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Heat transfer in a small gap between co-axial rotating cylinders
International Communications in Heat and Mass Transfer 33 (2006) 737 – 743 www.elsevier.com/locate/ichmt
Heat transfer in a small gap between co-axial rotating cylinders☆ Sheng-Chung Tzeng Department of Mechanical Engineering, ChienKuo Technology University, Changhua, Changhua 500, Taiwan, ROC Available online 27 March 2006
Abstract This paper investigates the local heat transfer of a co-axial rotating cylinder. In the inner flow field of the rotating cylinder, the dimensionless parameters include the rotational Reynolds number (ReΩ) and buoyancy parameter (Gr). The test rig is designed to make the rotating in the inner cylinder and stationary in the outer cylinder. The local temperature distributions of the inner and outer cylinder on axial direction were measured. Under the experimental condition, whereas the ranges of the rotational Reynolds number are 2400 ≤ ReΩ ≤ 45,000. Experimental results reveal that the rotational Reynolds number's increase is with the heat transfer coefficient distributions increase types. Finally, the local heat transfer rate on the wall are correlated and compared with that in the existing literature. © 2006 Elsevier Ltd. All rights reserved. Keywords: Co-axial rotating cylinder; Rotation Reynolds number; Buoyancy parameter
1. Introduction The conversion of many dynamics is achieved by rotation, whereas the heat transfer of rotating machines is one of the important reference factors for mechanical designers. So, in an effort to avoid damage of components arising from local high temperature due to imperfect design, the designers shall take into account the factors of heat transfer, and pay particular attentions to the shape, rotational speed and gap of the inner and outer cylinders of rotating machines. The space between two cylinders, rotational speed of cylinders and physical property of fluids are defined as the dimensionless parameters, referred to as the rotational Reynolds number. Some recent research in co-axial rotating cylinders with a variety of geometry, dimensions and dimensionless parameters are explored. Mohanty et al. [1–3] observed the heat transfer effect of the rotating cylinders with different aspect ratios under different rotational speeds and cross flows. Becker and Kaye [4,5] analyzed the influence of a radial temperature gradient on the instability of fluid flow in an annulus with an inner rotating cylinder. The results of the analysis show that heating of the inner rotating cylinder stabilizes the flow, while heating of the outer stationary cylinder destabilizes the flow. Jakoby et al. [6] correlated the convection heat transfer in annular channels with rotating inner cylinder. Their results present a study of the flow and heat transfer in annular channels with rotating inner cylinder and axial through-flow. The research of Gardiner and Sabersky [7] shows that the experimental research on heat transfer within the circular gap of two cylinders could provide a basis for the design of the cooling system ☆
Communicated by W.J. Minkowycz. E-mail address: [email protected].
0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.02.012
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S.-C. Tzeng / International Communications in Heat and Mass Transfer 33 (2006) 737–743
Nomenclature D Gr h Kf Nu Pr qin ReΩ T X
gap of the inner and outer cylinder, mm Grashof number, gβ (Tw − Tb)D3 / ν2 local heat transfer coefficient, h = qnet / (Tw − Tb), W/m2 K thermal conductivity of fluid Nusselt number, Nu = hD / Kf Prandtl number, Pr = ν / α wall heat flux per unit area, W/cm2 rotational Reynolds number, ReΩ = ΩD2 / ν temperature, K axial local position
Greek letters α thermal diffusivity β coefficient of thermal expansion ν kinematic viscosity Ω rotational speed Superscript ð Þ mean Subscript f fluid o stationary w wall b bulk
of high-power electric motor. Lee and Minkowycz [8] studied the heat transfer characteristics of the annulus of two co-axial cylinders with one cylinder rotating: a bigger space is helpful to heat transfer and pressure drop when the fluids flow through the annulus. Kataoka et al. [9] investigated the heat/mass transfer in the Taylor vortex flow with constant axial flow rates. When the Reynolds number of the axial flow is raised gradually at a fixed value of the rotational parameter of the rotating flow. As the driving mode of co-axial rotating cylinder with rotating inner cylinder and stationary outer cylinder is widely used for industrial applications, such as the design of high-speed main shaft of high-speed machines, or the design of rotating machines including the rotating blade couplers (RBC) of four-wheel drive vehicles, gap bearings, electric motors and turbine motors. Besides, it's also suitable for the winding design of rotating heat exchangers, paper-making and electroplating engineering, or even the manufacturing and design of semiconductor components, etc. Therefore, a deep understanding of the heat transfer within the flow field of co-axial rotating cylinders can help improve the performance of related components and offer baseline information for design purpose. This research aims to find out the impact of the rotating effect of rotating cylinders on heat transfer when the flow fields are available with different input power and rotational Reynolds numbers. 2. Experimental set-up and test section detailed This experiment is specifically designed for local heat transfer of the co-axial rotating cylinder, with experimental set-up and test section detailed as shown in Fig. 1. Most of the structures comprise of four parts: the rotating main shaft system, test section, data collection system and heating system. The major rotating power supply of testing the main shaft is a 5 Hp AC motor, in conjunction with the rotational speed and direction of the inveter control mainshaft. The dynamic of the AC motor is transferred by a V-type dual-groove belt pulley, and the measured rotational speed is read by a screened tachometer.
