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1
Thermal Analysis of Flat Plate Solar Collectors
Basic Flat Plate Energy Equation
When a certain amount of solar radiation falls on the surface of a collector, most of it is
absorbed and delivered to the transport fluid, and it is carried away as useful energy
As in all thermal systems, Heat Losses to the environment by various modes of heat
transfer are inevitable
Glazing
Casing
Back and side
Insulation
Absorber Plate with
Selective Coating
Heat Loss to Ambient
Reflection
Fluid Carrying
passage
Incident radiation
(visible range)
Emitted radiation by absorber(Infrared range)
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Thermal Analysis of Flat Plate Solar Collectors
Basic Flat Plate Energy Equation
Assumptions to simplify the Numerical Modeling and Calculations
Collector is in a steady state
Collector is of the header and riser type
fixed on a sheet with parallel tubes
Headers cover only a small area of thecollector and can be neglected
Heaters provide uniform flow to the riser
tubes
Flow through the back insulation is one
dimensional
Temperature gradients around tubes are neglected
Properties of materials are independent of temperature
Heat flow through the cover is one dimensional
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Thermal Analysis of Flat Plate Solar Collectors
Basic Flat Plate Energy Equation
Assumptions to simplify the Numerical Modeling and Calculations
Temperature drop through the cover is
negligible
Covers are opaque to infrared radiation
Same ambient temperature exists at thefront and back of the collector
Dust effects on the cover are negligible
There is no shading of the absorber plate
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Energy Absorbed By The Collectorper unit area of absorberS is equal to the differencebetween the incident solar radiation and the optical losses, as defined by the following Eq.
= + + +
4.2
Thermal performance of a collector can be calculated from a First-law Energy Balance.according to the first law of thermodynamics, for a simple flat-plate collector an
instantaneous steady-state energy balance is
Useful energy = Energy Absorbed Heat Loss To
Gain (Qu) By The Collector Surroundings4.1
Thermal Analysis of Flat Plate Solar Collectors
Basic Flat Plate Energy Equation
Thermal Energy Lost To Surroundings by
conduction, convection, and infrared radiation
can be represented as the product of a heattransfer coefficient UL times the differencebetween mean absorber plate temperatureTpmand the ambient TemperatureTa
QL = UL (Tpm - Ta) 4.3
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Thermal Analysis of Flat Plate Solar Collectors
Basic Flat Plate Energy Equation
In steady state the useful energy output Qu of a collector of area Ac is the differencebetween absorbed solar radiation and thermal loss: combining Eqs. (4.1, 4.2 and 4.3):
Qu = Ac [S - UL (Tpm - Ta)] 4.4
Problem with Eq. (4.4) is that the mean absorber temperatureTpm is difficult to calculate ormeasure since it is a function of the collector design, incident solar radiation, and the
entering fluid conditions. The Eq. will be reformulated so that useful energy gain can beexpressed in terms of fluid temperatureTf.
Collector efficiency is defined as the ratio of the useful gain over some specified time
period to the incident solar energy over the same time period
4.5
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Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
As in all thermal systems, heat losses to the environment by various modes of heat
transfer are inevitable
Thermal network is drawn for a
single-cover, flat-plate collector
in terms of conduction,
convection, and radiation and in
terms of the Resistance
Between Plates
Temperature of the plate is Tp,the collector back temperature is
Tb, and the absorbed solarradiation is S
Fig. 4.1
Various thermal losses from the
collector can be combined into a
simple resistance, RLso that theenergy losses from the collector
4.6
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Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
4.6
UL= Overall Heat Loss Coefficientbased on collector areaAc(W/m2-K)
Tp = Plate Temperature (C)
UL is a complicated function ofthe collector construction and
its operating conditions, given
by:
4.7
Ut = Top loss coefficient (W/m2-K)Ub= Bottom heat loss coefficient
(W/m2-K)Ue= Heat loss coefficient form the
collector edges (W/m2-K)
Fig. 4.1
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Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
Top Heat Loss/Top loss coefficient, Ut (W/m2
-K)
Fig. 4.1
Heat transfer upward from the absorber plate
at temperatureTpto the glass cover atTgandfrom the glass cover at Tg to ambient at Ta isby convection and infrared radiation
Heat loss from absorber plate to glass is givenby
Ac = collector area (m2)
hc, p-g = convection heat transfer coefficient between
the absorber plate and glass cover (W/m2-K)
hr, p-g = radiation heat transfer coefficient between
the absorber plate and glass cover (W/m2-K)
Qt,p-g
qt,p-g 4.8
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Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
Fig. 4.1
=
+
.
