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Mixed Convection Heat Transfer to Air Flowing Upwards Through a Vertical Plane Passage: Part 3
0263–8762/02/$10.00+0.00 # Institution of Chemical Engineers Trans IChemE, Vol 80, Part A, April 2002
MIXED CONVECTION HEAT TRANSFER TO AIR FLOWING UPWARDS THROUGH A VERTICAL PLANE PASSAGE: PART 3 J. WANG1 , J. LI2 and J. D. JACKSON1 1 School of Engineering, University of Manchester, UK. Department of Mechanical Engineering, UMIST, Manchester, UK.
2
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his is the third of three related papers presented in this volume on mixed convection heat transfer in vertical passages. In it an experimental study is reported of buoyancyin uenced convective heat transfer to air owing upwards and downwards under turbulent conditions through a vertical plane passage of relatively short length with a uniform heat input on one wall and an adiabatic condition on the opposite one. Thermal development and ow development occurred simultaneously and neither were complete. As in the case of the studies reported in the two preceding papers, the results presented here reinforce the existing picture of buoyancy-aided mixed convection in vertical passages, which hitherto has been based largely on experiments performed with uniformly heated circular tubes. However, the effects of buoyancy were slightly weaker than for circular tubes, onset of buoyancy-induced impairment of heat transfer was delayed and it occured more gradually. These trends are consistent with the ndings reported in the two preceding papers and can again be explained by the fact that only one surface was heated and, therefore, turbulence was only modi ed directly by buoyancy in one of the two boundary layers. Keywords: mixed convection; heat transfer; buoyancy in uence; plane passage; air.
INTRODUCTION
papers on mixed convection presented in this volume, has proved to be quite successful in describing the behaviour outlined above. It relates the ratio of the Nusselt number, Nu, for buoyancy-in uenced conditions to that for forced conditions Nuf to a buoyancy parameter, Bo*, which combines Grashof number, Gr*, Reynolds number, Re, and Prandtl number, Pr, in the form Gr*=Re3.425Pr0.8 so as to characterize the strength of buoyancy in uences. No information can be found in the literature on buoyancy-in uenced convective heat transfer for an arrangement such as the one used in the present investigation, a relatively short plane passage with one wall uniformly heated and the opposite one adiabatic. The aim of the study was to see to what extent the effects of buoyancy and variable properties in such a con guration would prove to be similar to those found with circular tubes.
Buoyancy-aided convective heat transfer to air owing in vertical tubes has been studied extensively, see for example, Steiner1, Byrne and Ejiogu2, Carr, Connor and Buhr3, Polyakov and Shindin4, Vilemas Poskas and Kaupas5, Li and Jackson6, Eckert and Diaguila7, Khosla, Hoffman and Pollock8, Brown and Gauvin9, Axcell and Hall10 and the review paper of Jackson, Cotton and Axcell11. In the case of upward ow, it has been found that, with, onset of buoyancy in uence, the effectiveness of heat transfer is lower than for buoyancy-free forced convection under corresponding conditions. Severe impairment of heat transfer develops as buoyancy in uences build up. This is due to the turbulence production being reduced, an effect that has been described as ‘buoyancy-induced laminarization’12. With further increase in buoyancy in uence, the effectiveness of heat transfer improves as a result of turbulence production recovering. When the buoyancy in uence is very strong, heat transfer can even become enhanced in relation to that for forced convection under corresponding conditions. In the case of downward ow, systematic enhancement of heat transfer occurs with increase of buoyancy in uence due to turbulence production being increased. The semi-empirical model of fully developed mixed convection heat transfer in circular tubes, described in the introductory section of the rst of the three related
EXPERIMENTAL FACILITY Figure 1 shows the test section and ow circuit used in the study. The test section (see Figure 1(a)) is a vertical passage of rectangular cross section 612 mm by 80 mm and height 4.0 m. The main part of it has one wall heated by twenty separate heaters and the opposite one and the side walls unheated. The walls are made of stainless steel sheet and the inside surfaces have a polished mirror-like nish. Their emissivity is very low (about 0.12) and radiative heat 252
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Figure 1. Experimental facility.