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Fig. 1. Experiment setup.
Fig. 2 plots the test section and detailed thermocouple positions, which are also classified into the rotating inner cylinder and stationary outer cylinder. Constructed of Teflon materials, the rotating inner cylinder features extremely low heat transfer coefficient, whereby it's possible to reduce the radial and axial heat transfer within the test section for a declined heat loss. The rotating inner test section is a hollow cylinder sized in 120 mm (both in diameter and length), which is axially built-in 12 thermocouples and heating power lines for temperature measurement and heating of inner test section. Constructed of bakelite materials, the stationary outer cylinder comprises of a front cover, outer test section and rear cover, of which the front cover has a bearing and leak-proof liner connected to the main shaft, and the outer test section is a
Fig. 2. Test section and detailed thermocouple positions.
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cylinder with an inner diameter 134 mm, outer diameter 175 mm and length 165 mm. The measuring position of the stationary cylinder includes 63 temperature measuring points for the inner wall sized in 134 mm, which cover the axial and radial local temperature distributions. There are 16 points uniformly distributed at axial locations θ = 0° and 180°, and 31 points at location θ = 90°. On radial location θ = 0∼360°, a point is measured every 30°, totalized in 12 points for radial local temperature distribution. In such case, the temperature measured at such points will display the local temperature at this area, whereby it's possible to analyze the heat transfer characteristics and cooling of the co-axial rotating cylinder. This research measures the temperature of the test section of co-axial cylinder via TT-T-30SLE high-precision thermocouples. Every temperature measuring point is fixed by a composite thermosetting heat glue (OMEGABOND 200), which features high thermal conductivity and electrical insulation, etc. The signal of measuring points input into the data recorder, wherein potential difference signal is converted into actually measured local temperature value. The thermocouples of the rotating inner cylinder and data recorders are interfaced via data slip ring to transmit the temperature values. In the experiment, 0.01 mm stainless sheets are affixed on the surface of the rotating inner cylinder as heating strips. The composite thermosetting heat glue featuring high thermal conductivity and electrical insulation is fitted between Teflon heating strips, serving as adhesives fixed onto the heating section. For DC power supply for heating purpose, different powers are adjusted as per experimental conditions to achieve the centrifugal buoyancy effect. The electric circuit is heated by linking internal wires via brush holders and power slip ring. 3. Data reduction and uncertainty analysis The local heat transfer coefficient h was evaluated as the ratio of the net wall heat flux qnet to the temperature difference between the local wall temperature Tw and the local coolant bulk temperature Tb i.e. h = qnet / (Tw − Tb). The net wall heat flux from the cylinder wall to the air flow was obtained by subtracting the external heat loss from the electric power supplied to the film heaters. The heat loss test at each measurement point of the test section is obtained under the no-flow condition. The average value of heat loss with horizontal orientations of the test cylinder were taken. The measured electrical power input and steady state temperature determine the amount of heat loss at the measured temperature. By varying the electrical power input, the relationship between heat loss and temperature can be obtained. The local Nusselt number was calculated from the local heat transfer coefficient, gap of the inner and outer cylinder and thermal conductivity of air as Nu ¼ hD=kf
ð1Þ