(.).
+
+ ( ) +
For tilt angles () up to 75, the convective
heat transfer coefficient, hc,p-g is calculatedfrom Nusselt No. correlation, which is given
by:
4.9
Plus sign represents positive values only
Rayleigh value, Ra, is
4.10
g = gravitational constant, = 9.81 m2/s= volumetric coefficient of expansion; for ideal gas,
= 1/TPr = Prandtl numberL =absorber to glass cover distance (m)= kinetic viscosity (m2/s)
Top Heat Loss/Top loss coefficient, Ut (W/m2
-K)
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Thermal Analysis of Flat Plate Solar Collectors
Fluid properties in Eq. (4.10) are evaluated at
the mean gap temperature (Tp + Tg)/2
4.10
Radiation Heat Transfer Coefficient term in Eq.
(4.8) can be linearized to give:
Fig. 4.1
4.11
Eq. 4.11 is for radiation heat exchange between two
flat and equal surfaces which is derived from the
general case of radiation heat exchange (Eq.4.12)b/w any two surfaces:
4.12
p= Infrared emissivity of absorber plateg= Infrared emissivity of glass cover
Collector Energy Losses (U-value Estimation)
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heat loss from glass cover to ambient is given
by:
Thermal Analysis of Flat Plate Solar Collectors
Fig. 4.1
4.14
Qt,g-a
4.15qt,g-a
hc,g-a= convection heat transfer coefficient
between the the glass cover and
ambient (W/m2-K)
hr, g-a = radiation heat transfer coefficient
between the glass cover and ambient
(W/m2-K)
Collector Energy Losses (U-value Estimation)
Top Heat Loss/Top loss coefficient, Ut (W/m2
-K)
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hc,g-a = hw= Convective heat transfer coefficientfor wind blowing over the
collector.(W/m2.oC) = 5.7+3.8Vair
Thermal Analysis of Flat Plate Solar Collectors
Fig. 4.1
Collector Energy Losses (U-value Estimation)
Vair = Average wind speed (m/sec)
4.17
4.16
Top Heat Loss/Top loss coefficient, Ut (W/m2
-K)
aggp
t
RRU
11
4.18
4.19
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In some cases, collectors are constructed
with two glass covers in an attempt to lower
heat losses
Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
Top Heat Loss/Top loss coefficient, Ut (W/m2
-K)
Another resistance Rg1-g2 will be added to thesystem shown to account for the heat transfer
from the lower to upper glass covers
By following a similar analysis, the heat
transfer from the lower glass at Tg2 to theupper glass atTg1is given by
Qt,g1-g2qt, g1-g2 4.20
Fig. 4.1
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Thermal Analysis of Flat Plate Solar Collectors
Fig. 4.1
Collector Energy Losses (U-value Estimation)
Top Heat Loss/Top loss coefficient, Ut (W/m2
-K)
The convection heat transfer coefficient hc,g2-g1can be obtained by Eqs. 4.9 and 4.10
Radiation heat transfer coefficient can be
obtained again from Eq. (4.11) and is given
by
4.21
4.22
Procedure of solving for the Ut using Eqs.(4.8) through (4.20) is an iterative process
A guess is made for the unknown cover
temperature Tg
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Thermal Analysis of Flat Plate Solar Collectors
Fig. 4.1
Collector Energy Losses (U-value Estimation)
Top Heat Loss/Top loss coefficient, Ut (W/m2
-K)A guess is made for the unknown cover
temperature Tg, from which the convectiveand radiative heat transfer coefficients b/w
parallel surfaces are calculated
Since the energy exchange between pates
must be equal to the overall heat loss, a new
set of cover temperatures can be calculated:
jirhjich
aTpTtU
iT
jT
,,
)(4.23
i and j refer to two adjacent flat
surfaces. E.g. absorber plate and glass
cover or glass cover-1 and 2. Iterative Process is repeated until the cover
temperatures do not change significantly
b/w successive iterations
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Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
Top Heat Loss/Top loss coefficient, Ut (W/m2
-K)Example 4.1:
Calculate the top loss coefficient for an absorber with a single glass cover having
following specifications:
o Plate to cover spacing: 25 mm
o Plate Emittance: 0.95
o Ambient Air Temperature = 10 oC
o Wind Speed = 3 m/sec
o Collector tilt = 45o
o Glass Emittance = 0.88
= + . (.).
