Heat Transfer Engineering, 31(3):159–167, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903304335 Similarities and Differences Between Flow Boiling in Microchannels and Pool Boiling SATISH G KANDLIKAR Mechanical Engineering Department, Rochester Institute of Technology, Rochester, New York, USA Recent literature indicates that under certain conditions the heat transfer coefficient during flow boiling in microchannels is quite similar to that under pool boiling conditions This is rather unexpected, as microchannels are believed to provide significant heat transfer enhancement under single-phase as well as flow boiling conditions This article explores the underlying heat transfer mechanisms and illustrates the similarities and differences between the two processes Formation of elongated bubbles and their passage over the microchannel walls have similarities to the bubble ebullition cycle in pool boiling During the passage of elongated bubbles, the longer duration between two successive liquid slugs leads to wall dryout and a critical heat flux that may be lower than that under pool boiling conditions A clear understanding of these phenomena will help in overcoming these limiting factors and in developing strategies for enhancing heat transfer during flow boiling in microchannels INTRODUCTION The nucleation criterion developed by Hsu [1] has been successful in predicting the onset of nucleation in pool boiling as well as in flow boiling The criterion was also shown to be quite accurate for flow boiling in microchannels by a number of investigators, including Zhang et al [2] and Kandlikar et al [3] The high single-phase heat transfer coefficient value prior to nucleation in flow boiling leads to nucleation cavity diameters that are smaller than those in pool boiling This link between pool boiling and flow boiling is an important factor in comparing the two boiling modes In the quest for improved heat removal rates, in general, pool boiling is considered to be more efficient (higher heat transfer coefficient) than single-phase liquid flow, while flow boiling provides the highest heat transfer coefficients However, recent data obtained with enhanced single-phase flow channels and flow boiling in microchannels indicate that this may not be necessarily true with the current status of these two modes of heat transfer Following the definition of Kandlikar and Grande [4], microchannels are defined as channels with hydraulic diameter Address correspondence to Professor Satish G Kandlikar, Mechanical Engineering Department, Rochester Institute of Technology, Rochester, NY 14623, USA E-mail: sgkeme@rit.edu (or the smallest flow passage width of a channel) between 10 µm and 200 µm Table shows a comparison of heat transfer coefficients and heat fluxes for four cases: single-phase flow in plain microchannels, single-phase flow in enhanced microchannels, pool boiling, and saturated flow boiling in plain and enhanced (with reentrant cavities) microchannels Due to the pressure drop constraints, the flow in microchannels is generally in laminar flow regime The single-phase heat transfer coefficients are therefore calculated for laminar flow (including the entrance region effect) The plain microchannels are unable to meet the high heat flux cooling requirement of 1000 W/cm2 (10 MW/m2 ) However, the microchannels enhanced with short offset strip fins provide a very high heat transfer coefficient In a practical system with this geometry, Colgan et al [5] employed multiple-inlet/-outlet regions with a flow length of only mm through the microchannels This configuration holds the most promise in meeting the future chip cooling challenges Results with single-phase flow [5, 6], pool boiling [7, 8], and flow boiling in microchannels under stable and unstable conditions [9, 10] are used in the comparison presented in Table The pool boiling mode at macroscale offers an efficient mode of heat transfer The saturated flow boiling heat transfer with plain microchannels [9] and that with reentrant cavities [10] both provide an improvement over plain microchannels, but fall substantially below the desired values Although these values 159 160 S G KANDLIKAR Table Comparison of heat transfer coefficients with water under different modes Heat transfer mode Single-phase flow in plain 100- to 200-µm square microchannels Single-phase flow in enhanced microchannels, 48 àm ì 256 àm, offset strip fins, 500 µm, Colgan et al [5], Steinke and Kandlikar [6] Pool boiling, flat surface, fully developed boiling, Nukiyama [7], microdrilled surface, Das et al [8] Saturated flow boiling in Dh ∼ 207 µm rectangular microchannels, Steinke and Kandlikar [9], with reentrant cavities, Kuo and Peles [10] h, kW/m2◦ C q , MW/m2 10–15 0.1–0.2 >500 5–10 ∼30–300 1.2–4.8 30–80 2–3.5 are higher than those under pool boiling, the increase is not significant In fact, employing enhanced pool boiling surfaces can improve the performance by a factor of 2–4, e.g., by microdrilling the heater surface with holes at 5–10 mm pitch as reported by Das et al [8] In comparison, a set of parallel microchannels with or without reentrant cavities yields much lower performance as compared to single-phase flow in enhanced microchannels or pool boiling on enhanced surfaces This has been a major concern in developing flow boiling systems to meet the needs in electronic cooling applications There have been a number of papers published exploring the effect of diameter on flow boiling heat transfer A comparison of two data sets obtained by Kenning and Cooper [11] for a 9.6 mm diameter circular tube and by Steinke and Kandlikar [9] for a 207 µm hydraulic diameter rectangular channel is shown in Figure Both data sets are obtained at close to atmospheric pressure and the Boiling numbers are around 1.5 × 10−4 in both cases The effect of diameter on the ratio of two-phase to liquid-only single-phase heat transfer coefficients is depicted Figure Effect of channel hydraulic diameter on the ratio hT P / hLO for Bo ≈ 1.5 × 104 during flow boiling of water near atmospheric pressure heat transfer engineering in Figure It is seen that this ratio is reduced considerably from a value of 9.5 for the 9.6 mm-diameter tube to for the 207 µm channel Further, considering the fact that hLO in the microchannel corresponds to laminar flow conditions, a compelling argument can be made for the dramatic reduction in heat transfer coefficient for the smaller diameter tube It has been suggested by a number of authors, including Lazarek and Black [12], that flow boiling in narrow channels can be predicted reasonably well with a pool boiling correlation Kew and Cornwell [13] compared the flow boiling data in narrow channels with an established pool boiling correlation by Cooper [14] with some degree of success The other correlations that were similarly