Fluid Flow and Heat Transfer of Combined Forced-Natural Convection around Vertical Plate Placed in Vertical Downward Flow of Water

Transcription

1 Advanced Experimental Mechanics, Vol.2 (2017), Copyright C 2017 JSEM Fluid Flow and Heat Transfer of Combined Forced-Natural Convection around Vertical Plate Placed in Vertical Downward Flow of Water Fumiyoshi KIMURA 1, Jyunji KIDA 2 and Kenzo KITAMURA 3 1 Department of Mechanical Engineering, University of Hyogo, Himeji , Japan 2 Department of Mechanical and Systems Engineering, University of Hyogo, Himeji , Japan 3 Department of Mechanical Engineering, Toyohashi University of Technology, Toyohashi , Japan (Received 10 January 2017; received in revised form 19 May 2017; accepted 19 May 2017) Abstract: Experimental investigations have been carried out on the fluid flow and heat transfer of opposing flows induced over a vertical heated plate placed in a uniform downward flow of water. The vertical plates of lengths L = 50 and 100 mm, heated with uniform heat flux were utilized in the experiments. The Reynolds and modified Rayleigh numbers based on the plate length were ranged as; Re L = and Ra L = , respectively. The flow fields around the heated plates were first visualized with dye. The result showed that the separation of the laminar boundary layer of forced convection appears first at the bottom edge of the plate and the separation point shifts from the bottom to the leading edge of the plate with the surface heat flux. We also found that the separations of flow at the bottom and upper edge can be predicted with the non-dimensional parameter as; ( /Re L 2.5 ) = 0.4 and 3, respectively, where the parameter ( /Re L 2.5 ) stands for the ratio of the buoyancy to the inertia force. The local heat transfer coefficients from the plates were subsequently measured with thermocouples. The result showed that the coefficients deviate from those of the pure-forced convection with the onset of flow separation at the bottom edge. We have also found that the overall Nusselt numbers from the plate show minima at around ( /Re L 2.5 ) = 1.0. Moreover, by comparing the overall Nusselt numbers to those of the forced and natural convections, the combined convection region was determined as; 0.35< ( /Re L 2.5 ) <4.0. Keywords: Combined convection, Forced convection, Natural convection, Heat transfer, Flow visualization, Flow separation, Vertical plate 1. Introduction Combined flows of forced and natural convections induced around a vertical heated plate appear in a wide variety of heat transfer devices and environmental situations. This leads to intensive investigations on their flow and heat transfer characteristics by both experiments and analyses. However, the most of them have been dealt with the aiding flows, while very few studies have been conducted with the opposing flows, where the buoyancy force acts opposite direction to the forced convective flow. This will make the flow unstable and complex. Thus, it is supposed difficult to treat the opposing flows analytically and experimentally as well. To the best of the author s knowledge, we can cite the analyses by Szewczyk [1], Markin [2] and Oosthuizen and Hart [3] and the experiment by Oosthuizen and Bassey [4], Kobus and Wedekind [5]. In the previous analyses [1, 2, 3], they have dealt with the case where the forced flow is dominant and the buoyancy exerts minor effects on the flow and heat transfer. While, in the previous experiments [4, 5], they have measured the overall heat transfer coefficients of the plates heated isothermally. However, it is difficult to derive comprehensive information on the flow and heat transfer from the results of the overall heat transfer coefficients. Taking account of the above state-of-the-art, the authors have carried out the flow visualizations and the heat transfer measurements on the opposing combined convection of air in the previous study [6]. The flow fields over the plates were visualized with smoke. The local heat transfer coefficients were also measured with thermocouples, where the test plates were heated with uniform heat fluxes. The results showed that the separation of the laminar boundary layer of the forced convection occurs over the vertical plate with increasing buoyancy. We also found that the heat transfer coefficients from the plate begin to deviate from those of the forced convection with the flow separation. In light of the prior results with air, we have carried out the experimental investigations on the opposing flows of water over a vertical heated plate placed in a uniform, downward flow. This is because water is widely used as a working fluid in many heat transfer devices. 