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General model for testing solar domestic hot water systems
Solar Energy Materials and Solar Cells 28 (1992) 93-102 North-Holland
Solar Energy Materials and Solar Cells
General model for testing solar domestic hot water systems W. Spirkl and J. Muschaweck Sektion Physik, Ludwig-Maximilians-Universitiit Miinchen, 14/-8000 Munich, Germany Received 20 May 1992
For modelling solar domestic hot water (SDHW) systems, a plug flow store model is presented. It is used in conjunction with the dynamic system testing algorithm to predict the long term performance of S D H W systems from short term test data. The basic property of a plug flow model is its capability of modelling drawoffs without any mixing inside the store. As an extension, the model developed in this paper covers the range between pure plug flow and full mixing. The degree of draw off mixing is characterized by a continuous parameter instead of a (discrete) n u m b e r of thermal nodes which actually is defined neither for a plug flow model nor for a real system. Less than ten model parameters characterize a system under test. It was found that the parameters can be identified in short time (two to four weeks outdoors) for a large class of systems, e.g. thermosyphon systems or integrated collector store (ICS) systems, systems with load side heat exchanger, auxiliary heater, and heat pipe collectors. A summary of experimental results for different systems is given.
I. Introduction At the University of Munich a method was developed to predict the long term performance of solar domestic hot water (SDHW) systems from a short term test. This method, called dynamic system testing (DST) method [1,2], was adopted by the International Energy Agency (lEA), and an lEA working group concerned solely with the DST method, the Dynamic System Testing Group (DSTG), worked from 1989 to 1991 [3]. This paper describes the progress made within the DSTG, with special emphasis on the plug flow model used in the DST method. Fig. 1 shows the input and output variables for a SDHW system under test. Only external quantities need to be measured. In ref. [1], an algorithm is presented which allows parameter identification from measured data using non-linear, dynamic models. Although the correlation model with two state variables (store temperatures) described there was found to be sufficiently accurate in many cases, Correspondence to: W. Spirkl, Sektion Physik, Ludwig-Maximilians-Universit~it Miinchen, Amalienstrasse 54, W-8000 Munich, Germany.
0927-0248/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
q4
IV. Spirkl, J. Muschaweek / Solar domestic hot water systems
~a ])(1 ll X
(C,~, TeA, .)
Pc
/
(Tcw, ds) Fig. 1. Input and output variables for a S D H W system. The system within the thick box is viewed as a black box. Thus, no intrusive m e a s u r e m e n t s are needed (like e.g. collector power or store temperatures).
a more detailed model had to be developed to include systems with integrated auxiliary heater a n d / o r high store stratification. Therefore we decided to retain the DST method, but to replace the correlation model by a plug flow model. This term means that the temperature distribution within the store is assumed to vary only in one dimension (i.e. in vertical direction), and that any flow through the store will just shift this temperature distribution upwards for load flow,downwards for collector loop flow (see fig. 2). In principle, a simulation program like TRNSYS [4], which also provides a plug flow model, could be used. But from a p a r a m e t e r identification point of view, a S D H W system modelled by T R N S Y S has too many free parameters. It is therefore impossible to identify the parameters from short term test data. In comparison to the TRNSYS model, we made simplifying assumptions; e.g. extra store losses at the bottom and the top are summarized in a global loss coefficient. On the other hand, the plug flow concept was extended to take into account mixing caused by load flow. The drawoff mixing effect is modelled using a diffusion equation.
[
Load outlet
t
T
[
h=l
h=0
T
Load inlet Fig. 2. Drawoff in the plug flow model.
W. Spirkl, Z Muschaweck / Solar domestic hot water .sTstems
95
2. The plug flow model for SDHW systems 2.1. The store The main assumptions on the store for the model presented here are listed below. For now, we restrict ourselves to systems with a solar loop heat exchanger immersed at the bottom of the store and no load side heat exchanger: (1) The state of the system at time t is characterized by the one-dimensional distribution T(t, h) of the store temperature, with the normalized height h in the range [0, 1], i.e. horizontal temperature gradients are neglected. This assumption implies neglecting of different effects such as convection caused by store losses as well as horizontal temperature gradients caused by the geometry of the solar loop heat exchanger. (2) The cold water is injected at height h = 0, the load is withdrawn at height h = 1. The remaining water in the store is shifted by an according value (plug flow). (3) The collector power is brought in at height h = 0 and is transported to water above by natural convection, see item (5). This is equivalent to a heat exchanger with no vertical extension immersed at the bottom of the store. (4) The auxiliary power is brought in at height h = 1 -faux and is transported to water above by convection, see item (5). (5) Local convection assures that ~T/Oh >10 holds. (6) Mixing or heat conduction do not occur for zero load capacitance rate (Cs = 0), except for fulfilling the relation mentioned in (5). (7) Cold water mixing is modelled by a diffusion term: 0T 0t
0h
+---.
