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## Journal of Heat Transfer Heat Transfer Coefficients in Concentric Annuli

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@MISC{Dirker_journalof,

author = {Jaco Dirker and Josua P Meyer},

title = {Journal of Heat Transfer Heat Transfer Coefficients in Concentric Annuli},

year = {}

}

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### Abstract

The geometric shape of a passage's cross-section has an effect on its convective heat transfer capabilities. For concentric annuli, the diameter ratio of the annular space plays an important role. The purpose of this investigation was to find a correlation that will accurately predict heat transfer coefficients at the inner wall of smooth concentric annuli for turbulent flow of water. Experiments were conducted with a wide range of annular diameter ratios and the Wilson plot method was used to develop a convective heat transfer correlation. The deduced correlation predicted Introduction Since the early nineteen hundreds many researchers have investigated heat transfer in annuli, particularly in order to find correlations that can describe the Nusselt number and convective heat transfer for a wide range of flow conditions and annular diameter ratios. Most of the proposed equations for calculating annular Nusselt numbers are functions of the annular diameter ratio, the Reynolds number and the Prandtl number and correspond with the Dittus-Boelter type form. When comparing the various correlations, applicable to the flow of water, over a wide range of annular diameter cases and Reynolds numbers, it is found that large differences, in the region of 25 percent in terms of the average, exist between predicted values. No literature was found that indicates the existence of an accurate heat transfer correlation for concentric annuli. It was thus the purpose of this investigation to deduce a correlation with which accurate predictions could be made of average Nusselt numbers at the inner annular wall under turbulent flow conditions of water. Experimental Facility and Data Eight different concentric tube-in-tube heat exchangers, each with a different annular diameter ratio, were used during the experimental investigation. Each heat exchanger had an effective heat transfer length of about 6 m and was operated in a counterflow arrangement with hot water in the inner tube and cold water in the annulus. The heat exchangers were constructed from harddrawn refrigeration copper tubing and were operated in a horizontal configuration. The inner tubes were kept in concentric positions by employing sets of radial supporting pins along the length of each heat exchanger at different intervals. The size and position of the supporting pins were carefully calculated to minimize possible sagging of the inner tube. The supporting structures occupied between 3.9 percent and at most 6.5 percent of the cross-sectional areas of the various test sections ͓11͔. Volumetric flow rates were measured by using semi-rotary circular-piston-type displacement flowmeters with a measuring accuracy of greater than 98 percent. Hot water supplied by an on-site hot-water storage tank (1000l), fitted with an electric resistance water heater, was pumped through the inner tube by means of a positive displacement pump and then returned to the storage tank. The hot-water flow rates were controlled with a hand-operated bypass system. Similarly, cold water was supplied from a cold-water storage tank (1000l) connected to a chiller and pumped through the annulus by means of two series connected centrifugal pumps to ensure high flow rates before being returned to the storage tank. Temperature measurements were facilitated by means of K-type thermocouples fixed on the outside surfaces of entry and exit regions of the heat exchangers. Temperature errors were usually less than 0.1 K. Measuring points were sufficiently insulated from the ambient. A high level of accuracy in the experimental data was maintained. More than 90 percent of all data points exhibited an energy balance error of less than 1 percent between the inner tube and annular heat transfer rates. A Reynolds number range, based on the annular hydraulic diameter, of 2 600 to 35,000 was covered. Processing of Data It was assumed from previous work ͑Table 1͒ that the internal and annular Nusselt numbers can be written in a Sieder-Tate ͓12͔ format, respectively: Õ Vol. 124, DECEMBER 2002 Copyright © 2002 by ASME Transactions of the ASME P, C i , and C o are added to account for geometry influences. The modified Wilson plot method developed by Briggs and Young ͓14͔ was used to determine these values for the different annular diameter ratios while for the inner tube the exponent of the Reynolds number was kept at 0.8 as suggested in literature ͓13,14͔. More than 95 percent of all data points were predicted within a 3 percent accuracy by the Wilson plot obtained correlations for the different heat exchangers. All Wilson plot correlations exhibited a median error of less than or in close proximity to 1 percent. Standard deviances for error values were less than 2 percent. Derivation of Correlation P and C o , from Eq. 2, showed a dependence on the annular diameter ratio. These heat exchangers were rebuilt and the experimental tests repeated. The values of