2. Experimental Apparatus and Measurement A schematic illustration of the present experiment apparatus is given in Figure 1. The apparatus consists of a low-speed water-tunnel and a test plate. The water-tunnel is composed of a settling chamber, a contraction nozzle, a test duct, a reservoir and a pump. Water at room-temperature was utilized as a test fluid. The water stored in the reservoir was first fed into the settling chamber by a pump and, then, rectified to the uniform vertical flow by honeycomb meshes and a contraction nozzle of 4:1 contraction ratio at the inlet of the test duct. The test duct has mm 2 cross-sectional area and was 600 mm-long. The flow passed through the duct returns to the reservoir. The flow rate through duct was measured with weight per unit time period. A preliminary experiment has been conducted to ascertain the uniformity of the forced flow velocities at the inlet of the test duct. For the sake of this, a fine nickel-wire was placed in the cross-section 15 mm downstream from the duct inlet and time-lines of hydrogen bubbles were generated by applying a high-voltage, direct-current through the wire with constant time period. The time-lines of the bubbles showed uniform profiles throughout the cross- 41

2 F. KIMURA, J. KIDA and K. KITAMURA section of the duct except in the near wall regions. A test plate fabricated with a 2 mm-thick, 295 mm-wide acrylic-resin plate and 30 μm-thick stainless steel foil heaters. The heaters were glued on the both surfaces of the acrylic plate and were connected in series. A constant heat flux condition was accomplished by supplying an alternating-current to the heaters. Since the test plate has a finite thickness of 2 mm, the leading edge of the plate may cause a flow separation. Thus, stainless steel pipes of 2 mmouter-diameter were flush mounted to the both edges of the plate to prevent the flow separation as shown in Fig. 1(b). The pipes also support the test plate vertically in the test section. The plate was placed at 50 mm-downstream from the inlet of test section. The vertical plates with different lengths of L = 50, 100 mm were utilized in the experiment. For the visualizations of the flow fields around the plate, two kinds of fluorescent dyes dissolved with water were utilized as the tracer, the one was uranine (green), the other Settling chamber ( mm 2 ) Honeycomb meshes Contraction nozzle Overflow was rhodamine B (orange). For the sake of heat transfer measurement, Chromel- Alumel thermocouples of 100 μm-diameter were spotwelded on the back of the heaters along the vertical centerline of the plate. These thermocouples measured the local surface temperatures of the plate, T wx. The thermocouples of the same diameter and material were placed at the inlet of the test section to measure the ambient temperature of water, T. Since the both surfaces of the plate were heated with identical heat flux, a conduction heat loss through plate is considered negligible. Hence, the surface heat flux q w was calculated as q w = Q/A, where Q and A stand for the electrical power input to the heaters and total surface area of the heaters, respectively. Then, by using the heat flux and the temperature difference between the surface and ambient water temperatures, (T wx T ), the local heat transfer coefficients, h x, were defined and calculated as: qw hx (1) T T wx The present experiments have covered the ranges of the Reynolds and modified Rayleigh numbers based on the plate length as; Re L (= u L/ν) = and Ra L (= gβq w L 4 / (λαν)) = , where the thermo-physical properties in those numbers were estimated at the film temperature, T f (= (T w +T )/2). 