(1)
It is assumed that the (time dependent) diffusion coefficient D is correlated with the load:
D ( t ) = D L C s / C s.
(2)
Fig. 3 shows the resulting step response for different values of the drawoff mixing constant D L. Note that, due to eq. (2), the step response does not depend on CsFor D L = 0, the drawoff accords to fig. 2 (pure plug flow). For D E > 0, each drawoff is associated with mixing (extended plug flow). (8) The loss coefficient is equally distributed over the height ( d U s / d h = 0). (9) The heat capacity is equally distributed over the height ( d C s / d h = 0). The capacity does not depend on the temperature (latent heat storage presumably cannot be modelled this way).
2.2. The collector loop The collector loop is modelled according to the Hottel-Whillier-Bliss equation [5] neglecting thermal capacitance and fluid flow time. The collector loop power
14/.Spirkl, J. Muschaweck / Solar domestic hot water ~vstems
96
1"c for solar irradiance Gt* in collector plane, collector ambient temperature TeA and inlet temperature T is modelled by Pc=A~[Gt*-u~.(T-TcA)]
+
'
where
[x]~={;
forx>0 otherwise"
(3)
For a collector with net area A c, absorptance-transmittance ( a z ) , heat removal factor F R and specific loss coefficient u c, the model parameters A~. and u~. are given by
A~ =AcFR(ar ) ,
u~. = U c / ( a z ) .
(4)
The loss coefficient is modelled as a linear function of the wind speed v: uc(,J)
= u
(O) + u , , , .
(5)
2.3. The partial differential equation In the sequel, the plug flow model is formulated in a partial differential equation for the special case of a heat exchanger immersed at the bottom of the store. The function T(t, h) is modelled as the limit of the solutions of the following parabolic partial differential equations for • ~ 0:
Cs aT(t,a~h) _ 8 , ( h ) A ~ [ G *
- u~(T-
TEA)] +
+ 3 , ( h + f , ox - 1)P~,u× -
Us(T- TsA)
+
+ a A h ) ( r c w - T) a (DICsaT)
+ Oh b exp
• ~
.
(6)
The meaning of the terms of the right side is: collector gain, auxiliary power, store losses, plug flow, diffusion and convection, respectively. Here, TCA and TSA denote ambient temperatures of the collector and the store, respectively; Paux denotes the auxiliary power. The model parameters U s and C s represent the total loss coefficient and the thermal capacity of the store. For e ~ 0, the function 6, converges to a Dirac distribution at zero: /
a , ( x ) = J _e _~/~
~0
e
for x > 0,
(7)
otherwise.
Convection is modelled by a diffusion process with a diffusion coefficient depend-
Hd Spirkl, J. Muschaweck / Solar domestic: hot water systems'
97
g 0.6
0.4
0 0
0.5
I
1.5
N o r m a l i z e d Load Volume
Fig. 3. Step response of the store to a change in the cold water temperature for different values of Dl.
(0, 0.01, 0.l, 1). ing on the temperature gradient (a and b are arbitrary positive constants). This ensures that in the limit •--+ 0 the requirement "OT/ah >10 is fulfilled exactly. Furthermore, with • --+ 0 the heat exchanger, the cold water inlet and the auxiliary heater are modelled with zero extension. We now go beyond the restrictions made at the beginning of this section. 2.4. Load side heat exchanger A load side heat exchanger is characterized by its thermal resistance R L. The capacitance rate Cs is reduced by the factor q~(RLCs), q~(x)= ( 1 - e - ~ ) / x , see fig. 4. 2.5. External collector loop heat exchanger An external heat exchanger is modelled using an additional p a r a m e t e r S c by a direct collector loop with fixed capacitance rate Cc: d c =At*u,*.~( I - e-~'c),
S c. > 0.
• Ts
cs
Tow
(8)
7
,Ts
0} 0s
Tow
Fig. 4. Modelling of a load side heat exchanger. Within the model, the heat exchanger is replaced by a mixing valve outside the store. The capacitance rate through the store is C~ = (;s~;(Rl C;s).
