energies
Article
A Thermal-Hydraulic Model for the Stagnation of Solar
Thermal Systems with Flat-Plate Collector Arrays
Ralph Eismann 1,*, Sebastian Hummel 2 and Federico Giovannetti 3
1 Institute of Sustainability and Energy in Construction, School of Architecture, Civil Engineering and
Geomatics, University of Applied Sciences Northwestern Switzerland FHNW, 4132 Muttenz, Switzerland
2 Technische Hochschule Nürnberg Georg Simon Ohm, 90121 Nuremberg, Germany;
sebastian.hummel@th-nuernberg.de
3 Institute for Solar Energy Research Hamelin (ISFH), 31860 Emmerthal, Germany; giovannetti@isfh.de
* Correspondence: ralph.eismann@fhnw.ch; Tel.: +41-(0)61-228-53-61
Abstract: Stagnation is the transient state of a solar thermal system under high solar irradiation
where the useful solar gain is zero. Both flat-plate collectors with selective absorber coatings and
vacuum-tube collectors exhibit stagnation temperatures far above the saturation temperature of the
glycol-based heat carriers within the range of typical system pressures. Therefore, stagnation is
always associated with vaporization and propagation of vapor into the pipes of the solar circuit. It
is therefore essential to design the system in such a way that vapor never reaches components that
cannot withstand high temperatures. In this article, a thermal-hydraulic model based on the integral
form of a two-phase mixture model and a drift-flux correlation is presented. The model is applicable
to solar thermal flat-plate collectors with meander-shaped absorber tubes and selective absorber
coatings. Experimental data from stagnation experiments on two systems, which are identical except
for the optical properties of the absorber coating, allowed comparison with simulations carried





 out under the same boundary conditions. The absorber of one system features a conventional
Citation: Eismann, R.; Hummel, S.; highly selective coating, while the absorber of the other system features a thermochromic coating,
Giovannetti, F. A Thermal-Hydraulic which exhibits a significantly lower stagnation temperature. Comparison of simulation results and
Model for the Stagnation of Solar experimental data shows good conformity. This model is implemented into an open-source software
Thermal Systems with Flat-Plate tool called THD for the thermal-hydraulic dimensioning of solar systems. The latest version of THD,
Collector Arrays. Energies 2021, 14, updated by the results of this article, enables planners to achieve cost-optimal design of solar thermal
733. https://doi.org/10.3390/ systems and to ensure failsafe operation by predicting the steam range under the initial and boundary
en14030733 conditions of worst-case scenarios.
Academic Editor: Maurizio De Lucia Keywords: solar thermal; flat-plate collector; stagnation; steam range; two-phase mixture model;
Received: 29 December 2020 thermal-hydraulic model
Accepted: 23 January 2021
Published: 30 January 2021
Publisher’s Note: MDPI stays neutral 1. Introduction
with regard to jurisdictional claims in
published maps and institutional affil- Solar thermal systems with pressurized circuits for heating, domestic hot water prepa-
iations. ration and process heat applications generally use water-glycol mixtures as a frost-resistant
heat carrier. In order to prevent the storage tank from overheating, the circulation pump is
switched off as soon as the temperature reaches a pre-defined maximum value. Hence, the
collector array is no longer actively cooled, and the process of stagnation begins.
Solar thermal systems for the above-mentioned applications are usually operated
Copyright: © 2021 by the authors.
at a system overpressure below three bar which corresponds to saturation temperatures
Licensee MDPI, Basel, Switzerland.
This article is an open access article below 140
◦C. The maximum stagnation temperature of flat-plate collectors with single
◦
distributed under the terms and glazing and selective absorber coatings can reach above 200 C. Therefore, stagnation is
conditions of the Creative Commons always associated with the vaporization of the heat carrier. Stagnation is a transient process
Attribution (CC BY) license (https:// during which several phenomena occur. After pump shutdown, solar irradiation heats
creativecommons.org/licenses/by/ the absorber, thus increasing its temperature and heat losses. The saturation temperature
4.0/). normally is reached near the top of the absorber. In this state, the absorbed solar irradiation
Energies 2021, 14, 733. https://doi.org/10.3390/en14030733 https://www.mdpi.com/journal/energies
Energies 2021, 14, 733 2 of 39
is still higher than the heat losses. As a result, the collector turns the excess absorbed energy
with a certain efficiency into vaporization and displacement of the liquid. The rate of these
processes depends on the optical and thermal properties of the collector, the boundary
conditions and the size of the absorber region at saturation. The major part of the liquid
content is displaced into the pipes of the circuit, mostly by volumetric displacement but
also by interfacial friction between the increasing steam flow and the liquid. The growing
vapor volume displaces an equivalent liquid volume from the circuit into the expansion
vessel. The residual liquid within the absorber steadily vaporizes and propagates into
the pipes of the circuit. Condensing vapor heats the exposed pipe walls from their initial
temperature up to saturation temperature. The vapor propagation into the circuit increases
until the enthalpy flow rate of vapor from the collector field drops below the heat losses of
the pipes.
The following historical overview (see also Eismann [1]) shows why stagnation be-
came a research topic relatively late, when solar thermal technology was already well
established. Three qualitative ranges of the stagnation phenomenon can be associated with
the corresponding collector technologies.
Up to the middle of the 1970s, the vast majority of flat-plate collectors were equipped
with nonselective absorbers. The maximum stagnation temperature was about 140 ◦C,
which is below the permissible operation temperature of water-glycol mixtures. This
allowed suppression of vaporization by setting the system pressure above saturation
pressure.
With the introduction of selective coatings like black-chrome and black-nickel, the
efficiency at elevated temperatures was significantly increased. As a result, flat-plate
collectors reached a maximum stagnation temperature of up to 180 ◦C. In most cases,
the system pressure was set to values in such a way that the saturation temperature was
significantly below the actual temperature during dry stagnation. Vaporization of the liquid
and compensation of the vapor volume by a sufficiently large expansion vessel was an
accepted strategy to limit the pressure rise during stagnation. Because of the low efficiency
in saturation conditions, the steam range was correspondingly small. For this reason
and because the market volume was small at the time, the number of failures due to the
excessive steam range was too low to merit scientific interest. Pioneering companies solved
stagnation problems based on their own expertise and subsequently devised measures, like
rules for the arrangement of check valves and the sizing of membrane expansion vessels.
Some suppliers proposed suppression of vaporization by a high concentration of glycol
and a sufficiently high system pressure.
The adoption of cost-effective, highly selective thin-film absorber coatings in the
1990s and the ensuing market growth led to a dramatic increase in failures caused by
the excessive steam range. Typical cases of damage are the loss of liquid caused by the
opening of the safety valve, pumps destroyed by the passage of vapor, deformed and thus
leaking membranes of expansion vessels due to excessive temperatures and water hammer
induced by rapid condensation within the heat exchangers. These phenomena initiated
a considerable amount of research dedicated to understanding the stagnation process,
to develop measures to limit the steam range and derive theoretical models capable of
predicting maximum pressure and steam range.
Terschueren [2] firstly described the stagnation phenomenon qualitatively and recom-
mended useful practical conclusions from observations. In the subsequent years, several
experimental studies were carried out. The main objective of these studies was to under-
stand and characterize the stagnation process qualitatively and to derive practical rules for
the design of collectors and solar thermal systems. Eismann and von Felten [3] conducted
experiments on a real system with 52 m2 flat-plate collectors. Stagnation was triggered
by manually switching off the pump. From the results, they deduced that stagnation is
manageable if the piping is routed in such a way that the steam forms a continuous volume
and propagates monotonously downward. They provided rules for pipe routing resulting
in the best possible emptying of the collector field. As another rule, they define the location
Energies 2021, 14, 733 3 of 39
of the check valve, which must be upstream from the connection point of the pressure
maintenance. This allows displacement of liquid from the collector field in both the supply
and return lines. Based on the measurement data from experiments on a real system with
two 22 m2 flat-plate collectors, Hausner and Fink [4] described stagnation as a sequence
of five successive phases as (1) liquid expansion, (2) displacement of liquid by steam (3)
emptying by boiling (phase with saturated steam), (4) emptying by boiling (phase with
saturated and superheated steam) and (5) refilling of the collectors. From experimental data
and qualitative considerations, they classified the hydraulic designs of collectors and their
connections by external pipes according to their emptying behavior. This and the practical
rules derived from it were published also by Hausner and Fink [5]. Streicher [6] addressed
the danger of water hammer induced by rapid condensation of steam during stagnation.
He explained, with examples, that those hydraulic designs in which the steam forms a
continuous volume are to be preferred. Lustig [7] conducted experiments on systems with
flat-plate collectors and vacuum-tube collectors focusing on the processes in the absorber.
He recognized that the displacement of the liquid and the spreading of the vapor takes the
form of a two-phase flow. For a collector with a meander type absorber tube, he identified
the various flow patterns in all the tube sections. He used a plug-flow model to simulate
the propagation of steam into a single pipe. The residual liquid was predefined as a fraction
of the absorber volume. By adjusting the residual amount and the absorber volume, a fairly
good agreement with the experiments could be achieved.
Good emptying behavior, characterized by a small quantity of residual liquid within
the absorbers, was identified as a decisive prerequisite for a small steam range. However,
there was no way to predict the steam range as a function of system properties and
boundary conditions. Hausner, et al. [8] were the first who came up with a correlation
for the maximum steam range, based on a comprehensive experimental study on the
effect of hydraulic collector design and collector arrangement on the steam range. The
location of the steam front was deduced from the signals of equidistant temperature sensors
attached to the pipe walls. They introduced the experimental quantity of steam production
power, defined as the product of vapor mass flow and the enthalpy of vaporization at
the moment of maximum steam range. Their results provided a valuable insight into the
influence of hydraulic design and pipe routing on the steam range. However, application
of their empirical correlation is limited to the design and operational conditions of the
arrangements investigated. In subsequent research projects the influence of collector
efficiency and hydraulic design on maximum steam range was further investigated by
Rommel, et al. [9] who used their indoor test facility to conduct experiments on a system
of vacuum tube collectors made entirely of glass by Schott. On some absorber tubes,
a narrow strip remained uncoated. As a result, the evaporation process was directly
observable. Rommel, et al. [10] discussed measures to limit the steam range and control
strategies to reduce stagnation events. They presented a sophisticated measurement
method to determine the steam range, steam volume, and the amount of liquid remaining
in the collector. Building on these results Scheuren et al. [11] designed and built an
experimental facility with collector fields of different hydraulic design with up to 25.2 m2
aperture area. Their experiments showed the strong influence of the emptying behavior
on the steam range. Based on these experiments Scheuren [12] derived a more general
empirical correlation for the steam range. He distinguished three classes of collector fields
characterized by good, average and poor emptying behavior and provided a corresponding
set of two parameters for the correlation. However, the class needs to be determined initially,
based on qualitative considerations and practical experience.
Harrison and Cruickshank [13] and Frank, et al. [14] provided a literature review on
measures to prevent excessive steam range, covering control strategies, condensers as well
as other devices capable of reducing stagnation temperature. They also provided practical
examples such as cooling devices, sunshades and collectors with automatic devices such as
thermochromic absorber coatings.
Energies 2021, 14, 733 4 of 39
Thus far, research has considered stagnation as a succession of five distinguishable
processes: heating-up, liquid displacement, vaporization of residual liquid, overheating of
the dried-out absorber regions and the re-filling of the collectors.
Eismann [1] used the thermal hydraulic system code TRACE, specifically developed
for the safety analysis of nuclear systems [15], to model a solar system consisting of eight
flat-plate collectors with meander-type absorbers. The representation of the two-phase
states is essentially based on the material data of water and steam. The two-fluid model of
TRACE [16] describes both the gas and liquid phases by three one-dimensional conservation
equations for mass, momentum and energy. This enables the simulation of two-phase
flows in thermodynamic imbalance, for example subcooled boiling and condensation.
Simulation results showed that the processes of liquid displacement, vaporization
and overheating of dried-out parts of the absorber are, in fact, overlapping. The simulation
showed that the phases of displacement and evaporation occur at the same time and
that the maximum steam range is reached in an early phase when far less than half
of the liquid has evaporated. It was concluded that the maximum steam range can be
approximately calculated based on the properties of water and steam, neglecting the effect
of fractioned distillation of the water propylene glycol mixture used in the real system.
Based on the insight gained, a simplified model was derived. Contrary to the findings of the
TRACE simulations, displacement and evaporation were considered as separate processes.
These simplifications made it possible to solve the differential equation describing the
propagation of steam into the pipes analytically. For the generation of the vapor, only the
residual amount of liquid that remained in the absorbers after displacement was considered.
This residual quantity was calculated beforehand by a drift-flux model. This model was
incorporated in the first version of the open source tool THD [17]. However, experimental
validation of the model was not possible at the time. Furthermore, the two-phase model
of TRACE is based on the properties of water and steam while the real heat carrier is
a mixture of water and propylene glycol. This leads to a considerable uncertainty in
the determination of the steam range which motivated further research resulting in the
present article. The aim of this article is to derive a model that foregoes the simplifications
undertaken earlier.
The new model accounts for the different heat capacities of the steam- and liquid-filled
parts of the absorbers and the rise of the saturation temperature as the water content of
the water-glycol mixture decreases during evaporation. The new model allows for the fact
that displacement and evaporation occur at the same time. The model was calibrated on
the basis of stagnation experiments on two real-size solar systems. Section 2 describes the
experimental facility and the models to calculate heat losses and state variables not covered
by measurements. In Section 3, the thermal-hydraulic model for the displacement of liquid
from the collector array and the propagation of vapor into the circuit is described. Care has
been taken to ensure that the derivations are fully comprehensible, without the need to
consult the literature cited.
2. Basic Models and Methods
2.1. Experimental Facility
At the Institute for Solar Energy Research in Hamelin (ISFH), two identically designed
solar thermal systems have been designed and built, dedicated to the experimental analysis
of stagnation [18]. The collectors are installed on the roof of an outdoor testing facility, as
shown in Figure 1. Both systems have an array of four identically constructed flat-plate
collectors, with different absorber coatings. The collectors of one system are equipped
with conventional highly selective absorbers, whose optical properties are almost not
temperature dependent. The absorber coatings of the other array feature thermochromic
behavior.
Energies 2021, 14, 733 5 of 42 
 
Energies 2021, 14, 733 5 of 42 
 
equipped with conventional highly selective absorbers, whose optical properties are al-
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mooscth nrotm tiecm bpeheraavtiuorre. d ependent. The absorber coatings of the other array feature ther-
mochromic behavior.  
 
Figure 1. Array of four flat-plate collectors with (a) thermochromic absorbers and (b) standard selective absorbe rs. 
Figure 1. Array of four flat-plate collectors with (a) thermochromic absorbers and (b) standard selective absorbers.
Figure 1. Array of four flat-plate collectors with (a) thermochromic absorbers and (b) standard selective absorbers. 
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mffrreoansttu.. red is compared to saturation temperature, which indicates the arrival of the vapor 
front. 
0
0 p
Metal hoses
p Supply line Metal hoses
Supply line Storage tank
1.5
0 Storage tank
1.5 h
0 1.5 C
3 15
Return line 4.5 h1.5 C
3 15
Return line 67.5 4.5
3 MEV
4.5
6 6
7.5 7.5 MEV
9 34.5
6 10.5
Gas
79.5
109.5 1120.5 Gas p
9
10.5
12 p
Temperature sensors: Distances in m Liquid Membrane
 
Temperature sensors: Distances in m Liquid Membrane
Fiigurre 2.. Hyyddrraaulliicc diiaagrraam off tthee cciirrcuiitt,, componentts and ttemperature sensor positions, valid for both sys tems. 
Figure 2. Hydraulic diagram of the circuit, components and temperature sensor positions, valid for both systems. 
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material properties of the pipes and insulation layers are listed in Appendix A. A complete
set of system data and material properties is given in spreadsheets of the simulation tool.
 
Appendix B offers a list of physical and mathematical quantities and their symbols.
 
