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Transcript

Scaling of Heat Transfer Coefficients Along Louvered Fins A. C. Lyman1, R. A. Stephan2, and K. A. Thole Virginia Tech Mechanical Engineering Department Blacksburg, VA 24061 L. W. Zhang and S. B. Memory Modine Manufacturing Company Research and Development 1500 DeKoven Avenue Racine, WI 53403-2552 Corresponding author: K. A. Thole Address: Virginia Tech, Mechanical Engineering Department Blacksburg, VA 24061 Phone/Fax: (540)231-7192 / (540)231-9100 E-mail: [email protected] Submission for review to the Experimental Thermal and Fluid Science 1 2 Present address: Intel Corporation, CH5-157, 5000 W. Chandler Blvd.,Chandler, AZ 85226 Present address: NASA Langley Research Center, M/S 431, 1 North Dryden Street, Hampton, VA 23681-2199 1 Abstract Louvered fins provide a method for improving the heat transfer performance of compact heat exchangers without a prohibitive increase in the pressure drop. Spatially resolved heat transfer coefficients along the louvers protruding from the fins are needed to understand the details of compact heat exchanger performance. Experiments were conducted in a number of large-scale louver models with varied fin pitch and louver angle over a range of Reynolds numbers. This paper presents a method for evaluating the spatially-resolved louver heat transfer coefficients using various reference temperatures, such as the bulk flow temperature and adiabatic wall temperature, to define the convective heat transfer coefficients. The results from this study indicate that the thermal field surrounding a particular louver is the overriding influence on the heat transfer from that louver. Keywords: compact heat exchangers, louvered fins, convective heat transfer Introduction Louvered fins, typically found in many compact heat exchanger designs, increase the average heat transfer by interrupting the boundary layer formation and by providing more surface area. In contrast to a continuous fin, which allows an uninterrupted growth of the boundary layer, a louvered fin provides a new start for the boundary layer resulting in overall higher averages of heat transfer coefficients. Louvered fin heat exchangers, such as the one shown in Figure 1a, are used predominantly in the automotive industry where space and weight are of the utmost importance. Figure 1b shows two fins with protruding louvers that are indicated by the line (A-A) shown in Figure 1a. Since over 85% of the total thermal resistance in a typical air-cooled heat exchanger occurs on the air-side, compact heat exchanger design is highly dependent on enhanced air-side heat transfer surfaces such as louvered fins to improve its performance for a given heat exchanger size. The heat transfer along an individual louver primarily depends on two factors. The first of these factors is the flow field surrounding a particular louver. The flow field dictates the boundary layer growth along the louver surface. The second of these factors is the thermal field since it sets up the local driving temperature potential between the louver surface and the fluid. This paper presents a detailed study of the heat transfer along louvers embedded in an array of louvered fins, which is representative of what occurs in a compact heat exchanger. The 2 boundary condition applied along the louvers, both experimentally and computationally, was that of a constant heat flux. While actual heat exchangers have neither a constant heat flux nor a constant surface temperature boundary condition, a constant heat flux insured that our objectives were met. The most important of those objectives was to gain an understanding of the flow and thermal field effects on the louver heat transfer. In particular, the question that will be addressed is whether the heat transfer coefficients can be collapsed to a single curve for a louvered fin considering the complexity of the flow and thermal fields. The latter part of this paper compares the heat transfer performance of nine different louvered fin geometries where the fin pitch and the louver angle were varied. Past Studies of Relevance Relative to the flow field of a louvered fin, one of the dominating factors is whether the flow is duct (axially) or louver (louver-angle) directed. Achaichia and Cowell [3] were the first to experimentally verify the relative importance of designing a louvered fin where the flow is louver directed rather than duct directed. Their study indicated an increase in heat transfer coefficients as the flow transitioned from being duct directed to being louver directed. In general, how the louver directs the flow depends on fin pitch, louver angle, and Reynolds numbers. The flow tends to be louver directed at high Reynolds numbers, lower louver angles, and larger fin pitches [4]. Although most compact heat exchangers are designed to have louverdirected flow, there is a development length required for the flow to transition from being axially-directed to being louver-directed. That development length can be as many as five or six streamwise louvers, which is generally more than a third of the total number of louvers along a fin [5]. Given that most louvered fin designs contain a flow reversal louver, there are a large number of louver passages that are not fully-developed. In their computational study of parallel plates, Zhang et al. [6] found that as the Reynolds numbers increased, the reduced velocity from upstream louver wakes become stronger and progress further downstream. Similar results were obtained by Springer and Thole [7], who performed flow field measurements in louvered fins, and by Kurosaki et al [8], who performed visualizations of thermal wakes for parallel surfaces. Kurosaki et al. [8] also pointed out the importance of the thermal wakes on the local heat transfer coefficients along a particular louver. 3 Thermal wakes surrounding a louver serve to reduce the temperature potential between the louver surface and surrounding flow thereby decreasing the heat transfer. Given the combinations of transitioning from duct to louver directed flow, flow and thermal wake effects, and Reynolds number effects, it is relatively important to perform row-byrow analyses for louvered fins. The only studies, to the authors’ knowledge, that have been completed analyses on a row-to-row basis for louvered fin geometries are those of Huihua and Xuesheng [9], Suga and Aoki [10], and Zhang and Tafti [11]. Although there have been heat transfer and flow field studies, there have been no attempts to separate out the relative importance of the thermal and flow field effects. Huihua and Xuesheng [9] determined that the flow became fully developed after the fifth louver position based on their mass transfer experiments. Their results, however, were obtained with only three fin rows whereby only one louver was coated with naphthalene during a particular test (equivalent to heating only one louver). As a result of coating one louver, only the effects of the flow field for a particular louver position could be deduced without the effects of the local thermal field. In a computational study by Suga and Aoki [10] to determine the optimal louver geometry, they