98
W. Spirkl, J. Mus('haweck / Solar domestic hot water xvstems
Table I I,ist of all model p a r a m e t e r s Symbol
Units
Range
Physical meaning
A~ u~ u~
me Wm Jm
~> 0 >0 >0
Us
WK
Cs
MJ/K
Effective collector a r e a Effective collector loss coefficient W i n d speed dependence o f u~. T o t a l store heat loss coefficient Total store heat capacity Fraction of the store volume used for auxiliary beating. Mixing constant, describing mixing effects during cold water inlet
2K i ~K i t
~0
>/0
L....
~ ]o, l]
l ) 1.
~
S~.
>~0
S t r a t i f i c a t k m p a r a m e t e r , S¢. = 0 is equivalent to a heat exchanger immersed at the bottom.
>~0
Thermal resistance of the load side heat exchanger (if any). A value of R I = 0 is e q u i v a l e n t to no load side heat exchanger.
(I
( D L - 0 for n o mixing).
Rl.
KkW
i
The height of the inlet is chosen such as to match the collector outlet and the according store temperature. A n immersed heat exchanger (S c = 0) is equivalent to (~c = o0. Table 1 shows the complete list of model parameters to be determined in a test.
2.6. Comparison with TRNSYS For D L = 0, the model is equivalent to the simulation program T R N S Y S [4] using the linear collector model, small temperature differences for the controller, variable collector inlet position and the plut flow tank (type 38) with - zero internal heat conduction, - zero losses at the bottom and top of the store. For D L >> 1 and S c = 0, the type 4 store with one node (fully mixed) is equivalent. Intermediate values of D L cannot be represented by T R N S Y S standard modules.
3. Numerical implementation The partial differential equation (6) could be solved e.g. in discrete height with a R u n g e - K u t t a algorithm using adaptive timestep [6]. However, the coupling constants due to convection would yield infinitely small time steps. A n o t h e r problem of fixed segments is mixing caused by numerical effects. To estimate this effect for a store with N h equally sized fixed segments we consider segments of height Ah = 1 / N h in an infinite store. W e further use a normalized time u = t C s / C s in N, steps Au = u / N , . Then the difference equation for the segment temperatures Tn in the case of plug flow is L(u
+ a.)
= (1 -
(9)
W. Spirkl, J. Muschaweck / Solar domestic hot water systems
99
where the function a ( A u / A h ) depends on the integration scheme, but always fulfills a ( A u / A h ) / ( A u / A h ) ~ 1 for A u / A h --* O. T,(u) depends on the temperatures at u = 0 according to T~(u) =
(1 - a )
'
a T~_i(0 ).
(10)
i=1
For an initial delta distribution Tn(0) = 8n0, this yields
rn(u)=(N,)(1-alN"-"a kn!
~.
(11)
Even for infinitely small timesteps, the shape of the initial distribution is not retained:
=
e ,/ah
for
--
---,0.
(12)
This is a Poisson distribution in n, with ( n ) = u / A h , and standard deviation ~r = ~ / ( n ) . The standard deviation ~rh, the mixing range, is ~rh = Ah o-, = v ~ .
(13)
For achieving a certain mixing range through the whole store, u = 1, the number of segments N h needed grows quadratically with 1 / % : 1
Nh
1
Ah - ~r,2, "
(14)
Hence, the fixed segment approach is rather ineffective for modelling plug flow. A mixing range of 1% requires 10000 segments! To overcome these problems, a numerical model with isothermal segments of variable size and position is used. The segments are shifted according to the collector and load flow, see fig. 2. This algorithm is especially suitable for the pure plug flow with D L = 0, see fig. 3. For this case, it allows the description of the load and collector flow without artificial numerical mixing. The algorithm is comparable to that of the type 38 store of TRNSYS [4]. For D L 4: 0, the concept of moving segments must be extended to take into account the diffusion term in eq. (6): aT at
a2T
-D
(15)
ah 2 •
It turns out that a second-order filter, composed of two first-order low pass filters applied in increasing and decreasing h-direction, is stable under all conditions and fast. For small timesteps it accurately models the diffusion. This is shown by extending the function T(h) to any real h symmetrically around zero and periodically with period two:
T(h)
= [ T(hT(-h)
2)
for h > 1, for h < 0,
(16)
~ Spirkl, J. Muschaweck / Solar domestic hot water ,tvstems
100
which according to ref. [7] is equivalent to have OT/Oh = 0, i.e. no heat flow through the bottom or the top. The transfer function in the frequency domain of the filter is 1/(1 + w2Dt). In the limit N ~ oo, the transfer function of N filter operations each using t/N as timestep converges to that of the diffusion equation:
w2Dt
e-~°2r~' = X--,~lim~1 +
( 17)
At the rim of large segments, small protection segments must be cut off to avoid an unwanted large distance mixing. Variable timesteps are used in order to minimize the computation time for given accuracy requirements.
