Heat Transfer Engineering, 31(13):1023–1033, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457631003639059 Two-Phase Flow Modeling in Microchannels and Minichannels M M AWAD and Y S MUZYCHKA Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St John’s, Newfoundland, Canada In this article, three different methods for two-phase flow modeling in microchannels and minichannels are presented They are effective property models for homogeneous two-phase flows, an asymptotic modeling approach for separated two-phase flow, and bounds on two-phase frictional pressure gradient In the first method, new definitions for two-phase viscosity are proposed using a one-dimensional transport analogy between thermal conductivity of porous media and viscosity in twophase flow These new definitions can be used to compute the two-phase frictional pressure gradient using the homogeneous modeling approach In the second method, a simple semitheoretical method for calculating two-phase frictional pressure gradient using asymptotic analysis is presented Two-phase frictional pressure gradient is expressed in terms of the asymptotic single-phase frictional pressure gradients for liquid and gas flowing alone In the final method, simple rules are developed for obtaining rational bounds for two-phase frictional pressure gradient in minichannels and microchannels In all cases, the proposed modeling approaches are validated using the published experimental data INTRODUCTION The pressure drop in two-phase flow through microchannels and minichannels constitutes an important parameter because pumping costs could be a significant portion of the total operating cost As a result, expressions are needed to predict the pressure drop in two-phase flow through microchannels and minichannels accurately Total pressure drop for two-phase flow in microchannels and minichannels has three different components They are frictional, acceleration, and gravitational components It is necessary to know the void fraction (the ratio of gas flow area to total flow area) to compute the acceleration and gravitational components To compute the frictional component of pressure drop, either the two-phase friction factor or the two-phase frictional multiplier must be known [1] There are two principal types of frictional pressure drop models in two-phase flow: the homogeneous model and the separated flow model In the first, both liquid and vapor phases move at the same velocity (slip ratio = 1) Consequently, the homogeneous model has also been called the zero-slip model The homogeneous model considers the two-phase flow as a single-phase flow having average fluid properties depending on mass quality Thus, the frictional pressure drop is calculated by assuming a constant friction coefficient between the inlet and outlet sections The prediction of the frictional pressure drop using the homogeneous model is reasonably accurate only for bubble and mist flows since the entrained phase travels at nearly the same velocity as the continuous phase In the second, two-phase flow is considered to be divided into liquid and gas streams Hence, the separated flow model has been referred to as the slip flow model The separated model is popular in the power plant industry Also, the separated model is relevant for the prediction of pressure drop in heat pump systems and evaporators in refrigeration The success of the separated model is due to the basic assumptions in the model, which are closely met by the flow patterns observed in the major portion of the evaporators In this study, three different methods for two-phase flow modeling in microchannels and minichannels are presented They are effective property models for homogeneous two-phase flows, an asymptotic modeling approach for separated two-phase flow, and bounds on two-phase frictional pressure gradient The literature review on two-phase flow modeling in microchannels and minichannels can be found in tabular form in a number of textbooks [2–6] PROPOSED METHODOLOGIES The authors acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) Homogeneous Property Modeling Address correspondence to Dr M M Awad, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St John’s, Newfoundland, Canada, A1B 3X5 E-mail: awad@engr.mun.ca In the first method, new definitions for two-phase viscosity are proposed [7] using a one-dimensional transport analogy 1023 1024 M M AWAD AND Y S MUZYCHKA between thermal conductivity of porous media [8] and viscosity in two-phase flow The series and parallel combination rules for thermal conductivity of porous media are analogous to existing rules proposed by McAdams et al [9], and Cicchitti et al [10] McAdams et al [9], introduced the definition of two-phase viscosity (µm ) based on the mass averaged value of reciprocals as follows: µm = x 1−x + µg µl (1 − x) −1 (1) µm = xµg + (1 − x)µl (2) They used the preceding definition of µm in place of the definition proposed by McAdams et al [9] The only reason for doing this, in addition to simplicity, was a reasonable agreement with experimental data Definitions for two-phase viscosity were generated by analogy to the effective thermal conductivity using the Maxwell–Eucken I and II models [12] Maxwell–Eucken I [12] is suitable for materials in which the thermal conductivity of the continuous phase is higher than the thermal conductivity of the dispersed phase (kcont > kdisp ), like foam or sponge In this case, the heat flow essentially avoids the dispersed phase In the case of momentum transport, this is akin to a bubbly flow, where the dominant phase is the liquid This definition for two-phase viscosity is: 2µl + µg − 2(µl − µg )x 2µl + µg + (µl − µg )x (3) Maxwell–Eucken II [12] is suitable for materials in which the thermal conductivity of the continuous phase is lower than the thermal conductivity of the dispersed phase (kcont < kdisp ), like particulate materials surrounded by a lower conductivity phase In this case, the heat flow involves the dispersed phase as much as possible In the case of momentum transport, this is akin to droplet flow, where the dominant phase is the gas This definition for two-phase viscosity is: µm = µg 2µg + µl − 2(µg − µl )(1 − x) 2µg + µl + (µg − µl )(1 − x) µ g − µm µl − µm +x =0 µl +2 µm µg +2 µm (5) which may be rewritten to be explicit for µm : They proposed their viscosity expression by analogy to the expression for the homogeneous density (ρm ) Equation (1) leads to the homogeneous Reynolds number (Rem ) being equal to the sum of the liquid Reynolds number (Rel ) and the gas Reynolds number (Reg ) In the realm of two-phase flow viscosity models, Collier and Thome [11] mentioned that the definition of µm proposed by McAdams et al [9], Eq (1), is the most common definition of µm Cicchitti et al [10] introduced the definition of two-phase viscosity (µm ) based on the mass averaged value as follows: µm = µl heterogeneous material in which the two components are distributed randomly, with neither phase being necessarily continuous or dispersed In the case of momentum transport, this averaging scheme seems reasonable given the unstable and random distribution of phases in a liquid/gas flow This definition for two-phase viscosity is: (4) A definition for two-phase viscosity was also generated by analogy to the effective thermal conductivity using the effective medium theory (EMT [13, 14]) The effective medium theory (EMT [13, 14]) is suitable for the structure that represents a heat transfer engineering µm = 1/4 (3x − 1)µg + [3(1 − x) − 1]µl + [(3x − 1)µg + (3{1 − x} − 1)µl ]2 + 8µl µg (6) Finally, a definition for two-phase viscosity was also proposed based on the arithmetic mean of Maxwell–Eucken I and II models [12] This is proposed here as a simple alternative to the effective medium theory: µm = 2µl + µg − 2(µl − µg )x µl 2µl + µg + (µl − µg )x + µg 2µg + µl − 2(µg − µl )(1 − x) 2µg + µl + (µg − µl )(1 − x) (7) It is clear that these new definitions satisfy the following two conditions: namely, (i) the two-phase viscosity is equal to the liquid viscosity at mass quality = 0% and (ii) the twophase viscosity is equal to the gas viscosity at mass quality = 100% These new definitions overcome the disadvantages of some definitions of two-phase viscosity such as the Davidson et al definition [15], Owens’s definition [16], and the Garc´ıa et al definition [17, 18] that not satisfy the condition at x = 1, µm = µg For example, Garc´ıa et al [17, 18] defined the Reynolds number of two-phase gas–liquid flow using the kinematic viscosity of liquid flow (νl ) instead of the kinematic viscosity of two-phase gas–liquid flow (νm ) They used this definition because the frictional resistance of the mixture was due mainly to the liquid This was equivalent to defining µm as µm = µl ρm ρl = µl ρg xρl + (1 − x)ρg (8) From Eq (8), it is clear that µm = µl ρg /ρl at x = This result leads to µm = 0.067µg at x = for an air–water mixture at atmospheric conditions These new definitions of two-phase viscosity can be used to compute the two-phase frictional pressure gradient using the homogeneous modeling approach It is desirable to express the two-phase frictional pressure gradient, (dp/dz)f , versus the total mass flux (G) in a dimensionless vol 31 no 13 2010 M M AWAD AND Y S MUZYCHKA form like the Fanning friction factor (fm ) versus the Reynolds number (Rem ) The Fanning friction factor (fm ) based on the homogeneous model (fm ) can be expressed as follows: fm = ρm (dp/dz) f d 2G x 1−x + ρg ρl ρm = (9) −1 (10) The Reynolds number based on the homogeneous model (Rem ) can be expressed as follows: Rem = Gd µm (11) Equations (10) and (11) represent the two-phase density based on the homogeneous model (ρm ) and Reynolds number based on the homogeneous model (Rem ) To satisfy a good agreement between the experimental data and well-known friction