Energies 2021, 14, 733 6 of 42 
 
and immediately above the two collector fields. The properties of the collectors and the 
material properties of the pipes and insulation layers are listed in Appendix A. A complete 
set of system data and material properties is given in spreadsheets of the simulation tool. 
Appendix B offers a list of physical and mathematical quantities and their symbols. 
Energies 2021, 14, 733 2.2. Collector Design 6 of 39
The model is derived for flat-plate collectors with meander-shaped absorbers tubes 
and integrated header tubes as shown in Error! Reference source not found.. The absorber 
2.2. Collector Design
consists of a copper sheet and meander- and header-tubes made of copper attached to its 
The model is derived for flat-plate collectors with meander-shaped absorbers tubes
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The total aperture area, A MC, of one collector is composed of the meander tube area,HA,tM andH ,b H
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are identical. VM and VH are the volumes of these regions. Four collectors are arranged in
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Top header
AH,t
AM
Meander
AH,b
Bottom header
Supply line Return line
 
Figure 3. Solar collector array with meander-shaped absorber tubes and integrated header tubes connected in parallel.
Figure 3. Solar collector array with meander-shaped absorber tubes and integrated header tubes connected in parallel. 
2.3. Collector Models
2.3. Collector Models 
The stationary useful solar gain of a flat-plate collector with an aperture area, AC,
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empirical collector model, Equation (1). Its parameters, the zero-loss efficiency factor,
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brackets represents the efficiency.
the zero-loss efficiency factor,[0 , and the linear and quadra]tic heat loss coefficients, a1  . a1 a2 2
and a2 , were determinQed= bGyA Ca ηs0ta−nda(TG rdmiz−eTda )te−stG p(rTomc−edTua)re [19] prior to the(1 )stagnation 
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only variable of state. The initial temperatures of the circuit components during normal
operation prior to stagnaQtionGarAe ca
lculatea  d1usTingthTismo
ad2el.TTheoTptic
2alp roperties of a (1) 
conventional selective absorber aCrepr0 actic

Gally cmonstaant wiGthin tmhe opaerational range and
up to stagnation temperature. Therefore, one set of parameters is sufficient to characterize
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requires a second sTet of parameters, η , aliquid temperature, , as the only va0,rTiCabl1e,TC
, and a
 of state.2,TC
. One set corresponds to the
optical properties ofma selective absorber whose emittan Tcehies ivneirtyiallo wte,me.pg.e, r5a%tu. rAebso ovfe the circuit 
compaotnraennstitsi odnutreimnpge nraoturmre,aTl Co,ptheeraetmioitnta pncreioisr mtou scthahginghaetiro, en.g a.,r3e5 c%a,lcreuslualtiendg uinsimnugc hthis model. 
The ohpigthicearlt hperrompaelrlotisesse so. fT ah eceofnficvieennctyiocnuarvl essedleecfitnievde baybtshoerebmerp iarircea lpmraocdteilcaanlldyt hceontwsotant within 
sets of parameters intersect approximately at the transition temperature (red dashed lines
the operational range and up to stagnation temperature. Therefore, one set of parameters 
in Figure 4).
is sufficient to characterize its efficiency. The characterization of collectors with thermo-
chromic absorbers, however, requires a second set of parameters, 0,TC , a1,TC , and 
a2,TC . One set corresponds to the optical properties of a selective absorber whose emit-
tance is very low, e.g., 5%. Above a transition temperature, TC , the emittance is much 
higher, e.g., 35%, resulting in much higher thermal losses. The efficiency curves defined 
 
Energies 2021, 14, 733 7 of 42 
 
by the empirical model and the two sets of parameters intersect approximately at the tran-
sition temperature (red dashed lines in Error! Reference source not found.). 
In normal operation, the heat flow within the absorber plate towards the absorber 
tubes causes a corresponding non-uniform temperature distribution, which results in a 
higher average absorber temperature and a higher convective heat loss compared to the 
isothermal case [20]. Consequently, the average fluid temperature at zero solar gain, TQ0
, is considerably lower than the stagnation temperature of a dry absorber, TS,dry , which 
can be considered locally isothermal. For the same reason, the heat capacity determined 
experimentally from the transient response is higher than the heat capacity determined 
from the masses and the corresponding specific heat capacities. On the other hand, vapor-
ization of the liquid also causes a corresponding heat flow within the absorber plate and 
thus, a temperature distribution different to the locally isothermal case. Therefore, the 
equilibrium temperature of an absorber section containing evaporating liquid lies be-
tween the extremal values for zero solar gain during normal operation and dry stagnation. 
This temperature is defined by a linear relationship, Equation 
Error! Reference source not found., of the average absorber temperature at zero gain, the 
stagnation temperature from the test report, and a parameter, KS . Based on the proce-
dure outlined in Section 0, a parameter value of KS  0.65 was determined. 
TS TQ 0 KS TS,test T Q0   (2) 
The derivation of the model in Section 3 requires the simplified representation of the 
efficiency by which a stagnating collector converts the irradiance into the evaporation of 
the liquid by Equation (3), which is a linear function of the difference between saturation 
and ambient temperature.  
U
  L0  Ts Ta   (3) G
It is based on the zero-loss efficiency factor, 0 , valid for T  TC  and a heat loss 
coefficient, UL , determined by evaluating Equation Error! Reference source not found. 
for   0 , 
0GS,test
UL   (4) TS,test Ta,test
Energies 2021, 14, 733 Figure 4 shows the efficiency curves for the collector with standard selectiv7eo fa3n9d 
thermochromic absorbers characterized by model parameters listed in Table A1, Appen-
dix A. 
0.8
Normal op. Stag. Zero gain, TC
0.7
Zero gain
0.6
0.5
Normal op., TC
0.4
Stag., TC
0.3
Dry stag. TC
0.2
Dry stag.
0.1
0
30 50 70 90 110 130 150 170 190 210
Temperature    C  
Figure 4. Collector efficiency curves for conventional selective coating (black lines) and thermochromic coating (red lines),
valid for G = 1000 W/m2 and Ta = 30 ◦C. 
In normal operation, the heat flow within the absorber plate towards the absorber
tubes causes a corresponding non-uniform temperature distribution, which results in a
higher average absorber temperature and a higher convective heat loss compared to the
isothermal case [20]. Consequently, the average fluid temperature at zero solar gain, T . ,
Q=0
is considerably lower than the stagnation temperature of a dry absorber, TS,dry, which
can be considered locally isothermal. For the same reason, the heat capacity determined
experimentally from the transient response is higher than the heat capacity determined from
the masses and the corresponding specific heat capacities. On the other hand, vaporization
of the liquid also causes a corresponding heat flow within the absorber plate and thus, a
temperature distribution different to the locally isothermal case. Therefore, the equilibrium
temperature of an absorber section containing evaporating liquid lies between the extremal
values for zero solar gain during normal operation and dry stagnation. This temperature is
defined by a linear relationship, Equation (2), of the average absorber temperature at zero
gain, the stagnation temperature from the test report, and a parameter, KS. Based on the
procedure outlined in Section 3.6, a paramete(r value of KS =) 0.65 was determined.
TS = T . + KS TS,test − T . (2)Q=0 Q=0
The derivation of the model in Section 3 requires the simplified representation of the
efficiency by which a stagnating collector converts the irradiance into the evaporation of
the liquid by Equation (3), which is a linear function of the difference between saturation
and ambient temperature.
U
η = η L0 − (TG s − Ta) (3)
It is based on the zero-loss efficiency factor, η0, valid for T < TC and a heat loss
coefficient, UL, determined by evaluating Equation (3) for η = 0,
η G
U 0= S,testL (4)TS,test − Ta,test
Figure 4 shows the efficiency curves for the collector with standard selective and ther-
mochromic absorbers characterized by model parameters listed in Table A1, Appendix A.
2.4. Heat Capacity of Different Absorber Regions
After the pump is shut down, the circulating liquid no longer cools the collector. The
useful solar gain is zero and the temperature of the absorber rises. Heat flow within the
absorber during heating-up is caused solely by the inhomogeneity of the heat capacity.
Therefore, the local variation of the absorber temperature is negligibly small. In con-
Efficiency   -
Energies 2021, 14, 733 8 of 39
sequence, the heat capacity governing the temperature rise of an absorber after pump
shutdown should be calculated solely from the material properties. The contribution of
insulation material and the collector casing to the heat capacity is negligible.
The heat capacity of the dry meander region, CM,0, and the dry top and bottom header
regions, CH,0 = CH,t,0 = CH,b,0 are calculated on the basis of the corresponding masses and
specific heat capacities. The heat capacity of the liquid-filled absorber is,
CC = CM,0 + 2CH,0 + ρlcl(VM + 2VH) (5)
During liquid displacement and vapor generation, the absorber regions contain resid-
ual liquid and saturated vapor. The reduction of steam volume by the very small volume
fraction of residual liquid is neglected. With this simplification the heat capacities of the
three absorber regions can be define(d as. )CX = CX,0 + mr,Xcl + ρgcgVX (6)
with index X as H,t, H,b and M, respectively.
2.5. Heat Loss Distribution
Due to convection within the gap between the absorber and the glass cover, the
cross-sectional average of the air temperature increases from bottom to top along the
slope direction, while the corresponding heat loss coefficient decreases. In addition, the
heat losses along the edges are higher than in the center region of the absorber. Both
effects are superimposed and result in a temperature distribution in which the maximum
is located within the upper third of the absorber. The exact location depends on the
width of the air gap, on the heat losses across the edge and on the inclination angle. One
collector in each collector array was equipped with four temperature sensors attached
to the rear side of the absorber sheet along the middle axis in slope-direction. Figure 5
shows the normalized temperature distribution during dry stagnation, valid both for both
selective and thermochromic coatings. Measured values are indicated by the black markers.
Extrapolated values at the positions of the upper and lower header tubes are indicated
Energies 2021, 14, 733 
 by transparent markers. Because the pressure decreases towards the top of the colle
9c otfo 4r2,  
evaporation starts a little above the location where the temperature is maximal.
1.00
0.98
0.96
0.94
0.92
0.90
Experimental data
0.88
Extrapolated
0.86
0 0.25 0.5 0.75 1
Normalized Distance from lower header axis   -  
FFiigguurree 55.. Norrmalliized ttemperratturree diissttrriibbuttiioon durriingg drryy ssttaaggnaattiioonn,, aass aa ffunccttiioon off tthee diissttanccee ffrrom tthee lloweerr heeaadeerr 
aaxxiiss.. TThhee bbllaacckk maarrkkeerrss iinnddiiccaattee vvaalluueess meeaassuurreedd.. 
The heat loss at the location with the highest temperature is defifined by Equation (3). 
The average heat lloss ccoeeffffiicciieenntt ffoorr tthhee mmeeaannddeer rrereggioionn isi scaclacluclualtaetde dusuinsign gthteh aevaevraergaeg oef 
the experimental values. The heat loss coefficients for the lower and upper header are 
calculated using the extrapolated values. 
T
U S
TS TS
L,H,b UL ; UL,M,0 UL ; UL,H,t UL  (7) TH,b TM TH,t
2.6. Supply and Return Lines 
The supply and return lines are nodalized according to the physical properties, di-
mensions and orientations of the pipe sections. The dimensions of each pipe section are 
characterized by their inner diameter, dt , length, lt , wall cross-section, At , and their 
orientation, sin . Subscripts indicate the section number of the supply line, k, and return 
line, q , starting with k q 1  at the collector field. The heat capacities are 
Cw,i  t,ict,i Aw,ilt ,i ; Cl,i  Vt ,ilcl ; i  k,q  (8) 
The pipes are insulated by rubber foam Armaflex®  HT tubes, whose properties are 
described in the datasheet [21]. Equivalent values are listed in Error! Reference source not 
found.. The inner diameter equals the outer diameter of the pipe, dt ,o . The outer diameter 
is denoted by, di . The heat capacity of the insulation is ignored. 
In practical application, the heat losses of the pipes can be represented by a simple 
model that essentially accounts for the heat conductivity of the pipe insulation. However, 
calibration of the stagnation model against measured data requires minimization of the 
uncertainties. For this reason, radiative and convection losses as well as the solar gain of 
the pipe insulation are modelled in detail. The heat losses of the pipes outside the building 
are influenced by the solar irradiation absorbed by the surface of the insulation layer and 
the convective and radiative heat losses to the environment. The outer surface is assumed 
to be a Lambertian body. The sky temperature, Tsky , is calculated using the model of 
Swinbank as presented in Adelard, et al. [22]. The environment below the horizon is as-
sumed to be a black body at ambient temperature. The contributions to the infrared radi-
ation heat transfer rate from the insulation surface to the environment are weighted by a 
viewing-factor,   0.375 , which accounts for shielding of the sky by the contour of the 
horizon as seen from the position of the pipe. 
The Rayleigh, Prandtl and Reynolds numbers are defined for ambient temperature. 
 
