering three types: plane tip, 2% recess tip, and 3% recess tip A two-dimensional (2D) cavity flow problem was used to validate the k-ω turbulence model These authors found two dominant flow structures in the recess region, which strongly affect the heat transfer rate, as shown in Figure 9, but no significant effect on the adiabatic efficiency was observed for these three tips Also, Ameri et al [16] studied the effects of tip clearance and casing recess on heat transfer and stage efficiency in axial turbines Their numerical study reconfirmed a linear relationship between the efficiency and the tip gap height Introduction of a recessed casing resulted in a drop in the rate of heat transfer on the pressure side, and a marked reduction of the heat load and peak values on the blade tip Ameri et al observed that the recessed casing has a small effect on the efficiency but can have a moderating effect on the flow underturning at smaller tip clearances Experimental Measurements for Tip Region Flowfield and Heat Transfer Bunker et al [17], and Ameri and Bunker [18] reported results of a combined experimental and simulation study designed to investigate the detailed distribution of the convective heat transfer coefficient on the flat tip surface with both sharp and rounded edges for a large power generation turbine This study showed good agreement between experiments and comheat transfer engineering Figure 10 Sharp and rounded edge tip heat transfer coefficients [18] putations Figure 10 presents a sample of these experimental and computed tip heat transfer coefficients for the sharp and rounded edge tips Ameri [19] also conducted experimental and numerical studies of detailed heat transfer coefficient distribution on the rounded blade tip of a gas turbine equipped with a mean-camberline strip Generally good agreement between experimental data and computations was achieved, as shown in Figure 11 Results showed that the mean-camberline strip could reduce the tip leakage flow but the total pressure loss was not reduced comparatively, and the sharp edge tip was better in vol 31 no 2010 532 B SUNDEN AND G XIE Figure 11 Heat transfer coefficient and flow pattern for blade tip with meancamberline strip [19] reduction of the tip leakage flow and tip heat transfer compared to the rounded edge tip Thorpe et al [20, 21] reported experimental measurements of time-mean/time-resolved heat transfer and static pressure on the over-tip casing of a transonic axial flow turbine They presented axial and circumferential distributions of the heat transfer rate as well as adiabatic wall temperature, Nusselt number, and static pressure They found that the rate of heat transfer to casing wall and the wall temperature varied strongly with axial position through the rotor, and the effects of the vane exit flow features were small Through assessments of the relative importance of different time varying phenomena to the casing heat load distribution, they concluded that up to half of the casing heat load was associated with the tip leakage flow Also, discussion about shroudless turbine design accounting for the high heat flux was addressed Thorpe et al [22] also experimentally studied the blade tip heat transfer and aerodynamics in a transonic turbine stage They observed high heat transfer rates near the nose of the blade tip and also in the region of high blade lift near the midaxial chord, and proposed three primary mechanisms: vane– shock interaction, relative total temperature fluctuations, and fluctuations in tip leakage flow speed and direction driving the unsteady heat transfer Chana and Jones [23] presented detailed experimental measurements of heat transfer and static pressure distributions on the shroudless rotor blade tip and casing with and without inlet nonuniform temperatures Also, a simple 2D model was developed to estimate the heat transfer rate to tip and casing as a function of Reynolds number Results showed that the overall heat load was reduced with inlet nonuniformity, that the highest heat transfer rate was on the pressure side of the blade where the highest random unsteadiness was marked, and that the average static pressures did not show significant difference between the two cases Camci et al [24] investigated experimentally aerodynamic characteristics of full and partial length squealer rims in an axial turbine Figure 12 shows a schematic picture of partial heat transfer engineering Figure 12 Geometries of partial squealer rims [24] squealer rims studied Results showed that the partial squealer rim could seal the tip effectively, and a mid-size partial rim was most effective in reducing the tip leakage flow Compared to the two studied channel arrangements having partial rims near the corners of the suction and pressure sides, the sealing performance of the mid-size rim on the suction side was even better This indicated that the partial squealer rims on the suction side were capable of reducing the exit total pressure loss by the tip leakage flow to a significant degree Key and Arts [25] studied the tip leakage flow characteristics for flat and squealer turbine tips The experiments were conducted at different Reynolds number and Mach number conditions for a fixed value of the tip gap in a nonrotating, linear cascade arrangement Oil flow visualization was used, as shown in Figure 13, and the static pressure and aerodynamic loss were measured These authors found that the squealer tip showed a significant decrease in velocity through the tip gap, and for the flat tip the increase of Reynolds number would cause an increase in the tip velocity level whereas for the squealer tip the sensitivity was not much Their data are valuable for validation of CFD computations, and in turn CFD can provide insight to some details of the flow physics in the tip region Azad et al [26, 27] and Teng et al [28] measured the heat transfer coefficient and static pressure distributions on gas turbine tips in a five-bladed stationary linear cascade Various regions of high and low heat transfer coefficients at the tip surface were observed The heat transfer coefficients increase with an increase of the inlet turbulence intensity Compared to the flat vol 31 no 2010 B SUNDEN AND G XIE 533 Figure 13 Flow visualization of squealer and flat tip [25] tip, the squealer tip showed a lower overall heat transfer coefficient Also, a reduced tip gap clearance resulted in a weaker unsteady wake effect on the blade tip heat transfer and a reduction in the heat transfer coefficient over the blade tip surface Azad et al [29] also studied the effect of the squealer geometry arrangement on a blade tip The detailed heat transfer coefficient distributions of six tip geometry cases were obtained It was shown that the suction-sided squealer could provide a better benefit compared to other cases, and the mid-chamber lined squealer behaved better than the pressure-sided squealer Also, a single squealer provided better performance in reducing the overall heat transfer than a double squealer Dhadwal and Kurkov [30] used a dual-laser probe integrated fiber optic system to measure the blade tip clearance in a rotating turbomachinery A symmetric configuration of the probe installation could offer better resolution The time-of-flight measurements were robust and reliable Saxena et al [31] presented a comprehensive investigation of the effect of various tip sealing geometries on the blade tip leakage flow and heat transfer of a scaled up high-pressure turbine They found that compared to other geometries, the tripped strips placed against the leakage flow (as shown in Figure 14a) led to the lowest heat transfer on the tips with a reduction of 10–15% The use of strips and pin-fins did not decrease the tip surface heat transfer coefficients Saxena and Ekkad [32] also experimentally investigated the effect of squealer tip geometries on the blade tip leakage and associated heat transfer in the same facility It was