factor models, assessment of the best definition of two-phase viscosity among the different definitions (old and new) is based on the definition that corresponds to the minimum root mean square (RMS) error The fractional error (e) in applying the model to each available data point is defined as: e= Predicted − Available Available (12) For groups of data, the root mean square error, eRMS , is defined as: eR M S = N 1/2 N e2K (13) K =1 For the case of microchannels and minichannels, the friction factor is calculated using the Churchill model [19], which allows for prediction over the full range of laminar–transition–turbulent regions The Fanning friction factor (fm ) can be predicted using the Churchill model [19] as follows: fm = Rem 12 + (am + bm )3/2 1/12 am = 2.457 ln 0.9 (7/Rem ) + (0.27ε/d) bm = 37530 Rem (14) 16 (15) 1025 Asymptotic Modeling In the second method, new two-phase flow modeling in microchannels and minichannels was proposed [20], based upon an asymptotic modeling method Two-phase frictional pressure gradient is expressed in terms of the asymptotic single-phase frictional pressure gradients for liquid and gas flowing alone Asymptotes appear in many engineering problems, such as steady and unsteady internal and external conduction, free and forced internal and external convection, fluid flow, and mass transfer Often, there exists a smooth transition between two asymptotic solutions [21–24] This smooth transition indicates that there is no sudden change in slope and no discontinuity within the transition region The asymptotic analysis method was first introduced by Churchill and Usagi [21] in 1972 After this time, this method of combining asymptotic solutions proved quite successful in developing models in many applications [24] Recently, it has been applied to two-phase flow in circular pipes, minichannels, and microchannels [20] Moreover, Awad and Butt have shown that the asymptotic method works well for petroleum industry applications for flows through porous media [25], liquid–liquid flows [26], and flows through fractured media [27] The main advantage of the asymptotic modeling method in two-phase flow is taking into account the important frictional interactions that occur at the interface between liquid and gas because the liquid and gas phases are assumed to flow in the same channel This overcomes the main disadvantage of the separate cylinders model [28] for two-phase flow Using the asymptotic analysis method, two-phase frictional pressure gradient (dp/dz)f can be expressed in terms of singlephase frictional pressure gradient for liquid flowing alone (dp/dz)f,l and single-phase frictional pressure gradient for gas flowing alone (dp/dz)f,g as follows: dp dz = f dp dz p + f,l dp dz l/ p p (17) f,g Equation (17) reduces to (dp/dz)f,l and (dp/dz)f,g as x = and 1, respectively If the two-phase frictional pressure gradient (dp/dz)f is presented in terms of the single-phase frictional pressure gradient for liquid flowing alone (dp/dz)f,l , then the model can be expressed using the Lockhart–Martinelli parameter (X) as follows: dp dz = f dp dz 1+ f,l X2 p l/ p (18) Equation (18) can be expressed in terms of a two-phase frictional multiplier liquid flowing alone (φl2 ) as follows: 16 (16) φl2 = + The Churchill model [19] is preferable since it encompasses all Reynolds numbers and includes roughness effects in the turbulent regime heat transfer engineering X2 p l/ p (19) On the other hand, if the two-phase frictional pressure gradient (dp/dz)f is presented in terms of the single-phase frictional vol 31 no 13 2010 1026 M M AWAD AND Y S MUZYCHKA pressure gradient for gas flowing alone (dp/dz)f,g , then the model can be expressed using the Lockhart–Martinelli parameter (X) as follows: dp dz dp dz = f [1 + (X ) p ]l/ p (20) The lower bound is based on the Ali et al correlation [33] for laminar–laminar flow This correlation is based on modification of a simplified stratified flow model derived from the theoretical approach of Taitel and Dukler [34] for the case of two-phase flow in a narrow channel The equations of the lower bound are f,g Equation (20) can be expressed in terms of a two-phase frictional multiplier for gas flowing alone (φ2g ) as follows: φ2g = [1 + (X ) p ]1/ p (23) φ2g = + X (24) or (21) In this method, p is chosen as the value, which minimizes the root mean square (RMS) error, eRMS (Eq (13)), between the model predictions and the available data X2 φl2 = + and dp dz = f,lower 2( f Re)G(1 − x)µl dh2 ρl Bounds ∗ 1+ In the third method, simple rules were developed for obtaining rational bounds for two-phase frictional pressure gradient in minichannels and microchannels [29] This approach is very useful in design and analysis, as engineers can then use the resulting average and bounding values in predictions of system performance The approach is also useful when conducting new experiments, since it provides a reasonable envelope for the data to fall within The bounds are intended to provide the most realistic range of data and not firm absolute limits Statistically, this is unreasonable as the upper and lower bounds would be far apart The bounds are not fit to capture all data but rather a majority of data points, as some outlying points are due to experimental error If a vast majority of data is within the bounds, then a reasonable expectation is realistically assured These bounds may be used to determine the maximum and minimum values that may reasonably be expected in a two-phase flow Further, by averaging these limiting values an acceptable prediction for the pressure gradient is obtained, which is then bracketed by the bounding values: dp dz ≤ f,lower dp dz ≤ f dp dz (25) + X X (26) φ2g = + 5X + X (27) φl2 = + or (22) heat transfer engineering µg µl The equations of the lower bound are equivalent to the Chisholm correlation [35] with C = The physical meaning of the lower bound (C = 0) is that the two-phase frictional pressure gradient is the sum of the frictional pressure of liquid phase alone and the frictional pressure of gas phase alone This means no pressure gradient caused by the phase interaction Although the data points are in laminar–laminar flow, they cover different flow patterns such as bubble, stratified, and annular As the mass flow rate of the gas in two-phase flow increases, the flow pattern changes from bubble until it reaches annular at a high mass flow rate of gas As mentioned in the literature, the Chisholm correlation [35] has a good accuracy for annular flow pattern This is why the upper bound is based on Chisholm correlation [35] for laminar–laminar flow The equations of the upper bound are f,upper The bounds model can be in the form of two-phase frictional pressure gradient versus mass flux at constant mass quality; they may also be presented in the form of a two-phase frictional multiplier, which is often useful for calculation and comparison needs For this reason, development of lower and upper bounds in terms of a two-phase frictional multiplier (φl and φg ) versus the Lockhart–Martinelli parameter (X) will also be presented Awad and Muzychka [30, 31] applied the bounds method for the case of turbulent/turbulent flow in large circular pipes because, in practice, both Rel and Reg are most often greater than 2,000 Faghri and Zhang [32] further commented that the use of bounds alleviates the uncertainty in the separated flow models In the present study, the method is applied for the case of laminar/laminar flow in minichannels and microchannels because, in practice, both Rel and Reg are most often lower than 2,000 ρl ρg x 1−x and dp dz = f,upper 2( f Re)G(1 − x)µl dh2 ρl ∗ 1+5 + x 1−x x 1−x ρl ρg 0.5 ρl ρg µg µl 0.5 µg µl 0.5 (28) A simple model may be developed by averaging the two bounds This is defined as follows: 2.5 φl2 = + + (29) X X vol 31 no 13 2010 M M AWAD AND Y S MUZYCHKA 1027 or φ2g = + 2.5X + X (30) and dp dz = f,av 2( f Re)G(1 − x)µl dh2 ρl ∗ + 2.5 + x 1−x x 1−x ρl ρg 0.5 µg µl ρl ρg 0.5 µg µl 0.5 (31) The equations of the mean model are equivalent to the Chisholm correlation [35] with C = 2.5 This model can be applied for circular shapes using tube diameter d, as well as using hydraulic diameter dh for noncircular shapes For noncircular shapes, the Hagen–Poiseuille constant (fRe) = 16 will be changed For example, for a rectangular channel with the aspect ratio of 0, the Hagen–Poiseuille constant (fRe) = 24, while for a rectangular channel with the aspect ratio of (square channel), the Hagen–Poiseuille constant (fRe) = 14.23 RESULTS AND DISCUSSION Comparisons of the two-phase frictional pressure gradient versus mass flux from published experimental studies in minichannels and microchannels are undertaken using the old and new definitions of two-phase viscosity, after expressing the data in dimensionless form as Fanning friction factor versus Reynolds number The published data include different working fluids such as R717, R134a, R410A, and propane (R290) at different diameters and different saturation temperatures Also, examples of two-phase frictional multiplier (φl and φg ) versus Lockhart–Martinelli parameter (X) using published data of different working fluids, such as air–water mixture and nitrogen–water mixture in laminar–laminar flow, from other experimental work are presented to validate the asymptotic model and the bounds model in dimensionless form Figures and show the Fanning friction factor (fm ) versus Reynolds number (Rem ) in minichannels and microchannels using one of the old definitions (McAdams et al [9]) and one of the new definitions (Maxwell–Eucken II [12]) of two-phase viscosity on log-log scale The sample of the published data includes Ungar and Cornwell’s data [36] for R 717 flow at Ts ≈ 74◦ F (165.2◦ C) in a smooth horizontal tube at d = 0.1017 inches (2.583 mm), the Tran et al data [37] for R134a flow