concluded that the optimal geometry would be one in which the thermal wake would pass directly between the smallest gap of the downstream louvers. In their analysis, they assumed that the thermal wakes were 100% louver directed. From the perspective of combined heat transfer and pressure drop, smaller louver angles were found to provide an overall better performance. The importance of considering the thermal wake effects on louver heat transfer is most clearly described by the computational study of Zhang and Tafti [11]. They describe an inter-fin interference for highly louver directed flow and an intra-fin interference for duct directed flows. Zhang and Tafti found that the thermal wake effects could be expressed as a function of the flow efficiency (measure of how duct or louver directed the flow is) and the fin to louver pitch ratio. Moreover, they state that neglecting thermal wake effects at low flow efficiencies can introduce errors as high as 100% in the heat transfer. Their results point to the necessity of heating both the louver of interest as well as the upstream and adjacent louvers when performing heat transfer studies in louvered fins. 4 Experimental and Computational Methodology The work that was completed for this study involves both experimental and computational simulations of a portion of a compact heat exchanger with louvered fins that represents the middle core (no endwall effects were evaluated). This paper presents spatiallyresolved heat transfer coefficients along scaled-up louver surfaces. Because the interest of this work is focused on understanding the details of the heat transfer for a louver in a particular louvered fin geometry, assessing the performance based on an effectiveness-NTU (number of transfer units) or on an LMTD (log-mean-temperature difference) is not applicable. Instead, this study evaluates different reference temperatures to scale the heat transfer coefficients to gain an understanding of the flow and thermal field effects for individual louvered fins. Experimental Methodology The experimental design was such to insure that meaningful, spatially-resolved heat transfer coefficients could be made along several streamwise louvers over a range of Reynolds numbers. To insure good measurement resolution, the louver model was scaled up by a factor of 20. The flow conditions were also scaled to match Reynolds numbers (ReLp) relevant to the range for operating compact heat exchangers, which were based on the louver pitch and the inlet velocity (Lp and Uin). A schematic of the open-loop test rig used for this study is shown in Figure 2. The inlet of the test rig consisted of honeycomb, a screen, and a contraction that together provided a uniform flow at the entrance to the test section. The contraction was designed using a commercial CFD package (described later in this section) and had a 16:1 area reduction for each of the models tested. Laser Doppler velocimeter measurements were made to verify the flow uniformity at the inlet to the louver test section. A variable speed centrifugal fan located at the exit of the test rig provided the flow through the test rig with the speed of the fan being controlled using an AC inverter. The flow rate was measured using a laminar flow element (LFE) located just downstream of the test section. Nine louver geometries were used in this study and are shown in Figure 3a-i for the entrance louver through the eighth louver position. These geometries were based on the optimum designs reported by [9] and variations of fin pitch and louver angle from those optimum designs. Louver heat transfer measurements were made along louvers 2 – 8 while a 5 constant heat flux boundary condition was applied along louvers 1 – 8. In all of these models, the louver pitch and fin thickness (t/Lp = 0.082) remained constant while the fin pitch and the louver angle were varied, as indicated by Figure 3a-i and in Table 1. The number of streamwise louvers for a particular fin also remained constant for each of the models with 17 streamwise louvers. The number of fin rows (louver passages) varied from nine rows for the model having the largest fin pitch to eleven rows for the model having the smallest fin pitch. A larger number of fins could be incorporated for smaller fin pitches and still only require a reasonable test section size. In all of the cases, a CFD study was conducted to insure enough fin rows were simulated in each model to maintain periodic conditions. Incorporated into the test section were styrofoam plates (shown in Figure 2) that matched the louver angle to remove any endwall effects such that a number of periodic louver passages were achieved. Styrofoam was also used to insulate both sidewalls of the test section to minimize any losses. Using the heating foils shown in Figure 4, a constant heat flux was provided for each of the louvers upstream of the flow reversal louver (measurements were only made on louvers upstream of the reversal louver). The core of the louvers was balsa wood, which minimized the conduction through the louvers. This core was sandwiched between manila file folder paper, which further insulated the louver and provided the correct louver thickness. The outside layers of the louver consisted of a stainless steel foil that served as the resistive heating element. Lead wires were connected using copper bus bars that were soldered to both ends of the stainless steel foils. The three heat flux values for the ReLp=230, 370, and 1016 cases were q ′′ = 60, 95, and 160 W/m2, respectively. The heat flux levels were chosen such that the temperature differences from the leading edge to the trailing edge were similar for each of the Reynolds numbers to reduce flow property effects. All of the louvers in the array were connected in series to be sure that the surface heat flux was the same for every louver. Because the surface area of the entrance louver was about two times (1.94 times) greater than that of the other louvers, however, a separate circuit for the entrance louvers was required. The electrical resistance of the louvers were measured to be 0.149 ± 2% Ohms for the main field louvers and 0.078 ± 2% Ohms for the entrance louvers and, as such, the power was constant for all of the louvers. The current through the circuits was determined by measuring the voltage drop across precision resistors located in 6 the circuits. The surface heat flux was then calculated using the heater resistance and the current through the circuit. The surface temperatures of the louvers were measured using thermocouples embedded in the balsa wood core, as shown in Figure 4, for two instrumented louvers. The thermal resistance of the foil was small relative to the convective resistance (Rfoil/Rconvection = 2x10-5 in most cases) and was therefore neglected. The two instrumented louvers included one that was used for the streamwise temperature measurements and one that was used for spanwise temperature measurements. These instrumented louvers were placed in different streamwise louver positions to acquire the data. The streamwise instrumented louver had 27 thermocouples on both the front and back sides along the spanwise center of the louver giving 54 