found that the suction-sided squealer rim might be favorable for reducing the heat transfer coefficients on the tip surface, whereas the pressure-sided squealer did not reduce the heat transfer and behaved like the plane tip Nasir et al [33] also investigated the effect of tip gap and squealer geometry on the detailed heat transfer engineering Figure 14 Blade tip geometries for test [31] heat transfer over a high pressure turbine rotor blade tip The squealer studied altered the tip gap flow significantly and hence resulted in lower heat transfer coefficient Also, experimental results showed that some partial burning of the squealers might be good for overall reduction in the heat transfer coefficient Rhee and Cho [34, 35] experimentally measured local heat/mass transfer characteristics on tip, shroud, and near-tip surface of a rotating blade in a low-speed annular cascade The effects of rotation and incoming flow incidence angle were examined Results showed that the heat transfer was complex with complicated flow patterns such as flow acceleration, laminarization, transition, separation, and tip leakage flow, and the blade rotation caused increased incoming flow turbulence intensity while the tip leakage flow was reduced Also, they found that the heat/mass transfer coefficients were about 1.7 times than those on the blade surface and shroud, and due to the reduced tip leakage flow under rotation the heat/mass transfer coefficients on the tip slightly decreased while they remained similar on the shroud With a positive incidence angle, more uniform and higher heat transfer rate were found on the tip because of the increased tip gap flow and high flow angle Rhee and Cho vol 31 no 2010 534 B SUNDEN AND G XIE Figure 15 Blade geometries for test [40] [36, 37] experimentally studied the effect of vane/blade position on heat transfer in a stationary blade and shroud in a low-speed wind tunnel They presented detailed mass transfer measurements, and the results showed that the mass transfer coefficients in the upstream region varied up to 25% due to the blockage effect as the blade position changed The size and level of the peak region were affected strongly Also, distinctly different patterns near the blade tip were observed due to the variation in the tip leakage flow Matsunuma [38] observed the effect of Reynolds number and freestream turbulence on turbine tip clearance flow Threedimensional flow fields at the exit of the turbine with and without tip clearance were measured Results indicated that variations heat transfer engineering in Reynolds number and freestream turbulence intensity did not affect the mass-averaged tip clearance loss Due to the strong interaction between the leakage vortex and tip-side passage vortex, the decrease in flow angle at lower Reynolds numbers was larger than that at higher Reynolds numbers Kwak and Han [39] and Kwak et al [40] conducted a series of measurements on the tip and near-tip region heat transfer coefficients of a turbine blade with flat or squealer tip, and the effects of rim location and height as well as tip clearance on heat transfer were measured The geometry is shown in Figure 15 The blade tip clearance was 1.0%, 1.5%, and 2.5% and the rim height was 2.1%, 4.2%, and 6.3% of the blade span, as shown in Figure 15 Experimental results showed that the heat transfer coefficients vol 31 no 2010 B SUNDEN AND G XIE on the tip surface were higher than those on the shroud and on the near-tip region of the pressure and suction sides, and with an increase of the tip clearance the heat transfer on the tip surface increased whereas heat transfer on the shroud and the suction side first increased and then decreased On the blade pressure side the heat transfer coefficient was kept constant They also found that higher rims could reduce the heat transfer coefficient on the tip and shroud, while on the pressure side and suction side the reduction was not significant The suction-sided rim could provide lower heat transfer coefficient on the tip and near-tip region than the double-sided rim case Kwak and co-workers [41, 42] also performed measurements on detailed heat transfer coefficients on the squealer tip and near-tip region of a turbine blade Results showed that the overall heat transfer coefficients on the squealer tip were higher than those on the shroud surface and the near-tip region of the pressure side and suction side Near the tip region the heat transfer coefficient showed no significant reduction Also, the suction-sided squealer tip revealed the lowest heat transfer coefficients on the blade tip and near tip Papa et al [43] investigated experimentally the effects of squealer or winglet-squealer tip and tip clearance on the average and local mass transfer coefficients for a large-scale gas turbine blade, and used the heat–mass analogy to obtain heat transfer coefficients Flow visualization on the tip surface was presented Compared to the winglet-squealer tip, the squealer tip provided a higher average mass/heat transfer coefficient Rehder and Dannhauer [44] studied the effect of the tip leakage flow on the three-dimensional (3D) flow field and end-wall heat transfer Results showed that when the leakage mass flow rate increased from 1% to 2%, significant changes in the secondary and end-wall heat transfer occurred The secondary flow was amplified as the leakage flow was ejected perpendicular to the main flow direction, whereas it was reduced significantly as the leakage flow was ejected tangentially Govardhan et al [45] investigated the 3D flow in a large deflection turbine cascade with tip clearance 0.08%, 1.5%, and 3.0% of the chord They found that there was a strong horseshoe vortex in front of the leading edge for 0.08% clearance, while for 3% clearance there was no vortex A small tip separation vortex was also observed on the tip surface, which made the flow from the pressure side to be accelerated The passage vortex did not diminish as the tip clearance increased Also, Govardhan et al [46] investigated the effect of endwall and tip clearance on the flow in a twodimensional turbine rotor blade cascade Five incidence angles were chosen: −10, −5, 0, 5, and 10◦ Results showed that as the tip clearance was increased the adverse pressure gradient upstream the leading edge was reduced, and with the increase of incidence angle the blade loading due to the static pressure gradient also increased Porreca et al [47] conducted experimental and numerical investigation on flow dynamics and performance of partially and fully shrouded axial turbines, as shown in Figure 16 Experimental results showed that for the partial shroud case a strong tip leakage vortex was developed from the first rotor and transheat transfer engineering 535 Figure 16 Shroud configuration and probe planes [41] ported through the downstream blade row CFD computational results showed a good agreement with the measured data at the midspan for the first stage The overall second stage efficiency for the full shroud case could be improved by 1% Newton et al [48] measured the heat transfer coefficient and pressure coefficient on the tip and near-tip region of a generic turbine blade in a five-blade linear cascade Two tip clearances of 1.6% and 2.8% of chord were considered and three tip geometries were studied: plane tip, suction-sided squealer, and cavity squealer They found that the flow separation at the pressure side edge dominated the flow through the plain gap, that the highest heat transfer was located in such a region that the flow reattached on the tip, and that the suction-sided and cavity squealers could reduce the heat transfer in the gap The suction-sided squealer provided an overall net heat flux reduction of 15%, while the cavity squealer revealed no net heat flux reduction Palafox et al [49] measured new detailed flow fields for a very large low-speed, high-pressure turbine rotor blade using particle image velocimetry (PIV) The interaction