Sizing and applicability considerations of solar combisystems

P.D. Lund
*
,1
Advanced Energy Systems, Helsinki University of Technology, P.O. Box 2200, FIN-02015 HUT (Espoo), Finland
Received 7 August 2003; received in revised form 9 February 2004; accepted 29 July 2004
Available online 11 September 2004
Communicated by: Associate Editor Claudio Estrada-Gasca
Abstract
Sizing and applicability of solar combisystems providing both space heating and domestic hot water are investigated
in this paper. The analysis is based on an analytical daily model to predict the average yearly system performance. The
study indicates that increasing the collector area in a solar combisystem for higher solar fractions could be economically
justied in average or older building stock in northern and central Europe but not yet in low energy or very energy
ecient buildings nor in a more southern climate. Increasing the size of the heat storage much beyond the daily capacity
proved not to be well justied in solar combisystems.
2004 Elsevier Ltd. All rights reserved.
Keywords: Solar combisystems; System sizing; Analytical model
1. Introduction
An important design paradigm of active solar heating
systems has traditionally been to target for applications in
which the solar output satisfactorily matches with the
heat load. Good cases for solar heating are such in which
the heat load is almost constant over the year or has its
maximum in the peak solar season, for example domestic
hot water, process heat or pool heating, whereas applica-
tions with a heat minimumdemand in the summer are not
optimal. Otherwise there would be little rationale for a so-
lar heating system excluding cases with eective long-
term heat storage. Consequently the solar collector area
is dimensioned in respect to the heat load and available
heat storage in such a way that overproduction of solar
heat during the peak solar season is avoided. One of the
basic reasons behind this has been the higher cost of solar
heat over traditional heating systems, which has not jus-
tied oversized solar collector arrays. The sizing recom-
mendations and rules for the above described solar
heating systems are well tried and established (Beckman
et al., 1977) and the available design tools for such sys-
tems (e.g. TRNSYS, 1990) are reliable.
Solar water and solar pool heating systems are over-
whelmingly the most important market segments of ac-
tive solar heating systems and these presently represent
the optimum applications from a solar matching point
of view. As a curiosity one may mention the seasonal
storage solar heating systems in which solar heat is
stored from the peak solar season to the peak heat de-
mand season, but these are still in a research phase,
not on the market and are meant for larger heating
applications mainly (Dalenback, 1990).
With an increasing solar collector market and
decreasing collector prices on one hand, and occurrence
0038-092X/$ - see front matter 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.solener.2004.07.008
*
Tel.: +358 9 451 3197; fax: +358 9 451 3195.
E-mail address: peter.lund@hut.
1
ISES member.
Solar Energy 78 (2005) 5971
www.elsevier.com/locate/solener
of low energy houses on the other hand, active solar
heating systems designed for non-optimum solar versus
load conditions and hence departing from the tradi-
tional design practice have occurred. Examples of these
applications are solar space heating and so-called solar
combisystems with short-term heat storage for both
hot water and space heating already known and built
for decades but in smaller amounts. Notably the solar
combisystems have regained much new interest in cen-
tral-Europe during the last years. The market for such
systems is predicated to grow indicated also by collector
manufacturers new eorts in this eld (Weiss, 2002). So-
lar combisystems are perceived as an important new
market segment in Europe as the solar water heating
market shows already stagnation in some regions.
The basic active solar heating design tools are capa-
ble of handling both solar space and solar combisystems
but the design basis and practices of these systems are
not that well established. International Energy Agencys
SHC Task 26 is by far the largest international eort in
Nomenclature
a annual
A
c
solar collector area (m
2
)
A
p
building perimeter area (m
2
)
A
s
building surface area (m
2
)
A
w
total building window area (m
2
)
A
w,s
south window area (m
2
)
c specic heat (J/kgK)
C
0
xed solar system cost (EUR)
C
c
solar collector unit cost (EUR/m
2
)
C
v
heat storage unit cost (EUR/m
3
)
C
sys
total solar heating system cost (EUR)
d thickness of building insulation (m)
dhw domestic hot water
DL length of day (h)
EUR euro (1 euro = 1.25 US$, January 2004)
F
0
collector eciency factor
F
R
collector heat removal factor
g
sol
usability factor of passive solar
h building height (m)
i day number
I radiation ux on the collector plane (W/m
2
)
I
vert
radiation ux on the south wall (W/m
2
)
L normalized daylength
m storage mass (kg)
q instantaneous or daily eect (W)
q
u
solar collector output (W)
q
load
heat load (W)
q
aux
auxiliary power (W)
q
loss
storage heat losses (W)
q
space
space heating demand (W)
q
dhw
domestic hot water demand (W)
Q
L
yearly heat load (J)
Q
u
yearly net solar output (J)
Q
0
u
yearly gross solar output (J)
r
m
solar variation
r
s
relative marginal collector output
sdhw solar domestic hot water
sf solar fraction
t time (s)
t
op
operational time of solar heating system (s)
T temperature (C)
T
a
ambient temperature (C)
T
cold
cold water temperature (C)
T
del
forward temperature to space heat load (C)
T
f,in
collector inlet temperature (C)
T
hot
hot water temperature (C)
T
in
indoor temperature (C)
T
min
minimum useable temperature of heat for
heat load (C)
T
op
average operational temperature (C)
T
ret
return temperature from space heating load
(C)
T
s
storage temperature (C)
U building heat loss factor (W/K)
U
L
collector heat loss factor (W/m
2
K)
U
0
internal heat gains (W)
U
w
window heat loss factor (W/m
2
K)
v ventilation rate (1/h)
V storage volume (m
3
)
Greek letters
d declination angle (deg) or incremental
change
D dierence
k thermal conductivity (W/mK)
/ latitude (deg)
g
0
optical eciency
g
ex
heat recovery eciency of ventilation
exhaust air
u form factor
x 2p/s
q density (kg/m
3
)
s length of the year (days)
Subscripts
0 reference system
air indoor air
i day number
vert vertical
60 P.D. Lund / Solar Energy 78 (2005) 5971
developing solar combisystems further. This task has
produced new modules for TRNSYS simulation pro-
gram resulting in improved performance predictions of
dierent system congurations (IEA, 2003).
Taking the increasing interest in active solar space
and combisystems, it was timely found to have a closer
look on the design basis of such systems considering also
the developments in the building technology that have a
strong link to performance of the active solar heating
system. The use of active solar heating in space or combi
heating applications with short-term heat storage will
mean reduced solar collector yields per m
2
but a higher
solar fraction of the total heat demand. A profound
analysis of the interaction between the solar output
and the heat demand is therefore highly motivated to
better understand the optimum conditions for the use
of solar combisystems.
The focus of this paper is on the dimensioning of so-
lar combisystem with short-term heat storage, analysis
of system performance and nding conditions for justi-
ed use of these systems. In this way, promising direc-
tions for further development may be identied. The
basic approach taken here is an analytical mathematical
analysis associated with numerical transient simulations
in cases where validation is necessary or an analytical
approach is limited. It was recognized that many exist-
ing numerical simulation tools would be fully capable
of producing accurate performance predictions and
these results could be used for producing general sizing
correlations or nomograms. Going for analytical formu-
lations was here highly motivated to understand more
profoundly the underlying thermal processes and inter-
actions in solar space and solar combisystems and to dis-
aggregate the inuence of dierent variables and system
parameters on system performance. Hence a central
theme of this paper along the analyses is the presenta-
tion of the analytical tool used.
2. Solar combisystems
Solar combisystems for single-family houses have in
practice collector areas between 8 and 30m
2
and storage
sizes of 3001500l compared to solar dhw systems with
48m
2
and 200500l, respectively. The solar combisys-
tems may in practice vary quite much in hydraulic de-
signs. A general system principle is illustrated in Fig. 1
showing the main components. Two possible placements
for the auxiliary heating system (1, 2) and collector re-
turn-to-storage point (a, b) are indicated.
In system case (1) all main energy ows go through
the heat storage and the operation of the active solar
heating system can be described by a simple energy bal-
ance equation:
mc
dT
s
dt
q
u
q
load
q
aux
q
loss
1
q
u
A
c
F
R
g
0
I U
L
T
f;in
T
a
2
q
load
q
space
q
dhw
3
q