Normalized stag. temperature   -
Energies 2021, 14, 733 9 of 39
of the experimental values. The heat loss coefficients for the lower and upper header are
calculated using the extrapolated values.
T T T
U = U S ; U S SL,H,b L T L,M,0
= UL 〈 〉 ; UL,H,t = UL (7)H,b TM TH,t
2.6. Supply and Return Lines
The supply and return lines are nodalized according to the physical properties, di-
mensions and orientations of the pipe sections. The dimensions of each pipe section are
characterized by their inner diameter, dt, length, lt, wall cross-section, At, and their orienta-
tion, sinφ. Subscripts indicate the section number of the supply line, k, and return line, q,
starting with k = q = 1 at the collector field. The heat capacities are
Cw,i = ρt,ict,i Aw,ilt,i ; Cl,i = Vt,iρlcl ; i = k, q (8)
The pipes are insulated by rubber foam Armaflex® HT tubes, whose properties are
described in the datasheet [21]. Equivalent values are listed in Table A2. The inner diameter
equals the outer diameter of the pipe, dt,o. The outer diameter is denoted by, di. The heat
capacity of the insulation is ignored.
In practical application, the heat losses of the pipes can be represented by a simple
model that essentially accounts for the heat conductivity of the pipe insulation. However,
calibration of the stagnation model against measured data requires minimization of the
uncertainties. For this reason, radiative and convection losses as well as the solar gain
of the pipe insulation are modelled in detail. The heat losses of the pipes outside the
building are influenced by the solar irradiation absorbed by the surface of the insulation
layer and the convective and radiative heat losses to the environment. The outer surface is
assumed to be a Lambertian body. The sky temperature, Tsky, is calculated using the model
of Swinbank as presented in Adelard, et al. [22]. The environment below the horizon is
assumed to be a black body at ambient temperature. The contributions to the infrared
radiation heat transfer rate from the insulation surface to the environment are weighted by
a viewing-factor, γ = 0.375, which accounts for shielding of the sky by the contour of the
horizon as seen from the position of the pipe.
The Rayleigh, Prandtl and Reynolds numbers are defined for ambient temperature.
gL3β(Ti − Ta)ρcp ρνc wLRa = ; Pr p= ; Re = (9)
νλ λ ν
According to Churchill [23], the Nusselt number for mixed natural and forced convec-
tion can be expressed as, ( )
Nu = Nu3 3
1/3
f ree + Nu f orced (10)
The Nusselt number of natural convection from a horizontal cylinder is given by
Churchill and Chu [24], valid for air[, ]2
Nuhor, f ree = 0.752 + 0.387(Ra · 0.401)1/6 (11)
According to Gnielinski [25], the Nusselt number for forced convection from a hori-
zontal cylinder can be expressed as a co√mbination of laminar and turbulent contributions,
Nu 2 2f orced = 0.3 + Nu f orced,lam + Nu f orced,turb (12)
where
√
1/3 0.037Re
0.8Pr
Nu f orced,lam = 0.664 RePr ; Nu f orced,turb = ( ) (13)
1 + 2.443Re−0.1 Pr2/3 − 1
Energies 2021, 14, 733 10 of 39
Tetsu and Haruo [26] derived a correlation for free convection from a vertical cylinder
of the length, lt, based on the Nusselt number for free convection from a vertical wall of the
same height,
l
Nu tvert, f ree = Nuplate, f ree + 0.435 (14)d
where [ ]
Nu 1/6
2
plate, f ree = 0.825 + 0.387(Ra0.345) (15)
which is also valid for air. The convective heat transfer coefficient is defined as the average
of the vertical and horizontal values, weighted by the sine of the inclination angle.
− Nusin 1 sin ; vertλ Nuα = α ϕ + α ( ϕ) α = ; α = horλc c,vert c,hor c,vert c,hor (16)lt di
The heat transfer from the fluid to the tube wall and the thermal conductivity of
the copper tubes are so high that the tube temperature can be considered equal to the
temperature of the tube contents. The surface temperature, Ti, of the insulating layer is
calculated in a simplified way. The net heat flux is the sum of absorbed solar irradiation,
the heat losses through the insulation layer and the heat flux from the surface to the
environment by convection and radiation. In a stationary state, the net heat flux at the
boundaries to the environment is equal to zero, as expressed by Equation (17).
Gαrdi[+( λ2πln d /d (Tt − T )− α( t cd π(T − Ta)i ,o) ) i
4 ( i i )] (17)−σεsdiπ γ Ti − T4sky + (1− γ) T4i − T4a = 0
This equation is solved iteratively for the surface temperature. Subsequently, the
heat conductivity of the insulation layer, defined as a function of the average temperature,
is calculated. Finally, the heat loss coefficient, Uk, for the pipe section, k, related to the
temperature difference between the pipe wall and the environment is calculated.
2.7. Pressure Maintenance
An expansion vessel with a diaphragm type of membrane and a constant gas content
(MEV) serves to maintain pressure. A detailed diagram of the vessel and its location within
the circuit is shown in Figure 2. The steel wall of the vessel has a thickness of s = 2.5 mm
and a mass of mw = 9.5 kg. The rim of the membrane is attached to the circumference of
the inner vessel wall. The diaphragm separates the gas content from the variable liquid
content below the membrane.
The pressure jump across the membrane due to its elasticity is considered negligible.
In consequence, the pressure of the gas volume and the liquid pressure below the rim
are assumed to be identical. The pressure of the water-glycol mixture, indicated in the
formulas by WG, at the connection of the MEV is interpreted as the pressure of the gas
volume above the membrane.
The nominal gas volume of the empty vessel was determined experimentally as
VN = 46.6 liters. The preset pressure, p0, related to a vessel temperature of T0 = 20 ◦C was
set prior to the filling of the circuit. At the time of pump shutdown, the state of the gas
content within the MEV is characterized by a temperature T and a pressure pWG. In this
state the vessel contains a liquid volume Vl , which is calculated using the model of the
ideal gas. ( )
p0,WGVN pWG(VN −Vl)= ⇒ pV = V 1− 0,WGT (18)
T l N0 T pWGT0
The pressure within the upper header of the collector is,
pH,WG = pWG − ρghH (19)
Energies 2021, 14, 733 11 of 39
The stagnation model is based on the properties of water and steam. It is therefore
necessary to transform the pressure measured at the top of the collector array, pH,WG, so that
vaporization of water begins at the same temperature as vaporization of the heat transfer
liquid TyfocorLS® used in the experiments. For this purpose, numerical routines for the
vapor pressure of water, pv(T), as a function of temperature and the saturation temperature
of TyfocorLS®, Ts,WG(p), as a function of saturation pressure were implemented. The
corresponding saturation pressure of steam is,
pH = pv[Ts,WG(pH,WG)] (20)
The corresponding pressure of the gas volume within the MEV is,
p = pH + ρghH (21)
Finally, the corresponding preset pressure for water, p0, at the reference temperature,
T0, is calculated to be,
pT0(VN −Vl)p0 = (22)VNT
With increasing vapor volume during stagnation, hot liquid from the circuit is dis-
placed into the MEV. As a result, the gas temperature increases. The gas temperature, Tg, is
calculated by a model using the following simplifications. The vessel is represented by a
sphere of the same nominal volume, VN , and a corresponding diameter which represents
the characteristic length, L. The liquid content and the associated lower half sphere of the
vessel wall are considered isothermal. An emittance of ε = 1 is assumed for both the MEV
and the environment. The emittances of the membrane and the inside of the steel wall are
defined as εM = 1 and εW = 0.5. The convective heat transfer between vessel wall and
ambient are calculated by Raithby and Hollands [27].
Nu = 0.56[(RaPr/(0.846 + Pr))]1/4 + 2 (23)
The convective heat transfer across the gas volume between membrane and the vessel
wall is estimated using the correlation of Dropkin and Somerscales [28] for 45◦ inclined
cavities heated from below.
Nu = 0.0059Ra0.33Pr0.074 (24)
Heat conduction between the two isothermal vessel shells is estimated using d/3 as
the equivalent length,
. λ
Q = sdπ(Tl − TW) (25)d/3
In a first step, the temperature change of the liquid content due to the ingress of hot
liquid from the expansion line is calculated. In a second step, the temperature changes due
to convective and radiative heat losses during the time interval, τ, are determined. Finally,
the average temperature of the gas content is calculated.
2.8. Properties of the Liquid and Gaseous Phase within the Circuit
The heat carrier liquid TyfocorLS® used in the experiments is a mixture of water,
propylene glycol and anti-corrosion additives with a mass fraction of water, xm = 0.58.
Figure 6 shows the phase diagram of a binary mixture of water and propylene glycol as a
function of the molar fraction, x, calculated for a total pressure of three bar. Apparently,
the vapor phase of the original mixture with a mass fraction of water, xm = 0.58 consists
practically only of the vapor of water. Even at the dry stagnation temperature of the
thermochromic collector, 167 ◦C, and the selective collector, 192 ◦C, the molar fraction of
propylene glycol is quite small. It can be concluded from the diagram that the collector
will never dry out entirely because the saturation temperature of pure propylene glycol
will never be reached.
EEnneregrgieises2 2002211, ,1 144, ,7 7333 1123o off3 492 
 
240
3 bar
Saturated vapor
220
2 bar
200
TS standard selectivity
180
TS thermochromic
160
140
Saturated liquid
120
Mass fraction 0.04 Mass fraction 0.12 Mass fraction 0.29 Mass fraction 0.58
100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Molar fraction water   -
 
FFiigguurree6 6.. PPhhaassee ddiiaaggrraam ooff waatteerr pprrooppyylleenneeg gllyyccoolla attt thhrreeeb baarra abbssoolulutetep prreesssuurere. . 
TThhee ffeeaattuurreess ssttaatteedda abboovveej ujussttiiffyyt thheef foolllloowwininggs simimpplilfiifcicaatitoionnss: : 
• TThhee eefffificciieennccyy ooff tthheea abbssoorrbbeerr rreeggiioonni issd deefifinneeddb byyt thhees saattuurraattiioonnt teemppeerraattuurreeo offt thhee 
rreessiidduuaalll liiqquuiidd,,T Ts,r, w, hwichhicihts ietslfeilsf ias fau fnucnticotnioonf othf es,r thwe awteartecro cnotenntet.nt. 
• The glycol content of the vapor is neglected. Henceforth, the gas phase is referred to
 The glycol content of the vapor is neglected. Henceforth, the gas phase is referred to 
as steam.
as steam. 
• Contribution of the overheated regions of the absorber to the energy balance is ne-
 Contribution of the overheated regions of the absorber to the energy balance is ne-
glected.
• glected. The pressure and the fluid properties of the original mixture define the saturation
 The pressure and the fluid properties of the original mixture define the saturation 
temperature of steam within the pipes.
temperature of steam within the pipes. 
The saturation temperature of the liquid increases as the water content vaporizes
durinTghseta sgantautriaotni.onT hteemcuprereranttumrea sosf frthacet iloiqnuoifdw inatcerreaissecsa lacus ltahteed wfraotmer tchoenitneintita lvraepsoidriuzaels 
mduasrsinagn sdtatghneamtiaosns. fTrhaect ciounrroenf tt hmeaosrsi gfriancatliomni xotfu wreatxer i,s calculated from the initial residual m0
mass and the mass fraction of the original mixture, xm0 , 
m
x r
mr
m = m = m 1−x (26)
x mr + mrr,G mr + mr0r m0m xm m m0
 
r r ,G 1 x (26) m m m0
The molar fraction needed for the calculationr of thr0e saxtmu0ration pressure is,
The molar fraction needmerd/ MfoWr the calculation of thxem /saMtuWration pressure is, x = = (27)
mr/MW + mG0/MG xm/MW + (1− xm0)/Mmr MW x
G
x   m
MW  
The saturation temperature of the liquid, T , is iteratively c (27) mr MW mG0 MG xms,r MW  1 xm0 aMlcuGlated by applying
Raoult’s law,
The saturation temperaturep so=f thxep vl,iWqu+id(,1 T−s,rx), pisv ,Giteratively calculated by apply(2in8)g 
Raoult’s law, 
2.9. Time Evolution of Circuit Temperatures
The time evolution of the tempeprsatuxrpevi,nWeach1pixpepsve,Gct ion is calculated in a three-s(t2ep8) 
procedure within a succession of equidistant time steps. In each time step, the liquid-filled
part of a pipe section is considered as isothermal. In a first step, the temperature change
d2u.9e. tToimthee Edvioslpultaiocne dofl iCqiurciduiits Tceamlcpuelraatteudre. sI f the general flow direction in normal operation
is fromThteh etibmoett oevmoltuottihoen toofp twheit theimn epaecrhatcuorlele icnto eraocfha pcioplele scetoctrioarnr aisy ,ctahlecnulaabteodu tinR Da =thr0e.e8-
osftetph eprflouciedducoren twenitthiisn dai ssupclacceesdsiovnia otfh eequreidtuisrtnanlitn teimdeu rsitnepgss. tIang enaacthio tnim, ree gstaerpd,l tehses loiqf uthide-
hfiyllderda uplaicrtd oefs iag np.ipTe hsiescrtieomna irsk caobnlesirdeesrueldt  wasa sisofothuenrdmeaxl.p Ienr iam feinrstta lsltyepb,y thHea utesmnepre,rFaitnukre,  
change due to the displaced liquid is calculated. If the general flow direction in normal 
operation is from the bottom to the top within each collector of a collector array, then 
 
Temperature    C
Energies 2021, 14, 733 13 of 39
Wagner, Riva and Hillerns [8] and confirmed by Eismann [1] via TRACE simulations. The
average temperatures of th[e liquid(filled pipe se)ction], q, of the return line at the time step, j,
Cw,q + Vq − RD∆Vj ρlcl Tq,j−1 + RD∆V ρT T j l
clTq−1,j−1
q,j = q,j−1 + (29)ρlclVq + Cw,q
are subsequently calculated, beginning with the pipe section, q = 1, connected to the inlet
of the collector array. The average temperatures of the supply line sections, k, are calculated
in the same way while RD is replaced by 1− RD. As shown by simulation results, the total
volume of liquid displaced into the supply line and the maximum steam volume in the
supply line are much less than the volume of the helical coil heat exchanger of about 10 L. It
is therefore assumed that the temperature of the liquid leaving the heat exchanger towards
the reference point is constant and equal to the initial temperature of the return line. The
average temperatures of the expansion line section and of the liquid content of the MEV are
calculated analogously. In a second step, new heat loss coefficients are determined. Finally,
the temperature change within the same time step due to heat losses into the ambient is
calculated.
2.10. Two-Phase Mixture Model
In this section a detailed two-phase mixture model is presented. The model is based
on the integral form of the energy and mass conservation Equations (30) and (31). The
following assumptions were taken: The temperature of the gas and liquid phase are the
same; phase change is caused only by heat transfer across the pipe wall; Kinetic and
potential energy are neglected so that the mass-specific total energy can be substituted
by the mass-specific inner energy, ul and ug, of the liquid and gas phase. In the three-
dimensional formulation, the state variables and the fluid properties are local quantities.
The phase indicator function, ε, indicates the presence of a gas phase, ε = 1, or liquid phase,
ε = 0. The energy[ conservation Equat]ion (30),t s [ → → ]∂ →
∂t ερgug + (1− ε)ρlul dV + ερgugwg + (1− ε)ρlul wl n dS
V
. s ( SC→ → )→ (30)
= ∑ Q− p εwg + (1− ε)wl n dS
S
consists of four parts. The volume integral accounts for the change of internal energy within
the collector volume. The surface integral on the left-hand side describes the flow of inner
energy across the boundary, i.e., the cross-section of the inlet and outlet, of the collector
field or a pipe section. The first term on the right-hand side represents the energy flux
exchanged with the ambient. The second term on the right-hand side is the displacement
power exercised at the inlet and outlet ports. T[he mass conservation]equation is defined as,y ∂ [ ] x → → →
ερg + (1− ε)ρl dV + ερgwg + (1− ε)ρl wl n dS = 0 (31)∂t
V S
Since the pressure losses are ignored, the momentum conservation equation can be
replaced by a static pressure balance. The pressure of the saturated steam is determined by
the temperature of the gas content, Tg,MEV , and the liquid volume, Vl , inside the MEV and
the height, hv, of the liquid column above the MEV.
p V
p = 0 N
Tg,MEV − ρgh (32)
VN −V
v
l T0
Energies 2021, 14, 733 14 of 39
Further simplifications introduced in the following sections allow the integration of
Equations (30) and (31). Integration transforms the phase indicator function, ε, into the
macroscopic void fraction, which is denoted by the same symbol.
Vg V: gε = = (33)
Vg + Vl V
2.11. Drift-Flux Correlation
In a mixture model, the effect of interfacial friction on the void fraction caused by
different phasic velocities must be modelled using an additional correlation. One of the
most successful approaches is the drift-flux model originally developed by Zuber and
Findlay [29], which is presented in a simplified way as follows. The local drift velocity, wgj,
is defined as the difference between the local velocity of the gas phase, wg, and the sum of
the superficial velocities of the gas and liquid phases, jg and jl.
( ) j ( )
wgj = wg − j j
g
g + l = − jg + jl (34)ε
Multiplying both sides by 〈the lo〉cal v〈oid〉 fra〈ct(ion and averaging over the cross-sectionof the pipe yields, )〉
εwgj = jg − ε jg + jl (35)
On both sides of the equation the average of products is replaced with the product
of averages, which requires the introduction of a phase distribution parameter, C0, and a
corrected average drift velocity, Ugj. 〈 〉 ( )
〈ε〉Ugj = jg − C0〈ε〉 jg + jl (36)
Solving Equation (36) for the cross-sectional average of the void fraction yields the
cross-sectional average of the void fraction, 〈 〉
j
〈ε〉 = 〈 g 〉 (37)
C0 jg + jl + Ugj
In the subsequent formulas, the brackets indicating averages are omitted. Many
correlations have been developed for various flow geometries and boundary conditions.
A comprehensive overview is given by Coddington and Macian [30]. The correlations
of Choi, et al. [31], applicable to pipe inclinations from upward-vertically to downward-
vertically oriented circular pipes, were found to be sufficiently suitable for the purpose of
this model. The phase distribution parameter is d√efined as,
2 1.2− 0.2 ρg/ρl(1− exp(−18ε))C0 = + (38)
1 + (Re/1000)2 1 + (1000/Re)2
The two-phase Reynolds number is, ( )
jg + jl dRe t= (39)
νl
For the average drift velocity, they derived an extended form of the Zuber and Findlay
correlation that accounts for the effect of the inclinati[on angle.( ) ]
− 1/4ρ ρ σg
Ugj = 0.0246 · cos ϕ + 1.606 · sin
l g
ϕ 2 (40)ρl
Energies 2021, 14, 733 15 of 39
The correlations of the phase distribution parameter and the average drift velocity are
calibrated by data from stationary experiments with nonzero superficial velocities. With an
appropriate correction presented later in Section 3.1.1, the model can be used to estimate
the void fraction at the limit of vanishing liquid superficial velocity. The liquid holdup is
complementary to the void fraction and a measure for the amount of residual liquid.
ε l = 1− ε (41)
The residual liquid mass of the water-glycol mixture is,
mr0,WG = ρl,WG ·V(1− ε) (42)
The residual mass of water is,
mr0 = xm0 ·mr0,WG (43)
3. Derivation of the Model
3.1. Residual Liquid
Calculation of the amount of residual liquid is decoupled from the transient model. It
is determined on the basis of state variables and boundary conditions that occur when the
saturation temperature is reached, and the following simplifying assumptions:
1. The meander and header tubes are completely wetted. All parts of the absorber
contribute to vaporization according to their efficiency.
2. Only steam leaves the wetted and steam-filled parts of the absorber. The residual
liquid is considered stationary, which is expressed by a liquid superficial velocity of
jl = 0 everywhere in the absorber.
3. The influence of two-phase pressure losses on the saturation temperature is neglected.
Consequently, each absorber within the collector array undergoes the same process
and the saturation temperature is the same everywhere.
4. The steam flow distributions in the upper and lower parts of the absorber are sym-
metrical.
Meander and header tubes are treated separately. The length and the volume of
the pipe bends in the meander tube are added to the straight parts. A meander with a
total length, lM, is nodalized into 2n straight, horizontal pipe sections. The numbering of
absorber pipe sections starts at the middle of the meander tube from k = 1 to nM in both
upstream and downstream direction. It follows from assumptions 2, 3 and 4 that the steam
flow distributions in the upper and lower parts of the absorber are symmetrical.
The superficial velocity of the steam flow is proportional to the section number, k, as
calculated from the mass and energy balance,
.
4m
j = k
4GA η
= M s,M
k
g,k (44)ρgdt,Mπ dt,Mπ2nMhvρg
The efficiency of the meander region is,
ηs,M = η0 −UL,M(Ts − Ta) = 0 (45)
Inserting into Equation (37) and using Equation (41) yields the liquid holdup of the
water-glycol mixture,
ε l,WG,k = (1− εWG,k) (46)
The total amount of residual water in the meander tube is calculated using the initial
mass fraction of water, xm0, and summing up over all the meander tube sections,
nM
mr0,M = xm0mr0,M,WG = xm02ρlVM ∑ (1− εWG,k) (47)
k=1
Energies 2021, 14, 733 16 of 39
Because the absorber area associated with a header is small compared to the meander
region, their contribution to the vaporization is also small. It is therefore sufficient to
describe each header tube by a single node. The superficial velocity in the kth upper header
tube results from the sum of steam flows in the upper half of one to k meanders and the
steam flows in one to k header tubes,
4G(A /2 + A )η k
jg,k =
M H s,H,t (48)
dt,Hπhvρg
Inserting Equation (48) into Equation (37) yields the void fraction. The residual mass
is calculated using Equation (47). The residual mass of the water-glycol mixture in the kth
bottom header tube is calculated analogously.
3.1.1. Corrections for the Residual Mass of Water
The simplifying assumptions two and four stated in the section above were a pre-
requisite for the calculation of the initial residual mass. However, the liquid content will
move during the stagnation process under the influence of gravity and friction between
the steam flow and the liquid. A fraction of the initial residual quantity will leave the
absorber in liquid form and therefore not contribute to the steam flow entering the pipes.
The magnitude of this fraction is estimated considering the following effects:
Eismann [1] found by thermal-hydraulic simulation of a system with flat-plate col-
lectors featuring meander absorbers with highly selective coatings that about 80% of the
displaced liquid flows downwards into the return line, which is in accordance with ex-
perimental results of Hausner, Fink, Wagner, Riva and Hillerns [8]. The complementary
fraction is displaced from the region above the origin of evaporation in upward direction
into the supply line. It can be concluded that with a relatively high efficiency at stagnation
conditions, the vaporization rate and the steam velocity soon become so high that interfa-
cial friction dominates, and the effect of gravity on the liquid content within the pipe bends
can be neglected.
At low efficiencies at stagnation conditions, e.g., in the case of thermochromic ab-
sorber coating, the vaporization rate hence the interfacial friction increase only slowly. In
consequence, a larger part of the liquid above the origin of evaporation is allowed to flow
against the upward flowing steam and collect in the lower parts of the meander where the
heat losses are higher. As a result, phase separation is more pronounced and nearly all
the displaced liquid leaves the collector array via the return line. During the much longer
evaporation phase a considerable amount of residual liquid does not evaporate but eventu-
ally flows into the return line, driven by the weak interfacial friction and gravity. Based
on this hypothesis and the procedure outlined in Section 3.6, the following distribution
function was derived, which describes the ratio of the liquid mass that evaporates during
stagnation to the initial residual mass as[calculate(d by the drift-flux co)r]relation.
δ = 0.02 + (0.7− 0.02) 1− exp −240 · (2.28+4ηM,0)ηM,0 (49)
The same distribution function is used to correlate the probable center of the residual
liquid with the corresponding local heat loss coefficient,
UL,M = UL,H,b + δ(UL,M,0 −UL,H,b) (50)
The relevant part of the residual water contributing to the steam range is,
mr0,M = δxm,0 ·mr0,M,WG (51)
3.2. Temperature Rise
After pump shutdown the absorber is heated up by the solar gain. The temperature
rise is not uniform, due to the increased heat losses across the edges of the collector and
Energies 2021, 14, 733 17 of 39
along the flow path. However, the model for the displacement phase, which is derived in
Section 3.2, requires the assumption of a linear absorber temperature distribution within
the liquid-filled part, which simplifies the modelling tremendously. It follows from this
simplification that boiling commences as soon as the outlet reaches saturation temperature,
Ts. The temperature rise during the heating-up period is described by Equation (52), where
T is the temperature of the absorber at an arbitrary location along the flow path.
When dimensioning solar thermal systems, the maximum steam range should be
determined for extremal but constant values of solar irradiation and ambient temperature.
The elapsed time between pump shutdown and the onset of boiling is irrelevant. For
comparison of simulations and experiments under varying boundary conditions, however,
onset of boiling should coincide with experimental data, because the boundary condi-
tions are time-dependent. A correction factor, Kh = 0.8, accounts for the fact that the
temperature within the meander region rises faster than the average temperature.
.
KhCCT + UL AC(T − Ta) = GACη0 (52)
It is sufficient to solve this equation numerically using the Euler method. For Gη0 >
UL(Ts − Ta), the temperature at the absorber outlet, Tω , reaches saturation temperature, Ts,
within a finite time interval.
With the onset of boiling three processes start to run in parallel. The growing steam
volume displaces the liquid content from the absorber into the connecting pipes. The
regions below saturation temperature are heated up by the solar gain and the displaced
liquid from regions with higher temperature. At the same time, steam leaves the absorber
and enters the pipes. Describing these combined processes using the two-phase mixture
model requires further simplifications. Time evolution of the state variables is assumed as
identical in each collector of the collector array. In consequence, the mass flows of displaced
liquid and of steam entering the adjoining pipes are proportional to the number, n, of col-
Energies 2021, 14, 733 
 lectors. It is adequate to describe liquid displacement and steam generation indepen
1d9e onft 4ly2 ,
based on the following definitions and simplifications illustrated by Figure 7.
Ts
T
T ,0
V
0
VF VF VF
Water Steam
Saturated water Residual water
 