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transfer is of only secondary importance. The ow circuit shown in Figure 1(b) is for upward ow in the test section. As can be seen, air from the laboratory is drawn into the test section at the bottom and passes upwards through a short unheated section of length 1.0 m. This contains some honeycomb material to straighten the ow. The air then passes through the heated section which is of length 2.5 m. Above this is a further short unheated section. On leaving the passage at the top, the air ows through an ori ce plate ow-metering section and then, after mixing with additional air from the laboratory to lower its temperature, it enters a blower and is nally discharged from the laboratory. The ow circuit arrangement shown in Figure 1(b) could be readily modi ed to give downward ow through the test section. The power supplied to each of the electrical heaters on the test section could be controlled automatically by means of a multi-channel computer-based system so as to enable prescribed axial distributions of temperature to be achieved on that wall and local measurements of the heat input made. Alternatively, prescribed distributions of heat input could be applied. In the present study it was used in this mode with uniform heat input. A 208 channel Intercole system was used in conjunction with a Pentium PC for data acquisition. Software was developed which combined the data logging, power control, measurement and data processing activities in one single package. A total of 143 thermocouples were used to measure temperature on the test section walls. There were 83 on the heated wall, 36 on the opposite adiabatic wall and 24 on the two side walls. An arbitrary upper limit of about 300± C was imposed on the temperature of the heated wall, mainly to avoid changing the emissivity of the surface. Specially-designed thermal insulation packs of high resistance were installed on the outside of both the unheated and heated walls of the test section. The heat losses through the insulation were calculated using the measured temperature differences across low conductivity panels of known thermal resistance. The radiative heat transfer within the test section was quite small in relation to convective heat transfer under most of the conditions of the present experiments. It was calculated knowing the emissivity of the surfaces and the measured distributions of wall temperature. The radiative heat transfer and the heat loss were then subtracted from the measured power input to each of the heaters to give local values of convective heat ux from the heated surface to the air owing through the passage. Knowing the wall temperature distribution and the local bulk temperature (obtained from an energy balance), local values of convective heat transfer coef cient could be reliably determined to an accuracy of better than 10%.
meters was twice the spacing between the heated and unheated walls. RESULTS AND DISCUSSION In order to quantify the extent to which heat transfer was affected by in uences of buoyancy in these experiments, it was rst necessary to establish a reliable, forced convection base. The authors chose to do this using an equation of the Dittus Boelter form combined with an axial development function and a wall to bulk temperature ratio correction term to account for the effects of uid property variations. Using results from the present study obtained under conditions where buoyancy in uences were negligible, the following variable property, forced convection correlation equation was established: ³ ´¡0:34 T Nu = Ctherm £ 0:0228Re0:79 Pr0:4 w (1) Tb in which the axial development function Ctherm is given by: " ³ ´¡0:7 #³ ´¡0:29 x 5520 x Ctherem = 1:0+ 0:69+ Re De De (2) ³ ´ x £exp ¡0:07 : De This function is of the same form as that used by Petukhov et al.13 to describe thermal development in the case of forced convection in circular tubes. However, the constants and indices have been adjusted so as to make it describe the axial development found in the present study, which in this case is due to both ow development and thermal development. Figure 2 shows the variation of this function with non-dimensional axial distance from the start of heating and also that given by the function of Petukhov et al. for circular tubes. As can be seen, they are very different. It is apparent that a fully developed condition was not achieved with the relatively short test section used in the present study.
EXPERIMENTAL INVESTIGATION A detailed programme of experiments was completed which covered conditions ranging from forced convection with negligible buoyancy in uence through to buoyancy dominated mixed convection. The Reynolds number Re was varied from about 44,000 down to 7000 and the Grashof number Gr* from 3 £ 108 to 9 £ 109, giving values of the buoyancy parameter Bo* from about 10¡7 to 10¡4. The characteristic dimension used in these dimensionless para-
Figure 2. Comparison between axial development in the present con guration and in a circular tube.