space
U T
in
T
a
U
0
g
sol
A
w;s
I
vert
q

space
P0
4
Eq. (1) is the storage balance equation. Eq. (2) is the
well-known HottelWhillierBliss relation giving the so-
lar collector output (Due and Beckman, 1991). The
load in Eq. (3) is composed both of the space heating
and hot water demand and includes the heat losses in
the piping system. Most of the variables above are time
dependent, for example the weather data (I, T
a
), load,
and storage temperature (T
s
). Eq. (4) describes the space
heating demand as a dierence of heat losses and heat
gains of the building (see Appendix A for more detailed
description of the individual terms).
solar
collectors
heat storage

cold water
hot water

return
forward


(1) auxiliary boiler


space heating


(2) auxiliary unit


(a) return

(b) return


(2) auxiliary unit
Fig. 1. Illustration of a solar combisystem.
P.D. Lund / Solar Energy 78 (2005) 5971 61
System case (2) diers from above in that the auxil-
iary power q
aux
in Eq. (1) is zero. The storage tempera-
ture determines the fraction of instantaneous load met
by solar energy. The auxiliary heat needed can be writ-
ten as
q
aux
q
space
1
T
s
T
ret
T
del
T
ret

q
dhw
1
T
s
T
cold
T
hot
T
cold

5
where T
s
= storage temperature, T
del
= forward temper-
ature to space heating, T
ret
= return temperature from
space heating, T
hot
= hot water temperature, and
T
cold
= cold water temperature. jxj
+
= 0 if x < 0.
The use of the auxiliary heating system is controlled
through the storage temperature T
s
. If T
s
drops under
a certain threshold then the boiler is turned on. This
applies both for having the auxiliary heating unit
charging the storage unit (system (1)) or using the aux-
iliary after the storage to upgrade preheated water (sys-
tem (2)).
The fraction of the instantaneous load met by solar
heat for a preheating type of system (case 2) is given by
sf
T
s
T
min
T
del
T
min
6
where T
s
= storage temperature, T
min
= minimum usea-
ble temperature for heat load (e.g. cold water tempera-
ture or return temperature), T
del
= heat delivery
temperature. The solar fraction needs to be separately
determined for the space heat and hot water because
of the dierent temperature requirements.
With a storage integrated auxiliary system, the stor-
age temperature is not an accurate indicator of the solar
fraction as the auxiliary heat is also stored and there will
always be heat available at adequate temperature from
the storage, i.e. sf = 1 in Eq. (6). However, if the storage
is operated stratied with a warmer upper layer and
colder lower layer, Eq. (6) may give some indication of
the solar fraction. In this case we need to use the non-
auxiliary intercepted part of the storage for determining
the T
s
. A constant collector outlet temperature matching
the heat delivery temperature may be achieved with a
variable mass ow (connection b in Fig. 1) and the stor-
age would then be closer to a two-layer model. If the
share of storage volume for the auxiliary is small then
Eq. (6) would describe the solar fraction of system 1
as well.
Solving the Eqs. (1)(4) over the year yields the
yearly solar output and the yearly solar fraction. Simu-
lation tools calculate the hour-by-hour performance and
give the yearly performance values as integrated values.
In this paper, some simplications are rst done in Eqs.
(1)(4) and then an analytical solution for the solar yield
is sought for.
3. Solar and load proles
The solar fraction of an active solar heating system is
dened as the ratio of delivered solar heat to heat load.
The usable amount of the gross solar heat output for the
load and hence also the overproduction of solar heat de-
pends on how the solar output and heat load match to-
gether. On an hourly to daily basis matching is often
handled through the heat storage whereas on a seasonal
scale the mismatch of average load and solar output pro-
les becomes the determining factor for overproduction
of solar heat. Actually, the seasonal matching of solar
the heat load prole is one of the main factors determin-
ing the collector yield and solar fraction.
To illustrate how the seasonal matching changes
when increasing the solar collector area, e.g. moving
from hot water to solar combisystems, four cases are ta-
ken for closer analysis. This is also helpful for the ana-
lytical expressions to be derived in next chapter, as the
net delivered solar output is non-linear. In the four cases
studied, the collector area is increased such that
A
c,I
> A
c,II
> A
c,III
> A
c,IV
. The storage volume is not in-
creased in same proportion and it represents a few days
or at most a weekly storage. The average contours or
line integrals are shown in Fig. 2 and the solar heat
meeting the load is given in Table 1. For simplicity, it
is assumed here that the year is symmetric so that only