4. Experimental experience The model was applied experimentally to about 50 different systems in national [8,9] and international [3] projects. The class of systems included e.g. thermosyphon systems, integrated collector store (ICS) systems, systems with load side heat exchanger, auxiliary heater, and heat pipe collectors. The main results are: (i) It is possible to describe systems whose design differs from the model assumptions. E.g. an ICS system is described by lumping together collector losses and store losses. (ii) The model is accurate to about 0.5 K in the thermal output. (iii) The prediction error is about 1 K, corresponding to an error of about 3% referred to 35 K load temperature difference.
5. Conclusions For the considered class of systems, with collector area less than 10 m 2 and store volume less than 1000 g, the model presented here is found to be adequate for the use in p a r a m e t e r identification from short term tests.
Acknowledgements Thanks are due to Professor R. Sizmann for helpful suggestions, and to the participants of the D S T G group for testing the D S T method program package [2]. This work was supported by the Bundesministerium fiir Forschung und Technologie, grant numbers 0328101 B, 0328768 A and 0328101 C.
Nomenclature Symbol
Units
Meaning
A c
m 2
At
m2
Collector area Effective collector area, A~ = F~(ar)A c. Parameter of the model
W. Spirkl, J. Muschaweck / Solar domestic hot water systems
Cs Cc
cs
MJ K - 1 W K -1 WK 1
O L
f ~ux G* h Paux
P,.(T) PL RE
Wm-2
W W W KkW
1
Sc t
S
T
oC oC oC oC oC
T,:A Tow Ts TsA uc
Wm-2 K-1 W m - 2 K -1
WK-~ /'/l'
jm-3K
U
m s -1
(a~') n! o)
i
101
Thermal capacity of the store. P a r a m e t e r of the model Thermal capacitance rate in the collector loop Load side capacitance rate through in the store (Cs ~< 6"e, 6"s = CL for no thermostat mixer) Drawoff mixing coefficient. Parameter of the model Auxiliary fraction of the store Incidence angle corrected solar irradiance in collector plane. Normalized vertical position inside the store Auxiliary power Collector loop power for given inlet temperature T Power delivered to the load, PL = ~'L(TL -- Tcw) Thermal resistance of the load side heat exchanger Solar loop stratification parameter. Parameter of the mode[ Time Temperature Ambient temperature of the collector loop Cold water temperature Outlet temperature of the store Ambient temperature of the store Specific loss coefficient of the collector Effective heat loss coefficient of the collector loop. For a collector without pipes or a heat exchanger, u~ = u c / ( a ~ - ) holds. Parameter of the model Overall heat loss coefficient of the store Wind speed dependence of u~. Parameter of the model Wind speed Effective transmission-absorption product Kronecker symbol; 6ij = 1 for i = j , ~i~ = 0 otherwise Faculty o f n , n ! = 1 × 2 × " " × ( n - 1 ) × n Frequency in the normalized height h with period two
References [1] W. Spirkl, J. Sol. Energy Eng., Trans. ASME 112 (1990) 98. [2] InSitu Scientific Software, Dynamic System Testing Program (Version 2.0), c / o Manfred Klein, Mozartstrasse 13, W-8000 Muni/zh 2, Germany, 1992. [3] Dynamic Testing of Solar Domestic Hot Water Systems, Technical Report, International Energy Agency, TNO Building and Construction Research, Delft, The Netherlands, 1992. [4l S.A. Klein, W.A. Beckman and P.I. Cooper, TRNSYS: A Transient System Simulation Program, Version 12.2. Solar Energy Laboratory, Madison Wisconsin, 1988. [5] J.A. Duffle and W.A. Beckman, Solar Engineering of Thermal Processes (Wiley, New York, 1983). [6] W. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes (Cambridge Univ. Press, Cambridge 1986). [7] J. Crank, The Mathematics of Diffusion (Clarendon Press, Oxford, 1956).
1(12
14. Spirkl. J. Muschaweck / Solar domestic hot water systems
[8] W. Spirkl, Dynamische Vermessung von Solaranlagen zur Warmwasserbereitung, Fortschrittsberichte der VDI-Zeitschriften, Reihe 6. Energietechnik, Nr. 241 (VDl-Verlag, Diisseldorf, 1990). [9] VELS, Verbundforschung zur Ermittlung der Leistungsf:,ihigkeit von Solaranlagen (VELS). 1¢;881990. BMFW-Projekt Nr. 0328768A.