The errors in the temperature reading can be obtained from the calibration of the thermocouples. The uncertainty was ± 0.1 °C from the readout of the data recorder. The calibration of flow rate was carried out with compressed air at 1 atmospheric pressure. The rotational speed was detected by a tachometer with a bit of oscillation. The maximum error of the rotational speed was 1%. The uncertainties of the air thermo-physical properties were included in the analysis. Uncertainties in the parameters were estimated by using the root–sum–square method of Kline and McClintock [10], and Moffat [11]. The measured value and its uncertainty can be expressed as R = R ± δR. The uncertainties of the experimental parameter are listed in Table 1, respectively. 4. Results and discussion Fig. 3 plots the local temperature distributions on the inner cylinder for the stationary condition (ReΩ = 0). For the steady-state temperature distribution of the rotating cylinder measured by the thermocouple in different heating conditions, the profile presents a
Table1 Experimental parameters and uncertainty analysis Experimental parameters
Uncertainty
Test sectional area (A) Temperature (T) Input power (Q) Rotational speed (n) Rotational Reynolds number (ReΩ) Nusselt number (Nu)
±0.36% ±1.52% ±1.68% ±1.0% ±3.35% ±2.45%
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Fig. 3. Local temperature distributions on the inner cylinder for the stationary condition.
temperature distribution under different heat flux. In the case of heat flux qin = 559.15 W/m2, the stabilized average temperature is 55.45 °C. In the case of heat flux qin = 1193.64 W/m2, the stabilized average temperature is 98.72 °C, showing a difference of 43.27 °C. In the stationary condition, no rotating effect of centrifugal force occurs, so the heat will be accumulated onto the rotating cylinder, leading to subsequent damage. However, given the fact of middle section with maximum temperature distribution. Due to an open space of 15 mm between the side wall and rotating cylinder, which is enough for thermal diffusion, the temperature distribution will decline at both sides, but rise sharply at the middle section due to a limited space of gap 7 mm for heat diffusion. The accuracy of the temperature measurement in this experiment can be demonstrated from the geometrical analysis of the test section. The Nusselt number with variations of ReΩ at rotating condition is plotted in Fig. 4. During rotation of the inner cylinder, the mean Nusselt number for the rotating condition will increase with the rising rotational Reynolds number, when comparing the
Fig. 4. Nusselt number with variations of the rotational Reynolds number.
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Fig. 5. Correlation on the mean Nusselt number ratio (to Nuo) with variations of Gr × ReΩ.
experimental results with the research findings of scholars Mohanty et al. [1] and Corne et al. [12]. Mohanty et al. [1] explored the local heat transfer on the rotating cylinder with length-to-diameter ratios varying from 3.8 to 6.5 under rotating condition. The rotational speed varied from 25 to 1400 rpm. The maximum values of rotational parameter attained were 4.6 × 104 and 2.7 × 104, the transport rate could be summarized as Nu ¼ 0:592 ðReX Þ0:515 . Cornet et al. [12] experimented with a 38.1 mm cylinder, 368.3 mm long at 43 rpm covering a Reynolds number range of 2 × 103 to 1 × 105 and summarized the results as Nu ¼ 0:234 ðReX Þ0:632 . In this research, the inner rotating test section is a cylinder sized in 120 mm (both in diameter and length), with a rotational speed of 63∼1170 rpm. Rotational Reynolds number ranges from 2.4 × 103 to 4.5 × 104. The empirical equation, NuX ¼ 7:764 ðReX Þ0:262 , can be obtained from the experimental data. As the working fluid is air in a small gap between the co-axial rotating cylinders, the property of the cooling fluid is also taken into account of this empirical equation. The empirical equation deducted from the experimental results is as follows: NuX ¼ 8:854Pr0:4 ðReX Þ0:262
ð2Þ