the rst half is shown here. The second-half of the year
is a mirror picture of the rst-half.
The contour lines represent the gross solar output.
Using Fig. 2 and Table 1, the following expression can
now be written for the yearly net solar output or usable
solar heat Q
u
(1/2 is due to the symmetry).
tB
A
B
D
I
II
IV
dhw
space
heat
time
solar gross output
F
E
III
0
C
t0 tA tC tE= tD
solar output
< heat load
solar output
> heat load
E1
E2
E3
E4
0
t = 0
Fig. 2. Solar output and heat load proles (half-year,
symmetric).
62 P.D. Lund / Solar Energy 78 (2005) 5971
1
2
Q
u;I

_
B
0
0
q
u;I
dt
_
C
B
q
space
q
dhw
dt
_
E
C
q
dhw
dt
7
1
2
Q
u;II

_
D
0
0
q
u;II
dt
_
E
D
q
dhw
dt 8
1
2
Q
u;III

_
E
0
0
q
u;III
dt 9
1
2
Q
u;IV

_
F
0
0
q
u;IV
dt 10
For the yearly heat load Q
L
, we have
1
2
Q
L

_
E
0
q
load
dt
_
C
0
q
space
dt
_
E
0
q
dhw
dt 11
The gross solar output in the above cases is given by
1
2
Q
0
u;k

_
E
k
0
0
q
u;k
dt; k I; IV 12
The solar output exceeding the heat load must be
dumped and is thus unusable. Dividing Q
u
with Q
L
yields the solar fraction (f). The fraction of unusable
or dumped solar heat is equal to
1
Q
u;k
Q
0
u;k
13
4. Daily averaging
Averaged values are used as performance indicators
in the analyses to follow. This in turn enables using peri-
odic functions for the climatic variables that drive both
the solar and load processes and eases considerably a
closed analytical formulation.
The two basic weather variables of interest is the
ambient temperature and solar insolation for which we
use the following expressions:
T
a;i
T
a
DT
a
cosx i 14
I
i
I DI cosx i 15
where T
a
= ambient temperature, I = solar insolation
and x
2p
365
[day
1
].
The averages used above are yearly average values.
The index i refers to the day number and is equivalent
to time (t). Dening r
m
as the ratio of maximum
monthly to minimum monthly values (in practice often
June or July to December or January in the northern
hemisphere), we can write for the variation DI
DI
r
m
1
r
m
1
I 16
A similar relation can be written for DT
a
by just
changing insolation to ambient temperature in the above
equation.
Perers and Zinko (1984); Perers and Karlsson (1990)
have shown that the daily collector energy output ts
well the equation
Q
u;i
A
c
g
0
I
i
U
L
hDTi cons: 17
In a more sophisticated version, a form factor and
the daylength were incorporated to the loss factor U
L
in Eq. (17) (Perers, 1995) reecting the operational time
t
op
and conditions of the solar heating system:
Q
u;i
% A
c
F
0
g
0
I
i
24 U
L
hDTi t
op
u 18
where hDTi = T
op
T
a,i
, T
op
= average operational tem-
perature, u = form factor.
Based on regression analysis, IEA SHC Task 6
(Schreitmu ller, 1992) showed that daily insolation on
collector plane, mean operational temperature dierence
and length of operational time or length of day (DL)
proved to be important parameters of inuence when
predicting daily collector output. Some higher order
terms such as DL hDTi or DL hDTi
2
appears in these
regressions but are more of secondary importance.
The operational time t
op
correlates closely to the
length of day (DL). The mean operational temperature
dierence hDTi is eected both by the storage and ambi-
ent temperature. The mean daily ambient temperature
can well be approximated by a simple sinusoidal func-
tion, whereas nding T
op
would require solving the en-
ergy balance equation (1). To get around this and to
be able to provide analytical expressions but also to en-
sure that the produced solar heat is temperaturewise use-
able for the heat load, T
op
will be set equal to the
delivery temperature (=T
max
) which is a conservative
estimate. During the summertime, this approximation
is well justied due to high storage temperature, but
the lower storage temperatures during the o-solar sea-
son need to be compensated.
The form factor u that adjusts for energy losses out-
side operational time was used here also to provide cor-
rection for the operational temperature approximation
used. Literature gives u = 0.3 when using daily or sea-
sonally varying T
op
as input (Perers, 1995). We found
Table 1
Delivered solar heat for dierent solar heating system types
Case Solar system type Delivered solar heat
(prole contours)
I Solar combisystem 0
0
A + AB + BC + CE
II Overdimensioned sdhw
or underdimensioned
solar combisystem
0
0
D + DE
III Solar dhw 0
0
E
IV Underdimensioned
solar dhw
0
0
F
P.D. Lund / Solar Energy 78 (2005) 5971 63
that a time dependent form factor equal to DL/24 pro-
vided the best t for our case with T
op
= constant.
Depending on the location and time of the year, u
may vary from 0.1 to 0.5. u gets lower values in winter
and higher in summer.
With above assumptions, daily collector output (in J)
can nally be written in the following form:
Q
u;i
% A
c
F
0
g
0
I
i
24 DL
i

