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THE DYNAMIC BEHAVIOR OF A BAYONET-TYPE THERMAL DIODE
Pergamon
PII:
Solar Energy Vol. 64, Nos 4–6, pp. 257–263, 1998 1998 Elsevier Science Ltd S 0 0 3 8 – 0 9 2 X ( 9 8 ) 0 0 0 7 5 – 9 All rights reserved. Printed in Great Britain 0038-092X / 98 / $ - see front matter
THE DYNAMIC BEHAVIOR OF A BAYONET-TYPE THERMAL DIODE KUAN CHEN*,†, PEERAPONG CHAILAPO*, WONGEE CHUN**, SIN KIM** and KYUNG JIN LEE** *Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112, U.S.A. **Department of Nuclear and Energy Engineering, Cheju National University, Cheju, Korea Received 14 August 1997; revised version accepted 29 June 1998 Communicated by BRIAN NORTON
Abstract—The time-dependent heat transfer and temperature variation of a bayonet-type thermal diode are theoretically investigated. A one-dimensional analytical model is developed to predict the dynamic behavior of the diode during heating, and a lumped-system approximation is employed for the cooling phase. Results of the analytical model and FLUENT are compared with experimental data of a bayonet thermal diode tested in Utah. Measured solar flux, ambient and room temperatures for a 24-hour period are used as the input data for the analytical model and the numerical calculation. The diode temperature variations predicted by the analytical model and FLUENT are in good agreement with experimental data. The simple analytical model is therefore capable of predicting the diode performance under real weather conditions. 1998 Elsevier Science Ltd. All rights reserved.
analytical model was developed by him for predicting the diode’s temperature variation and performance during the warm-up and cool-down phases. A good agreement was obtained between Jones’ analytical solutions and experimental data. Jones’ diode design was later modified by Chen et al. (1995) for better insulation between the tank and the outdoor air, and for variable direction of heat transfer. The new design of the diode, which is essentially a loop thermosyphon, is called a bayonet diode because the cross-section of the diode resembles a bayonet, as depicted in Fig. 1. The bayonet diode involves a large water tank and a small vertical cavity connected by an inclined rectangular channel. When the tank is above the cavity and the diode is heated from below, heat is transferred effectively from the cavity to the tank by convection. If the fluid in the tank is warmer than that in the cavity, heat transfer between the tank and the cavity is by conduction alone. A guide vane is inserted in the inclined channel for better insulation between the ascending and descending flows. When the diode is turned upside down, the direction of heat flow can be reversed. Alternatively if several bayonet diodes are stacked on top of one another and the tanks and cavities of adjacent diodes are connected by a valve for controlling the flow direction in the inclined channel, the direction of heat transfer can be altered (Chen et al., 1995). Thus, the bayonet diode has the advantage of being used for both
1. INTRODUCTION
A variety of thermal diode designs have been proposed and investigated for improving the energy efficiency of building envelopes. Although they are not as effective as heat-pipe thermal diodes, thermosyphon-type diodes (e.g., Jasinsky and Buckley, 1977; Jones, 1986; Chen, 1988; Chen et al., 1995; Chun et al., 1996) are very attractive for space heating applications because of their simple construction and operation. The working fluid of a thermosyphon-type diode remains in one phase during operation, making it easy to construct and seal the diode. The heat transfer and flow characteristics of liquid-filled loops have been investigated in some detail in the past (e.g., references cited in Chen, 1985a,b; Jones and Cai, 1993). A bidirectional thermal diode which consists of several waterfilled rectangular loops was proposed and studied by Chen, Chun and co-workers (Chen et al., 1995; Chun et al., 1996) for both winter heating and summer cooling applications. By tilting the loops up or down, the direction of heat transfer of the loops can be changed. The dynamic behavior of a tank-and-tongue liquid diode was analyzed and tested by Jones (1986) for space heating. A two-dimensional †Author to whom correspondence should be addressed. Tel.: 11 801 5816441; fax: 11 801 5818692. 257
258
K. Chen et al.
Fig. 1. Configuration and coordinate system of the bayonet diode.