successful had heat flux as the primary variable This realization, brought about by the success of the pool boiling correlations in predicting the flow boiling in microchannels, is really the precursor to the present article This similarity is further explored using the available literature on experimental data and theoretical models The discussion is focused on water as the working fluid, but by no means is this study intended to be restrictive in this regard The broad availability of experimental data with water makes it possible to present a more comprehensive comparison between the pool boiling and microchannel flow boiling NUCLEATE BOILING AND CONVECTIVE BOILING CONTRIBUTIONS The contributions from nucleate boiling and convective boiling during flow boiling are well recognized The nucleate boiling contribution is dependent on the heat flux, in a manner similar to the pool boiling with an exponent of around 0.7 The convective boiling component is independent of the heat flux and varies with the mass flux For the conventional large diameter tubes, the mass flux dependence was identified with an exponent of 0.8, which is in agreement with the turbulent single-phase flow relationship A flow boiling map proposed by Kandlikar [15] showed these contributions with Boiling number Bo and density ratio, ρL /ρG , as parameters The map was developed with hT P / hLO versus x using the Kandlikar [16] correlation The map was instrumental in explaining the different dependencies observed in the two-phase heat transfer data as a function of quality The nucleate boiling component is adversely affected with an increase in quality, while the convective boiling term increases with quality due to the higher specific volume of vapor being produced The relative contributions from these components are governed by Bo and ρL /ρG A higher density ratio causes a larger increase in the overall flow velocity upon vaporization, leading to a greater increase in the heat transfer coefficient, while a low value of density ratio causes the convective contribution to increase only moderately A combination of low Bo and high ρL /ρG causes hT P /hLO to increase with an increase in x, while a combination of high Bo and low ρL /ρG causes hT P /hLO to decrease with an increase in x vol 31 no 2010 S G KANDLIKAR Table Comparison of two-phase flow structures in the two boiling modes Flow boiling Pool boiling Bubble inception as the nucleation criterion is met for specific cavities under single-phase liquid flow Elongated bubble covering the channel walls Liquid slug being pushed between the two consecutive elongated bubbles Liquid slugs are intensely mixed with vapor in a churn flow Bubble inception as the nucleation criterion is met for specific cavities under natural convection with liquid Growing bubbles covering the heater surface Liquid circulation around the nucleating bubbles as a result of the individual bubble ebullition cycles Liquid surrounding bubbles (undergoing ebullition cycles) is intensely mixed with vapor at high heat fluxes under fully developed boiling conditions These trends, as described by Kandlikar [15], are further affected by laminar flow occurring in small-diameter channels Depending on the single-phase liquid Reynolds number, the flow may be in the laminar region, where the single-phase liquid heat transfer coefficient under fully developed flow conditions is independent of the mass flux This is one of the reasons why the two-phase heat transfer coefficient is dramatically altered in microscale channels Another effect of the small channel dimensions arises due to the changes occurring in the flow patterns The nucleating vapor bubbles are confined in the small channels and grow as elongated bubbles, forming alternate liquid slugs and elongated bubbles The two-phase flow structures during flow boiling resemble the respective pool boiling characteristics as shown in Table The single-phase heat transfer in microchannels is generally under laminar flow conditions due to the pressure drop limitations and the small channel dimensions As pointed out earlier, the convective contribution from the single-phase liquid flow needs to be considered using the laminar flow equation The dependence of the convective contribution is thus altered from the conventional channel trends since the Nusselt number in laminar flow is independent of the flow rate These effects are accounted for in the flow boiling correlation proposed by Kandlikar and Balasubramanian [17] The correlation is rewritten in terms of the density ratio and Boiling number as follows: For 400 ≤ ReLO ≤ 1600: hT P = larger of hLO hT P ,N BD hLO hT P ,CBD hLO 161 hT P ,CBD = 1.136(ρL /ρG )0.45 x 0.72 (1 − x)0.0.08 hLO + 667.2 Bo0.7 (1 − x)0.8 FF l (3) The single-phase heat transfer coefficient hLO is calculated from the laminar flow equation instead of the Gnielinski [18] correlation for the turbulent region In the region of Re from 1600 to 3000, a linear interpolation is recommended For the low Reynolds number range 100 ≤ ReLO < 400, the heat transfer coefficient is found to be always nucleate boiling dominant (NBD) Thus: hT P hT P ,NBD = = 0.6683(ρL /ρG )0.1 x 0.16 (1 − x)0.64 hLO hLO + 1058.0Bo0.7 (1 − x)0.8 FF l (4) In the range Re < 100, the convective component in the above NBD term is reduced further and hT P depends on the nucleate boiling component alone: hT P = 1058.0Bo0.7 (1 − x)0.8 FF l hLO (5) Equations (1)–(5) are used to generate a flow boiling map for microchannels Three values of density ratio, 10, 100, and 1000, and two values of Bo∗ , 10−4 and 10−3 , are used to generate the plots The modified Boiling number Bo∗ is defined as follows: Bo∗ = Bo × (FF l )1/0.7 (6) Figures 2–4 show the plots generated for different Re ranges Figure shows the variation of the ratio hT P / hLO with x with different values of Bo for 400 ≤ ReLO ≤ 1600 This plot is same as the one for large-diameter tubes, but the actual heat transfer coefficient will be different since the single-phase (1) where hT P ,NBD = 0.6683 (ρL /ρG )0.1 x 0.16 (1 − x)0.64 hLO + 1058.0 Bo0.7 (1 − x)0.8 FF l (2) heat transfer engineering Figure Flow boiling map for microchannels in the range 400 ≤ ReLO ≤ 1600, Bo∗ = Bo × (FF l )1/0.7 vol 31 no 2010 162 S G KANDLIKAR HEAT TRANSFER PROCESSES DURING POOL AND FLOW BOILING Factors Responsible for Heat Transfer Degradation in Microchannels Figure Flow boiling map for microchannels in the range 100 ≤ ReLO < 400, Bo∗ = Bo × (FF l )1/0.7 coefficient hLO will be derived from the laminar flow equations Figure shows the flow boiling map for 100 ≤ ReLO < 400 Here the nucleate boiling component begins to play a major role as seen by the continuous decrease in h with x throughout the range In other words, the increased flow velocity at higher x does not provide the expected benefits in terms of improved