at saturation pressure of 365 kPa and x ≈ 0.73 in a smooth horizontal pipe at d = 2.46 mm, the Cavallini et al data [38] for 410A flow at Ts = 40◦ C and x = 0.74 in smooth multi-port minichannels at hydraulic diameter of 1.4 mm, and Field and Hrnjak data [39] heat transfer engineering Figure fm versus Rem in microchannels and minichannels using McAdams et al [9] definition for propane (R 290) flow at reduced pressure of 0.23 and G ≈ 330 kg/m2-s in a smooth horizontal pipe at hydraulic diameter of 0.148 mm The literature data represented a wide range of fluid properties, across R717, R134a, R410A, and propane (R290) Equation (9) defines the measured Fanning friction factor, while Eqs (14)–(16) represent the predicted Fanning friction factor Table presents eRMS % values based on measured Fanning friction factor and predicted Fanning friction factor using the six different definitions of two-phase viscosity for this sample of the published data It can be seen that two-phase viscosity based on the Maxwell–Eucken II model [12] gives the best agreement between the published data and the Churchill model [19] with a root mean square error (eRMS ) of 16.47% In Figure 2, it is interesting to observe that the fluids with the higher vapor–liquid density ratios, which were supposed to be more appropriate for the Maxwell–Eucken II model of homogeneous viscosity definition [12], might be thought to have better agreement It can be seen from Figure and Table that the definition of effective viscosity based on the Maxwell–Eucken II model [12] appears to be more appropriate for defining twophase flow viscosity in microchannels and minichannels On Figure fm versus Rem in microchannels and minichannels using Maxwell–Eucken II [12] definition vol 31 no 13 2010 1028 M M AWAD AND Y S MUZYCHKA Table eRMS % Values based on measured Fanning friction factor and predicted Fanning friction factor in microchannels and minichannels using different definitions of two-phase viscosity Definition eRMS McAdams et al [9] Cicchitti et al [10] Maxwell–Eucken I [12] Maxwell–Eucken II [12] Effective medium theory (EMT [13,14]) Arithmetic mean of Maxwell–Eucken I and II [12] 20.76% 31.06% 24.78% 16.47% 23.60% 17.98% Table Values of the asymptotic parameter (p) in microchannels and minichannels at different conditions Author Lee and Lee [40] Chung and Kawaji [41] Kawaji et al [42]+ Kawaji et al [42]++ Ohtake et al [43] d (mm) p eRMS eRMS p = 1/2 0.78∗ 1/1.75 1/1.7 1/2.15 1/2.55 1/1.55 11.7% 13.44% 10.39% 11.65% 19.56% 16.08%∗∗ 14.07% 16.09% 11.34% 17.36% 24.16% ∗∗ 18.24% 0.1 0.1 0.1 0.32∗ 0.42∗ 0.49∗ ∗ Hydraulic diameter two lower points are not taken into account +Gas in the main channel and liquid in the branch ++ Liquid in the main channel and gas in the branch ∗∗ The the basis of the data considered, a nominal 5–6% gain in accuracy can be achieved using the homogeneous flow modeling approach When one considers the nature of the Maxwell–Eucken II definition, whereby the dominant phase is the lower viscosity phase, i.e., the gas, it is clear that this definition is most appropriate for liquid/gas mixtures that have very high density ratios Thus, even for small mixture qualities, a significant portion of the flow volume is occupied by gas, making the Maxwell–Eucken II definition most appropriate Figure shows φl versus Lockhart–Martinelli parameter (X) for laminar–laminar flow for different working fluids in smooth microchannels and minichannels of different diameters at different conditions using the present asymptotic model and the bounds model with the first three data sets in Table Equation (18) represents the present asymptotic model with different values of p as shown in Table Equation (23) represents the lower bound and Eq (26) represents the upper bound, while Eq (29) represents the average Figure shows φg versus Lockhart–Martinelli parameter (X) for laminar–laminar flow for different working fluids in smooth microchannels and minichannels at different conditions using the present asymptotic model and the bounds model with the last two data sets in Table Equation (21) represents the present asymptotic model with different values of p as shown in Table Equation (24) represents the lower bound and Eq (27) represents the upper bound, while Eq (30) represents the average To have a robust model, one value of the fitting parameter (p) is chosen as p = 1/2 Choosing p = 1/2 is physically meaningful In fact, p = 1/2 in the asymptotic model corresponds to C = in the bound model When p = 1/2, the root mean square (RMS) error eRMS is 17.14%, or 15.69% if the two lower points of Ohtake et al data [43] are not taken into account Figure shows φl versus X for the first three data sets in Table 2, while Figure shows φg versus X for the last two data sets in Table with p = 1/2 On the basis of the experimental data shown in Figures and 4, it is clear that the experimental points set in a form, when X → 0, φl → ∞, and φg → and when X → ∞, φl → 1, and φg → ∞ in line with the expected asymptotic behavior of the Lockhart-Martinelli correlation [44] It can be seen that there is a good agreement between the present asymptotic model and the different data sets in Figures and The mean model predicts the first three data sets in Table of φl with the root mean square (RMS) error of 17.91%, 19.29%, and 10.49%, respectively while the asymptotic model gives the root mean square (RMS) error of 14.07%, 16.09%, and 11.34%, respectively The mean model predicts the last two data sets in Table of φg with the root mean square (RMS) error of 14.87%, and 28.04%, respectively while the asymptotic model gives the root mean square (RMS) error of 17.36% and 24.16%, Figure φl versus X for different sets of data Figure φg versus X for different sets of data heat transfer engineering vol 31 no 13 2010 M M AWAD AND Y S MUZYCHKA 1029 Figure φl versus X for Saisorn and Wongwises’s data [45] with various mass flux values at d = 0.53 mm Figure φl versus X for Saisorn and Wongwises’s data [47] with various mass flux values at d = 0.22 mm respectively In Figure 4, if the the two lower points of Ohtake et al data [43] are not considered, the root mean square (RMS) error will be 21.77% instead of 28.04%, while the asymptotic model gives the root mean square (RMS) error of 18.24% instead of 24.16% These outlying points are likely affected by experimental error The second method (the asymptotic model (p = 1/2 or C = 2)) and the third method (the bounds model (C = 2.5)) are also validated against the recent data sets of Saisorn and Wongwises [45–47] that were published in 2008 and 2009 for twophase air–water flow in circular microchannels of d = 0.53, 0.15, and 0.22 mm, respectivley Figures 5–7 show φl versus Lockhart–Martinelli parameter (X) for Saisorn and Wongwises’s data [45–47] with various mass flux values for laminar–laminar two-phase air-water flow in circular microchannels of d = 0.53, 0.15, and 0.22 mm, respectivley Figures 8–10 show φl versus Lockhart–Martinelli parameter (X) for Saisorn and Wongwises’s data [45–47] with various flow patterns for laminar–laminar two-phase air–water flow in circular microchannels of d = 0.53, 0.15, and 0.22 mm, respectivley The observed flow patterns in a 0.53-mm-diameter channel include slug flow, throat-annular flow, churn flow, and annular–rivulet flow The observed flow patterns in a 0.15-mm-diameter channel include liquid unstable annular alternating flow (LUAAF), liquid/annular alternating flow (LAAF), and annular flow The observed flow patterns in a 0.22-mm-diameter channel include throat-annular flow, annular flow, and annular–rivulet flow Equation (18) represents the present asymptotic model with p = 1/2 Equation (23) represents the lower bound and Eq (26) represents the upper bound, while Eq (29) represents the average In Figures 5–10, Saisorn and Wongwises’s data [45–47] are also compared with the Mishima and Hibiki correlation [48] (C = 21(1 − e−319d)) and the English and Kandlikar correlation [49] (C = 5(1 − e−319d)) It should be noted that the Saisorn and Wongwises correlations (φi2 = + (6.627/X 0.761)) [45] for d = 0.53 mm and (φi2 = + (2.844/X 1.666)) [46] for d = 0.15 mm neglect the 1/X term, which represents the limit of primarily gas flow in the Lockhart–Martinelli [44] formulation Neglecting this term ignores this important limiting case, which is an essential contribution As a result, at low values of X, the proposed correlations Figure φl versus X for Saisorn and Wongwises’s data [46] with various mass flux values at d = 0.15 mm Figure φl versus X for Saisorn and Wongwises’s data [45] with various flow patterns at d = 0.53 mm heat transfer engineering vol 31 no 13 2010 1030 M M AWAD AND Y S MUZYCHKA Figure φl versus X for Saisorn and Wongwises’s data [46] with various flow patterns at d = 0.15 mm undershoot the trend of the data, limiting their use in the low X range [50] From Figures 3–10, it clear that the greatest departure of bounds from the mean occurs at X = From the relation that the Lockhart–Martinelli parameter (X) for laminar–laminar flow is equal to ((1−x)/x)0.5(ρg /ρl )0.5(µl /µg )0.5, this greatest departure of bounds from the mean corresponds to x = 6.28% for the air–water mixture at the atmospheric pressure while it corresponds to x = 50% at the critical state (ρg = ρl and µg = µl ) for any working fluid Figure 11 shows C parameter versus the channel diameter (dh ) for laminar–laminar flow using the asymptotic model (p = 1/2 or C = 2), the bounds model (C = 2.5), the Chisholm correlation [35] (C = 5), the Mishima and Hibiki correlation [48] (C = 21(1 − e−319dh)), and the English and Kandlikar correlation [49] (C = 5(1 − e−319dh)) It is found that the asymptotic model (p = 1/2 or C = 2) is equivalent to the Mishima and Hibiki correlation [48] at dh = 0.314 mm and equivalent to the English and Kandlikar