thermocouples in the entire louver. The spanwise instrumented louver had ten thermocouples placed along the span of the louver at streamwise locations that were 40% and 80% of the louver length. The spanwise instrumented louver was used to insure two dimensionality along the louver surface. Due to the complexity of the velocity and thermal fields that develop as air flows through the louver array, the temperature distributions on the front and back sides of the louvers were not identical. Prior to calculating the heat transfer coefficients, a conduction correction was made due to a temperature difference between the top and bottom surfaces. The temperature differences between the two surfaces were minimal in most cases, but compensation was necessary where there were temperature differences across the louver thickness. The thermocouples used for this study were type E and were accurately calibrated relative to one another. Corrections were made for voltage biases that occurred for any of the thermocouples. Since only temperature differences were required for these studies, this calibration method was appropriate. Data acquisition hardware used to acquire the thermocouple voltages consisted of a National Instruments SCXI-1000 chassis into which three SXCI-1102 modules were inserted. An SXCI-330 terminal block was inserted into each of the modules. An energy balance was performed for each of the experiments to determine if the expected bulk flow temperature of the flow was in agreement with the measured flow rate through the test section and the measured amount of heat added by the louvers. At ReLp = 230, the maximum difference between the computed exit temperature and the measured, spatiallyaveraged exit temperature for all the models was 10.7%, which occurred for the θ = 27°, Fp/Lp = 0.76 model. At ReLp = 1016, the maximum difference between the computed exit temperature 7 and the measured, spatially-averaged exit temperature was 5%. Note that the measured exit temperature was taken by traversing a thermocouple across the exit of the test section. Some of the reported differences between the computed and spatially-averaged exit temperatures may be due to the differences between a bulk (energy calculation) and spatially-averaged (rather than bulk-averaged) temperature averages. Also considered was the effect of natural convection as compared to forced convection. In the measurement region upstream of the reversal louver, a number of experiments were conducted whereby the instrumented louver was placed in vertical positions other than in the center of the louver array. These tests were conducted for the third and seventh streamwise louver positions. At ReLp = 1016, the measured heat transfer coefficients were identical at three different vertical louver positions for both the third and seventh louver position. At ReLp = 230 for the θ = 27°, Fp/Lp = 0.76 model, there was a 9.5% effect on the heat transfer coefficient as the instrumented louver was raised two fin rows above the measurement position for the seventh louver. Uncertainty Estimates The uncertainties of experimental quantities were computed by using the method presented by Moffat [4]. The uncertainty was calculated by acquiring the derivatives of the desired variable with respect to individual experimental quantities and applying known uncertainties. The combined precision and bias uncertainty of the individual temperature measurements was ± 0.15 ºC, which dominated the other uncertainties. The uncertainties in the Colburn factor were the highest for the ReLp = 230 condition at the leading edge of the second louver where the temperature differences between the surface and fluid were small. At this location, the uncertainties ranged as high as 11.4% using the bulk flow temperature and 9.0% using the adiabatic wall temperature for the reference temperatures. Uncertainties decreased along the louver where more representative numbers were 6.0% using the bulk flow temperature and 5.8% using the adiabatic wall temperatures. For the ReLp = 1016 flow condition, the uncertainties in the Colburn factors using the bulk flow and adiabatic wall temperatures were 4.2% along most of the louvers. A representative uncertainty in the nondimensional adiabatic wall temperatures was 4.4% for a level of η = 0.94 at ReLp = 230 along the seventh louver and was 13.4% for a level of η = 0.57 at ReLp = 230 along the second louver. 8 Computational Methodology Computational simulations were done using the commercial package Fluent 5.5 (1998). The CFD predictions were obtained by solving the momentum and energy equations using second-order discretization. The flow was assumed to be two-dimensional, laminar, and steady. The computational and experimental models consisted of a single fin row with 17 streamwise louvers including one entrance louver, one exit louver, and one flow reversal (turning) louver. To computationally simulate an infinite stack of louvers, periodic boundary conditions were used while a constant velocity was applied at three louver pitches upstream of the entrance louver. An outflow boundary condition (negligible streamwise gradients) was applied at seven louver pitches downstream of the exit louver. A constant heat flux was assigned to the front and back side of each louver in the array. The value of the heat flux was set to match the different heat flux values for each Reynolds number as stated for the experiments. The mesh for the model consisted of a triangular grid for the majority of the louver passage with a quadrilateral grid placed along the surface boundaries of the louvers. The solution-insensitive grid, verified through a number of grid studies, consisted of approximately 505,000 cells. The computations were performed in parallel on an SGI Origin 2000 and required nominally 1100 iterations to insure convergence. Scaling of Heat Transfer Coefficients All studies that involve convective heat transfer are made difficult by having to define a relevant reference temperature needed for the heat transfer coefficient. The classical definition of the heat transfer coefficient is typically written as h= q′′ Tw − Tref (1) In the preceding equation q ′′ represents the surface heat flux, Tw is the surface temperature of the object being considered, and Tref is the reference “driving” temperature. The question arises as to what temperature should be used as the reference temperature in a complex heat transfer problem such as the one being considered in the present study. For low speed external flow 9 applications the freestream temperature is taken as the reference temperature, and for internal flow applications the bulk flow temperature is taken as the reference temperature. Moffat [1] addressed the particular question regarding what reference temperature should be used in defining the heat transfer coefficient for various complex thermal fields such as that for louvered fins and for complex electronic cooling configurations. One reference temperature suggested by Moffat was the adiabatic wall temperature, which is the temperature of the surface with no applied heat flux. The adiabatic wall temperature indicates the local driving temperature of the fluid in