winter heating and summer cooling. It has been tested for building energy conservation year round in the United States and Korea (Chen et al., 1995; Chun et al., 1996). The two-dimensional analytical model Jones developed for his liquid-convective diode is not applicable to the bayonet diode because of the flow in the connecting channel. Although CFD (Computational Fluid Dynamics) codes are available for an accurate simulation of natural convection in the bayonet diode, it would be expensive and time-consuming to solve the natural convection equations numerically under real operating conditions since the weather conditions change continuously with time. In the present study a simple heat transfer model was developed for predicting the dynamic behavior and performance of the bayonet diode. A one-dimensional, internal flow analysis was performed to predict the fluid dynamics and heat transfer of the diode during heating, and a lumped-system approximation was utilized for the cooling phase. The temperature differences in the tank and the cavity were assumed to be negligible. Fully developed laminar flow was assumed and axial conduction was included for the fluid in the connecting channel because of the low velocity for natural convection with a small temperature difference. At the beginning of the heating phase, the fluid in the cavity is colder than the fluid in the tank and the inclined channel. Natural convection from the cavity to the tank is not induced until the cavity temperature exceeds the temperature of the
fluid above it. The analytical model predicts small temperature differences in the diode after natural convection from the cavity to the tank is established. The calculated velocity based on the analytical model for a constant heat flux on the cavity surface is slightly lower than the result of a commercial CFD code. The heat transfer rate is not strongly dependent upon the natural convection velocity under normal operating conditions. Both the analytical and the numerical solutions agree well with experimental data for a bayonet diode tested in Utah. 2. THE ANALYTICAL MODEL FOR THE BAYONET-TYPE DIODE
The attention of the present investigation is focused on winter heating applications. For winter heating, the cavity of the diode is below the water tank and the desired direction of heat flow is from the outdoors to the indoors. The diode geometry and dimensions and the coordinate system employed in the analytical model are depicted in Fig. 1. The analytical model developed here can be easily extended to summer cooling applications for which the tank is below the cavity and the desired direction of heat transfer is from the indoors to the outdoors. The time-dependent heat transfer process of the diode for winter heating is divided into two phases: a warm-up phase during which the cavity is heated by the sun, and a cool-down phase after sunset when the cavity is cooled by the outdoor air. The following assumptions which are com-
The dynamic behavior of a bayonet-type thermal diode
monly employed for the heat transfer analysis of solar water heaters and solar houses are made in the present investigation: 1. All fluid properties except for density in the buoyancy term are constant and evaluated at the room temperature (about 208C). This assumption is valid for water experiencing a small temperature variation. 2. The heat transfer coefficients associated with natural convection along the indoor- and outdoor-facing surfaces are uniform and do not vary with time. 3. Heat loss through the insulation is negligible. 4. Change in internal energy of the insulating material and the walls of the diode are negligible in comparison with the internal energy change of water. The heating phase is investigated first. This phase is further divided into two intervals. The first interval is a short, start-up period during which the water in the cavity is heated by the sun, but the water temperature in the cavity is lower than the water tank temperature. When sun starts shining on the cavity surface, the fluid behind it is heated and small convection cells start to appear in the cavity. At the beginning of the heating process, the fluid behind the heated plate and in the connecting channel do not have sufficient buoyancy force to penetrate to the water tank. The fluid in these regions is broken into many small cells and a large, weak convection cell exists in the tank. The temperature of the fluid below the tank is fairly uniform, but little heat is transferred to the fluid in the tank. The weak convection cell in the tank is driven by conduction through the connecting channel and convection along the indoor-facing surface. This type of flow structure was observed in Jones (1986) experiment of a tank-and-tongue diode and in Behnia et al. (1987) experiment of inclined open thermosyphons. The assumption of a one-dimensional flow in the connecting channel is not valid during this startup period. After the fluid in the cavity becomes sufficiently warm, the convection cells can enter the tank and a large cell circulating around the vane is formed. The one-dimensional flow assumption in the passageway is valid from this point after until heating ceases. As the buoyancy force of the ascending flow from the cavity increases, the top of the cell will eventually reach the top of the tank and a strong convection cell appears in the tank. Because of the strong convection from the heated cavity surface to the tank, the temperature of the fluid in the diode is very uniform.
259
Neglecting the fluid in the connecting channel, the time for the cold fluid below the tank to be heated up to the water tank temperature can be estimated from the following energy-balance equation applied to a lumped system: m c C dT c / dt 5 A c [q 2 Uo (T c 2 T `,o )]
(1)
where m c is the mass of fluid in the cavity and A c is the area of the cavity surface facing the outdoors. Convection between the tank and the cavity ceases when the cavity temperature is lower than the tank temperature. Therefore the equation for fluid temperature variation in the cavity during the cool-down period is given by m c C dT c / dt 5 2 A cUo (T c 2 T `,o )
(2)
T c was assumed to be uniform during start-up and cool-down periods. Temperature variation of the fluid in the tank during start-up and cool-down phases can be calculated from m t C dT t / dt 5 2 A tUi (T t 2 T `,i )
(3)
where A t is the area of the tank surface facing indoors. The temperature of the fluid in the tank, T t , was assumed to be uniform at any instant in the analytical model. Except for the start-up period, the flow in the connecting channel was modeled as a one-dimensional flow; that is, the axial velocity component was assumed to be much greater than the other velocity components. Applying Newton’s second law of motion and the conservation of energy principle to a differential control volume in the flow passageway yields the following equations: Momentum equation: r dV/ dt 5 2 ≠p / ≠s 2 r gb (T 2 T r ) cos a 2 2t /d
(4)
where T r is the reference temperature at which fluid properties are evaluated and a is the angle between the vertical and the flow direction. For a fully developed laminar flow between two infinite parallel plates separated by a distance d, the wall shear stress can be expressed as:
t 5 6rV 2 / Re d
(5)
where Re d is the Reynolds number based on the height of the flow passageway. Energy equation: r Cd(≠T / ≠t 1V ≠T / ≠s) 5 kd ≠ 2 T / ≠s 2 1 q
(6)
where q50 for s 5 0 to j and s 5 m to L . 5 qs 2 Uo (T 2 T `,o ) for s 5 k to m
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K. Chen et al.