Solar Energy Materials and Solar Cells
General model for testing solar domestic hot water systems W. Spirkl and J. Muschaweck Sektion Physik, Ludwig-Maximilians-Universitiit Miinchen, 14/-8000 Munich, Germany Received 20 May 1992
For modelling solar domestic hot water (SDHW) systems, a plug flow store model is presented. It is used in conjunction with the dynamic system testing algorithm to predict the long term performance of S D H W systems from short term test data. The basic property of a plug flow model is its capability of modelling drawoffs without any mixing inside the store. As an extension, the model developed in this paper covers the range between pure plug flow and full mixing. The degree of draw off mixing is characterized by a continuous parameter instead of a (discrete) n u m b e r of thermal nodes which actually is defined neither for a plug flow model nor for a real system. Less than ten model parameters characterize a system under test. It was found that the parameters can be identified in short time (two to four weeks outdoors) for a large class of systems, e.g. thermosyphon systems or integrated collector store (ICS) systems, systems with load side heat exchanger, auxiliary heater, and heat pipe collectors. A summary of experimental results for different systems is given.
I. Introduction At the University of Munich a method was developed to predict the long term performance of solar domestic hot water (SDHW) systems from a short term test. This method, called dynamic system testing (DST) method [1,2], was adopted by the International Energy Agency (lEA), and an lEA working group concerned solely with the DST method, the Dynamic System Testing Group (DSTG), worked from 1989 to 1991 [3]. This paper describes the progress made within the DSTG, with special emphasis on the plug flow model used in the DST method. Fig. 1 shows the input and output variables for a SDHW system under test. Only external quantities need to be measured. In ref. [1], an algorithm is presented which allows parameter identification from measured data using non-linear, dynamic models. Although the correlation model with two state variables (store temperatures) described there was found to be sufficiently accurate in many cases, Correspondence to: W. Spirkl, Sektion Physik, Ludwig-Maximilians-Universit~it Miinchen, Amalienstrasse 54, W-8000 Munich, Germany.
0927-0248/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
q4
IV. Spirkl, J. Muschaweek / Solar domestic hot water systems
~a ])(1 ll X
(C,~, TeA, .)
Pc
/
(Tcw, ds) Fig. 1. Input and output variables for a S D H W system. The system within the thick box is viewed as a black box. Thus, no intrusive m e a s u r e m e n t s are needed (like e.g. collector power or store temperatures).
a more detailed model had to be developed to include systems with integrated auxiliary heater a n d / o r high store stratification. Therefore we decided to retain the DST method, but to replace the correlation model by a plug flow model. This term means that the temperature distribution within the store is assumed to vary only in one dimension (i.e. in vertical direction), and that any flow through the store will just shift this temperature distribution upwards for load flow,downwards for collector loop flow (see fig. 2). In principle, a simulation program like TRNSYS [4], which also provides a plug flow model, could be used. But from a p a r a m e t e r identification point of view, a S D H W system modelled by T R N S Y S has too many free parameters. It is therefore impossible to identify the parameters from short term test data. In comparison to the TRNSYS model, we made simplifying assumptions; e.g. extra store losses at the bottom and the top are summarized in a global loss coefficient. On the other hand, the plug flow concept was extended to take into account mixing caused by load flow. The drawoff mixing effect is modelled using a diffusion equation.
[
Load outlet
t
T
[
h=l
h=0
T
Load inlet Fig. 2. Drawoff in the plug flow model.
W. Spirkl, Z Muschaweck / Solar domestic hot water .sTstems
95
2. The plug flow model for SDHW systems 2.1. The store The main assumptions on the store for the model presented here are listed below. For now, we restrict ourselves to systems with a solar loop heat exchanger immersed at the bottom of the store and no load side heat exchanger: (1) The state of the system at time t is characterized by the one-dimensional distribution T(t, h) of the store temperature, with the normalized height h in the range [0, 1], i.e. horizontal temperature gradients are neglected. This assumption implies neglecting of different effects such as convection caused by store losses as well as horizontal temperature gradients caused by the geometry of the solar loop heat exchanger. (2) The cold water is injected at height h = 0, the load is withdrawn at height h = 1. The remaining water in the store is shifted by an according value (plug flow). (3) The collector power is brought in at height h = 0 and is transported to water above by natural convection, see item (5). This is equivalent to a heat exchanger with no vertical extension immersed at the bottom of the store. (4) The auxiliary power is brought in at height h = 1 -faux and is transported to water above by convection, see item (5). (5) Local convection assures that ~T/Oh >10 holds. (6) Mixing or heat conduction do not occur for zero load capacitance rate (Cs = 0), except for fulfilling the relation mentioned in (5). (7) Cold water mixing is modelled by a diffusion term: 0T 0t
0h
+---.