the near-wall region. The method of using the adiabatic wall temperature as a reference temperature is commonly applied to film-cooling applications on gas turbine airfoils. The reason for using the adiabatic wall temperature as the reference temperature in film-cooling applications is because it is a three temperature problem that includes the wall temperature, the coolant temperature, and the freestream temperature. The most important benefit in using the adiabatic wall temperature to define the heat transfer coefficient is that the coefficients are independent of the surrounding thermal field. One can also use the adiabatic wall temperature measured along a louver to provide details on the surrounding thermal field. For example consider the non-dimensional adiabatic wall temperature defined as η= Taw − Tin Tb − Tin (2) where Tb is the bulk temperature of the fluid and Tin is the temperature of the air at the inlet to the compact heat exchanger. A η equal to unity results for a completely mixed out thermal field where the fluid temperature is at the bulk flow temperature. An η value greater than unity results if the surrounding fluid is above the bulk flow temperature, due to heated wakes from upstream louvers, and an η value less than unity occurs if the fluid is cooler than the bulk flow temperatures. Increased heat transfer performance would be expected at low η values. Other reference temperatures that one considers in defining the local heat transfer coefficient for a louvered fin application include the inlet temperature to the compact heat exchanger and the local bulk flow temperature to a particular louver passage. The inlet temperature to the compact heat exchanger is not a particularly useful reference temperature since it does not allow a meaningful one-to-one comparison of different louver geometries. The 10 local bulk flow temperature for a louver passage, however, contains information as to the amount of heat added upstream of a particular louver passage and is obtained through an energy balance as shown below: Tb = Tin + q ′′A & cp m (3) The heat transfer coefficients that use the bulk flow temperature as the reference temperature in equation 1 are dependent on both the surrounding thermal field and flow field. Alternatively, the heat transfer coefficients that use the adiabatic wall temperature as the reference temperature in equation 1 are only dependent on the local flow field that is affecting the boundary layer growth along the louver. Since one of the goals of this study was to discern the flow and thermal field effects, which also provides information on scaling the heat transfer coefficients, heat transfer coefficients with two reference temperatures were compared. Those two reference temperatures included the computed bulk air flow temperature at the entrance to the particular louver passage (Tb) whereby hb is defined, and the local adiabatic wall temperature of the louver (Taw) whereby haw is defined. Using the bulk flow temperature of the flow as a reference temperature allows the effects of heated wakes from upstream louvers to appear in the heat transfer coefficient data. For instance, if a hot wake directly impacts a downstream louver, the surface temperature of that louver will be increased with respect to the bulk flow temperatures causing hb to decrease. Heat transfer coefficients using the adiabatic wall temperature along the louver, referred to as haw, were achieved by performing two experiments. In the first experiment all of the louver surfaces were heated and the surface temperatures of a particular louver were measured (Tw). In the second experiment a zero heat flux boundary condition was applied to the particular louver of interest and the surface temperatures of that louver were measured (Taw). Using the adiabatic temperature as a reference temperature removes the thermal field effects of heated wakes from upstream louvers and allows any flow field effects that might be present to be distinguished. To compare the different geometries, the Colburn factor, a non-dimensional version of the heat transfer coefficient was used. Note that both of the two heat transfer coefficients (hb and haw) were used in the Colburn factors (jb and jaw). Colburn factors were used since it scales the heat transfer coefficient with the maximum mass velocity such that comparisons can be appropriately made between models having different fin pitches. 11 Spatially Resolved Heat Transfer Measurements The first section of the paper will discuss the scaling of the heat transfer coefficients using the two reference temperatures for one of the louver models (θ = 27º, Fp/Lp = 0.76). Note that the results of this louver model are representative of the other models except when noted. Following the discussion on the heat transfer scaling, results of all nine models will be compared in how the thermal field and flow field influences the heat transfer performance. Scaling of the Heat Transfer Coefficients Consider the θ = 27º, Fp/Lp = 0.76 model for comparing the heat transfer coefficients using two different reference temperatures. Note that for this particular geometry, the angled portion of the entrance louver was directly aligned with louvers 4 and 7 (Figure 3d). This is particularly important, as will be illustrated throughout this section, because the entrance louver puts out two times the amount of energy as compared with the downstream louvers (louver length was two times that of the downstream louvers and was set at the same heat flux level). Consequently, the flow leaving the entrance louvers created a significantly hotter thermal wake than the wakes exiting the downstream louvers. These higher wake temperatures resulted in higher surface temperatures along the louvers in the flow path downstream of the entrance louvers. Figure 5a shows the non-dimensional adiabatic wall temperatures for ReLp = 230. Recall Taw is the local adiabatic surface temperature of the unheated louver of interest, Tin is the temperature at the inlet to the test section, and Tb is the bulk flow temperature at the entrance to the local louver passage. The adiabatic wall temperature is representative of the local fluid temperature surrounding the louver. Louver 2 shows low η values, which is an indication that it is exposed to a stream of cool inlet air as in agreement with the predictions made using CFD shown in Figure 5b. Alternatively, the hot wake from the entrance louver progressed in the louver direction through the array and impinged on louver 4 giving values of η > 1. While the bulk flow temperature still met the energy balance requirements at this location, the louver adiabatic wall temperatures were above unity because the local fluid temperatures were above the bulk flow temperature. 