between the tip leakage vortex and passage vortex was clearly characterized, and the effect on the tip leakage vortex was examined Results showed that a separation bubble under the tip significantly affected the leakage flow, and the end-wall movement influenced the shape and size of the bubble distinctly, while the relative blade vol 31 no 2010 612 M BACHIRI AND A BOUABDALLAH Figure Similarity temperature profiles along a heated vertical plate for various Pr values where √ √ a π Q(η) = − − 3/2 erf ( bη) + 4b + 1 η+ 2c c √ a π + 3/2 η 2c3 2b exp(−cη) − c4 (24) Figure shows the evolution of the temperature profiles determined analytically via the two preceding methods, and compared with the numerical results for five Prandtl number values In Figure we notice two major results First, in the case of Pr ≥ 1, the two temperature profiles of Eqs (21) and (23) are almost equal on the whole domain [0, ∞] with maximum gap less than 0.02, and the comparison of these profiles with numerical results gives good agreements with the maximum variations that not exceed 0.01, in general Second, in the case of Pr 1, the heat transfer engineering two temperature profiles of Eqs (21) and (23) are completely different, and it appears that only Eq (23) agrees with the numerical results At this stage, the approximation cannot satisfy the equilibrium of the boundary layer equations for smaller Prandtl number values In order to generalize the approximation for a wide range of Pr values, we established two correlations: the first one between ηmax and Pr and the second between G(ηmax ) and ηmax Thus, by trial and error we obtained the following correlations: ηmax = σ1 + σ2 ln( Pr) (25) where σ1 = 1.3390 and σ2 = 0.2150, and G(ηmax ) = a1 − a2 exp(a3 ηmax ) where a1 = 1.2215, a2 = 1.8110, and a3 = −0.6993 vol 31 no 2010 (26) M BACHIRI AND A BOUABDALLAH 613 Figure Analytical determinations of the velocity and temperature profiles We note here that the correlation of Eq (25) is not applicable for all fluids and its applications are limited to Pr ≥ 0.25 However, the correlation of Eq (26) can be used for all fluids By another way, we show that the approximation is not satisfied for all fluids and its utilizations will be made for all liquids and for few gases with 0.25 ≤ Pr < Using the calculation program (synoptic diagram) presented in Figure 5, we can solve analytically the steady-state natural convection problem of a vertical plate heated at uniform temperature This synoptic diagram permits us to establish the analytic formulas of the velocity and temperature profiles and all others derived formulas for the fluids with Pr ≥ 0.25 The results are shown in Figures 6a and b · Figure Resolution of the natural convection along a heated vertical plate by the analytic approach with Pr ≥ 0.25: (a) velocity profiles and (b) temperature profiles HEAT TRANSFER COEFFICIENT The Nusselt number is defined by hy N uy = k where h and k are the heat transfer coefficient and thermal conductivity coefficient, respectively The local heat transfer coefficient predicted by the similarity solution is written as N uy = − dθ dη Ray1/4 (27) η=0 Using the analytical expression of θ (η), we have exp(Q(η )) dθ = − η∞ dη exp(Q(η )dη 1/4 Ray η∞ exp (Q (η ))dη ∞ [ exp(Q(η ))dη ]−1 1/4 N uy = N¯ uoy + 0.665Ray [1 + (0.492/Pr)9/16 ]4/9 (29) where N¯ uoy is the limit-conductive rate when Ray → Churchill and Chu [20] had established an accurate correlation representing the experimental data of the average Nusselt number for an isothermal vertical plate The correlation given Table Similarity solution heat transfer results by analytical approach for natural convection boundary layer flow with Pr ≥ 0.25 on a vertical isothermal plate Starting from Eq (24), Q(0) equals to zero So, we have N uy = The average Nusselt number expression established by our analytic approach for the natural convection boundary layer flow with Pr ≥ 0.25 on an isothermal vertical plate is given as (28) The numerical coefficient is, in general, a function of the Prandtl number, as is shown in Table heat transfer engineering Pr −1/4 Nu Ray 0.25 0.72 10 102 103 104 0.32 0.384 0.399 0.449 0.46 0.483 0.495 0.508 vol 31 no 2010 614 M BACHIRI AND A BOUABDALLAH Figure Comparison of the present work Nusselt numbers with experimental correlation of [20] and the theoretical results of [11] for a wide range of Prandtl number values Figure Local shear stress rate in laminar natural convection along a heated vertical plate is as follows: 1/4 Nuy = 0.68 + 0.67Ray [1 + (0.492/Pr)9/16 ]4/9 (30) We notice that the average Nusselt number expression obtained by our approximation is almost equal to the expression given by experimental correlation of [20] Furthermore, Figure shows a good agreement between our analytic approach and the theoretical [11] and experimental results [20] for all fluids with Pr ≥ 0.25 SHEAR STRESS RATE As we know the analytical form of the dimensionless velocity profile G(η), we can calculate the shear stress rate τo (y) related to the skin friction Starting from Eq (8), we have y 1/2 αRay v = G (η) (31) ∂v and by definition τ0 (y) = µ0 ( ∂x )x=0 , which represents the wall shear stress rate evaluated on the vertical wall (x = 0), and µ0 indicates dynamic viscosity at constant temperature The preceding expression can be brought back to dimensionless form of wall shear stress rate, τ0 (y) 3/4 αµ0 Ray /y = dG (η) dη (32) η=0 Figure shows that the shear stress rate increases when the Prandtl number increases However, this is not the same case in regard to the Ostrach’s similarity solution [10], who found that the shear stress rate decreases when the Prandtl number increases In fact, these results constitute the best indicator that 1/2 the Ostrach’s vertical velocity scale [(ν/y)Gry ] did not characterize the Pr limits heat transfer engineering CONCLUSIONS In this article, we have established an analytic approximation of the coupled nonlinear differential equations system in the problem of the steady-state natural convection evolving along a semi-infinite vertical plate with an isothermal surface condition Using the scale analysis and the similarity transformation given by Bejan, we have studied the phenomenon analytically by proposing an ad hoc analytic expression of the velocity profile On the basis of this, we have determined the analytic expressions of the temperature profile via both momentum and energy equations The agreements are quite satisfactory when we compare the analytic formulas of the velocity and temperature with the numerical results However, it was found that there is no agreement of the temperature profile determined via the first method with the numerical results for fluids with Pr < 0.25 A generalization of this approximation is given by proposing two correlations that permit one to solve analytically the problem of the steady-state natural convection boundary layer flow with Pr ≥ 0.25 on a vertical plate The Nusselt number obtained by our approximation verifies the Bejan’s scale analysis and agrees well with the numerical and experimental results As the aim of this work, we have shown that we can calculate directly the principal functions of this problem, such as the stream function, the velocity profile, the temperature profile, the Nusselt number, and the shear stress rate Finally, these results can be used advantageously, with the entire phenomenon bringing into play the velocity–temperature coupling From this point of view, we seek to apply this analytic solution to interesting physical situations, in particular the calculation of the plate fin exchangers, the study of the crystal growth, and the influence of temperature on the polarographic probes when the diffusion coefficient varies according to temperature vol 31 no 2010 M BACHIRI AND A BOUABDALLAH NOMENCLATURE a, a1 , a2 , a3 b, c G g0 Gry F H h k Nuy Pr Q Ray T Tw T∞ u ν x, y constants constants dimensionless velocity profile acceleration due to