(1)
It is assumed that the (time dependent) diffusion coefficient D is correlated with the load:
D ( t ) = D L C s / C s.
(2)
Fig. 3 shows the resulting step response for different values of the drawoff mixing constant D L. Note that, due to eq. (2), the step response does not depend on CsFor D L = 0, the drawoff accords to fig. 2 (pure plug flow). For D E > 0, each drawoff is associated with mixing (extended plug flow). (8) The loss coefficient is equally distributed over the height ( d U s / d h = 0). (9) The heat capacity is equally distributed over the height ( d C s / d h = 0). The capacity does not depend on the temperature (latent heat storage presumably cannot be modelled this way).
2.2. The collector loop The collector loop is modelled according to the Hottel-Whillier-Bliss equation [5] neglecting thermal capacitance and fluid flow time. The collector loop power
14/.Spirkl, J. Muschaweck / Solar domestic hot water ~vstems
96
1"c for solar irradiance Gt* in collector plane, collector ambient temperature TeA and inlet temperature T is modelled by Pc=A~[Gt*-u~.(T-TcA)]
+
'
where
[x]~={;
forx>0 otherwise"
(3)
For a collector with net area A c, absorptance-transmittance ( a z ) , heat removal factor F R and specific loss coefficient u c, the model parameters A~. and u~. are given by
A~ =AcFR(ar ) ,
u~. = U c / ( a z ) .
(4)
The loss coefficient is modelled as a linear function of the wind speed v: uc(,J)
= u
(O) + u , , , .
(5)
2.3. The partial differential equation In the sequel, the plug flow model is formulated in a partial differential equation for the special case of a heat exchanger immersed at the bottom of the store. The function T(t, h) is modelled as the limit of the solutions of the following parabolic partial differential equations for • ~ 0:
Cs aT(t,a~h) _ 8 , ( h ) A ~ [ G *
- u~(T-
TEA)] +
+ 3 , ( h + f , ox - 1)P~,u× -
Us(T- TsA)
+
+ a A h ) ( r c w - T) a (DICsaT)
+ Oh b exp
• ~
.
(6)
The meaning of the terms of the right side is: collector gain, auxiliary power, store losses, plug flow, diffusion and convection, respectively. Here, TCA and TSA denote ambient temperatures of the collector and the store, respectively; Paux denotes the auxiliary power. The model parameters U s and C s represent the total loss coefficient and the thermal capacity of the store. For e ~ 0, the function 6, converges to a Dirac distribution at zero: /
a , ( x ) = J _e _~/~
~0
e
for x > 0,
(7)
otherwise.
Convection is modelled by a diffusion process with a diffusion coefficient depend-
Hd Spirkl, J. Muschaweck / Solar domestic: hot water systems'
97
g 0.6
0.4
0 0
0.5
I
1.5
N o r m a l i z e d Load Volume
Fig. 3. Step response of the store to a change in the cold water temperature for different values of Dl.
(0, 0.01, 0.l, 1). ing on the temperature gradient (a and b are arbitrary positive constants). This ensures that in the limit •--+ 0 the requirement "OT/ah >10 is fulfilled exactly. Furthermore, with • --+ 0 the heat exchanger, the cold water inlet and the auxiliary heater are modelled with zero extension. We now go beyond the restrictions made at the beginning of this section. 2.4. Load side heat exchanger A load side heat exchanger is characterized by its thermal resistance R L. The capacitance rate Cs is reduced by the factor q~(RLCs), q~(x)= ( 1 - e - ~ ) / x , see fig. 4. 2.5. External collector loop heat exchanger An external heat exchanger is modelled using an additional p a r a m e t e r S c by a direct collector loop with fixed capacitance rate Cc: d c =At*u,*.~( I - e-~'c),
S c. > 0.
• Ts
cs
Tow
(8)
7
,Ts
0} 0s
Tow
Fig. 4. Modelling of a load side heat exchanger. Within the model, the heat exchanger is replaced by a mixing valve outside the store. The capacitance rate through the store is C~ = (;s~;(Rl C;s).