In this experiment, there are two influential factors for centrifugal buoyancy force: one is the centrifugal force triggered by rotation, the other one is the buoyancy effect arising from the temperature difference between the inner wall surface of the outer cylinder and heating inner cylinder. In this research, the Grashof number for buoyancy force effect and rotational Reynolds number for centrifugal force effect will be the key factors decisive to the inner thermal behavior of the entire co-axial rotating cylinder. Fig. 5 shows the correlation on mean Nusselt number ratio (to Nuo) with variations of rotating buoyancy parameter (Gr × ReΩ) of the three heat fluxes under the ten rotating speeds. To present more easily the thermal effect arising from rotation, all heat transfer coefficients were divided by those of the stationary condition, thereby separating the thermal buoyancy effect of the stationary flow field. In this figure, mean ratios of the Nusselt number will increase with the rising centrifugal buoyancy force, and the correction equation established by the power law type for all experimental data is as follows: NuX ¼ 0:375ðGr ReX 1011 Þ0:328 Nuo
ð3Þ
5. Conclusion This research is intended to measure the temperature distributions against different rotational speeds and heat flux, as well as the impact of the rotating number on heat transfer under a co-axial rotating cylinder with the rotating inner cylinder and stationary outer cylinder. It can be found from the case of rotation of the inner cylinder, the heat transfer of
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the flow field will increase with the rising rotational Reynolds number under the interaction of the centrifugal buoyancy force. This research has successfully explored the thermal behavior in a small gap between the co-axial rotating cylinders, and established an empirical formula for the experiment covering the relationship between the parameters of centrifugal force/buoyancy force and heat transfer coefficient, which may be referenced by the designers or engineers in the field of rotating cylinder and components. Acknowledgement The author would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC 94-2212-E-270-002. References [1] A.K. Mohanty, A.A. Tawfek, B.V.S.S.S. Prasad, Heat transfer from a rotating cylinder in crossflow, Previews of Heat and Mass Transfer 21 (1995) 54–61. [2] B.V.S.S.S. Prasad, A.A. Tawfek, A.K. Mohanty, Heat transfer from a circular cylinder rotating about an orthogonal axis in quiescent air, Experiments in Fluids 10 (1991) 267–272. [3] A.A. Tawfek, B.V.S.S.S. Prasad, A.K. Mohanty, Pressure measurements around a rotating cylinder with and without crossflow, ASME Journal of Fluids Engineering 115 (1993) 526–528. [4] K.M. Becker, J. Kaye, The influence of a radial temperature gradient on the instability of fluid flow in an annulus with an inner rotating cylinder, Transaction of ASME Journal of Heat Transfer (1962) 106–110. [5] K.M. Becker, J. Kaye, Measurements of adiabatic flow in an annulus with an inner rotating cylinder, ASME Journal of Heat Transfer (1962) 97–105. [6] R. Jakoby, S. Kim, S. Wittig, Correlations of the convection heat transfer in annular channels with rotating inner cylinder, Transaction of ASME Journal of Engineering for Gas Turbines and Power 121 (1999) 670–677. [7] S.R.M. Gardiner, R.H. Sabersky, Heat transfer in an annular gap, International Journal of Heat Mass Transfer 21 (1977) 1459–1466. [8] Y.N. Lee, W.J. Minkowycz, Heat transfer characteristics of the annulus of two-coaxial cylinders with one cylinder rotating, International Journal of Heat Mass Transfer 32 (1989) 711–722. [9] K. Kataoka, H. Doi, T. Komai, Heat/mass transfer in Taylor vortex flow with constant axial flow rates, International Journal of Heat Mass Transfer 20 (1977) 57–63. [10] S.J. Kline, F.A. Mcclintock, Describing uncertainties in single-sample experiments, Mechanical Engineering (1953) 3–8. [11] R.J. Moffat, Contributions to the theory of single-sample uncertainty analysis, Transaction of ASME Journal of Fluids Engineering 104 (1986) 250–260. [12] I. Cornet, R. Greif, J.T. Teng, F. Roehler, Mass transfer to rotating rods and plates, International Journal of Heat Mass Transfer 23 (1980) 805–811.