convective heat transfer Figure shows the flow boiling map for very low values of ReLO < 100 The convective component is completely blocked off; the density ratio has no effect on h Here the suppression effects are overriding and the heat transfer exhibits similar characteristics as in nucleate boiling with the increased suppression effects at higher qualities The flow boiling maps depicted in Figures 2–4 are based on the experimental data and provide a visual tool to illustrate the effects of flow parameters on heat transfer coefficient As the ReLO is reduced, it is seen that the nucleate boiling becomes the dominant mode, with its decreasing trend in h versus x Further decreases in Reynolds number cause the heat transfer to deteriorate, with the elimination of the convective contribution term in Eq (5) The boiling instabilities experienced in microchannels are another major cause for heat transfer reduction These instabilities occur at lower mass fluxes as the inertia of the incoming liquid is insufficient to prevent the liquid from rushing back The reasons for instabilities and methods for preventing them have been discussed in a number of publications, including Serizawa et al [19], Steinke and Kandlikar [9], Hetsroni et al [20], Kandlikar et al [3], and Kuo and Peles [21] As a result of the instabilities, the walls of the microchannels remain exposed to the expanding vapor bubble, creating local dry patches on the wall and causing heat transfer deterioration Another method to avoid the instabilities is to change the operating conditions with increased mass fluxes Dong et al [22] conducted experiments with R-141b in 60 àm ì 200 àm parallel rectangular microchannels for mass fluxes of 400 to 980 kg/m2 -s Boiling was initiated within the channels with subcooled liquid inlet Pressure drop oscillations were not observed and stable boiling was attained The stable results obtained under such conditions were shown to agree quite well with the Kandlikar and Balasubramanian [17] correlation, whereas the unstable data observed in the Steinke and Kandlikar correlation showed a marked deterioration with increasing quality as shown in Figure The results of Dong et al [22] are shown in Figure Although a higher mass flux is beneficial for heat transfer, the resulting pressure drop could be prohibitively large Similarities Between Pool Boiling and Microchannel Flow Boiling Mechanisms Figure Flow boiling map for microchannels in the range ReLO < 100, Bo∗ = Bo × (FF l )1/0.7 heat transfer engineering Some of the recent publications provide an insight into the reasons for this dramatic reduction in h with x, even under stable conditions Using the elongated bubble flow pattern description, Kandlikar [23] pointed out the similarities between the microchannel flow boiling and pool boiling heat transfer As a bubble grows, the downstream interface represents the receding liquid–vapor interface of a growing nucleating bubble, whereas the upstream interface of the elongated bubble is similar to the advancing liquid–vapor interface of a nucleating bubble as its base shrinks and the bubble begins to depart from the heated vol 31 no 2010 S G KANDLIKAR 163 Figure Flow boiling results from Steinke and Kandlikar [9] for water, showing dramatic reduction in heat transfer performance at increased qualities due to instabilities; q is heat flux, W/m2 surface in pool boiling Figure 7a depicts the respective interfaces as elongated bubbles are formed in a microchannel These two interfaces were experimentally studied in a moving meniscus on a heated surface by Kandlikar et al [24] and numerically by Mukherjee and Kandlikar [25] Their studies showed the important roles played by transient conduction as the liquid interface advances over the heater surface The microconvection caused by the liquid flow behind the advancing liquid interface for a moving meniscus is shown in Figure 7b, and during a nucleate boiling bubble ebullition cycle is shown in Figure 7c The receding interface provides a phase change surface where the liquid superheat is dissipated and cooled liquid becomes avail- Figure Elongated bubbles in microchannels presenting advancing and receding interfaces in (a) that are similar to interface movements of a moving meniscus (b) and a nucleating bubble during a bubble ebullition cycle in pool boiling shown in (c) Figure Experimental data for flow boiling of R-141b by Dong et al [22] and predictions from Kandlikar and Balasubramanian [17] at G = 500 kg/m2 s under stable operation, FF l = 1.8, Bo* = [q/(Ghfg )] × FF l = 1.2 × 10−3 (lower line) and 1.6 × 10−3 (upper line) able for the transient conduction process The advancing and receding interfaces seen around an elongated bubble are shown in Figure 7a In the model proposed by Jacobi and Thome [26], the heat transfer in the liquid slug region is assumed to be by laminar steady-state convection, and its contribution is quite small compared to that from microlayer evaporation However, the numerical simulation and the results from transient conduction model described by Mukherjee and Kandlikar [25] and Kandlikar et al [24] indicate that transient conduction and microconvection modes contribute significantly in the evaporating meniscus geometry Mukherjee and Kandlikar [27] simulated the bubble growth and elongated flow pattern development in microchannels and concluded that the transient conduction and the subsequent convection behind theevaporating liquid–vapor interface were the major contributors to the total heat transfer process in microchannel flow boiling as well heat transfer engineering vol 31 no 2010 164 S G KANDLIKAR Role of Microlayer Evaporation during Elongated Bubble Flow Pattern Comparing pool boiling and the microchannel flow boiling processes, the three distinct modes of heat transfer that can be identified in both cases are: Transient conduction heat transfer resulting from the motion of the liquid–vapor interface over the heated surface The heat transfer is enhanced due to the cooler liquid being brought in contact with the heater surface as a result of interface movement Microconvection heat transfer resulting from the increased convection from the interface movement It is combined with the transient conduction contribution effect described above, since it is difficult to identify and isolate their individual effects Microlayer evaporation resulting from the evaporation of a thin layer of liquid left on the heater by the receding liquid– vapor interface Relative contributions from these three mechanisms have been a topic of intense research in pool boiling Myers et al [28] used silicon chips with heaters and sensors to determine the localized heat fluxes and surface temperatures around nucleating bubbles Figure shows the relative contributions from these three mechanisms for water It can be seen that