correlation [49] for laminar–laminar flow at dh = 1.601 mm Moreover, the bounds model (C = 2.5) is equivalent Figure 10 φl versus X for Saisorn and Wongwises’s data [47] with various flow patterns at d = 0.22 mm heat transfer engineering Figure 11 C parameter versus the channel diameter (dh ) to the Mishima and Hibiki correlation [48] at dh = 0.397 mm and equivalent to the English and Kandlikar correlation [49] for laminar–laminar flow at dh = 2.173 mm SUMMARY AND CONCLUSIONS First, using a one-dimensional transport analogy between thermal conductivity in porous media and viscosity in twophase flow, new definitions for two-phase viscosity are examined These new definitions for two-phase viscosity satisfy the following two conditions: (i) µm = µl at x = and (ii) µm = µg at x = These new definitions of two-phase viscosity can be used to compute the two-phase frictional pressure gradient using a homogeneous modeling approach Expressing two-phase frictional pressure gradient in dimensionless form as Fanning friction factor versus Reynolds number is also desirable in many applications The models are verified using published experimental data for two-phase frictional pressure gradient in microchannels and minichannels after expressing these in a dimensionless form as Fanning friction factor versus Reynolds number The published data include different working fluids such as R717, R134a, R410A, and propane (R290) at different diameters and different saturation temperatures To provide good agreement between the experimental data and well-known friction factor models such as the Churchill model [19], selection of the best definition of two-phase viscosity is based on the definition that corresponds to the minimization of the root mean square error (eRMS ) From eRMS % values based on measured Fanning friction factor and predicted Fanning friction factor using the six different definitions of two-phase viscosity, it is shown that one of the new definitions of two-phase viscosity (Maxwell–Eucken II [12]) gives the best agreement between the experimental data and well-known friction factor models in microchannels and minichannels These new definitions of two-phase viscosity can be used to analyze the experimental data of two-phase frictional pressure gradient in microchannels and minichannels using the homogeneous model vol 31 no 13 2010 M M AWAD AND Y S MUZYCHKA Second, new two-phase flow modeling in microchannels and minichannels is proposed, based upon an asymptotic modeling method The main advantage of the asymptotic modeling method in two-phase flow is taking into account the important frictional interactions that occur at the interface between liquid and gas, because the liquid and gas phases are assumed to flow in the same channel This overcomes the main disadvantage of the separate cylinders model for two-phase flow The only unknown parameter in the asymptotic modeling method in two-phase flow is the fitting parameter (p) The value of the fitting parameter (p) is chosen to correspond to the minimum root mean square (RMS) error eRMS for any data set To have a robust model, one value of the fitting parameter (p) is chosen as p = 1/2 Third, simple expressions are presented for obtaining bounds for two-phase frictional pressure gradient in minichannels and microchannels The lower bound is based on the Ali et al correlation [33] for laminar–laminar flow This correlation is based on modification of a simplified stratified flow model derived from the theoretical approach of Taitel and Dukler [34] for the case of two-phase flow in a narrow channel The upper bound is based on Chisholm correlation [35] for laminar–laminar flow The mean model is based on the arithmetic mean of lower bound and upper bound The model is verified using published experimental data of two-phase frictional pressure gradient in circular and noncircular shapes The bounds models are presented in a dimensionless form as two-phase frictional multiplier (φl and φg ) versus Lockhart–Martinelli parameter (X) for different working fluids such as the air–water mixture and nitrogen–water mixture The present model is very successful in bounding two-phase frictional multiplier (φl and φg ) versus Lockhart–Martinelli parameter (X) well for different working fluids over a wide range of mass fluxes, mass qualities, and diameters The proposed mean model provides a simple prediction of two-phase flow parameters Finally, the first method (homogeneous property modeling) is recommended in predicting the two-phase frictional pressure drop in microchannels and minichannels if we use the homogeneous model It is reasonably accurate only for bubble and mist flows since the entrained phase travels at nearly the same velocity as the continuous phase The second and third methods (asymptotic modeling and bounds) are recommended in predicting the two-phase frictional pressure drop in microchannels and minichannels if we use the separated model that originated from the classical work of Lockhart and Martinelli [44] NOMENCLATURE a b C d e f f Re Churchill parameter Churchill parameter Chisholm constant pipe diameter, m error Fanning friction factor Hagen–Poiseuille constant heat transfer engineering 1031 mass flux, kg/m2.s index for summation thermal conductivity, W/m2.K number of data points pressure gradient, Pa/m Reynolds number = Gd/µ temperature, ◦ C Lockhart–Martinelli parameter mass quality G K k N dp/dz Re T X x Greek Symbols ε ρ φ2g φl2 µ ν pipe roughness, m density, kg/m3 two-phase frictional multiplier for gas alone flow two-phase frictional multiplier for liquid alone flow dynamic viscosity, kg/m.s kinematic viscosity, m2/s Subscripts av cont disp f g h l lower m RMS s upper average continuous phase dispersed (discontinuous) phase frictional gas hydraulic liquid lower bound homogeneous mixture root mean square saturation upper bound Superscript p fitting parameter REFERENCES [1] ASHRAE, Two-Phase Flow, in Handbook of Fundamentals, Chap 4, ASHRAE, Atlanta, GA, p 12, 1993 [2] Celata, G P., Heat Transfer and Fluid Flow in Microchannels, Begell House, Redding, CT, 2004 [3] Kandlikar, S G., Garimella, S., Li, D., Colin, S., and King, M R., Heat Transfer and Fluid Flow in Minichannels and Microchannels, Elsevier, Oxford, UK, 2006 [4] Crowe, C T., Multiphase Flow Handbook, CRC/Taylor & Francis, Boca Raton, FL, 2006 [5] Ghiaasiaan, S M., Two-Phase Flow, Boiling and Condensation in Conventional and Miniature Systems, Cambridge University Press, New York, 2008 vol 31 no 13 2010 1110 C DEPCIK ET AL described to account for this density change in an aftertreatment device To illustrate the effects of compressibility, a reaction rate mechanism for the EAFR process is formulated for use in the model When mole fractions are used for the reaction mechanism, there is not any difference between the classical and compressible versions because of the model formulation and the fact that mole fractions are density independent To account for compressibility, the kinetics should use concentrations similar to homogeneous kinetic mechanisms used in gas-phase combustion models Optimization of both types of kinetic models (mole fraction and concentration based) illustrated a slight improvement in using concentration-based kinetics; however, future research must be accomplished to state this result for certain The 10-reaction mechanism formulated has significant error with respect to the modeling the temperature profile and the hydrogen production trend There is considerable room for improvement involving EAFR kinetics; however, because of the dearth of data, this will be hard to accomplish This first effort in modeling the EAFR kinetic process provides a good background in the reaction mechanisms and, more importantly, formulates a catalyst model that properly accounts for the density change through the device In addition, an important find is the use of concentrations instead of mole fractions in the reaction mechanism While the results of the simulation were reasonable at best, the lessons learned in this paper will help the modeler take the next step in formulating a more complete kinetic model Future work will look at modeling the EAFR process involving a simpler inlet fuel, like methane, while also including homogeneous phase kinetics that can be significant In addition, the researchers will explore autothermal reformer literature to find other data that are suitable for investigating kinetics and the compressible model solver One unintended benefit is the formulation of a transient, variable-property gas dynamics solver that models the source terms more effectively NM p R R Ru Sh S T t u W x X NOMENCLATURE REFERENCES A cm cp C¯ d D E Ea fF Ga Gca h hg H km kinetic pre-exponent [varies] monolith specific heat [J/kg/K] constant pressure specific heat [J/kg/K] molar species concentration [mol/m3 ] monolith channel diameter [m] overall catalyst diameter [m] total internal energy [J/kg] activation energy [kJ/mol] Fanning friction factor [—] geometric surface area per unit volume [m2 /m3 ] catalytic surface area per unit volume [m2 /m3 ] 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is an assistant professor at the University of Kansas (KU) in Lawrence, Kansas, USA Prior to joining KU, he worked at the University of Michigan (UM) as a postdoctoral research fellow He received his Ph.D in mechanical engineering from UM in 2003, as well as an M.S in aerospace engineering in 2002 and an M.S in mechanical engineering in 1999 He received his B.S in mechanical engineering from the University of Florida in 1997 His graduate research interests include catalytic aftertreatment modeling, exhaust energy recovery, diesel particulate filter modeling, first- and second-generation biofuels, fuel reforming, hydrogen combustion, hybrid vehicles, and sustainable engineering His undergraduate students