RELATION BETWEEN HEAT TRANSFER AND FRICTIONAL PRESSURE DROP OF GAS-LIQUID TWO-PHASE FLOW IN SMALL BORE TUBES

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1 ISP-16, 005, PRAGUE 16 H INERNAIONAL SYMPOSIUM ON RANSPOR PHENOMENA RELAION BEWEEN HEA RANSFER AND FRICIONAL PRESSURE DROP OF GAS-LIQUID WO-PHASE FLOW IN SMALL BORE UBES Masuo KAJI*, oru SAWAI*, adanobu UEDA* *Department of Mechanical Engineering, School of B. O. S.., Kinki University, Uchita Wakayama, Japan Corresponding author: Masuo KAJI kaji@aka.kindai.ac.jp Phone , fax Keyords: o-phase flo, Heat transfer, Frictional pressure drop, small bore tube Abstract Heat transfer, pressure drop and void fraction ere simultaneously measured for upard heated air-ater to-phase flo in about 1 and mm I.D. tubes to investigate the analogy beteen heat transfer and fluid friction of tophase flo in small bore tubes. When the liquid superficial velocity j l as lo, frictional pressure drop agreed ith the experimental correlation of Mishima-Hibiki for small bore tube, hereas agreed ell ith Chisholm-Laird correlation hen j l as relatively high. Heat transfer coefficients agreed ith the frictional pressure drop correlation of Chisholm-Laird except hen j l as very lo. heoretical calculations of annular liquid film flo model ere performed by applying experimental values of all shear stress and liquid holdup, and satisfactory results ere obtained except hen j l as very lo. 1 Introduction hermo-hydrodynamic characteristics of gas-liquid to-phase flo in mini-scale channels are important to design and operate miniature heat exchangers, cooling systems of high poer electronic devices, heat rejection systems of spacecraft, biotechnology systems and so on. he hydrodynamic characteristics of gas-liquid to-phase flo in mini channels have been studied by many investigators. Experimental studies on pressure drop and void fraction of air-ater to-phase flo in circular tubes ere conducted by Sugaara et al. [1], Fukano and Kariyasaki [], Bao et al. [3], Mishima and Hibiki [4] and riplett et al. [5,6], and the flo pattern, void fraction and frictional pressure drop characteristics have been clarified to some extent. As for the heat transfer characteristics, almost experimental studies are concerned ith the onset of nucleate boiling and critical heat flux, and the forced convection heat transfer characteristics of to-phase flo have not been clarified. he authors [7] investigated the analogy beteen the momentum and heat transfer of gas-liquid to-phase flo by experiment for a relatively large bore tube and found to hold the folloing relation in the annular flo region. h K h P l 1 = 1 α here h P and h l are the heat transfer coefficients of to-phase and liquid component flos, respectively. α is the void fraction, (dp/dz) f the frictional pressure gradient and Φ l the to-phase flo multiplier by Lockhart- Martinelli. K is an experimental factor. If K=1 the velocity and temperature fields are analogous ith each other. fp fl = Φ l (1) 1

2 In this study the analogy beteen heat transfer and fluid friction of gas-liquid tophase flo in small bore tube is experimentally investigated. he heat transfer coefficient, the frictional pressure loss and the void fraction ere simultaneously measured for vertical upard non-boiling heated air-ater flo in about 1 and mm I.D. tubes. he relation beteen the experimental data of frictional pressure loss and heat transfer coefficient is compared ith Eq. (1). A theoretical calculation is performed for an annular liquid film flo model using experimental values of all shear stress and liquid holdup. Experimental Apparatus Figure 1 shos a schematic diagram of the experimental apparatus. Water is fed by a gear pump from a separator tank through a rotameter to the test section and circulated. Air is fed from a compressor through a mass flo meter. he