FFiigguurree 77. .TThheerrmmooddyynnaammicic ssttaatteess wwiitthhiinn aa ssttaaggnnaattiinngg ccoolllleeccttoorr ffiieelldd dduurriinngg ddiissppllaacceemmeenntt.. 
Liquid displacement happens simultaneously in the upper header, the lower header
The temperature gradient within the region below saturation temperature results 
and the meander tube. It is therefore not necessary to distinguish between these three
from normal operation prior to pump shutdown, 
regions. This allows the description of the processes of displacement and steam generation
in one dimension as a function of the vodlTumeT,V. T
The temperature gradient within thdeVre
gionVbelow
 saturation temperature results f(r5o3m) 
F
normal operation prior to pump shutdown,
and is defined as a constant during the whole displacement process. The sizes of three 
volume fractions and the correspondindgT a Tω −=bsorber aTrαeas are defined using parameters, (53)
and  , between 0 and 1. dV VF
The volume part, VF , contains steam and residual water. The solar gain of the cor-
responding absorber area,  AF , causes vaporization and, if the pressure increases, a cor-
responding increase in temperature. Only saturated steam generated in this region con-
tributes to the steam flow from the collector into the adjoining pipes. 
The volume part, VF , contains saturated water. Steam generated in this region of 
the absorber only contributes to the displacement of liquid. The simplifying assumption 
of a constant heat loss allows the definition of the parameter,  , as a function of the time 
dependent inlet temperature, T . 
 Ts,r T 
 ;     1 
Ts,r T ,0 
 :    (54) 
 Ts,r T1 ;       1
 Ts,r T ,0 
The temperature rises linearly along the coordinate, V, from the actual temperature, 
T , at the collector inlet, V  0 , to saturation temperature at V  VF . 
1  ;    1
     (55) 
 0 ;    1
By using the definitions stated by Equation (54) and Equation (55), Equation (30) can 
be represented by two coupled equations describing liquid displacement and steam gen-
eration as follows.  
 
Temperature
Energies 2021, 14, 733 18 of 39
and is defined as a constant during the whole displacement process. The sizes of three
volume fractions and the corresponding absorber areas are defined using parameters, µ
and β, between 0 and 1.
The volume part, βVF, contains steam and residual water. The solar gain of the
corresponding absorber area, βAF, causes vaporization and, if the pressure increases, a
corresponding increase in temperature. Only saturated steam generated in this region
contributes to the steam flow from the collector into the adjoining pipes.
The volume part, µVF, contains saturated water. Steam generated in this region of
the absorber only contributes to the displacement of liquid. The simplifying assumption
of a constant heat loss allows the definition of the parameter, µ, as a function of the time
dependent inlet temperature, T{α. T −T }s,r α ; µ + β < 1
µ := Ts,r−Tα,0 (54)
1− β ; Ts,r−TαT −T + β > 1s,r α,0
The temperature rises linearly along the coordinate, V, from the actual temperature,
Tα, at the collector inlet, V = 0, t{o saturation temperature at V} = λVF.
1− β− µ ; β + µ < 1
λ = (55)
0 ; β + µ = 1
By using the definitions stated by Equations (54) and (55), Equation (30) can be
represented by two coupled equations describing liquid displacement and steam generation
as follows.
3.3. Liquid Displacement
Equation (56) is the one-dimensional representation of Equation (30) describing liquid
displacement.
µ∫VF [ ( ) ] ( )∂ . ∂ . p
εµρt g
ug + 1− εµ ρlul dx + m∂ lul = µ GAFηs − nC (CCTs) −m∂t l (56)ρl
0
The first term on the right-hand side represents the part of the solar gain transferred
to the saturated liquid within the region, µVF, of the absorber content, which causes steam
generation. By application of the one-dimensional form of the mass conservation equation,
µ∫VF
∂ [ ( ) ] .
εµρg + 1− εµ ρl dx + ml = 0 (57)∂t
0
the derivative of the void fractio[n can be replaced by th](e mass fl)ow of displaced liquid.
. µ GAFηs − n
∂
C ∂t (CCTs) ρl − ρg
ml = (58)ρghv
In order to integrate this equation analytically over short time periods, the heat
capacity of the collector is set as a constant and the derivative of the saturation temperature
of steam is replaced by the finite difference ratio of values from the preceding time step.
∂ T − T
∆C : n C T ' n s,j s,j−1C,j = C ( )∂t C s CCC ts,j −
(59)
ts,j−1
Energies 2021, 14, 733 19 of 39
The growth rate of the overall void fraction r can be obtained from the mass balance
applied on the whole absorber.
.
. m
ε = ( l ) (60)
VF ρg − ρl
Integration over time yields the void fraction and, finally, the actual steam volume,
Vv = εVF. The parameter, β, describing the steam-filled part of the collector array is defined
as a function of the steam volume and the total residual mass of water:
Vv ε m: rβ = − = ; ε = (61)(1 εr)VF 1−
r
εr VCρl
3.4. Steam Generation
The model for steam generation is based on the following considerations. The va-
.
porization rate, mr, of residual liquid depends on the solar gain at saturation temperature
and the change of enthalpy of the absorber due to change of the saturation temperature of
the residual liquid, Ts,r. The time evolutions of the vaporization rate within the absorber
and the header tubes are different because the initial residual masses and the respective
absorber areas are not the same. It is therefore necessary to calculate the vaporization rate
for the meander and the upper and lower headers separately, while the same parameter, β,
is used for all regions. In order to simplify mathematical expressions, the model will be
derived for an unspecified region, AX , of one single absorber.
Displacement of liquid is usually accompanied by an increase in pressure, hence an
increase in saturation temperature. Therefore, part of the absorbed energy within the
region, βVX , is used for the heating-up of the absorber. The steam generation rate within
the absorber region depends on the current mass of residual water within this region,
mr,X. If mr,X = mr0,X, the absorber region is assumed to be fully wetted and the whole
steam-filled region contributes to steam generation. Steam generation decreases with a
decreasing residual mass of water. It is therefore reasonable to quantify the fraction of
the steam-filled region contributing to steam generation as a function of the current mass
related to the initial mass of the residual water, (mr,X/mr0,X)
α. The value of the exponent,
α, is defined later. Thus, the to(tal power attributed to v)ap(orizatio)n is,
. ∂ m α
Qv,X = GA
r,X
Xηs,r − (C∂t XTs,r) β (62)mr0,X
The goal is to find a formula for the steam power, i.e., the enthalpy flow of steam
leaving the absorber, as a function of time. By definition, steam generated within the
volume fraction of the region, βVX, contributes only to the outflow of steam from the
collector field. Therefore, the surface integral on the left-hand side of Equation (30) reduces
. .
to mgug and the surface integral on the right-hand side of the same equation to mg p/ρg.
Thus, the one-dimensional form of Equation (30) becomes,
β∫VX . [ ] .
( εX ρgug − ρlul)dx(+ mg),Xug0 (63)α
= GAXηX − ∂∂t (C T
mr,X
X s,X) β m −
.
m p
r0,X g,X ρg
Integration and replacing specific inner energies by specific enthalpies, u = h− p/ρ,
yields, ( ) ( )α
.( ) . ∂ m
V h r,Xβ Xε ρg g − ρlhl + mg,Xhg = GAXηX − (C∂t XTs,X) β (64)mr0, X
Energies 2021, 14, 733 20 of 39
The void fraction and its derivative are defined by the residual mass and the mass
fraction of water. .
εX = 1−
mr,X ⇒ . mr,XεX = − (65)ρl βVXxm,X ρl βVXxm,X
Since the densities of water and propylene glycol above 100 ◦C differ by less than 2%,
it is sufficient to base the void fraction on the density of the saturated water. The mass flow
of steam can be expressed by the derivative of the residual liquid by applying the mass
balance, Equation (31), to the region, X. ( )
. . ρ
m −m l
− ρg
g,X = r,X (66)ρl
Inserting into Equation (64) yields(the differential equati)on for the residual liquidmass.
. β GAXηX −
∂
∂t (CXTs,X)
m αr,X = −mr,X (67)hvmαr0,X
Since the saturation temperature varies only slowly with pressure, the equation can
be integrated over a sufficiently short time interval, tj − tj−1. The derivative of stored heat
is replaced by the finite difference with values from the previous interval. The specific
enthalpies of steam are evaluated at the saturation temperature of the liquid.
( )( ) ( )
∂ CX,0 + mr,X,jcl TX,j − TX,j−1 + VX ρg,jhg,j − ρg,j−1hg,j−1
(C
∂t X
Ts,X) ' ∆CX,j = − (68)tj tj−1
It is reasonable to assume that dry out is reached within a finite time interval, which
requires an exponent α < 1. Solving Equation (67) by the separation of variables and
integration yields,
[ ( )] ( )1 β GA η − ∆C
mr,X(t) = m1−αr,X,j − −
1−α X X1 Xj( α)RX t− tj ; with RX = α (69)hvmr0,X
Differentiation and using Equation (64) and multiplication by the enthalpy of vapor-
ization results in the enthalpy flow of satur[ated steam from the region, X.. ρl − ρ ] αP g ( )v,X = mg,Xhv = R m1−α 1−αρ X r,j − (1− α)RX t− tj (70)l
The total enthalpy flow from the collector array is,
.
Pv = mghv = nC(Pv,M + Pv,TH + Pv,BH) (71)
Figure 8 shows the normaliz(ed enthal)py flow of steam in the isothermal case as afunction of the dimensionless time. The choice of α = 0.5 is justified as follows.
For a short time interval of tj−1 − tj < 1 s, the second term within the square
brackets of Equation (70) could be neglected because the magnitude of R 3X is 10 times
smaller than the initial residual mass. This would allow the differential equation for
the steam propagation derived in the next section to be solved analytically and result in
Equation (80). However, the choice of the exponent, α, is not critical, as will be explained in
Section 3.6. For α = 0.5 Equation (70) is a linear function of time, which allows to solve
Equation (80) analytically for any time interval, as long as the saturation temperature can
be considered as a constant. This is beneficial in two ways. In practical applications of this
model, where the steam range is determined under constant but extremal conditions, the
time interval can be considerably extended, which results in a short simulation time. Due
to the analytical solution, no convergence issues occur.
EEnneregrgieise s2022012,1 1,41,4 7,3733 3 23 of 42 21 of 39
 