have successfully recycled a 1974 Volkswagen Super-Beetle into a fuel neutral, series hybrid vehicle running on 100% biodiesel created from used campus cooking oil heat transfer engineering 1113 Arkadiusz Kobiera is an assistant professor at the Warsaw University of Technology (WUT) in Poland He received his Ph.D in mechanical engineering from WUT in 2004, as well as an M.S in mechanical engineering in 1999 After his Ph.D., he worked at the University of Michigan (UM) as a postdoctoral research fellow (2005–2007) The scholarship at UM was partially supported by the Dekaban Foundation His scientific interests are focused on the modeling of combustion and flow phenomena in propulsion systems His recent work involves the development and numerical simulation of a new concept of detonation engine; i.e., rotating detonation engine Dennis Assanis is the Jon R and Beverly S Holt Professor of Engineering at the University of Michigan, where he is also the director of the Automotive Research Center He received his B.Sc degree in marine engineering from the University of Newcastle-uponTyne, UK, in 1980 He has received four graduate degrees from the Massachusetts Institute of Technology: M.S in naval architecture and marine engineering (1982), M.S in mechanical engineering (1982), Ph.D in power and propulsion (1985), and M.S in management (1986) Prior to joining the University of Michigan in 1994, he was an assistant professor (1985–1990) and an associate professor (1990–1994) of the mechanical engineering Department at the University of Illinois Urbana– Champaign His research interests include modeling and computer simulation of internal combustion engine processes and systems; experimental studies of engine heat transfer, combustion, and emissions; and automotive systems design optimization He has published more than 150 articles in journals and refereed conference proceedings, and he is a Fellow of the Society of Automotive Engineers vol 31 no 13 2010 Heat Transfer Engineering, 31(13):1114–1124, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457631003640453 Shape Optimization of a Dimpled Channel to Enhance Heat Transfer Using a Weighted-Average Surrogate Model ABDUS SAMAD, KI-DON LEE, and KWANG-YONG KIM Department of Mechanical Engineering, Inha University, Incheon, Korea Shape optimization of a rectangular channel with the opposite walls roughened by staggered arrays of dimples has been performed not only to enhance turbulent heat transfer but also to reduce friction loss The dimpled channel shape is defined by three geometric design variables, and the design points within design space are selected using Latin hypercube sampling The shape of the channel is optimized with three-dimensional (3-D) Reynolds-averaged Navier–Stokes analysis and surrogate approximation methods A weighted-sum method for multi-objective optimization is applied to integrate multiple objectives related to heat transfer and friction loss into a single objective A weighted-average surrogate model is employed for this optimization By the optimization, the objective function value is improved largely and heat transfer rate is increased much higher than pressure loss increase due to shape deformation The optimum design results in lower channel height, wider dimple spacing, and deeper dimple The flow mechanism shows the heat transfer rate is increased mainly in the rear portion of the dimple INTRODUCTION To enhance convective heat transfer in internal flow passage of turbine blades, various heat transfer augmentation devices such as dimples, ribs, pin-fins, etc are used Among these, dimples delay development of thermal boundary layer and increase production of turbulent kinetic energy, and thus enhance turbulent heat transfer as reviewed by Ligrani et al [1] Dimpled channels can effectively be used in other heat transfer areas like nozzles, combustor liners, etc Inevitably, the surface roughened by dimples increases pressure loss with the increase of heat transfer The shape optimization of the dimpled surface thus is indispensable, to optimize between enhancement of heat transfer and reduction of pressure loss In this article, a dimpled channel shape is optimized using a surrogate model This work was supported by the National Research Foundation of Korea (NRF) grant no 20090083510 funded by the Korean government (MEST) through Multi-phenomena CFD Engineering Research Center Address correspondence to Professor Kwang-Yong Kim, Department of Mechanical Engineering, Inha University, 253 Yonghyun-Dong, Nam-Gu, Incheon, 402-751, Korea E-mail: kykim@inha.ac.kr Several works have been contributed to the analysis of heat transfer over dimpled surfaces by experimental and numerical studies Schukin et al [2] reported on heat transfer augmentation for a heated plate downstream of a single hemispherical cavity in a diffuser channel and also in a convergent channel The influences of the turbulence intensity level and the angles of divergence on heat transfer augmentation were investigated Laminar flows (Re = 1000–5000, based on hydraulic diameter) in parallel channels with hemicylindrical cavities on opposite walls were studied by Ridouane and Campo [3], and they presented that 30% relative heat transfer can be increased as compared to smooth surface while pressure loss increased 19% Dimples on surfaces of heat sinks of microelectronic cooling were reported by Silva et al [4], and they predicted that heat transfer can be increased by 2.5 times for laminar and transition flow However, most of the works have been performed for turbulent heat transfer Experimental work for dimples and protrusions in an internal cooling channel was reported by Hwang and Cho [5] They presented that having dimples on both surfaces of the channel is better than dimples on single surface or a protruded surface in terms of pressure drop or Nusselt number enhancement Mahmood et al [6] presented heat transfer characteristics when dimples and protrusions were placed 1114 A SAMAD ET AL on opposite walls Experimental study of staggered arrays of dimple on single surface was reported by Burgess and Ligrani [7] and Burgess et al [8] Seven different dimpled depressions on surface were studied by Park and Ligrani [9] Flow structure on the dimpled surface was reported by Park et al [10] Mahmood and Ligrani [11] presented experimental study on combined influences of flow parameters A study by Moon et al [12] shows that the friction factor is relatively independent of the Reynolds number and channel height for a single surface roughened by staggered dimples The works just described report that the dimples change flow pattern and as a result they increase turbulent heat transfer as well as pressure drop Obviously, the modification of shape and arrangement of dimples in the channel affects the performance with respect to cooling of channel Recent development of high-speed computers has made system design easier Numerical work with optimization techniques has given a boost to design a better performing system Kim and Choi [13] performed shape optimization of a rectangular channel with a single surface roughened by dimples in an inline array to enhance turbulent heat transfer using a response surface method Samad et al [14] optimized the shape of a channel with dimples on a single surface using different surrogates and improved heat transfer and friction loss-based objective function value by approximately 20% To reduce the burden of computationally expensive simulations to optimize a design, numerical approximation models called surrogate models [15] are used The response surface model is a basic surrogate model that has been used most widely so far The surrogate models being used in multidisciplinary optimizations have advantages in two important aspects: computational economy that requires as few data points as possible for constructing a surrogate model, and accuracy in representing the characteristics of the design space Queipo et al [15] and Li and Padula [16] reviewed various surrogate-based models used in aerospace applications Goel et al [17] developed a weightedaverage surrogate model, using response surface approximation (RSA), kriging (KRG), and radial-basis neural network (RBNN) models The weighting factors for the weighted average model are determined according to cross-validation (CV) errors The larger the error in prediction by any surrogate, the less is the weight assigned to the surrogate constructing the weightedaverage surrogate These authors concluded that the weightedaverage surrogate model gives more reliable prediction method than individual RSA, KRG, or RBNN surrogates Samad et al [14, 18] reported the performances of several surrogate models optimizing a heat transfer surface and turbomachinery blade shape, respectively, and they showed that the weighted-average surrogate model developed by Goel et al [17] is reliable in those applications In this paper, use of staggered arrays of dimples printed on opposite walls of a channel to enhance heat transfer compromising with pressure drop is formulated numerically and an optimal design of the channel is presented The procedure to reduce computational expenses is also explained heat transfer engineering 1115 Figure Dimple structure and arrangement PROBLEM DESCRIPTION AND NUMERICAL PROCEDURE Staggered arrays of dimples printed on opposite surfaces of a channel shown in Figure are considered in this work, and three-dimensional Reynolds-averaged Navier–Stokes (RANS) analyses of fluid flow and convective heat transfer have been performed using ANSYS-CFX 11.0 [19], which employs an unstructured grid system Modifications of source terms in streamwise momentum and energy equations have been done to adopt the periodic boundary conditions to calibrate the gradual changes of temperature and pressure, respectively, as