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Heat transfer correlations for vertical mantle heat exchangers
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Solar Energy Vol. 69(Suppl.), Nos. 1–6, pp. 157–171, 2000 2001 Elsevier Science Ltd S 0 0 3 8 – 0 9 2 X ( 0 1 ) 0 0 0 3 9 – 1 All rights reserved. Printed in Great Britain 0038-092X / 00 / $ - see front matter
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HEAT TRANSFER CORRELATIONS FOR VERTICAL MANTLE HEAT EXCHANGERS LOUISE JIVAN SHAH † Department of Civil Engineering, Technical University of Denmark DK-2800 Kgs. Lyngby, Denmark Received 10 July 2000; revised version accepted 23 February 2001 Communicated by ERICH HAHNE
Abstract—This paper investigates heat transfer in vertical mantle heat exchangers for application in low flow solar domestic hot water systems. Two new heat transfer correlations for vertical mantle heat exchangers with top entry port and bottom exit ports are developed. The correlations are based on computational fluid dynamic modelling of whole vertical mantle tanks. The correlations are combined with a heat storage model in a simulation program that predicts the yearly thermal performance of low flow solar domestic hot water systems based on mantle tanks. The model predictions of energy gains and temperatures are compared with outdoor measurements and the model is found to give reliable results. 2001 Elsevier Science Ltd. All rights reserved.
1.1. Advantages of the mantle tank system
1. INTRODUCTION
To optimise the design of components in SDHW systems, simulation tools are often needed. These tools must be able to simulate heat transfer and fluid dynamics in detail. Computational fluid dynamics (CFD) would be the obvious simulation choice as the Navier– Stokes and energy equations are solved with CFD. Thus, detailed solutions can be obtained for the convection and heat transfer characteristics. On the other hand, CFD computations are extremely CPU-time demanding — often so demanding that it is impossible to carry out the investigations wanted. However, if the numerous CFD results could somehow be extracted, sorted, and used in a less detailed model that is less CPU-time demanding it could be possible to use the CFD information in a less time demanding way. This is what is tried in this project. The case study concerns low flow solar domestic hot water (SDHW) systems based on mantle tanks. Low flow systems in northern Europe, especially Holland and Denmark, are often based on a vertical mantle tank as illustrated in Fig. 1. The heat exchange between the solar collector fluid and the consumption water takes place in the mantle. A control system and a pump are used in the solar collector loop, but the system can also be used as a thermosiphoning system. †
1.1.1. Thermal stratification. The mantle tank system is one of the simplest ways of producing high heat exchanger effectiveness while promoting thermal stratification. The mantle configuration provides a large heat transfer area and effective distribution of the collector loop flow over the wall of the tank. Most of the incoming mantle fluid seeks the thermal equilibrium level in the mantle, and thermal stratification in the mantle and the inner tank is not disturbed (Shah et al., 1999). 1.1.2. Cost. The cost of the tank is cheap compared to the increased performance initiated by the fine thermal stratification (IEA Task 14). Further, Furbo (1993) showed that the mantle tank system had the best cost / performance ratio compared to systems with external heat exchangers and side-arm and compared to systems with internal heat exchanger coils. 1.2. Disadvantages of the mantle tank system
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1.2.1. Size limitation. The mantle tank design is not suitable for large low flow SDHW systems, as the heat transfer area gets too small for tanks with volumes over 800–1000 l (Shah, 1999). As the heat storage has proved to be the most important system component (Furbo, 1998; Furbo and Shah, 1997), there has been an emphasis on this component in the case study. Therefore, CFD models are used to take mantle tanks into calculation and detailed information is obtained for the
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Fig. 2. Heat flux distribution over mantle heat transfer area based on one-dimensional heat transfer models.
Fig. 1. Simplified sketch of low flow SDHW system used in Denmark and Holland (Shah, 1999).
convection and heat transfer characteristics of both mantle and inner tank. With CFD results, general correlations are developed for heat transfer between: • the solar collector fluid and the inner tank wall • the solar collector fluid and the mantle wall • the inner tank walls and the domestic hot water. These correlations are used in a heat storage model in a simulation program that predicts the yearly thermal performance of low flow SDHW systems based on mantle tank. 2. PREVIOUS STUDIES OF HEAT TRANSFER IN VERTICAL ANNULI
2.1. Vertical annuli in solar applications Several SDHW simulation programs can take SDHW systems based on mantle tanks into calculations. In TRNSYS (Klein et al., 1996), two approaches have been used to model mantle flow. Baur et al. (1993) assumed that the heat transfer in the mantle passageway could be estimated from a correlation for flow between two flat plates as reported by Mercer et al. (1967):
S S
exchange rate (UA) is an input value to the model. The UA value can be modelled either as a constant or as a function of the mantle inlet temperature, the store temperature and the mantle flow rate. This model is very suitable for parameter identification during storage testing; however, it does not provide the heat transfer information and is thus less usable if a non-tested storage is to be modelled. For another simulation program Shah and Furbo (1998) evaluated heat transfer in mantle heat exchangers using numerical simulation of the flows in the mantle and inside the storage tank. For low flow rates, they showed that the flow was dominated by buoyancy effects and proposed that the heat transfer in the mantle flow could be correlated as:
S
x ] Nu(x) 5 0.46 ? Ra x ? ] Dh
D
0.28
(2)
Comparison of the distribution of heat flux over the mantle surface and the variation of the mantle fluid temperature predicted by the three approaches is shown in Figs. 2 and 3 for a mantle flow rate of 0.5 l / min, mantle inlet temperature of 508C, a uniform tank wall temperature of 208C and annulus gap of 3 cm. A constant heat transfer
D D
D 1.2 ]h 0.0606 ? Re ? Pr ? ] x Nu(x) 5 4.9 1 ]]]]]]]]] Dh 0.1 1 1 0.0909 ? Re ? Pr ? ] Pr x (1) ¨ and Pausinger (1997) Also for TRNSYS, Druck developed the Multiport model where the heat
Fig. 3. Mantle fluid temperature variation down the mantle based on one-dimensional heat transfer models.
Heat transfer correlations for vertical mantle heat exchangers
coefficient of 200 W/ m 2 K has been used for the curves presenting the approach used in the Multiport model. The natural convection model (Shah et al.) predicts substantially higher heat flux in the top section of the mantle and thus much faster cooling of the flow in the mantle. As recent studies have shown that natural convection is dominant in the mantle during low flow operation (Shah et al., 1998, 1999) the ‘buoyant approach’ is continued in this study.
2.2. Other numerical studies of vertical annuli Khan and Kumar (1989) conducted a numerical investigation to evaluate the effects of diameter ratio, k 5 r outer /r inner , and aspect ratio, A 5 (height of annulus) /(r outer 2 r inner ), in natural convection of gases in vertical annuli. The inner cylinder was at a constant surface heat flux, and the outer cylinder was at a constant temperature. The cases under consideration were 1# A#10 and 1# k #15. They obtained the following Nusselt-number correlation:
Nu 5 1.02 ? Ra 0.28
for 100 # Ra # 600
(5)
In this equation, they used the annulus gap as length scale. Dhimdi and Bolle (1997) determined heat transfer coefficients numerically for natural convection in vertical annuli for a large Grashof number using water as the heat transfer medium. Heat transfer results were related to radius ratios and aspect ratios using constant temperature boundaries. Analysing the curvature effect, they found the following Nusselt number equation: log (Nu) 5 2 0.17 ? log (K)1.82 1 2.573 for Gr 5 8 3 10 8
(6)
(3)
where L, the length of the annulus, is used as length scale. Also, they obtained a simpler correlation for A.5 and k .5 using the outer diameter of the inner cylinder as length scale: Nu di 5 0.455 ? Ra 0.202 di
dynamic stability of mixed convection in an annulus with a constant heat flux boundary applied to the inner annulus and an insulated outer boundary. For this case, they found that the parallel flow assumption could not be used. They presented a Nusselt–Rayleigh number equation for data in the region between linear stability and Ra 5 600:
where K is the radius ratio (K 5 r outer /r inner ) and the radius gap is used as the length scale.
Nu L 5 0.239 ? Ra L0.29 ? k ( 0.303 / k )10.316 ? A20.05 for Ra L , 10 6
159
for Ra L , 10 6
(4)
El-Shaarawi and Al-Nimr (1990) presented analytical solutions for fully developed natural convection in open-ended vertical annuli. They analysed the four fundamental boundary conditions: (1) constant temperature boundaries, where one of the walls is kept at the inlet ambient fluid temperature. (2) One boundary is kept at a constant heat flux, and the other boundary is perfectly insulated. (3) One boundary is kept at a constant temperature flux, and the other boundary is perfectly insulated. (4) One boundary is kept at a constant temperature, and the other boundary is kept at a constant heat flux. They obtained local Nusselt-number equations for these four cases. Due to the large amount and complexity of their findings, their results will not be described in further detail in this section. Rogers and Yao (1990) studied the hydro-
3. OVERVIEW OF THE CFD STUDIES
3.1. Model description A body-fitted three-dimensional grid model of a mantle tank is developed. Fig. 4 shows the model of the tank and the grid. Only half a tank is modelled as symmetry is assumed in the centre plane through the inlet and outlet to and from the mantle. The model is further simplified by modelling the top and the bottom of the tank with horizontal walls instead of dome-shaped walls. Also, the top and bottom mantle walls are modelled with horizontal walls instead of sloping walls. The computational domain consists of the total heat storage, i.e. the inner tank, the mantle with the inlet and the outlet and the tank insulation. The number of nodes along the tank axis is I 5 42 and in radial and circumferential directions the numbers are J 5 25 and K 5 10. The model is supplied with input data for the water, tank material, tank insulation and the solar collector fluid (water). The boundary conditions for the model are the ambient temperature, mantle inlet temperature, outer convective heat transfer coefficients and solar collector fluid flow rate.