+
+ ( ) +
jirhjich
aTpTtU
iT
jT
,,
)(
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Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
Top Heat Loss/Top loss coefficient, Ut (W/m2
-K)
As the iterations required are tedious and time consuming, especially for the case of
multiple-cover systems, straightforward evaluation of Ut is given by the followingempirical equation developed by Klein (1975) with sufficient accuracy for design purposes
4.23
Where;
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Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
Top Heat Loss/Top loss coefficient, Ut (W/m2
-K)Example 4.2:
Repeat Example 4.1 using the empirical Eq. 4.23 and compare the results.
Example 4.3:
Estimate the top heat loss coefficient of a collector that has the following specifications:
Collector slope = 35, Number of glass covers = 2,
Thickness of each glass cover = 4 mm, Thickness of absorbing plate = 0.5 mm, Space
between glass covers = 20 mm, Space between inner glass cover and absorber = 40 mm
Thickness of back insulation = 50 mm,
Back insulation thermal conductivity = 0.05 W/m-K.
Mean absorber temperature,Tp = 80C = 353 K, Ambient air temperature = 15C = 288 K.Absorber plate emissivity, p = 0.10, Glass emissivity, g = 0.88.Wind velocity = 2.5 m/s.
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Fig. 4.1
Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
Bottom Heat Loss Coefficient, Ut (W/m2
-K)
Energy loss from the bottom of the collector
is first conducted through the insulation and
then by a combined convection and infrared
radiation transfer to the surrounding
ambient air
1
Magnitudes ofRp-b and Rb-a are such that it isusually possible to assume Rb-a is zero andall resistance to heat flow is due to the
insulationBack loss coefficient is:
tb = thickness of back insulation (m)kb = conductivity of back insulation (W/m-K)
heat loss from the back of the plate rarely
exceeds 10% of the upward loss
Typical values of the back surface heat loss
coefficient are 0.30.6 W/m2-K
4.24
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Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
Edge Heat Loss Coefficient, Ue (W/m2
-K)Heat transfer coefficient for the heat loss from the collector edges can be obtained from
4.25
Where,te= thickness of edge insulation (m)ke= conductivity of edge insulation (W/m-K)hc,e-a= convection heat loss coefficient from edge to ambient (W/m
2-K)Typical values of the edge heat loss coefficient are 1.52.0 (W/m2-K)
Evaluation of edge losses is complicated, therefore in a well-designed system, the edge
loss should be small so that it is not necessary to predict it with great accuracy
Losses through the edge should be referenced to the collector area, If the edge loss
coefficient area product is (UA)edge then edge loss coefficient
c
edge
eA
UAU
)( 4.25A
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Thermal Analysis of Flat Plate Solar Collectors
Collector Energy Losses (U-value Estimation)
Overall Heat Loss Coefficient, Ue (W/m2
-K)Collectors overall heat loss coefficient is:
U= Ut +Ub + Ue 4.25B
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Fig. 4.3
Thermal Analysis of Flat Plate Solar Collectors
Temperature Distribution in Flat-Plate
Collectors and Collector Efficiency Factor
Energy transferred to fluid will heat the fluid, causing a temperature gradient to exist in the
direction of flow
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Temperature difference b/w tubes can be derived if we assume the temperature gradient in
the flow direction is negligible
Thermal Analysis of Flat Plate Solar Collectors
Temperature Distribution in Flat-Plate
Collectors and Collector Efficiency Factor
Analysis can be performed by considering the sheet-tube configuration, where the
distance between the tubes is W, the tube diameter is D, and the sheet thickness is tab
Region between the center line separating the tubes and the tube base can be considered
as a classical Fin Problem
An elemental region of width, dx, and length L in the flow direction are shown
Length of the fin is (W-D)/2
Fig. 4.4
dx
tabtab
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x
(W-D)/2
Tb
tab
qx qx+dx
dqc
dxFig. 4.5
Thermal Analysis of Flat Plate Solar Collectors
Temperature Distribution in Flat-Plate
Collectors and Collector Efficiency Factor
4.26
By solving energy balance on this element
Eq. (4.26) gives the
temperature distribution in thex direction at any given y
Energy conducted to the region
of tube per unit length in the flow
direction is:
4.27
or with the help ofFin Efficiency, F
4.28
F = Standard Fin Efficiency for straight fins with arectangular profile
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Thermal Analysis of Flat Plate Solar Collectors
Temperature Distribution in Flat-Plate
Collectors and Collector Efficiency Factor
tab
Useful gain of the collector
also includes the energy
collected above the tube
region:
This energy ultimately must be transferred to the fluid, which can be expressed in terms of
two resistances as
Cb
is the bond conductance, which can be estimated from
knowledge of the bond thermal conductivity, kb, theaverage bond thickness, , and the bond width, b
useful energy gain per unit length
in the direction of the fluid flow is:
4.29
4.30
4.31
hfi= heat transfer coefficient between the fluid and the tube wall
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Thermal Analysis of Flat Plate Solar Collectors
Temperature Distribution in Flat-Plate
Collectors and Collector Efficiency Factor
oF represents the ratio of the actual useful energy gain to the useful energy gain that
would result if the collector absorbing surface had been at the local fluid temperatureoFrepresents the effect of the temperature drop between the absorber plateTp and the
fluid in the pipeTf
oThe numerator of Eq. (4.32) is the heat transfer resistance from absorber plate to
ambient and denominator is the heat transfer resistance from the fluid to the ambient
air.