 
Figure 8. Relative enthalpy flow of steam as a function of dimensionless time.
Figure 8. Relative enthalpy flow of steam as a function of dimensionless time. 
3.5. Steam Propagation into the Circuit
3.5. Steam Propagation into the Circuit 
Steam propagation into the circuit is described by the energy Equation (30), applied
toSatecaomnt roplrovpoalugmatieond efiinnteod bthyea ficinrcitueits ecist iodne, skc,roibfepdi peb.yT htheei ndeenxe,rgky,  isEoqmuiatttieodn where
Errpoors! sRibelfee.reHnecaet scoounrdceu cntoiot nfoiunnadx.i, aalpdpilrieecdt itoon ai sconnetgrolel cvteodlu.mAes dheafripnepdh bays eab foinuinted saercy- across
tion, k , of pipe. The index, k , is omitted where possible. Heat conduction in axial direc-
the pipe cross-section is assumed. The flow across any pipe cross-section is therefore either
tion is neglected. A sharp phase boundary across the pipe cross-section is assumed. The 
steam or liquid. The liquid holdup, ε , of the steam-filled part of a pipe section is considered
flow across any pipe cross-section is thelrefore either steam or liquid. The liquid holdup, 
as zero. The void fraction of a pipe section is interpreted as the dimensionless location, x,
 l , of the steam-filled part of a pipe section is considered as zero. The void fraction of a of a virtual steam front.
pipe section is interpreted as the dimensionless Vlogcatioxn, x, of a virtual steam front. 
ε = = (72)
V V lg x
With these simplifications, the left-hand si de of Equation (30) can be integ(r7a2t)e d and
rearranged as follows: V l
With these simplifi[ca(tions, the left)-hand s]ide of Equa[tion (30) can be in]tegrated and t s →
rearranged as follow∂ s: → →
∂t (ε ρgug − ρluVk .  )
l + ρ(lul dV + ε ρgu)gwg − ρlul wl d SSk (73)→ →
=Ax ρguu  g gg−ρllull +Alul udgVρgwg−uglρu t    lgwlg  lulwl dS
V  k Sk
The first term of the right-hand side of Equation (30), (73) 
. −Ax. gug− lul− Auggwg u∑ Q = xC l
lwl   ∂
k(Ts Tk) xk · dkπUk(Ts − Ta)− xk (CkTs) (74)∂t
The first term of the right-hand side of Equation (30), 
consists of three parts. The first part describes the enthalpy flow into the pipe wall as it is
exposed by the moQvingsxtCeamTfroTnt. TxhisdterUm contributes on
ly to positive velocities of the
steam front. The second pkartsaccokuntsk fokr thk
T T x C T   (74) 
e hesat laossesk tot thek asmbient. The third part
describes the enthalpy flow due to change of saturation temperature. The time derivative
consists of three parts. The first part describes the enthalpy flow into the pipe wall as it is 
expoofstehde beyn tthhea lmpoyviisnrge sptelaacme dfrobnyt.t hTehifis nteitremd cioffnetrreibnucteesa nonallyo gtoo upsostiotivEeq vuealtoicointie(s6 8o)f. thTeh e heat
stecaamp afcroitnyt.p Tehr eu nseitcolenndg tpharotf aaccpoipunetsse fcotiro tnh,eC hke, adte lpoesnsedss toon ththe easmpbeiceinfitc. hTehaet tchaiprda cpitairets of the
despcirpibeeasn tdhet heentshtaelapmy .flow due to change of saturation temperature. The time derivative 
of the enthalpy is replaced by the finiCtek d=iffAerweρntccet  +anAaltoρggocugs to Equation (68). The heat (75)
capacityT pheerh uenaittc laepnagcthit yofo af  tphiepien sseucltaiotinn,g Claky,e dr eispiegnndosr oedn .tTheh espseecciofnicd hteeartm caopnatchiteiersi gohf t-hand
thes ipdipeeo afnEdq uthaet isotena(m30. ) describes the rate of work exercised by the steam entering and the
liquid leaving the control volumCe,k  Awtct A(tgcg  ) (75) x . ..The heat capac−ity∑ oWf th=e i−nsulati→pwng→n dlaSye=r iAs igpnwore−d. wThe =secpomndg, kte m−rmp onl, kthe right-k g l (76)
hand side of Equation (30) describSes the rate of work exercised by th
ρeg steam eρnltering and 
the liquid leaving the control volume, 
Combining Equations (73), (74) and (76), and substituting the phasic velocities by the
.
mass flows m = A f ρw and the inner energies with enthalpies, u = h− p/ρ, leads to the
 following representation of the energy equation.
Energies 2021, 14, 733 22 of 39
. ( ) . .
Akx ρgug − ρlul −mg,khg + ml,khl
= − .xCt,k(Ts − Tk)− x · dkπkk(Ts − Ta)−
(77)
x∆Ck
From the mass conservation applied to the same control volume one gets an expression
for the liquid mass flow across the boundary.
. . . ( )
ml,k = mg,k − xAk ρg − ρl (78)
Substituting into Equation (77) and replacing the enthalpies of saturated steam and
water with the enthalpy of vaporization, hv, results in a differential equation for the location,
x, of the steam [front within the contro]l volume defined by the pipe section, k.. .
x Akρghv + Ck(Ts − Tk) + x[dkπUk(Ts − Ta)− ∆Ck] = mg,khv (79)
.
The enthalpy flow, mg,khv, of saturated steam entering the pipe section, k, is equal to
.
the enthalpy flow of steam leaving the collector, Pv = mghv, with the mass flow of steam
from Equation (71), reduced by the heat flux, c, dissipated from the steam into the pipe
walls.
. .
mg,khv = Pv − c = mghv − c (80)
Inserting this equation into Equation (79) and substituting Equation (71) for the steam
power results in a differential equation for the l(ocatio)n, x, of the steam front..
xa + xb = e− f t− tj − c (81)
The quantities, e, and, f, are defined for the whole collector array consisting of nC
collectors and regions X ∈ {Ht, M, Hb}[as follows: ]
ρl − ρg ( )( )m 1/2e = nCβG∑ AXηX − ∆C r,XX,j (82)ρl X m[( r0,)X ]
ρ − ρg n β2l C GA
2
f X
ηX − ∆CX,j
=
ρl 2h
∑
v m1/2
(83)
X r0,X
Because the pressure losses and the influence of check valves are ignored, the height
levels of the steam front in the supply and return lines are equal. If the phase boundaries
of both the supply and return line are located in horizontal pipe sections, both steam fronts
are assumed to propagate at the same speed. These rules are formally implemented by
parameters, δk = 0, 1 and δq = 0, 1, according to Table 1 and the coefficients, a, b and c,
defined as follows: [ ] [ ( )]
a = δk Akρghv + Ck(Ts − Tk) + δq Aqρghv + Cq Ts − Tq
b = dkπUk(Ts − Ta) + ∆Ck + dqπUq(Ts − Ta) + ∆Cq
k−1 q−1 (84)
c = ∑ li[diπUi(Ts − Ta) + ∆Ci] + ∑ li[diπUi(Ts − Ta) + ∆Ci]
i=1 i=1
Table 1. Propagation parameters and pipe direction with respect to the inlet and outlet of the collector
field.
Supply Line Return Line δk δq
horizontal horizontal 1 1
horizontal downward 1 0
downward horizontal 0 1
downward downward 1 1
Energies 2021, 14, 733 23 of 39
An increase in steam volume usually accompanies an increase in pressure. Therefore,
all coefficients become time-dependent. However, within a sufficiently short time step the
coefficients can be considered as constant, which allows the derivation of an analytical
solution. The solution of the homogeneous p[art of Equation (81) is,
b ( )]
xh = C1 exp − t− tj (85)a
The solution of the inhomogeneous equatio(n has)the form,
xp = C2 + C3 t− tj (86)
Comparing of coefficients yiel(ds, )
1 f a f
C2 = e− c + ; C3 = − (87)b ( ) b ( ) b
Inserting the initial condition, xh tj + xp tj = xj, yields the constant, C1, of the
homogeneous solution.
C1 = xj − C2 (88)
Finally, the location of the(steam front,)starti[ng at the ini]tial time, tj, is,( ) ( )
x t− tj = xj −
Pv − c − b − Pv − cexp t t + (89)
b a j b
.
Of interest are also the heat losses from the pipes to the ambient, Q , and the total
. loss
dissipated(heat fr)om the steam to the pipes, Qdiss, which [are calculated using Equ]ation (84)..
Qdiss xk, xq = c +[xk[dkπUk(Ts − Ta) + ∆Ck] + xq dqπUq(Ts − Ta]) + ∆Cq (90)
. ( ) k−1 q−1
Qloss xk, xq = ∑ lidiπUi + lkdkπUk + ∑ lidiπUi + lqdqπUq (Ts − Ta) (91)
i=1 i=1
.
The change rate of heat, Qaccum, accumulate(d by the)steam filled parts of the collectorsis,
. m α
Q r,Xacc = β∑ ∆CX (92)
X mr0,X
3.6. Numerical Procedure
The equations describing the states of the MEV, the steam-filled parts of the circuit and
the absorber regions are formally coupled by shared state variables. Therefore, stepwise
calculation can lead to oscillations of these variables. Some of these oscillations, like
the pressure transients observed in reality [8], p. 67 are caused by the physical processes.
Others, which could be limited by a sufficiently short time step, are caused by the numerical
procedure. A time step of 0.2 s proved to be a good compromise between computing time
and time resolution. Inexplicable peaks and oscillations are sufficiently canceled out by
taking the average of new and previous values of pressure, saturation temperature and
the change rates of heat, where the previous values are weighted by a factor of 20. The
consequently resulting infringement of the law of conservation of energy is negligible.
The residual mass of the system with standard selective absorbers is calculated based
on a 5 min average of solar irradiation. For the system with thermochromic absorbers the
solar irradiation is averaged over 10 min. The starting point of the time series is taken as
the moment when saturation conditions are reached.
Eleven experimental data sets for both solar thermal systems were available. Ac-
cording to the properties of the model only four data sets from experiments with inactive
Energies 2021, 14, 733 26 of 42 
 

 mr ,X 
Qacc  CX    (92)  
X  mr0,X 
3.6. Numerical Procedure 
The equations describing the states of the MEV, the steam-filled parts of the circuit 
and the absorber regions are formally coupled by shared state variables. Therefore, step-
wise calculation can lead to oscillations of these variables. Some of these oscillations, like 
the pressure transients observed in reality [8], p. 67 are caused by the physical processes. 
Others, which could be limited by a sufficiently short time step, are caused by the numer-
ical procedure. A time step of 0.2 s proved to be a good compromise between computing 
time and time resolution. Inexplicable peaks and oscillations are sufficiently canceled out 
by taking the average of new and previous values of pressure, saturation temperature and 
the change rates of heat, where the previous values are weighted by a factor of 20. The 
consequently resulting infringement of the law of conservation of energy is negligible. 
The residual mass of the system with standard selective absorbers is calculated based 
on a 5 min average of solar irradiation. For the system with thermochromic absorbers the 
Energies 2021, 14, 733 solar irradiation is averaged over 10 min. The starting point of the time series is taken2 4aosf 39
the moment when saturation conditions are reached. 
Eleven experimental data sets for both solar thermal systems were available. Accord-
ing to the properties of the model only four data sets from experiments with inactive check 
valcvheesc wk evrael vuessewd eforer umsoedeflo cramlibordaetliocanl.i bTrhaet iuon.kTnhowe unn pkanroawmnetpearsr awmeertee rdsewteerrmeidnetde rimteirn-ed
ativiteelrya tbiyv eal yqubayliataqtuivaeli tpartoivcedpurroec eddisuprleadyeisdp liany FeidguinreF 9ig. ure 9.
Figure 9. Procedure to determine model parameters.
 
Figure 9.A Pnroicnedituiarle vtoa dlueeterfmorinteh me opdaerla pmareatmere, tKerSs,. for the effective stagnation temperature in
Equation (2), was chosen. The distribution parameter, δ, was varied until the measured
anAdns iimniutilaalt evdasluteea mforr athnge epsamraamtcehteedr,. TKhe ,c foorrr etshpeo enfdfeincgtiveeffi sctiaegnncyatoiofnth teempeaenradteurrree ignio n,S
EquηaMti,owna (s2s),t owreads cfohrocsuenrv. eThfiett idnigst.rTibhuettiiomn epeavroamluteitoenr,s of ,t hweams veaasruierded uanntidl tshime mulaeatesdurseteda m
andra snimgeuslwateedre sctoemamp arraendg.eIsf tmheatgchraeddi.e Tnhtse ocfotrhreesspimonudlaintegd esfftiecaimencryan ogfe t,hwe hmicehanddepere nrde-on
giotnh,e efficM , 
iewnacsy ,swtoerreedt ofoors cmuarlvle,  tfhiettivnaglu. eThoef  tthime pe aervaomluetteior,nKs So, fw tahse imnceraesauserdeda nanddv isciemvue-rsa.
This procedure was repeated until the steam ranges and the time evolutions matched.
lateFdin satlelay,mth reanpgareasm weetreers coofmthpeardeidst. rIifb tuhteio gnrfaudniecntitosn o, fE tqhuea stiiomnu(l4a9t)e,dw setreeamde trearnmgein, ewdh, ibcahs ed
depoenntdh eonfi nthael  veaffliuceiesnc(y, w).ere too small, the value of the parameter, δ η KM S
, was increased 
and vice versa. This procedure was repeated until the steam ranges and the time evolu-
tion4s.  Rmeastuclhtesda.n FdinDailslycu, sthsieo pnarameters of the distribution function, Equation (49), were 
determiTnhede, ibnaitsieadl  sotnat tehseo ffinthale vtawluoesso la(rtMhe)r.m al systems were determined by running in
normal operation prior to pump shutdown. The pumps of both systems were switched
off manually at the same time. The subsequent process of stagnation is discussed in the
 following sections as a comparison between measured and simulated data.
4.1. Comparison of Experimental Data and Simulations
Both solar thermal systems were simulated using the initial and boundary conditions
from data sets of eleven experiments. Figure 10 shows measured and simulated total
maximum steam ranges, i.e., the sums of maximum steam ranges in the supply and return
line. The confidence limits are based on the assessment of uncertainties in Section 4.2.
In seven experiments, the check valves were active (transparent markers). As a result,
a height difference of the liquid columns within the return and supply line establishes,
which roughly corresponds to the opening pressure of two serial check valves. As soon as
the steam range reduces, the circuit is refilled via the return line. If the solar gain and/or
released heat from the tube walls are sufficient to vaporize the entering liquid, the steam
volume within the collector and the steam range in the supply line are maintained beyond
the time span considered.
In four experiments, the check valves were manually opened after pump shutdown
(filled-out markers). As expected, the steam ranges in the system with thermochromic
absorbers (red markers) are considerably smaller than in the system with conventional
selective absorbers (black markers). A higher system pressure, based on a preset overpres-
sure of two bar, also leads to smaller steam ranges (rhombic markers) than with a system
overpressure of only one bar (circles), based on a preset pressure of one bar. The circles
Energies 2021, 14, 733 27 of 42 
 
4. Results and Discussion 
The initial states of the two solar thermal systems were determined by running in 
normal operation prior to pump shutdown. The pumps of both systems were switched 
off manually at the same time. The subsequent process of stagnation is discussed in the 
following sections as a comparison between measured and simulated data. 
4.1. Comparison of Experimental Data and Simulations 
Both solar thermal systems were simulated using the initial and boundary conditions 
from data sets of eleven experiments. Figure 10 shows measured and simulated total max-
imum steam ranges, i.e., the sums of maximum steam ranges in the supply and return 
line. The confidence limits are based on the assessment of uncertainties in Section 4.2. 
In seven experiments, the check valves were active (transparent markers). As a result, 
a height difference of the liquid columns within the return and supply line establishes, 
which roughly corresponds to the opening pressure of two serial check valves. As soon as 
the steam range reduces, the circuit is refilled via the return line. If the solar gain and/or 
released heat from the tube walls are sufficient to vaporize the entering liquid, the steam 
volume within the collector and the steam range in the supply line are maintained beyond 
the time span considered. 
In four experiments, the check valves were manually opened after pump shutdown 
(filled-out markers). As expected, the steam ranges in the system with thermochromic ab-
sorbers (red markers) are considerably smaller than in the system with conventional se-
Energies 2021, 14, 733 lective absorbers (black markers). A higher system pressure, based on a preset ove2r5porfe3s9-
sure of two bar, also leads to smaller steam ranges (rhombic markers) than with a system 
overpressure of only one bar (circles), based on a preset pressure of one bar. The circles 
marked with a white cross correspond to the dataset “08−16” which was used to determine 
marked with a white cross correspond to the dataset “08−16” which was used to determine
the uncertainties in Section 4.2. The circle marked with a star corresponds to the dataset 
the uncertainties in Section 4.2. The circle marked with a star corresponds to the dataset
““0077−−2255”” wwhhiicchh iiss ddiissccuusssseedd iinn SSeeccttiioonn 44..11..44.. 
22
p0=1bar
20 CV active
TC, p =2bar p0=1bar
18 0
CV active CV deactivated
16 TC, p0=2bar TC, p0=1bar
CV deactivated CV active
14
TC, p0=1bar
12 CV deactvated
p0=2bar
10 CV active
8
Confidence limit
6
4
p0=2bar
2 CV deactivated
0
0 2 4 6 8 10 12 14 16 18 20 22
Measured steam range   m
  
FFiigguurree1 100.. CCoomppaarriissoonn ooff ttoottaall sstteeaam rraannggeess ffrroome exxppeerriimeennttaalld daattaaa anndds siimuullaattiioonnssf foorr tthhees syysstteem wiitthh tthheerrmoocchhrroomiicc 
aabbssoorrbbeerrss( (rreedd))a annddc coonnvveenntitoionnaalls seelelecctitviveea abbssoorrbbeerrss( b(blalacckk).).M Maarrkkeerrssw witihthw whhitietec crroossseessc coorreressppoonnddt otot htheed daatataseset t0 088−−16.. 
TThheem maarrkkeerrw witihtha ar reedds statarrc coorrrreessppoonnddsst otot htheed daatatasseet t0 077−−25.. 
Thee numeerriiccaall daattaa arree lliisstteed iin Taabbllee2 2,,t tooggeetthheerr wiitthht thhee meeaassuurreedds soollaarri irrrraaddiiaannccee 
aand tthe resiiduall mass of tthe waatteerr--ggllyyccooll mixixttuurree. .TThhe eavaevreargaegde dirirrardaidainacnec edadtat laisltiesdte idn 
iTnaTbaleb l2e d2ifdfiefrf efrorf otwr tow roearseoansos.n Ts.hTe hseysstyesmte wmitwhi tthetrhmeromchorcohmroicm aibcsaobrsboerrbse (rTsC(T) rCe)arcehaecsh tehse 
the onset of evaporation later than the system with standard selective absorbers (SC). The
averages are taken over the approximate period, τ, of liquid displacement, which was
estimated as τSC = 300 s, and τTC = 600 s.
 