DL
i
24
_ _
U
L
T
max
T
a;i

_ _
19
Dividing Eq. (19) with 24h gives for the daily average
collector output (in W):
q
u;i

Q
u;i
24
A
c
F
0
g
0
I
i
L
2
i
U
L
T
max
T
a;i

_
; q
u;i
> 0
20
where
L
i

length of day
24

DL
24
21
The length of the day is given by (Due and Beck-
man, 1991)
DL
i

2
15
cos
1
tan/tand 22
where d = declination angle of day i and / = latitude.
DL ts well to a sinusoidal function which enables us
to write L
i
in the form
L
i
% L DL cosx i 23
Finally, the average solar and load proles can be
written in the following form
q
u;i
a
0
a
1
cosx i a
2
cos
2
x i
a
3
cos
3
x i 24
q
load;i
k
0
0
k
0
1
cosx i 25
where the coecients a
j
and k
j
are described in Appen-
dix B. These equations are used in Eqs. (7)(12) to solve
analytically the solar yield, dumped solar heat, and solar
fraction.
5. Verication of the model
Before using the averaging model for more detailed
analysis, validation of the model against an accurate
numerical model was made. The numerical hour-by-
hour simulations were performed with a transient
TRYNSYS-type program EUROSOL (Lund, 1995).
The input parameters used were the same in both cases.
Hourly reference weather data was used in the numerical
simulations and averaged data from the hourly reference
weather data was used in the analytical model as de-
scribed in Table 2. The cross-checking of the validity
of the analytical model was made against several factors:
type system, climate, collector area, and seasonal
behavior.
The rst set of validation is a comparison of analyt-
ical versus numerical performance predictions for a
standard solar domestic hot water system in dierent cli-
matic conditions. The system comprises 6m
2
of collec-
tors and a 500l water tank for a 180l/day hot water
demand. The solar collectors are of standard type with
selective surface (g
0
= 0.82, U
L
= 3.8W/m
2
K) readily
found on the market. The system temperature is 60C.
The collector inclination is 30.
The results of this comparison are shown in Fig. 3 for
the 4260 N latitude range. Taking rst the total solar
output over the time period shown (symmetric situa-
tion), the agreement between the two models is within
4%. The largest deviations are found in Helsinki
+3.7% and Rome 2.0%, respectively. On a monthly
basis the dierences increase, as the averaging process
smoothens out the irregularities in the weather pattern.
For example, the analytical model gives +20% for Diex
in May or 45% for Milan in February compared with
the hour-by-hour simulations. Summarizing, the model
seems to be more suitable for yearly performance predic-
tions as may be expected from the averaging process.
Table 2
Climatic parameters used in the model validation
Site Country Latitude ( N) Length of day Solar radiation Ambient
temperature
L 24 (h) DL 24 (h) I 2s, slope = 30
(kWh/m
2
, a)
Solar variation, r
m
T
a
(C) DT
a
(C)
Rome Italy 41.8 11.80 2.90 1883 3 15 10
Milan Italy 45.0 11.78 3.24 1358 5 12 11
Diex Austria 46.8 11.77 3.47 1420 3 6 10
Vienna Austria 48.3 11.75 3.67 1210 5 10 10
Copenhagen Denmark 55.8 11.66 4.92 1165 9 8 9
Helsinki Finland 60.2 11.59 5.94 1169 50 5 12
64 P.D. Lund / Solar Energy 78 (2005) 5971
A limitation of the model not shown above occurs at
very high latitudes beyond 63 N where the seasonal
uctuation becomes very large or DL ! L. In this case
the assumption for the form factor underestimates the
solar yield even by 2030% on a yearly basis. This has
not very large practical signicance as these regions
are often much dispersed and the solar utilization is
not signicant. In this case numerical models would be
necessary.
Next two dierent system types are taken for closer
scrutiny: the sdhw system used above and a solar com-
bi-system for both dhw and space heating in an average
house. The collector area is varied keeping the storage
volume constant to see the eects of the solar and load
proles matching. The standard dhw system with 500l
storage tank is taken as the starting point and the collec-
tor area is gradually increased from 2m
2
of collectors
(highly under-dimensioned solar system) to 30m
2
Site: Rome 42 N
0
100
200
300
400
500
600
1 2 3 4 5 6 7
Month
A
v
e
r
a
g
e
d

c
o
l
l
e
c
t
o
r

o
u
t
p
u
t
,

W numerical
analytical
Site:Helsinki 60 N
0
100
200
300
400
500
600
1 2 3 4 5 6 7
Month
A
v
e
r
a
g
e
d

