Heat Transfer to Falling Water Film on a Vertical Surface

Transcription

1 Journal of NUCLEAR SCIENCE and TECHNOLOGY, 12[101 pp (October 1975). Heat Transfer to Falling Water Film on a Vertical Surface Keisuke YOSHIOKA and Shu HASEGAWA Department of Nuclear Engineering, Kyushu University* Recerived March 29, 1975 An experimental study was performed on the heat transfer to a turbulent water film which is falling down the outside surface of a heated vertical tube. The test section was made of stainless steel tube 13 mm in outside diameter and 1,500 mm long. It was found that a nearly constant heat transfer coefficient was obtained for the lower half of the test section covering a length of about 1 m, which was direct-heated by electric current. The resulting values of the Nusselt number in this region were correlated to the Reynolds number, and the plots fell in a region intermediate between those given by Wilke and by McAdams. An analytical model taking account of the undulation on the film surface is proposed. The predictions from the model give fairly good agreement with the experimental data. KEYWORDS: heal transfer, turbulent water film, falling water film, maximum wave height, heat transfer coefficient, Nusselt number, heating, fluid flaw, water I. INTRODUCTION Heat transfer to a falling liquid film would occur in the unlikely event of high loss-ofcoolant accident in a light water reactor. Predictions on heat transfer coefficients have been made by McAdams(1) and Wilke(2) for turbulent region. A wide discrepancy, however, exists between these predictions not only in their absolute values but also in their tendencies in varying with Re. Furthermore, a generally applicable theoretical approach has yet to be established for heat transfer analyses taking account of undulation on the film surface in turbulent flow. The present work covers measurements of the heat transfer coefficient made on a vertical tube transferring heat at a uniform rate to a water film falling down its outer surface. The Nusselt number obtained along the test section shows a nearly constant value in the lower half of the test section covering a length of about 1 m, which was direct-heated by electric current. The values obtained were slightly lower than those given by Wilke's but higher than McAdams'. Moreover, it was lower than the result obtained by the authors' calculation in which the free surface of falling film is replaced by the flat surface constituted by the mid-plane between two parallel plates constituting a channel for turbulent flow. A simple mode(3) representing the undulation on the surface, based on experimental evidence, has been adopted to describe the heat transfer to the turbulent film flow. The values predicted by the proposed model agree well with the experimental results. II. EXPERIMENTAL The apparatus employed is illustrated schematically in Fig. 1. Two sets of test sections were used. These test sections were made of stainless steel tube 13 mm in outside diameter with 1 mm thick wall. One was 1,500 mm long, and the other 1,000 mm. The shorter test section was provided with an approach section of the same diameter and 1,000 mm long, to which the test section was joined by brazing, and which served to establish a hydrodynamically developed flow in the falling film before it reached the test section. The D.C. power supplied to the test section to provide the Joule heating was calculated from voltage drops across the test section and * Hakozaki, Higashi-ku, Fukuoka-shi. 18

2 Vol. 12, No. 10 (Oct. 1975) 619 Fig. 1 Schematic drawing of experimental loop a known resistance rigged as shunt in the circuit. The wall temperature was determined by 0.2 mm Chromel-Alumel thermocouples fixed by pressing against the inner wall surface with springs. Mica membranes 0.01 mm thick were inserted between the thermocouples and the tube wall to avoid any effect of stray e.m.f. that might be generated by the heating current. The location of the thermocouples as shown in Fig. 2. Their number was limited by the capacity of installation. Tap water from a reservoir was led through a calibrated orifice to the distributor, which, with careful adjustment, ensured a uniform distribution of the water in a film of even thickness around the periphery of the test tube. The clearances between the tube and distributor were adjusted for a given flow rates by exchanging several pieces of distributors. In each test run, measurements were made of the wall temperature and the water temperature at the inlet and the exit after the water temperature had reached steady state. The inlet and exit temperature were used to determine the increment of the bulk mean temperature. In addition, these data were compared with the electrical input, and it was ascertained that the heat loss to the ambient atmosphere was very small. Evaluation of the local heat transfer coefficient was based on the difference between Fig. 2 Details of thermocouple locations the wall temperature and the bulk mean temperature. The wall temperature was calculated by subtracting the temperature drop through the thickness of the wall. The amount of temperature drop was estimated from the applied heat flux(4), and was found to be about 10% or more in the resultant temperature difference. III. RESULTS The data obtained from the experiment were reduced to the local Nusselt numbers for which the film thickness evaluated by the correlation given by Brauer(5) was adopted as reference length. Typical examples of the resulting plots are presented for the whole range of the test section in Fig. 3, where the local film Reynolds number is used as parameter. It is seen that the plots register no significant change in level for the test sections. The anomalously high plots appearing around 750 mm in the figure occurred in a number of runs with the longer test section, but rarely with the shorter tube provided with approach section. It is likely that this irregular rise caused by a local thickening of the film, is which was discerned by measurements with - 19