12 The effects of the thermal fields surrounding the louver were much more pronounced for the ReLp = 1016, as shown in Figure 6a, since the progression of the wakes were further downstream and the thermal wakes were not as diffuse as for ReLp = 230. Here, η was greater than one for louvers 4 and 7 and η was less than one for louvers 2 and 3. The predicted thermal contour plot for the ReLp = 1016 case in Figure 6b indicates wakes from the louvers to be more narrow at ReLp = 1016 than at ReLp = 230 (Figure 5b). As a result, the heat added to the flow by a given louver is contained in a narrower, more concentrated wake at higher Reynolds numbers. As discussed previously, the surrounding thermal field temperatures appear in the local heat transfer coefficients (Colburn factors) when the bulk flow temperature is used as the reference temperature. Figures 7a-b show the jb and jaw values for the Fp/Lp = 0.76 model at ReLp = 1016. Figure 7a shows that louver 4 exhibits reduced jb values, particularly on the back side of the louver, while louver 2 exhibits increased jb values, particularly on the front side, relative to the other louver locations. The reduced values can be explained by considering, as previously discussed, that louver 4 is aligned with the entrance louver. Because of this alignment, the measured wall temperature is relatively hotter with respect to the bulk flow temperature when in the presence of a heated wake resulting in a larger temperature difference and a lower jb value. Figure 7b shows the Colburn factors for each louver position in which the adiabatic wall temperature was used as the reference temperature for heat transfer coefficients. It is not surprising that when using the adiabatic wall temperature as a reference temperature the data collapses to a single curve for all streamwise louver positions. The only exception to this collapse, is the back side of louver two at ReLp = 1016. The anomaly for the back side of louver two can be explained by the fact that at this location the flow is not completely louver directed and therefore the flow field is influencing the local heat transfer. Figure 7b indicates that once the local thermal field surrounding the louvers is taken into account along the louver, there are no significant differences between louver positions with respect to the flow field (except for louvers very near the entrance), which dictates the boundary layer growth. These results are typical of what was found by the experiments on most of the louver models (exceptions to this will be discussed later in this paper). 13 Effects on the Louver Thermal Field Louver temperatures measured with an adiabatic boundary condition provide information as to the thermal field surrounding a particular louver, as was illustrated for the θ = 27º, Fp/Lp = 0.76 model. Previous studies have pointed out the importance of the flow efficiency for a particular louver geometry, which is defined as the ratio of the angle of the velocity vector to the angle of the louver [4,11]. While this ratio is clearly important for determining the trajectory of a thermal wake, it is also important to consider the streamwise progression and diffusion of the wake. The diffusion of the thermal wakes determines how many downstream louvers are affected. The experimental results obtained in this study show that the thermal field surrounding a particular louver is highly dependent upon the geometry and on the Reynolds number. Consider the η values for the lowest louver angle of 20° as shown in Figures 8-10. At the smallest fin pitch, the thermal fields are relatively uniform beyond the third louver row with η values slightly greater than unity. Values greater than unity indicate that the louvers are surrounded by fluid warmer than the bulk temperature. The low η values for the first two louvers are primarily because of the interaction with cool fluid entering the passage. As the fin pitch increases for ReLp = 230 (Figures 8a, 9a, and 10a), the thermal field becomes increasingly less uniform with η > 2 occurring for the largest fin pitch. The high η values on the second streamwise louver for the Fp/Lp = 1.52 are a result of relatively hot fluid from the entrance louver engulfing the second louver. As expected for this large fin pitch, these high η values are an indication that the flow is mostly duct (axially) directed. The discontinuity in the η values between the front and back of the louvers indicate that the leading edge of the front side of the louver is hotter than the leading edge of the back side. Based on the geometry, this can be expected (Figure 3c). For the Fp/Lp = 1.52 geometry the thermal fields surrounding the louvers are cooler than at smaller fin pitches with the coolest being the fifth louver. The thermal field of the fifth louver is cool because it was surrounded by cool fluid that entered the test rig and bypassed the entrance louver. A dramatic effect in the uniformity of the thermal fields occurs at ReLp = 1016 (Figures 8b, 9b, and 10b). For all of the geometries considered, there are large variations in the thermal field. The highest η values occur for the θ = 20°, Fp/Lp = 0.91 model at the fourth louver position. If one considers that the flow is primarily louver directed, the flow passing over the 14 fourth louver is fluid convected downstream from the entrance louver. This hot wake effect did not occur at the ReLp = 230, which is consistent with the fact that wakes progress further downstream at higher Reynolds numbers. For the Fp/Lp = 0.54 model, the highest η values occurred for the fourth louver and for the Fp/Lp = 1.52 model the highest η values occurred for the seventh louver. Because the flow is more duct directed at the larger fin pitch spacing of Fp/Lp = 1.52 the fourth louver, which is also in-line with the entrance louver, was actually surrounded with relatively cool fluid. To compare the relative thermal fields for each louver geometry at each louver position, louver-averaged η values were calculated and are shown in Figures 11-13. These plots show that every thermal field pattern is unique and provide an indication of where a particular louver is located relative to the heated wakes. The peak η value for each of the models corresponds to the louver position in which the heated entrance louver wake impacts the louver. The louver position with the peak η does not always agree with what would be expected if one considered a purely louver directed flow. A prime example of this phenomena is if one considers the θ = 20°, Fp/Lp = 1.52 geometry. Although the fourth louver is aligned with the entrance louver, the peak η value occurs for the seventh louver. The seventh louver is also aligned with the entrance louver, but is downstream of the fourth louver. Except for the largest fin pitch at 20°, the second louvers have low η values indicating these louvers are mostly surrounded by relatively cooler inlet fluid. For all louver angles, the models having the smallest fin pitch are the least affected by the heated wake from the entrance louver. For the θ = 27°, Fp/Lp = 0.91 geometry a wavelike η pattern occurs, which decreases in amplitude as one progresses downstream and increases in amplitude as the Reynolds number increases (Figure 12). The highest temperatures surrounding a louver occurred at ReLp = 1016 for the θ = 27°, Fp/Lp = 0.91 geometry where three of the streamwise louvers (louvers 3, 5, and 7) had η values much larger than unity, as shown in Figure 12. In contrast, for this same model the thermal fields were relatively uniform at values of η ~ 1 at ReLp = 230 except for louver 3 which is immediately downstream of the entrance louver. The model-averaged η values as a function of Reynolds number are given in Figure 14. The lowest η values occurred for the θ = 27°, Fp/Lp = 1.52 geometry at all Reynolds numbers while the highest η values occurred for different models depending upon Reynolds numbers. 