Axial conduction was included in the energy equation because the flow velocity may be very low at low energy input rates. In convection analyses axial conduction and convection are equally important at low Peclet numbers. Integration of the differential-form momentum equation from s 50 to L yields the integral momentum equation:
temperature variation in the lower portion of the cavity during heating is m c,b CdT c,b / dt 5 r CVwd(T s 5j 2 T s 5k ) 1 A c,b [qs 2 Uo (T c,b 2 T `,o )] (9)
where T s 5j and T s 5k are the temperatures of the incoming and outgoing streams. The integral of r L dV/ dt 5 ( ps 50 2 ps 5L ) the buoyancy term in the momentum equation 2 r gb (T 2 T r )cosa ds 2 2Lt /d must be broken into two parts in this case.
E
(7) The first two terms on the right-hand side of Eq. (7) are the reduced pressures ( p5thermodynamic pressure minus the buoyancy force) at s50 and L. Since the fluid temperature in the tank was assumed to be uniform and these two points are at the same depth in the tank, these two pressure terms can be dropped out in Eq. (7). After a one-dimensional flow is established in the connecting channel, the temperature of the fluid in the tank can be calculated from: m t CdT t / dt 5 r CVwd(T s 5L 2 T s 50 ) 2 A tUi (T t 2 T `,i )
(8)
Conduction through the inclined channel is neglected in Eqs. (3) and (8) because of the small channel cross-sectional area, wd, in comparison with the surface areas of the tank and the cavity. The computational procedure for the coupled momentum and energy equations are outlined below: 1. The energy equation was first solved based on the initial conditions or velocity and temperature distribution computed in the preceding time step. 2. The calculated temperature distribution was substituted into the momentum equation which was then solved for the mean velocity in the flow passageway. The trapezoidal rule was employed for numerical integration of the buoyancy term. 3. With the mean velocity determined, the water tank temperature was updated using Eq. (8). 4. A new value of time was assigned and the computation started from step (1) again when the explicit analytical model was used. In the implicit scheme, steps (1) through (3) were iterated using the updated velocity and temperatures until computation converged. For a short vane which was not extended fully to the bottom of the cavity, the equation for fluid
3. COMPARISON OF ANALYTICAL AND NUMERICAL RESULTS WITH EXPERIMENTAL DATA
A commercial CFD code called FLUENT was employed to validate the analytical model we developed. The grid generated by FLUENT version 4.23 is 14321 for the tank, 638 for the connecting channel, and 6311 for the cavity. The water in the diode was initially assigned a uniform temperature of 278C. During the heating process, the outdoor-facing surface of the cavity was heated by a uniform heat flux, but lost heat to the outdoors by convection and radiation. The indoor and outdoor air temperatures were maintained at 278C and 08C respectively during the heating process. The first analytical model we developed was explicit which became unstable at high energy input rates. As a result a very low heat flux (qs |0.02 kW/ m 2 ) was assumed for the heating phase. The overall heat transfer coefficients for heat transfer between the diode surfaces and the indoor and outdoor air were assumed to be Ui 5 Uo 51.8 W/ m 2 8C. These small heat transfer coefficients were used because the small heat flux caused only about 28C increase of the water tank temperature during heating. The outdoor-facing surface was covered by a glass sheet in the winter and therefore the convection heat transfer from this surface was also very low. The guide vane which was assumed to be well insulated prevents the hot and cold streams in the flow passageway from mixing. Therefore, a higher buoyancy force can be generated which results in a higher flow velocity in the diode than the one without a vane. A longer vane in the cavity increases frictional losses under steady flow conditions due to the increase in length of the flow passageway. However it has the advantage of a higher buoyancy force. In addition, the start-up time of a longer vane was found to be shorter
The dynamic behavior of a bayonet-type thermal diode
because of the smaller amount of water in the heating section of the cavity. Results of FLUENT 4.23 and the explicit analytical model are compared for 8 hours of heating followed by 8 hours of cooling. Except for the short start-up period, FLUENT’s results show the flow behind the vane in the cavity and in the connecting channel is in one direction during heating and thus can be modeled as a fullydeveloped channel flow. On the other hand, flow reversal was observed in front of the vane in the cavity in FLUENT’s results (see Fig. 2). Thus the analytical model which employs the friction coefficient for fully developed channel flow underestimates frictional losses in this region. This discrepancy, together with the neglect of frictional losses in the tank and in the lower portion of the cavity, resulted in a 30% increase in mean velocity when the analytical solutions were compared with FLUENT’s results. The difference in velocity calculations was found to have little effect on diode temperature calculations, as shown in Fig. 3. This is because
Fig. 2. Velocity vector plot of FLUENT for a constant heat flux (qs 50.02 kW/ m 2 , t54 hours, max53.9310 23 m / s).