Trans IChemE, Vol 80, Part A, April 2002
MIXED CONVECTION HEAT TRANSFER: PART 3 Figure 3 shows comparisons between axial distributions of Nusselt number from four experiments with negligible buoyancy in uence (Bo* · 5 £ 10¡7) and those calculated using equation (1). It should be noted that the uid properties in Nu, Re and Pr were evaluated at the local bulk temperature. As can be seen, the agreement between the experimental results and the calculated distributions is excellent. On the basis of such comparisons the authors are con dent that the forced convection correlation equation provides a reliable base against which the mixed convection results can be compared. Figures 4 to 8 show distributions of Nusselt number from experiments with upward ow for a series of Reynolds
numbers from about 30,000 down to 7000 in which buoyancy in uences were progressively more signi cant. In each case, results are presented for three values of Grashof number. Also shown are the corresponding distributions for conditions of forced convection (negligible buoyancy in uence) calculated using equation (1). It can be seen from Figure 4 that, even at a relatively high value of Reynolds number of 30,000, the effectiveness of heat transfer deteriorates noticeably with increase of x=De due to the in uence of buoyancy for the two higher values of Grashof number. At a Reynolds number of about 20,000 (see Figure 5), such effects are even more apparent.
Figure 3. Distributions of Nusselt number of conditions of negligible buoyancy in uence, Bo* · 5 £ 10¡ 7.
Figure 4. Distributions of Nusselt number with upward ow for Re º 30,000.
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Figure 5. Distributions of Nusselt number with upward ow for Re º 20,000.
Figure 6. Distributions of Nusselt number with upward ow for Re º 15,000.
When the Reynolds number is reduced to about 15,000 (see Figure 6), there are signs of recovery of heat transfer in the case of the highest value of Grashof number. This is more clearly evident in the case of the results for a Reynolds number of about 10,000 shown in Figure 7. There, some enhancement of heat transfer in relation to that for forced convection is seen to be developing near the downstream end of the test section at the highest value of Grashof number. When the Reynolds number is decreased further to about 7000 (see Figure 8), this enhancement occurs in the case of the intermediate value of Grashof number. In the case of the experiment with the highest value Grashof number, strong enhancement of heat transfer occurs, even in the upstream region. Figures 9, 10 and 11 show distributions of Nusselt number from experiments with buoyancy-in uenced down-
ward ow. Also shown are the corresponding distributions for conditions of forced convection (negligible buoyancy in uence) calculated using equation (1). It can be seen that downstream of the thermal development region heat transfer is systematically enhanced with increase of buoyancy in uence and a fully developed condition is approached. CORRELATION OF DATA Figure 12 shows the experimental data for upward and downward ow presented in terms of Nusselt number ratio, Nu=Nuf, and buoyancy parameter, Bo*. The axial location is x=De = 14.4. In the case of upward ow, the values of normalized Nusselt number ratio decrease gradually with increase, of buoyancy parameter beyond a value of about 10¡6. A minimum is reached when the buoyancy parameter Trans IChemE, Vol 80, Part A, April 2002
MIXED CONVECTION HEAT TRANSFER: PART 3
Figure 7. Distributions of Nusselt number with upward ow for Re º 10,000.
Figure 8. Distributions of Nusselt number with upward ow for Re º 7000.
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Figure 9. Nusselt number distribution with downward ow for Bo* = 2.84 £ 10¡ 6.
Figure 10. Nusselt number distribution with downward ow for Bo* = 2.84 £ 10¡ 5.
Figure 11. Nusselt number distribution with downward ow for Bo* = 1.44 £ 10¡ 4.