c
o
l
l
e
c
t
o
r

o
u
t
p
u
t
,

W
numerical
analytical
Site:Copenhagen 56 N
0
100
200
300
400
500
600
1 2 3 4 5 6 7
Month
A
v
e
r
a
g
e
d

c
o
l
l
e
c
t
o
r

o
u
t
p
u
t
,

W
numerical
analytical
Site:Diex, 47 N
0
100
200
300
400
500
600
1 2 3 4 5 6 7
Month
A
v
e
r
a
g
e
d

c
o
l
l
e
c
t
o
r

o
u
t
p
u
t
,

W
numerical
analytical
Site:Milan 45N
0
100
200
300
400
500
600
1 2 3 4 5 6 7
Month
A
v
e
r
a
g
e
d

c
o
l
l
e
c
t
o
r

o
u
t
p
u
t
,

W
numerical
analytical
Site:Vienna 48 N
0
100
200
300
400
500
600
1 2 3 4 5 6 7
Month
A
v
e
r
a
g
e
d

c
o
l
l
e
c
t
o
r

o
u
t
p
u
t
,

W
numerical
analytical
Fig. 3. Comparison of the analytical model with a numerical model in dierent climates. The reference system is a solar domestic hot
water system (A
c
= 6m
2
, V = 500l, U
L
= 3.8W/m
2
K).
P.D. Lund / Solar Energy 78 (2005) 5971 65
(highly overdimensioned solar system). The eect in the
solar output prole would correspond to shifts
IV ! III ! II ! I illustrated in Fig. 2. The analytical
model does not have storage volume as parameter and
assumes a short-term storage capacity to be available
to compensate for daily uctuations in solar output
and heat load.
The results of the system comparison are shown in
Fig. 4. The yearly useful solar collector output is used
as the performance indicator corresponding to solution
of Eqs. (7)(10) in the analytical model. Two climatic
cases (42 and 60 N) are shown as the solar yields
for latitudes 4860 N are quite close each other.
Increasing the collector area decreases the solar output
per m
2
as the relative share of usable solar heat de-
creases. As may be expected, the solar yield between
a solar dhw and combisystem shows minor dierence
at lower collector areas but after a certain value the
combisystem performs better. Increasing the collector
area will rst increase the solar contribution for dhw,
but after a certain threshold it starts also to match
with the space heating which gives the dierence be-
tween the sdhw and combisystems shown. With a high
solar collector area, the solar combisystem would deli-
ver 2030% more heat than a sdhw for the same collec-
tor area.
The dierences between the analytical and numeri-
cal model are quite small, or a few percent, and within
the accuracy of the input parameters used. At small
collector area to storage volume ratios, e.g. A
c
< 4m
2
at 60 N, or for undersized solar heating systems,
the discrepancies increase and the analytical model is
conservative which is explained by the decrease of
the true solar system temperature well below the deliv-
ery temperature of heat to load. In the analytical
model T
s
= constant whereas in the numerical model
it is oating which explains the dierence. However,
if the storage volume would be decreased correspond-
ingly in the numerical model to keep A
c
/V > 0.01m
2
/l,
or reducing the system temperature in the analytical
model from 60 to 35C in this case, the dierences
disappear.
All above calculations were done with a xed storage
volume. Increasing the storage capacity would level out
short term uctuations in solar output and store some of
the dumped solar heat in summer for the autumn. A
storage tank can provide seasonal storage capacity equal
to its storage capacity. The storage capacity of water is
1.16kWh/m
3
per C, or 58kWh/m
3
for DT = 50C. This
means that solving the solar overproduction when
increasing the collector area would necessitate large stor-
age volumes, or A
c
/V < 0.002m
2
/l. For example in
Rome, increasing the collector area of a solar combisys-
tem from 10 to 15m
2
, will decrease the collector output
from 480 to 370kWh/m
2
per annum. Keeping the yearly
unit solar output (kWh/m
2
, a) unchanged, i.e. storing
the solar overproduction, would require a storage tank
close to 10,000l in this case, or 15 the original tank
volume.
Numerical hour-by-hour transient simulations with
EUROSOL indicated that increasing the storage volume
from the original value to up to 6 original value in-
creased the useful yearly solar output by maximum
5%. Scaling this to the storage capacity or size, corre-
sponds to an increase in the solar output equivalent to
12 storage capacity where the lower value is valid
for a higher volume increase and the higher value for a
smaller storage volume increase, respectively. This
means that in addition to seasonal storage (=1 storage
volume), a storage tank could provide up to 1 its
capacity short-term compensation in weather or load
uctuations. Comparing this with Fig. 4 means that even
a 6-fold increase in the storage size would mean at most
an extra solar output corresponding to 1m
2
of yearly
collector output, which would not be economically justi-
ed. In conclusion, the range of dierent storage vol-
umes used in solar combisystems in practice would
cause a few percent uncertainty in the results of the ana-
lytical model.
Site: Helsinki 60 N
0
100
200
300
400
500
600
0 10 20 30
collector area, sq.m.

c
o
l
l
e
c
t
o
r

y
i
e
l
d
,

k
W
h
/
s
q
.
m
,
a
a-dhw
a-combi
n-dhw
n-combi
Site:Rome 42 N
0
200
400
600
800
1000
1200
0 10 20 30
collector area, sq.m.