261
Fig. 3. Diode temperature variations predicted by the explicit analytical model and FLUENT.
the heat losses from the cavity surface by convection and radiation are very small in comparison with convection heat transfer between the tank and the cavity. Different velocities result in different temperature distributions along the cavity surface, but the rates at which energy is transferred from the heated cavity to the tank are almost the same if little heat loss takes place at the cavity surface. Therefore, the calculated water tank temperatures differ only slightly. The difference in velocity calculations may have a stronger effect on the water tank temperature if convection losses on the cavity surface are higher. Applications of the explicit analytical model were limited to very low heat input rates. To overcome this deficiency, the model was later modified to employ an implicit scheme for the momentum equation calculation. The mean velocity was determined from the integral-form momentum equation by iteration in the modified model. The implicit analytical model is stable for a heat flux of 0.4 kW/ m 2 . This heat flux value is typical for solar irradiation on a vertical surface in the winter. Results of the implicit analytical model and FLUENT were compared with experimental data collected from a weather test. A bayonet diode with configuration and dimensions similar to those employed in the theoretical investigation was constructed and installed on a large storage shed. Its performance was tested in Salt Lake City, Utah, in 1993–1994. A detailed discussion of the weather test can be found in Chen et al.’s (1995) report. Diode temperatures measured on April 21, 1994 were selected for comparison. The elevation angle of sun was high in April in Utah. The tank temperature increased only slightly during heating because of the small amount of solar energy received. At sunrise, the temperature
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K. Chen et al.
of the fluid in the cavity was about 48C lower than the water tank temperature. The fluid below the tank was heated to the temperature of the fluid in the tank 3 hours after sunrise. Then the two temperatures increased at almost the same rate, but with a phase-lag about 2 hours because of the large thermal inertia of the water in the tank. After heating stopped, the tank was cooled by the indoor air while the cavity was cooled by the outdoor air. They were cooled down at different rates, indicating little heat transfer between the tank and the cavity during the cool-down phase. The predicted and measured diode temperature variations are presented in Fig. 4. In this comparison the overall heat transfer coefficients Uo and Ui in the analytical model were assigned to be 6.5 and 10 W/ m 2 8C, respectively. The same values were used for FLUENT 4.23. These coefficients were determined by matching the fluid cooling rates predicted by the analytical model with experimental data. They are slightly larger than the natural convection coefficient of a vertical isothermal plate. The high indoor convection coefficient may be the consequence of the use of fins in the experiments. Generally speaking, the predictions of the analytical model and FLUENT were in good agreement with experimental measurements. Possible causes for the discrepancy between theoretically predicted and experimentally measured temperatures are: 1. Water properties were assumed in theoretical predictions, but the fluid was a water / antifreeze mixture in the experiments. 2. The heights of the flow passageway in the inclined channel and the cavity were slightly
different for the diodes used in the weather test. 3. One-dimensional flow was assumed in the inclined channel. 4. CONCLUSION
In conclusion, the present analytical model slightly overpredicted the mean velocity during heating because of flow reversal behind the heated surface and the neglect of frictional losses in the tank. Fluid temperature variations and heat transfer rates computed from the analytical model and FLUENT were in good agreement. Both agreed well with experimental results when experimentally determined heat transfer coefficients were used for convection from the diode surfaces. These results suggest the simple analytical model can be employed to predict the dynamic behavior of the bayonet diode without recourse to complicated and time-consuming CFD codes. It may be applicable to other diodes of similar designs and operating conditions. Results of the analytical model and FLUENT both suggest a long guide vane is preferred. Extending the vane to the full length of the cavity eliminates mixing of the hot and cold fluids in the cavity and therefore increases the buoyancy force of the diode. NOMENCLATURE A C d g k
area specific heat height of the flow passageway gravitational acceleration thermal conductivity of the working fluid
Fig. 4. Measured and predicted diode temperature variations.
The dynamic behavior of a bayonet-type thermal diode
T U t V w a b r t
total length of the flow passageway mass reduced pressure heat flux one-dimensional coordinate system employed in the analytical model temperature overall heat-transfer coefficient time mean velocity in the flow passageway diode width angle between the vertical and the flow direction thermal expansion coefficient density wall shear stress
Subscripts b c i o r s t `
refers to the lower portion of the diode cavity refers to the cavity of the diode refers to the indoors refers to the outdoors reference properties solar radiation refers to the tank of the diode indoor or outdoor air
L m p q s
Acknowledgements—The first two authors thank the support of USAF Wright Laboratory under contract number F08635-93C-0029.