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L. J. Shah Table 1. Standard conditions and boundaries for the CFD simulations Inner tank volume
170 l
Tank material
Steel: l 560 W/ m K r 57820 kg / m 3 c p 5460 J / kg K Thickness: 0.003 m
Insulation
Mineral wool: l 50.045 W/ m K r 515 kg / m 3 c p 5800 J / kg K Thickness: top: 0.05 m; sides: 0.05 m bottom: 0.03 m Water 258C 0.13 m 2 K / W
Collector fluid Ambient temperature Outer thermal resistance
Fig. 4. Three-dimensional model of the mantle tank. Top, model outline with the mantle inlet and outlet. The model outline is slightly tilted. Bottom, computational mesh in the vertical plane.
3.1.1. Solving strategy. Turbulence is modelled with the k– ´ model and buoyancy is modelled with the Boussinesq assumption. Under relaxation parameters are used on velocities and viscosity to achieve convergence. The velocity–pressure coupling is treated using the SIMPLE algorithm and QUICK discretisation is used for convection terms. The time step in the transient simulations is 0.5 s. The standard boundary conditions are given in Table 1.
a step change of inlet temperature to 688C for another 2 h approximately. Then again, the mantle inlet temperature is dropped to 308C. In terms of the flow patterns in the tank, this case is similar to the mantle inlet temperature in solar water heaters on a ‘typical’ day. The irradiance is low in the morning, high at midday and low again in the afternoon. At the times t56000 s (1.67 h), t515,000 s (4.16 h) and t521,000 s (5.83 h) results for the analysis are extracted from the simulation (see Fig. 5). In this way, typical operation conditions are used for the analysis. To investigate how the tank height / diameter ratio, the mantle gap and the volume flow rate influence the convective heat transfer in the mantle and in the inner tank, a number of parameter variations are carried out with CFD modelling. The parameter variations are described in Table 2. In all the different models, the tank volume over the mantle is the same. Furthermore, only one parameter is varied at a time. Hereby the influence of the specific parameter is directly determined. However, from Table 2 it appears that at the height–diameter variations both the
3.2. Numerical test sequence The CFD model is used to simulate an |5-hlong heating sequence for an initially mixed inner tank. The mantle is supplied with constant flow and inlet temperature (308C) for |2 h followed by
Fig. 5. The data used for the heat transfer analysis are extracted at three specific hours.
Heat transfer correlations for vertical mantle heat exchangers
161
Table 2. Description of the parameter variations. The bold numbers describe the standard tank and the parameters varied with respect to the standard tank Model
Standard
Small mantle gap
Large mantle gap
Low flow rate
High flow rate
Small H /D ratio
Large H /D ratio
Tank volume, (m 3 ) Tank height, (m) Tank diameter (m) Mantle height, (m) Heat transfer area, (m 2 ) Mantle gap, (m) Mass flow rate, (kg / min)
0.170 1.449 0.386 0.700 0.862 0.036 0.4
0.170 1.449 0.386 0.700 0.862 0.018 0.4
0.170 1.449 0.386 0.700 0.862 0.054 0.4
0.170 1.449 0.386 0.700 0.862 0.036 0.2
0.170 1.449 0.386 0.700 0.862 0.036 0.6
0.170 0.644 0.579 0.311 0.572 0.036 0.4
0.170 2.174 0.315 1.050 1.059 0.036 0.4
height and diameter are changed. This is necessary to hold the inner tank volume at a constant.
3.3. Model validation Shah and Furbo (1998) validated the CFD model with measured data and a good degree of similarity between measured and simulated results was obtained. 4. OBTAINING HEAT TRANSFER CORRELATIONS
The outputs from a CFD-simulation are numerous. For each control volume temperature, enthalpy, three velocity components, viscosity, turbulence etc. are indicated. Among other things, also shearing forces and heat fluxes at the surfaces are given. To be able to control this amount of information it is important to keep in view which moments are to be investigated, as well as which method of analysis is required. In this study, the convective heat transfer between fluids and walls is to be analysed. To generalise the result processing, the simulation results are analysed by means of dimensionless heat transfer theory. For the heat transfer from the collector fluid to the inner tank wall, a brief review of the method of analysis follows below. The principles of this method are also used to identify the convective heat transfer coefficient from mantle fluid to outer mantle wall and from tank wall to consumption water. The required determination is the convective coefficient of heat transfer, h, from the solar collector fluid to the tank wall at any height in the mantle, cf. Fig. 6. The convective coefficient of heat transfer can be determined from the equation below: qxd h x 5 ]]]]]]] (7) T mantle fluid, x 2 T tank wall, x From the above equation, it appears that the
convective transition coefficient is analysed as a function of the x-direction, i.e. as a function of the distance from the top of the mantle. This means that the differences in the tangential direction are not taken into account. Values for q and T can be determined from the results of the CFD-simulation as the mantle is modelled by a number of control volumes (cf. Fig. 7) where these quantities are calculated. The heat flux from the solar collector fluid to the tank wall, q, is given directly in the CFD-results for each of the surfaces of the control volumes that faces the mantle fluid. The average heat flux for each layer in the direction of the x-axis is calculated in the following way:
O [q(x, z) ? A(x, z)] q 5 q 5 ]]]]]] O A(x, z) n
z 51
n
x
(8)
z 51
In a similar way the mean wall temperature, T wall , is calculated for each layer in the direction of the x-axis:
O [T(x, z) ? A(x, z)] 5 ]]]]]] O A(x, z) n
T wall
z 51
n
(9)
z 51
Fig. 6. The convective coefficient of heat transfer from solar collector fluid to tank wall is required.
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T mantle fluid, x 2 T tank wall, x , is rewritten to the dimensionless Rayleigh number, Ra x : g ? bx ? (T mantle fluid, x 2 T tank wall, x ) ? x 3 Ra x 5 ]]]]]]]]]]] ? Pr x n x2 (13) Thus, by means of these rewritings it is possible, for a given tank and with a given solar collector fluid, to determine the Nusselt number as a function of the Rayleigh number: Nu x 5 f(Ra x , . . . )
However, the above theory is only usable if it is the natural convection and not the forced convection that dominates in the mantle. This applies if the Rayleigh number is much larger than the squared Reynolds number, Re x , i.e. Ra x 4 Re x2 (Mills (1992)), where the Reynolds number is given as: u?x Re x 5 ]] (15) nx
Fig. 7. Sketch of the design of the mantle.
The temperature of the solar collector fluid is determined as a mean value of the temperatures in the control volumes with solar collector fluid. That is, for each layer in the direction of the x-axis a mean temperature for the solar collector fluid is calculated and this temperature is the average in both the radial and the tangential direction. Radial direction:
O [T(x, y, z) ?V(x, y, z)] T(x, z) 5 ]]]]]]] O V(x, y, z) m
y 51
m
(10)
Tangential direction: n
T(x) 5 T mantle fluid
z 51
n
(11)
z 51
In order to increase the use of the results of the analysis, the dimensionless heat transfer theory is now introduced. The convective coefficient of heat transfer, h x , is rewritten to the dimensionless Nusselt number, Nu x : hx ? x Nu x 5 ]] lx
Fig. 8 shows the squared Reynolds number as a function of the Rayleigh number at the three moments illustrated in Fig. 5. It appears that the Rayleigh number is much larger than the squared Reynolds number. Consequently, the natural convection is dominating and the chosen method of analysis is usable.
4.1. Heat transfer from mantle fluid to inner tank wall To correlate the data concerning heat transfer from the mantle fluid to the inner tank wall (cf. Fig. 6), the Nusselt number is plotted as a function of the Rayleigh number and as a function of the distance from the top of the mantle: Nu x 5 f(Ra x , C )
y 51
O [T(x, z) ?V(x, z)] 5 ]]]]]] O V(x, z)
(14)
(12)
Similarly, the driving force of the heat flow,
the distance from the top of the mantle C 5 ]]]]]]]]]]] the total mantle height x 5] H
(16)
(17)
Figs. 9–11 show the Nusselt number as a function of Rayleigh number and as a function of the dimensionless height in the mantle, for different mantle gaps, different tank height–diameter ratios and for different flow rates, respectively. The points in the graph are the calculated values, whereas the lines are plotted from the correlations given in the figure. The correlations are calculated by the least squared fit through the points using the equation Nu 5 C1 ? (Ra ? C )C 2 where C1 and C2 are constants.
Heat transfer correlations for vertical mantle heat exchangers
163
Fig. 8. The squared Reynolds numbers as a function of the Rayleigh number. The values are based on data from CFD simulations with different flow rates in the mantle. Reynolds numbers are much smaller than Rayleigh numbers; therefore the natural convection is dominating.