gravity Grashof number, g0 β Ty3 /ν2 dimensionless stream function reference length heat transfer coefficient heat conductivity Nusselt number, hy/k Prandtl number, ν/α a function Rayleigh number, g0 β Ty3 /αν temperature wall temperature ambient temperature velocity component in x direction velocity component in y direction Cartesian coordinates along the surface and normal to it, respectively Greek Symbols α β η µ0 ν θ θ0 , θ1 ρ τ0 T σ1 , σ ψ thermal diffusivity thermal expansion coefficient independent similarity variable dynamic viscosity kinematic viscosity dimensionless temperature profile integration constants density wall shear stress temperature difference constants stream function Subscripts max maximum Superscripts , , differentiation with respect to η REFERENCES [1] Gebhart, B., Heat Transfer, 2nd ed., McGraw-Hill, New York, pp 327–330, 1971 heat transfer engineering 615 [2] Eckert, E R G., and Drake, R M., Jr., Analysis of Heat and Mass Transfer, International Student Edition, McGraw-Hill, New York, 1972 [3] Jaluria, Y., Natural Convection Heat and Mass Transfer, Pergamon, Oxford, UK, 1980 [4] Gebhart, B., Jaluria, Y., Mahajan, R L., and Sammakia, B., Buoyancy-Induced Flows and Transport, Hemisphere, New York, 1988 [5] Burmeister, L C., Convective Heat Transfer, 2nd ed, Wiley, New York 1993 [6] Bejan, A., Convection Heat Transfer, 2nd ed., John Wiley & Sons, New York, 1995 [7] Martynenko, O G., and Khramtsov, P., Free-Convective Heat Transfer, Springer Verlag, Berlin, Germany, 2005 [8] Goldstein R J., et al., Heat Transfer—A Review of 2003 Literature, International Journal of Heat and Mass Transfer, vol 49, pp 451–534, 2006 [9] Eckert, E R G., Introduction to the Transfer of Heat and Mass, McGraw-Hill, New York, 1950 [10] Ostrach, S., An Analysis of Laminar Free-Convection Flow and Heat Transfer about a Flat Plate Parallel to the Direction of the Generating Body Force, NACA TN 2635, 1952 [11] Lefevre, E J., Laminar Free Convection From a Vertical Plane Surface, in Ninth International Congress of Applied Mechanics, Brussels, pp 1–168, 1956 [12] Kuiken, H K., Free Convection at Low Prandtl Numbers, Journal of Fluid Mechanics, vol 37, pp 785–798, 1969 [13] Merkin, J H., A Note on the Similarity Solutions for Free Convection on a Vertical Plate, Journal of Engineering Mathematics, vol 19, pp 189–201,1985 [14] Merkin, J H., Free Convection on a Heated Vertical Plate: The Solution for Small Prandtl Number, Journal of Engineering Mathematics, vol 23, pp 273–282, 1989 [15] Merkin, J H., Pop, I., and Mahmood, T., Mixed Convection on a Vertical Surface with a Prescribed Heat Flux: The Solution for Small and Large Prandtl Numbers, Journal of Engineering Mathematics, vol 25, pp 165–190, 1991 [16] Ramanaiah, G., and Malarvizhi, G., Unified Treatment of Free Convection Adjacent to a Vertical Plate with Three Thermal Boundary Conditions, Warme Stoffubertrag, vol 27, pp 393–396, 1992 [17] Sparrow, E M., and Gregg, J L., Laminar Free Convection from a Vertical Plate with Uniform Surface Heat Flux, Trans ASME, vol 78, pp 435–440, 1956 [18] Squire, H B., Integral Solution Published in S Goldstein, ed., Modern Developments in Fluid Dynamics, Dover, New York, vol 2, pp 641–643, 1965 [19] Churchill, S W., and Usagi, R., A General Expression for Correlation of Rates of Transfer and Other Phenomena, AIChE Journal, vol 18, pp 1121–1128, 1972 [20] Churchill, S W., and Chu, H H S., Correlation Equations for Laminar and Turbulent Free Convection From Vertical Plates, International Journal of Heat and Mass Transfer, vol 18, pp 1323–1329, 1975 [21] Lin, W., Armfield, S W., and Morgan, P L., Unsteady Natural Convection Boundary-Layer Flow Along a Vertical Isothermal Plate in a Linearly Stratified Fluid with Pr > 1, International Journal of Heat and Mass Transfer, vol 45, pp 451–459, 2002 vol 31 no 2010 616 M BACHIRI AND A BOUABDALLAH [22] Lin, W., and Armfield, S W., Unsteady Natural Convection on an Evenly Heated Vertical Plate for Prandtl Number Pr < 1, Physics Review E 72, 066309, 2005 [23] Lin, W., Armfield, S W., and Patterson, J C., Unsteady Natural Convection Boundary-Layer Flow of a Linearly-Stratified Fluid With Pr < on an Evenly Heated Semi-Infinite Vertical Plate, International Journal of Heat and Mass Transfer, vol 51, pp 327– 343, 2008 [24] Armfield, S W., Lin, W., and Patterson, J C., Direct and Scaling Investigation of the Natural Convection Boundary Layer on an Evenly Heated Plate, International Journal of Heat and Mass Transfer, vol 50, pp 1595–1602, 2007 [25] Shapiro, A., and Fedorovich, A E., Unsteady Convectively Driven Flow along a Vertical Plate Immersed in a Stably Stratified Fluid, Journal of Fluid Mechanics, vol 498, pp 333–352, 2004 [26] Shapiro, A., and Fedorovich, E., Prandtl Number Dependence of Unsteady Natural Convection along a Vertical Plate in a Stably Stratified Fluid, International Journal of Heat and Mass Transfer, vol 47, pp 4911–4927, 2004 heat transfer engineering Mohamed Bachiri received his magister of science in energy and fluid mechanics engineering from University of Sciences and Technology Houari Boumedi`ene (USTHB) in Algiers, Algeria He is currently a doctoral student at the Laboratory of Thermodynamics and energetic Systems in the Faculty of Physics in USTHB, in Algiers, Algeria Ahcene Bouabdallah is a professor of fluid mechanics and thermodynamics at the University of Sciences and Technology Houari Boumedi`ene (USTHB) in Algiers, Algeria He is presently the director of the Laboratory of Thermodynamics and Energetical Systems and head of interfacial dynamics in the Continued Flow Research Group of the Laboratory His research interests lie in the physics of turbulence, instability, and transition to turbulence in the rotating flow systems He has more than 100 publications, including full-length proceeding papers He is a member of several national and international scientific committees vol 31 no 2010 Heat Transfer Engineering, 31(7):617–624, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903425940 Three-Phase Linear Motor Heat Transfer Analysis Using the Finite-Element Method DE-SHAU HUANG,1 JIYE-SIANG SHIH,2,3 HUNG-CHIH HSIA,2 and MING-TZER LIN2 Graduate School of Vehicle & Mechatronic Industry, Nan-Kai University of Technology, Nantou, Taiwan, Republic of China Institute of Precision Engineering, National Chung Hsing University, Taichung, Taiwan, Republic of China Hiwin Technologies and MiKrosystem Corporation, Taichung, Taiwan, Republic of China As the need for speed and precision in modern manufacturing technologies increases, linear motors are attracting considerable research attention Heat management and thermal deformation associated with linear motors comprise a major concern tied to the demand for machining accuracy Here we describe a three-dimensional finite-element simulation analysis method that utilizes a three-phase linear synchronous motor for thermally characterizing a feed-driven system We also present results from simulations and experiments aimed at verifying system thermal time constants and temperature distributions These results demonstrate that the proposed analytical method is capable of making accurate and useful predictions of temperature and thermal behavior distributions for linear motors in terms of conventional heat generation in various transient and steady-state phases It is our hope that these results will support future efforts to design modern feed-driven systems equipped with linear motors INTRODUCTION Remarkable improvements in position actuator technology are supporting the development of large-scale, cost-effective, high-resolution, and high-speed drive systems Due to their simplicity and short dynamic response times, linear actuator motors are currently employed in a broad range of precision engineering applications Since they use direct-drive mechanisms without transmission elements, linear actuator motors are capable of producing high position accuracy without