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REFERENCES Behnia M., Morrison G. L. and Paramasivam S. (1987) Heat transfer and flow in inclined open thermosyphons, Proceedings of the 1987 ASME /JSME Thermal Engineering Joint Conference, pp. 7–14. Chen K. (1985) The optimum configuration of natural convection loops. Solar Energy 34, 4 / 5, 407–416. Chen K. (1985) On the oscillatory instability of closed-loop thermosyphons. ASME Journal of Heat Transfer 107, 4, 826–832. Chen K. (1988) Design of a bi-directional thermal diode. ASME Journal of Solar Energy Engineering 110, 299–305. Chen K., Shorthill R. W., Chu S. S., Chailapo P. and Narasimhan S. (1995) An Energy-efficient Construction Module of Variable Direction of Heat Flow, Heat Capacity, and Surface Absorption, USAF WL-TR-95-3045. Chun W., Lee Y. J., Lee J. Y., Chen K., Kim H. T. and Lee T. K. (1996) Application of the thermal diode concept for the utilization of solar energy. Proceedings of 31 st Intersociety Energy Conversion Engineering Conference 3, 1709–1714. Jasinsky T. and Buckley S. (1977) Thermosyphon Analysis of a Thermic Diode Solar Heating System, ASME Paper No. 77-WA / Sol-9. Jones G. F. (1986) Heat transfer in a liquid connective diode. ASME Journal of Solar Energy Engineering 108, 163–171. Jones G. F. and Cai J. (1993) Analysis of a transient asymmetrically heated / cooled open thermosyphon. ASME J. of Heat Transfer 115, 621–630.
PII:
Solar Energy Vol. 64, Nos 4–6, pp. 257–263, 1998 1998 Elsevier Science Ltd S 0 0 3 8 – 0 9 2 X ( 9 8 ) 0 0 0 7 5 – 9 All rights reserved. Printed in Great Britain 0038-092X / 98 / $ - see front matter
THE DYNAMIC BEHAVIOR OF A BAYONET-TYPE THERMAL DIODE KUAN CHEN*,†, PEERAPONG CHAILAPO*, WONGEE CHUN**, SIN KIM** and KYUNG JIN LEE** *Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112, U.S.A. **Department of Nuclear and Energy Engineering, Cheju National University, Cheju, Korea Received 14 August 1997; revised version accepted 29 June 1998 Communicated by BRIAN NORTON
Abstract—The time-dependent heat transfer and temperature variation of a bayonet-type thermal diode are theoretically investigated. A one-dimensional analytical model is developed to predict the dynamic behavior of the diode during heating, and a lumped-system approximation is employed for the cooling phase. Results of the analytical model and FLUENT are compared with experimental data of a bayonet thermal diode tested in Utah. Measured solar flux, ambient and room temperatures for a 24-hour period are used as the input data for the analytical model and the numerical calculation. The diode temperature variations predicted by the analytical model and FLUENT are in good agreement with experimental data. The simple analytical model is therefore capable of predicting the diode performance under real weather conditions. 1998 Elsevier Science Ltd. All rights reserved.
analytical model was developed by him for predicting the diode’s temperature variation and performance during the warm-up and cool-down phases. A good agreement was obtained between Jones’ analytical solutions and experimental data. Jones’ diode design was later modified by Chen et al. (1995) for better insulation between the tank and the outdoor air, and for variable direction of heat transfer. The new design of the diode, which is essentially a loop thermosyphon, is called a bayonet diode because the cross-section of the diode resembles a bayonet, as depicted in Fig. 1. The bayonet diode involves a large water tank and a small vertical cavity connected by an inclined rectangular channel. When the tank is above the cavity and the diode is heated from below, heat is transferred effectively from the cavity to the tank by convection. If the fluid in the tank is warmer than that in the cavity, heat transfer between the tank and the cavity is by conduction alone. A guide vane is inserted in the inclined channel for better insulation between the ascending and descending flows. When the diode is turned upside down, the direction of heat flow can be reversed. Alternatively if several bayonet diodes are stacked on top of one another and the tanks and cavities of adjacent diodes are connected by a valve for controlling the flow direction in the inclined channel, the direction of heat transfer can be altered (Chen et al., 1995). Thus, the bayonet diode has the advantage of being used for both
1. INTRODUCTION
A variety of thermal diode designs have been proposed and investigated for improving the energy efficiency of building envelopes. Although they are not as effective as heat-pipe thermal diodes, thermosyphon-type diodes (e.g., Jasinsky and Buckley, 1977; Jones, 1986; Chen, 1988; Chen et al., 1995; Chun et al., 1996) are very attractive for space heating applications because of their simple construction and operation. The working fluid of a thermosyphon-type diode remains in one phase during operation, making it easy to construct and seal the diode. The heat transfer and flow characteristics of liquid-filled loops have been investigated in some detail in the past (e.g., references cited in Chen, 1985a,b; Jones and Cai, 1993). A bidirectional thermal diode which consists of several waterfilled rectangular loops was proposed and studied by Chen, Chun and co-workers (Chen et al., 1995; Chun et al., 1996) for both winter heating and summer cooling applications. By tilting the loops up or down, the direction of heat transfer of the loops can be changed. The dynamic behavior of a tank-and-tongue liquid diode was analyzed and tested by Jones (1986) for space heating. A two-dimensional †Author to whom correspondence should be addressed. Tel.: 11 801 5816441; fax: 11 801 5818692. 257
258
K. Chen et al.
Fig. 1. Configuration and coordinate system of the bayonet diode.