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Figure 12. In uences of buoyancy on heat transfer at location x=De = 14.4.
exceeds 10¡5. The impairment of heat transfer is then about 35%. Beyond this value of buoyancy parameter there is a systematic recovery of heat transfer, with enhancement of heat transfer occurring for values above 5 £ 10¡5. In the case of downward ow there is a systematic enhancement of heat transfer with increase of buoyancy parameter. The values of Nusselt number ratio for upward and downward ow approach each other at the highest values of buoyancy parameter achieved. It is apparent that, with a relatively short plane channel of the kind used in the present study, having one wall heated and the opposite one adiabatic, heat transfer can be signi cantly affected by buoyancy. It is either impaired or enhanced depending on the strength of the buoyancy in uence and the ow direction. Also shown in the Figure 12 is the prediction of the semiempirical model of fully developed mixed convection in vertical circular tubes, referred to earlier in this paper, and described in the introduction of the rst of the three related papers on mixed convection appearing in this volume. As can be seen, the mixed convection behaviour in the present experiments is broadly similar to that found with circular tubes but there are some important differences. The onset of the buoyancyeffects is delayed. In the case of upward ow, the effects develop more gradually than with circular tubes and the maximum impairment of heat transfer is not as great. These trends are closely consistentwith what was found in the investigations reported in the two preceding papers.
broadly reinforce the existing picture of the effects buoyancy-aided, mixed convection heat transfer in vertical passages which has hitherto been largely based on experiments with uniformly heated circular tubes. However, as in the case of the related studies reported in the two preceding papers, there are some clear differences between the behaviour observed and that found with circular tubes. The onset of buoyancy in uence on heat transfer is delayed. In the case of upward ow impairment of heat transfer occurs much more gradually and the maximum impairment is reduced. NOMENCLATURE *
Bo cp De Gr* Nu Nuf Pr qc Re Tb Tw ub x
buoyancy parameter, Gr*=Re3.425Pr0.8 speci c heat capacity characteristic dimension of passage Grashof number based on wall heat ux, Gr* = bgDe4qc=ln2 local Nusselt number, aD=l forced convection Nusselt number Prandtl number, mcp=l convective heat ux Reynolds number, ubDe=n bulk temperature wall temperature bulk velocity axial distance from start of heating
Greek symbols a heat transfer coef cient, a = qc=(Tw 7 Tb) b coef cient of volume expansion l thermal conductivity m dynamic viscosity n kinematic viscosity
CONCLUSIONS For the con guration studied in the present investigation, a relatively short plane passage with one wall uniformly heated and the opposite one adiabatic, the results obtained Trans IChemE, Vol 80, Part A, April 2002
REFERENCES 1. Steiner, A. A., 1971, On the reverse transition of turbulent ow under the action of buoyancy forces, J Fluid Mech, 47: 71–75.
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2. Byrne, J. E. and Ejiogu, E., 1971, Combined free and forced convection heat transfer in a vertical pipe, Symp Heat Mass Trans Combined Forc Nat Conv, C118=71. 3. Carr, A. D., Connor, M. A. and Buhr, H. O., 1973, Velocity, temperature and turbulence measurements in air pipe ow with combined free and forced convection, Trans ASME J Heat Trans, 95: 445–452. 4. Polyakov, A. F. and Shindin, S. A., 1988, Development of heat transfer along vertical tubes in the presence of mixed air convection, Int J Heat Mass Trans, 31: 987–992. 5. Vilemas, J. V., Poskas, P. S. and Kaupas, V. E., 1992, Local heat transfer in a vertical gas-cooled tube with turbulent mixed convection and different heat uxes, Int J Heat Mass Trans, 35: 2421–2428. 6. Li, J. and Jackson, J. D., 1998, Buoyancy-in uenced variable property turbulent heat transfer to air owing in a uniformly heated vertical tube, Proc 2nd EF Inter Conf Turbulent Heat Trans. 7. Eckert, E. R. and Diaguila, A. J., 1954, Convective heat transfer for mixed, free, and forced ow through tubes, Trans ASME, 76: 497–504. 8. Khosla, J., Hoffman, T. W. and Pollock, K. G., 1974, Combined forced and natural convective heat transfer to air in a vertical tube, Proc 5th Int Heat Trans Conf. 9. Brown, C. K. and Gauvin, W. H., 1965, Combined free and forced convection heat transfer in opposing ow, Can J Chem Eng, 313–318. 10. Axcell, B. P. and Hall, W. B., 1978, Mixed convection to air in a vertical pipe, 6th Inter Heat Trans Conf, 1: 37–42.