P and C o were reaffirmed. From the experimental results, the behavior of P and C o can be described relatively precisely for annular diameter ratios below 3.2. For ratios greater than this, it is unfortunately not the case and more experimental data are needed. Data points between annulus ratios of 3.2 and 5 are difficult to obtain as tube sizes which would give these ratios are not readily available. Using results for annular ratios of below 3.2, it was possible to describe the trend mathematically by evaluating different curve fits. Equations ͑3͒ and ͑4͒ exhibited the best accuracies and are indicated in Figs. 1 and 2 as dotted lines. Pϭ1.013e Ϫ0.067a By substituting Eqs. ͑3͒ and ͑4͒ into Eq. ͑2͒ a correlation for the prediction of the Nusselt number is produced. The validity of the resulting correlation for the prediction of Nusselt numbers was tested with experimental data from all heat exchangers having an annular diameter ratio of less than 3.2. All predictions were within 3 percent of experimentally obtained values ͑Fig. 3͒. The correlation was also compared to correlations in literature ͑Table 1͒ for an arbitrary thermal condition over a wide range of annular diameter ratios and Reynolds numbers. For a case where the Reynolds number is 8 000 and the Prandtl number is 3.36, the result is shown in For small annular diameter ratios, up to about 2.5, the predictions correspond well with the correlation by Dittus and Boelter ͓8͔, and an equation by McAdams ͓2͔. In the region of an annular ratio of 3.5, a close agreement exists with the correlation of Stein and Begell ͓9͔. Conclusion As was expected, it was found that the convective heat transfer correlation for an annulus is dependent on the annular diameter ratios. A correlation was deduced from experimental results that predicts Nusselt numbers accurately for water within 3 percent from the measured values for diameter ratios between 1.7 and 3.2 and a Reynolds number range of 4 000 to 30,000. Nomenclature A Thermocapillary Mechanism for Lateral Motion of Bubbles A hypothesis about a transport mechanism that promotes coalescence of bubbles during subcooled nucleate boiling is presented. The hypothesis is that adjacent bubbles entrain each other in thermocapillary flow surrounding them during nucleate boiling of subcooled liquids. The entrainment manifests itself as motion of the bubbles toward each other, which promotes their coalescence. The discussion and calculations provided in this contribution are offered in support of the hypothesis. Consider the circumstances appearing in The bubble on the left of The bubble on the left of This motion has been observed in electrolytic gas evolution and in model experiments. Sides and Tobias ͓3͔ observed lateral motion of oxygen bubbles on transparent tin oxide electrodes in electrolytic gas evolution; Sides and co-workers ͓4͔ formulated the thermocapillary pumping hypothesis and investigated the thermocapillary pumping mechanism theoretically and experimentally in a model system ͓5͔ where only viscous transport of momentum was allowed. The subject of the hypothesis of the present work is The heated surface of The question is the extent to which this phenomenon might be apparent during boiling. Evidence both of thermocapillary flow and of boiling bubble motion driven by forces other than buoyancy exists in the literature. McGrew et al. ͓6͔ heated a suspension of small particles in n-butanol and other liquids. The particles traced the flow in the vicinity of bubbles that appeared during nucleate boiling of the test liquid. Their comments are worth quoting ͓6͔: ''When boiling was established, we consistently observed a streamline type of flow and a rapid circulation of liquid around This was one of the earliest observations of thermocapillary flow during boiling. Elsewhere in this article, the authors record the following observation: ''Some of the bubbles could be observed moving along the heating surface, and a tendency for the bubbles to move toward each other was apparent.'' McGrew et al. ͓6͔ ascribed the aggregation to thermophoretic motion along the horizontal temperature gradients that exist because the low conductivity bubbles disturb the heat flux paths that would otherwise be normal to the heated surface. Ervin et al. ͓7͔ observed that bubbles moved on the surface of their heater and coalesced. They noted that they met at the warmest part of the heater, which echoes the lateral thermophoresis argument of McGrew et al. ͓6͔. Qiu et al. ͓8͔ observed coalescence of bubbles grown side by side from prepared nucleation sites on silicon, but the aggregation of bubbles seemed to depend more on proximity and size than on directed motion toward each other. The identification of thermocapillary flow in boiling of subcooled liquids led primarily to investigation of both its enhancement of heat transfer and its mechanism. Previously, the substantial improvement of heat transfer by boiling was ascribed to the agitation associated with bubble growth, coalescence, and departure, but investigators have turned their attention to the consequences of thermocapillary flow for heat transfer by boiling in subcooled liquids. Marek and Straub ͓9͔ describe this effect. Their vision of nucleate boiling into subcooled liquid in the presence of noncondensibles appears in Recent reporting ͓10-12͔ on visual observation of growing bubbles during boiling has provided additional evidence both for the concerted motion of small bubbles to coalesce with large ones and for the aggregation of equal-sized bubbles, evidence that is strikingly similar to the observations of Sides and Tobias ͓3͔. Kim et al. ͓10,11͔ flew a micromosaic heater in a reduced gravity environment and found that the large bubble that formed was ''fed by smaller satellite bubbles that surround it.'' Thus the phenomenon of thermocapillary driven aggregation of bubbles during boiling is plausible but it remains to examine whether it might be observable for the relatively large bubbles produced in that process. The circumstances are quite different since the small bubbles of electrolysis of aqueous solutions allowed assumption of flow dominated by low Reynolds number flow, while that assumption is clearly not appropriate for boiling. Second, it is not clear whether the possible thermocapillary flow velocities would be sufficient to generate an observable flow. Estimation of Flow Rate Due to the Thermocapillary Mechanism We perform a scaling analysis of the equations of motion including both diffusion and convection of momentum for the purpose of estimation of the strength of the flow and hence the potential lateral migration velocity of the bubble. Consider a vertical mobile interface in cartesian coordinates, as shown in in which d␥/dT is the variation of surface tension with temperature, ⌬T is the temperature difference over the length scale a, and is the liquid viscosity. The z component of the steady equation of motion and the continuity equation appropriate for this case are Õ Vol. 124, DECEMBER 2002 Transactions of the ASME where v denotes velocity, x and z are cardinal directions, and v is the kinematic viscosity. Unlike the analysis for electrolysis ͓4,5͔, the flow equation above includes convective terms. The goal is to scale these equations appropriately and in the process deduce a characteristic velocity in the x direction. Defining x o , v xo as characteristic length and velocity normal to the interface, we substitute them into Eq. ͓2͔ along with a and v zo . where is velocity and ϭx/x o and ϭz/a are dimensionless distance in the x and z directions, respectively. Convection of momentum in the direction parallel to the interface is obviously an important term, so we scale it to O(1) by dividing the equation through by the coefficient of the second term on the lhs, which gives The thermocapillary flow at the interface is extended into the bulk fluid by diffusion of momentum, so the first term on the rhs is important. Scaling it to O(1), one obtains a formula for a characteristic distance in the lateral direction, x o ϵͱa/v zo . The coefficient of the second term on the rhs is O(10 Ϫ4 ) so diffusion of momentum in the z direction can be neglected. The reference velocities and distances are now inserted into the continuity equation. The terms balance if v xo ϵͱv zo /a. Using ͑1͒, one obtains from this process an estimate of the lateral velocity to be expected for a given temperature difference over the distance a. Taking the distance a to represent the radius of the bubble and ⌬T to represent the temperature difference between the base of the bubble and its apex, we obtain an estimate of the lateral velocity of the liquid, where is its density. Equation The observability of motion due to thermocapillary pumping is given by the ratio of the time scale for bubble release divided by the time scale for motion of the bubble. Its meaning is that an observer can hope to record the lateral motion of bubbles due to the proposed thermocapillary mechanism before the bubble departs from the surface. As an example of the use of this equation, consider the results of Ibrahim and Judd ͓14͔ who boiled water on a copper surface and recorded a bubble growth time of 6 ms for bubbles that grew to 1.8 mm in radius in liquid subcooled by 10 K. Using Ϫ1.8 •10 Ϫ4 N/(m•K) as the derivative of surface tension for water near the boiling point ͓15͔, one calculates the observability to be 0.12, which means that thermocapillary motion was not obvious in their experiment performed in 1 g. The observability calculated above is not so far from unity that bubble motion due to mutual thermocapillary entrainment must always be negligible; instances where bubbles remain on surfaces, such as downward facing surfaces in earth gravity or any heated surface in microgravity, might reveal the effect. Closure The principal contributions of this note are the statement of a hypothesis concerning a possible thermocapillary pumping mechanism for lateral bubble motion on a heated surface during nucleate boiling, and the presentation of equations for estimating the strength of the effect. The phenomenon will be most evident in circumstances where bubbles are retained on surfaces, such as nucleate boiling from subcooled liquids in microgravity or on the underside of horizontal heaters. Acknowledgment This work was supported under the microgravity fluid physics program of NASA, Grant NAG3-2159 Õ Vol. 124, DECEMBER 2002 Transactions of the ASME Introduction In a recent paper, Roy et al. ͓1͔, we reported local measurements in the liquid and vapor phases of turbulent subcooled boiling flow. The vapor residence time fraction and