test section consists of an air-ater mixer, a calming section, a void fraction measuring section and a heated section. he airater mixer is made of a co-axial nozzle. Air is supplied from the inner tube and ater is supplied from the outer tube. he cross section of the air-ater mixer is conically diminished at the outlet. he calming section is made of a transparent PCV tube or Pyrex glass tube ith the same inside diameter as the test section in order to observe the flo pattern. he heated sections are made of stainless steel tubes of hich dimensions are shon in able 1. he inside diameter of the test section is the measured mean value of the inlet and outlet of each test section. he test section as heated by an alternate current. Inlet air and ater, and outlet mixture temperatures are measured by K type sheath thermocouples of 1mm O.D.. Outside all temperatures of the heated test section are measured by E type thermocouples of 0.mm O.D. at several locations along the axis ith an interval of 10 mm for 1mm I.D. tube or 15mm for mm I.D. tube. hese thermocouples ere calibrated to have an accuracy of 0.1 K. Outlet pressure of the heated test section as measured by a Bourdon-tube Heating test section P DP Void fraction measuring section Air ater mixer Cooling ater Bus-bar A.C. gauge and pressure drop as measured by a pressure difference transducer. he void fraction measurement section as manufactured by sandich arrangement of five acrylic resin and four brass plates bonded Air Separator Water Rota-meter Pump Air Mass flo meter Fig.1 Schematic diagram of experimental apparatus able 1: est tube dimensions Inside Outside Heated Diffrential pressure diameter diameter length measurement lengh Acrylic 1.0t Acrylic Acrylic φ 1.0 or.0 0.8t Brass 0.3t Brass Fig. Detail of void fraction measuring section

3 RELAION BEWEEN HEA RANSFER AND FRICIONAL PRESSURE DROP OF GAS-LIQUID WO-PHASE FLOW IN SMALL BORE UBES together as shon in Fig.. he measurement method is illustrated in Fig. 3. he measurement section and the single-phase liquid section are electrically connected in a series and a direct current is supplied. Current loss in the electric circuit as verified to be negligible by the preliminary experiment. Assuming that the electric conductance of to-phase mixture is proportional to the average cross section area occupied by ater, the average liquid holdup η (=1-α) is determined by the folloing relation. η = V / V ) /( V / V ) () ( P0 P L0 L V P0 and VL l0 are measured in the preliminary experiment here the ater flos alone. Liquid holdup is determined from the voltage drop ratios in the to-phase section V P0 /V P and the liquid single phase section V L0 /V L. o make conductive the distilled ater, NaCl as dissolved in about 0.04% concentration. But the output signal from the liquid holdup probe as not affected by an alternate current for heating the air-ater mixture floing in the test section. Heat flux as selected by a criterion that the subcooled boiling did not occur. 3 Experimental Results 3.1 Frictional Pressure Drop Frictional pressure loss is determined by subtracting the static and acceleration losses from the measured total pressure drop. he static loss is estimated by the measured void fraction. he acceleration loss is estimated by the changes of gas density, since the phase change does not occur, and the quality and the void fraction do not change throughout the test section. he change of gas density is so small that the acceleration loss is very small compared to the other components. he frictional pressure drop is correlated by the relation beteen Lockhart-Martinelli parameter X and to-phase multiplier Φ l. X and Φ l are defined by using the frictional pressure gradient of to-phase flo (dp/dz) fp, and those of gas and liquid component flos (dp/dz) fg and (dp/dz) fl as follos: X Φ = l = fl fp (3) In