Heat transfer in a small gap between co-axial rotating cylinders☆ Sheng-Chung Tzeng Department of Mechanical Engineering, ChienKuo Technology University, Changhua, Changhua 500, Taiwan, ROC Available online 27 March 2006
Abstract This paper investigates the local heat transfer of a co-axial rotating cylinder. In the inner flow field of the rotating cylinder, the dimensionless parameters include the rotational Reynolds number (ReΩ) and buoyancy parameter (Gr). The test rig is designed to make the rotating in the inner cylinder and stationary in the outer cylinder. The local temperature distributions of the inner and outer cylinder on axial direction were measured. Under the experimental condition, whereas the ranges of the rotational Reynolds number are 2400 ≤ ReΩ ≤ 45,000. Experimental results reveal that the rotational Reynolds number's increase is with the heat transfer coefficient distributions increase types. Finally, the local heat transfer rate on the wall are correlated and compared with that in the existing literature. © 2006 Elsevier Ltd. All rights reserved. Keywords: Co-axial rotating cylinder; Rotation Reynolds number; Buoyancy parameter
1. Introduction The conversion of many dynamics is achieved by rotation, whereas the heat transfer of rotating machines is one of the important reference factors for mechanical designers. So, in an effort to avoid damage of components arising from local high temperature due to imperfect design, the designers shall take into account the factors of heat transfer, and pay particular attentions to the shape, rotational speed and gap of the inner and outer cylinders of rotating machines. The space between two cylinders, rotational speed of cylinders and physical property of fluids are defined as the dimensionless parameters, referred to as the rotational Reynolds number. Some recent research in co-axial rotating cylinders with a variety of geometry, dimensions and dimensionless parameters are explored. Mohanty et al. [1–3] observed the heat transfer effect of the rotating cylinders with different aspect ratios under different rotational speeds and cross flows. Becker and Kaye [4,5] analyzed the influence of a radial temperature gradient on the instability of fluid flow in an annulus with an inner rotating cylinder. The results of the analysis show that heating of the inner rotating cylinder stabilizes the flow, while heating of the outer stationary cylinder destabilizes the flow. Jakoby et al. [6] correlated the convection heat transfer in annular channels with rotating inner cylinder. Their results present a study of the flow and heat transfer in annular channels with rotating inner cylinder and axial through-flow. The research of Gardiner and Sabersky [7] shows that the experimental research on heat transfer within the circular gap of two cylinders could provide a basis for the design of the cooling system ☆
Communicated by W.J. Minkowycz. E-mail address: [email protected].
0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.02.012
738
S.-C. Tzeng / International Communications in Heat and Mass Transfer 33 (2006) 737–743
Nomenclature D Gr h Kf Nu Pr qin ReΩ T X
gap of the inner and outer cylinder, mm Grashof number, gβ (Tw − Tb)D3 / ν2 local heat transfer coefficient, h = qnet / (Tw − Tb), W/m2 K thermal conductivity of fluid Nusselt number, Nu = hD / Kf Prandtl number, Pr = ν / α wall heat flux per unit area, W/cm2 rotational Reynolds number, ReΩ = ΩD2 / ν temperature, K axial local position
Greek letters α thermal diffusivity β coefficient of thermal expansion ν kinematic viscosity Ω rotational speed Superscript ð Þ mean Subscript f fluid o stationary w wall b bulk
of high-power electric motor. Lee and Minkowycz [8] studied the heat transfer characteristics of the annulus of two co-axial cylinders with one cylinder rotating: a bigger space is helpful to heat transfer and pressure drop when the fluids flow through the annulus. Kataoka et al. [9] investigated the heat/mass transfer in the Taylor vortex flow with constant axial flow rates. When the Reynolds number of the axial flow is raised gradually at a fixed value of the rotational parameter of the rotating flow. As the driving mode of co-axial rotating cylinder with rotating inner cylinder and stationary outer cylinder is widely used for industrial applications, such as the design of high-speed main shaft of high-speed machines, or the design of rotating machines including the rotating blade couplers (RBC) of four-wheel drive vehicles, gap bearings, electric motors and turbine motors. Besides, it's also suitable for the winding design of rotating heat exchangers, paper-making and electroplating engineering, or even the manufacturing and design of semiconductor components, etc. Therefore, a deep understanding of the heat transfer within the flow field of co-axial rotating cylinders can help improve the performance of related components and offer baseline information for design purpose. This research aims to find out the impact of the rotating effect of rotating cylinders on heat transfer when the flow fields are available with different input power and rotational Reynolds numbers. 2. Experimental set-up and test section detailed This experiment is specifically designed for local heat transfer of the co-axial rotating cylinder, with experimental set-up and test section detailed as shown in Fig. 1. Most of the structures comprise of four parts: the rotating main shaft system, test section, data collection system and heating system. The major rotating power supply of testing the main shaft is a 5 Hp AC motor, in conjunction with the rotational speed and direction of the inveter control mainshaft. The dynamic of the AC motor is transferred by a V-type dual-groove belt pulley, and the measured rotational speed is read by a screened tachometer.