15 The highest η values at ReLp = 230 occurred for the θ = 20°, Fp/Lp = 0.54 geometry. At ReLp = 370 and 1016, the highest η values occurred for the θ = 27°, Fp/Lp = 0.91. The θ = 27°, Fp/Lp = 0.91 geometry is one of the most interesting geometries in that the louvers are staggered with every other louver row being aligned with one another. This geometry is in contrast with other geometries in which there is a larger streamwise spacing between the alignment of the louvers. Considering that a heat exchanger operates over a range of Reynolds numbers, the louvers having the lowest adiabatic wall temperatures (an indication of the highest heat transfer) is the θ = 27°, Fp/Lp = 1.52 model. The highest adiabatic wall temperatures over the Reynolds number range that was considered occurred for the θ = 27°, Fp/Lp = 0.91 model. Effects on Louver Boundary Layer Development The reference temperature as the adiabatic wall for the heat transfer coefficients resulted in a consistent scaling, for the most part, of the Colburn factors (jaw) for each of the models. Figures 15a-c show the jaw values for louver 6 at ReLp = 1016 for each of the fin spacings and louver angles. Except for two models (θ = 20°, Fp/Lp = 0.54 and θ = 39°, Fp/Lp = 0.91), the data collapsed to a single curve for each of the angles. This collapse in the data between the models occurred for each louver position, as well, with the exception being the second louver position. These results indicate that the fin pitch does not have a large effect on the boundary layer growth along the louvers. For the two models in which larger effects on the boundary layer growth did occur, as shown in Figure 15a-c, those models need to be examined more closely. The θ = 20°, Fp/Lp = 0.54 model represents the closest fin spacing studied relative to all of the nine models tested. Because of the resulting geometry and close fin spacing, it is surmised that the flow is being blocked by the downstream louver. This blockage (and small gap between adjacent louvers) redirects the flow towards the upper portion of a particular louver passage such that it flows along the back side of the downstream louver. The data is consistent with a slightly separated boundary layer along the front side of the louvers (lower heat transfer coefficients indicated in Figure 16a). As a result of the nearly-aligned θ = 39°, Fp/Lp = 0.91 geometry, the front side of the louver is not affected by any upstream flow field wakes. The measurements indicate higher Colburn factors based on the adiabatic wall temperature difference, which indicate higher heat 16 transfer on the front side of the model relative to the other models. Because the boundary layer along the front side of the louvers is not affected by any wakes, the boundary layer is thinner resulting in higher heat transfer coefficients. Also what must be considered for this model, however, is the back side where there are lower jaw values. The results presented in Figure 15a-c are typical of what was found by the experiments at all of the Reynolds numbers tested. While the data collapsed for most of the louver positions in all of the models, the only exception was for the second louver position. The reason for the difference occurring for the second louver position was primarily because the flow was not completely louver directed at this position thereby influencing the flow field surrounding the louver. The differences were more pronounced at the higher Reynolds numbers (ReLp = 1016) and larger fin pitches. Because the flow was not fully turned by the second louver position, separation occurred off the back side of the second louver resulting in lower jaw values. The effect of the louver angle gave a slightly larger effect on the boundary layer development along the louver as compared with the fin pitch at the sixth louver position. The largest differences, however, were at the Fp/Lp = 0.91 and can be explained by the fact that for the 39° model the louvers were nearly aligned. Combined Thermal and Flowfield Effects Based on Colburn Factors Using the bulk temperature as the reference temperature in the heat transfer coefficient combines the effects of the boundary layer development and surrounding thermal field. Figure 16 displays the model-averaged Colburn factors based on the bulk temperature reference for all of the models. The model-averaged Colburn factors, for the most part, agree with the modelaveraged non-dimensional adiabatic wall temperatures given in Figure 14 with the highest jb values occurring at the lowest η values. Again this indicates the strong influence of the local thermal field. The spread in the data for the ReLp = 230 case is much larger than for ReLp = 1016 case because the heat transfer is strongly influenced by the flow direction, which is quite variable at the low Reynolds numbers. The flow direction sets the trajectory of the thermal wake. Recall that a larger number of streamwise louvers are needed at the lower Reynolds numbers to transition the flow to being fully louver directed, which influences the averages presented in Figure 16. In general, higher averaged Colburn factors occur for the lower Reynolds numbers at 17 larger fin pitches and higher louver angles. Noticeably, the worst performer at ReLp = 230 and 370 is the model having the smallest fin pitch. Table 1 presents the Reynolds number-averaged Colburn factors based on the bulk temperature reference and non-dimensional adiabatic wall temperatures for each of the models. These overall averages of jb inversely correlate with η, as discussed earlier. The averages presented in Table 1 indicate the overall best performer to be the θ = 27°, Fp/Lp = 1.52 geometry for the entire Reynolds number range studied. This overall “best” heat exchanger geometry was dictated by the fact that it performed the best at ReLp = 230. Practical Significance of Results The results presented in this paper provide a method for evaluating spatially-resolved heat transfer in a complicated geometry such as along a louvered fin that is relative to compact heat exchanger designs. While the focus of these studies, is on comparing the spatially-resolved louver heat transfer, methods to do so are complicated by the choice for the reference temperature in the convective heat transfer coefficient. This paper presents analyses that evaluate different reference temperatures. Based on the method of using measured adiabatic surface temperatures, it is possible to distinguish thermal field from flow field effects on heat transfer from a louvered fin. Moreover, it is possible to track the thermal wakes from upstream louvers, which provide an indication as to whether the flow is duct-directed or louver-directed. The ability to distinguish flow and thermal field effects is significant particularly in understanding the physics of various heat transfer enhancement methods. While one important aspect is clearly