the preliminary experiment, friction factor of single phase flo as verified to agree ell ith the folloing equations of Hagen- Poiseuille for laminar flo and Blasius for turbulent flo. In the gas-liquid to-phase flo, transition from laminar to turbulent is considered to occur hen the Reynolds number Re becomes greater than λ = Mixer 64 / Re : Re < 1000 λ = Re Air 0.5 D.C. Supply V P Chisholm and Laird [8] proposed to correlate Φ l ith X by the folloing equation. here C is a parameter hich depends upon hether the gas and liquid component flos are laminar or turbulent, respectively. hey recommended C=1 hen the both component flos are turbulent. V L fl fg : Re Water Fig.3 Illustration of void fraction measurement method 1000 (4) (5) Φ l = 1 C / X 1/ X (6) 3

4 In Figs. 4(a) and (b) the present experimental data are plotted by the relation beteen Φ l and 1/X ith an parameter of liquid superficial velocity j l. Equation (6) ith C=1 is shon by a solid line in each figure. For small diameter tubes, Mishima and Hibiki [4] measured the pressure drop of air-ater to-phase flo and found that the parameter C became small as the tube diameter decreased. hey proposed to modify the correlation of Chisholm and Laird by using the folloing parameter C: 0.333d C = 1(1 e ) (7) here d is the inside tube diameter in mm. he Mishima-Hibiki correlation is shon by a dotted line in each figure. Comparing the present experimental data ith these correlations, almost the data are found to agree ith the Chisholm- Laird correlation at relatively high liquid flo rates. In Fig. 4(a), the experimental data at j l =0.105 m/s agree ith the Mishima-Hibiki correlation shon by a dotted line. When j l becomes greater than m/s, almost the data agree ith the Chisholm-Laird correlation except for a fe data at j l =0.301 m/s. he liquid superficial velocities of j l =0.105 m/s and m/s correspond to the liquid phase Reynolds numbers of about 300 and 800, respectively. Although the liquid component flo is laminar considering the value of Re l, transition from laminar to turbulent may occur in the liquid phase flo as the gas flo rate is increased. Similar tendency can be seen in Fig. 4(b) for d=1.03mm. As j l decreases, the pressure drop becomes smaller than the Chisholm-Laird correlation and tends to approach the Mishima- Hibiki correlation. When j l is lo and 1/X is small, hoever, the present data give higher values compared ith Mishima-Hibiki correlation, particularly, as can be seen in Fig. 4(b). In theses cases, the flo patterns are bubbly or slug flo. It is considered that the frictional pressure drop characteristics of gas-liquid tophase flo depend on the flo pattern rather than the flo condition hether the flo is laminar or turbulent, hen the tube diameter is small and the gas and liquid flo rates are lo. Φ l Φ l 10 3 d=.01mm (Vertical heated flo) (a) d=.01 mm 10 3 d=1.03mm (Vertical heated flo) Φ l =11/X1/X Φ l =110.3/X1/X Φ l =11/X1/X Φ l =16.104/X1/X (b) d=1.03 mm Fig. 4 Comparison of experimental frictional pressure loss ith the correlations of Chisholm-Laird and Mishima-Hibiki Making a comparison of these results ith our previous experiment [9] hich as conducted ith adiabatic horizontal flos in Pyrex glass tubes of approximately the same diameters, similar results ere obtained. he influences of the tube material, flo direction and heating on the frictional pressure drop characteristics could not be found. If the pressure drop in the gas-liquid tophase flo is assumed to be mainly caused by the liquid flo, the to-phase flo multiplierφ l is expressed by the folloing equation. 4