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739
Fig. 1. Experiment setup.
Fig. 2 plots the test section and detailed thermocouple positions, which are also classified into the rotating inner cylinder and stationary outer cylinder. Constructed of Teflon materials, the rotating inner cylinder features extremely low heat transfer coefficient, whereby it's possible to reduce the radial and axial heat transfer within the test section for a declined heat loss. The rotating inner test section is a hollow cylinder sized in 120 mm (both in diameter and length), which is axially built-in 12 thermocouples and heating power lines for temperature measurement and heating of inner test section. Constructed of bakelite materials, the stationary outer cylinder comprises of a front cover, outer test section and rear cover, of which the front cover has a bearing and leak-proof liner connected to the main shaft, and the outer test section is a
Fig. 2. Test section and detailed thermocouple positions.
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cylinder with an inner diameter 134 mm, outer diameter 175 mm and length 165 mm. The measuring position of the stationary cylinder includes 63 temperature measuring points for the inner wall sized in 134 mm, which cover the axial and radial local temperature distributions. There are 16 points uniformly distributed at axial locations θ = 0° and 180°, and 31 points at location θ = 90°. On radial location θ = 0∼360°, a point is measured every 30°, totalized in 12 points for radial local temperature distribution. In such case, the temperature measured at such points will display the local temperature at this area, whereby it's possible to analyze the heat transfer characteristics and cooling of the co-axial rotating cylinder. This research measures the temperature of the test section of co-axial cylinder via TT-T-30SLE high-precision thermocouples. Every temperature measuring point is fixed by a composite thermosetting heat glue (OMEGABOND 200), which features high thermal conductivity and electrical insulation, etc. The signal of measuring points input into the data recorder, wherein potential difference signal is converted into actually measured local temperature value. The thermocouples of the rotating inner cylinder and data recorders are interfaced via data slip ring to transmit the temperature values. In the experiment, 0.01 mm stainless sheets are affixed on the surface of the rotating inner cylinder as heating strips. The composite thermosetting heat glue featuring high thermal conductivity and electrical insulation is fitted between Teflon heating strips, serving as adhesives fixed onto the heating section. For DC power supply for heating purpose, different powers are adjusted as per experimental conditions to achieve the centrifugal buoyancy effect. The electric circuit is heated by linking internal wires via brush holders and power slip ring. 3. Data reduction and uncertainty analysis The local heat transfer coefficient h was evaluated as the ratio of the net wall heat flux qnet to the temperature difference between the local wall temperature Tw and the local coolant bulk temperature Tb i.e. h = qnet / (Tw − Tb). The net wall heat flux from the cylinder wall to the air flow was obtained by subtracting the external heat loss from the electric power supplied to the film heaters. The heat loss test at each measurement point of the test section is obtained under the no-flow condition. The average value of heat loss with horizontal orientations of the test cylinder were taken. The measured electrical power input and steady state temperature determine the amount of heat loss at the measured temperature. By varying the electrical power input, the relationship between heat loss and temperature can be obtained. The local Nusselt number was calculated from the local heat transfer coefficient, gap of the inner and outer cylinder and thermal conductivity of air as Nu ¼ hD=kf
ð1Þ