vapor bubble time-mean axial velocity ͑taken to be the mean propagation velocity in the axial direction of the front interface of the bubble͒ were among the quantities measured at six different experimental conditions. A two-sensor fiberoptic probe ͑Photonetics͒ with 50 m sensor tip size, The New Two-Sensor Fiber Optic Probe Results The six experiments ͑tp1 through tp6͒ reported in ͓1͔ were repeated with the new probe. For brevity, we show the results of only three experiments ͑tp1, tp2, and tp5͒. Data from the other Transactions of the ASME The uncertainty in bubble axial velocity at the outermost location was larger ͑Ϯ10% of value͒ mainly because very few bubbles were present. Furthermore, these bubbles were smaller in size and more susceptible to turbulent velocity fluctuations of the liquid phase than the larger bubbles. Numerical simulation results for ␣ G and U G were presented in ͓1͔ and are not shown here. Concluding Remarks The altered orientation of the sensors in the new two-sensor fiberoptic probe yielded measurements of the bubble time-mean axial velocity that were more repeatable and with less scatter in the outer low vapor fraction region of the boiling layer compared to our earlier ͓1͔ measurements. In our opinion, this is because of the more sharply defined bubble piercing action of the sensor tips. Acknowledgment This work was partially funded by Electricité de France. Introduction Single turbulent plane and offset wall jets are of great engineering importance, and consequently have been studied extensively ͓1-3͔. Applications include burners and boilers, film-cooling of lining walls within gas turbine combustors, fuel-injection systems, and heating and air-conditioning systems. However, far fewer investigations into the behavior of multiple parallel jets appear in the literature. In addition to the applications mentioned above, the study of multiple jets may be particularly important in the design of pollutant exhaust stacks. Specifically, relative to a single exhaust stack, the close grouping of stacks to form parallel jets may be employed as a means to increase the exhaust plume trajectory and consequently decrease the impact of exhaust pollutants ͓4͔. Flow patterns for two parallel plane jets have previously been reported in the literature ͓c.f. ͓5-11͔͔. The earliest studies were those of Tanaka ͓5,6͔ in which the basic flow patterns and entrainment mechanisms of parallel jets were described. In particular, Tanaka identified three relevant regions of the flowfield in the axial direction. The first may be termed the converging region, which begins at the nozzle exit and extends to the point where the inside shear layers of the jets merge ͑denoted the merge point͒. The merging of the jets is due to the asymmetric nature of the entrainment rates which results in a region of sub-atmospheric pressure between the jets. The jets are consequently deflected toward each other; at their merge point the velocity on the symmetry plane is equal to zero. The intermediate, or merge region is that existing between the merge point and the combine point, where the combine point is defined as that point along the symmetry plane at which the velocity is a maximum. Finally, the combined region is that downstream of the combine point where the two jets begin to resemble a self-similar single jet. The general characteristics of the flow field are illustrated in Anderson and Spall ͓11͔ recently presented experimental and numerical results for isothermal, plane parallel jets at spacings S/dϭ9, 13, and 18.25 ͑where S is the spacing between jet centerlines and d is the jet width͒. Values of the merge and combine points computed using both the standard kϪ and a differential Reynolds stress model were compared with experimentally measured values. Good agreement between numerical and experimental results was obtained. Furthermore, no significant differences between the results of the two turbulence models was observed. The author has found no published literature concerning the behavior of buoyant, plane parallel jets. However, experiments for free convection over two parallel heat sources were first carried out by Rouse et al. ͓12͔ in the early 1950s. They observed that the two plumes quickly merged so that the maximum velocities were located along the vertical symmetry plane. Subsequently, Pera and Gebhart ͓13͔ looked at both plane, parallel plumes and axisymmetric plumes, and observed that the plane, parallel plumes interacted more strongly than did the axisymmetric plumes at the same spacing. Gebhart et al. ͓14͔ further investigated the interaction of unequal plane plumes. In the present work, the findings of Anderson and Spall ͓11͔, and the work of Rouse et al. ͓12͔ are extended by investigating numerically the evolution of buoyant, plane parallel jets. Mathematical Model and Numerical Method The governing equations consist of the incompressible Reynolds averaged momentum, continuity and energy equations, and equations for turbulence closure. The equations were solved using the pressure-based, structured-grid, finite-volume code Fluent ͑Version 4.4, Fluent, Inc., Lebanon, NH͒. The governing equations are well known, and hence for purposes of brevity are not listed. However, a brief description of the modeling assumptions regarding density variations and turbulence