3. Results and Discussion Stainless steel foil heaters (30 μm-thick) L Test duct ( mm 2 ) Heated plate Control valve Reservoirs Acrylic resin plate (2 mm-thick) (a) low speed water tunnel Stainless steel pipes ( 2) (b) test plate 295 Fig.1 Experimental apparatus 600 Pump x L 3.1 Visualization of flow around plate In order to obtain comprehensive information on the flow fields over the test plate, we have first carried out the visualization of flow using the fluorescent dye. For the sake of this, small holes were drilled on the side of the stainless-steel pipes with spanwise pitch of 50 mm. Then, the pipes were glued onto the upper and lower edges of the plate. The dyes of different colors green (uranine) and orange (rhodamine B) were issued slowly from the holes of upper and lower pipes, respectively. Figure 2 shows representative results for the plate of L = 50 mm, where main stream velocity of forced convection was fixed at u = 1.6 cm/s. The photos were taken from the side of the plate. A metal halide light sheet was used to illuminates the movement of the dyes in the plane parallel to the flow direction. In those photos, the main flow of forced convection directs from the top to the bottom. Figure 2(a) represents the flow field around the non-heated vertical plate. The green dye issued from the upper leading edge flows along the plate film-wisely. While, the orange dye issued from the bottom edge flows toward downstream. The result depicts that a laminar boundary layer of the forced convection develops over the plate. The boundary layer at the bottom edge becomes thick and the separation of the flow begins to appear at the bottom edge of the plate, when the heat flux q w = 280 W/m 2 is imposed as shown in Fig. 2(b). The separation point shifts from the bottom to the leading edge of the plate with an increase in the heat flux as are shown in Figs. 2(c) and (d). With further 42

3 Advanced Experimental Mechanics, Vol.2 (2017) (a) q w = 0 W/m 2 (b) q w = 280 W/m 2 (c) q w = 1000 W/m 2 (d) q w = 1500 W/m 2 (e) q w = 2350 W/m 2 (f ) q w = W/m 2 Fig.2 Visualized flow fields around plate (L = 50 mm, u = 1.6 cm/s) Present exp. ; water (Pr = 7) 10 9 L = 50 [mm] L = 100 [mm] 10 8 /Re L 2.5 = 3 /Re L 2.5 = /Re L 2.5 = Previous exp. ; air (Pr = 0.72) Gr L /Re 2.5 L = Re L Fig.3 Conditions of flow separation increase in the heat flux, the separation points reach to the leading edge of the plate, and the separation bubble grows markedly as are shown in Figs. 2(e) and (f). Although these photos show the dye movements in the cross-section along a vertical centerline of the plate, we have observed that the dyes issued from the holes other than the plate center show identical movements, indicating that the twodimensional flow field is attained over the plate. Meanwhile similar flow separations as above have been observed in the previous paper of the present authors [6], where the opposing flows of air were visualized with smoke. The authors have reported that the onset of the flow separation at the bottom edge of the plate and the separation at the top edge of the plate can be predicted with the non-dimensional parameter ( /Re L 2.5 ), where and Re L stand for the modified Ralyeigh number and the Reynolds number based on the plate length L. In light of these results, we have next measured the condition of the flow separation at the bottom and at the top of the vertical plate. The results are presented in Figure 3, where the previous data for air are plotted together with the dotted lines for comparison. The present data for water show that the plots for the separations at the bottom and at the top of the plate gather around the solid lines in the figure, and that the separations at the bottom and at the top of the plate occur when ( /Re L 2.5 ) = 0.4 and 3, respectively. The figure also indicates that the separations of flow from the bottom edge occur with almost identical values of ( /Re L 2.5 ) = between air and water. While the conditions for the flow separation at the top show difference between water and air. The discrepancy will be attributed to the difference in Prandtl number of the test fluid. 3.2 Heat transfer characteristics In light of the above results of the flow visualization, we have subsequently carried out the quantitative measurements of the local heat transfer coefficients from the vertical plates using thermocouples. The measurements have been performed in the wide ranges of Reynolds and modified 43