described by Kim and Choi [13] The shear stress transport (SST) turbulence model with automatic wall treatment [20] (no damping function in the near wall region) is used as a turbulence closure The SST model combines the advantages of the k-ε and k-ω models with a blending function The k-ω model is activated at the near wall region and the k-ε model is used at rest of the region Bardina et al [21] showed that the SST model captures separation under adverse pressure gradient well vol 31 no 13 2010 1116 A SAMAD ET AL DESIGN VARIABLES AND OBJECTIVE FUNCTIONS Figure shows definitions of the geometric variables: dimple depth, Hd ; dimple print diameter, d; pitch of dimple, p; distance between dimples, S; and channel height, H From these variables, the nondimensional numbers H /d, Hd /d, d/S, and S/p can be found However, in this work, S/p is set to 1.73 to reduce the number of variables Hence, three design variables, H /d, Hd /d, and d/S, are selected as design variables To maximize the performance of the channel, two objective functions FNu and Ff based on Nusselt number and friction factor, respectively, are selected The Nusselt number-based objective function that is responsible for enhancement of the heat transfer rate is defined as FNu = 1/N ua (1) where N ua = Figure Example of computational grids heat transfer engineering N u/N uo dA (2) Ad Here, Nu and Nua are local and average Nusselt numbers, respectively, and, Nuo is the Nusselt number for the fully developed turbulent flow in a smooth pipe Ad is area of heat transfer surface, i.e., actual area of the dimpled surface On the other hand, the pressure loss-related objective function is defined as Ff = f f0 1/3 (3) where f0 is a friction factor for fully developed flow in a smooth pipe and f is defined as f = pDh 2ρUb2 P (4) 250 50000 10000 200000 350000 200 150 Nu compared to other eddy viscosity models Thus, it predicts well near-wall turbulence that plays a vital role in the accurate prediction of turbulent heat transfer The numerical model of Lai and So [22] is adopted for modeling turbulent heat flux The geometric parameters and computational domain are shown in Figure The computational domain is composed of one single dimple at the center and four one-fourth dimples at corners of the domain Periodic boundary conditions are applied to inlet-to-outlet and side-to-side surfaces An unstructured tetrahedral grid system is used with the hexahedral at the wall region to resolve the high-velocity gradient as shown in Figure First grid points are placed at 0.002H to satisfy y+ less than so that the SST model can be implemented properly In the present calculation, uniform heat flux (600 w/m2 ) is specified on the dimpled surfaces For the convergence criteria, root mean square (RMS) relative residual values of all flow parameters are set to 1.0E-6 and imbalances of both mass and energy in the entire computational domain are each less than 1.0E-2% The solver finishes a single simulation by approximately 1,000 iterations The inlet turbulence intensity (Tu) and Reynolds number based on hydraulic diameter (ReDh ) are set to 0.05 and 10,000, respectively The computations are performed by an Intel Pentium IV CPU, 3.0 GHz PC The computation time is typically 10–15 hours and the time of computations depends on geometry considered and convergence rate Grid independence testing for the reference geometry of the dimpled channel was performed, and the result is presented in Figure Among different numbers of grids, approximately 200,000 is selected as an optimum number of grids With the variation of the shape of dimpled channel, attempts are made to refine the grids adjacent to the wall to satisfy y+ less than Ad 100 50 0 0.25 0.5 x/P Figure Grid dependency test vol 31 no 13 2010 0.75 A SAMAD ET AL 1117 construct an approximate surrogate model and a better performing system is designed The beauty of the surrogate approach is that it reduces computational expenses Problem dependency of surrogates produces different data fitting and hence produces different optimal designs As the PRESS-based averaging (PBA) model was robust in previous work [14, 18], the design predicted by PBA is used in this work A brief description of the methods used in this paper is given next (Problem setup) Objective functions & Design variables (Deciding of design space) Lower and upper bound of variables are set (Design of experiments) Selection of design points by LHS Weighted Sum of Objective Functions (Numerical Analysis) Determination of the value of objective functions of the designs by RASN solver The weighted sum of objective functions method, which is also known as the “naive approach,” is frequently used in heat transfer problems [13, 24] In this problem, objectives FNu and Ff are linearly combined with a weighting factor wf to constitute a mono-objective, F , by the naive approach, and the final objective (F ) is defined by (Construction of surrogate) Surrogate model construction using the evaluated objectives F = FNu + wf Ff (Search for optimal point) Optimal point search from constructed surrogate using optimization algorithm The weighting factor wf is selected as per design requirement In general, it is the “designer’s choice.” No Is optimal point within design? (5) Yes Optimal Design Figure Surrogate-based optimization procedure where p, Dh , ρ, Ub , and P are pressure drop, channel hydraulic diameter, fluid density average axial velocity, and dimple pitch, respectively Hence, the main motive of the optimizations is to minimize the objective functions, FNu and Ff A weighted-sum approach of multi-objective optimization is applied to transform the biobjective problem into mono-objective problem and THE surrogate approach is introduced OPTIMIZATION METHODOLOGY The optimization procedure is shown in the flow chart in Figure Initially, the variables are selected and the design space is decided for the improvement of system performance The design space is defined by the lower and upper limits of variables The design points within design space are selected with the help of Latin hypercube sampling (LHS) [23] of design of experiments (DOE) The objective function values at these design points are evaluated using a flow solver A weighted sum of objective functions method [24] is applied to make multiple objectives into a single objective and a surrogate-based approximation procedure is applied Limited numbers of designs are evaluated to heat transfer engineering Surrogate Approach The PRESS (predicted error sum of squares)-based averaging (PBA) model, which was originally the WTA3 model [17] and was renamed by Samad et al [14, 18], is adopted in this investigation This model is basically a weighted average of the basic surrogates: second-order polynomial response surface approximation (RSA), kriging (KRG), and radial basis neural network (RBNN) The predicted response is defined as follows for the PBA model: NSM Fˆ wt.avg (x) = wi (x) Fˆ i (x) (6) i where NSM is the number of basic surrogate models used to construct a weighted-average model The ith surrogate model at design point x produces weight wi (x) and Fˆ i (x) is the predicted response by the ith surrogate model In simplified form, the function can be written as Fwt.avg = wRSA FRSA + wKRG FKRG + wRBNN FRBNN (7) where FRSA , FKRG , or FRBNN is constructed using RANS evaluated responses Brief descriptions of the basic surrogates (RSA, KRG, and RBNN) are given in the appendix Weights are decided such that surrogates that produce high error have low weight and thus low contribution toward the final weighted average surrogate and vice versa In this work, global weights are calculated from each basic surrogates using a generalized mean square cross-validation error (GMSE) that is a global data-based measure of goodness In cross-validation (CV), the data is divided into k subsets (k-fold CV) of vol 31 no 13 2010 1118 A SAMAD ET AL approximately equal size A surrogate model is constructed k times, each time leaving out one of the subsets from training and using the omitted subset to compute the error measure of interest The generalization error estimate is computed using the k error measures obtained (e.g., average) If k equals the sample size, this approach is called the leave-one-out CV In the present problem, the value of k is equal to the number of designs considered for optimization The weighting scheme used in PBA surrogate is given as follows: wi∗ = Ei /Eavg + α β , wi = wi∗ / wi∗ i NSM Eavg = Ei /NSM ; β < 0, α < (8) i=1 Ei = GMSEi , i = 1, 2, , NSM with the two constants α and β chosen as α = 0.05 and β = –1 [17] After constructing the PBA model, a gradient-based search algorithm, sequential quadratic programming is used to search the optimal point from the PBA model Since the algorithm is dependent on initial guess of optimal point, a series of trials is performed before getting the final optimal point from any surrogate RESULTS AND DISCUSSION Initially, the numerical results are validated in comparison with the experimental data obtained by Hwang and Cho [5] as shown in Figure Figure 5a shows normalized average Nusselt numbers and friction factors at several different Reynolds numbers based on hydraulic diameter The RANS results match well with experimental results for Re = 10,000 As shown in this figure, the Nusselt number and friction factor are almost constant with the variation of Reynolds number This is also supported by the literature [7, 8, 12] Figure 5b shows local Nusselt number distributions along the dimple diagonal in streamwise direction The calculated local Nusselt number distribution also shows good agreement with the experimental one The design space that is constituted by the lower and upper limits of the variables is shown in Table The design variables are presented in nondimensional form The ranges of