the transient conduction/microlayer convection together are the largest contributor to the total heat flux during a bubble ebullition cycle The microlayer contribution was seen to be quite small, around 20% These results are in agreement with the numerical work by Son et al [29] Recent work by Moghaddam and Kiger [30] showed similar results for FC-72 The microlayer contribution has received considerable attention in recent flow boiling studies in microchannels It is Figure Relative contributions from different mechanisms during pool boiling Redrawn using data from Myers et al [28] Figure Equivalent convective coefficient for films under a steady-state conduction model very difficult to measure the microlayer thickness in the microchannel flows Calculating from the experimental data, a film thickness on the order of 10–20 µm has been estimated by Jacobi and Thome [26] from a parametric study The initial film S G KANDLIKAR 165 Table Similarities and differences between pool boiling and microchannel flow boiling processes Process Similarities Pool boiling Microchannel flow boiling Nucleation The same nucleation criterion by Hsu [1] is applicable for both processes Bubble growth Transient conduction and microconvention heat transfer processes are similar in the liquid slug At higher flow rates, the two-phase flow characteristics of large-diameter tubes appear and the microchannel flow boiling becomes distinctly different from pool boiling Role of microlayer evaporation is relatively limited in both cases, accounting for only 20–25% of the total heat transfer Nucleation cavities and bubble departure sizes are larger The low h in single-phase flow prior to nucleation allows nucleation at lower wall superheats The bulk liquid is not highly superheated prior to onset of nucleation, causing the bubbles to grow predominantly near the heater surface Nucleation cavities and bubble departure sizes are smaller The high h value during single-phase liquid flow prior to nucleation introduces large wall superheat Bulk liquid also reaches a high degree of superheat causing explosive bubble growth following nucleation The microlayer thickness is on the order of 1–3 µm, Koffman and Plesset [34] The high frequency in the bubble ebullition cycle limits the occurrence, extent, and duration of dry patches from microlayer depletion from evaporation at high heat fluxes Smaller bubbles coalesce prior to CHF as the liquid interface retracts away The microlayer under bubbles in flow boiling are thicker and impede heat transfer The lower frequency of elongated bubble passage allows longer time for microchannel wall dryout Microlayer Critical heat flux (CHF) CHF condition results from the inability of the advancing liquid front to rewet the dry patches Heat transfer enhancement Altering nucleation characteristics will provide significant heat transfer enhancements in both cases Providing early nucleation by introducing cavities of right sizes and geometries is successfully implemented in pool boiling Comparison Between the Pool Boiling and Microchannel Flow Boiling Processes The underlying heat transfer mechanisms in the two processes have many similarities, with transient conduction, microconvection, and microlayer evaporation playing similar roles in both The essential differences between the two processes emerge from the presence of strong inertia forces in the bulk flow and the large shear stress present at the wall These forces affect the nucleation and other flow characteristics directly Heat transfer processes are also affected Table summarizes the main features that are common between the two processes The role of gravity is critical in pool boiling, but this effect is negligible in microchannels, where the interface motion is mainly governed by evaporation momentum and inertia forces Although the forces are different, the resulting interface movement leads to similarities in the underlying heat transfer mechanisms in the two cases The effect on critical heat flux is also described, and some enhancement strategies are outlined Avoiding microlayer dryout and avoiding or delaying the elongated bubble formation are seen as ways to improve the heat transfer and CHF in microchannels Microbubbles seem to have great promise in improving the heat transfer They may be generated in microchannels by using localized heating elements heat transfer engineering The dry patches formed during long duration of elongated bubble flow are heated to a high temperature before the arrival of the liquid front, leading to the CHF condition New ideas need to be developed Microbubble generation to avoid or delay formation of large elongated bubbles may lead to higher heat transfer rates Local heating elements driven by pulsed currents, vibrations, or dissolved gases may be used to generate the microbubbles that are supplied with pulsed electric supply Introducing vibrations using piezoelectric elements is also seen as a promising technique to generate microbubbles Although dissolved gases will also lead to generation of microbubbles, their overall effect on the interfacial heat transfer and system performance needs to be investigated Experimental results from Steinke and Kandlikar [36] indicate an increase in the subcooled flow boiling heat transfer at the nucleation, but the heat transfer was reduced as the bubbles formed a thin insulating layer Effective removal of bubbles is important Further research on these topics is warranted CONCLUSIONS The similarities between the pool boiling and microchannel flow boiling processes are discussed The roles of transient conduction, microconvection, and microlayer evaporation during elongated bubble flow patterns in microchannel flow boiling are similar to those in pool boiling Avoiding liquid film dryout, and delaying the formation of elongated bubble flow pattern by introducing microbubbles are proposed as some of the ways to enhance the heat transfer and critical heat flux (CHF) As the flow velocity increases, the microchannel flow vol 31 no 2010 166 S G KANDLIKAR boiling is expected to resemble the flow boiling in minichannels and conventional-sized channels (>3 mm) with the presence of churn flow pattern The resulting high pressure drop needs to be considered while operating under such high flow conditions Shorter flow lengths and improved header arrangements are needed to alleviate the pressure drop limitations Microbubbles are seen as an effective way to improve heat transfer by avoiding or delaying the formation of elongated bubble flow pattern NOMENCLATURE Bo Bo* Dh FF l G h hfg q Re x Boiling number = q /(Ghfg ), dimensionless modified Boiling number, defined by Eq (6), dimensionless hydraulic diameter, m fluid surface parameter, dimensionless mass velocity, kg/m2 -s Heat transfer coefficient, W/m2 -K latent heat of vaporization, J/kg heat