98
W. Spirkl, J. Mus('haweck / Solar domestic hot water xvstems
Table I I,ist of all model p a r a m e t e r s Symbol
Units
Range
Physical meaning
A~ u~ u~
me Wm Jm
~> 0 >0 >0
Us
WK
Cs
MJ/K
Effective collector a r e a Effective collector loss coefficient W i n d speed dependence o f u~. T o t a l store heat loss coefficient Total store heat capacity Fraction of the store volume used for auxiliary beating. Mixing constant, describing mixing effects during cold water inlet
2K i ~K i t
~0
>/0
L....
~ ]o, l]
l ) 1.
~
S~.
>~0
S t r a t i f i c a t k m p a r a m e t e r , S¢. = 0 is equivalent to a heat exchanger immersed at the bottom.
>~0
Thermal resistance of the load side heat exchanger (if any). A value of R I = 0 is e q u i v a l e n t to no load side heat exchanger.
(I
( D L - 0 for n o mixing).
Rl.
KkW
i
The height of the inlet is chosen such as to match the collector outlet and the according store temperature. A n immersed heat exchanger (S c = 0) is equivalent to (~c = o0. Table 1 shows the complete list of model parameters to be determined in a test.
2.6. Comparison with TRNSYS For D L = 0, the model is equivalent to the simulation program T R N S Y S [4] using the linear collector model, small temperature differences for the controller, variable collector inlet position and the plut flow tank (type 38) with - zero internal heat conduction, - zero losses at the bottom and top of the store. For D L >> 1 and S c = 0, the type 4 store with one node (fully mixed) is equivalent. Intermediate values of D L cannot be represented by T R N S Y S standard modules.
3. Numerical implementation The partial differential equation (6) could be solved e.g. in discrete height with a R u n g e - K u t t a algorithm using adaptive timestep [6]. However, the coupling constants due to convection would yield infinitely small time steps. A n o t h e r problem of fixed segments is mixing caused by numerical effects. To estimate this effect for a store with N h equally sized fixed segments we consider segments of height Ah = 1 / N h in an infinite store. W e further use a normalized time u = t C s / C s in N, steps Au = u / N , . Then the difference equation for the segment temperatures Tn in the case of plug flow is L(u
+ a.)
= (1 -
(9)
W. Spirkl, J. Muschaweck / Solar domestic hot water systems
99
where the function a ( A u / A h ) depends on the integration scheme, but always fulfills a ( A u / A h ) / ( A u / A h ) ~ 1 for A u / A h --* O. T,(u) depends on the temperatures at u = 0 according to T~(u) =
(1 - a )
'
a T~_i(0 ).
(10)
i=1
For an initial delta distribution Tn(0) = 8n0, this yields
rn(u)=(N,)(1-alN"-"a kn!
~.
(11)
Even for infinitely small timesteps, the shape of the initial distribution is not retained:
=
e ,/ah
for
--
---,0.
(12)
This is a Poisson distribution in n, with ( n ) = u / A h , and standard deviation ~r = ~ / ( n ) . The standard deviation ~rh, the mixing range, is ~rh = Ah o-, = v ~ .
(13)
For achieving a certain mixing range through the whole store, u = 1, the number of segments N h needed grows quadratically with 1 / % : 1
Nh
1
Ah - ~r,2, "
(14)
Hence, the fixed segment approach is rather ineffective for modelling plug flow. A mixing range of 1% requires 10000 segments! To overcome these problems, a numerical model with isothermal segments of variable size and position is used. The segments are shifted according to the collector and load flow, see fig. 2. This algorithm is especially suitable for the pure plug flow with D L = 0, see fig. 3. For this case, it allows the description of the load and collector flow without artificial numerical mixing. The algorithm is comparable to that of the type 38 store of TRNSYS [4]. For D L 4: 0, the concept of moving segments must be extended to take into account the diffusion term in eq. (6): aT at
a2T
-D
(15)
ah 2 •
It turns out that a second-order filter, composed of two first-order low pass filters applied in increasing and decreasing h-direction, is stable under all conditions and fast. For small timesteps it accurately models the diffusion. This is shown by extending the function T(h) to any real h symmetrically around zero and periodically with period two:
T(h)
= [ T(hT(-h)
2)
for h > 1, for h < 0,
(16)
~ Spirkl, J. Muschaweck / Solar domestic hot water ,tvstems
100
which according to ref. [7] is equivalent to have OT/Oh = 0, i.e. no heat flow through the bottom or the top. The transfer function in the frequency domain of the filter is 1/(1 + w2Dt). In the limit N ~ oo, the transfer function of N filter operations each using t/N as timestep converges to that of the diffusion equation:
w2Dt
e-~°2r~' = X--,~lim~1 +
( 17)
At the rim of large segments, small protection segments must be cut off to avoid an unwanted large distance mixing. Variable timesteps are used in order to minimize the computation time for given accuracy requirements.