how the thermal wake affects the downstream louvers (thermal field effect), another important aspect is how the local flow field affects the louver boundary layer growth. While there are many past studies on compact heat exchangers, the focus as of late is on spatially-resolved heat transfer characteristics along a louver. Given that the results of this study show the dominance of the thermal fields, it becomes imperative that upstream thermal wakes are simulated in louvered fin heat transfer studies. This is in agreement with the computational results by Zhang and Tafti [3]. The results presented in this study indicate that the performance of a louvered fin design is dependent upon the particular operating range for the heat exchanger. At low Reynolds 18 numbers, the performance is highly influenced by how quickly the flow can become fully or partially louver directed. After the flow becomes louver directed, the thermal fields are quite uniform since the thermal wake disappears quicker than that at the higher Reynolds numbers. Larger fin pitches and higher louver angles were better performers at the lower Reynolds numbers. At high Reynolds numbers, the performance is highly influenced by how many louvers downstream the heated wakes influence. Louver geometries in which the louvers are nearly aligned do not provide a good design from a heat transfer perspective, particularly for high Reynolds numbers. The results from this study also indicate that the design of the entrance louver should be carefully considered. Since this louver is nominally two times the length of the other louvers, the heated wake that leaves this louver is detrimental to the heat transfer from the downstream louvers. Similar considerations should be given to the reversal louver since it is also quite long relative to the other louvers. Conclusions Louvers provide a method for increasing the heat transfer of fins in compact heat exchangers. To further increase the benefit of having louvers, it is particularly important to understand the physics of the air side heat transfer since it represents the largest thermal resistance. Understanding the physics requires having spatially-resolved measurements along each louver and properly analyzing those results relative to actual heat exchanger designs. The results of this study provide a method for analyzing heat transfer data from large-scale experiments. The large-scale experiments included nine different louver models, which included three louver angles with each louver angle having three different fin pitches. One of the goals of this study was to determine a method to discern the flow and thermal field effects on the louver heat transfer. To discern these effects, the louver heat transfer coefficients were analyzed in which bulk and adiabatic wall temperatures were used as reference temperatures. Non-dimensional adiabatic wall temperatures along the louvers were also compared, which provides information as to the surrounding thermal field and thus the local driving temperature for each louver surface. The non-dimensional adiabatic wall temperatures indicated a unique pattern for each louver arrangement. At higher Reynolds numbers the heated wakes were more concentrated and 19 convected further downstream while at lower Reynolds numbers the heated wakes were more diffuse. More important at the lower Reynolds numbers is whether the flow was louver or duct directed and what streamwise distance was required for the flow to become louver directed. As a result of these differences, there were large variations in the performance of the various louver models at the lower Reynolds numbers relative to the higher Reynolds numbers. The heat transfer coefficients using the adiabatic wall as the reference temperature collapsed the data along the louver to a single curve for most models. This collapse occurred for each louver position for a single louver model as well as the data from models with different louver arrangements. The only exceptions to this occurred when there was a significant flow field effect such as along the second louver, where separation occurred off the back side, and for louver arrangements which had a large fin pitch or aligned louvers. A collapse of the heat transfer coefficients indicated that the boundary layer development along each louver surface was not a strong function of the particular louver geometry. The heat transfer coefficients based on the bulk flow temperature for a particular louver did correlate closely with its local thermal field indicating the dominant importance of the thermal field approaching a particular louver. Acknowledgments The authors gratefully acknowledge Modine Manufacturing Company for sponsoring the work that was presented in this paper. References [1] Kays, W. M. and A. L. London Compact Heat Exchangers (McGraw-Hill: New York) 1964. [2] Achaichia, A. and T. A. Cowell, Heat Transfer and Pressure Drop Characteristics of Flat Tube and Louvered Plate Fin Surfaces. Experimental Thermal and Fluid Science, 1988, 1, 147-157. [3] Webb, R. L. and P. Trauger, Flow Structure in the Louvered Fin Heat Exchanger Geometry. Experimental Thermal and Fluid Science, 1991, 4, 205-214. [4] Springer, M. E. and K. A. Thole, Entry Region of Louvered Fin Heat Exchangers. Experimental Thermal and Fluid Science, 1999, 19, 223-232. [5] Zhang, L. W., S. Balachandar, D. K. Tafti, and F. M. Najjar, Heat Transfer Enhancement mechanisms in Inline and Staggered Parallel-Plate in Heat Exchangers. International Journal of Heat and Mass Transfer, 1997, 40, 2307-2325. [6] Springer, M. E., and K.A. Thole, Experimental Design for Flowfield Studies of Louvered Fins. Experimental Thermal and Fluid Science, 1998, 18, 258-269. 20 [7] Kurosaki, Y., T. Kashiwagi, H. Kobayashi, H. Uzuhashi, and S. Tang, Experimental Study on Heat Transfer from Parallel Louvered Fins by Laser Holographic Interferometry. Experimental Thermal and Fluid Science, 1988, 1, 59-67. [8] Huihua, Z. and L. Xuesheng, The Experimental Investigation of Oblique Angles and Interrupted Plate Lengths for Louvered Fins in Compact Heat Exchangers. 1989, 2, 100-106. [9] Suga, K. and H. Aoki, Numerical Study on Heat Transfer and Pressure Drop in Multilouvered Fins. ASME/JSME Thermal Engineering Proceedings, 1991, 4, 361-368. [10] Zhang, X. and D. K. Tafti, Classification and Effects of Thermal Wakes on Heat Transfer in Multilouvered Fins. International Journal of Heat and Mass Transfer, 2001, 44, 2461-2473. [11] Moffat, R. J., Describing the Uncertainties in Experimental Results. Experimental Thermal and Fluid Science, 1988, 1, 3-17. [12] Moffat, R. J. What’s New in Convective Heat Transfer? International Journal of Heat and Fluid Flow, 1998, 19, 90-101. 