flux, W/m2 Reynolds number, dimensionless vapor quality, dimensionless [8] [9] [10] [11] density, kg/m3 [12] Subscripts CBD G L LO NBD TP V [6] [7] Greek Symbols ρ [5] [13] convective boiling dominant gas liquid entire flow as liquid nucleate boiling dominant two-phase vapor [14] [15] [16] REFERENCES [17] [1] Hsu, Y Y., On the Size Range of Active Nucleation Cavities on a Heating Surface, Journal of Heat Transfer, vol 84, pp 207–216, 1962 [2] Zhang, L., Wang, E N., Goodson, K E., and Kenny, T W., Phase Change Phenomena in Silicon Microchannels, International Journal of Heat and Mass Transfer, vol 48, pp 1572–1582, 2005 [3] Kandlikar, S G., Kuan, K., Willistein, D A., and Borrelli, J., Stabilization of Flow Boiling in Microchannels Using Pressure Drop Elements and Fabricated Nucleation Sites, Journal of Heat Transfer, vol 128, pp 389–396, 2006 [4] Kandlikar, S G., and Grande, W., Evolution of Microchannel Flow Passages—Thermohydraulic Performance and Fabrication heat transfer engineering [18] [19] [20] Technology, Heat Transfer Engineering, vol 24, no 1, pp 3–17, 2003 Colgan, E G., Furman, B., Gaynes, M., Graham, W., LaBianca, N., Magerlein, J H., Polastre, R J., Rothwell, M B., Bezama, R J., Choudhary, R., Marston, K., Toy, H., Wakil, J., Zitz, J., and Schmidt R., A Practical Implementation of Silicon Microchannel Coolers for High Power Chips, IEEE Transactions on Components and Packaging Technologies, vol 30, pp 218–225, 2007 Steinke, M E., and Kandlikar, S G., Single-Phase Liquid Heat Transfer in Plain and Enhanced Microchannels, Proceedings of the ASME 4th International Conference on Nanochannels, Microchannels and Minichannels, ICNMM2006, pp 943–951, 2006 Nukiyama, S., Translation—The Maximum and Minimum Values of Heat Q Transmitted from Metal to Boiling Water Under Atmospheric Pressure, International Journal of Heat and Mass Transfer, vol 9, pp 1419–1433, 1966 Original article in Japanese Society of Mechanical Engineers, vol 37, pp 367–374, 1934 Das, S K., Das, P K., and Saha, P., Nucleate Boiling of Water from Plain and Structured Surfaces, Experimental Thermal and Fluid Science, vol 31, pp 967–977, 2007 Steinke, M E., and Kandlikar, S G., An Experimental Investigation of Flow Boiling Characteristics of Water in Parallel Microchannels, Journal of Heat Transfer, vol 126, 518–526, 2004 Kuo, C.-J., and Peles, Y., Local Measurement of Flow Boiling in Structured Surface Microchannels, International Journal of Heat and Mass Transfer, vol 50, pp 4513–4526, 2007 Kenning, D B R., and Cooper, M G., Saturated Flow Boiling of Water in Vertical Tubes, International Journal of Heat and Mass Transfer, vol 31, pp 445–458, 1988 Lazarek, G M., and Black, S H., Evaporative Heat Transfer, Pressure Drop and Critical Heat Flux in a Small Vertical Tube with R-113, International Journal of Heat and Mass Transfer, vol 25, pp 945–960, 1982 Kew, K., and Cornwell, K., Correlations for Prediction of Boiling Heat Transfer in Small Diameter Channels, Applied Thermal Engineering, vol.17, pp 705–715, 1997 Cooper, M G., Saturated Nucleate Pool Boiling—A Simple Correlation, 1st UK National Heat Transactions, Conference, IChemE Symposium Series, vol 86, no 2, pp 785–793, 1984 Kandlikar, S G., Development of a Flow Boiling Map for Subcooled and Saturated Flow Boiling of Different Fluids inside Circular Tubes, Journal of Heat Transfer, vol 113, no 1, pp 190–200, 1991 Kandlikar, S G., A General Correlation for Two-Phase Flow Boiling Heat Transfer inside Horizontal and Vertical Tubes, Journal of Heat Transfer, vol 112, no 1, pp 21–228, 1990 Kandlikar, S G., and Balasubramanian, P., An Extension of the Flow Boiling Correlation to Transition, Laminar and Deep Laminar Flows in Minichannels and Microchannels, Heat Transfer Engineering, 25, no 3, pp 86–93, 2004 Gnielinski, V., New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow, International Chemical Engineering, vol 16, pp 359–368, 1976 Serizawa, A., Feng, Z., and Kawara, Z., Two-Phase Flow in Microchannels, Experimental Thermal and Fluid Science, vol 26, pp 703–714, 2002 Hetsroni, G., Mosyak, A., Pogrebnyak, E., and Segal, Z., Explosive Boiling in Parallel Microchannels, International Journal of Multiphase Flow, vol 31, pp 371–392, 2005 vol 31 no 2010 S G KANDLIKAR [21] Kuo, C.-J., and Peles, Y., Flow Boiling Instabilities in Microchannels and Means for Mitigation by Reentrant Cavities, Journal of Heat Transfer, vol 130, article 072402, 10 pp., 2008 [22] Dong, T., Yang, Z., Bi, Q., and Zhang, Y., Freon R141b Flow Boiling in Silicon Microchannel Heat Sinks: Experimental Investigation, Heat Mass Transfer, vol 44, pp 315–324, 2008 [23] Kandlikar, S G., Scale Effects of Flow Boiling in Microchannels: A Fundamental Perspective, Keynote paper presented at the 7th International Conference on Boiling Heat Transfer, Florianopolis, Brazil, May 2–7, 2009 [24] Kandlikar, S G., Kuan, W K., and Mukherjee, A., Experimental Study of Heat Transfer in an Evaporating Meniscus on a Moving Heated Surface, Journal of Heat Transfer, vol 127, pp 244–252, 2005 [25] Mukherjee, A., and Kandlikar, S G., Numerical Study of an Evaporating Meniscus on a Moving Heated Surface, Journal of Heat Transfer, vol 128, pp 1285–1292, 2006 [26] Jacobi, A M., and Thome, J R., Heat Transfer Model for Evaporation of Elongated Bubble Flows in Microchannels Journal of Heat Transfer, vol 124, pp 1131–1136, 2002 [27] Mukherjee, A., and Kandlikar, S G., Numerical Simulation of Growth of a Vapor Bubble During Flow Boiling in a Microchannel, Microfluidics and Nanofluidics, vol 1, vol 2, pp 137–145, 2005 [28] Myers, J G., Yerramilli, V K., Hussey, S W., Yee, G F., and Kim, J., Time and Space Resolved Wall Temperature and Heat Flux Measurements during Nucleate Boiling With Constant Heat Flux Boundary Conditions, International Journal of Heat and Mass Transfer, vol 48, pp 2429–2442, 2005 [29] Son, G., Dhir, V K., and Ramanujapu, N., Dynamics and Heat Transfer Associated With a Single Bubble During Nucleate Boiling on a Horizontal Surface, Journal of Heat Transfer, vol 121, pp 623–629, 1999 [30] Moghaddam, S., and Kiger, K., Physical Mechanisms of Heat Transfer during Single Bubble Nucleate Boiling of FC-72 Under Saturation Conditions—I Experimental Investig(al)sT Inaanismkad7ahB1erTand 167 L M AL-HADHRAMI At large X/d where the impingement channel is wide and the crossflow starts to develop, the jets impinge completely on the target surface Then the crossflow strength increases as more jets impinge on the target surface Moreover, because of the inclined target surface toward the exit, the distance between the orifice jet and the target surface is reduced, in which case the crossflow is congested in the narrow region of the impingement channel, which increases the crossflow strength further This causes the jets at small X/d where the impingement channel is narrow to be