c
o
l
l
e
c
t
o
r

y
i
e
l
d
,

k
W
h
/
s
q
.
m
,
a
a-dhw
a-combi
n-dhw
n-combi
Fig. 4. Comparison of the collector output of dierent system
types between the analytical and numerical model. a = analyt-
ical model, n = numerical simulation.
66 P.D. Lund / Solar Energy 78 (2005) 5971
6. Performance of solar combisystems in dierent
load situations
The seasonal matching of the solar and load proles
have a signicant eect on the performance of a solar
combisystem. A more detailed analysis was made with
dierent house types attached to the active solar heating
system. Four dierent representative space heating load
types covering an important part of perceived practical
cases were considered. These and the input parameters
are described in Table 3. The internal gains are the same
in all cases and the oor area A
s
= 120m
2
. Table 4 shows
the total yearly heat load (space + dhw + pipe losses) for
the four house types in three dierent climate zones. The
passive house combines direct solar gain and good en-
ergy eciency. This house type is at present the most en-
ergy conscious design available on the market.
Using the heat loads dened in Tables 3 and 4, the cal-
culated performance indicators of the solar combisystems
for two climate zones are shown in Fig. 5. More southern
climates were excluded here as the space heating load be-
comes negligible. The solar usability gives the share of so-
lar heat utilized of the total available solar heat. The share
of dumped solar heat is equal to one minus usability.
Increasing the collector area will increase the solar frac-
tion but decreases the usability of solar heat as the match-
ing of solar output versus load worsens. In a sdhw mode,
the usability is close to 100% and the solar fraction ranges
from 520% (60 N) to 1030% (48 N). In a solar combi
with a 50% usability, the solar fraction is 2030% in Hel-
sinki (60 N) and 2540% in Vienna (48 N), respectively.
The solar usability is poorer for more energy ecient
houses as the main heating period moves towards the
non-solar season. For example, the passive house in a Vien-
nese climate has a very small space heating load or 6% of
the total heat load. FromFig. 5 one can see in this case, that
the solar yield beyond 10m
2
stagnates. With A
c
= 10m
2
we
have sf = 50% and with A
c
= 30m
2
, sf = 60%.
7. Trade-o situations
Without adequate heat storage capacity, the solar
collector output and the solar utilizability will decrease
with increasing solar collector area. This in turn means
that the cost of produced solar heat would be higher
in a solar combisystem compared to a solar dhw system.
As described in the previous chapter, having a higher so-
lar cost may be motivated through a high fraction of
non-purchased energy. The cost of a solar collector per
m
2
is cheaper than the solar system cost per m
2
which
means that if having already decided for a solar combi-
system, there may be economic grounds to add some
more collectors. In the next, a more detailed analysis
on the economic rationale is described.
Let us start with a solar combisystem having a collec-
tor area of A
c,0
yielding a total yearly solar output of
Q
u,0
. The total system or investment cost C
SYS,0
can be
written as
C
SYS;0
C
0
C
c
A
c;0
C
v
V 26
where C
0
= xed system costs, C
c
= unit cost of solar
collectors (EUR/m
2
) and C
v
= unit cost of heat storage
(EUR/m
3
).
Increasing the collector area by dA
c
and keeping the
remaining system unchanged, will increase the costs by
C
c
dA
c
and the solar output by (dQ
u
/dA
c
)
0
dA
c
. Increasing
the collector area from A
c,0
to A
c,0
+ dA
c
is justied only
if the benet to cost ratio (bcr) of the increase is equal or
larger than the bcr of the original setup, or simply
oQ
u
oAc
_ _
0
dA
c
C
c
dA
c
P
Q
u;0
C
SYS;0
27
where subscript 0 refers to the reference system with
A
c
= A
c,0
.
Eq. (27) can be written in the following form:
Table 3
Case buildings and their input parameters
Case Load prole Building envelope
U-value (W/m
2
K)
Window U-value
(W/m
2
K)
Ventilation
rate (1/h)
Heat recovery
eciency of exhaust
air (ventilation)
Comments
Old Old house 0.37 3.0 0 0
a
Average Average house 0.24 1.65 0.3 0.7
a
Low e Low energy house 0.13 1.0 0.3 0.75
a
Passive Passive low energy house 0.09 0.8 0.5 0.75
b
a
T
in
= 21C, A
w,s
/A
w
= 0.1, dhw = 3820kWh/a + 20% losses.
b
T
in
= 20C, A
w,s
/A
w
= 0.2, dhw = 3000kWh/a + 10% losses.
Table 4
Yearly total heat load (MWh/a) of the case buildings in
dierent climates (space heating demand is in parenthesis)
Case Helsinki Vienna Rome
Old 25.2 (20.6) 19.5 (14.9) 11.2 (6.6)
Average 16.3 (11.7) 12.8 (8.2) 7.4 (3.8)
Low e 10.0 (5.4) 8.2 (3.6) 5.1 (0.5)
Passive 7.5 (2.9) 4.9 (0.3) 4.6 (0.0)
Dhw 4.6 4.6 4.6
P.D. Lund / Solar Energy 78 (2005) 5971 67
r
s