design variables is set considering the results of the previous experimental and numerical works [4, 5, 7, 11, 12, 13, 19] Before deciding the final ranges of variables, it is checked that the lower and upper limits of the variables produce feasible geometries during design of experiments (DOE) To shorten the optimization iteration loop (Figure 4), selection of proper design space is necessary Single iteration produced the optimum point inside the design space in current problem heat transfer engineering Figure Validation of numerical results: (a) average Nusselt number, and (b) local Nusselt number with experiments [5] Latin hypercube sampling (LHS) is used to generate 20 designs in the design space These designs are evaluated by the RANS solver and the evaluated results are used for surrogate analysis through a multi-objective procedure (Eq (4)) The basic surrogates (RSA, KRG, and RBNN) used to construct the PBA surrogate are constructed and weights are calculated for each basic surrogate Finally, the PBA surrogate Table Design space Variables Limits H /d Hd /d d/S Lower Upper 0.2 1.5 0.1 0.3 0.30 0.57 vol 31 no 13 2010 A SAMAD ET AL 1119 Table Reference and optimal shape with wf = 0.09 Variables Objective function values with wf = 0.09 Objectives Shapes H /d Hd /d d/S Nua Ff Fsurrogate FRANS Fexperimental Reference (Exp [5]) Reference (RANS) Optimal (by PBA) 1.155 1.155 0.219 0.289 0.289 0.264 0.499 0.499 0.318 2.250 2.279 5.065 1.587 1.710 2.224 — — 0.3909 — 0.5927 0.3975 0.5974 — — is constructed as shown in Eq (6) and the optimum design is found Table shows the reference and optimal shapes and their objective function values The geometry presented by Hwang and Cho [5] is used as a reference shape in the present work The RANS analysis result for the reference matches well with the experimental result The objective function (F in Eq (5)) value obtained by RANS analysis with the weighting factor wf = 0.09 differs from the experimental one with an error less than 1% The optimal design shows the ratio of Nusselt numbers (Nua ) is largely increased (by 122.0%), and the friction factor (Ff ) is also increased (by 30.0%) but much less as compared to Nua Value of objective function (F ) is reduced by 32.9% From Eqs (1), (2), and (4), this is regarded as quite good enhancement of heat transfer performance The variable H /d, which is the ratio of channel height to dimple diameter, shows a lower value in the design space A finding by Mahmood and Ligrani [11] shows that the lower value of channel height produces higher heat transfer rate This value supports also the optimum designs of inline arrays [13] and staggered arrays [14] for a single surface dimple depression Hd /d ratio is coming near the upper limit in design space and it contradicts the laminar flow in the dimpled channel [3], which shows a higher Nusselt number ratio at a lower value of Hd /d But a deeper dimple (higher value of Hd /d) produces a higher heat transfer rate because of vortex pair shedding with different strength and different shear layer formation, development, and reattachment as presented by Burgess and Ligrani [7] They reported the result for turbulent flow with Hd /d = 0.1, 0.2, and 0.3 An optimum value of the d/S ratio is coming to be near the lower bound in design space As stated by Silva et al [4], the high thermal coefficient area increases with the distance between dimples Although the optimum design shows some similarity with previous works [13, 14], the combination of variables (optimum design) does not match exactly with previous results The objective function value of optimal design predicted by the PBA model is evaluated by the RANS solver to check the prediction accuracy of PBA surrogate model The predicted value of objective function (Fsurrogate = 0.3909) shows only 1.66% error as compared to the RANS result (FRANS = 0.3975) (Table 2) As the PBA model is constructed using RSA, KRG, and RBNN surrogates, initially the PRESS (Ei ) is calculated and weights (wi ) are obtained using Eq (8) The PRESS values and weights shown in Table are assigned to each surrogate Obviously, in terms of contribution to construct the PBA model, RSA heat transfer engineering is highest and RBNN has the lowest contribution As RBNN has the highest error (PRESS), the weight assigned to RBNN to the construction is lowest This global error estimate gives robustness in prediction of the PBA surrogate In Figure 6a, the values of optimal design variables are given when the weighting factor (wf ) is varied from to 0.15 The variable values are normalized in the range of to Here, the total six designs are predicted by the surrogate to check the trends of variables with the variation of wf The ratio of channel height to dimple diameter (H /d) increases largely with the increase of wf However, the ratios of dimple depth to dimple diameter (Hd /d) and dimple diameter to channel width (d/S) decrease with increase of wf Similar results were found by Kim and Choi [13] for inline arrays of dimples on a single surface of a channel Figure 6b shows the predicted optimum objective function value with the variation of wf and compared with the RANS result for reference shape The improvement of objective function due to optimization decreases with the increase of weighting factor The weighting factor is assumed to be 0.09 in the following discussions Thermal performance of optimized shape is increased by 70.8% as compared to reference geometry, as shown in Figure The experimental result [5] is also shown in this figure Thermal performance (Th) is defined as follows: T h = (N ua )/(f/fo )1/3 (9) Burgess and Ligrani [7] presented that the higher value of Th is produced when Hd /d is high and this is similar to the present optimum design, which also has a deeper dimple Local Nusselt number contours on dimpled walls are shown in Figure for reference and optimum geometries It is found that heat transfer is enhanced especially at the rear part of the dimpled surface At the inside surface of the dimple cavity, highvelocity fluids impinge and produce a higher mixing zone and hence higher heat transfer rate Just downstream of the front rim of the dimple, the heat transfer rate is suddenly decreased due to flow separation Downstream of the dimple, heat transfer Table PRESS and weight of respective surrogate to construct PBA model Model PRESS wi RSA KRG RBNN 0.069 0.072 0.088 0.363 0.348 0.289 vol 31 no 13 2010 1120 A SAMAD ET AL Figure Nusselt number contours on dimpled surface: (a) reference geometry, and (b) optimized geometry Figure Optimum values with different weighting factors: (a) optimum design variables, and (b) optimum objective function and percent improvement Figure 0.09 Thermal performances of reference and optimized shape with wf = heat transfer engineering increases due to the pairs of vortices These effects are higher for optimized geometry Figure shows local Nusselt number distributions along the centerline The distributions for optimum geometry indicate that improvement of heat transfer occurs throughout the domain The small dotted circles in Figure represent the dimple rim Figure Local Nusselt number distributions along centerline at z/P = 0.5 (wf = 0.09) vol 31 no 13 2010 A SAMAD ET AL Figure 10 Velocity vectors Figure 11 Vorticity contours near dimpled surface: (a) reference geometry, and (b) optimized geometry with wf = 0.09 heat transfer engineering 1121 locations The optimum geometry shows the higher values of peaks of local Nusselt number and the highest peak is located at the rear rim of the dimple Figure shows that heat transfer improvement due to the dimple begins after the leading edge of the dimple The heat transfer rate builds up through the cavity, reaching a maximum at the trailing edge, and reduces drastically just downstream Figure 10 shows the velocity vector plots on streamwise cross section, including diagonal of the dimple At the rear rim location of the optimum channel, the up-wash is much stronger than in the reference channel The optimum shape shows shorter length of reattachment in the dimple than the reference shape These are related to the rapid increase in heat transfer rate on the dimpled surface, shown in Figure and Figure 9a Figure 11 shows the normalized vorticity distribution in the streamwise direction near the dimpled surface for the reference and the optimized geometry Here, the higher intensity of vorticity (positive or negative) is found for the optimal shape The higher vorticity produces the higher turbulence kinetic energy, and thus is related to the higher heat transfer augmentation Figures 12 and 13 show the normalized turbulence kinetic energy distributions on streamwise cross section, including diagonal of the dimple and near the dimpled surface, respectively The optimum geometry shows a much higher level of turbulence kinetic energy in comparison with that of the reference geometry The distributions commonly show highest turbulent kinetic energy in the region near the rear rim of the dimple, Figure 12 Turbulence kinetic energy distributions on streamwise cross section including diagonal of the dimple: (a) reference geometry, and (b) optimized geometry with wf = 0.09 vol 31 no 13 2010 1122 A SAMAD ET AL to dimple diameter increases, while ratios of dimple depth to dimple diameter and dimple diameter to dimple pitch reduce for optimal shape The optimum shape shows stronger vorticity in the dimple and shorter length of reattachment on dimple surface From this research, it is concluded that the surrogate approach can safely be used, reducing computational and experimental expenses to design the better performing dimpled surfaces NOMENCLATURE Ad CV d Dh DOE Ei f f0 F GMSE H Hd KRG LHS Nu NSM Nua Nuo Figure 13 Turbulence kinetic energy distributions near dimpled surface: (a) reference geometry, and (b) optimized