swept away from the target surface, reducing the Nusselt number distribution along the target surface as shown in Figure The highest values of Nusselt number were reported in the vicinity of large X/d = 83.4 to X/d = 95.3 This is an indication that the jet impingement penetrability reached a maximum because the crossflow level is weak At the beginning of large X/d = 110 to X/d = 114.6, the flow emerging from the first two jets was trapped and created a recirculation zone where the flow formed a blanket that caused low Nusselt number values The Nusselt number distribution along the target surface is unaffected by the feed channel aspect ratio except for H/d = This may be attributed to the high feed channel aspect ratio providing more volume for the compressed air to stream and merge through the orifice jets As the channel aspect ratio is reduced further, the effect is almost negligible Figure shows the Nusselt number distribution for case where the flow orientation opposes the entry flows In this case, the exit flow direction is counterflowing to the inlet For this reason, the crossflow effect is stronger at large X/d locations where the impingement channel is wide The effect of the crossflow is to push the jets away from the target surface In addition, the inclination is putting the target surface away from the jets’ impingement This is evidenced by the Nusselt number distribution The jet impingement would be expected to be strongest at low X/d because of small crossflow effect The inclined target surface has added some benefits because the impingement is accruing in the narrow region of the channel The first jet at small X/d expected a low Nusselt number value As mentioned previously, this is due to some fluid being trapped and creating a recirculation zone, working as a blanket to reduce the Nusselt number value Figure presents the Nusselt number distribution for case 3, where the outflow exits in both directions The Nusselt number distribution is almost flat except at large X/d This is because the crossflow is passing in both directions The low Nusselt number distribution at large X/d may be due to the downstream cross-sectional area of the impingement channel being 175% (3.5 cm/2.0 cm) of the upstream area where the target surface is away from the jets The effect of feed channel aspect ratio is not very much except for H/d = 7, which shows low Nusselt number distribution by 6% compared to H/d = 5, heat transfer engineering 35 Case-1 Case-2 Case-3 30 Nuavg 240 25 H/d=9, Staggered 20 15 10 6000 9000 12000 Rej 15000 18000 21000 Figure Average Nusselt number distribution for different jet Reynolds numnber and different outflow orientations AVERAGE NUSSELT NUMBER DISTRIBUTIONS Figure presents the effect of the Reynolds number on the averaged Nusselt number distributions for all three flow orientations The average Nusselt number is calculated based on: Nuavg = 13 i=1 Nui=1/13 (14) For all three flow orientations, the Nusselt numbers increase with increasing averaged jet Reynolds numbers The comparison shows that for the flow case and case 3, the span-wise averaged Nusselt numbers are similar for the two target plates However, there is a drop for case For case and case 3, the crossflow effect is less than the crossflow of case because of the exit flow opposing the entry flow Figure 10 compares the overall averaged Nusselt number for each case and the Reynolds number with existing published correlations [18, 21, 22] Kercher and Tabakoff [21] studied impingement on a target surface with a strong crossflow effect similar to that in case Based on all geometrical parameters, they proposed a correlation of the form Nusselt number = f(Reynolds number, Prandtl number, Figure 10 Comparison of average Nusselt number of present study with archival results for different jet Reynolds number and different outflow orientations vol 31 no 2010 L M AL-HADHRAMI jet height to diameter ratio, spanwise location to diameter ratio, and transverse location to jet diameter ratio) Florschuetz et al [22] proposed a correlation for jet impingement for minimum crossflow as Nusselt number = f(Reynolds number, Prandtl number, jet height to diameter ratio, spanwise location to diameter ratio, and transverse location to jet diameter ratio) Huang et al [18] developed correlations for three outflow orientations as follows, Nusselt number = f (Reynolds number) Comparing the various correlations to the present study, the Nusselt number variation with jet Reynolds number for all flow orientation is lower than the correlations This is because all reported data and correlations were developed based on different jet impingement configurations and the mean temperature or adiabatic wall temperature Moreover, in the present study one single row of jets is considered with large spacing between the jets, 8d CONCLUSION The effect of an inclined target surface on heat transfer with different feed channel aspect ratios for different flow orientations has been investigated The results are compared to the available literature This is the first study focused on the effect of an inclined surface in the impingement channel with different feed channel aspect ratios The conclusions are the following: Different exit flow orientations produce different heat transfer distributions on the target surface Results show that case (outflow coincident with entry flow) where the exit flow is coincident with the entry flow and case (outflow exiting in both directions) produce the highest Nusselt number compared to case (outflow opposing the entry flow) because crossflow effects are smaller for case and case compared to that of case (outflow orientation opposing the entry flow) For case and case 3, Nusselt number distributions are not significantly affected by the presence of an inclined impingement surface The highest Nusselt number for case occurs at large X/d where jets completely impinge on the target surface and the crossflow effects are small For case the highest Nusselt number occurs at low X/d where the crossflow effect is minimal and the target surface is closer to the orifice jet because of the inclination For outflow to both sides in case 3, the relatively maximum Nusselt number occurs in the middle because of less crossflow effect Inclined target surface affected the Nusselt number distribution across the target surface For case -1, the exit is located at the end of the narrow region in which the crossflow is congested and has a stronger effect that affected the penetrability of the jets and skewed them away from the target surface As a result of this, there is low heat transfer at the small X/d heat transfer engineering 241 region However, for case 2, the exit flow is located at the end of the wide region of the impingement channel in which the target surface is not reached by the jets, resulting in