The errors in the temperature reading can be obtained from the calibration of the thermocouples. The uncertainty was ± 0.1 °C from the readout of the data recorder. The calibration of flow rate was carried out with compressed air at 1 atmospheric pressure. The rotational speed was detected by a tachometer with a bit of oscillation. The maximum error of the rotational speed was 1%. The uncertainties of the air thermo-physical properties were included in the analysis. Uncertainties in the parameters were estimated by using the root–sum–square method of Kline and McClintock [10], and Moffat [11]. The measured value and its uncertainty can be expressed as R = R ± δR. The uncertainties of the experimental parameter are listed in Table 1, respectively. 4. Results and discussion Fig. 3 plots the local temperature distributions on the inner cylinder for the stationary condition (ReΩ = 0). For the steady-state temperature distribution of the rotating cylinder measured by the thermocouple in different heating conditions, the profile presents a
Table1 Experimental parameters and uncertainty analysis Experimental parameters
Uncertainty
Test sectional area (A) Temperature (T) Input power (Q) Rotational speed (n) Rotational Reynolds number (ReΩ) Nusselt number (Nu)
±0.36% ±1.52% ±1.68% ±1.0% ±3.35% ±2.45%
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Fig. 3. Local temperature distributions on the inner cylinder for the stationary condition.
temperature distribution under different heat flux. In the case of heat flux qin = 559.15 W/m2, the stabilized average temperature is 55.45 °C. In the case of heat flux qin = 1193.64 W/m2, the stabilized average temperature is 98.72 °C, showing a difference of 43.27 °C. In the stationary condition, no rotating effect of centrifugal force occurs, so the heat will be accumulated onto the rotating cylinder, leading to subsequent damage. However, given the fact of middle section with maximum temperature distribution. Due to an open space of 15 mm between the side wall and rotating cylinder, which is enough for thermal diffusion, the temperature distribution will decline at both sides, but rise sharply at the middle section due to a limited space of gap 7 mm for heat diffusion. The accuracy of the temperature measurement in this experiment can be demonstrated from the geometrical analysis of the test section. The Nusselt number with variations of ReΩ at rotating condition is plotted in Fig. 4. During rotation of the inner cylinder, the mean Nusselt number for the rotating condition will increase with the rising rotational Reynolds number, when comparing the
Fig. 4. Nusselt number with variations of the rotational Reynolds number.
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Fig. 5. Correlation on the mean Nusselt number ratio (to Nuo) with variations of Gr × ReΩ.
experimental results with the research findings of scholars Mohanty et al. [1] and Corne et al. [12]. Mohanty et al. [1] explored the local heat transfer on the rotating cylinder with length-to-diameter ratios varying from 3.8 to 6.5 under rotating condition. The rotational speed varied from 25 to 1400 rpm. The maximum values of rotational parameter attained were 4.6 × 104 and 2.7 × 104, the transport rate could be summarized as Nu ¼ 0:592 ðReX Þ0:515 . Cornet et al. [12] experimented with a 38.1 mm cylinder, 368.3 mm long at 43 rpm covering a Reynolds number range of 2 × 103 to 1 × 105 and summarized the results as Nu ¼ 0:234 ðReX Þ0:632 . In this research, the inner rotating test section is a cylinder sized in 120 mm (both in diameter and length), with a rotational speed of 63∼1170 rpm. Rotational Reynolds number ranges from 2.4 × 103 to 4.5 × 104. The empirical equation, NuX ¼ 7:764 ðReX Þ0:262 , can be obtained from the experimental data. As the working fluid is air in a small gap between the co-axial rotating cylinders, the property of the cooling fluid is also taken into account of this empirical equation. The empirical equation deducted from the experimental results is as follows: NuX ¼ 8:854Pr0:4 ðReX Þ0:262
ð2Þ