4 F. KIMURA, J. KIDA and K. KITAMURA Rayleigh numbers so as to realize the forced, combined, natural convective flows over the plate. Figure 4 shows the representative results for the vertical plate of L = 50 mm, where the velocity of the forced flow was fixed at u = 1.6 cm/s. The local heat transfer coefficients are plotted in terms of the stream-wise distance x from the top to the bottom edges of the plate. For comparison, the analytical coefficients for the laminar forced convection [7] and for the laminar natural convection [8] are presented with solid and dotted lines, which were calculated from the following 0 x [mm] L = 50 [mm], u = 1.6 [cm/s] q w [W/m 2 ] Natural convection Forced convection h [W/m 2 K] x Fig.4 Local heat transfer coefficients (L = 50 mm, u = 1.6 cm/s) 10 Forced convection Combined convection Natural convection 10 /n ( /Re L 2.5 ) = 0.35 ( /Re L 2.5 ) = 4.0 /f 1 1 Flow separation L = 50 [mm] /n /f L = 100 [mm] /n /f /Re L Fig. 5 Non-dimensional plots for overall Nusselt numbers from plates 44

5 Advanced Experimental Mechanics, Vol.2 (2017) equations: For laminar forced convection; 1 / 2 x Re x Nu (2) For laminar natural convection; 1 / 5 Nu Ra (3) where denotes the stream-wise distance from the bottom to top edges of the plate. Fig. 4 shows that the coefficients for the smallest heat flux q w = 200 W/m 2 coincide fairly well with those of the forced convection. While when the flow separation begins to occur at the bottom edge of the plate, as for the case of q w = 280 W/m 2, the coefficients deviate from those of the forced convection. With further increase in the heat flux, the coefficients show a minimum at certain location of the plate as shown for the case of q w = 1000 W/m 2. Comparing the result with the visualizations in Fig. 2(c), we will see that the separation point almost coincides with those of the minimum coefficients. Moreover, we will find the coefficients downstream the minimum asymptote to those of the natural convection. The region gradually enlarges with further increase in the heat flux and, finally the coefficients agree well with those for the natural convection as shown for the case of q w = W/m 2. Based on the above local heat transfer data, we have next calculated the overall Nusselt numbers from the plate. The results are represented in Figure 5, where the overall Nusselt numbers, (= h m L/λ) normalized with those of the forced and natural convections, f and n, are plotted with the parameter ( /Re L 2.5 ). As is obvious from Fig. 5, the whole plots for the plates of different length L = 50 mm and L = 100 mm gather within narrow bands. The ratios ( /f ), ( /n ) show minima at around ( /Re L 2.5 ) = 1.0. One will also find that the ratios ( /f ) show unity when ( /Re L 2.5 ) is less than Moreover, the ratios ( /n ) are unity when the parameter ( /Re L 2.5 ) is larger than 4.0. The results indicate that ( /Re L 2.5 ) = 0.35 and 4.0 give thresholds for the forced and natural convection, respectively. On the other hand, in the intermediate region of 0.35< ( /Re L 2.5 ) <4.0, the overall Nusselt numbers do not coincide with those of the forced and natural convections. Thus, the region can be referred as the combined convection region. In Fig.5, the conditions of flow separation at the bottom and top edges of the plate are marked with arrows, we will see from Fig. 5 that the separations of flow at the bottom and the top of the plate occur in the combined convection region, 0.35< ( /Re L 2.5 ) <4.0, determined from the heat transfer data. The results mentioned in the above will afford useful information on the heat transfer and fluid flow of opposing, combined convective flows over vertical plates. 