There is a significant difference in the correlations with different mantle gaps and height–diameter ratios (Figs. 9 and 10), whereas the mantle flow rate variations do not have a unique influence on the heat transfer (Fig. 11). The constants C1 and C2 in the Nusselt– Rayleigh correlations for the heat transfer from the solar collector fluid in the mantle to the wall of the inner tank are summarised in Table 3. From the table, it appears that the constant C1 differs for the different parameter variations, whereas the C2 ,
the slope of the correlation, is the same for all variations. Thus, one can find a power law correlation of the form: Nu x 5 C3 ? (F )C 4 ? (Ra x ? C )0.28
(18)
where C3 and C4 are two new constants and F is a new parameter. In this case, the correlation is developed with: ro 2 ri F 5 ]] ri
(19)
Fig. 9. Nusselt number as a function of Rayleigh number and as a function of the dimensionless height in the mantle, for different mantle gaps. The lines are correlations based on the values in the graph.
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Fig. 10. Nusselt number as a function of Rayleigh number and as a function of the dimensionless height in the mantle, for different tank height–diameter ratios. The lines are correlations based on the values in the graph.
Fig. 11. Nusselt number as a function of Rayleigh number and as a function of the dimensionless height in the mantle, for different flow rates in the mantle. The lines are correlations based on the values in the graph.
Table 3. The constants used in the correlations Variation
C1
C2
Mantle gap50.018 m Mantle gap50.036 m Mantle gap50.054 m Height–diameter ratio51.11 Height–diameter ratio53.75 Height–diameter ratio56.90 Mantle flow rate50.2 kg / min Mantle flow rate50.4 kg / min Mantle flow rate50.6 kg / min
1.36 0.86 0.63 0.95 0.86 0.67 0.93 0.86 0.86
0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28
Fig. 12. Definition of r i and r o .
Heat transfer correlations for vertical mantle heat exchangers
165
Fig. 13. The ratio Nu x /(Ra x ? x /H )0.28 versus (r o 2 r i ) /r i for heat transfer from the mantle fluid to the inner tank wall.
where r i and r o are inner and outer annulus radii, respectively, see Fig. 12. In Fig. 13 C1 5 Nu x /(Ra x ? x /H )0.28 is plotted versus F. The values of C1 are found in Table 3 and the corresponding value of F is calculated from the equation above. From the figure it appears that C3 5 0.28 and C4 5 2 0.63. Thus, the final equation becomes:
S
ro 2 ri Nu x 5 0.28 ? ]] ri
D
20.63
S
x ? Ra x ? ] H
D
0.28
(20)
10
which is valid for Ra x , 10 . Fluid properties are evaluated at the film temperature: T fluid 1 T wall T film 5 ]]]]. 2
(21)
4.2. Heat transfer from mantle fluid to outer mantle wall
could be used also for the heat transfer from the mantle fluid to the outer mantle wall (cf. Fig. 14), it would simplify the numerical tank model, in which the correlations are to be implemented. Shah (1999) showed that the developed correlation is usable also for determining heat transfer between the mantle fluid and the mantle wall. However, now the Rayleigh number is based on the temperature difference between the mantle fluid and the outer mantle wall: g ? bx ? (T mantle fluid, x 2 T outer mantle wall, x ) ? x 3 Rax 5 ]]]]]]]]]] ? Prx . n 2x
(22)
4.3. Heat transfer from tank wall to consumption water
If the correlation found in the previous section
To correlate the data concerning heat transfer from the inner tank wall to the consumption water (cf. Fig. 15) the Nusselt number is plotted as a
Fig. 14. The convective heat transfer coefficient from the solar collector fluid to the outer mantle tank wall is required.
Fig. 15. The convective heat transfer coefficient from inner tank wall to the consumption water is required.
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Fig. 16. Nusselt number as a function of Rayleigh number and as a function of the dimensionless tank height for different tank height–diameter ratios. The lines are correlations based on the values in the graph.
function of the Rayleigh number and as a function of the distance from the tank bottom Nu x 5 f(Ra x , C )
Nu x 5 F(F ) ? (Ra x ? C )0.19
(25)
(23)
where C3 and C4 are two new constants and F is a new parameter. In this case, the correlation is developed with
(24)
H F 5] D
the distance from the inner bottom of the tank x C 5 ]]]]]]]]]] 5 ]. the total inner tank height H
Fig. 16 shows the Nusselt number as a function of Rayleigh number and as a function of the dimensionless height in the mantle (C ) for different tank height–diameter ratios. The points in the graph are the calculated values, whereas the lines are plotted from the correlations given in the figure. The correlations are calculated by the least squared fit through the points using the equation Nu 5 C1 ? (Ra ? C )C 2 where C1 and C2 are constants. The constants C1 and C2 in the Nusselt– Rayleigh correlations for the heat transfer from the inner tank wall to the consumption water are summarised in Table 4. From the table, it appears that the constant C1 differs for the different parameter variations, whereas C2 , the slope of the correlation, is the same for all variations. Thus, one can find a new correlation of the form:
Table 4. The constants used in the correlations Variation
C1
C2
Height–diameter ratio51.11 Height–diameter ratio53.75 Height–diameter ratio56.90
4.01 3.71 1.70
0.19 0.19 0.19
(26)
where H and D are inner tank height and tank diameter, respectively. In Fig. 17 C1 5 Nu x /(Ra xC )0.19 is plotted versus F. The values of C1 are found in Table 4 and the corresponding value of F is calculated from the equation above. From the figure it appears that C3 5 0.28 and C4 5 2 0.63 3.103 F(F ) 5 4.501 2 ]] F
(27)
Thus, the final equation becomes:
S
DS
3.103 ? D x Nu x 5 4.501 2 ]]] ? Ra x ? ] H H
D
0.19
(28)
which is valid for Ra x , 10 11 . Fluid properties are evaluated at the film temperature: T fluid 1 T wall T film 5 ]]]] 2
(29)
In Eq. (28) only the height and the diameter of the tank and Rayleigh’s number are included. Other parameter studies were performed (as in the previous section) but no correlation was achieved with these parameters.
Heat transfer correlations for vertical mantle heat exchangers
167
Fig. 17. The ratio Nu x /(Ra x ? x /H )0.19 versus H /D for heat transfer from inner tank wall to the domestic hot water.
5. THE SIMPLIFIED NUMERICAL MODEL
The mantle tank model was originally developed by Berg (1990) and later modified by Shah and Furbo (1996). The changes made in this study only concern the convective heat transfer coefficients. The model is developed in cylindrical co-ordinates, which simplifies it to a two-dimensional problem. The model is divided into NZ 3 vertical layers to determine the vertical temperature stratification. Fig. 18 illustrates the numerical mantle tank model. In order to determine the horizontal temperature stratification, each layer is further divided
into two or four control elements: one element for the domestic water, one for the tank wall and in the mantle layers one further element for the mantle fluid and one for the mantle wall. Each element is assumed to have a uniform temperature. The energy balance in each layer, i, can be presented as: DQ i 5 Q sol 1 Q mix 1 Q cond 1 Q loss 1 Q aux 1 Q tap (30) Q mix means energy transport caused by mixing when a lower layer has a higher temperature than a higher layer. The temperature gradient along the vertical direction is calculated by modelling the inner tank with NZ 3 elements. The inner tank contains consumption water and the energy balances must therefore include eventual draw-off. The correlations derived in the previous sections enter the energy balance as a part of Q loss as energy transfer through the walls is treated as losses (positive or negative) for each control element. 6. COMPARISON WITH AN OUTDOOR MEASURED SYSTEM
Fig. 18. Numerical mantle tank model.
The described system is a Danish commercial marketed system. The system is based on a Danlager, 2000 mantle tank (Otto and Clausen, 1996), which is connected to two solar collector panels with a total transparent area of 4.02 m 2 . An electric heating element is placed inside the tank to heat up the consumption water if the required
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temperature is not reached with the solar heating alone. The solar collector panels are of the type Nordsol 1 (Olesen and Clausen, 1997) and they consist of a cover of 3-mm hardened iron-free glass and an absorber based on a double plate absorber. The solar collector is placed on a 458tilted roof facing south. The solar collector is connected to the storage tank through 17.9 m foam-insulated outlet pipe and 15.1 m foam-insulated return pipe. The pipes are 18 / 16 mm copper pipes. The solar collector loop is equipped with a Grundfos circulation pump (type UPS 25-40) which has been running at stage 1 throughout the measuring period. The circulation pump is controlled by a differential thermostat, which measures the temperature difference between the outlet from the solar collector and the bottom of the storage tank. The differential thermostat has a start / stop set point at 6 / 2 K. During a period, the power consumption of the
Fig. 19. The collectors on the roof.
circulation pump and the control system was metered at 30 W and 2 W, respectively. The power consumption of the electric heating element was metered at 1200 W. Table 5 summarises the system data. Fig. 19 shows the collector panel on the roof and Fig. 20 shows the insulated storage in a cabinet.