Table 2. Average irradiation on the system with the standard selective (SC) and thermochromic (TC) absorber coating.
Experimental and simulation results of the total steam range and simulation results of the residual mass. Active check
valves (CV) are indicated by “X”.
Dataset CV p0 <G> Residual Mass Total Steam Range
mm−dd - Bar W/m2 W/m2 kg kg m m
SC TC SC TC SC TC SC TC
Exp. Exp. Sim. Sim. Exp. Exp. Sim. Sim.
07−24 X 1 960 960 1.61 2.24 19.0 4.5 17.5 6.0
07−25 X 1 1014 1014 1.52 1.74 20.5 9.0 20.2 12.4
07−26 X 1 960 956 1.60 2.16 17.5 2.3 18.0 6.3
07−27 X 1 902 903 1.63 2.17 20.5 6.5 19.6 7.5
08−02 X 2 960 960 2.03 6.59 13.5 0.8 11.1 0.9
08−03 X 2 988 988 1.78 3.72 15.8 0.8 15.0 1.6
08−06 X 2 997 997 1.89 5.15 13.5 0.8 12.5 0.9
08−16 − 1 986 990 1.59 2.40 18.0 4.5 18.9 5.2
09−04 − 1 892 890 1.82 6.90 16.5 4.5 14.9 3.4
09−05 − 1 948 950 1.61 2.70 16.5 3.0 19.1 4.0
09−18 − 2 948 948 2.39 6.83 7.5 0.0 5.9 0.0
The comparison of experimental and simulated data shown in Figure 10 and Table 2
leads to the following conclusions.
Steam range by simulation   m
Energies 2021, 14, 733 26 of 39
• For the system with standard selective absorbers, the simulated values for the total
steam ranges lie well within the confidence limits. No significant difference between
active and open check valves can be determined. It follows that the effect of the
check valves on displacement and evaporation processes within the collector array is
negligible.
• For the system with thermochromic absorbers, the steam ranges with open check
valves are within the confidence range, whereas the model tends to overestimate
steam ranges in the cases with active check valves.
Energies 2021, 14, 733 29 of 42 
 It should be noted that information on the total steam range is not sufficient. In the
next sections the considerable effect of the check valves on the individual steam ranges in
the supply and return lines will be discussed.
In the simulation, at about the same time as in the experiment, the beginning vapor-
4.i1z.a1t.iCono ncvauensetiso sntaelaSme lteoc teivneteAr bthseo rsbueprps,lyD eliancet.i vTahte dcoCnhtiencukoVuasl vbelasck line shows the corre-
spoFnidgiunrge h1e1igshhot wofs tthhee lhiqeiugihdt lpevroefil.l eAsso sfotohne saus pthpely satenadmr erteuarcnhelisn eths.e Ilteavleslo osfh tohwe slothweer 
mhaexaimdearl (sdteaasmhedra bnlgaecsk olifnteh)e tshyes stetemamw ietxhpcaonndvse anltsioo ninaltos etlheec trievteuarnb sloinrbe.e Brseccoaurrsees pthoen mdiondgel 
todtoheesd naotta asectco0u8−nt1 f6o.rI nprtehsesuexrep elorismseesn, tt,hteh leiqliuqiudi dlelveevlesl asrdei tfhfeer sbaym0e.2 a3nmd .thTeh cisonistivneuryoulist talend 
anddasjuhestdifi belsactkh eliansessu omvpertiloapn. tThhate tshiempurleastseudr seteloasmse rsacnagne breeancehgelse cat emda. xCimonusmeq u24e nmtliyn, tahfeter 
liqpuimdple svheulstdinowthne, swimhuiclha tcioninacrideeesq quuail.te well with the experiment.  
 
Figure 11. Height profiles of the supply and return lines.
Figure 11. Height profiles of the supply and return lines. 
Figure 12a shows the steam range within the supply and return lines of the system with
conventional selective absorbers for the dataset 08−16. The check valves were manually
opened at the same time as the pump was switched off. The experimental data are
interpreted as follows. About 13 min after pump shutdown, steam starts to leave the
collector and first enters the supply line, as expected. Even before the steam front reaches
level the lower header, steam enters the return line. A small hydrostatic imbalance is
established, which is compensated by pressure losses within the collector array. Between
21 and 25 min after pump shutdown the steam range is maximal in both the supply- and
return line.
 
Energies 2021, 14, 733 27 of 39
Energies 2021, 14, 733 30 of 42 
 
a) 12 3
Return line exp. / sim.
10 2.5
8 Liquid level 2
above MEV
6 1.5
Supply line exp. / sim.
4 1
2 0.5
0 0
0 5 10 15 20 25 30 35 40
Time after pump stop   min
b) 1.6 0.8
Steam power
1.4
Solar irrad.
1.2 0.6
1
0.8 0.4
Total dissip. heat
0.6
Heat losses
0.4 0.2
0.2
0 0
c) 350 2
300
Steam filled part of coll.
250 1.5
200
Liq. filled part of coll.
150 1
Steam filled part of pipes
100
50 0.5
0
-50 0
d) 5.0 14
Total steam vol.
12
4.0
Pressure exp. / sim. 10
3.0 8
2.0 6
Pipes Coll. array 4
1.0
2
0.0 0
e) 170 70
Meander
160 60
150 50
140 40
Orig. mixture
130 30
MEV
120 20
110 10
0 5 10 15 20 25 30 35 40
Time after pump stop   min  
FFiigguurree 1122.. EExxppeerriimmeennttaall aanndd ssiimmuullaattiioonn rreessuullttss ffoorr tthhee ssyysstteemm wwiitthh ssttaannddaarrdd sseelleeccttiivvee aabbssoorrbbeerrss.. OOppeenneedd cchheecckk vvaallvveess. .
CCaassee 0088−−116.6 .(a()a S)tSeatemam rarnagneg aenadn ldevleevl ehlehigehigt,h (tb, )( bS)teSatmea pmowpoewr, ehre, ahte laotssloess saensda nirdraidrriaatdioiant,i o(cn), C(ch)aCnhgae nrgaetersa otef shoeaf th, e(dat), 
Overpressure and steam volume, (e) Saturation temperature and gas temperature inside MEV. 
(d) Overpressure and steam volume, (e) Saturation temperature and gas temperature inside MEV.
UInp tthoe 3s0im muilna taioftne,r aptuambopu sthtuhtedsoawmne, tthime esiamsuilnattehde reexspueltrsi mareen tc,lothsee tboe gthinen einxpgevria--
mpoernitzaalt ivoanluceasu. sAefstesrt etahmat,t tohee nstiemruthlaetisounp rpelsyulltisn efo. rT thhee csuonptpinlyu loinues dbleaccrkealsine,e wshhoerweasst hine 
tchoer reexsppeornimdienngth tehieg hstteoafmth setaliyqsu icdonlesvtaeln.tA wsistohoinn tahset hcoenstseidamererdea tcihmees tshpealne.v Tehl oerfet haerelo twweor 
 
Sat. temperature    C Overpressure   bar Change rate of heat W Irradiation   kW/m
2 Steam range   m
Gas temperature MEV   Steam volume   l Change rate of heat kW Steam power and heat Height level   m
 C losses   kW
Energies 2021, 14, 733 28 of 39
header (dashed black line) the steam expands also into the return line. Because the model
does not account for pressure losses, the liquid levels are the same and the continuous and
dashed black lines overlap. The simulated steam range reaches a maximum 24 min after
pump shutdown, which coincides quite well with the experiment.
Up to 30 min after pump shutdown, the simulated results are close to the experimental
values. After that, the simulation results for the supply line decrease, whereas in the
experiment the steam stays constant within the considered time span. There are two main
reasons for the deviation of the simulation from the experiment. (1) From the beginning of
the vaporization process, steam leaving the meander tube penetrates a layer of residual
liquid in the upper header. If the velocity of the steam drops below a certain value, gravity
starts to dominate the interfacial friction and liquid trickles into the meander tube where
it vaporizes within a short distance from the junction. As a consequence, an asymmetric
pressure loss distribution of the steam flow within the meanders is established, and the
height of the liquid level within the supply line is correspondingly lower. This leads to a
higher steam range in the supply line. (2) The increasing concentration of glycol leads to
an increasing concentration of the glycol content in the vapor phase. Both effects are not
reflected by the model.
Figure 12b shows the simulated steam power, the heat losses and the thermal power
dissipated into the steam- filled parts of the supply and return lines. The difference between
steam power and dissipated thermal power corresponds to the heating-up of the pipe walls
from their initial temperature to saturation temperature as the steam front propagates. The
difference is quite high because the initial temperatures of the supply and return lines
are only 49 ◦C and 38 ◦C. Because the pressure and the saturation temperature increase
while the steam range increases (see Figure 12d,e), the heat flow dissipated into the pipes
is higher than the heat losses into the ambient. The difference is accumulated in the pipes.
The steam range increases until the heat losses, the dissipated heat flow, and the steam
power coincide.
Figure 12c shows the simulated enthalpy change of the liquid-filled part of the collector
during heating-up, which becomes zero as soon as the liquid mass flow displaced from the
collector array into the supply and return lines becomes zero. Also shown is the enthalpy
change of the steam-filled regions of the collector field and the pipes due to pressure change.
If the pressure and thus the saturation temperature increase, the steam-filled parts of the
pipes accumulate heat. With decreasing steam range, the pressure decreases, too, and heat
is released again. In the collectors, however, the saturation temperature of the residual
liquid increases because the water content of the mixture decreases. Therefore, in this case,
the enthalpy change rate is always positive. This can also be concluded from the monotonic
growth of the saturation temperature of the residual liquid shown in Figure 12e. Also
shown in the same figure is the temperature course of the gas volume within the MEV.
It is essential to take the heat capacities of the pipe walls and the collectors into
account. Otherwise, a much faster process and a considerably higher steam range would
result. The following effect is also caused by heat capacity and the pressure transient.
About 22 min after pump stop the steam front leaves the long, slightly downwardly-sloped
pipe section and enters the short, nearly vertical connector that leads to the next long,
downwardly-sloped pipe section. This is visible by the sudden decrease in level height in
Figure 12a. As a result, pressure and saturation temperature also increase, which causes
the heat transfer rate into the steam-filled parts of the collectors and pipes to increase as
well. This is indicated by the sudden increase in the dissipated heat flow into the pipes
visible in the corresponding curves in Figure 12b,c and the sharp decrease in the steam
power in Figure 12b.
Figure 12d shows the pressure transients and the development of steam volume in the
collector field and the pipes. The pressure depends only on the state of the MEV, i.e., the
liquid volume transferred from the circuit into the MEV by the growing steam volume, on
the temperature of the gas volume and on the height of the liquid column above the MEV.
Energies 2021, 14, 733 29 of 39
4.1.2. Conventional Selective Absorbers, Check Valves Active
The check valves are of the spring-loaded type, which require a pressure of 0.04 bar to
open in flow-direction. Therefore, a corresponding difference of liquid levels occurs, and
the steam displaces the liquid mainly via the return pipe into the MEV.
Figure 13a shows the steam ranges from simulation and experiment for the dataset
07−26. The experimental results show the expected asymmetry. A height difference of
about 1 m is established, which corresponds to the head loss of two serially-connected
check valves. The model does not reflect this effect. The fact that the total steam range
from the simulation nevertheless corresponds closely to the experimental value can be
explained: During the early stages of liquid displacement, the two-phase pressure loss
within the collector array causes a pressure difference across the check valves in normal
flow direction. Apparently, the pressure difference is higher than the opening pressure of
the check valves. As a result, the displacement process is basically the same as in the case
with open check valves. This is confirmed by the fact that, at the very beginning of steam
expanding into the circuit, steam enters the supply line first.
In later stages of stagnation, the pressure loss is much smaller than the opening
pressure of the check valves, which results in the different steam ranges as can be seen in
the experimental values.
The initial pipe temperatures of the supply and return lines immediately before pump
stop are 94 ◦C and 88 ◦C for the dataset 07−26 compared to only 49 ◦C and 38 ◦C for the
dataset 08−16. There is less steam power needed to heat the pipe walls up to saturation
temperature. Furthermore, the average solar irradiation for the dataset 07−26 is about 3%
higher than for 08−16. Based on these data alone one would expect a significantly higher
total steam range for 07−26. However, the total steam ranges are practically the same. This
is explained by the following countering effect. Because of liquid expansion due to the
higher circuit temperature the initial pressure in the case 07−27, Figure 13d is about 0.2
bar higher than the initial pressure for 08−16, Figure 12d. Consequently, the saturation
temperature is higher, which results in a lower collector efficiency. In this example, the
steam range is practically independent from the initial temperature. However, with a larger
MEV or even a compressor type of pressure maintenance system, where the pressure is
kept within the range of a small hysteresis, the maximum steam range would increase with
increasing initial temperature. This must be kept in mind when designing a solar thermal
system.
4.1.3. Thermochromic Absorbers—Deactivated Check Valves
Figure 14a shows the development of the steam range in the system with ther-
mochromic absorbers for the dataset 08−16. The steam ranges are slightly overestimated,
but the time evolution is qualitatively consistent. As expected, steam enters in the supply
line first. In the experiment, the steam range in the return line starts to develop about 5 min
later than in the simulation, which is interpreted as follows: Due to the lower efficiency, the
vaporization rate is lower than with standard selective absorbers. As a result, interfacial
friction is so weak that liquid within the pipe bends flows downwards against the steam
flow. This results in a more pronounced separation of the steam region, which occupies the
top of the meander, from the liquid filled region below. In consequence, the liquid is almost
entirely displaced into the return line. This process is mainly balanced by the pressure
losses within the meander tube which leads to a corresponding difference of the liquid
levels. Compared to the system with standard selective absorbers, where a symmetrical
steam flow distribution is a reasonable assumption, the origin of evaporation lies at a
location much closer to the top of the meander. In consequence, a larger proportion of the
residual mass accumulates in the lower regions of the absorber and, driven by gravity and
the weak interfacial friction exercised by the co-current steam flow, enters the lower header
and finally the return line. Figure 14b shows the corresponding steam power and pipe heat
losses.
Energies 2021, 14, 733 30 of 39
Energies 2021, 14, 733 33 of 42 
 
a) 18 3
Return line sim. / exp.
16
2.5
14
Supply line sim. / exp.
12 2
10 Liquid level
above MEV 1.5
8
6 1
4
0.5
2
0 0
0 5 10 15 20 25 30 35 40 45 50
Time after pump stop   min
b) 1.6 0.8
Steam power
1.4
Solar irrad.
1.2 0.6
1
0.8 0.4
Total dissip. heat
0.6
Heat losses
0.4 0.2
0.2
0 0
c) 350 2
Steam filled part of coll.
300
250 1.5
200
Steam filled part of pipes
150 1
Liq. filled part of coll.
100
50 0.5
0
-50 0
d) 5.0 14
Total steam vol.
12
4.0
Coll. array
Pressure exp. / sim. 10
3.0 8
2.0 6
Pipes 4
1.0
2
0.0 0
e) 170 70
Meander
160 60
Orig. mixture
150 50
140 40
130 30
MEV
120 20
110 10
0 5 10 15 20 25 30 35 40 45 50
Time after pump stop   min  
Fiigurree 1133.. Expeerriimeennttaall aand ssiimullaattiioonn rreessullttss fforr tthee ssysstteem wiitth ssttaandaarrd sseelleeccttiivee aabssorrbeerrss.. Accttiivvee cchheecckk vvaallvveess. .
DDaattaasseett 0077−−262.6 (.a()a S)tSetaemam rarnagneg aenadn ldevleevl ehlehigehigt,h (tb, )( bS)teSatmea mpowpoewr, ehre, ahte laotslsoesss aensda nirdraidrriaadtiioanti, o(cn), C(ch)aCnhgaen rgaetersa otef shoeaf th, e(dat), 
O(dv)eOrpvreerspsruerses uarneda sntdeasmte avmoluvmoleu,m (ee), S(ea)tuSraatutiroant itoenmtpemerpateurraetu arneda ngadsg taesmtpemerpateuraretu irnesiidnesi MdeEMV.E V.
In later stages of stagnation, the pressure loss is much smaller than the opening pres-
sure of the check valves, which results in the different steam ranges as can be seen in the 
experimental values. 
 