4. Conclusion The fluid flow and heat transfer of opposing combined convective flows over a vertical heated plate were investigated experimentally. The experiments were carried out with water and the test plate was heated with constant heat flux. The test plates of different lengths L = 50 mm and 100 mm enabled the experiments in the ranges of Reynolds and modified Rayleigh numbers as, < Re L < and <Ra L < The flow around the plate was first visualized with dyes, and, then, the local heat transfer coefficients from the plates were measured with thermocouples. The following results were obtained from the present experiments. (1) The laminar boundary layer of forced convection develops over the plate when the buoyancy force is small enough. With increasing the heat flux of the plate, the boundary layer begins to separate from the bottom edge of the plate, then, the separation point shifts toward upstream and, finally, reaches to the top edge of the plate. With the separation at the top, large separation bubble appears over the plate. (2) The separations of flow at the bottom and top edges of the plate can be predicted with the non-dimensional parameter as ( /Re L 2.5 ) = 0.4 and 3, respectively, where the parameter ( /Re L 2.5 ) stands for the ratio of the buoyancy to the inertia force. (3) The local heat transfer coefficients from the plate begin to deviate from those of the pure-forced convection with the onset of flow separation at the bottom edge. (4) The overall Nusselt numbers from the plate show minima at around ( /Re L 2.5 ) = 1.0. Moreover, by comparing the overall Nusselt numbers to those of the forced and natural convections, the combined convection region was determined as; 0.35< ( /Re L 2.5 ) <4.0. Nomenclatures A surface area [m 2 ] g gravity acceleration [m/s 2 ] Gr L modified Grashof numbers (= gβq w L 4 /(λν 2 )) h x local heat transfer coefficient [W/(m 2 K)] h m overall heat transfer coefficient [W/(m 2 K)] L length of plate [mm] overall Nusselt numbers (= h m L/λ) f overall Nusselt numbers for forced convection n overall Nusselt numbers for natural convection Nu x local Nusselt numbers (= h x x/λ) Nu ξ local Nusselt numbers (= h x ξ/λ) Pr Prandtl number (= ν/α) Q electrical power input [W] q w surface heat flux [W/m 2 ] Ra L modified Rayleigh numbers (= gβq w L 4 /(λαν)) Ra ξ local modified Rayleigh numbers (= gβq w ξ 4 /(λαν)) Re L Reynolds numbers (= u L/ν) Re x local Reynolds numbers (= u x/ν) T wx local surface temperature [K] T f film-temperature [K] T ambient fluid temperature [K] u main stream velocity of forced convection [m/s] x distance from leading edge to bottom edge of plate [mm] α thermal diffusivity [m 2 /s] β volumetric expansion coefficient [1/K] 45

6 F. KIMURA, J. KIDA and K. KITAMURA λ thermal conductivity [W/(mK)] ν kinematic viscosity [m 2 /s] ρ density [kg/m 3 ] ξ distance from bottom edge to leading edge of plate [mm] References [1] Szewczyk, A. A.: Combined forced and free-convection laminar flow, Transactions of the American Society of Mechanical Engineers, Series C, Journal of Heat Transfer, 86-4 (1964), [2] Markin, J. H.: The effect of buoyancy forces on the boundary-layer flow over a semi-infinite vertical flat plate in a uniform free stream, Journal of Fluid Mechanics, 35-3 (1969), [3] Oosthuizen, P. H. and Hart, R.: A numerical study of laminar combined convective flow over flat plates, Transactions of the American Society of Mechanical Engineers, Series C, Journal of Heat Transfer, 95-1 (1973), [4] Oosthuizen, P. H. and Bassey, M.: An experimental study of combined forced- and free-convection heat transfer from flat plates to air at low Reynolds numbers, Transactions of the American Society of Mechanical Engineers, Series C, Journal of Heat Transfer, 95-1 (1973), [5] Kobus, C. J. and Wedekind, G. L.: Modelling the local and average heat transfer coefficients for an isothermal vertical flat plate with assisting and opposing combined forced and natural convection, International Journal of Heat and Mass Transfer, (1996), [6] Kitamura, K., Yamamoto, M. and Kimura, F.: Fluid flow and heat transfer of opposing mixed convection adjacent to vertical plates, Transactions of the Japan Society of Mechanical Engineers, Series B, (2004), (in Japanese) [7] Lighthill, M. J.: Contributions to the theory of heat transfer through a laminar boundary layer, Proceedings of the Royal Society of London, Series A, (1950), [8] Fujii, T.: Fundamental of Free Convection Heat Transfer, Advances in Heat Transfer, 3 (1976), Yokendo, (in Japanese) 46