1.1. Description of an STSC
An STSC is the result of the interception of the open circular area
with a small side section of a parabola, and this directs the solar radiation
to a fixed focal point where a Stirling engine is installed and secured
by a fixed support structure. It used a 2-axis tracking system,
which consists of a daily tracking mechanism that moves the reflector
mounted on a carriage in proportion to solar and other timetracking
mechanisms, which provides for rotation of the reflector
that is synchronised with the movement of the sun during the
day. This angular movement of the reflector is made around an axis
oriented to maintain a fixed focal point normal to the incidence of
the aperture opening area of the reflector, thus concentrating the solar
radiation in the cavity that receives and heats a gas (helium,
hydrogen or air) at high temperatures [23]. Later, the Stirling engine
converts the heat into electricity, as observed in Fig. 1.
2. Materials and methods
2.1. Mathematical model
In this section, the mathematical model for attaching an STSC to
a Stirling engine is developed using the mathematical estimation of
Nomenclature
A area (m2
)
a normal distribution segment (–)
b constant approximation to normal distribution (–)
Cgeo geometric concentration (–)
d diameter (m)
f focal length (m)
Gr Grashof number (–)
h convective heat transfer coefficient (W/m2 K)
I direct solar irradiance (W/m2
)
K fluid Thermal Conductivity (W/(m K))
L thickness length (m)
p distance from concentrator surface to focal point (m)
Pr Prandtl number (–)
n number of segment reflectors (–)
Nu Nusselt number (–)
Q energy flux density (W/m2
)
r radius (m)
Ra Rayleigh number (–)
Re Reynolds number (–)
Sp spacing (m)
S separation (m)
T temperature (K)
t normal distribution variable (–)
w width of the focal image (m)
Greek symbols
aeff effective absorbance of the cavity
e subtended angle of the sun
e
⁄ emissivity
r standard deviation (mrad)
r⁄ Stefan–Boltzmann constant
g efficiency (–)
q surface reflectance (–)
u intercept factor (–)
h inclination angle of the cavity ()
t wind speed (m/s)
w rim angle ()
Subscripts
abs absorber
amb ambient
ap aperture
ins insulator
cond conduction
conv convection
cav cavity
for forced
Eff effective
ext outside
int inside
rad radiation
ref reflector
rec receiver
nat natural
tub pipe
254 J. Ruelas et al. / Applied Energy 101 (2013) 253–260
the intercept factor for an STSC, which considers the optical and
geometric models and incorporates the thermal model of a cavity
receiver in accordance with the following considerations: the distribution
of the solar image at the focal point corresponds to a normal
distribution; the temperature inside the receiving cavity is
evenly distributed; the heat transfer analysis is performed under
stable conditions and is one-dimensional; the material properties
remain constant; and the mathematical model begins by using
Eq. (1), described by Duffie and Beckman [16], which estimates
the amount of energy captured by a cavity receiver.
Qrec ¼ IdAap;refqu ð1Þ
In Eq. (1), all of the terms are known, except for the intercept factor
of the STSC, which is necessary to consider the geometric model of
the STSC. Eq. (2) corresponds to the estimation of the intercept factor,
and the total energy incident on the reflector (denominator) is
easily established by substituting the value of the direct radiation
and the aperture opening area. However, to determine the amount
of energy intercepted (numerator), it is necessary to develop the
1.1. Description of an STSCAn STSC is the result of the interception of the open circular areawith a small side section of a parabola, and this directs the solar radiationto a fixed focal point where a Stirling engine is installed and securedby a fixed support structure. It used a 2-axis tracking system,which consists of a daily tracking mechanism that moves the reflectormounted on a carriage in proportion to solar and other timetrackingmechanisms, which provides for rotation of the reflectorthat is synchronised with the movement of the sun during theday. This angular movement of the reflector is made around an axisoriented to maintain a fixed focal point normal to the incidence ofthe aperture opening area of the reflector, thus concentrating the solarradiation in the cavity that receives and heats a gas (helium,hydrogen or air) at high temperatures [23]. Later, the Stirling engineconverts the heat into electricity, as observed in Fig. 1.2. Materials and methods2.1. Mathematical modelIn this section, the mathematical model for attaching an STSC toa Stirling engine is developed using the mathematical estimation ofNomenclatureA area (m2)a normal distribution segment (–)b constant approximation to normal distribution (–)Cgeo geometric concentration (–)d diameter (m)f focal length (m)Gr Grashof number (–)h convective heat transfer coefficient (W/m2 K)I direct solar irradiance (W/m2)K fluid Thermal Conductivity (W/(m K))L thickness length (m)p distance from concentrator surface to focal point (m)Pr Prandtl number (–)n number of segment reflectors (–)Nu Nusselt number (–)Q energy flux density (W/m2)r radius (m)Ra Rayleigh number (–)Re Reynolds number (–)Sp spacing (m)S separation (m)T temperature (K)t normal distribution variable (–)w width of the focal image (m)Greek symbolsaeff effective absorbance of the cavitye subtended angle of the sune⁄ emissivityr standard deviation (mrad)r⁄ Stefan–Boltzmann constantg efficiency (–)q surface reflectance (–)u intercept factor (–)h inclination angle of the cavity ()t wind speed (m/s)w rim angle ()Subscriptsabs absorberamb ambientap apertureins insulatorcond conductionconv convectioncav cavityfor forcedEff effectiveext outsideint insiderad radiationref reflectorrec receivernat naturaltub pipe254 J. Ruelas et al. / Applied Energy 101 (2013) 253–260the intercept factor for an STSC, which considers the optical andgeometric models and incorporates the thermal model of a cavityreceiver in accordance with the following considerations: the distributionof the solar image at the focal point corresponds to a normaldistribution; the temperature inside the receiving cavity isevenly distributed; the heat transfer analysis is performed understable conditions and is one-dimensional; the material propertiesremain constant; and the mathematical model begins by usingEq. (1), described by Duffie and Beckman [16], which estimatesthe amount of energy captured by a cavity receiver.Qrec ¼ IdAap;refqu ð1ÞIn Eq. (1), all of the terms are known, except for the intercept factorof the STSC, which is necessary to consider the geometric model ofthe STSC. Eq. (2) corresponds to the estimation of the intercept factor,and the total energy incident on the reflector (denominator) iseasily established by substituting the value of the direct radiationand the aperture opening area. However, to determine the amountof energy intercepted (numerator), it is necessary to develop the
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