backlash or transmission parts that can break down This leads to reduced unit power consumption and increased durability [1, 2] Due to their direct feed and drive, straightforward load, and movement power transformation characteristics, linear motors can be used in systems that require large amounts of thrust, fast We are grateful to the Taiwan National Center for High-Performance Computing for access to their facilities This work was supported by the Hiwin Technologies and Mikrosystem Corporation, Taichung, Taiwan; we thank Hiwin employees for their technical assistance Partial support was provided by the Republic of China Ministry of Education under the ATU program Address correspondence to Professor Ming-Tzer Lin, Institute of Precision Engineering, National Chung Hsing University, Taichung 40249, Taiwan, Republic of China E-mail: mingtlin@nchu.edu.tw acceleration, sustained speed and accuracy, quick dynamic response, and high servo rigidity [1] As demands for speed and precision positioning increase in modern processing technology, linear actuator motors are replacing many traditional motor models The need for precision and transient response in linear motor-driven systems at adjustable speeds makes it critical to enhance velocity and position control efficiency and accuracy Achieving these goals requires addressing issues associated with the thermal characteristics of feed and drive mechanisms For instance, high-speed cutting machines that use feed-driven linear motors require fast material removal rates with minimum tool wear and near-perfect surface finishing Furthermore, there is a strong requirement for positioning accuracy in processes that use linear motor systems [2] Standard linear motor coil generators and frictional parts can suffer significant damage due to heat generation While the primary requirement of linear motors is speed, associated heat and thermal deformation can adversely affect overall performance and accuracy Excess heat generated by the linear motor coil (usually the main heat source) can cause thermal stress and deformation during the drive stage A quick inspection of the current literature shows that thermal characterization studies of machine tools equipped with feed and drive mechanisms have mostly focused on ball-screw tools 617 618 D.-S HUANG ET AL [3, 4] To our knowledge, only a handful of researchers have focused on linear motor thermal behavior and optimization Kim et al [6, 7] performed heat transfer analyses of linear motor machine tools and presented their findings on the heat produced by the motor coil and deformation friction Yun et al and Wu et al [3, 4] discuss heat deformation error in the structure of feed drive systems and its effect on linear positioning These researchers have addressed the thermal behavior characteristics of linear motors as applied to advanced manufacturing tasks Most of these studies are machine-specific (e.g., the Kim et al [7] study of the Daewoo FH500 horizontal machining center, which uses 1FN3 and SIEMENS linear motors [8]), meaning the results may not be applicable to other products Coil- and friction-generated heat in a horizontal machine can differ from that generated by conventional three-phase linear motors; therefore, the latter case requires additional research considerations and assumptions We used a three-dimensional thermomechanical finiteelement analysis approach to study the thermal behavior of a feed-driven stage system equipped with a conventional threephase linear motor, and established linear motor temperature and thermal behavior in both transient and steady-state stages Experiments were conducted to verify our simulation results and the system’s thermal time constant The methodology used in this study can be applied to other problems involving complex structures and operations FINITE-ELEMENT METHOD (FEM) MODEL FOR LINEAR MOTORS Figure shows a conventional stage equipped with a linear motor and heat-generating coil assembly, which can affect both stage position and motor performance A geometric model was established using finite-element analysis (FEA) with ANSYS 11 for estimating the temperature distribution and thermal effect on the feed-driven system A three-dimensional geometric sketch of the feed-driven stage was created using computer-assisted drafting (CAD) (length 200 mm, width 138 mm, height 60 mm) (Figure 2) The stage consists of a linear motor with a platform attached near its top, with convection heat transfer occurring via air gaps on each side For computation purposes, the geometric configuration of the feed-driven stage was meshed to tetrahedral elements; mesh size independence analyses were performed prior to the simulation Comparisons were made for 180,438 nodes and 90,994 elements; a temperature difference of 0.1◦ C was observed As shown in Figure 3, we used 90,994 elements and 133,290 nodes for the feed-driven stage In order to avoid singularity issues and to obtain more accurate simulation results, a large number of nodes and a smaller number of elements were meshed in the corner sections surrounding the heat generator Detailed descriptions of material properties for each feed-driven stage section are given in Table heat transfer engineering Figure A practical feed-driven stage with a linear motor: (a) top view and (b) side view Heat Transfer Governing Equation Feed-driven stage power is generated by a coil assembly inside the linear motor Most heat is generated by the coil assembly and dissipated from the stage The heat transfer governing equation for a feed-driven stage is expressed as ∂ ∂x k ∂T ∂x + ∂ ∂y k ∂T ∂y + ∂ ∂T k ∂z ∂z + q˙ = ρC ∂T ∂t (1) where k is the thermal conductivity of the material, ρ the density of the material, C the heat capacity of the material, and q˙ the rate of energy generated per unit volume For steady-state conditions vol 31 no 2010 D.-S HUANG ET AL 619 Table The characteristic of the materials of the linear motor Material Epoxy PTFE Silicon steel Cu Al alloy 6061 k (W/m-◦ C) ρ (kg/m3 ) C (J/kg-◦ C) α (×10−6 ◦ C) ν E (GPa) 0.19 0.25 66.1 385 130 1100 2170 7700 8900 2700 1050 1050 434 380 896 81 126 18 0.17 23.6 0.3 0.46 0.22 0.35 0.33 2.41 0.4 200 115 69 case are k Figure The geometric dimension of a feed-driven stage k ∂T ∂x + ∂ ∂y k ∂T ∂y k + ∂ ∂T k ∂z ∂z + q˙ = (2) Under normal operating conditions, the feed-driven stage moves at a speed of 1.2 m/s, resulting in an estimated Reynolds number of approximately 15,000 under laminar flow conditions In addition, researchers must consider forced convection conditions that surround the linear motor and stage on all four sides A heat transfer correlation equation [9] at a flat plate for laminar flow was used to estimate a heat transfer coefficient as follows: St Pr2/3 = 0.332 Re−0.5 (4) at the contact surface and the equation can be simplified to ∂ ∂x ∂T ∂T ∂T +k +k =q ∂x ∂y ∂z (3) In the present study, heat conduction was suggested between the top of the linear motor and the bottom of the platform A very thin gas gap exists at the bottom of the feed-driven stage, making heat convection the given condition Boundary conditions in this ∂T ∂T ∂T +k +k = h · (Ts − T∞ ) ∂x ∂y ∂z (5) in the surrounding area In Eq (5), h is the convection heat transfer coefficient, Ts the surface temperature, and T∞ the surrounding temperature For transient situations, the initial feed-driven stage temperature is equal to the surrounding temperature In these cases the initial and boundary conditions are expressed as Ti (x, y, z, 0) = T∞ −k ∂T ∂S = h · (Ts − T∞ ) (6) (7) (xS ,yS ,zs ,t) A conventional finite-element method with tetrahedral elements derived from the Galerkin approach for governing heat transfer was then applied using ANSYS 11 Heat Generation Equation Since the major heat source in a linear motor is the coil assembly, we purposefully focused on that assembly instead of on friction Heat generated from a coil assembly is the result of coil resistance (which varies with temperature), the electromagnetic eddy current, and hysteresis Heat generation is a complex phenomenon Electromagnetic and thermal analyses of a linear motor coil assembly are described in detail in [10] In linear motor designs for industrial applications, electrical current is the main source of heat generation For the present study we assumed that heat generation from a linear motor can be expressed as P = I 2R Figure Mesh configurations of the feed-driven stage and the linear motor heat transfer engineering (8) since the continuous electrical current generates power to move the stage In the steady-moving feed-driven stage, heat in the coil assembly is generated by continuous electrical current and vol 31 no 2010 620 D.