Solving Eq. (4.31) forTb, substituting it into Eq. (4.30), and solving the resultant equationfor the useful Heat Gain of the Collectorin terms of local fluid temperature is:
4.33
where Fis the Collector Efficiency Factor, given by
4.32
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oFconsiders the heat transfer resistance from absorber to fluid in the pipe
due to fin conduction,
due to the conduction through the contact bond between absorber and pipe, and
due to the forced convection between the pipe inner wall and the flowing fluid
Thermal Analysis of Flat Plate Solar Collectors
Temperature Distribution in Flat-Plate
Collectors and Collector Efficiency Factor
oFdecreases with increased tube center-to-center distances and increases with increasein both material thicknesses and thermal conductivity
4.34
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Thermal Analysis of Flat Plate Solar Collectors
Temperature Distribution in Flat-Plate
Collectors and Collector Efficiency Factor
Example 4.4:
For a collector having the following characteristics and ignoring the bond resistance,
calculate the fin efficiency and the collector efficiency factor:
Overall loss coefficient = 6.9 W/m
2
-CTube spacing = 120 mm
Tube outside diameter = 15 mm
Tube inside diameter = 13.5 mm
Plate thickness = 0.4 mm
Plate material = copper
Heat transfer coefficient inside the tubes = 320 W/m2-C
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Thermal Analysis of Flat Plate Solar Collectors
Temperature Distribution in Flow Direction
Consider an infinitesimal length y of the tube as shown in Figure. Useful energydelivered to the fluid is quy
Fluid enters the collector at temperatureTfi and increases in temperature until at the exit itisTfo
By solving energy balance on this element
Fig. 4.6
4.33B
p
L
Lafi
Laf
CmynWFU
USTTUSTT
.
'
exp// 4.33A
h l l f l l l ll
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Thermal Analysis of Flat Plate Solar Collectors
Collector Heat Removal Factor, Flow Factor , and Thermal
Efficiency
Heat Removal Factor FR represents the ratio of the actual useful energy gain that wouldresult if the collector-absorbing surface had been at the inlet fluid temperature,Tfi
It is usually desirable to express the collector total useful energy gain Qu in terms of thefluid inlet temperature,Tfi
4.34
4.34
Rearranging yields:
4.35
h l l i f l l S l C ll
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Thermal Analysis of Flat Plate Solar Collectors
Collector Heat Removal Factor, Flow Factor, and Thermal
Efficiency
Another parameter usually used in the analysis of collectors is the Flow FactorF,whichis defined as the ratio ofFR to F
4.36
FR is equivalent to the Effectiveness of a Conventional Heat Exchangerwhich is defined asthe ratio of the actual heat transfer to the maximum possible heat transfer
Maximum possible useful energy gain in a solar collector occurs when the whole collector
is at the inlet fluid temperature,Tfi; heat losses to the ambient are then at a minimum
4.35
collector flow factor F is a function of only a single variable, the dimensionlesscollector capacitance rate, mcp /AcULF
Th l A l i f Fl Pl S l C ll
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FR times this maximum possible useful energy is the actual useful energy:
Qu = Ac FR[S - UL (Tfi - Ta)]
oThis is same as Eq. (4.4), with the difference that the inlet fluid temperature (Tfi)replaces the average plate temperature (Tp) with the use of the FR
4.37
oEq. (4.37), is a convenient representation when analyzing the solar energy systems,
since the inlet fluid temperature (Tfi) is usually known
Qu = Ac [S - UL (Tpm - Ta)] 4.4
Thermal Analysis of Flat Plate Solar Collectors
Collector Heat Removal Factor, Flow Factor, and Thermal
Efficiency
Th l A l i f Fl t Pl t S l C ll t
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Calculate the useful gain and efficiency of an
array of 10 solar collector modules installed
in parallel, at a slope of 60o and a surface
azimuth of 0o. The hourly radiation on the
plane of the plane of the collector IT, thehourly radiation absorbed by the absorber
plate S, and the hourly ambient temperatureTa, are given in the table. For the collectorassume UL to be 8.0 W/m
2 oC, and the plate
efficiency factorF to be 0.841. The water flow
rate through each 1 2-m collector panel is0.03 kg/s and the inlet water temperature
remains constant at 40 oC. Assume a
controller turns off the water flow whenever
the outlet temperature is less than the inlet
temperature.