11. Jackson, J. D., Cotton, M. A. and Axcell, B. P., 1989, Studies of mixed convection in vertical tubes, Int J Heat and Fluid Flow, 10: 2–15. 12. Hall, W. B. and Jackson, J. D., 1969, Laminarization of a turbulent pipe ow by buoyancy forces, ASME Paper, No. 69-HT-55. 13. Petukhov, B. S., Kurganov, V. A. and Gladuntsov, A. I., Turbulent heat transfer in tubes to gases with variable physical properties, Heat Mass Trans, 1: 117–127.
ACKNOWLEDGEMENT The work reported here was supported by research contract TE=G=00458=N (Analysis of air ow round RPV gas ducts), funded by BNFL Magnox Generation.
ADDRESS Correspondence concerning this paper should be addressed to Emeritus Professor J. D. Jackson, School of Engineering, University of Manchester, Oxford Road, Manchester, M13 9PL, UK. E-mail: [email protected] The manuscript was received 1 November 2001 and accepted for publication 11 January 2002.
Trans IChemE, Vol 80, Part A, April 2002
MIXED CONVECTION HEAT TRANSFER TO AIR FLOWING UPWARDS THROUGH A VERTICAL PLANE PASSAGE: PART 3 J. WANG1 , J. LI2 and J. D. JACKSON1 1 School of Engineering, University of Manchester, UK. Department of Mechanical Engineering, UMIST, Manchester, UK.
2
T
his is the third of three related papers presented in this volume on mixed convection heat transfer in vertical passages. In it an experimental study is reported of buoyancyin uenced convective heat transfer to air owing upwards and downwards under turbulent conditions through a vertical plane passage of relatively short length with a uniform heat input on one wall and an adiabatic condition on the opposite one. Thermal development and ow development occurred simultaneously and neither were complete. As in the case of the studies reported in the two preceding papers, the results presented here reinforce the existing picture of buoyancy-aided mixed convection in vertical passages, which hitherto has been based largely on experiments performed with uniformly heated circular tubes. However, the effects of buoyancy were slightly weaker than for circular tubes, onset of buoyancy-induced impairment of heat transfer was delayed and it occured more gradually. These trends are consistent with the ndings reported in the two preceding papers and can again be explained by the fact that only one surface was heated and, therefore, turbulence was only modi ed directly by buoyancy in one of the two boundary layers. Keywords: mixed convection; heat transfer; buoyancy in uence; plane passage; air.
INTRODUCTION
papers on mixed convection presented in this volume, has proved to be quite successful in describing the behaviour outlined above. It relates the ratio of the Nusselt number, Nu, for buoyancy-in uenced conditions to that for forced conditions Nuf to a buoyancy parameter, Bo*, which combines Grashof number, Gr*, Reynolds number, Re, and Prandtl number, Pr, in the form Gr*=Re3.425Pr0.8 so as to characterize the strength of buoyancy in uences. No information can be found in the literature on buoyancy-in uenced convective heat transfer for an arrangement such as the one used in the present investigation, a relatively short plane passage with one wall uniformly heated and the opposite one adiabatic. The aim of the study was to see to what extent the effects of buoyancy and variable properties in such a con guration would prove to be similar to those found with circular tubes.