Table 5. System data Tank design Inner tank Total volume 0.265 m 3 Inner height 1.38 m Inner diameter 0.494 m Tank wall thickness 0.003 m Auxiliary energy volume: 0.099 m 3 . T set 550.58C Mantle Mantle height Mantle gap Tank wall thickness
0.7 m 0.0105 m 0.003 m
Insulation Material Top Side, over / under mantle Side, mantle Bottom
PUR foam 0.065 m 0.05 m 0.035 m 0.03 m
Thermal bridges Tank top Tank bottom
0 W/ K 0 W/ K
Collector Area Start efficiency Heat loss coefficient Incident angle modifier (tangens eqn.) Heat capacity Tilt Orientation
4.02 m 2 0.78 5.09 W/ m 2 K a54.2 1010 J / m 2 K 458 South
Solar collector loop Pipe material Outer diameter Inner diameter Insulation thickness (PUR foam) Pipe length, storage to collector, indoor Pipe length, storage to collector, outdoor Pipe length, collector to storage, outdoor Pipe length, collector to storage, indoor
Copper 0.010 m 0.008 m 0.01 m 4.6 m 13.3 m 10 m 5.1 m
Fig. 20. The insulated storage.
Heat transfer correlations for vertical mantle heat exchangers
To make sure that the simulation model behaves as expected; all energy quantities in the SDHW system must be evaluated. Thus, comparisons of solar energy transferred to the fluid in the collector evaluate the collector model. Comparisons of measured and calculated solar energy transferred to the storage, energy tapped from storage tank and auxiliary energy supplied to the tank evaluate the tank model. Comparisons of measured and calculated net utilised solar energy defined as energy tapped from the storage minus auxiliary energy supplied to the storage evaluate the total system simulation. Not only the energy quantities are important for the simulation model evaluation. In addition, the temperature levels are of relevance. The consumption water temperature at the top of the tank has a significant importance combined with the auxiliary energy supply. If this temperature is modelled either too high or too low it will have a major effect on the modelled auxiliary energy supply and thus on the net utilised solar energy. Figs. 21–24 show measured and calculated solar energy transferred to the storage, measured and calculated auxiliary energy supply, measured and calculated energy tapped from the storage and measured and calculated net utilised solar energy. The figures clearly show that a good agreement
169
Fig. 23. Measured and calculated daily energy tapped from the storage.
Fig. 24. Measured and calculated daily net utilised solar energy.
Fig. 21. Measured and calculated daily solar energy to storage.
Fig. 25. Measured and calculated temperatures at the top of the tank in System 3.
Fig. 22. Measured and calculated daily auxiliary energy to storage.
Fig. 26. Measured and calculated mantle inlet and outlet temperatures for three randomly selected days. The horizontal line (T3) indicates that the solar collector is out of operation.
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Table 6. Measured and calculated energy quantities Energy quantity
Calculated (kWh)
Measured (kWh)
Difference (%)
Solar energy to storage Auxiliary energy to storage Energy tapped from storage Net utilised solar energy
155 54 186 132
159 53 187 134
2.5 21.9 0.5 1.5
between measured and calculated quantities is achieved. The results are summarised in Table 6 where it also appears that the differences lie below the measuring inaccuracy. Comparisons of measured and calculated temperatures are shown in Figs. 25 and 26. Fig. 25 shows the measured and calculated temperatures at the top of the tank and it appears that a good agreement is achieved. Fig. 26 shows comparisons of calculated and measured mantle inlet and outlet temperatures for three randomly selected days in the test period. From the figures, it is seen that a good agreement of measured and calculated temperatures is achieved. 7. CONCLUSION
In this paper heat transfer in vertical mantle heat exchangers for application in low flow solar domestic hot water systems is investigated. CFD models are used to take mantle tanks into the calculation and detailed information is obtained for the convection and heat transfer characteristics of both mantle and inner tank. Two new heat transfer correlations for vertical mantle heat exchangers with top entry port and bottom exit port are developed based on computational fluid dynamic modelling of whole vertical mantle tanks. The correlations are developed for heat transfer between (1) the solar collector fluid and the inner tank wall, (2) the solar collector fluid and the mantle wall and (3) the inner tank walls and the domestic hot water. The correlations are combined with a heat storage model in a simulation program that predicts the yearly thermal performance of low flow solar domestic hot water systems based on mantle tanks. The model predictions of energy gains and temperatures are compared with outdoor measurements and the model is found to give reliable results. NOMENCLATURE A(x, z) A
area of each control volume of the current layer in the x-direction (m 2 ) height of annulus (m)
Dh g h K m n q qx q(x, z) DQ i Q mix Q cond Q sol Q loss Q aux Q tap T T(x, z) V(x, z)
V(x, y, z)
Pr u b
k l n
hydraulic diameter (m) acceleration due to gravity (m / s 2 ) convective heat transfer coefficient (W/ m 2 K) outer annulus diameter divided by inner annulus diameter (–) number of control volumes in the radial direction (–) number of control volumes in the tangential direction (–) heat flux (W/ m 2 ) is the average heat flow at a specific level in the x-direction (W/ m 2 ) heat flux at the surfaces of each control volume at a specific level in the x-direction (W/ m 2 ) energy stored in the layer i (J) convective energy due to mixing, entering layer no. i (J) diffusive energy, entering layer no. i (J) solar energy, entering layer no. i (J) is the heat loss from the layer no. i (J) is the auxiliary energy supplied to the layer no. i (J) is the energy amount tapped from the layer no. i (J) temperature (K) mean temperature at a single tangential position in the current level in the x-direction (K) sum of the volume of the control volumes at the single tangential positions in the current layer in the x-direction (m 3 ) volume of the control volumes at the single tangential and radial positions in the current layer in the x-direction (m 3 ) Prandtl number of the solar collector fluid (–) vertical velocity of the solar collector fluid (m / s) thermal volume expansion coefficient of the mantle fluid (1 / K) outer annulus diameter divided by inner annulus diameter (–) thermal conductivity for the solar collector fluid (W/ m K) viscosity of the solar collector fluid (m 2 / s)
REFERENCES Baur J. M., Klein S. A. and Beckman W. A. (1993) Simulation of water tanks with mantle heat exchangers. Proceedings ASES Annual Conference, Solar93, 286–291. Berg P. (1990). Højtydende solvarmeanlæg med sma˚ volumenstrømme, Technical University of Denmark, Thermal Insulation Laboratory, Report 209. Dhimdi S. and Bolle L. (1997) Natural convection: the effect of geometrical parameters. In Proceedings of the 4 th National Congress on Theoretical and Applied Mechanics, Leuven, Belgium, pp. 43–46.
Heat transfer correlations for vertical mantle heat exchangers ¨ H. and Pausinger T. (1997). MULTIPORT Store–Model Druck ¨ Stuttgart. Institut fur ¨ for TrnSys. Type 140, Universitat ¨ Thermodynamik und Warmetechnik. El-Shaarawi M. A. I. and Al-Nimr M. A. (1990) Fully developed laminar natural convection in open-ended vertical concentric annuli. Int. J. Heat Mass Transfer 33(9), 1873– 1889. Furbo S. (1998). Ydelser af solvarmeanlæg under laboratoriemæssige forhold, Technical University of Denmark, Department of Buildings and Energy, Note U-18. Furbo S. and Shah L. J. (1997) Laboratory tests of small SDHW systems. In Proceedings NorthSun’97, Espoo, Finland, Vol. I, pp. 153–159, ISBN 951-22-3567-6. Furbo S. (1993) Optimum design of small DHW low flow solar heating systems. In Proceedings, ISES Solar World Congress, Budapest, Hungary. Khan J. A. and Kumar R. (1989) Natural convection in vertical annuli: a numerical study for constant heat flux on the inner wall. J. Heat Transfer 111, 909–915. Klein S. A. et al. (1996). TRNSYS 14.1, User Manual, University of Wisconsin Solar Energy Laboratory. Mercer W. E., Pearce W. M. and Hitchcock J. E. (1967) Laminar forced convection in the entrance region between parallel flat plates. ASME J. Heat Transfer V89, 251–257. Mills A. F. (1992). Heat Transfer. International Student Edition, Irwin, ISBN 0-256-12817-0.
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˚ Olesen M. and Clausen I. (1997). Maling af solfangereffektivitet. Nordsol A /S. Nordsol1. Prøvningsrapport D2115 B, Prøvestationen for Solenergi. Otto W. and Clausen I. (1996). Beholderprøvning. Nilan A /S. Danlager 2000. Prøvningsrapport D3070, Prøvestationen for Solenergi. Rogers B. B. and Yao L. S. (1990) The effect of mixed convection instability of heat transfer in a vertical annulus. Int. J. Heat Mass Transfer 33(1), 79–90. Shah L. J. and Furbo S. (1998) Correlation of experimental and theoretical data for mantle tanks used in low flow SDHW systems. Solar Energy V64, 245–256. Shah L. J. (1999). Investigation and modelling of low flow SDHW systems. Report R-034, Department of Buildings and Energy, Technical University of Denmark, ISBN 87-7877035-1. Shah L. J. and Furbo S. (1996) Optimisation of mantle tanks for low flow solar heating systems. In Proceedings EuroSun ’96, Freiburg, September, Vol. I, pp. 369–375. Shah L. J., Morrison G. L. and Behnia M. (1998) Modelling mantle tanks for SDHW systems using PIV and CFD. In ˆ Slovenia, Vol. 2, pp. Proceedings EuroSun 98, Portoroz, III3.1–6. Shah L. J., Morrison G. L. and Behnia M. (1999) Characteristics of vertical mantle heat exchangers for solar water heaters. In Solar99 ISES Israel.