low heat transfer distribution The feed channel aspect ratio effect is minimal on the heat transfer distribution over the target surface NOMENCLATURE Acp,i Atotal d hi H I kair kwood L Nui Nuavg q” Qcp,i Qactual,i Qcond,i Qrad,i Qtotal Rej R t Tin Ts,i Tsurr Tw V Vavg ∀ W X area of each copper plate area of each copper plate area of all copper plate diameter of the orifice jet local convective heat transfer coefficient height of the feed channel current supplied to heater thermal conductivity of air thermal conductivity of wood length of the copper plate local Nusselt number for each copper plate average Nusselt number heat flux from the heater heat input for each copper plate actual heat released from target surface heat lost due to conduction heat lost due to radiation total heat input jet Reynolds number resistance of the heater thickness of wood block behind the heater inlet temperature surface temperature temperature of the surroundings wood block temperature voltage supplied to the heater average velocity of all jets volume flow rate width of the feed channel distance in the x direction Subscripts cp i j w copper plate index number for each copper plate jet wood vol 31 no 2010 [m2 ] [m2 ] [cm] [W/m2 -K] [cm] [A] [W/m-K] [W/m-K] [cm] [W/m2 ] [W] [W] [W] [W] [W] [ohm] [m] [◦ C] [◦ C] [◦ C] [◦ C] [V] [m/s] [m3 /s] [cm] [m] 242 L M AL-HADHRAMI Greek Symbols ε σ θ µ ρ emissivity Stefan–Boltzmann constant inclination angle dynamic viscosity density [13] [W/(m2 -K4 )] [1.5◦ ] [kg/(ms)] [kg/m3 ] [14] [15] REFERENCES [1] Dong, L L., Leung, C W., and Cheung, C S., Heat Transfer Characteristics of Premixed Butane/Air Flame Jet Impinging on an Inclined Flat Surface, Heat and Mass Transfer, vol 39, no 1, pp 19–26, 2002 [2] Rasipuram, S C., and Nasr, K J., A Numerically-Based Parametric Study of Heat Transfer off an Inclined Surface Subject to Impinging Air Flow, International Journal of Heat and Mass Transfer, vol 47, no 23, pp 4967–4977, 2004 [3] Beitelmal, A H., Saad, M A., and Patel, C D., Effect of Inclination on the Heat Transfer between a Flat Surface and an Impinging Two-Dimensional Air Jet, International Journal of Heat and Fluid Flow, vol 21, no 2, pp 156–163, 2000 [4] Roy, S., and Patel, P., Study of Heat Transfer for a Pair of Rectangular Jets Impinging on an Inclined Surface, International Journal of Heat and Mass Transfer, vol 46, no 3, pp 411–425, 2003 [5] Ekkad, S., Huang, Y., and Han, J C., Impingement Heat Transfer Measurements Under an Array of Inclined Jets, Journal of Thermophysics and Heat Transfer, vol 14, no 2, pp 286–288, 2000 [6] Tawfek, A A., Heat Transfer Studies of the Oblique Impingement of Round Jets Upon a Curved Surface, Heat and Mass Transfer, vol 38, no 6, pp 467–475, 2002 [7] Seyedein, S H., Hasan, M., and Mujumdar, A S., Laminar Flow and Heat Transfer From Multiple Impinging Slot Jets with an Inclined Confinement Surface, International Journal of Heat and Mass Transfer, vol 37, no 13, pp 1867–1875, 1994 [8] Yang, Y., and Shyu, C H., Numerical Study of Multiple Impinging Slot Jets With an Inclined Confinement Surface, Numerical Heat Transfer; Part A: Applications, vol 33, no 1, pp 23–37, 1998 [9] Yan, X., and Saniei, N., Heat Transfer From an Obliquely Impinging Circular Air Jet to a Flat Plate, International Journal of Heat and Fluid Flow, vol 18, no 6, pp 591–599, 1997 [10] Hwang, J J., Shih, N C., and Cheng, C, S., Jet-Spacing Effect on Impinged Heat Transfer in a Triangular Duct with a Tangential Jet-Array, International Journal of Transport Phenomena, vol 5, pp 65–74, 2003 [11] Ramirez, C., Murray, D B., and Fitzpatrick, J A., Convective Heat Transfer of an Inclined Rectangular Plate, Experimental Heat Transfer, vol 15, no 1, pp 1–18, 2002 [12] Stevens, J., and Webb, B W., Effect of Inclination on Local Heat Transfer Under an Axisymmetric Free Liquid Jet, International heat transfer engineering [16] [17] [18] [19] [20] [21] [22] Journal of Heat and Mass Transfer, vol 34, no 4, pp 1227–1236, 1991 Hwang, J J., and Cheng, C S., Impingement Cooling in Triangular Ducts Using an Array of Side-Entry Wall Jets, International Journal of Heat and Mass Transfer, vol 44, no 5, pp 1053–1063, 2001 Hwang, J J., and Cheng T T., Augmented Heat Transfer in a Triangular Duct by using Multiple Swirling Jets, Journal of Heat Transfer, vol 121, pp 683–690, 1999 Hwang, J J., and Chang, Y., Effect of Outflow Orientation on Heat Transfer and Pressure Drop in a Triangular Duct with an Array of Tangential Jets, Journal of Heat Transfer, vol 122, pp 669–678, 2000 Ekkad, S., Huang, Y., and Han, J C., Impingement Heat Transfer on a Target Plate with Cooling Holes, Journal of Thermophysics and Heat Transfer, vol 13, no 4, pp 522–528, 1999 Obot, N T., and Trabold, T A., Impingement Heat Transfer Within Arrays of Circular Jets: Part 1—Effects of Minimum, Intermediate, and Complete Crossflow for Small and Large Spacings, Journal of Heat Transfer, vol 109, pp 872–879, 1987 Huang, Y., Ekkad, S V., and Han, J., Detailed Heat Transfer Distributions Under an Array of Orthogonal Impinging Jets, Journal of Thermophysics and Heat Transfer, vol 12, no 1, pp 73–79, 1998 San, J.-Y., Tsou, Y.-M., and Chen, Z.-C., Impingement heat transfer of staggered arrays of air jets confined in a channel, International Journal of Heat and Mass Transfer, vol 50, no 19-20, pp 3718–3727, 2007 Taylor, B N., and Kuyatt, C E., Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, U.S Department of Commerce National National Institute of Standards and Technology, NIST Technical Note 1297, 1994 Kercher, D M., and Tabakoff, W., Heat Transfer by a Square Array of Round Airjets Impinging Perpendicular to a Flat Surface Including the Effect of Spent Air, Journal of Engineering for Power, vol 92, no 1, pp 73–82,1970 Florschuetz, L W., Truman, C R., and Metzger, D E., Streamwise flow and heat transfer distributions for jet impingement with cross flow Journal of Heat Transfer, vol 103, no 2, pp 337–342, 1981 Luai M Al-Hadhrami is assistant professor of thermo-fluids at King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia He received his Ph.D in 2002 from Texas A&M University, College Station, TX His main research interests are experimental heat transfer and fluid mechanics He has published 15 articles in well-recognized journals and proceedings He is the center director for the Center for Engineering Research, Research Institute at KFUPM vol 31 no 2010 Heat Transfer Engineering, 31(3):243–249, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903304673 Experimental Study of Parameters Affecting the Nusselt Number of Generator Rotor and Stator S J E MAHDAVIFAR,1 M NILI-AHMADABADI,2 and A HASHEMI1 Power Plant Mechanical System Department, Niroo Research Institute, Tehran, Iran Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran In this research, the parameters affecting