oQ
u
oAc
_ _
0
Q
u;0
A
c;0
P
C
c
C
SYS;0
A
c;0
28
Eq. (28) can be interpreted as follows: increasing the
solar collector area is justied if the increase in the solar
output per m
2
achieved compared to the original output
per m
2
is larger than the collector to system price per m
2
.
System costs are typically 22.5 collector cost, which
yields a minimum target value of 0.40.5 for r
s
. This
means that increasing the collector area in a solar com-
bisystem would be economically justied as long as the
decrease of the unit collector output is less than 50
60% from the original value Q
u,0
/A
c,0
.
In Fig. 6 we have calculated the r
s
value based on the
collector performance calculations (left hand side of Eq.
(27)) for dierent load proles and sites (Tables 3 and 4).
The smaller the space heating demand, the lower the r
s
value. Less energy ecient houses have large heat loads
which match better with the solar output and hence
show higher r
s
values in general. The r
s
values of solar
combisystems are clearly higher than for just dhw.
Starting with the reference system setup with 6m
2
of
solar collectors and 500l water tank, it is observed from
Fig. 6 that both in a northern and central European cli-
mate, increasing the collector area in an old or average
house type would yield r
s
around 0.4 over a large range
of collector areas (here up to 60m
2
). Taking a low en-
ergy house, r
s
drops to 0.20.25 which means that
increasing the collector area would not yet be economi-
Site: Vienna 48 N
0
20
40
60
80
100
0 10 20 30
collector area, sq.m.
%
old
19.5 MWh
avg
12.8 MWh
low
8.2 MWh
passive
4.9 MWh
= solar fraction
= solar utilizability
Site: Helsinki 60 N
0
20
40
60
80
100
0 10 20 30
collector area, sq.m.
%
old
25.2 MWh
avg
16.3 MWh
low
10 MWh
passive
7.5 MWh
= solar fraction
= solar utilizability
Fig. 5. Eect of the building heat load (load prole) on the
solar fraction and solar usability. The numbers shown in the
legend are the yearly total heat load values.
Site: Helsinki 60 N
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50 60
collector area, sq.m.
(
d
Q
u
/
d
A
c
)
0
:
(
Q
u
/
A
c
)
0
dhw
old
average
low e
Site: Rome 42N
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50 60
collector area, sq.m.
(
d
Q
u
/
d
A
c
)
0
:
(
Q
u
/
A
c
)
0
dhw
old
average
low e
Site: Vienna 48N
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50 60
collector area, sq.m.

(
d
Q
u
/
d
A
c
)
0
:
(
Q
u
/
A
c
)
0
dhw
old
average
low e
Fig. 6. The eect of increasing the collector area on the
marginal collector output (r
s
-values) for dierent load and
climate conditions.
68 P.D. Lund / Solar Energy 78 (2005) 5971
cally justied with present cost conditions. For a south-
ern climate, an old non-energy ecient house gives
r
s
> 0.4 when A
c
< 20m
2
whereas for improved houses
r
s
is clearly below 0.3. For solar dhw systems, r
s
< 0.4
with A
c
> 4m
2
(Rome) and A
c
> 67m
2
(Helsinki, Vien-
na). These collector areas correspond well to common
design practices of sdhw in these climates, i.e. going be-
yond these values is not well justied with present cost
ratios.
Another trade-o question is between storage volume
and collector area. Increasing the storage instead of the
collector area is justied if the following relation holds:
oQ
u
oV
_ _
0
dV
C
V
dV
P
oQ
u
oA
c
_ _
0
dA
c
C
c
dA
c
29
Recalling from the previous sections that increasing
the storage volume would provide an addition in solar
heat output of up to twice the capacity of the volume,
gives oQ
u
/oV % 116kWh/m
3
. Eq. (29) can then be writ-
ten in the form
C
v
C
c
6
116
oQ
u
oAc
_ _
0

116
Q
u
Ac
_ _
0
r
s
30
For a single-family house, the unit cost of a collector
is typically C
c
= 100200 EUR/m
2
and for additional
storage volume we have C
v
= 300500 EUR/m
3
which
gives for the ratio C
v
/C
c
% 23. For solar-combisystems
in apartment houses with the same collector area and
storage volume per residence as for a single-family case,
the larger overall storage volume would result in a lower
unit cost or C
v
= 200300 EUR/m
3
which gives C
v
/
C
c
% 1.52. The right hand side of Eq. (30) depends on
the performance of the solar combisystem and its value
is between 0.71.4 (Helsinki) and 0.50.7 (Rome) where
the lower value refers to a combisystem with smaller col-
lector areas and the higher to larger ones, respectively.
Comparing the C
v
/C
c
ratio to these values shows that
in neither case would it be justied to improve the solar
yield through increasing the storage volume over the
typically used. A much reduced unit price of storage
would be necessary and such a situation may be found
when moving to seasonal storage systems.
8. Discussion and conclusions
In this paper, sizing and applicability aspects of solar
combisystems with short-term heat storage have been
investigated. First, a simple analytical model was de-
rived to study the eects of dierent sizing parameters
and driving factors of solar combisystems. The model
predicts average yearly solar system performance with
reasonable accuracy (<10%) compared to transient
numerical models.
Comparing a solar dhw and solar combisystem at
higher collector areas indicate that the space heating
could increase the collector output by 2030% over the
solar dhw output only for the solar collector area.
Increasing the collector area will increase the solar frac-
tion but decreases the solar usability, i.e. the solar out-
put per m
2
reduces. A solar combisystem means often
oversizing the collector area in this respect. The study
indicates that with present and near term cost structure
of solar heating systems a 5060% decrease in the collec-
tor output could still be economically justied when
increasing the collector area over normal design values.
Comparing this to the calculated solar combisystem per-
formance values in northern and central Europe with
dierent heat load proles showed that in average and
in older building stock with a space heating demand
extending more toward the summer the solar output
reduction would be within an acceptable economic
range. However, in case of low energy and very energy
ecient buildings, oversizing of the solar system would
lead to negative economic outcome and implies there-
fore careful sizing of the system to avoid solar dumping.
In low energy houses, going for solar combisystems and
collector oversizing would require halving of the collec-
tor unit cost compared to the system cost which may be
possible on a longer term.
The eect of increasing the storage size to reduce the
summertime overproduction of solar heat and hence
increasing the solar output was briey analyzed. The re-
sults show that increasing the storage in a solar combi-
system would deliver only marginally more solar heat,
or up to 2 the storage capacity. Increasing the storage
volume much beyond a few days capacity would not
thus be well justied.
Future work will include comprehensive sensitivity
analysis of the solar combisystem sizing against a multi-
tude of parameters included in the analytical model and
also numerical simulations of some non-linear eects.
Appendix A. Analytical expression of the load and
solar proles
The space heating demand is calculated by subtract-
ing building heat losses from the heat gains. The thermal
losses are basically due to conductive and ventilation or
inltration losses. The heat gains are from passive solar
and internal heat gains (e.g. from appliances).
The space heating demand can be written in the fol-
lowing form:
q