5 RELAION BEWEEN HEA RANSFER AND FRICIONAL PRESSURE DROP OF GAS-LIQUID WO-PHASE FLOW IN SMALL BORE UBES 1 Φ l = (8) Figure 5 shos a comparison of Eq. (8) ith the present experimental data. Reasonable agreement can be seen for d=1.03mm, but the calculated values are slightly greater than the experimental data. For d=.01mm, Eq. (8) gives remarkably smaller values compared ith the experimental data. his means that the effect of gas flo on the pressure drop cannot be neglected hen the tube diameter becomes large. 3. Heat ransfer Coefficient o evaluate the heat transfer coefficient, the inside all temperature of the tube as calculated from the measured outside all temperature by solving Fourier's equation for heat conduction, considering thermally insulated condition at the outside tube all. Local bulk temperature b as determined from the heat balance of the inlet fluid enthalpy and the electric poer input. If the evaporation of liquid can be neglected, the bulk temperature is calculated from the enthalpy changes in gas and liquid. Hoever, hen the liquid flo rate is relatively lo, the effect of absolute humidity change on the enthalpy change in air-ater mixture cannot be neglected. he local bulk enthalpy as estimated on the assumption of saturated humidity condition. he heat transfer coefficient of to-phase flo is usually expressed by the ratio to that of single-phase flo here the liquid component flos alone, as h P /h Lf. h Lf is evaluated by Dittus-Boelter s equation for turbulent flo hen Reynolds number Re is greater than 1000, because the turbulence is considered to be increased due to the interaction beteen gas and liquid phases. For Re<1000, the correlation for laminar flo is employed as shon in the folloing. Nu = 1.6{Re Pr( d / z)} Nu = 0.03Re ( 1 α ) 0.8 Pr 0.4 1/3 : Re < 1000 : Re 1000 (9) Φ l cal h P / h Lf h P / h Lf 10 3 d=1.03mm 50% 10 d=.01mm Fig. 5 Comparison of experimental data of frictional pressure drop ith the calculation values by Eq. (8) 10 d=.01mm d=1.03mm d=8.03mm (a) d=.01 mm d=8.03mm Φ l exp h P /h Lf =.5(1/X) /3 h P /h Lf =.5(1/X) / (b) d=1.03 mm -50% Fig. 6 Relation beteen heat transfer ratio h P /h Lf and Lockhart-Martinelli parameter 1/X 5

6 In Figs. 6(a) and (b) the data of h P /h Lf is plotted against the reciprocal of Lockhart- Martinelli parameter 1/X hich corresponds to the ratio of gas to liquid flo rate, comparing ith our previous experimental data for d=8.03 mm tube [10]. Since the values of h P /h Lf are almost constant along the axis, only a representative value is plotted for each run. In the case of flo boiling, heat transfer characteristics are classified into three regions, i.e. subcooled boiling, saturated nucleate boiling and forced convective evaporation regions. he solid line represents the empirical correlation of Sekoguchi et al. [11] for the flo boiling of ater in the forced convective evaporation region. From a comparison beteen Figs. 6(a) and (b), it is found that the relation beteen h P /h Lf and 1/X agree ell ith each other, but the data of the present study sho values to or three times as much as the correlation of flo boiling of ater. For the case of d=8.03mm, hoever, heat transfer is significantly augmented in the slug and froth flo regions corresponding to 0.1<X<0.5. Since the liquid velocity vigorously fluctuates along the axis and reversed flo occurs very often, heat transfer may be improved in a large bore tube. On the contrary, in a small bore tube, heat transfer augmentation due to reversed flo of liquid film may not occur. 