Buoyancy-aided convective heat transfer to air owing in vertical tubes has been studied extensively, see for example, Steiner1, Byrne and Ejiogu2, Carr, Connor and Buhr3, Polyakov and Shindin4, Vilemas Poskas and Kaupas5, Li and Jackson6, Eckert and Diaguila7, Khosla, Hoffman and Pollock8, Brown and Gauvin9, Axcell and Hall10 and the review paper of Jackson, Cotton and Axcell11. In the case of upward ow, it has been found that, with, onset of buoyancy in uence, the effectiveness of heat transfer is lower than for buoyancy-free forced convection under corresponding conditions. Severe impairment of heat transfer develops as buoyancy in uences build up. This is due to the turbulence production being reduced, an effect that has been described as ‘buoyancy-induced laminarization’12. With further increase in buoyancy in uence, the effectiveness of heat transfer improves as a result of turbulence production recovering. When the buoyancy in uence is very strong, heat transfer can even become enhanced in relation to that for forced convection under corresponding conditions. In the case of downward ow, systematic enhancement of heat transfer occurs with increase of buoyancy in uence due to turbulence production being increased. The semi-empirical model of fully developed mixed convection heat transfer in circular tubes, described in the introductory section of the rst of the three related
EXPERIMENTAL FACILITY Figure 1 shows the test section and ow circuit used in the study. The test section (see Figure 1(a)) is a vertical passage of rectangular cross section 612 mm by 80 mm and height 4.0 m. The main part of it has one wall heated by twenty separate heaters and the opposite one and the side walls unheated. The walls are made of stainless steel sheet and the inside surfaces have a polished mirror-like nish. Their emissivity is very low (about 0.12) and radiative heat 252
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Figure 1. Experimental facility.
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transfer is of only secondary importance. The ow circuit shown in Figure 1(b) is for upward ow in the test section. As can be seen, air from the laboratory is drawn into the test section at the bottom and passes upwards through a short unheated section of length 1.0 m. This contains some honeycomb material to straighten the ow. The air then passes through the heated section which is of length 2.5 m. Above this is a further short unheated section. On leaving the passage at the top, the air ows through an ori ce plate ow-metering section and then, after mixing with additional air from the laboratory to lower its temperature, it enters a blower and is nally discharged from the laboratory. The ow circuit arrangement shown in Figure 1(b) could be readily modi ed to give downward ow through the test section. The power supplied to each of the electrical heaters on the test section could be controlled automatically by means of a multi-channel computer-based system so as to enable prescribed axial distributions of temperature to be achieved on that wall and local measurements of the heat input made. Alternatively, prescribed distributions of heat input could be applied. In the present study it was used in this mode with uniform heat input. A 208 channel Intercole system was used in conjunction with a Pentium PC for data acquisition. Software was developed which combined the data logging, power control, measurement and data processing activities in one single package. A total of 143 thermocouples were used to measure temperature on the test section walls. There were 83 on the heated wall, 36 on the opposite adiabatic wall and 24 on the two side walls. An arbitrary upper limit of about 300± C was imposed on the temperature of the heated wall, mainly to avoid changing the emissivity of the surface. Specially-designed thermal insulation packs of high resistance were installed on the outside of both the unheated and heated walls of the test section. The heat losses through the insulation were calculated using the measured temperature differences across low conductivity panels of known thermal resistance. The radiative heat transfer within the test section was quite small in relation to convective heat transfer under most of the conditions of the present experiments. It was calculated knowing the emissivity of the surfaces and the measured distributions of wall temperature. The radiative heat transfer and the heat loss were then subtracted from the measured power input to each of the heaters to give local values of convective heat ux from the heated surface to the air owing through the passage. Knowing the wall temperature distribution and the local bulk temperature (obtained from an energy balance), local values of convective heat transfer coef cient could be reliably determined to an accuracy of better than 10%.