21 Nomenclature A Louver surface area Minimum cross sectional free flow area Aff Specific heat cp Fin pitch Fp & /Aff G Maximum mass velocity, G = m hb Convective heat transfer coefficient based on bulk flow reference temperature, hb = q″/(Tw-Tb) Convective heat transfer coefficient based on adiabatic wall reference temperature, haw = haw q″/(Taw-Tb) Colburn factor based on bulk flow reference temperature, jb = hb·Pr2/3/(G·cp) jb jaw Colburn factor based on adiabatic wall reference temperature, jaw = haw·Pr2/3/(G·cp) k Thermal conductivity Lp Louver pitch, length of louver & Mass flow rate m q″ Applied heat flux boundary condition ReLp Reynolds number based on louver pitch, ReLp = Uin·Lp/ν t Louver thickness Louver wall temperature with adiabatic boundary condition Taw Bulk flow flow temperature from energy balance Tb Tw Louver wall temperature with heated boundary condition Inlet face velocity to test section Uin x* Distance along the streamwise direction of the louver surface Greek η Non-dimensional adiabatic wall temperature, η = (Taw − Tin ) /(Tb − Tin ) θ Louver angle Superscripts ¯ Louver averaged value Model averaged value 22 Table 1. Geometric Parameters and Averages for the Scaled Up Louvered Fin Models. Louver Model a b c d e f g h i θ Ratio of fin pitch to louver pitch (Fp/Lp) Reynolds Number Averaged 20° 20° 20° 27° 27° 27° 39° 39° 39° 0.54 0.91 1.52 0.76 0.91 1.52 0.91 1.22 1.52 0.93 0.86 0.83 0.85 1.1 0.66 0.90 0.91 0.88 η Reynolds Number Averaged 0.045 0.052 0.056 0.055 0.049 0.060 0.052 0.057 0.051 jb 23 a) A A b) Section A-A Figure 1. Typical compact heat exchanger with louvered fins: (a) overall view, and (b) section A-A. 24 a) Styrofoam A Test Section PVC Pipe Blower Inlet Nozzle A Laminar Flow Element Honeycomb and Screen Motor Controller b) Section A-A Sidewalls of Test Rig Insulation Figure 2. Schematic of the test facility used for the heat transfer measurements along the scaled-up louvers: (a) overall view, and (b) section A-A. 25 1 2 3 c b a 4 5 6 7 8 d e g h f i Figure 3a-i. Louver arrangements for geometries with θ = 20° for a a) Fp/Lp = 0.54, b) Fp/Lp = 0.91, and c) Fp/Lp = 1.52; with θ = 27° for a d) Fp/Lp = 0.76, e) Fp/Lp = 0.91, and f) Fp/Lp = 1.52; and with θ = 39° for a g) Fp/Lp = 0.91, h) Fp/Lp = 1.22, and i) Fp/Lp = 1.52. 26 Copper Jump Wire Copper Sheet Stainless Steel Foil Paper Balsa Wood 20 AWG Wire Balsa Wood Thermocouple Bead Paper Omega 2000 Paste Figure 4. Schematic of the heated louver design (top) and an instrumented louver (bottom). 27 2.0 Louver Louver Louver Louver 1.5 η 2 3 4 5 Louver 6 Louver 7 Louver 8 1.0 0.5 0.0 -1 Back Side 0 x*/L Front Side 1 p Figure 5a. Non-dimensional adiabatic wall temperatures at each louver position in the θ = 27º, Fp/Lp = 0.76 model at ReLp = 230. (_____ T-Tin ) Tout Figure 5b. Normalized predicted temperature contours for the θ = 27º, Fp/Lp = 0.76 geometry at ReLp = 230. 28 2.0 Louver Louver Louver Louver 2 3 4 5 Louver 6 Louver 7 Louver 8 1.5 1.0 η 0.5 0.0 -1 Back Side 0 x*/L Figure 6a. Front Side 1 p Non-dimensional adiabatic wall temperatures at each louver position in the θ = 27º, Fp/Lp = 0.76 model at ReLp = 1016. (_____ T-Tin ) Tout Figure 6b. Normalized predicted isothermal temperature contours for the θ = 27º, Fp/Lp = 0.76 geometry at ReLp = 1016. 29 0.08 Louver 2 0.07 Re Lp Louver 3 = 1016 Louver 4 Louver 5 0.06 Louver 6 Louver 7 0.05 Louver 8 j b 0.04 0.03 0.02 0.01 0.00 -1 0 Back Side 1 Front Side x*/L p 0.08 Louver 2 Re 0.07 Lp = 1016 Louver 3 Louver 4 Louver 5 0.06 Louver 6 Louver 7 0.05 Louver 8 j aw 0.04 0.03 0.02 0.01 0.00 -1 Back Side 0 x*/L Front Side 1 p Figure 7a-b. Colburn factor measurements based on the adiabatic wall reference temperature at each louver position for ReLp = 1016 in the θ = 27°, Fp/Lp = 0.76 model. 30 Louver Louver Louver Louver 2 3 4 5 Louver 6 Louver 7 Louver 8 3.0 θ = 20, F p /L p = 0.54 Re Lp = 230 2.5 2.0 η 1.5 1.0 0.5 0.0 -1.0 -0.5 Louver Louver Louver Louver 0.0 x */Lp 2 3 4 5 0.5 1.0 Louver 6 Louver 7 Louver 8 3.0 θ = 20, F p /L p = 0.54 Re Lp = 1016 2.5 2.0 η 1.5 1.0 0.5 0.0 -1.0 -0.5 0.0 x */Lp 0.5 1.0 Figure 8a-b. Non-dimensional adiabatic wall temperatures at each louver position in the θ = 20º, Fp/Lp = 0.54 model at a) ReLp = 230 and b) ReLp = 1016. 31 Louver Louver Louver Louver 3.0 2 3 4 5 Louver 6 Louver 7 Louver 8 θ = 20, F p /L p = 0.91 Re Lp = 230 2.5 2.0 η 1.5 1.0 0.5 0.0 -1.0 -0.5 Louver Louver Louver Louver 0.0 x */L p 2 3 4 5 0.5 1.0 Louver 6 Louver 7 Louver 8 3.0 θ = 20, F p /L p = 0.91 2.5 Re Lp = 1016 2.0 η 1.5 1.0 0.5 0.0 -1.0 -0.5 0.0 x */L p 0.5 1.0 Figure 9a-b. Non-dimensional adiabatic wall temperatures at each louver position in the θ = 20º, Fp/Lp = 0.91 model at a) ReLp = 230 and b) ReLp = 1016. 32 Louver 2 Louver 3 3.0 Louver 6 Louver 7 Louver 8 Louver 4 Louver 5 θ = 20, F p /L p = 1.52 Re = 230 2.5 2.0 η 1.5 1.0 0.5 0.0 -1.0 3.0 -0.5 Louver Louver Louver Louver 0.0 x */L p 2 3 4 5 0.5 1.0 Louver 6 Louver 7 Louver 8 θ = 20, F p /L p = 1.52 Re = 1016 2.5 2.0 η 1.5 1.0 0.5 0.0 -1.0 -0.5 0.0 x */L 0.5 1.0 p Figure 10a-b. Non-dimensional adiabatic wall temperatures at each louver position in the θ = 20º, Fp/Lp = 1.52 model at a) ReLp = 230 and b) ReLp = 1016. 33 Re = 230 Re = 370 Re = 1016 3 θ = 20 F /L = 1.52 2 p p 1 0 η θ = 20 F /L = 0.91 2 p p 1 0 θ = 20 F /L = 0.54 2 p p 1 0 2 3 4 5 6 7 8 Louver Position Figure 11. Louver-averaged non-dimensional adiabatic wall temperatures at each louver position in the θ = 20º models at all Reynolds numbers that were tested. 34 Re = 230 Re = 370 Re = 1016 3 θ = 27 F /L = 1.52 2 p p 1 0 η θ = 27 F /L = 0.91 2 p p 1 0 θ = 27 F /L = 0.76 2 p p 1 0 2 3 4 5 6 7 8 Louver Position Figure 12. Louver-averaged non-dimensional adiabatic wall temperatures at each louver position in the θ = 27º models at all Reynolds numbers that were tested. 35 Re = 230 Re = 370 Re = 1016 3 θ = 39 F /L = 1.52 2 p p 1 0 η θ = 39 F /L = 1.22 2 p p 1 0 θ = 39 F /L = 0.91 2 p p 1 0 2 3 4 5 6 7 8 Louver Position Figure 13. Louver-averaged non-dimensional adiabatic wall temperatures at each louver position in the θ = 39º models at all Reynolds numbers that were tested. 36 F /L = 0.54, θ = 20 F /L = 0.76, θ = 27 F /L = 0.91, θ = 39 F /L = 0.91, θ = 20 F /L = 0.91, θ = 27 F /L = 1.22, θ = 39 F /L = 1.52, θ = 20 F /L = 1.52, θ = 27 F /L = 1.52, θ = 39 p p p p p p p p p p p p p p p p p p 1.4 1.3 1.2 1.1 η 1 0.9 0.8 0.7 0.6 200 400 600 Re 800 1000 1200 Lp Figure 14. Model-averaged non-dimensional adiabatic wall temperatures for all models. 37 0.08 F /L = 0.54 θ = 20 0.07 Re Lp p = 1016 p F /L = 0.91 p 0.06 p F /L = 1.52 p p 0.05 j aw 0.04 0.03 0.02 0.01 0.00 -1 0 x*/L Back Side 1 Front Side p 0.08 F /L = 0.76 θ = 27 0.07 Re Lp p = 1016 p F /L = 0.91 p 0.06 p F /L = 1.52 p p 0.05 j aw 0.04 0.03 0.02 0.01 0.00 -1 0 Back Side x*/L Front Side 1 p Figure 15a-b. Colburn factors based on the adiabatic wall reference temperature at louver position 6 for ReLp = 1016 in the a) θ = 20°, Fp/Lp = 0.54, 0.91, and 1.52 and b) θ = 27°, Fp/Lp = 0.76, 0.91, and 1.52 models. 38 0.08 F /L = 0.91 θ = 39 0.07 Re Lp p p F /L = 1.22 = 1016 p 0.06 p F /L = 1.52 p p 0.05 j aw 0.04 0.03 0.02 0.01 0.00 -1 Back Side 0 x*/L Front Side 1 Figure 15c. Colburn factors based on the adiabatic wall reference temperature at louver position 6 for ReLp = 1016 for the θ = 39°, Fp/Lp = 0.91, 1.22, and 1.52 models. 39 F /L = 0.54, θ = 20 F /L = 0.76, θ = 27 F /L = 0.91, θ = 39 F /L = 0.91, θ = 20 F /L = 0.91, θ = 27 F /L = 1.22, θ = 39 F /L = 1.52, θ = 20 F /L = 1.52, θ = 27 F /L = 1.52, θ = 39 p p p p p p p p p p p p p p p 0.1 0.08 0.06 j b 0.04 0.02 0 200 400 600 Re 800 1000 1200 Lp Figure 16. Reynolds number dependency of jb in all of the models tested. 40 p p p