Thermal Analysis of Flat Plate Solar Collectors
Collector Heat Removal Factor, Flow Factor, and Thermal Efficiency
Example 4.5:
Qu = Ac FR[S - UL (Tfi - Ta)]
Th l A l i f Fl t Pl t S l C ll t
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Thermal Analysis of Flat Plate Solar Collectors
Collector Heat Removal Factor, Flow Factor, and Thermal Efficiency
Example 4.5:--contd--
Th l A l i f Fl t Pl t S l C ll t
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Thermal Analysis of Flat Plate Solar Collectors
Collector Heat Removal Factor, Flow Factor, and Thermal Efficiency
To evaluate Collector performance, It is necessary to know ULand hfi
Mean Fluid and Plate Temperature
Both UL and hfi are to some degree functions of temperatures
Mean Fluid Temperature can be found by integrating Eq. 4.33A from zero to L
4.38
Solving this integration and substituting FR from Eq. (4.35),Tfm is:
4.39
Solving Eqs. (4.37) and (4.4), for the mean Plate Temp. (Tpm):
Qu = Ac FR[S - UL (Tfi - Ta)] Qu = Ac [S - UL (Tpm - Ta)]
4.40
Th l A l i f Fl t Pl t S l C ll t
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Thermal Analysis of Flat Plate Solar Collectors
Liquid Heater Plate Geometries
Here only one basic collector design Sheet and tube solar water heater with parallel tubes
on the back of the plate is analyzed
There are many designs ofFlat-plate Collectors
Fortunately, it is not necessary to develop a completely new analysis for each situation
Generalized relationships for the tube and sheet case apply to most collector designs
It is necessary to derive the appropriate form of the Collector Efficiency FactorFand Eqs.(4.35 - 4.37) then can be used to predict the thermal performance
Fig. 4.7a
Under some circumstances, ULwillhave to be modified slightly
Thermal Analysis of Flat Plate Solar Collectors
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Thermal Analysis of Flat Plate Solar Collectors
Liquid Heater Plate Geometries
Fig. 4.7b Fig. 4.7c
Fig. 4.7d
Fig. 4.7e
In Figs. 4.7d, e, narrow, flat absorbers
are mounted inside evacuated glass
tubes
Configuration of Fig. 4.7d is similar to
type a but with a single riser Type e collecor is down and back
with a U-tube joining the two conduits
Thermal Analysis of Flat Plate Solar Collectors
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D t t f M h i l E i i 39
Thermal Analysis of Flat Plate Solar Collectors
Collector Heat Removal Factor, Flow Factor , and Thermal
Efficiency
Radiation level where the absorbed solar radiation and loss term are equal
Critical Radiation Level
This is obtained by setting the term in the right-hand side of Eq. (4.37) equal to 0
Qu = Ac FR[GT() - UL (Tfi - Ta)] 4.37
Critical Radiation Level, Gtc, is given by
4.38
Collector can provide useful output only when the
available radiation Gavis higher than the criticalone Gtc
Qu = Ac FR ()av(GT GTc)+ 4.39
Collector output can be written in terms of the critical radiation level:
o Eq. 4.39 indicate that for the collector to produce useful out put, i.e, Qu> 0, absorbedradiation must exceed the thermal losses and GT must be greater than GTC
oThis implies that there is a Controlleron the collector that shutts off the flow of fluid
when the value in parentheses is not positive