winter heating and summer cooling. It has been tested for building energy conservation year round in the United States and Korea (Chen et al., 1995; Chun et al., 1996). The two-dimensional analytical model Jones developed for his liquid-convective diode is not applicable to the bayonet diode because of the flow in the connecting channel. Although CFD (Computational Fluid Dynamics) codes are available for an accurate simulation of natural convection in the bayonet diode, it would be expensive and time-consuming to solve the natural convection equations numerically under real operating conditions since the weather conditions change continuously with time. In the present study a simple heat transfer model was developed for predicting the dynamic behavior and performance of the bayonet diode. A one-dimensional, internal flow analysis was performed to predict the fluid dynamics and heat transfer of the diode during heating, and a lumped-system approximation was utilized for the cooling phase. The temperature differences in the tank and the cavity were assumed to be negligible. Fully developed laminar flow was assumed and axial conduction was included for the fluid in the connecting channel because of the low velocity for natural convection with a small temperature difference. At the beginning of the heating phase, the fluid in the cavity is colder than the fluid in the tank and the inclined channel. Natural convection from the cavity to the tank is not induced until the cavity temperature exceeds the temperature of the
fluid above it. The analytical model predicts small temperature differences in the diode after natural convection from the cavity to the tank is established. The calculated velocity based on the analytical model for a constant heat flux on the cavity surface is slightly lower than the result of a commercial CFD code. The heat transfer rate is not strongly dependent upon the natural convection velocity under normal operating conditions. Both the analytical and the numerical solutions agree well with experimental data for a bayonet diode tested in Utah. 2. THE ANALYTICAL MODEL FOR THE BAYONET-TYPE DIODE
The attention of the present investigation is focused on winter heating applications. For winter heating, the cavity of the diode is below the water tank and the desired direction of heat flow is from the outdoors to the indoors. The diode geometry and dimensions and the coordinate system employed in the analytical model are depicted in Fig. 1. The analytical model developed here can be easily extended to summer cooling applications for which the tank is below the cavity and the desired direction of heat transfer is from the indoors to the outdoors. The time-dependent heat transfer process of the diode for winter heating is divided into two phases: a warm-up phase during which the cavity is heated by the sun, and a cool-down phase after sunset when the cavity is cooled by the outdoor air. The following assumptions which are com-
The dynamic behavior of a bayonet-type thermal diode
monly employed for the heat transfer analysis of solar water heaters and solar houses are made in the present investigation: 1. All fluid properties except for density in the buoyancy term are constant and evaluated at the room temperature (about 208C). This assumption is valid for water experiencing a small temperature variation. 2. The heat transfer coefficients associated with natural convection along the indoor- and outdoor-facing surfaces are uniform and do not vary with time. 3. Heat loss through the insulation is negligible. 4. Change in internal energy of the insulating material and the walls of the diode are negligible in comparison with the internal energy change of water. The heating phase is investigated first. This phase is further divided into two intervals. The first interval is a short, start-up period during which the water in the cavity is heated by the sun, but the water temperature in the cavity is lower than the water tank temperature. When sun starts shining on the cavity surface, the fluid behind it is heated and small convection cells start to appear in the cavity. At the beginning of the heating process, the fluid behind the heated plate and in the connecting channel do not have sufficient buoyancy force to penetrate to the water tank. The fluid in these regions is broken into many small cells and a large, weak convection cell exists in the tank. The temperature of the fluid below the tank is fairly uniform, but little heat is transferred to the fluid in the tank. The weak convection cell in the tank is driven by conduction through the connecting channel and convection along the indoor-facing surface. This type of flow structure was observed in Jones (1986) experiment of a tank-and-tongue diode and in Behnia et al. (1987) experiment of inclined open thermosyphons. The assumption of a one-dimensional flow in the connecting channel is not valid during this startup period. After the fluid in the cavity becomes sufficiently warm, the convection cells can enter the tank and a large cell circulating around the vane is formed. The one-dimensional flow assumption in the passageway is valid from this point after until heating ceases. As the buoyancy force of the ascending flow from the cavity increases, the top of the cell will eventually reach the top of the tank and a strong convection cell appears in the tank. Because of the strong convection from the heated cavity surface to the tank, the temperature of the fluid in the diode is very uniform.
259
Neglecting the fluid in the connecting channel, the time for the cold fluid below the tank to be heated up to the water tank temperature can be estimated from the following energy-balance equation applied to a lumped system: m c C dT c / dt 5 A c [q 2 Uo (T c 2 T `,o )]
(1)
where m c is the mass of fluid in the cavity and A c is the area of the cavity surface facing the outdoors. Convection between the tank and the cavity ceases when the cavity temperature is lower than the tank temperature. Therefore the equation for fluid temperature variation in the cavity during the cool-down period is given by m c C dT c / dt 5 2 A cUo (T c 2 T `,o )
(2)
T c was assumed to be uniform during start-up and cool-down periods. Temperature variation of the fluid in the tank during start-up and cool-down phases can be calculated from m t C dT t / dt 5 2 A tUi (T t 2 T `,i )
(3)
where A t is the area of the tank surface facing indoors. The temperature of the fluid in the tank, T t , was assumed to be uniform at any instant in the analytical model. Except for the start-up period, the flow in the connecting channel was modeled as a one-dimensional flow; that is, the axial velocity component was assumed to be much greater than the other velocity components. Applying Newton’s second law of motion and the conservation of energy principle to a differential control volume in the flow passageway yields the following equations: Momentum equation: r dV/ dt 5 2 ≠p / ≠s 2 r gb (T 2 T r ) cos a 2 2t /d
(4)
where T r is the reference temperature at which fluid properties are evaluated and a is the angle between the vertical and the flow direction. For a fully developed laminar flow between two infinite parallel plates separated by a distance d, the wall shear stress can be expressed as:
t 5 6rV 2 / Re d
(5)
where Re d is the Reynolds number based on the height of the flow passageway. Energy equation: r Cd(≠T / ≠t 1V ≠T / ≠s) 5 kd ≠ 2 T / ≠s 2 1 q
(6)