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Solar Energy Vol. 69(Suppl.), Nos. 1–6, pp. 157–171, 2000 2001 Elsevier Science Ltd S 0 0 3 8 – 0 9 2 X ( 0 1 ) 0 0 0 3 9 – 1 All rights reserved. Printed in Great Britain 0038-092X / 00 / $ - see front matter
www.elsevier.com / locate / solener
HEAT TRANSFER CORRELATIONS FOR VERTICAL MANTLE HEAT EXCHANGERS LOUISE JIVAN SHAH † Department of Civil Engineering, Technical University of Denmark DK-2800 Kgs. Lyngby, Denmark Received 10 July 2000; revised version accepted 23 February 2001 Communicated by ERICH HAHNE
Abstract—This paper investigates heat transfer in vertical mantle heat exchangers for application in low flow solar domestic hot water systems. Two new heat transfer correlations for vertical mantle heat exchangers with top entry port and bottom exit ports are developed. The correlations are based on computational fluid dynamic modelling of whole vertical mantle tanks. The correlations are combined with a heat storage model in a simulation program that predicts the yearly thermal performance of low flow solar domestic hot water systems based on mantle tanks. The model predictions of energy gains and temperatures are compared with outdoor measurements and the model is found to give reliable results. 2001 Elsevier Science Ltd. All rights reserved.
1.1. Advantages of the mantle tank system
1. INTRODUCTION
To optimise the design of components in SDHW systems, simulation tools are often needed. These tools must be able to simulate heat transfer and fluid dynamics in detail. Computational fluid dynamics (CFD) would be the obvious simulation choice as the Navier– Stokes and energy equations are solved with CFD. Thus, detailed solutions can be obtained for the convection and heat transfer characteristics. On the other hand, CFD computations are extremely CPU-time demanding — often so demanding that it is impossible to carry out the investigations wanted. However, if the numerous CFD results could somehow be extracted, sorted, and used in a less detailed model that is less CPU-time demanding it could be possible to use the CFD information in a less time demanding way. This is what is tried in this project. The case study concerns low flow solar domestic hot water (SDHW) systems based on mantle tanks. Low flow systems in northern Europe, especially Holland and Denmark, are often based on a vertical mantle tank as illustrated in Fig. 1. The heat exchange between the solar collector fluid and the consumption water takes place in the mantle. A control system and a pump are used in the solar collector loop, but the system can also be used as a thermosiphoning system. †
1.1.1. Thermal stratification. The mantle tank system is one of the simplest ways of producing high heat exchanger effectiveness while promoting thermal stratification. The mantle configuration provides a large heat transfer area and effective distribution of the collector loop flow over the wall of the tank. Most of the incoming mantle fluid seeks the thermal equilibrium level in the mantle, and thermal stratification in the mantle and the inner tank is not disturbed (Shah et al., 1999). 1.1.2. Cost. The cost of the tank is cheap compared to the increased performance initiated by the fine thermal stratification (IEA Task 14). Further, Furbo (1993) showed that the mantle tank system had the best cost / performance ratio compared to systems with external heat exchangers and side-arm and compared to systems with internal heat exchanger coils. 1.2. Disadvantages of the mantle tank system
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1.2.1. Size limitation. The mantle tank design is not suitable for large low flow SDHW systems, as the heat transfer area gets too small for tanks with volumes over 800–1000 l (Shah, 1999). As the heat storage has proved to be the most important system component (Furbo, 1998; Furbo and Shah, 1997), there has been an emphasis on this component in the case study. Therefore, CFD models are used to take mantle tanks into calculation and detailed information is obtained for the
157
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Fig. 2. Heat flux distribution over mantle heat transfer area based on one-dimensional heat transfer models.
Fig. 1. Simplified sketch of low flow SDHW system used in Denmark and Holland (Shah, 1999).
convection and heat transfer characteristics of both mantle and inner tank. With CFD results, general correlations are developed for heat transfer between: • the solar collector fluid and the inner tank wall • the solar collector fluid and the mantle wall • the inner tank walls and the domestic hot water. These correlations are used in a heat storage model in a simulation program that predicts the yearly thermal performance of low flow SDHW systems based on mantle tank. 2. PREVIOUS STUDIES OF HEAT TRANSFER IN VERTICAL ANNULI
2.1. Vertical annuli in solar applications Several SDHW simulation programs can take SDHW systems based on mantle tanks into calculations. In TRNSYS (Klein et al., 1996), two approaches have been used to model mantle flow. Baur et al. (1993) assumed that the heat transfer in the mantle passageway could be estimated from a correlation for flow between two flat plates as reported by Mercer et al. (1967):
S S
exchange rate (UA) is an input value to the model. The UA value can be modelled either as a constant or as a function of the mantle inlet temperature, the store temperature and the mantle flow rate. This model is very suitable for parameter identification during storage testing; however, it does not provide the heat transfer information and is thus less usable if a non-tested storage is to be modelled. For another simulation program Shah and Furbo (1998) evaluated heat transfer in mantle heat exchangers using numerical simulation of the flows in the mantle and inside the storage tank. For low flow rates, they showed that the flow was dominated by buoyancy effects and proposed that the heat transfer in the mantle flow could be correlated as:
S
x ] Nu(x) 5 0.46 ? Ra x ? ] Dh
D
0.28
(2)
Comparison of the distribution of heat flux over the mantle surface and the variation of the mantle fluid temperature predicted by the three approaches is shown in Figs. 2 and 3 for a mantle flow rate of 0.5 l / min, mantle inlet temperature of 508C, a uniform tank wall temperature of 208C and annulus gap of 3 cm. A constant heat transfer
D D
D 1.2 ]h 0.0606 ? Re ? Pr ? ] x Nu(x) 5 4.9 1 ]]]]]]]]] Dh 0.1 1 1 0.0909 ? Re ? Pr ? ] Pr x (1) ¨ and Pausinger (1997) Also for TRNSYS, Druck developed the Multiport model where the heat
Fig. 3. Mantle fluid temperature variation down the mantle based on one-dimensional heat transfer models.
Heat transfer correlations for vertical mantle heat exchangers
coefficient of 200 W/ m 2 K has been used for the curves presenting the approach used in the Multiport model. The natural convection model (Shah et al.) predicts substantially higher heat flux in the top section of the mantle and thus much faster cooling of the flow in the mantle. As recent studies have shown that natural convection is dominant in the mantle during low flow operation (Shah et al., 1998, 1999) the ‘buoyant approach’ is continued in this study.
2.2. Other numerical studies of vertical annuli Khan and Kumar (1989) conducted a numerical investigation to evaluate the effects of diameter ratio, k 5 r outer /r inner , and aspect ratio, A 5 (height of annulus) /(r outer 2 r inner ), in natural convection of gases in vertical annuli. The inner cylinder was at a constant surface heat flux, and the outer cylinder was at a constant temperature. The cases under consideration were 1# A#10 and 1# k #15. They obtained the following Nusselt-number correlation:
Nu 5 1.02 ? Ra 0.28
for 100 # Ra # 600
(5)
In this equation, they used the annulus gap as length scale. Dhimdi and Bolle (1997) determined heat transfer coefficients numerically for natural convection in vertical annuli for a large Grashof number using water as the heat transfer medium. Heat transfer results were related to radius ratios and aspect ratios using constant temperature boundaries. Analysing the curvature effect, they found the following Nusselt number equation: log (Nu) 5 2 0.17 ? log (K)1.82 1 2.573 for Gr 5 8 3 10 8
(6)
(3)
where L, the length of the annulus, is used as length scale. Also, they obtained a simpler correlation for A.5 and k .5 using the outer diameter of the inner cylinder as length scale: Nu di 5 0.455 ? Ra 0.202 di
dynamic stability of mixed convection in an annulus with a constant heat flux boundary applied to the inner annulus and an insulated outer boundary. For this case, they found that the parallel flow assumption could not be used. They presented a Nusselt–Rayleigh number equation for data in the region between linear stability and Ra 5 600:
where K is the radius ratio (K 5 r outer /r inner ) and the radius gap is used as the length scale.
Nu L 5 0.239 ? Ra L0.29 ? k ( 0.303 / k )10.316 ? A20.05 for Ra L , 10 6
159
for Ra L , 10 6
(4)
El-Shaarawi and Al-Nimr (1990) presented analytical solutions for fully developed natural convection in open-ended vertical annuli. They analysed the four fundamental boundary conditions: (1) constant temperature boundaries, where one of the walls is kept at the inlet ambient fluid temperature. (2) One boundary is kept at a constant heat flux, and the other boundary is perfectly insulated. (3) One boundary is kept at a constant temperature flux, and the other boundary is perfectly insulated. (4) One boundary is kept at a constant temperature, and the other boundary is kept at a constant heat flux. They obtained local Nusselt-number equations for these four cases. Due to the large amount and complexity of their findings, their results will not be described in further detail in this section. Rogers and Yao (1990) studied the hydro-
3. OVERVIEW OF THE CFD STUDIES
3.1. Model description A body-fitted three-dimensional grid model of a mantle tank is developed. Fig. 4 shows the model of the tank and the grid. Only half a tank is modelled as symmetry is assumed in the centre plane through the inlet and outlet to and from the mantle. The model is further simplified by modelling the top and the bottom of the tank with horizontal walls instead of dome-shaped walls. Also, the top and bottom mantle walls are modelled with horizontal walls instead of sloping walls. The computational domain consists of the total heat storage, i.e. the inner tank, the mantle with the inlet and the outlet and the tank insulation. The number of nodes along the tank axis is I 5 42 and in radial and circumferential directions the numbers are J 5 25 and K 5 10. The model is supplied with input data for the water, tank material, tank insulation and the solar collector fluid (water). The boundary conditions for the model are the ambient temperature, mantle inlet temperature, outer convective heat transfer coefficients and solar collector fluid flow rate.