meters was twice the spacing between the heated and unheated walls. RESULTS AND DISCUSSION In order to quantify the extent to which heat transfer was affected by in uences of buoyancy in these experiments, it was rst necessary to establish a reliable, forced convection base. The authors chose to do this using an equation of the Dittus Boelter form combined with an axial development function and a wall to bulk temperature ratio correction term to account for the effects of uid property variations. Using results from the present study obtained under conditions where buoyancy in uences were negligible, the following variable property, forced convection correlation equation was established: ³ ´¡0:34 T Nu = Ctherm £ 0:0228Re0:79 Pr0:4 w (1) Tb in which the axial development function Ctherm is given by: " ³ ´¡0:7 #³ ´¡0:29 x 5520 x Ctherem = 1:0+ 0:69+ Re De De (2) ³ ´ x £exp ¡0:07 : De This function is of the same form as that used by Petukhov et al.13 to describe thermal development in the case of forced convection in circular tubes. However, the constants and indices have been adjusted so as to make it describe the axial development found in the present study, which in this case is due to both ow development and thermal development. Figure 2 shows the variation of this function with non-dimensional axial distance from the start of heating and also that given by the function of Petukhov et al. for circular tubes. As can be seen, they are very different. It is apparent that a fully developed condition was not achieved with the relatively short test section used in the present study.
EXPERIMENTAL INVESTIGATION A detailed programme of experiments was completed which covered conditions ranging from forced convection with negligible buoyancy in uence through to buoyancy dominated mixed convection. The Reynolds number Re was varied from about 44,000 down to 7000 and the Grashof number Gr* from 3 £ 108 to 9 £ 109, giving values of the buoyancy parameter Bo* from about 10¡7 to 10¡4. The characteristic dimension used in these dimensionless para-
Figure 2. Comparison between axial development in the present con guration and in a circular tube.
Trans IChemE, Vol 80, Part A, April 2002
MIXED CONVECTION HEAT TRANSFER: PART 3 Figure 3 shows comparisons between axial distributions of Nusselt number from four experiments with negligible buoyancy in uence (Bo* · 5 £ 10¡7) and those calculated using equation (1). It should be noted that the uid properties in Nu, Re and Pr were evaluated at the local bulk temperature. As can be seen, the agreement between the experimental results and the calculated distributions is excellent. On the basis of such comparisons the authors are con dent that the forced convection correlation equation provides a reliable base against which the mixed convection results can be compared. Figures 4 to 8 show distributions of Nusselt number from experiments with upward ow for a series of Reynolds
numbers from about 30,000 down to 7000 in which buoyancy in uences were progressively more signi cant. In each case, results are presented for three values of Grashof number. Also shown are the corresponding distributions for conditions of forced convection (negligible buoyancy in uence) calculated using equation (1). It can be seen from Figure 4 that, even at a relatively high value of Reynolds number of 30,000, the effectiveness of heat transfer deteriorates noticeably with increase of x=De due to the in uence of buoyancy for the two higher values of Grashof number. At a Reynolds number of about 20,000 (see Figure 5), such effects are even more apparent.
Figure 3. Distributions of Nusselt number of conditions of negligible buoyancy in uence, Bo* · 5 £ 10¡ 7.
Figure 4. Distributions of Nusselt number with upward ow for Re º 30,000.
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Figure 5. Distributions of Nusselt number with upward ow for Re º 20,000.
Figure 6. Distributions of Nusselt number with upward ow for Re º 15,000.
When the Reynolds number is reduced to about 15,000 (see Figure 6), there are signs of recovery of heat transfer in the case of the highest value of Grashof number. This is more clearly evident in the case of the results for a Reynolds number of about 10,000 shown in Figure 7. There, some enhancement of heat transfer in relation to that for forced convection is seen to be developing near the downstream end of the test section at the highest value of Grashof number. When the Reynolds number is decreased further to about 7000 (see Figure 8), this enhancement occurs in the case of the intermediate value of Grashof number. In the case of the experiment with the highest value Grashof number, strong enhancement of heat transfer occurs, even in the upstream region. Figures 9, 10 and 11 show distributions of Nusselt number from experiments with buoyancy-in uenced down-
ward ow. Also shown are the corresponding distributions for conditions of forced convection (negligible buoyancy in uence) calculated using equation (1). It can be seen that downstream of the thermal development region heat transfer is systematically enhanced with increase of buoyancy in uence and a fully developed condition is approached. CORRELATION OF DATA Figure 12 shows the experimental data for upward and downward ow presented in terms of Nusselt number ratio, Nu=Nuf, and buoyancy parameter, Bo*. The axial location is x=De = 14.4. In the case of upward ow, the values of normalized Nusselt number ratio decrease gradually with increase, of buoyancy parameter beyond a value of about 10¡6. A minimum is reached when the buoyancy parameter Trans IChemE, Vol 80, Part A, April 2002
MIXED CONVECTION HEAT TRANSFER: PART 3
Figure 7. Distributions of Nusselt number with upward ow for Re º 10,000.