geometry with wf = 0.09 where the heat transfer rate also shows the highest peak, as in Figure CONCLUSIONS Staggered dimple arrays on opposite walls of a channel are numerically optimized using RANS analysis Twenty designs for three nondimensional design variables are selected using LHS method The optimization is performed to increase the heat transfer rate, compromising with pressure drop A reliable weighted-average surrogate, the PBA model, is used to predict the optimum design By the optimization, for a weighting factor in a weighted sum of objective functions, a large improvement in objective function and thermal performance has been obtained With the increase in contribution of friction loss to the final objective function, the ratio of channel height heat transfer engineering P PBA PRESS RANS RBNN Re ReDh RMS RSA SST S TKE Th Tu u, v, w Ub u2rms vrms wrms wf area of heat transfer surface cross-validation dimple print diameter channel hydraulic diameter design of experiments error of ith surrogate model (PRESS) friction factor friction factor for fully developed flow in a smooth pipe objective function generalized mean square cross-validation error channel height dimple depth kriging Latin hypercube sampling local Nusselt number number of basic surrogate models average Nusselt numbers Nusselt number for the fully developed turbulent flow in a smooth pipe dimple pitch PRESS-based averaging predicted error sum of squares Reynolds-averaged Navier–Stokes radial basis neural network Reynolds number Reynolds number based on hydraulic diameter root mean square response surface approximation shear stress turbulence distance between dimples nondimensional turbulence kinetic energy (u2 +v2 +w2 ) (= /2 rms Ubrms rms ) thermal performance turbulence intensity velocity components in x, y, and z directions, respectively average axial velocity root mean square fluctuation of u velocity root mean square fluctuation of w velocity root mean square fluctuation of v velocity weighting factor vol 31 no 13 2010 A SAMAD ET AL y y+ distance from the wall y in law of the wall coordinate Greek Symbols p ρ pressure drop fluid density REFERENCES [1] Ligrani, P M., Oliveira, M M., and Blaskovich, T., Comparison of Heat Transfer Augmentation Techniques, AIAA Journal, vol 41, no 3, pp 337–362, 2003 [2] Schukin, A V., Kozlov, A P., and Agachev, R S., Study and Application of Hemispherical Cavities for Surface Heat Transfer Augmentation, ASME Paper 95-GT-59, 1995 [3] Ridouane, E H., and Campo, A., Heat Transfer and Pressure Drop Characteristics of Laminar Air Flows Moving in a Parallel Plate Channel With Transverse Hemi-Cylindrical Cavities, International Journal of Heat and Mass Transfer, vol 50, issue 19–20, pp 3913–3924, 2007 [4] Silva, C., Marotta, E., and Fletcher, L., Flow Structure and Enhanced Heat Transfer in Channel Flow With Dimpled Surfaces: Application to Heat Sinks in Microelectronic Cooling, Journal of Electronic Packaging, vol 129, pp 157–166, 2007 [5] Hwang, S D., and Cho, H H., Heat Transfer Enhancement of Internal Passage Using Dimple/Protrusion, 13th International Heat Transfer Conference, Sydney, Australia, HTE24, 2006 [6] Mahmood, G, I., Mounir, Z., Sabbagh, M Z., and Ligrani, P M., Heat Transfer in a Channel With Dimples and Protrusions on Opposite Walls, Journal of Thermophysics and Heat Transfer, vol 15, no 3, pp 275–283, 2001 [7] Burgess, N K., and Ligrani, P M., Effects of Dimple Depth on Channel Nusselt Numbers and Friction Factors, Journal of Heat Transfer, vol 127 pp 839–847, 2005 [8] Burgess, N K., Oliveira, M M., and Ligrani, P M., Nusselt Number Behavior on Deep Dimpled Surfaces Within a Channel, Journal of Heat Transfer, vol 125, pp 11–18, 2003 [9] Park, J and Ligrani, P M., Numerical Predictions of Heat Transfer and Fluid Flow Characteristics for Seven Different Dimpled Surfaces in a Channel, Numerical Heat Transfer, Part A, vol 47, pp 209–232, 2005 [10] Park, J., Desam, P R., and Ligrani, P M., Numerical Predictions of Flow Structure Above a Dimpled Surface in a Channel, Numerical Heat Transfer, Part A, vol 45, pp 1–20, 2004 [11] Mahmood, G I., and Ligrani, P.M., Heat Transfer in a Dimpled Channel: Combined Influences of Aspect Ratio, Temperature Ratio, Reynolds Number, and Flow Structure, International Journal of Heat and Mass Transfer, vol 45, pp 2011–2020, 2002 [12] Moon, H K., Connell, T O., and Glezer, B., Channel Height Effect on Heat Transfer and Friction in a Dimpled Passage, Journal of Engineering for Gas Turbines and Power, vol 122, pp 307–313, 2000 [13] Kim, K Y., and Choi, J Y., Shape Optimization of a Dimpled Channel to Enhance Turbulent Heat Transfer, Numerical Heat Transfer, Part A, vol 48, no 9, pp 901–915, 2005 heat transfer engineering 1123 [14] Samad, A., Shin, D Y., Kim, K Y., Goel, T., and Haftka, R T., Surrogate Modeling for Optimization of a Dimpled Channel to Enhance Heat Transfer Performance, Journal of Thermophysics and Heat Transfer, vol 21, no 3, pp 667–670, 2007 [15] Queipo, N V., Haftka, R T., Shyy, W., Goel, T., Vaidyanathan, R., and Tucker, P K., Surrogate-Based Analysis and Optimization, Progress in Aerospace Sciences, vol 41, pp 1–28, 2005 [16] Li, W., and Padula, S., Approximation Methods for Conceptual Design of Complex Systems, in Eleventh International Conference on Approximation Theory, eds C Chui, M Neaumtu, and L Schumaker, Gatlinburg, TN, pp 241–278, 2004 [17] Goel, T., Haftka, R T., Shyy, W., and Queipo, N., Ensemble of Surrogates, Structural and Multidisciplinary Optimization, vol 33 no 3, pp 199–216, 2007 [18] Samad, A., Kim, K Y., Goel, T., Haftka, R T., and Shyy, W., Multiple Surrogate Modeling for Axial Compressor Blade Shape Optimization, Journal of Propulsion and Power, vol 24, no 2, pp 302–310, 2008 [19] Ansys CFX-11.0, Ansys, Inc., 2006 [20] Menter, F R., Kuntz, M., and Langtry, R., Ten Years of Industrial Experience With the SST Turbulence Model, Turbulence, Heat and Mass Transfer 4, Begell House Inc., Redding, CT, pp 625– 632, 2003 [21] Bardina, J E., Hung, P G., and Coakley, T., Turbulence Modeling Validation, AIAA Paper 97-2121, 1997 [22] Lai, Y G., and So, R M C., Near-Wall Modeling of Turbulent Heat Fluxes, International Journal of Heat and Mass Transfer, vol 33, pp 1429–1440, 1990 [23] JMP R 5.1, SAS Institute, Inc., Cary, NC, 2004 [24] Collette, Y., and Siarry, P., Multiobjective Optimization, Principles and Case Study, Springer-Verlag, New York, 2003 [25] Myers, R H., and Montgomery, D C., Response Surface Methodology—Process and Product Optimization Using Designed Experiments, John Wiley & Sons, New York, 2005 [26] Orr, M J L., Introduction to Radial Basis Neural Networks, Center for Cognitive Science, Edinburgh University, Scotland, UK, 1996 Available at: http://anc.ed.ac.uk/RBNN [27] MATLAB R , The Language of Technical Computing, Release 14, The MathWorks, Inc., Natick, MA, 2004 [28] Martin, J D., and Simpson, T W., Use of Kriging Models to Approximate Deterministic Computer Models, AIAA Journal, vol 43, no 4, pp 853–863, 2005 APPENDIX: RSA, KRG, AND RBNN SURROGATES In the RSA method [25], a second-order polynomial function is fitted to get the approximation function The RBNN model [26] is a two-layer network that consists of a hidden layer of radial basis function and a linear output layer The parameters for fitting this surrogate model are a spread constant (SC) and a user-defined error goal (EG) The allowable error goal is decided from the allowable error from the mean input responses In MATLAB [27], newrb is the function for RBNN design KRG model [28] is an interpolating meta-modeling technique that employs a trend model f (x) to capture large-scale variations and a systematic departure Z(x) to capture small-scale variations Kriging postulation is the combination of global model vol 31 no 13 2010 1124 A SAMAD ET AL and departures of the following form: Ki-Don Lee is a Ph.D student at the Computational Fluids Engineering Laboratory, Inha University, Korea He received his bachelor’s and master’s degrees in 2007 and 2009, respectively, at Inha University Numerical analysis and optimization for cooling passages in gas turbine blades have been his major research interests Currently, he is working on design optimization of shaped holes for film cooling using surrogate models Fˆ (x) = f (x) + Z(x) where Fˆ (x) represents the unknown function, f (x) is the global model, and Z(x) represents the localized deviations Z(x) is the realization of a stochastic process with mean zero and nonzero covariance A linear polynomial function is used as a trend model and the systematic departure terms follow Gaussian correlation function Abdus Samad received his bachelor’s and master’s degrees in mechanical engineering from Aligarh Muslim University, India He received his Ph.D degree from Inha University, Korea, in 2008 His research area for his Ph.D was turbomachinery, blade shape and heat transfer augmentation, shape optimization, and modeling of multiple surrogate-based optimization techniques Presently he is working as a research and development engineer at the University of Aberdeen, United Kingdom, and is involved with design of down-hole pumps He has published 45 journal and conference articles heat transfer engineering Kwang-Yong Kim received his B.S degree from Seoul National University in 1978, and his M.S and Ph.D degrees from the Korea Advanced Institute of Science and Technology (KAIST), Korea, in 1981 and 1987, respectively He is currently a professor and the head of the School of Mechanical Engineering of Inha University, Incheon, Korea Professor Kim is also the current editor-in-chief of the International Journal of Fluid Machinery and Systems (IJFMS), and the chief vice-president of the Korean Fluid Machinery Association (KFMA) He is also a fellow of the American Society of Mechanical Engineers (ASME) vol 31 no 13 2010