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L. J. Shah Table 1. Standard conditions and boundaries for the CFD simulations Inner tank volume
170 l
Tank material
Steel: l 560 W/ m K r 57820 kg / m 3 c p 5460 J / kg K Thickness: 0.003 m
Insulation
Mineral wool: l 50.045 W/ m K r 515 kg / m 3 c p 5800 J / kg K Thickness: top: 0.05 m; sides: 0.05 m bottom: 0.03 m Water 258C 0.13 m 2 K / W
Collector fluid Ambient temperature Outer thermal resistance
Fig. 4. Three-dimensional model of the mantle tank. Top, model outline with the mantle inlet and outlet. The model outline is slightly tilted. Bottom, computational mesh in the vertical plane.
3.1.1. Solving strategy. Turbulence is modelled with the k– ´ model and buoyancy is modelled with the Boussinesq assumption. Under relaxation parameters are used on velocities and viscosity to achieve convergence. The velocity–pressure coupling is treated using the SIMPLE algorithm and QUICK discretisation is used for convection terms. The time step in the transient simulations is 0.5 s. The standard boundary conditions are given in Table 1.
a step change of inlet temperature to 688C for another 2 h approximately. Then again, the mantle inlet temperature is dropped to 308C. In terms of the flow patterns in the tank, this case is similar to the mantle inlet temperature in solar water heaters on a ‘typical’ day. The irradiance is low in the morning, high at midday and low again in the afternoon. At the times t56000 s (1.67 h), t515,000 s (4.16 h) and t521,000 s (5.83 h) results for the analysis are extracted from the simulation (see Fig. 5). In this way, typical operation conditions are used for the analysis. To investigate how the tank height / diameter ratio, the mantle gap and the volume flow rate influence the convective heat transfer in the mantle and in the inner tank, a number of parameter variations are carried out with CFD modelling. The parameter variations are described in Table 2. In all the different models, the tank volume over the mantle is the same. Furthermore, only one parameter is varied at a time. Hereby the influence of the specific parameter is directly determined. However, from Table 2 it appears that at the height–diameter variations both the
3.2. Numerical test sequence The CFD model is used to simulate an |5-hlong heating sequence for an initially mixed inner tank. The mantle is supplied with constant flow and inlet temperature (308C) for |2 h followed by
Fig. 5. The data used for the heat transfer analysis are extracted at three specific hours.
Heat transfer correlations for vertical mantle heat exchangers
161
Table 2. Description of the parameter variations. The bold numbers describe the standard tank and the parameters varied with respect to the standard tank Model
Standard
Small mantle gap
Large mantle gap
Low flow rate
High flow rate
Small H /D ratio
Large H /D ratio
Tank volume, (m 3 ) Tank height, (m) Tank diameter (m) Mantle height, (m) Heat transfer area, (m 2 ) Mantle gap, (m) Mass flow rate, (kg / min)
0.170 1.449 0.386 0.700 0.862 0.036 0.4
0.170 1.449 0.386 0.700 0.862 0.018 0.4
0.170 1.449 0.386 0.700 0.862 0.054 0.4
0.170 1.449 0.386 0.700 0.862 0.036 0.2
0.170 1.449 0.386 0.700 0.862 0.036 0.6
0.170 0.644 0.579 0.311 0.572 0.036 0.4
0.170 2.174 0.315 1.050 1.059 0.036 0.4
height and diameter are changed. This is necessary to hold the inner tank volume at a constant.
3.3. Model validation Shah and Furbo (1998) validated the CFD model with measured data and a good degree of similarity between measured and simulated results was obtained. 4. OBTAINING HEAT TRANSFER CORRELATIONS
The outputs from a CFD-simulation are numerous. For each control volume temperature, enthalpy, three velocity components, viscosity, turbulence etc. are indicated. Among other things, also shearing forces and heat fluxes at the surfaces are given. To be able to control this amount of information it is important to keep in view which moments are to be investigated, as well as which method of analysis is required. In this study, the convective heat transfer between fluids and walls is to be analysed. To generalise the result processing, the simulation results are analysed by means of dimensionless heat transfer theory. For the heat transfer from the collector fluid to the inner tank wall, a brief review of the method of analysis follows below. The principles of this method are also used to identify the convective heat transfer coefficient from mantle fluid to outer mantle wall and from tank wall to consumption water. The required determination is the convective coefficient of heat transfer, h, from the solar collector fluid to the tank wall at any height in the mantle, cf. Fig. 6. The convective coefficient of heat transfer can be determined from the equation below: qxd h x 5 ]]]]]]] (7) T mantle fluid, x 2 T tank wall, x From the above equation, it appears that the
convective transition coefficient is analysed as a function of the x-direction, i.e. as a function of the distance from the top of the mantle. This means that the differences in the tangential direction are not taken into account. Values for q and T can be determined from the results of the CFD-simulation as the mantle is modelled by a number of control volumes (cf. Fig. 7) where these quantities are calculated. The heat flux from the solar collector fluid to the tank wall, q, is given directly in the CFD-results for each of the surfaces of the control volumes that faces the mantle fluid. The average heat flux for each layer in the direction of the x-axis is calculated in the following way:
O [q(x, z) ? A(x, z)] q 5 q 5 ]]]]]] O A(x, z) n
z 51
n
x
(8)
z 51
In a similar way the mean wall temperature, T wall , is calculated for each layer in the direction of the x-axis:
O [T(x, z) ? A(x, z)] 5 ]]]]]] O A(x, z) n
T wall
z 51
n
(9)
z 51
Fig. 6. The convective coefficient of heat transfer from solar collector fluid to tank wall is required.
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L. J. Shah
T mantle fluid, x 2 T tank wall, x , is rewritten to the dimensionless Rayleigh number, Ra x : g ? bx ? (T mantle fluid, x 2 T tank wall, x ) ? x 3 Ra x 5 ]]]]]]]]]]] ? Pr x n x2 (13) Thus, by means of these rewritings it is possible, for a given tank and with a given solar collector fluid, to determine the Nusselt number as a function of the Rayleigh number: Nu x 5 f(Ra x , . . . )
However, the above theory is only usable if it is the natural convection and not the forced convection that dominates in the mantle. This applies if the Rayleigh number is much larger than the squared Reynolds number, Re x , i.e. Ra x 4 Re x2 (Mills (1992)), where the Reynolds number is given as: u?x Re x 5 ]] (15) nx
Fig. 7. Sketch of the design of the mantle.
The temperature of the solar collector fluid is determined as a mean value of the temperatures in the control volumes with solar collector fluid. That is, for each layer in the direction of the x-axis a mean temperature for the solar collector fluid is calculated and this temperature is the average in both the radial and the tangential direction. Radial direction:
O [T(x, y, z) ?V(x, y, z)] T(x, z) 5 ]]]]]]] O V(x, y, z) m
y 51
m
(10)
Tangential direction: n
T(x) 5 T mantle fluid
z 51
n
(11)
z 51
In order to increase the use of the results of the analysis, the dimensionless heat transfer theory is now introduced. The convective coefficient of heat transfer, h x , is rewritten to the dimensionless Nusselt number, Nu x : hx ? x Nu x 5 ]] lx
Fig. 8 shows the squared Reynolds number as a function of the Rayleigh number at the three moments illustrated in Fig. 5. It appears that the Rayleigh number is much larger than the squared Reynolds number. Consequently, the natural convection is dominating and the chosen method of analysis is usable.
4.1. Heat transfer from mantle fluid to inner tank wall To correlate the data concerning heat transfer from the mantle fluid to the inner tank wall (cf. Fig. 6), the Nusselt number is plotted as a function of the Rayleigh number and as a function of the distance from the top of the mantle: Nu x 5 f(Ra x , C )
y 51
O [T(x, z) ?V(x, z)] 5 ]]]]]] O V(x, z)
(14)
(12)
Similarly, the driving force of the heat flow,
the distance from the top of the mantle C 5 ]]]]]]]]]]] the total mantle height x 5] H
(16)
(17)
Figs. 9–11 show the Nusselt number as a function of Rayleigh number and as a function of the dimensionless height in the mantle, for different mantle gaps, different tank height–diameter ratios and for different flow rates, respectively. The points in the graph are the calculated values, whereas the lines are plotted from the correlations given in the figure. The correlations are calculated by the least squared fit through the points using the equation Nu 5 C1 ? (Ra ? C )C 2 where C1 and C2 are constants.
Heat transfer correlations for vertical mantle heat exchangers
163
Fig. 8. The squared Reynolds numbers as a function of the Rayleigh number. The values are based on data from CFD simulations with different flow rates in the mantle. Reynolds numbers are much smaller than Rayleigh numbers; therefore the natural convection is dominating.