space;i
UT
in
T
a;i
U
0
g
sol
A
w;s
I
vert;i
q

space;i
P0
A:1
U A
p
A
w

k
d
A
w
U
w
_ vA
s
hq
air
c
air
g
ex
A:2
P.D. Lund / Solar Energy 78 (2005) 5971 69
A
p
2A
s
3h

2A
s
_
A:3
Here we assumed that the non-south orientated win-
dows have a negligible contribution to the solar gain.
The average solar radiation on a vertical surface fol-
lows the functional form / cos(2x i) with one maxi-
mum in spring (t = s/4) and one in autumn (t = 3s/4),
and one local minimum around midsummer (t = s/2).
However, as the space heating prole only is considered
here, the solar gains and hence only the vertical radia-
tion up to the point q
space
= 0 are relevant. In practice
this point occurs around the maximum monthly vertical
insolation point which is April or May for the latitudes
considered here (4060 N). From January to April and
for symmetry reasons also from September to Decem-
ber, the average vertical radiation can be approximated
by a functional form cos(x i). A good t to the vertical
radiation (t 2 0, s/4, 3s/4, s) is obtained with the func-
tion I
vert
DI
vert
cosx i, where DI
vert
= (2I
vert,max

2I
vert,min
)/3 and I
vert
2I
vert;max
I
vert;min
/3, where
I
vert,max
= maximum monthly solar radiation on vertical
surface (April or May) and I
vert,min
= minimum monthly
solar radiation on vertical surface (January).
We may write now Eq. (A1) further into the form:
q

space;i
k
0
k
1
cosx i A:4
where
k
0
UT
in
UT
a
U
0
g
sol
A
w
I
vert
A:5
k
1
UDT
a
g
sol
A
w
DI
vert
A:6
The explanation of the symbols is given in the
nomenclature.
Appendix B. Closed solution for intersection points of
load and solar proles
The average solar output is given by the Eq. (20)
q
u;i
A
c
F
0
bg
0
I
i
L
2
i
U
L
T
op
T
a;i
c B:1
which can be written in full form as
q
u;i
a
0
a
1
cosx i a
2
cos
2
x i
a
3
cos
3
x i B:2
where
a
0
A
c
F
0
L
2
U
L
T
op
T
a
g
0
I
a
1
A
c
F
0
g
0
DI DL U
L
T
op
T
a
U
L
L
2
DT
a
2U
L
LDLDT
a
a
2
A
c
F
0
2U
L
LDLDT
a
U
L
DL
2
T
op
T
a

a
3
A
c
F
0
U
L
DL
2
DT
a
B:3
The load prole consists of both the space and dhw
parts and is given by
q
load;i
k
0
0
k
0
1
cosx i B:4
where
k
0
0
k
0
q
dhw
k
0
1
k
1
Dq
dhw
B:5
This load prole is valid from t = 0 to t
B
.
The rst point of interest is t
0
. This is solved from Eq.
(B.1) by letting q
u
(t
0
) = 0 and solving t
0
from the 3-de-
gree polynoms. The second point of interest is t
B
which
is solved from q
u
(t
B
) = q
load
(t
B
).
The 3-degree polynoms can be solved as follows.
Here we seek rst a solution for t
B
. From Eqs. (B.1)
and (B.4) we have
a
0
a
1
cosx i a
2
cos
2
x i a
3
cos
3
x i
k
0
0
k
0
1
cosx i B:6
which can be written in the form
cos
3
x i
a
2
a
3
cos
2
x i
a
1
k
0
1

a
3
cosx i

a
0
k
0
0
a
3
0 B:7
Denoting rst
a
a
2
a
3
; b
a
1
k
0
1
a
3
; c
a
0
k
0
0
a
3
B:8
and making then the substitution z cosx i
a
3
will
turn Eq. (B.7) into the form
z
3
pz q 0 B:9
where p b
a
2
3
and q
2a
3
27

ab
3
c.
Using next the Cardanon formula
D
p
3
9

q
4
2
; A

q
2

D
p
3
_
; B

q
2

D
p
3
_
B:10
will yield a solution for z:z
1
= A + B which in turn ena-
bles to solve t
B:
t
B

1
x
cos
1
z
1

a
3
_ _
B:11
Next t
0
is obtained from above equations by just let-
ting k
0
0
and k
0
1
0 in Eq. (B.6). In a similar way, t
A
is
solved by setting k
0
= k
1
= 0 in Eq. (B.5); t
D
is obtained
from above equations by just letting k
0
and k
1
= 0 in Eq.
(B.8). The intersection point t
C
is solved from q
spa-
ce = qdhw
which yields
t
C

1
x
cos
1
q
dhw
k
0
Dq
dhw
k
1
_ _
B:12
t
E
is the middle point of the year and is equal to
365/2.
70 P.D. Lund / Solar Energy 78 (2005) 5971
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Design by the f-chart Method. Wiley Interscience, New
York.
Dalenback, J.-O., 1990. Central solar heating plants with
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