3.3 Relation beteen Heat ransfer and Pressure Drop Figures 7(a) and 7(b) sho the experimental results represented by the relation of Eq. (1) assuming that K=1. he abscissa is 1/X and the ordinate is h P /h Lf /(1-α). When 1/X as small, observed flo patterns ere bubble or slug flos. As 1/X increases at a given liquid flo rate, flo pattern changes from bubble to slug and annular flos. In the figures, correlations of Φ l by Mishima-Hibiki and Chisholm-Laird are shon to compare ith the present data. Almost the data of h P /h Lf /(1-α) agree ell ith the Chisholm-Laird correlation as a hole, but the agreement ith the Mishima-Hibiki correlation is poor for lo liquid flo rate conditions. h P / h Lf / (1-α) h P / h Lf / (1-α) 10 3 d=.01mm Φ l =11/X1/X Φ l =110.3/X1/X d=1.03mm (a) d=.01 mm Φ l =11/X1/X Φ l =16.104/X1/X (b) d=1.03 mm Fig. 7 Relation beteen h P /h Lf /(1-α) and Lockhart-Martinelli parameter 1/X When the liquid flo rate is very lo, h P /h Lf /(1-α) is greater than Φ l at around 1/X=0.5. his might be caused by the heat transfer augmentation due to the slug flo pattern. In Fig. 7(b) h P /h Lf /(1-α) is smaller than Φ l hen 1/X is less than 0.1. From the above discussion, the analogy beteen heat transfer and fluid friction is considered to hold roughly for the gas-liquid to-phase flo in small bore tubes. 6

7 RELAION BEWEEN HEA RANSFER AND FRICIONAL PRESSURE DROP OF GAS-LIQUID WO-PHASE FLOW IN SMALL BORE UBES 4 heoretical Calculation According to the above discussion, a theoretical calculation is performed by applying an annular liquid film flo model in a vertical tube as illustrated in Fig. 8. he liquid film flos adjacent along the tube all and the gas flos in the core region. he liquid film thickness is assumed to be constant as a mean value y i because the heat transfer is considered to be mainly affected by the sublayer of the liquid film. Physical properties of the liquid and gas are considered not to change across the radius. According to the turbulent mixing model, the momentum and energy equations can be expressed as here u and y are the folloing dimensionless velocity and distance from the all. u = u / τ / ρ here is the dimensionless temperature. τ du = ρ( ν ε ) τ dy q 1 ε h d = ( ) q Pr ν dy y = y ρ c = τ / ρ ν P ( q ) τ / ρ In the calculation, Karman's three layers model is applied to the velocity profile and is assumed that ε h =ε in Eq. (11). ε /ν is given by the folloing equations. Radial heat flux distribution across the liquid film is assumed to be uniform. Liquid thickness y i is estimated by the liquid hold-up η. (10) (11) (1) (13) (14) ε / ν = 0 : y 5 ε / ν = y /5 1 : 5 < y 30 (15) ε / ν = y /.5 1 : y > 30 emperature profile in the liquid film is determined by integrating Eq. (11). y q / q = dy 0 1/ Pr ε / ν (17) he mean temperature of the liquid film flo is obtained by the folloing equation. m = yi πρcpu ( r 0 0 yi πρcpu ( r 0 0 y y (18) he bulk mean temperature of gas-liquid tophase flo is calculated as, here i is the dimensionless gas-liquid interface temperature. Heat transfer coefficient h P can be calculated by the folloing equation. h P y = ( 1 1 η ) (16) i r o ) ) dy dy GlcPlm GgcPgi b = (19) GlcPl GgcPg q ρ c P τ = = ρ (0) b y i u Fig. 8 Annular liquid film flo model b u i i r o y 7

8 In the calculation, the physical properties of air and ater ere assumed to be constant and estimated at a mean temperature of the inlet and outlet bulk fluid temperatures. he experimental values of all shear stress τ and the liquid holdup η ere employed. When the liquid flo rate is lo, the liquid film flo is considered not to be turbulent but laminar. he Karman's three layers model cannot be applied and eddy diffusivity vanishes (ε /ν =0). rial calculations ere made to find the transition Reynolds number of liquid film flo Re l by comparing the calculated values ith the experimental data and reasonable results ere obtained for Re l =600. Flo conditions here Re l is less than 600 are j l =0.105m/s for d=.01mm and j l =0.00m/s for d=1.03mm. Figure 9 shos a comparison of the calculated values of to-phase heat transfer coefficient h Pcal ith the present experimental data h Pex. Almost the plots fall in a theoretical line ithin 0% deviations and satisfactory results are obtained. For the cases of j l =0.105, 0.301m/s for d=.01mm and j l =0.00, 0.53m/s for d=1.03mm, hoever, calculated results sho remarkably higher values. In these cases, disagreement of liquid flo rate beteen the calculation and experimental values as very large. his may be caused by unsuitability of the velocity profile in the liquid flo and inaccuracy of the void fraction measurement. 