There is a significant difference in the correlations with different mantle gaps and height–diameter ratios (Figs. 9 and 10), whereas the mantle flow rate variations do not have a unique influence on the heat transfer (Fig. 11). The constants C1 and C2 in the Nusselt– Rayleigh correlations for the heat transfer from the solar collector fluid in the mantle to the wall of the inner tank are summarised in Table 3. From the table, it appears that the constant C1 differs for the different parameter variations, whereas the C2 ,
the slope of the correlation, is the same for all variations. Thus, one can find a power law correlation of the form: Nu x 5 C3 ? (F )C 4 ? (Ra x ? C )0.28
(18)
where C3 and C4 are two new constants and F is a new parameter. In this case, the correlation is developed with: ro 2 ri F 5 ]] ri
(19)
Fig. 9. Nusselt number as a function of Rayleigh number and as a function of the dimensionless height in the mantle, for different mantle gaps. The lines are correlations based on the values in the graph.
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L. J. Shah
Fig. 10. Nusselt number as a function of Rayleigh number and as a function of the dimensionless height in the mantle, for different tank height–diameter ratios. The lines are correlations based on the values in the graph.
Fig. 11. Nusselt number as a function of Rayleigh number and as a function of the dimensionless height in the mantle, for different flow rates in the mantle. The lines are correlations based on the values in the graph.
Table 3. The constants used in the correlations Variation
C1
C2
Mantle gap50.018 m Mantle gap50.036 m Mantle gap50.054 m Height–diameter ratio51.11 Height–diameter ratio53.75 Height–diameter ratio56.90 Mantle flow rate50.2 kg / min Mantle flow rate50.4 kg / min Mantle flow rate50.6 kg / min
1.36 0.86 0.63 0.95 0.86 0.67 0.93 0.86 0.86
0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28
Fig. 12. Definition of r i and r o .
Heat transfer correlations for vertical mantle heat exchangers
165
Fig. 13. The ratio Nu x /(Ra x ? x /H )0.28 versus (r o 2 r i ) /r i for heat transfer from the mantle fluid to the inner tank wall.
where r i and r o are inner and outer annulus radii, respectively, see Fig. 12. In Fig. 13 C1 5 Nu x /(Ra x ? x /H )0.28 is plotted versus F. The values of C1 are found in Table 3 and the corresponding value of F is calculated from the equation above. From the figure it appears that C3 5 0.28 and C4 5 2 0.63. Thus, the final equation becomes:
S
ro 2 ri Nu x 5 0.28 ? ]] ri
D
20.63
S
x ? Ra x ? ] H
D
0.28
(20)
10
which is valid for Ra x , 10 . Fluid properties are evaluated at the film temperature: T fluid 1 T wall T film 5 ]]]]. 2
(21)
4.2. Heat transfer from mantle fluid to outer mantle wall
could be used also for the heat transfer from the mantle fluid to the outer mantle wall (cf. Fig. 14), it would simplify the numerical tank model, in which the correlations are to be implemented. Shah (1999) showed that the developed correlation is usable also for determining heat transfer between the mantle fluid and the mantle wall. However, now the Rayleigh number is based on the temperature difference between the mantle fluid and the outer mantle wall: g ? bx ? (T mantle fluid, x 2 T outer mantle wall, x ) ? x 3 Rax 5 ]]]]]]]]]] ? Prx . n 2x
(22)
4.3. Heat transfer from tank wall to consumption water
If the correlation found in the previous section
To correlate the data concerning heat transfer from the inner tank wall to the consumption water (cf. Fig. 15) the Nusselt number is plotted as a
Fig. 14. The convective heat transfer coefficient from the solar collector fluid to the outer mantle tank wall is required.
Fig. 15. The convective heat transfer coefficient from inner tank wall to the consumption water is required.
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L. J. Shah
Fig. 16. Nusselt number as a function of Rayleigh number and as a function of the dimensionless tank height for different tank height–diameter ratios. The lines are correlations based on the values in the graph.
function of the Rayleigh number and as a function of the distance from the tank bottom Nu x 5 f(Ra x , C )
Nu x 5 F(F ) ? (Ra x ? C )0.19
(25)
(23)
where C3 and C4 are two new constants and F is a new parameter. In this case, the correlation is developed with
(24)
H F 5] D
the distance from the inner bottom of the tank x C 5 ]]]]]]]]]] 5 ]. the total inner tank height H
Fig. 16 shows the Nusselt number as a function of Rayleigh number and as a function of the dimensionless height in the mantle (C ) for different tank height–diameter ratios. The points in the graph are the calculated values, whereas the lines are plotted from the correlations given in the figure. The correlations are calculated by the least squared fit through the points using the equation Nu 5 C1 ? (Ra ? C )C 2 where C1 and C2 are constants. The constants C1 and C2 in the Nusselt– Rayleigh correlations for the heat transfer from the inner tank wall to the consumption water are summarised in Table 4. From the table, it appears that the constant C1 differs for the different parameter variations, whereas C2 , the slope of the correlation, is the same for all variations. Thus, one can find a new correlation of the form:
Table 4. The constants used in the correlations Variation
C1
C2
Height–diameter ratio51.11 Height–diameter ratio53.75 Height–diameter ratio56.90
4.01 3.71 1.70
0.19 0.19 0.19
(26)
where H and D are inner tank height and tank diameter, respectively. In Fig. 17 C1 5 Nu x /(Ra xC )0.19 is plotted versus F. The values of C1 are found in Table 4 and the corresponding value of F is calculated from the equation above. From the figure it appears that C3 5 0.28 and C4 5 2 0.63 3.103 F(F ) 5 4.501 2 ]] F
(27)
Thus, the final equation becomes:
S
DS
3.103 ? D x Nu x 5 4.501 2 ]]] ? Ra x ? ] H H
D
0.19
(28)
which is valid for Ra x , 10 11 . Fluid properties are evaluated at the film temperature: T fluid 1 T wall T film 5 ]]]] 2
(29)
In Eq. (28) only the height and the diameter of the tank and Rayleigh’s number are included. Other parameter studies were performed (as in the previous section) but no correlation was achieved with these parameters.
Heat transfer correlations for vertical mantle heat exchangers
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Fig. 17. The ratio Nu x /(Ra x ? x /H )0.19 versus H /D for heat transfer from inner tank wall to the domestic hot water.
5. THE SIMPLIFIED NUMERICAL MODEL
The mantle tank model was originally developed by Berg (1990) and later modified by Shah and Furbo (1996). The changes made in this study only concern the convective heat transfer coefficients. The model is developed in cylindrical co-ordinates, which simplifies it to a two-dimensional problem. The model is divided into NZ 3 vertical layers to determine the vertical temperature stratification. Fig. 18 illustrates the numerical mantle tank model. In order to determine the horizontal temperature stratification, each layer is further divided
into two or four control elements: one element for the domestic water, one for the tank wall and in the mantle layers one further element for the mantle fluid and one for the mantle wall. Each element is assumed to have a uniform temperature. The energy balance in each layer, i, can be presented as: DQ i 5 Q sol 1 Q mix 1 Q cond 1 Q loss 1 Q aux 1 Q tap (30) Q mix means energy transport caused by mixing when a lower layer has a higher temperature than a higher layer. The temperature gradient along the vertical direction is calculated by modelling the inner tank with NZ 3 elements. The inner tank contains consumption water and the energy balances must therefore include eventual draw-off. The correlations derived in the previous sections enter the energy balance as a part of Q loss as energy transfer through the walls is treated as losses (positive or negative) for each control element. 6. COMPARISON WITH AN OUTDOOR MEASURED SYSTEM
Fig. 18. Numerical mantle tank model.
The described system is a Danish commercial marketed system. The system is based on a Danlager, 2000 mantle tank (Otto and Clausen, 1996), which is connected to two solar collector panels with a total transparent area of 4.02 m 2 . An electric heating element is placed inside the tank to heat up the consumption water if the required
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temperature is not reached with the solar heating alone. The solar collector panels are of the type Nordsol 1 (Olesen and Clausen, 1997) and they consist of a cover of 3-mm hardened iron-free glass and an absorber based on a double plate absorber. The solar collector is placed on a 458tilted roof facing south. The solar collector is connected to the storage tank through 17.9 m foam-insulated outlet pipe and 15.1 m foam-insulated return pipe. The pipes are 18 / 16 mm copper pipes. The solar collector loop is equipped with a Grundfos circulation pump (type UPS 25-40) which has been running at stage 1 throughout the measuring period. The circulation pump is controlled by a differential thermostat, which measures the temperature difference between the outlet from the solar collector and the bottom of the storage tank. The differential thermostat has a start / stop set point at 6 / 2 K. During a period, the power consumption of the
Fig. 19. The collectors on the roof.
circulation pump and the control system was metered at 30 W and 2 W, respectively. The power consumption of the electric heating element was metered at 1200 W. Table 5 summarises the system data. Fig. 19 shows the collector panel on the roof and Fig. 20 shows the insulated storage in a cabinet.