the Nusselt number of a generator rotor and stator under varying heat transfer rate are experimentally studied In spite of the stator having no grooves, the rotor has four large triangular grooves The temperature and then heat transfer rate of the rotor and stator are experimentally measured in three longitudinal and two angular positions First, the effect of axial Reynolds number and rotor rotational speed on the rotor and stator Nusselt number with constant heat transfer rate ratio is studied The range of the axial Reynolds number and rotational speed used is from 4000 to 30,000 and from 300 to 1500 rpm, respectively Next, the effect of stator to rotor heat transfer rate ratio on the Nusselt number at constant axial Reynolds number and rotational speed is investigated Three experiments were conducted at three heat transfer rate ratios (3, 5, and 8), defined as the ratio of heat transfer rate of the stator to the rotor The results show that the higher the heat transfer rate ratio, the lower is the stator mean Nusselt number and the higher the rotor mean Nusselt number INTRODUCTION according to the following relation for small axial Reynolds numbers, the critical Taylor number (Tacr ) increases Electrical current passing through generators heats the coils, so that a cooling system is needed to prevent the generator from overheating and burning its coil Different methods are used for cooling the generators One of these methods is an air cooling system In this method, air flow supplied with a fan passes through the air gap between the rotor and stator of the generator Therefore, the effect of different parameters on the forced convective heat transfer coefficient of the rotor and stator surfaces is an important factor for designing cooling systems Pai [1] experimented with laminar and turbulent annular flow between two concentric cylinders His results showed that the rotation of inner cylinder induces vortexes in both laminar and turbulent flow regimes so that in the laminar flow regime the axis of vortex and rotation is orthogonal, whereas in the turbulent flow regime it is not Chandrasekhar [2] reported that mixing axial flow with rotational flow increases the annular flow stability Therefore, Address correspondence to Mr S J E Mahdavifar, Power Plant Mechanical System Department, Niroo Research Institute, PO Box 14665517, Poonak Bakhtari Blvd., Shahrak Gharb, Tehran 14686, Iran E-mail: j mahdavifar@yahoo.com Tacr = Tacr |Rez =0 + 26.5Re2z (1) ¯ z (Do − Di )/ν where Rez = V Also, the combination of axial and rotational flow in the annular air gap has been studied experimentally by Kaye and Elgar [3] They showed that for Rez < 2000, there are four flow regimes in the air gap: (a) laminar flow, (b) laminar flow with Taylor vortexes, (c) turbulent flow, and (d) turbulent flow with Taylor vortexes Gu and Fahidy [4] observed that in the low-speed axial flows, Taylor vortexes kernels are generated in a round shape With increasing axial velocity, they are progressively destroyed Gazley [5] experimentally studied the heat transfer in annular flow with 0.43 and 8.1 mm air gap sizes and rotor radius of 63.5 mm in both grooved and non-grooved rotors The ranges of the rotor speed and air mean axial velocity were < ω < 4700 ¯ z < 90 m/s, respectively The results indicated rpm and < V that the relation between mean Nusselt number and effective Reynolds number is as follows: 243 Nu ≈ Re0.8 eff (2) 244 S J E MAHDAVIFAR ET AL where the effective velocity in the effective Reynolds number ¯ z )2 + (Vφ /2)2 Lee [6] experimentally is defined as Veff = (V studied the heat transfer and pressure loss between grooved and non-grooved coaxial cylinders one of which was rotating The ranges of rotational and axial Reynolds number were 103 < Reϕ < × 107 and 50 < Rez < 1000, respectively He showed that when the inner cylinder is grooved, the increment of Taylor number has the most effect on the increment of the outer cylinder Nusselt number Kuzay [7] experimentally investigated the turbulent flow heat transfer between two coaxial cylinders with rotating inner cylinder The surface of the inner cylinder was completely insulated, and the surface of the outer cylinder had uniform heat flux The ranges of axial Reynolds numbers and the ratio of rotational to mean axial velocity were 15,000 < Rez < 65,000 and ¯ z < 2.8, respectively The results showed that the < Vϕ /V rotation of the inner cylinder increases and decreases the temperature of the inner and outer cylinder surfaces, respectively, so the radial temperature profile is fixed sensibly Therefore, the Nusselt number of combinatory flow rises by increasing the rotational velocity of the inner cylinder The rotation effect of each cylinder in the annular flow between two coaxial cylinders on the velocity distribution, temperature profile, and heat transfer coefficient of the outer cylinder surface has been reported by Pfitzer and Beer [8] In their work, a fully developed turbulent flow theorem was investigated by using the corrected Prandtl mixing length model The outer cylinder diameter, inner to outer diameter ratio, and length to the air gap hydraulic diameter ratio were Do = 180 mm, Di /Do = 0.8575, and L/Dh = 60.94, respectively The inner cylinder was insulated and the outer cylinder had a uniform heat flux The range of axial Reynolds number and the ratio of the rotational speed to the air mean axial velocity were 3000 ¯ z < 4, respectively The results < Rez < 30,000 and < Vϕ /V showed that the rotation of the inner cylinder is more effective than that of the outer cylinder to raise the Nusselt number of the outer cylinder surface Smyth and Zurita [9] numerically analyzed forced convective heat transfer on a rotating cylinder The analysis was performed bilaterally and axisymmetrically They showed that Nusselt number depends on the Reynolds number to the power of 0.8 Kendous [10] presented an approximate solution for calculating heat transfer rate of laminar boundary layer on a rotating cylinder In his work, an approximate relation for the mean Nusselt number was obtained as: Nu = 0.6366(Rez Pr)0.5 thermography via infrared radiation, and then the analysis of flow construction between the rotor and stator was performed by particle image velocimetry (PIV) Also, the numerical solution of the steady flow energy equation was performed to determine the local heat transfer coefficient In their work, the ranges of rotational Reynolds number and the air gap width to the rotor diameter ratio were 58700 < Reϕ < 176,000 and 0.005 < Dh /Di < 0.085, respectively Ozerdem [12] researched forced convective heat transfer from a rotating cylinder in static air experimentally In his experiments, the mean heat transfer coefficient was measured by a radiative pyrometer He obtained the following relation: Nu = 0.318(Reϕ )0.571 for 2000