where q50 for s 5 0 to j and s 5 m to L . 5 qs 2 Uo (T 2 T `,o ) for s 5 k to m
260
K. Chen et al.
Axial conduction was included in the energy equation because the flow velocity may be very low at low energy input rates. In convection analyses axial conduction and convection are equally important at low Peclet numbers. Integration of the differential-form momentum equation from s 50 to L yields the integral momentum equation:
temperature variation in the lower portion of the cavity during heating is m c,b CdT c,b / dt 5 r CVwd(T s 5j 2 T s 5k ) 1 A c,b [qs 2 Uo (T c,b 2 T `,o )] (9)
where T s 5j and T s 5k are the temperatures of the incoming and outgoing streams. The integral of r L dV/ dt 5 ( ps 50 2 ps 5L ) the buoyancy term in the momentum equation 2 r gb (T 2 T r )cosa ds 2 2Lt /d must be broken into two parts in this case.
E
(7) The first two terms on the right-hand side of Eq. (7) are the reduced pressures ( p5thermodynamic pressure minus the buoyancy force) at s50 and L. Since the fluid temperature in the tank was assumed to be uniform and these two points are at the same depth in the tank, these two pressure terms can be dropped out in Eq. (7). After a one-dimensional flow is established in the connecting channel, the temperature of the fluid in the tank can be calculated from: m t CdT t / dt 5 r CVwd(T s 5L 2 T s 50 ) 2 A tUi (T t 2 T `,i )
(8)
Conduction through the inclined channel is neglected in Eqs. (3) and (8) because of the small channel cross-sectional area, wd, in comparison with the surface areas of the tank and the cavity. The computational procedure for the coupled momentum and energy equations are outlined below: 1. The energy equation was first solved based on the initial conditions or velocity and temperature distribution computed in the preceding time step. 2. The calculated temperature distribution was substituted into the momentum equation which was then solved for the mean velocity in the flow passageway. The trapezoidal rule was employed for numerical integration of the buoyancy term. 3. With the mean velocity determined, the water tank temperature was updated using Eq. (8). 4. A new value of time was assigned and the computation started from step (1) again when the explicit analytical model was used. In the implicit scheme, steps (1) through (3) were iterated using the updated velocity and temperatures until computation converged. For a short vane which was not extended fully to the bottom of the cavity, the equation for fluid
3. COMPARISON OF ANALYTICAL AND NUMERICAL RESULTS WITH EXPERIMENTAL DATA
A commercial CFD code called FLUENT was employed to validate the analytical model we developed. The grid generated by FLUENT version 4.23 is 14321 for the tank, 638 for the connecting channel, and 6311 for the cavity. The water in the diode was initially assigned a uniform temperature of 278C. During the heating process, the outdoor-facing surface of the cavity was heated by a uniform heat flux, but lost heat to the outdoors by convection and radiation. The indoor and outdoor air temperatures were maintained at 278C and 08C respectively during the heating process. The first analytical model we developed was explicit which became unstable at high energy input rates. As a result a very low heat flux (qs |0.02 kW/ m 2 ) was assumed for the heating phase. The overall heat transfer coefficients for heat transfer between the diode surfaces and the indoor and outdoor air were assumed to be Ui 5 Uo 51.8 W/ m 2 8C. These small heat transfer coefficients were used because the small heat flux caused only about 28C increase of the water tank temperature during heating. The outdoor-facing surface was covered by a glass sheet in the winter and therefore the convection heat transfer from this surface was also very low. The guide vane which was assumed to be well insulated prevents the hot and cold streams in the flow passageway from mixing. Therefore, a higher buoyancy force can be generated which results in a higher flow velocity in the diode than the one without a vane. A longer vane in the cavity increases frictional losses under steady flow conditions due to the increase in length of the flow passageway. However it has the advantage of a higher buoyancy force. In addition, the start-up time of a longer vane was found to be shorter
The dynamic behavior of a bayonet-type thermal diode
because of the smaller amount of water in the heating section of the cavity. Results of FLUENT 4.23 and the explicit analytical model are compared for 8 hours of heating followed by 8 hours of cooling. Except for the short start-up period, FLUENT’s results show the flow behind the vane in the cavity and in the connecting channel is in one direction during heating and thus can be modeled as a fullydeveloped channel flow. On the other hand, flow reversal was observed in front of the vane in the cavity in FLUENT’s results (see Fig. 2). Thus the analytical model which employs the friction coefficient for fully developed channel flow underestimates frictional losses in this region. This discrepancy, together with the neglect of frictional losses in the tank and in the lower portion of the cavity, resulted in a 30% increase in mean velocity when the analytical solutions were compared with FLUENT’s results. The difference in velocity calculations was found to have little effect on diode temperature calculations, as shown in Fig. 3. This is because
Fig. 2. Velocity vector plot of FLUENT for a constant heat flux (qs 50.02 kW/ m 2 , t54 hours, max53.9310 23 m / s).