-S HUANG ET AL Figure The schematic of the experimental setup fixed electrical resistance Temperature distribution can be calculated using FEM post processing based on three assumptions: (a) The heat generation rate of the coil assembly is defined as a constant per volume, and the temperature gradient is uniform in the magnet base; (b) heat transfer is calculated using conduction between the linear motor and stage, neglecting the contact thermal resistance effect; and (c) the thermal radiation effect is not considered Figure Temperature distribution of the linear motor terminal block with 30 pins for signal connections, and a 68-pin I/O connector for use with E Series and S Series multifunction DAQ devices Data were collected with a PC equipped with a National Instrument PCI-6014 card RESULTS AND DISCUSSION VALIDATION Heat Transfer Simulation The setup for our validation experiment is shown in Figure Verification tests consisted of retrieving temperature data from various measurement positions in the thermocouple converter via a connector block, RS232 connector, DAQ card, and LabView programmed PC interface Retrieved data were stored for purposes of monitoring system temperature distributions and validating results from finite-element simulations The machine tool stage used in this experiment was provided by Hiwin Technologies and Mikrosystem (Hsinchu, Taiwan) The stage consisted of a three-phase linear motor, linear motor stage, and linear motor drivers The main system stage connection package included a motor power cable, signal transmission arrays, position feedback arrays, driver and controller pulse lines, and transmission interfaces Thermocouple sensors were mounted at three different positions to simultaneously measure temperatures from the feed and drive operations The original position was in the bottom left-hand corner of the feed-driven stage As shown in Figure 2, positions 1, 2, and were on the right-hand, top-center, and front sides of the drive stage, respectively Cartesian coordinates for the original bottom left-hand corner positions were (in mm) (100, 30,138), (100, 60, 69), and (11, 30, 69) Our connector was a National Instrument BNC-2110 block, consisting of a desktop and a DIN rail-mountable BNC adapter that allowed for a connection to a data acquisition (DAQ) device The BNC-2110 assembly contains 15 BNC connectors, a Results for linear motor steady-state temperature distribution are shown in Figure They indicate that the maximum temperature (86.7◦ C) occurred at the bottom of the motor, near the coil assembly We observed a 2◦ C maximum-to-minimum temperature difference in the linear motor, suggesting poor dissipation from the motor bottom—in other words, the thin gap at that location does not significantly contribute to heat dissipation Lower temperatures were observed at the top corners heat transfer engineering Figure Temperature distribution of the feed-driven stage vol 31 no 2010 D.-S HUANG ET AL 621 Figure The transient temperature distribution in (a) 3,000 s, (b) 6,000 s, (c) 9,000 s, and (d) 12,000 s of the motor, suggesting that heat conduction from the linear motor to the adjacent stage may be the main contributor to heat dissipation According to the feed-driven stage temperature distribution shown in Figure 6, the highest temperature was approximately 86◦ C, recorded near the coil assembly at the topcenter part of the stage The temperature on the linear motor sides was 84.7◦ C; lower temperatures in other sections were the result of conduction and convection The 1◦ C range across the entire stage indicates a uniform temperature distribution In transient situations, feed-driven stage temperatures were recorded at the beginning of operation and at 3,000, 6,000, 9,000, and 12,000 s (Figure 7) At all times the highest temperatures were observed at the bottom of the linear motor, near the coil assembly Heat was subsequently diffused through the center of the linear motor to its outer parts The recorded temperature range at the top center part of the motor increased from 54◦ C to 82◦ C as operation time increased During the feedheat transfer engineering driven stage, we noted a maximum-to-minimum temperature difference of approximately 2.3◦ C during each successive time frame (3,000, 6,000, 9,000, and 12,000 s) On the top surface of the feed-driven stage we observed a higher temperature in the top-center position, indicating that core high temperature areas expanded over time Those core high temperature areas maintained the same shape until steady-state conditions were achieved Simulation-Experiment Comparisons Experiments were performed under various transient conditions until a steady state was reached Results from our comparison of simulation and experimental data for temperature values at points 1, 2, and in Figure versus operation time are presented as Figures 8a–c They show that temperatures vol 31 no 2010 622 D.-S HUANG ET AL approached a steady state at approximately 12,000 seconds of operation Compared to the simulation results, all three experimental temperatures were approximately 7◦ C lower than temperatures calculated prior to 7,000 seconds of operation This finding is in agreement with those previously reported by Kim et al [6], who reported simulation temperatures that were several degrees lower than those from their experiments Note that under transient conditions, the combination of the convention coefficient of air flowing around the system and the heat capacity of the linear motor and stage materials explains the observed changes in temperature over time Both factors may have influenced our simulation results, as well as the slight difference between those results and the experimental data that we observed over time Still, the convention heat transfer coefficient from the correlation equation was close to the actual data Due to the presence of compound materials in the feed-driven stage, our results were likely affected by their heat capacities and thermal contact resistance characteristics Overall, the simulation results were in close agreement with the experimental data for system temperature distribution, actual temperature increment data, and time required to reach thermal equilibrium Thermal Time Constant Evaluation In manufacturing processes that use a combination of feeddriven linear motors with stages, knowledge of actual thermal behavior during transient situations is considered very useful This is especially true for the thermal time constant, considered an important feed-driven stage design parameter because it influences stage operation stability and accuracy The transient conduction equation is expressed as θ(t) = T (t) − Ti −t = 1−eτ Tsteady − Ti (9) with θ representing a temperature difference ratio and τ a thermal time constant In Eq (9), the thermal time constant is associated with surface area, material density and volume, heat capacity, and a convection heat transfer coefficient Accordingly, in conventional manufacturing processes it is difficult to accurately estimate the thermal time constant of a linear motor/stage Figure Results of comparison between measurement and FEA on (a) point 1, (b) point 2, and (c) point heat transfer engineering Figure Thermal time constant evaluation on point of feed-driven stage with a linear motor vol 31 no 2010 D.