Sat. temperature    C Overpressure   bar Change rate of heat W Irradiation   kW/m
2 Steam range   m
Gas temperature MEV   Steam volume   l Change rate of heat kW Steam power and heat Height level   m
 C losses   kW
Energies 2021, 14, 733 31 of 39
Energies 2021, 14, 733 35 of 42 
 
a) 3 3
Return line exp. / sim. Supply line exp. / sim.
2.5 2.5
2 Liquid level 2
above MEV
1.5 1.5
1 1
0.5 0.5
0 0
0 10 20 30 40 50 60
Time after pump stop   min
b) 1.4 0.35
1.2 0.3
Solar irrad.
1 0.25
Steam power
0.8 0.2
0.6 0.15
0.4 0.1
0.2 Total dissip. heat 0.05
Heat losses
0 0
c) 40 0.5
Steam filled part of coll.
30 0.4
Steam filled part of pipes
20 0.3
10 0.2
0 0.1
Liq. filled part of coll.
-10 0
d) 5.0 10
Total steam vol.
4.0 8
3.0 6
Pressure exp. / sim. Coll. array
2.0 4
Pipes
1.0 2
0.0 0
e) 170 70
160 60
Meander
150 50
140 40
Orig. mixture
130 30
120 20
MEV
110 10
0 10 20 30 40 50 60
Time after pump stop   min  
FFiiggurree 1144.. EExxppeerriimeenntatal laanndd sismimuulaltaitoino nrerseuslutslt fsofro trheth seysstyesmte mwitwh itthhetrhmeormchorcohmroicm aibcsoabrbseorrsb.e rs. Inactive check valves.
DInaatcatsievte0 c8h−ec1k6 .v(aal)vSetse. aDmatraasnegt e08a−n1d6.l e(av)e lShteeaimgh rt,a(nbg)eS atenadm lepvoewl heer,ighheta, t(blo)s Ssetesaamn dpoirwraedr,i ahteioant ,lo(cs)seCsh aanndg eirrraatdeisatoifonh,e a(ct), 
(Cdh) aOnvgeer praretesss uorfe haenadt, s(dte)a Omvveorpluremsseu, (ree) aSnadtu srtaetaimon vtoemlupmeer,a (teu)r eSaatnudragtaiosnt etmemppereartauturerein asnidde gMasE tVem. perature inside MEV. 
As can be seen in Figure 15, the simulation not only considerably overpredicts the 
total maximum steam range. In addition, the progressions of experimental and simulated 
steam ranges do not even correspond qualitatively. 
 
Sat. temperature    C Overpressure   bar Change rate of heat Irradiation   kW/m
2 Steam range   m
W
Gas temperature MEV   Steam volume   l Change rate of heat Steam power and Height level   m
 C kW heat losses   kW
Energies 2021, 14, 733 32 of 39
4.1.4. Thermochromic Absorbers—Active Check Valves
In the experiments where the check valves are active, this effect is even more pro-
nounced. The liquid is almost completely displaced into the return pipe. Due to the
imbalance caused by the opening pressure of the check valves, the steam generally flows
downwards driving the residual liquid towards the lower regions. As a result, the total
steam range is lower compared to situations where the check valves are open, which corre-
sponds to the results of dedicated experiments on a solar system with the same hydraulic
design [3]. This is the reason why the steam ranges for collectors with thermochromic ab-
sorbers and operational check valves, datasets 07−24 to 07−27, are generally overestimated
by the model, as displayed in Figure 10 (red circles). The effect is much less pronounced
for the datasets 08−03, 08−04 and 08−06, where the collector efficiency is very low.
As can be seen in Figure 15, the simulation not only considerably overpredicts the
Energies 2021, 14, 733 
 total maximum steam range. In addition, the progressions of experimental and simulated
36 of 42 
steam ranges do not even correspond qualitatively.
9 3
Return line exp. / sim.
8
2.5
7
6 2
5
1.5
4
3 1
Liquid level
2 above MEV 0.5
1
Supply line exp. / sim.
0 0
0 5 10 15 20 25 30 35 40 45 50
Time after pump stop   min  
Figure 15F.i gSutreea1m5.  Srtaeanmger afnrgoemfro emxepxepreirmimeenntt aanndds imsiumlautiloantifoornt hfeosry sttheem swyisthtetmhe rmwoicthr otmheicrmabsoocrhberrosm. Aicct iavbe schoercbkevrasl.v Aes.ctive check 
valves. DDaatatasseett 0077−−2255.. 
4.2. U4.2n.cUerntcaerintatiinetsie s
The distance between the temperature sensors is ∆x = 1.5 m in both the feed and the
reTthuren dliinseta. nAcceco brdeitnwgeteonS cthheeu treenm[p12e]r,athtueruen sceerntasionrtys iosf Δthxe s=t e1a.m5 mfro innt lboocatthio tnhde ufeeed and the 
retutront hlieneq. uAidcicstoarndt ilnocga ttion Socfhseeunsroerns c[1an2]b, ethdet eurnmcineretdauinsitnyg othf ethreela stitoenamof Kfriorknutp loancdation due to 
the eFqreunikdeils[t3a2n]:t location of sensors can b√e determined using the relation of Kirkup and 
Frenkel [32]: σ∆x = ∆ x/ 3 = 0.87 m (93)
Due to the small slope angle, ϕ, of the long pipe sections the phase boundary occupies
about sin(ϕ)dt ' 0.5 m of pipe length.Wxithinxthis r3egio0n.,8t7he amver age temperature of both (93) 
the liquid and the pipe are below saturation as the steam range increases. Furthermore,
thDeuheig thoc tohned uscmtivailtly solfotphe caonpgpleer,tub, eosfp trhevee nlotsnsgh aprippceh asnegcteisoonfst etmhpe eprahtuarse. Tbhoeusne dary occu-
effects add about σϕ = 0.25 m to the uncertainty in the steam range.
pies aboTuht e suinncertaidntties0o.f5boumnd aorfy pcoipndei lteionngstahn.d Wmiattheirnia lthprios preergtiieosna,n tdhteh eaivreerffaegcteo tnemperature 
of bothteh sttheaem lirqaunigde wanasdd tehteer pmiipneed abrye sbimeluolwati osnatbuarsaedtioon tahse tdhaeta ssetet a08m− 1ra6.nAgcec oinrdcirnegatsoes. Further-
morKe,o vthaces h[3i3g]h, a cnounndceurtcatiinvtiytyof o2f% thfoer tchoepmpeears utruebdesso lparreirvraednitasn csehaanrdpa cnhuanncegretasi notfy toefmperature. 
4% for the collector efficiency at saturation conditions are assumed. The uncertainty of
Thesthee epfrfeescettsp aredsdsu areboofutht eMEVd0e.2p5endsmontot htehuen ucenrctaeirnttayionftyth einp rtehsesu srteegaamug rearneagdein. gs
oTf h0.e0 5ubnacr.eArtsasiunmtiiensg oafn buonucenrdtaainrtyy coofnthdeitgiaosntse manpedr amtuarteewriiathl ipnrtohpe eMrtEiVeso af n10dK thaet ir effect on 
the sthteeatmim eraonf gseet twinagst hdeepterersmetinpreedss burye sriemsuultlsaitnioanf ubrathseerdu onnce rtthaein dtyaotaf saebto u0t8−0.1065.b Aarc. cording to 
The uncertainty of the heat conductivity of the pipe insulation was estimated at 10%. The
Kovuanccse [r3ta3i]n,t ayno fulonccaelratmaibnietynt otef m2%pe rfaotru rtehaen md tehaeswurinedd vseololacirt yirarraednioatnccoen saindedr eadn.  Tuhnecertainty of 
4% fuonrc tehrtea icnotilelsecatnodrt ehfefiriceifefencctyo natth seasttueraamtiroang ceoanredliitsitoendsin aTraeb aless3u. med. The uncertainty of the 
preset pressure of the MEV depends on the uncertainty of the pressure gauge readings of 
0.05 bar. Assuming an uncertainty of the gas temperature within the MEV of 10 K at the 
time of setting the preset pressure results in a further uncertainty of about 0.05 bar. The 
uncertainty of the heat conductivity of the pipe insulation was estimated at 10%. The un-
certainty of local ambient temperature and the wind velocity are not considered. The un-
certainties and their effect on the steam range are listed in Table 3. 
Table 3. Uncertainties of material properties and boundary conditions and their effect on the simulated steam range, based 
on data set 08−16. 
Quantity Uncertainty Effect on Steam Range M 
  Standard Thermochromic 
Solar irradiance ±2% +1.06/−1.01 +0.87/−0.90 
Collector efficiency ±4% +0.49/−0.49 +0.17/−0.28 
Preset gauge pressure 1bar ±0.1 bar +0.46/−0.51 +0.22/−0.34 
Heat conductivity of pipe insulation ±10% −0.8/+0.96 −0.3/+0.32 
Based on these results, a total average uncertainty of ±4 m for the system with stand-
ard selective absorbers and ±2.9 m for the system with thermochromic absorbers was de-
termined. When simulating the steam range during system dimensioning, an increase the 
simulated steam ranges by the relative uncertainties listed in Table 4 is suggested.  
 