3 620 J. Nucl. Sci. Technol., a capacito-meter, but the origin of this sporadic thickening is yet to be established. of heating (nn) Fig. 3 Local overall Nusselt number The plots in Fig. 3 indicate particularly stable values of Nu in the lower half of the approximately 1 m long heated test section. It has been pointed out by Wilke that the length of the approach section required for establishing a fully developed temperature profile was slightly below 0.8 m or less for when the water, flow and the heating were initiated simultaneously. Our main interest in the present work lies in the region of nearly constant Nu. The correlation between Nu and Re is shown in Fig. 4. For reference, the predictions of Wilke for Pr-=7 and =5 which are of relevance to the present experiment are indicated in the same figure by the solid line. It is seen that the present plots all fall below Wilke's lines, which have been obtained by correlating the data of Nu averaged over a range of nearly constant heat transfer coeffine Fig. 4 Nu vs. Re, in constant heat transfer region comparison with other published results cient. It can be surmised from Wilke's data on measured wall temperature and bulk temperature that his experiment was made under the boundary condition of constant heat rate. In his experiment, he measured the temperature distribution in the thin liquid film by means of a sampling device, and from it he obtained the mixed mean temperature and the applied heat flux. Brauer has given a recommended curve for the heat transfer coefficient from his measurements using a short electrically heated wall installed in a test section. The bulk temperature of the film was estimated from the temperature measured at the outer edge of the thermal boundary layer. The results obtained by calculation with Brauer's recommended curve using the properties estimated at 25-C and based on the correlation of Mc- Adams for Pr=5 and =7 are also shown in Fig. 4 (chain line), together with McAdams' data (dashed line). It is seen that the present results fall in the zone roughly intermediate between that of McAdams and those of Wilke and Brauer. W. DISCUSSION It has been pointed out by Brauer(5) and Ishigai et al.(6) that undulation appears on the surface of the falling water film, whose maximum wave height increases consistently with progress downstream. Yet the data presented in Fig. 3 indicates a nearly constant value of Nu regardless of the distance from the distributor, and a similar behavior has been observed by Wilke. The maximum film thickness was measured by probing needle. The results are shown in Fig. 5, together with those reported by Brauer and Ishigai et al. Particular attention was paid to determine whether any difference would appear between the cases with and without heating. It proved that only a slight difference could be discerned between the two cases. The undulation which has a wave length L, defined, by the distance between two succeeding large-amplitude of undulations, is treated in the following manner. A typical pattern is depicted in Fig. 6 together with 20

4 Vol. 12, No. 10 (Oct. 1975) 621 Fig. 5 Maximum film thickness Fig. 6 Growing amplitude waves, and wave shape model representing this configuration the schematized model adopted in the present work. A discussion of this shape has been given in the previous report(3). A brief review of the undulation is presented below. Unlike simple harmonic motion, the wave height increases successively from one wave to the next until it reaches a certain limit, after which the wave height is abruptly diminished and the cycle is recommenced (left-hand drawing of Fig, 6). In the previous report, the calculated frequency of this sawtooth pattern cycle became roughly constant, with a value that agreed with experiment, when the change of the film thickness d was approximated by parabolas truncated at a length L and repeated in train (right-hand drawing, Fig. 6). The same model is adopted in the present instance. Leonard et al.(7) has calculated the Nusselt number based on a laminar flow on which a sinusoid shaped surface was superposed. The maximum value of Nu was found to coincide with the maximum film thickness reached during one period. The adoption of a Nusselt number averaged over one span of the wave length was considered to represent the condition of transition to turbulent heat transfer. They did not take into account the effect of differences in the Prandtl number on the transition, since their approach was made from a laminar flow model. Wilke & Ishigai et al.(8) have observed independently of each other that the transition was strongly affected by Pr. The Nusselt number for constant heat rate at the wall can be obtained by integrating the energy equation in which the film is regarded as a uniform directional flow and the eddy diffusivity is suitably assumed. The result calculated by this treatment is higher than given by Wilke, as shown in Fig. 8, where the velocity profile of Karman, the film thickness by Brauer, and equality of eddy diffusivity between the energy and the momentum are all assumed. Ishigai et al.(8) performed the calculation by adopting Spalding's expression for the turbulent velocity profile, and the result is roughly the same. Dukler(9) adopted the assumption that the dimensionless film thickness is lowered by 26 for turbulent film flow in consideration. His prediction gives a smaller value when applied to the present work. Limberg(10) using his eddy diffusivity succeeded in obtaining agreement with the result of Wilke. In the present work, it was observed that the flow rate varied with the saw-tooth pattern cycle, with a nearly constant frequency of about 25 Hz, as it was observed also in the previous study. A simple order-of-magnitude estimation of the energy equation showed that the transient term is less effective caused by a very thin film thickness in comparison with the wave length referred to successive large amplitude. Nevertheless, the two-dimensional treatment should be undertaken for the convection term in the energy equation. The meaning of an exact treatment, however, is practically lost by the simplification adopted of replacing the saw-tooth pattern succession of undulations by a smooth parabola. Accordingly, we shall obtain a first-approximation solution. 21