-S HUANG ET AL Figure 10 Thermal deformation of the feed-diven stage combination, since the convection heat transfer coefficient has wide variation Based on the transient temperature distribution from the experimental data, the thermal time constant can be found by fitting its curve The example shown in Figure was computed using MATLAB software We used the temperature measurement results (Figure 2, point 2) over successive time frames to quantify this behavior, and used the following equation to determine curves with the best fit: T2 (t) − 28.0 −t (10) = − e 2690 θ2 (t) = 54.7 According to the procedure just described, the thermal time constants for the other two measurement points in Figure were 2,758 and 2,317 s, respectively, meaning that the average thermal time constant for the feed-driven stage was 2,588 seconds Thermal Deformation Temperature change in linear motors and stages introduces thermal deformation, which can cause position errors After solving a stage thermal deformation problem, temperature controllers and error curves can be used to correct position errors To evaluate the thermal deformation of a feed-driven stage caused by linear motor-generated heat, thermal strain estimation simulations can be performed using ANYSIS 11 with constraining conditions along the bottom rail of the feed-driven stage The stage deformation simulation results shown in Figure 10 indicate larger thermal deformation in the surrounding stage and linear motor Maximum deformation (0.15 mm) was noted at the front and rear edges of the stage, closest to the linear motor CONCLUSIONS In this study we analyzed the heat transfer of a stage equipped with a linear motor, in both steady and transient states A finiteheat transfer engineering 623 element geometric model and a simple heat generation model were established to calculate the thermal behavior of a threephase linear motor FEM results were used to predict the temperature distributions of the feed-driven stage and linear motor, and a measurement experiment was conducted to verify the simulation results Results from a simulation based on a simple heat generation model agreed with the experiment results, indicating that the highest temperature occurred at the bottom of the stage—the result of poor convective heat transfer from a thin gap at that location Lower temperatures were observed at the sides and top-center position of the linear motor—the result of a combination of conductive heat transfer from the platform and convention heat transfer to ambient surroundings According to an analysis of transient temperature distribution and thermal deformation results, the estimated thermal time constant was 2,588 s Maximum thermal deformation (0.15 mm) occurred at the stage edges NOMENCLATURE C E h I k P Pr q˙ q R Re St S t T x, y, z specific heat (J/kg-◦ C) Young’s modulus (N/m2 ) convection heat transfer coefficient (W/m2 -◦ C) electrical current (Amp) thermal conductivity (W/m-◦ C) heat generation (W) Prandtl number (dimensionless) rate of energy generation per unit volume (W/m3 ) heat flux (W/m2 ) coil electrical resistance (ohm) Reynolds number (dimensionless) Stanton number (dimensionless) surface of the material time (s) temperature (◦ C) rectangular coordinates (m) Greek Symbols α ρ τ θ ν thermal expansion coefficient (1/◦ C) density (kg/m3 ) thermal time constant (s) a ratio of temperature difference (dimensionless) Poisson’s ratio Subscripts i s x y z ∞ initial surface rectangular coordinate system in the x-direction rectangular coordinate system in the y-direction rectangular coordinate system in the z-direction surroundings vol 31 no 2010 624 D.-S HUANG ET AL REFERENCES [1] Techn, S C., and Peter, H C., The Application of Linear Motor, Proc Power Electronics and Motion Control Conf., Beijing, vol 3, pp 1336–1341, 2000 [2] Gordon, S., and Hillery, M T., Development of a High CNC Cutting Machine Using Linear Motor, Journal of Materials Processing Technology, vol 166, pp 321–329, 2005 [3] Yun, W S., Kim, S K., and Cho, D W., Thermal Error Analysis for a CNC Lathe Feed Drive System, International Journal of Machine Tools & Manufacture, vol 39, pp 1087–1101, 1999 [4] Wu, C H., and Kung, Y T., Thermal Analysis for the Feed Drive System of a CNC Machine Center, International Journal of Machine Tools & Manufacture, vol 43, pp 1521–1528, 2003 [5] Widdowson, G P., Youyong, L., and Gaunekar, A S., Design of a High Speed Linear Motor Driven Gantry Table, Proc Power Electronic Drives and Energy Systems for Industrial Growth Conf., Perth, Australia, vol 2, pp 936–941, 1998 [6] Kim, J Y., Kim, Y J., and Kim, J O., Heat Transfer Analysis and Simplified Thermal Resistance Modeling of Linear Motor Driven Stage for SMT Applications, IEEE Transactions on Components and Packaging Technologies, vol 26, no 3, pp 532–540, 2003 [7] Kim, J J., Jeong, Y H., and Cho, D W., Thermal Behavior of a Machine Tool Equipped With Linear Motors, International Journal of Machine Tools & Manufacture, vol 44, pp 749–758, 2004 [8] Linear motors 1FN1/1FN3 (Overview), Order no 6ZB54110BD02-0BA0, 2002, Siemens Available online at: http://www sea.siemans.com/us/Internet/MachinToolsComm/General/Docs/ siemens linear motors 1FN1 1FN3.pdf [9] Kays, W M., and Crawford, M E., Convection Heat and Mass Transfer, 2nd ed., McGraw-Hill, New York, 1984 [10] Abdou, G., and Tereshkovich, W., Performance Evaluation of a Permanent Magnet Brushless DC Linear Drive for High-Speed Machining Using Finite-Element Analysis, Finite Elements Analysis and Design, vol 35, pp 169–188, 2000 De-Shau Huang is currently an assistant professor in the Graduate School of Vehicle & Mechatronic Industry at Nan-Kai University of Technology in Taiwan He received his B.S degree in vehicle engineering from Chung-Cheng Institute of Technology in 1987, M.S degree in mechanical engineering from National Taiwan University in 1993, and Ph.D degree in mechanical engineering from Lehigh University, Bethlehem, PA, in 2003 He had worked in the Ordnance Readiness and Development Center (ORDC) since 1993, served as a professional engineer He was the director of R&D depart- heat transfer engineering ment before leaving ORDC His research interests include heat transfer in two-phase flow, heat transfer analysis in electronic devices, and computational fluid dynamics to internal flows Jiye-Siang Shih is currently a professional engineer for Hiwin Technologies & Mikrosystem Company, Taichung Industrial Park, Taiwan He received his B.S degree in the Department of Bio-industrial Mechatronics Engineering from National Chung Hsing University, Taiwan, in 2002 and M.S degree from the Institute of Precision Engineering at National Chung Hsing University, Taiwan, in 2006 He has worked at Hiwin Technologies & Mikrosystem since 2006 His job function is related to heat transfer analysis in machine tools and manufacturing, mechanical, and electrical magnetic behavior of linear motor, actuator, and electronic devices Hung-Chih Hsia is currently a graduate student at the Institute of Precision Engineering, National Chung Hsing University, Taichung, Taiwan He received his B.S degree from the Department of Mechanical Engineering of National Chin-Yi University of Technology, Taiwan, in 2006 His research includes heat transfer and finite-element analysis in machine tools and electronic devices Ming-Tzer Lin is currently an associate professor at the Institute of Precision Engineering, National Chung Hsing University, Taichung, Taiwan He earned his B.S degree in mechanical engineering from National Cheng Kung University, Taiwan, in 1994 and his M.S and Ph.D degrees in mechanical engineering from Lehigh University, Bethlehem, PA, in May 1999 and January 2004, respectively He subsequently became a postdoctoral scholar in the Materials Science and Engineering Department, Lehigh University, Bethlehem, PA In August 2004 he joined the