Steam range   m
Height level   m
Energies 2021, 14, 733 33 of 39
Table 3. Uncertainties of material properties and boundary conditions and their effect on the simulated steam range, based
on data set 08−16.
Quantity Uncertainty Effect on Steam Range M
Standard Thermochromic
Solar irradiance ±2% +1.06/−1.01 +0.87/−0.90
Collector efficiency ±4% +0.49/−0.49 +0.17/−0.28
Preset gauge pressure 1bar ±0.1 bar +0.46/−0.51 +0.22/−0.34
Heat conductivity of pipe insulation ±10% −0.8/+0.96 −0.3/+0.32
Based on these results, a total average uncertainty of ±4 m for the system with
standard selective absorbers and ±2.9 m for the system with thermochromic absorbers was
determined. When simulating the steam range during system dimensioning, an increase
the simulated steam ranges by the relative uncertainties listed in Table 4 is suggested.
Table 4. Relative uncertainty to be used in system dimensioning.
Absorber Type Steam Range Average Uncertainty Relative Uncertainty
m m -
Conventional selective 18 4.0 0.22
Thermochromic 4.5 2.9 0.63
For two reasons, application of the drift flux model poses the most fundamental
uncertainty of the model. On the one hand, this correlation was neither validated nor
intended to be used at the limit of vanishing liquid mass flow. On the other hand, it is only
strictly applicable for constant superficial velocities in straight pipes. However, for absorber
tubes with diameters equal to or smaller than 9 mm the error is sufficiently low. For larger
pipe diameters, flow-pattern independent drift-flux correlations, e.g., Chexal, et al. [34]
and Bhagwat and Ghajar [35], or a combination with flooding correlations might be better
suited.
A more detailed analysis of uncertainties is not possible because the number of data
sets and the number of quantities affected by uncertainties are of the same magnitude.
5. Conclusions
A transient model for the stagnation of solar thermal systems was developed based
on the integral form of a two-phase mixture model. Based on scientific articles, and the
author’s own experience that fast pressure transients with large amplitudes do not occur,
the momentum conservation equation was replaced with a simple pressure balance and
a drift flux model. With successive simplifications, it was possible to derive at a one-
dimensional representation of the mixture model. The resulting differential equation for
the steam range could be analytically integrated over short time periods. Thus, stability
problems were avoided, and efficient computing was achieved.
The fact that displacement and steam generation occur at the same time has been
considered by defining two time-dependent volume fractions, one completely liquid-filled,
the other containing steam and residual liquid. Steam generation within the liquid-filled
volume fraction is attributed to liquid displacement, whereas evaporation of the residual
mass is linked to steam generation. This made it possible to formulate liquid displacement
and steam expansion into the circuit by two sets of energy and mass conservation equations.
Heat losses from collectors and pipes to the ambient are modelled by well-established
correlations for convective and radiative heat transfer. A transient model for the gas
temperature inside the membrane expansion vessel was also derived. The model leaves
only four parameters unknown, which were determined by comparison of experimental
and simulated data. One parameter, in Equation (2), accounts for the fact that the effective
stagnation temperature of a collector lies between the temperature of dry stagnation and
the temperature where the solar gain at normal operating conditions is zero. The other three
Energies 2021, 14, 733 34 of 39
parameters are numerical values within Equation (49), which defines the ratio of residual
mass relevant for steam expansion into the circuit to the residual mass calculated by the drift
flux model. The model is applicable to solar systems with flat-plate collectors and meander-
type absorbers with tube diameters ≤9 mm. Application to other hydraulic concepts and
larger diameters of absorber tube requires different correlations for the residual mass and
calibration experiments with collectors of the particular hydraulic concept.
In order to calibrate the model, an experimental facility was designed and built,
dedicated to the performance of well-defined stagnation tests under real conditions. The
facility consists of two solar thermal systems, the one with standard selective absorbers,
the other with thermochromic absorbers. Two spring-loaded check valves prevent natural
circulation in the circuit and protect the pump from hot liquid and/or steam during
stagnation. Experiments with active and deactivated check valves were carried out. Because
the model does not account for the effects of the check valves, only experiments with
deactivated check valves were used to determine the unknown parameters.
In the experiments where the check valves were deactivated, i.e., opened immediately
after pump shutdown, the experimental values of the liquid levels in the supply and return
lines were almost equal. Therefore the simulated progression of the steam ranges in the
supply and return lines correspond well to the experimental values.
The experiments with active check valves show a distinct difference of the liquid
levels due to the opening pressure of the check valves. The simulation result of the system
with standard selective absorbers correspond well to data from experiments with active
as well as deactivated check valves. Thus, check valves have practically no effect on the
total steam range. It can be concluded that the magnitude of the pressure losses within the
meander tube is larger than the opening pressure of the check valves.
The simulation results of the system with thermochromic absorbers correspond well
to data from experiments with deactivated check valves. With activated check valves, both
the progression of the steam range and its total maximum differ considerably. Due to the
much lower velocities of steam and liquid in the system with thermochromic absorbers,
the associated pressure losses are lower than the opening pressure of the check valves.
Therefore, phase separation is more pronounced, and a larger amount of liquid is displaced
via the return line, resulting in a smaller amount of residual liquid and, in consequence, a
smaller steam range. Because the model does not account for the effects of the check valves,
the simulation tends to overestimate the steam range.
It is important to note that the effect of the check valves must be considered when
dimensioning a solar system. It is suggested that the steam range in the supply and
return line should be corrected in accordance to Equations (94), which are functions of the
inclination angle and the number, n, of the check valves and their opening pressure, ∆p.
n∆p − n∆p∆lv,SL = ; ∆lv,RL = (94)2ρg sin(ϕ) 2ρg sin(ϕ)
In general, the model is capable of predicting the maximum steam range well within
the accuracy needed for system design. Due to the complexity of the experimental facility,
which represents two real systems, as well as the transient ambient conditions and the
limited number of sensors, it is not possible to attribute experimental data accurately to
certain effects.
The model is implemented into the open-source simulation tool THD, dedicated to the
thermal-hydraulic dimensioning of solar systems up to about 100 m2 collector area [17,36].
The actual size is limited mainly by the validity of the assumption that pressure losses
during stagnation are negligible. Tools of this kind will become increasingly important as
the key to designing both cost effective and operationally safe solar systems.
Energies 2021, 14, 733 35 of 39
Author Contributions: Derivation of the model and programming, R.E.; concept and design of the
experimental facility, F.G. and S.H.; building, instrumentation and calibration of the experimental
facility, S.H.; experimental work including data acquisition and post-processing, S.H.; writing—
original draft preparation, R.E.; writing—review and editing, F.G. and S.H. All authors have read
and agreed to the published version of the manuscript.
Funding: The theoretical part of this research was funded partially by the Swiss Federal Office of
Energy, grant number Sl/501722-01, and by funds from the research program Future Energy Efficient
Buildings and Districts (FEEB&D) of the Swiss Competence Center for Energy Research (SCCER),
grant number CTI.2014.0119. The experimental part, carried out in the frame of the project “Process
technology, quality assessment and system solutions for thermochromic absorbers in solar thermal
collectors (ProTASK)”, was funded by the German Federal Ministry of Economic Affairs and Energy
based on a decision of the German Federal Parliament (reference numbers 0325858 A and B).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The software tool and the datasets are available at https://sourceforge.
net/projects/thd/.
Acknowledgments: The authors would like to thank Nick Moon and Peter Fry for proofreading.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
The coefficients of the collector model, Equation (1), and the heat capacities of the
meander- and header regions in dry state are listed in Table A1.
Table A1. Model characteristics of collectors.
Quantity Unit Standard Thermochromic
Critical temperature Tc ◦C - 68
Temperature range T ≤ Tc T > Tc
zero-loss efficiency factor - η0 = 0.785 η0 = 0.757 η0,TC = 0.83
Heat loss coefficient W/Km2 a1= 4.19 a1 = 4.27 a1,TC = 6.17
Heat loss coefficient W/K2m2 a2 = 0.0135 a2 = 0.0065 a2,TC = 0.0103
Stagnation temperature ◦C 192.1 167
Dry meander region heat capacity J/K 2793
Meander volume l 0.924
Dry header region heat capacity J/K 566
Header volume l 0.660
Aperture area m2 2.33
Hydraulic elements Length m di m Number of nodes -
Absorber tube 21.88 0.0082 88
Header tube 1.1 0.02 1
The material properties of the insulation layer are shown in Table A2. The optical prop-
erties have been estimated. The heat conductivity is modelled by a 2nd order polynomic
based on the data-sheet [21],
λ = λ0 + aϑ + bϑ2 (A1)
Energies 2021, 14, 733 36 of 39
Table A2. Material properties of the pipe insulation.
Heat Conductivity W/Km W/◦C2m W/◦C3m
0.03764 7.7 × 10−05 8 × 10−07
Absorption coefficient - Emissivity - Inner diameter mm Outer diameter mm
0.9 0.95 22 60
The properties of the pipe material used for the supply and return lines are listed in
Table A3.
Table A3. Properties of copper tubes.
Inner Diameter Outer Diameter Density Specific Heat Conductivity
mm mm kg/m3 J/kgK W/Km
20 22 8600 382 300
Properties of the Heat Carrier
A chemical analysis of the heat carrier TyfocorLS® yielded a mass fraction, xm0 = 0.579,
for the system with conventional selective absorbers and, xm0 = 0.557, for the system with
thermochromic absorbers. The saturation temperature of the mixture is calculated based
on the simplifying assumption of an ideal two-component mixture of water and propylene
glycol using properties listed in Table A4. Within the temperature range considered here,
the vapor pressure is sufficiently well described by a power function,
p bv = aϑ (A2)
The coefficients listed in Table A4 are based on data from Steele, W. V. et al. [37] for
propylene glycol, from D’Ans J., Lax E., Blachnik R. [38] for water and from the data sheet
of TyfocorLS® [39].
Table A4. Properties of water and propylene glycol.
Molar Mass Coefficients for Vapor Pressure in Pa
Component kg/kmol a b
Water 18.02 0.7517·10−3 4.047
Propylene-glycol 76.1 1.8745·10−8 5.601
TyfocorLS - 0.7517·10−3 4.225
The specific heat capacity, density and viscosity are approximated by functions based
on D’Ans et al. [38] for water and the data sheet of TyfocorLS® [39]. The surface tension of
TyfocorLS® is modelled based on experimental data presented by Chang, et al. [40] for a
binary mixture of propylene glycol with a molar fraction x = 0.85 of water.
σ = 0.0373− 4.08 · 10−5ϑ N/m (A3)
Energies 2021, 14, 733 37 of 39
Appendix B
Table A5. Symbols and indices.
Symbol Unit Symbol Unit
A m2 Area V m3 Volume
.
a1 W/Km2 Heat loss coefficient W W Work rate
a2 W/K2m2 Heat loss coefficient w m/s Velocity
c J/kgK Specific heat capacity wgj m/s Drift velocity
C J/K Heat capacity x m Location of steam front
.
C0 - Distr. parameter x m/s Velocity of steam front
d m Diameter x - Molar fraction
D m Outer diameter xm - Mass fraction
G W/m2 Solar irradiation Greek symbols
hC m Height difference α - Steam power exponent
h J/kg Specific enthalpy α - Absorption coefficient
hv J/kg Spec. evap. enthalpy αc W/Km2 Conv. heat transfer coeff.
j m/s Superficial velocity α 2r W/Km Rad. heat transfer coeff.
K - Correction factor β 1/K Expansion coefficient
L m Char. length β - Steam-filled fraction
l m Length γ - View factor
m kg Mass δ - Distribution parameter
.
m kg/s Mass flow ε - Emissivity
n - Number of nodes η - Efficiency
nC - Number of collectors λ W/Km Heat conductivity
ν m2/s Kinemat. viscosity
Pv W Steam power ρ Kg/m3 Density
p Pa Pressure σ Pa Surface tension
.
Q W Enthalpy change, gain τ s Interval
RD - Displacement ratio ϕ rad Inclination angle
S m2 Surface Dimensionless numbers
T K, ◦C Temperature Nu - Nusselt number
UL W/Km2 Heat loss coefficient Pr - Prandtl number
Ugj m/s Average drift velocity Re - Reynolds number
u J/kg Specif. inner energy Ra - Rayleigh number
Subscripts
a Ambient i,k,q Indices
C Collector
F Field, array S Stagnation
G Glycol s Saturation
g,l Gas, liquid phase t Tube, pipe
H,b Bottom header v Vapor
H,t Top header W Water
i Insulation WG Water-Glycol mixture
M Meander α Inlet
m Mean value ω Outlet
Constants
g Acceleration of gravity 9.81 m/s2
σ Stefan-Boltzmann constant 5.67037·10−8 W/K4m2
References
1. Eismann, R. Thermohydraulik von Solaranlagen. Ph.D. Thesis, Eidgenössische Technische Hochschule ETH Zürich, Zürich,
Switzerland, 2014.
2. Terschueren, K.-H. Solaranlagen zur Brauchwassererwärmung, Anlagenkomponenten, Installation, Inbetriebnahme, Betrieb-
sverhalten, Teil 2. IKZ-Haustechnik, Ausgabe 4. 1996. Available online: https://www.ikz.de/ikz-archiv/1996/04/9604118.php
(accessed on 18 November 2020).
Energies 2021, 14, 733 38 of 39
3. Eismann, R.; von Felten, P. Stillstandsverhalten von Solaranlagen, Publikation 195200; 195200; Swiss Federal Ofice of Energy SFOE:
Bern, Switzerland, 1998.
4. Hausner, R.; Fink, C. Stagnation behaviour of solar thermal systems. In Proceedings of the ISES Europe Solar Congress,
Copenhagen, Denmark, 19–22 June 2002.
5. Hausner, R.; Fink, C. Stagnation Behaviour of Solar Thermal Systems—A Report of IEA SHC—Task 26—Solar Combisystems; Interna-
tional Energy Agency IEA: Paris, France, 2002.
6. Streicher, W. Minimising the risk of water hammer and other problems at the beginning of stagnation of solar thermal plants—A
theoretical approach. Solar Energy 2001, 69 (Suppl. 6), 187–196. [CrossRef]
7. Lustig, K. Experimentelle Untersuchungen zum Stillstandsverhalten thermischer Solaranlagen. Ph.D. Thesis, University Karl-
sruhe, Karlsruhe, Germany, 2002.
8. Hausner, R.; Fink, C.; Wagner, W.; Riva, R.; Hillerns, F. Entwicklung von Thermischen Solarsystemen mit Unproblematischem
Stagnationsverhalten; Bundesministerium für Verkehr, Innovation und Technologie: Wien, Austria, 2003.
9. Rommel, M.; Siems, T.; Schüle, K.; Mehnert, S.; Becker, R. Wieviel Dampf produziert ein Kollektor im Stillstandsfall? In
Proceedings of 15th Symposium Thermische Solarenergie, Kloster Banz, Germany, 7–9 May 2014.
10. Rommel, M.; Siems, T.; Schüle, K.; Mehnert, S.; Thoma, C. Schlussbericht zum Teilprojekt: Entwicklung von Techniken zur Beherrschung
des Stillstandsbetriebes; Fraunhofer Gesellschaft für Solare Energiesysteme ISE: Freiburg, Germany, 2007.
11. Scheuren, J.; Kirchner, M.; Eisenmann, W. Reduction of Stagnation Load of Large-Scale Collector Arrays. In Proceedings of the
EuroSun, Glasgow, UK, 27–30 June 2006.
12. Scheuren, J. Untersuchungen zum Stagnationsverhalten Solarthermischer Kollektorfelder. Ph.D. Thesis, Universität Kassel,
Fachbereich Maschinenbau, Kassel University Press, Kassel, Germany, 2008.
13. Harrison, S.; Cruickshank, C.A. A review of strategies for the control of high temperature stagnation in solar collectors and
systems. Energy Procedia 2012, 30, 793–804. [CrossRef]
14. Frank, E.; Mauthner, F.; Fischer, S. Overheating Prevention and Stagnation Handling in Solar Process Heat Applications—Technical
Report A.1.2 of IEA SHC—Task 49—Solar Process Heat for Production and Advanced Applications; International Energy Agency IEA:
Paris, France, 2015.
15. Freixa, J.; Kim, T.-W.; Manera, A. Thermal-hydraulic analysis of an intermediate LOCA test at the ROSA facility including
uncertainty evaluation. Nucl. Eng. Design 2012, 249, 97–103. [CrossRef]
16. U.S.NRC. TRACE V5. 0 Theory Manual. Field Equations, Solution Methods, and Physical Models; Division of Risk Assessment and
Special Projects, Office of Nuclear Regulatory Research, US Nuclear Regulatory Commission: Washington, DC, USA, 2007.
17. Eismann, R.; Föller, F.; Witzig, A. Programm THD: Thermohydraulisches Dimensionierungsprogramm für Solaranlagen. Schlussbericht;
Bundesamt für Energie BFE: Bern, Switzerland, 2017.
18. Müller, S.; Reineke-Koch, R.; Giovannetti, F.; Hafner, B. Experimental Investigations on the Stagnation Behavior of Thermochromic
Flat Plate Collectors. In Proceedings of the EuroSun, Rapperswil, Switzerland, 10–13 September 2018.
19. CEN. EN-12975-2, Thermische Solaranlagen und ihre Bauteile—Kollektoren. In Teil 2: Prüfverfahren; Beuth Verlag GmbH: Berlin,
Germany, 2006.
20. Eismann, R. Accurate analytical modeling of flat plate solar collectors: Extended correlation for convective heat loss across the air
gap between absorber and cover plate. Solar Energy 2015, 122, 1214–1224. [CrossRef]
21. Armacell. HT/ArmaFlex—The Flexible Insulation for High Temperature Applications. Available online: https://local.armacell.
com/fileadmin/cms/uk/products/en/HTArmaFlexRangeUKROI.pdf (accessed on 18 November 2020).
22. Adelard, L.; Pignolet-Tardan, F.; Mara, T.; Lauret, P.; Garde, F.; Boyer, H. Sky temperature modelisation and applications in
building simulation. Renew. Energy 1998, 15, 418–430. [CrossRef]
23. Churchill, S.W. A comprehensive correlating equation for laminar, assisting, forced and free convection. AIChE J. 1977, 23, 10–16.
[CrossRef]
24. Churchill, S.W.; Chu, H.H. Correlating equations for laminar and turbulent free convection from a horizontal cylinder. Int. J. Heat
Mass Transf. 1975, 18, 1049–1053. [CrossRef]
25. Gnielinski, V. Berechnung mittlerer Wärme- und Stoffübergangskoeffizienten an laminar und turbulent überströmten Einzelkör-
pern mit Hilfe einer einheitlichen Gleichung. Forsch Ing. 1975, 41, 145–153. [CrossRef]
26. Tetsu, F.; Haruo, U. Laminar natural-convective heat transfer from the outer surface of a vertical cylinder. Int. J. Heat Mass Transf.
1970, 13, 607–615. [CrossRef]
27. Raithby, G.D.; Hollands, K.G.T. A General Method of Obtaining Approximate Solutions to Laminar and Turbulent Free Convection
Problems. Adv. Heat Transf. 1975, 11, 265–315. [CrossRef]
28. Dropkin, D.; Somerscales, E. Heat transfer by natural convection in liquids confined by two parallel plates which are inclined at
various angles with respect to the horizontal. J. Heat Transfer. 1965, 87, 77–82. [CrossRef]
29. Zuber, N.; Findlay, N.A. Average Volumetric Concentration in Two-Phase Systems. Trans. ASME Ser. C J. Heat Transf. 1965, 87,
453–468. [CrossRef]
30. Coddington, P.; Macian, R. A study of the performance of void fraction correlations used in the context of drift-flux two-phase
flow models. Nucl. Eng. Design 2002, 215, 199–216. [CrossRef]
31. Choi, J.; Pereyra, E.; Sarica, C.; Park, C.; Kang, J. An Efficient Drift-Flux Closure Relationship to Estimate Liquid Holdups of
Gas-Liquid Two-Phase Flow in Pipes. Energies 2012, 5, 5294–5306. [CrossRef]
Energies 2021, 14, 733 39 of 39
32. Kirkup, L.; Frenkel, R. An Introduction to Uncertainty in Measurement: Using the GUM (Guide to the Expression of Uncertainty in
Measurement); Cambridge University Press: Cambridge, UK, 2006.
33. Kovacs, P. A Guide to the Standard EN 12975; SP—Technical Research Institute of Sweden: Borås, Sweden; European Solar Thermal
Industry Federation (ESTIF): Brussels, Belgium, 2012.
34. Chexal, B.; Lellouche, G.; Horowitz, J.; Healzer, J. A void fraction correlation for generalized applications. Prog. Nucl. Energy 1992,
27, 255–295. [CrossRef]
35. Bhagwat, S.M.; Ghajar, A.J. A flow pattern independent drift flux model based void fraction correlation for a wide range of
gas–liquid two phase flow. Int. J. Multiph. Flow 2014, 59, 186–205. [CrossRef]
36. Eismann, R. Cost reduction of solar thermal plants by advanced thermalhydraulic design methods. In Proceedings of the CISBAT
2019, Lausanne, Switzerland, 4–6 September 2019.
37. Steele, W.V.; Chirico, R.D.; Knipmeyer, S.E.; Nguyen, A. Measurements of Vapor Pressure, Heat Capacity, and Density along the
Saturation Line for ε-Caprolactam, Pyrazine, 1,2-Propanediol, Triethylene Glycol, Phenyl Acetylene, and Diphenyl Acetylene.
J. Chem. Eng. Data 2002, 47, 689–699. [CrossRef]
38. D’Ans, J.; Lax, E.; Blachnik, R. Taschenbuch für Chemiker und Physiker: Band 3: Elemente, anorganische Verbindungen und Materialien,
Minerale: Bd. III; Springer: Berlin/Heidelberg, Germany, 1998.
39. TYFOROP. Technical Information: TYFOCOR LS. Available online: https://www.tyfo.de/uploads/TI/Ti_TYFOCOR-LS_gb.pdf
(accessed on 8 July 2020).
40. Chang, C.-W.; Hsiung, T.-L.; Lui, C.-P.; Tu, C.-H. Densities, surface tensions, and isobaric vapor–liquid equilibria for the mixtures
of 2-propanol, water, and 1,2-propanediol. Fluid Phase Equilibria 2015, 389, 28–40. [CrossRef]