Table 5. System data Tank design Inner tank Total volume 0.265 m 3 Inner height 1.38 m Inner diameter 0.494 m Tank wall thickness 0.003 m Auxiliary energy volume: 0.099 m 3 . T set 550.58C Mantle Mantle height Mantle gap Tank wall thickness
0.7 m 0.0105 m 0.003 m
Insulation Material Top Side, over / under mantle Side, mantle Bottom
PUR foam 0.065 m 0.05 m 0.035 m 0.03 m
Thermal bridges Tank top Tank bottom
0 W/ K 0 W/ K
Collector Area Start efficiency Heat loss coefficient Incident angle modifier (tangens eqn.) Heat capacity Tilt Orientation
4.02 m 2 0.78 5.09 W/ m 2 K a54.2 1010 J / m 2 K 458 South
Solar collector loop Pipe material Outer diameter Inner diameter Insulation thickness (PUR foam) Pipe length, storage to collector, indoor Pipe length, storage to collector, outdoor Pipe length, collector to storage, outdoor Pipe length, collector to storage, indoor
Copper 0.010 m 0.008 m 0.01 m 4.6 m 13.3 m 10 m 5.1 m
Fig. 20. The insulated storage.
Heat transfer correlations for vertical mantle heat exchangers
To make sure that the simulation model behaves as expected; all energy quantities in the SDHW system must be evaluated. Thus, comparisons of solar energy transferred to the fluid in the collector evaluate the collector model. Comparisons of measured and calculated solar energy transferred to the storage, energy tapped from storage tank and auxiliary energy supplied to the tank evaluate the tank model. Comparisons of measured and calculated net utilised solar energy defined as energy tapped from the storage minus auxiliary energy supplied to the storage evaluate the total system simulation. Not only the energy quantities are important for the simulation model evaluation. In addition, the temperature levels are of relevance. The consumption water temperature at the top of the tank has a significant importance combined with the auxiliary energy supply. If this temperature is modelled either too high or too low it will have a major effect on the modelled auxiliary energy supply and thus on the net utilised solar energy. Figs. 21–24 show measured and calculated solar energy transferred to the storage, measured and calculated auxiliary energy supply, measured and calculated energy tapped from the storage and measured and calculated net utilised solar energy. The figures clearly show that a good agreement
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Fig. 23. Measured and calculated daily energy tapped from the storage.
Fig. 24. Measured and calculated daily net utilised solar energy.
Fig. 21. Measured and calculated daily solar energy to storage.
Fig. 25. Measured and calculated temperatures at the top of the tank in System 3.
Fig. 22. Measured and calculated daily auxiliary energy to storage.
Fig. 26. Measured and calculated mantle inlet and outlet temperatures for three randomly selected days. The horizontal line (T3) indicates that the solar collector is out of operation.
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Table 6. Measured and calculated energy quantities Energy quantity
Calculated (kWh)
Measured (kWh)
Difference (%)
Solar energy to storage Auxiliary energy to storage Energy tapped from storage Net utilised solar energy
155 54 186 132
159 53 187 134
2.5 21.9 0.5 1.5
between measured and calculated quantities is achieved. The results are summarised in Table 6 where it also appears that the differences lie below the measuring inaccuracy. Comparisons of measured and calculated temperatures are shown in Figs. 25 and 26. Fig. 25 shows the measured and calculated temperatures at the top of the tank and it appears that a good agreement is achieved. Fig. 26 shows comparisons of calculated and measured mantle inlet and outlet temperatures for three randomly selected days in the test period. From the figures, it is seen that a good agreement of measured and calculated temperatures is achieved. 7. CONCLUSION
In this paper heat transfer in vertical mantle heat exchangers for application in low flow solar domestic hot water systems is investigated. CFD models are used to take mantle tanks into the calculation and detailed information is obtained for the convection and heat transfer characteristics of both mantle and inner tank. Two new heat transfer correlations for vertical mantle heat exchangers with top entry port and bottom exit port are developed based on computational fluid dynamic modelling of whole vertical mantle tanks. The correlations are developed for heat transfer between (1) the solar collector fluid and the inner tank wall, (2) the solar collector fluid and the mantle wall and (3) the inner tank walls and the domestic hot water. The correlations are combined with a heat storage model in a simulation program that predicts the yearly thermal performance of low flow solar domestic hot water systems based on mantle tanks. The model predictions of energy gains and temperatures are compared with outdoor measurements and the model is found to give reliable results. NOMENCLATURE A(x, z) A
area of each control volume of the current layer in the x-direction (m 2 ) height of annulus (m)
Dh g h K m n q qx q(x, z) DQ i Q mix Q cond Q sol Q loss Q aux Q tap T T(x, z) V(x, z)
V(x, y, z)
Pr u b
k l n
hydraulic diameter (m) acceleration due to gravity (m / s 2 ) convective heat transfer coefficient (W/ m 2 K) outer annulus diameter divided by inner annulus diameter (–) number of control volumes in the radial direction (–) number of control volumes in the tangential direction (–) heat flux (W/ m 2 ) is the average heat flow at a specific level in the x-direction (W/ m 2 ) heat flux at the surfaces of each control volume at a specific level in the x-direction (W/ m 2 ) energy stored in the layer i (J) convective energy due to mixing, entering layer no. i (J) diffusive energy, entering layer no. i (J) solar energy, entering layer no. i (J) is the heat loss from the layer no. i (J) is the auxiliary energy supplied to the layer no. i (J) is the energy amount tapped from the layer no. i (J) temperature (K) mean temperature at a single tangential position in the current level in the x-direction (K) sum of the volume of the control volumes at the single tangential positions in the current layer in the x-direction (m 3 ) volume of the control volumes at the single tangential and radial positions in the current layer in the x-direction (m 3 ) Prandtl number of the solar collector fluid (–) vertical velocity of the solar collector fluid (m / s) thermal volume expansion coefficient of the mantle fluid (1 / K) outer annulus diameter divided by inner annulus diameter (–) thermal conductivity for the solar collector fluid (W/ m K) viscosity of the solar collector fluid (m 2 / s)
REFERENCES Baur J. M., Klein S. A. and Beckman W. A. (1993) Simulation of water tanks with mantle heat exchangers. Proceedings ASES Annual Conference, Solar93, 286–291. Berg P. (1990). Højtydende solvarmeanlæg med sma˚ volumenstrømme, Technical University of Denmark, Thermal Insulation Laboratory, Report 209. Dhimdi S. and Bolle L. (1997) Natural convection: the effect of geometrical parameters. In Proceedings of the 4 th National Congress on Theoretical and Applied Mechanics, Leuven, Belgium, pp. 43–46.
Heat transfer correlations for vertical mantle heat exchangers ¨ H. and Pausinger T. (1997). MULTIPORT Store–Model Druck ¨ Stuttgart. Institut fur ¨ for TrnSys. Type 140, Universitat ¨ Thermodynamik und Warmetechnik. El-Shaarawi M. A. I. and Al-Nimr M. A. (1990) Fully developed laminar natural convection in open-ended vertical concentric annuli. Int. J. Heat Mass Transfer 33(9), 1873– 1889. Furbo S. (1998). Ydelser af solvarmeanlæg under laboratoriemæssige forhold, Technical University of Denmark, Department of Buildings and Energy, Note U-18. Furbo S. and Shah L. J. (1997) Laboratory tests of small SDHW systems. In Proceedings NorthSun’97, Espoo, Finland, Vol. I, pp. 153–159, ISBN 951-22-3567-6. Furbo S. (1993) Optimum design of small DHW low flow solar heating systems. In Proceedings, ISES Solar World Congress, Budapest, Hungary. Khan J. A. and Kumar R. (1989) Natural convection in vertical annuli: a numerical study for constant heat flux on the inner wall. J. Heat Transfer 111, 909–915. Klein S. A. et al. (1996). TRNSYS 14.1, User Manual, University of Wisconsin Solar Energy Laboratory. Mercer W. E., Pearce W. M. and Hitchcock J. E. (1967) Laminar forced convection in the entrance region between parallel flat plates. ASME J. Heat Transfer V89, 251–257. Mills A. F. (1992). Heat Transfer. International Student Edition, Irwin, ISBN 0-256-12817-0.
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˚ Olesen M. and Clausen I. (1997). Maling af solfangereffektivitet. Nordsol A /S. Nordsol1. Prøvningsrapport D2115 B, Prøvestationen for Solenergi. Otto W. and Clausen I. (1996). Beholderprøvning. Nilan A /S. Danlager 2000. Prøvningsrapport D3070, Prøvestationen for Solenergi. Rogers B. B. and Yao L. S. (1990) The effect of mixed convection instability of heat transfer in a vertical annulus. Int. J. Heat Mass Transfer 33(1), 79–90. Shah L. J. and Furbo S. (1998) Correlation of experimental and theoretical data for mantle tanks used in low flow SDHW systems. Solar Energy V64, 245–256. Shah L. J. (1999). Investigation and modelling of low flow SDHW systems. Report R-034, Department of Buildings and Energy, Technical University of Denmark, ISBN 87-7877035-1. Shah L. J. and Furbo S. (1996) Optimisation of mantle tanks for low flow solar heating systems. In Proceedings EuroSun ’96, Freiburg, September, Vol. I, pp. 369–375. Shah L. J., Morrison G. L. and Behnia M. (1998) Modelling mantle tanks for SDHW systems using PIV and CFD. In ˆ Slovenia, Vol. 2, pp. Proceedings EuroSun 98, Portoroz, III3.1–6. Shah L. J., Morrison G. L. and Behnia M. (1999) Characteristics of vertical mantle heat exchangers for solar water heaters. In Solar99 ISES Israel.