5 Conclusions Heat transfer coefficient, pressure drop and void fraction ere simultaneously measured for vertical upard air-ater to-phase flo in heated tubes ith internal diameters of about 1 and mm in order to investigate the analogy beteen heat transfer and fluid friction of tophase flo in minichannels,. he results obtained are summarized in the folloing: (1) he present data of frictional pressure loss agreed ith the Misima-Hibiki correlation for small bore tube at lo liquid flo rate and the liquid component flo as laminar, hereas the present results agreed ith the Chisholm-Laird correlation for larg bore tube hen the liquid flo rate as large and the liquid component h Pcal, W/m K % 10 4 flo as turbulent. () he heat transfer coefficient ratio h P /h Lf agreed very ell beteen 1 and mm I.D. tubes and also agreed ith our previous experimental results for 8.03 mm I.D. tube for the annular flo region. he relation of momentum and heat transfer analogy as proved to hold roughly for the to-phase flo in small bore tubes. But, satisfactory results ere not obtained for very lo liquid flo rate conditions. (3) A theoretical calculation of heat transfer as performed by applying the annular liquid film flo model and compared ith the present experimental results. Satisfactory agreement could be obtained except hen the liquid flo rate as very lo. Acknoledgments he authors ish to thanks to Messrs. M. Sahara,. Morita, and. Nojima for their assistances in the experimental ork. References -0% d= mm h Pex, W/m K Fig. 9 Comparison of calculated values of heat transfer coefficient ith the present experimental data [1] Sugaara, S., Katsuta, K., Ishihara, I. and Muto,. Consideration on the pressure loss of to-phase flo in small diameter tubes. Proc. 4th National Heat ransfer Symp. of Japan, pp , 1967 (in Japanese). 8

9 RELAION BEWEEN HEA RANSFER AND FRICIONAL PRESSURE DROP OF GAS-LIQUID WO-PHASE FLOW IN SMALL BORE UBES [] Fukano,. and Kariyasaki, A. Characteristics of gasliquid to-phase flo in a capillary tube. Nuclear Engineering and Design, Vol.141, pp 59-68, [3] Bao, Z. Y., Bosnich, M. G., and Hayens, B. S. Estimation of void fraction and pressure drop for to-phase flo in fine passages. rans. Inst. Chem. Eng., Vol.7, pp 65-63, [4] Mishima, K., Hibiki,. Some characteristics of airater to-phase flo in small diameter vertical tubes. Int. J. Multiphase Flo, Vol., pp , [5] riplett, K. A., Ghiaasiaan, S. M., Abdel-Khalik, S. I. and Sadoski, D. L. Gas-liquid to-phase flo in micro-channels, Part I: o-phase flo pattern. Int. J. Multiphase Flo, Vol.5, pp , [6] riplett, K. A., Ghiaasiaan, S. M., Abdel-Khalik, S. I. LeMouel, A. and McCord, B. N. Gas-liquid tophase flo in micro-channels, Part II: Void fraction and pressure drop. Int. J. Multiphase Flo, Vol.5, pp , [7] Kaji, M., Saai,. and Mori, K. Analogy beteen heat transfer and fluid friction of gas-liquid tophase flo in minichannel, hermal Science and Engineering, Vol.11, No.6, pp 59-66, 003. [8] Chisholm, D. and Laird, A. D. K. o-phase flo in rough tubes, rans. ASME, Vol.80, pp [9] Kaji, M., Saai,. and Ueda,. Frictional pressure drop characteristics of gas-liquid to-phase flo in small bore tubes, Memories of the School of B.O. S.. of Kinki University, No.15, pp 65-74, 005. [10] Kaji, M., Sekoguchi, K. and Mori, K. An experimental study of momentum and heat transfer analogy in gas-liquid to-phase flo. Proc. Int. Symp on Heat and Mass ransfer, Kyoto, pp , [11] Sekoguchi, K., Han, Z. X., Kaji, M., Imasaka,. and Sumiyoshi, Y. An analogy beteen heat transfer and pressure drop in forced convective boiling flo. Dynamics of o-phase Flos, CRC Press, Inc., pp ,