National Chung Hsing University faculty as an assistant professor at the Institute of Precision Engineering and was promoted to associate professor with tenure in 2009 He has published 15 international peer-reviewed articles and conference papers in the last years His research involved the finite-element analysis of temperature and time-dependent mechanical behavior of materials, processing and mechanical engineering aspects of MEMS, small-scale structures, machine tools, and manufacturing, with an emphasis on the mechanical behavior of materials for the application of electrical engineering and microelectronics technology vol 31 no 2010 Heat Transfer Engineering, 31(7):625–626, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903543353 complete line of sensors for all of their HVAC needs Setra’s humidity offering will meet the quality, delivery, and cost needs of its customers while providing the best solution for all of their sensor requirements For more information on the 3100 Series or Setra’s complete product line, contact Setra Systems, Inc.,159 Swanson Road, Boxborough, MA 01719; call 1-800-257-3872, fax: 978-264-0292 or e-mail: sales@setra.com new products and services SETRA SYSTEMS INTRODUCES ITS NEW SRH HUMIDITY SERIES PRODUCT 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tips are easily removed in all three models, allowing the end user to replace the sensors on-site This essentially eliminates time consuming and costly factory calibration, while reducing downtime during service intervals Additionally, the duct mount probe is easily accessed by removing the front cover, pulling out the sensor board assembly, and replacing the sensor tip This further contributes to a more user-friendly, lower cost product line that is focused on customer needs and ease of use Setra’s humidity sensors are designed to complement its pressure and current products, thereby, providing customers with a GEMS Introduces Ultrasonic Continuous Liquid Level Sensor for Challenging Fluid Measurement R Gems Sensors & ControlsTM (Gems), a global leader in liquid level, pressure, and flow sensors, miniature solenoid valves, and fluidic systems has launched the UCL-520 Series of Ultrasonic Level Sensors The UCL-520 Series is suitable for non-contact applications such as industrial water treatment, wastewater management chemical storage as well as many other challenging fluid applications The UCL-520 is a two-wire ultrasonic transmitter for measurement up to 26.2 (8m) and is built with a rugged PVDF transducer for challenging ultra pure, corrosive or waste liquids The highly accurate sensor is easy to install within limited space applications and the minimum dead band allows for maximum fill of media The UCL-520 offers a (7.6cm) minimum beam width for restricted space applications and the selectable display indicates level in air gap or liquid height The UCL-520 is push button calibrated with 6-segment LCD display and is broadly selected for corrosive media including atmospheric bulk storage, day tank and waste sump applications 625 626 “Gems’ new line of ultrasonic level sensors offers our customers an innovative non-contact solution for both small and large tank applications These highly accurate sensors provide flexibility and control for challenging fluid measurement applications” says Kevin Castonguay, Product Marketing Manager for Gems Sensors & Controls R Watlow Introduces CAST-X 3000 Circulation Heater For Reliable Indirect Heating R Watlow , a designer and manufacturer of electric heaters, controllers and temperature sensors, introduces the CAST-X 3000 circulation heater—an economically priced heater offering high flow volume at the required temperature through a single compact product Typical applications include steam generation and superheating for process applications and applications where cleanliness is a major concern These include heating volatile materials in cases where contamination could be a problem such as deionized (DI) water and in food and beverage applications The CAST-X 3000 is also available with optional explosion proof enclosure for use in hazardous environments This newest edition to Watlow’s CAST-X family of heaters consists of two helical coiled tubes and tubular elements cast into a robust aluminum body which serves as the heat transfer media between the tubular element and the process tubes The use of heavy wall 316L seamless process tubing assures high reliability and high pressure operation The CAST-X 3000 features a fluid path independent of the heater sheath preventing fluid contamination, an integrated thermostat housing allowing the heater to run dry and the dual tube construction allowing water flow rates to 20 GPM when run in parallel Its non-welded construction minimizes potential leakage, allows highpressure operation and is self-draining when mounted vertically For additional information call your nearest Watlow representative: Watlow, Phone: 1+(800) WATLOW2, 1+ (314) 878-4600; Fax: 1+ (817) 893-1005, 1+ (314) 878-6814; Internet: www.watlow.com; Email: info@watlow.com Chromalox “Super Heater” Meets Unprecedented Demands for Power and Durability Chromalox, a leading manufacturer of electric heat and control products, announced that it has improved the heating capability and corrosion resistance of its horizontal electric vaporizers to create “superheater” technology to support production of materials for alternative energy systems The enhancements allow the production process to vaporize materials rather than heating them in a furnace “We are always working with our customers to engineer our solutions to ensure they meet their specific requirements,” said Mark Wheeler, manager, packaged systems for Chromalox “This application demanded more from a Chromalox vaporizer than any application we’d ever encountered and proved to be an opportunity to demonstrate the functionality of our basic vaporizer technology in an extreme application.” Chromalox horizontal electric vaporizers are designed to transfer heat at high temperatures and low pressures and are preengineered, pre-wired, and pre-piped for flexible, dependable and heat transfer engineering efficient operation The recent enhancements to the technology open up new applications for these vaporizers, including hydrocarbon vaporizing For more information, go to www.chromalox.com or contact Barbara Lee, Chromalox, Inc., 412-967-3803; barbara.lee@ chromalox.com SETRA SYSTEMS, INC INTRODUCES NEW MODEL 201 PRESSURE TRANSDUCER FOR MEASURING VERY LOW DIFFERENTIAL OF GAUGE PRESSURE: Technology Based on Setra’s Patented Variable Capacitance Sensor Design Setra Systems, Inc., a leading designer and manufacturer of pressure measurement devices, introduces the Model 201 Pressure Transducer Its rugged design, wide operating temperature range (−40◦ F to +175◦ F), and high 45 psi overpressure capability makes it ideal for the most demanding applications, including vapor recovery systems, exhaust gas control systems and industrial scrubbers The Model 201’s all-welded, no o-ring construction results in a leak-free design and is ideal for critical low range applications Based on Setra’s patented variable capacitance sensor design, this unit is used to measure very low differential or gauge pressures in ranges as low as ±2.5 in W.C up to 20 psi, and provides a to 20 mA output that is linear with applied pressure R The Model 201 features an Inconel diaphragm and an insulated electrode, which forms a variable capacitor As the sensor pressure increases or decreases, the capacitance changes This change in capacitance is detected and converted to a fully conditioned linear current output signal For further details on the new Model 201 or Setra’s complete product line, contact Setra Systems at 159 Swanson Road, Mail Stop 14-26, Boxborough, MA 01719; call 1-800-257-3872; fax: 508264-0292 or e-mail: sales@setra.com Visit Setra on the World Wide Web at http://www.setra.com vol 31 no 2010