In this experiment, there are two influential factors for centrifugal buoyancy force: one is the centrifugal force triggered by rotation, the other one is the buoyancy effect arising from the temperature difference between the inner wall surface of the outer cylinder and heating inner cylinder. In this research, the Grashof number for buoyancy force effect and rotational Reynolds number for centrifugal force effect will be the key factors decisive to the inner thermal behavior of the entire co-axial rotating cylinder. Fig. 5 shows the correlation on mean Nusselt number ratio (to Nuo) with variations of rotating buoyancy parameter (Gr × ReΩ) of the three heat fluxes under the ten rotating speeds. To present more easily the thermal effect arising from rotation, all heat transfer coefficients were divided by those of the stationary condition, thereby separating the thermal buoyancy effect of the stationary flow field. In this figure, mean ratios of the Nusselt number will increase with the rising centrifugal buoyancy force, and the correction equation established by the power law type for all experimental data is as follows: NuX ¼ 0:375ðGr ReX 1011 Þ0:328 Nuo
ð3Þ
5. Conclusion This research is intended to measure the temperature distributions against different rotational speeds and heat flux, as well as the impact of the rotating number on heat transfer under a co-axial rotating cylinder with the rotating inner cylinder and stationary outer cylinder. It can be found from the case of rotation of the inner cylinder, the heat transfer of
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the flow field will increase with the rising rotational Reynolds number under the interaction of the centrifugal buoyancy force. This research has successfully explored the thermal behavior in a small gap between the co-axial rotating cylinders, and established an empirical formula for the experiment covering the relationship between the parameters of centrifugal force/buoyancy force and heat transfer coefficient, which may be referenced by the designers or engineers in the field of rotating cylinder and components. Acknowledgement The author would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC 94-2212-E-270-002. References [1] A.K. Mohanty, A.A. Tawfek, B.V.S.S.S. Prasad, Heat transfer from a rotating cylinder in crossflow, Previews of Heat and Mass Transfer 21 (1995) 54–61. [2] B.V.S.S.S. Prasad, A.A. Tawfek, A.K. Mohanty, Heat transfer from a circular cylinder rotating about an orthogonal axis in quiescent air, Experiments in Fluids 10 (1991) 267–272. [3] A.A. Tawfek, B.V.S.S.S. Prasad, A.K. Mohanty, Pressure measurements around a rotating cylinder with and without crossflow, ASME Journal of Fluids Engineering 115 (1993) 526–528. [4] K.M. Becker, J. Kaye, The influence of a radial temperature gradient on the instability of fluid flow in an annulus with an inner rotating cylinder, Transaction of ASME Journal of Heat Transfer (1962) 106–110. [5] K.M. Becker, J. Kaye, Measurements of adiabatic flow in an annulus with an inner rotating cylinder, ASME Journal of Heat Transfer (1962) 97–105. [6] R. Jakoby, S. Kim, S. Wittig, Correlations of the convection heat transfer in annular channels with rotating inner cylinder, Transaction of ASME Journal of Engineering for Gas Turbines and Power 121 (1999) 670–677. [7] S.R.M. Gardiner, R.H. Sabersky, Heat transfer in an annular gap, International Journal of Heat Mass Transfer 21 (1977) 1459–1466. [8] Y.N. Lee, W.J. Minkowycz, Heat transfer characteristics of the annulus of two-coaxial cylinders with one cylinder rotating, International Journal of Heat Mass Transfer 32 (1989) 711–722. [9] K. Kataoka, H. Doi, T. Komai, Heat/mass transfer in Taylor vortex flow with constant axial flow rates, International Journal of Heat Mass Transfer 20 (1977) 57–63. [10] S.J. Kline, F.A. Mcclintock, Describing uncertainties in single-sample experiments, Mechanical Engineering (1953) 3–8. [11] R.J. Moffat, Contributions to the theory of single-sample uncertainty analysis, Transaction of ASME Journal of Fluids Engineering 104 (1986) 250–260. [12] I. Cornet, R. Greif, J.T. Teng, F. Roehler, Mass transfer to rotating rods and plates, International Journal of Heat Mass Transfer 23 (1980) 805–811.