4. Experimental experience The model was applied experimentally to about 50 different systems in national [8,9] and international [3] projects. The class of systems included e.g. thermosyphon systems, integrated collector store (ICS) systems, systems with load side heat exchanger, auxiliary heater, and heat pipe collectors. The main results are: (i) It is possible to describe systems whose design differs from the model assumptions. E.g. an ICS system is described by lumping together collector losses and store losses. (ii) The model is accurate to about 0.5 K in the thermal output. (iii) The prediction error is about 1 K, corresponding to an error of about 3% referred to 35 K load temperature difference.
5. Conclusions For the considered class of systems, with collector area less than 10 m 2 and store volume less than 1000 g, the model presented here is found to be adequate for the use in p a r a m e t e r identification from short term tests.
Acknowledgements Thanks are due to Professor R. Sizmann for helpful suggestions, and to the participants of the D S T G group for testing the D S T method program package [2]. This work was supported by the Bundesministerium fiir Forschung und Technologie, grant numbers 0328101 B, 0328768 A and 0328101 C.
Nomenclature Symbol
Units
Meaning
A c
m 2
At
m2
Collector area Effective collector area, A~ = F~(ar)A c. Parameter of the model
W. Spirkl, J. Muschaweck / Solar domestic hot water systems
Cs Cc
cs
MJ K - 1 W K -1 WK 1
O L
f ~ux G* h Paux
P,.(T) PL RE
Wm-2
W W W KkW
1
Sc t
S
T
oC oC oC oC oC
T,:A Tow Ts TsA uc
Wm-2 K-1 W m - 2 K -1
WK-~ /'/l'
jm-3K
U
m s -1
(a~') n! o)
i
101
Thermal capacity of the store. P a r a m e t e r of the model Thermal capacitance rate in the collector loop Load side capacitance rate through in the store (Cs ~< 6"e, 6"s = CL for no thermostat mixer) Drawoff mixing coefficient. Parameter of the model Auxiliary fraction of the store Incidence angle corrected solar irradiance in collector plane. Normalized vertical position inside the store Auxiliary power Collector loop power for given inlet temperature T Power delivered to the load, PL = ~'L(TL -- Tcw) Thermal resistance of the load side heat exchanger Solar loop stratification parameter. Parameter of the mode[ Time Temperature Ambient temperature of the collector loop Cold water temperature Outlet temperature of the store Ambient temperature of the store Specific loss coefficient of the collector Effective heat loss coefficient of the collector loop. For a collector without pipes or a heat exchanger, u~ = u c / ( a ~ - ) holds. Parameter of the model Overall heat loss coefficient of the store Wind speed dependence of u~. Parameter of the model Wind speed Effective transmission-absorption product Kronecker symbol; 6ij = 1 for i = j , ~i~ = 0 otherwise Faculty o f n , n ! = 1 × 2 × " " × ( n - 1 ) × n Frequency in the normalized height h with period two
References [1] W. Spirkl, J. Sol. Energy Eng., Trans. ASME 112 (1990) 98. [2] InSitu Scientific Software, Dynamic System Testing Program (Version 2.0), c / o Manfred Klein, Mozartstrasse 13, W-8000 Muni/zh 2, Germany, 1992. [3] Dynamic Testing of Solar Domestic Hot Water Systems, Technical Report, International Energy Agency, TNO Building and Construction Research, Delft, The Netherlands, 1992. [4l S.A. Klein, W.A. Beckman and P.I. Cooper, TRNSYS: A Transient System Simulation Program, Version 12.2. Solar Energy Laboratory, Madison Wisconsin, 1988. [5] J.A. Duffle and W.A. Beckman, Solar Engineering of Thermal Processes (Wiley, New York, 1983). [6] W. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes (Cambridge Univ. Press, Cambridge 1986). [7] J. Crank, The Mathematics of Diffusion (Clarendon Press, Oxford, 1956).
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14. Spirkl. J. Muschaweck / Solar domestic hot water systems
[8] W. Spirkl, Dynamische Vermessung von Solaranlagen zur Warmwasserbereitung, Fortschrittsberichte der VDI-Zeitschriften, Reihe 6. Energietechnik, Nr. 241 (VDl-Verlag, Diisseldorf, 1990). [9] VELS, Verbundforschung zur Ermittlung der Leistungsf:,ihigkeit von Solaranlagen (VELS). 1¢;881990. BMFW-Projekt Nr. 0328768A.