5 622 J. Nucl. Sci. Technol., The conservation of mass is accounted for the wavy flow by integrating the flow rate over one saw-tooth cycle in reference to a fixed point on the wall. The change in flow rate during cycle is derived by assuming the pattern of change of the film thickness and of the velocity distribution, and we shall call the resulting diagram a "map of flow" as depicted in Fig. 7. Even with the Reynolds number averaged over the period kept at a constant value Re, the map of flow will still vary with the maximum film thickness. In the figure, the upper drawing corresponds to a maximum film thickness of 4.7 mm and the lower to 3.8 mm, both for Re=6,500. On the other hand, the Nusselt number has been observed to hold a nearly constant value throughout the region. To explain this, we suppose that a thermal entrance section with a fully developed flow appears in a small division of the span of one cycle. The span of one cycle is given by ( 3 ) ( 4 ) From the result of the previous work, we shall assume that the period T is 1/25 sec. It is assumed that the film thickness varies with as follows e ( 5 ) The terminology is as noted in Fig. 6. By neglecting the acceleration due to surface curvature, the w where g : Gravitational acceleration. ( 6 ) Using the experimental value of the maximum film thickness, the velocity profile due to and tw, given by Eq. ( 6 ), the local Karman mean stream velocity u and the local Reynolds number Re* are obtained. In the calculation, the unknown parameter hi/a, is so determined as to satisfy Eq. ( 1 ) by considering the time occupied by a mass of fluid having a velocity n to pass by the fixed point on the wall. The map of flow thus derived is depicted in Fig. 7. The finely dotted line appearing in the lower diagram representing a surface shape obtained by the experimental data in the Fig. 7 Flow map (flow structure of model) of one wave First, the map of flow is derived as follows : The mass flow passing over a fixed point on the wall during the period is expressed by ( 1 ) ( 2 ) previous work. If curves of higher order than the parabola were adopted, they would indicate a steeper than actual reduction of film thickness near the crest, which should render their agreement poorer than the parabola. For the second stage of the analysis, the span L is divided into 40 intervals of e on the map of flow, to obtain the temperature distribution through the span, by applying the theory of entrance region for fully developed flow. It is assumed that the flow rate is invariant in each division. A given temperature is assumed at e=l. The temperature distri- - 22

6 Vol. 12, No. 10 (Oct. 1975) 623 bution at e=l- e is obtained by eigenfunction and eigenvalue. Thus, the temperature distribution is obtained successively from e= L to e=-0. The eigenvalue problem relevant to the present case is already treated by Hatton et a(11), but their boundary condition was not the same. Accordingly, the authors remade the calculations to obtain the eigenvalues and the eigenfunctions for the present problem under applicable boundary conditions : The energy equation for each division is written in dimensionless form Then The equation of e, is (11) (12) (13) (7) where The boundary conditions are (14) ( 8 ) q: Heat flux, u : Velocity l : Thermal conductivity z: Coordinate along the flow direction y: Coordinate normal to the flow direction eh : Thermal eddy diffusivity Pr : Prandtl number. The boundary conditions are ( 9 ) In the calculation, the assumption has been adopted that the momentum eddy diffusivity near the outer edge of the boundary layer is constant, and takes the value em=em(e=0.6) for 0.6, and that the thermal eddy diffusivity e>= is equal to the momentum eddy diffusivity. The solution is obtained in two parts (a) the fully developed temperature =Dev profile and (b) =1 the entrance region profile which disappears with increasing z1. The equation for =Dev is The boundary conditions are (10) (15) and the initial value =1,(e, 0) of each division is equal to =1, at the rear end of the preceding division. The origin of z1, is placed at the edge of the division on which the temperature profile is known. The solution of =1 is obtained by the method of separation of variables. Details of eigenvalue calculation will be reported elsewhere. The solution is where T1 (e, 0) : Known temperature profile along the edge of division Mn n-th order eigenvalue : Eigenfunction belong to M The Nusselt number is given by where a : Heat transfer coefficient. The Nusselt number derived from experiment, on the other hand, is calculated from the heat transfer coefficient which is related (16) (17) (18) (19) 23