Figure 8. Distributions of Nusselt number with upward ow for Re º 7000.
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Figure 9. Nusselt number distribution with downward ow for Bo* = 2.84 £ 10¡ 6.
Figure 10. Nusselt number distribution with downward ow for Bo* = 2.84 £ 10¡ 5.
Figure 11. Nusselt number distribution with downward ow for Bo* = 1.44 £ 10¡ 4.
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Figure 12. In uences of buoyancy on heat transfer at location x=De = 14.4.
exceeds 10¡5. The impairment of heat transfer is then about 35%. Beyond this value of buoyancy parameter there is a systematic recovery of heat transfer, with enhancement of heat transfer occurring for values above 5 £ 10¡5. In the case of downward ow there is a systematic enhancement of heat transfer with increase of buoyancy parameter. The values of Nusselt number ratio for upward and downward ow approach each other at the highest values of buoyancy parameter achieved. It is apparent that, with a relatively short plane channel of the kind used in the present study, having one wall heated and the opposite one adiabatic, heat transfer can be signi cantly affected by buoyancy. It is either impaired or enhanced depending on the strength of the buoyancy in uence and the ow direction. Also shown in the Figure 12 is the prediction of the semiempirical model of fully developed mixed convection in vertical circular tubes, referred to earlier in this paper, and described in the introduction of the rst of the three related papers on mixed convection appearing in this volume. As can be seen, the mixed convection behaviour in the present experiments is broadly similar to that found with circular tubes but there are some important differences. The onset of the buoyancyeffects is delayed. In the case of upward ow, the effects develop more gradually than with circular tubes and the maximum impairment of heat transfer is not as great. These trends are closely consistentwith what was found in the investigations reported in the two preceding papers.
broadly reinforce the existing picture of the effects buoyancy-aided, mixed convection heat transfer in vertical passages which has hitherto been largely based on experiments with uniformly heated circular tubes. However, as in the case of the related studies reported in the two preceding papers, there are some clear differences between the behaviour observed and that found with circular tubes. The onset of buoyancy in uence on heat transfer is delayed. In the case of upward ow impairment of heat transfer occurs much more gradually and the maximum impairment is reduced. NOMENCLATURE *
Bo cp De Gr* Nu Nuf Pr qc Re Tb Tw ub x
buoyancy parameter, Gr*=Re3.425Pr0.8 speci c heat capacity characteristic dimension of passage Grashof number based on wall heat ux, Gr* = bgDe4qc=ln2 local Nusselt number, aD=l forced convection Nusselt number Prandtl number, mcp=l convective heat ux Reynolds number, ubDe=n bulk temperature wall temperature bulk velocity axial distance from start of heating
Greek symbols a heat transfer coef cient, a = qc=(Tw 7 Tb) b coef cient of volume expansion l thermal conductivity m dynamic viscosity n kinematic viscosity
CONCLUSIONS For the con guration studied in the present investigation, a relatively short plane passage with one wall uniformly heated and the opposite one adiabatic, the results obtained Trans IChemE, Vol 80, Part A, April 2002
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ACKNOWLEDGEMENT The work reported here was supported by research contract TE=G=00458=N (Analysis of air ow round RPV gas ducts), funded by BNFL Magnox Generation.
ADDRESS Correspondence concerning this paper should be addressed to Emeritus Professor J. D. Jackson, School of Engineering, University of Manchester, Oxford Road, Manchester, M13 9PL, UK. E-mail: [email protected] The manuscript was received 1 November 2001 and accepted for publication 11 January 2002.
Trans IChemE, Vol 80, Part A, April 2002