closure follow. Density variations were taken into account using the Boussinesq approximation for which the density was treated as a constant value in all solved equations except for the buoyancy term in the momentum equations, which was treated as where 0 and T 0 are the far field reference density and temperature respectively,  is the thermal expansion coefficient, and ḡ is the gravity vector. Results presented in Anderson and Spall ͓11͔ revealed little difference between the merge points obtained using kϪ or differential Reynolds stress turbulence models. Consequently, for this study a kϪ model was employed for turbulence closure. The transport equations solved for the turbulence kinetic energy k and dissipation rate in the present work are given as The rate of production of turbulence kinetic energy, G k , is defined as and the generation of turbulence due to buoyancy, G b , as Õ Vol. 124, DECEMBER 2002 Copyright © 2002 by ASME Transactions of the ASME The empirical constants in the model were taken as C 1 ϭ1.44, C 2 ϭ1.92, C 3 ϭ1.0, C ϭ0.09, k ϭ1.0, ϭ1.3, and h ϭ0.85. Note that h is defined as t C p /k t ͑where k t is the effective thermal conductivity and C p is the specific heat͒. In terms of the solution procedure, interpolation to cell faces for the convection terms was performed using a bounded QUICK scheme ͓15͔; second-order central differencing was used for viscous terms. Pressure-velocity coupling was based on the SIM-PLEC procedure ͓16͔. Solutions obtained using the segregated solver were considered converged when residuals for each of the equations ͑based on an L2 norm͒ were reduced by a minimum of four to five orders of magnitude. Additional iterations were then performed to confirm iterative convergence. Geometry and Boundary Conditions The computational domain is identical in size to that employed in Anderson and Spall ͓11͔, defined by a rectangular region discretized using a Cartesian grid covering one half of the flow field. A symmetry boundary condition was defined along the yϭ0 plane, whereas constant pressure boundary conditions were specified on the yϭy max and xϭx max planes ͑see The relevant Reynolds number for the problem was defined as Reϭ(Vd)/ ͑where is the density, V the inlet velocity, and is the dynamic viscosity͒. Air was employed as the working fluid, and the variables defining the Reynolds number were chosen such that Reϭ75,000. ͑We note that experimental data from previous isothermal studies indicates that the location of the merge and combine points are nearly independent of the Reynolds number.͒ The turbulence intensities at the inlet were set to 5 percent, from which the turbulence kinetic energy distribution was obtained. The dissipation rate inlet boundary condition was derived from the relationship C k 1.5 /L where the turbulence length scale L was taken as 0.07d. Three different grid resolutions were employed for each S/d spacing. For the cases defined by S/dϭ9 and 13, grids consisting of 71ϫ152, 140ϫ300 and 278ϫ597 cells were used, whereas for the case S/dϭ18.25 the three grids contained 91ϫ152, 180 ϫ300, and 358ϫ597 cells. Across the jet inlet, the coarse, medium, and fine grids contained 10, 20, and 40 cells, respectively. In all cases, cells were clustered toward the (yϭ0) symmetry plane and the (xϭ0) wall. The importance of buoyancy in mixed convection flows is indicated by the ratio of the Grashof number to the square of the Reynolds number as (⌬gd)/(V 2 ). In general, when this ratio approaches unity, one may expect strong contributions from buoyancy. This ratio is also referred to as the Archimedes number ͑Ar͒, and when the density variation is accounted for by the Boussinesq approximation may be expressed as where ⌬TϭT inlet ϪT 0 . ͑Note that fluid at temperature T 0 may be entrained across the xϭx max and yϭy max constant pressure boundaries.͒ Parameters were set to provide values of Arϭ0, 1/16, 1/8, 1/4, and 1/2. The maximum value of ⌬T was limited to ϳ15°C so that the assumption of small temperature variations inherent in the Boussinesq approximation would not be violated. Results Shown in Contours of constant dimensionless temperature ((T ϪT 0 )/(T inlet ϪT 0 )) are shown in Õ Vol. 124, DECEMBER 2002 Transactions of the ASME in distance to the merge point ͑and consequently, the size of the recirculation zone͒ as the Archimedes number is increased only modestly from zero. In addition, the results indicate that for Ar у1/4, the location of the merge point becomes nearly independent of the jet spacing, S/d. Results shown in Conclusions The results reveal that the trajectory of plane parallel jets is strongly influenced by what may be considered as moderate levels of the ratio of buoyancy to inertial forces. This conclusion has implications in areas such as the ganging of smoke stacks, in which the close grouping of stacks may be utilized to combine non-manifolded exhausts into a single jet. The results also indicate that cooled jets are not prone to merging unless the Archimedes number is quite small. Although this is not likely to present itself as an application in the ganging of smoke stacks, it may be of significance in other engineering applications. The author plans to consider in future work the fully three-dimensional interaction between arrays of both isothermal and buoyant round jets. References