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Fig. 3. Diode temperature variations predicted by the explicit analytical model and FLUENT.
the heat losses from the cavity surface by convection and radiation are very small in comparison with convection heat transfer between the tank and the cavity. Different velocities result in different temperature distributions along the cavity surface, but the rates at which energy is transferred from the heated cavity to the tank are almost the same if little heat loss takes place at the cavity surface. Therefore, the calculated water tank temperatures differ only slightly. The difference in velocity calculations may have a stronger effect on the water tank temperature if convection losses on the cavity surface are higher. Applications of the explicit analytical model were limited to very low heat input rates. To overcome this deficiency, the model was later modified to employ an implicit scheme for the momentum equation calculation. The mean velocity was determined from the integral-form momentum equation by iteration in the modified model. The implicit analytical model is stable for a heat flux of 0.4 kW/ m 2 . This heat flux value is typical for solar irradiation on a vertical surface in the winter. Results of the implicit analytical model and FLUENT were compared with experimental data collected from a weather test. A bayonet diode with configuration and dimensions similar to those employed in the theoretical investigation was constructed and installed on a large storage shed. Its performance was tested in Salt Lake City, Utah, in 1993–1994. A detailed discussion of the weather test can be found in Chen et al.’s (1995) report. Diode temperatures measured on April 21, 1994 were selected for comparison. The elevation angle of sun was high in April in Utah. The tank temperature increased only slightly during heating because of the small amount of solar energy received. At sunrise, the temperature
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K. Chen et al.
of the fluid in the cavity was about 48C lower than the water tank temperature. The fluid below the tank was heated to the temperature of the fluid in the tank 3 hours after sunrise. Then the two temperatures increased at almost the same rate, but with a phase-lag about 2 hours because of the large thermal inertia of the water in the tank. After heating stopped, the tank was cooled by the indoor air while the cavity was cooled by the outdoor air. They were cooled down at different rates, indicating little heat transfer between the tank and the cavity during the cool-down phase. The predicted and measured diode temperature variations are presented in Fig. 4. In this comparison the overall heat transfer coefficients Uo and Ui in the analytical model were assigned to be 6.5 and 10 W/ m 2 8C, respectively. The same values were used for FLUENT 4.23. These coefficients were determined by matching the fluid cooling rates predicted by the analytical model with experimental data. They are slightly larger than the natural convection coefficient of a vertical isothermal plate. The high indoor convection coefficient may be the consequence of the use of fins in the experiments. Generally speaking, the predictions of the analytical model and FLUENT were in good agreement with experimental measurements. Possible causes for the discrepancy between theoretically predicted and experimentally measured temperatures are: 1. Water properties were assumed in theoretical predictions, but the fluid was a water / antifreeze mixture in the experiments. 2. The heights of the flow passageway in the inclined channel and the cavity were slightly
different for the diodes used in the weather test. 3. One-dimensional flow was assumed in the inclined channel. 4. CONCLUSION
In conclusion, the present analytical model slightly overpredicted the mean velocity during heating because of flow reversal behind the heated surface and the neglect of frictional losses in the tank. Fluid temperature variations and heat transfer rates computed from the analytical model and FLUENT were in good agreement. Both agreed well with experimental results when experimentally determined heat transfer coefficients were used for convection from the diode surfaces. These results suggest the simple analytical model can be employed to predict the dynamic behavior of the bayonet diode without recourse to complicated and time-consuming CFD codes. It may be applicable to other diodes of similar designs and operating conditions. Results of the analytical model and FLUENT both suggest a long guide vane is preferred. Extending the vane to the full length of the cavity eliminates mixing of the hot and cold fluids in the cavity and therefore increases the buoyancy force of the diode. NOMENCLATURE A C d g k
area specific heat height of the flow passageway gravitational acceleration thermal conductivity of the working fluid
Fig. 4. Measured and predicted diode temperature variations.
The dynamic behavior of a bayonet-type thermal diode
T U t V w a b r t
total length of the flow passageway mass reduced pressure heat flux one-dimensional coordinate system employed in the analytical model temperature overall heat-transfer coefficient time mean velocity in the flow passageway diode width angle between the vertical and the flow direction thermal expansion coefficient density wall shear stress
Subscripts b c i o r s t `
refers to the lower portion of the diode cavity refers to the cavity of the diode refers to the indoors refers to the outdoors reference properties solar radiation refers to the tank of the diode indoor or outdoor air
L m p q s
Acknowledgements—The first two authors thank the support of USAF Wright Laboratory under contract number F08635-93C-0029.
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