7 624 J. Nucl. Sci. Technol., to the observed wall temperature. Since the wall temperature is a time averaged value, the experimental value of heat transfer coefficient (20) In the case of constant heat rate, <tb> is independent of time. (21) By referring to the definition of dimensionless temperature =, <tw> is expressed by The difference between the two values estimated at 700 mm and at 1,400 mm is negligibly small, from which it can be considered that differences in the maximum film thickness do not significantly affect heat transfer. This due to the fact that the fraction of the cycle period taken by the large flow portion of the cycle span to pass by a fixed location is fairly small in the parabolic wave model that has been adopted. And also, this result agrees with the present experiment, which indicated a nearly constant heat transfer over the length of the test section. (22) Therefore, where =w,, : Dimensionless wall temperature. The Nuexp is defined by (23) Substituting Eq. (23) to Eq. (24). (24) (25) Since we consider the heat transfer at a fixed point, the integration should be performed with respect to time over one period. In reference to the present model however, the period may be replaced by the cycle span, in which case the integration is made with respect to e. Using the Brauer's prediction for the mean film thickness, (26) Wilke has, on the other hand, postulated that heat transfer was affected by the undulation when Re>Reu, where In the present calculation, Wilke's postulation (27) is adopted as the limiting value d of the turbulent flow region. Using the data given in Fig. 5, the Nusselt numbers were calculated for Re=4,500 and =6,500 at the positions 700 and 1,400 mm. The results are shown in Fig. 8. Fig. 8 Comparison between local Nusselt;number predictions and experimental results Figure 8 presents all the predictions by previous workers, together with the zone occupied by the present experimental results. Moreover, the curves of the Nusselt number for turbulent fully developed plane surface flow are also drawn for Pr=5 and =7, which have been calculated by assuming that the film surface is flat, as represented, for instance by the mid-plane between two parallel plates forming a channel for turbulent flow. As already seen in Fig. 4, the authors' results lie halfway between those of Wilke and of Mc- Adams. It might be remarked that the undulations on the film flow tend to reduce the flow rate in a fraction of the cycle span, and also to lower the heat transfer compared with turbulent conduit flow. 24

8 Vol. 12, No. 10 (Oct. 1975) 625 V. CONCLUSION The heat transfer to a turbulent water film falling down a heated vertical wall has been examined by experiment, and the results are discussed on the basis of a simple model. The result is correlated in the usual dimensionless form. The estimation based on this undulation proposed gives a satisfactory explanation to the authors' experimental result of nearly constant heat transfer over the length of the test section despite significant increase in the maximum film thickness observed over the same length. Further refinement of the model for eddy diffusivity of film flow should be necessary to improve the heat transfer prediction. ACKNOWLEDGMENT The authors would like to thank Mr. Y. Tanaka for his help in carrying out the experiment. The calculation was made at the Computation Center in Kyushu University. REFERENCES (1) McADAMS, W.H. : "Heat Transmission", (3rd ed.), p.245 (1954). (2) WILKE, W. : Warmeubergang an Rieselfilme, VDI-Forsch.,-Heft 490, Dtisseldorf (1962). (3) YOSHIOKA, K., HASEGAWA, S.: Frequency of undulations on a falling water film, J. Nucl. Sci. Technol., 12[9], 543 (1975). (4) HASEGAWA, S. : Remarks on the calculating method for the temperature drop across the electrical heated tube wall, Trans. JSME, (in Japanese), 31[ (1965). (5) BRAUER, H.: Stromung und Warmeubergang bei Rieselfilmen, VDI-Forsch.,-Heft 457, Dusseldorf (1956). (6) ISHIGAI, S., NAKANISHI, S., KOIZUMI, T., et al.: Hydrodynamic and heat transfer of vertical falling liquid films, Part 1, Classification of flow regimes, Trans. JSME, (in Japanese), 37[301] 1708 (1971). (7) LEONARD, W.K., ESTRIN, J. : Heat transfer theory on a wavy film on a vertical surface, A.I.Ch.E. J., 18[2], 439 (1972). (8) ISHIGAI, S., NAKANISHI, S., TAKEHARA, M., et al.: Hydrodynamics and heat transfer of vertical falling liquid film, Part 2, Analysis by using heat transfer data, Trans. JSME, (in Japanese), 39[321], 1620 (1973). (9) DUKLER, A.E.: Fluid mechanics and heat transfer in vertical falling film systems, Chem. Eng. Progr., 56[30], 1 (1960). (10) LIMBERG, H.: Warmeubergang an turbulent und rieselfilme, Int. J. Heat Mass Transfer, 16[9], 1691 (1973). (11) HATTON, A.P., QUARMBY, A.: The effect of axially varying and unsymmetrical boundary conditions on heat transfer with turbulent flow between parallel plates, ibid., 6[10], 903 (1963). 25