Title: Monte Carlo Method applied to Radiation Heat Transfer of a Solar Parabolic Collector
1Monte Carlo Method applied to Radiation Heat
Transfer of a Solar Parabolic Collector
2Proposal and Goals
- Investigate an alternate approach to
approximating radiative heat transfer of a solar
collector for a given geometry. - The Monte Carlo Method can be used calculate the
geometric configuration factor - Validate results to the analytical approach.
- Attempt optimization via parametric study
- MC numerical method written in FORTRAN code.
3Background
- Parabolic Solar collectors have been used for
over 30 years - Practice varies from domestic use to large scale
power generation in the Southwestern states. - Example Solels Mojave Solar Park (MSP-1)
becomes operational in 2011 with 553 MW capacity.
4Solar One -towers absorb energy reflected by
Heliostat mirrors
Solar Energy Generating plants utilize parabolic
collectors to heat pipes
5The Parabolic Solar Collector
- Mirrors used to reflect sunlight
- Concentrates energy
- Energy heats a thermal fluid flowing through the
pipe - Thermal fluid interfaces with heat exchanger to
create high pressure steam - Steam drives turbine generators.
Fluid in pipe
Solar energy
Parabolic mirror
6Solar collectors substitute for nuclear process
for generating electricity
7Using the Monte Carlo Method
- Requires physical governing equations of physical
phenomena. - Applies probability distribution combining
behavior of individual players. - Use a pseudo-random number generator (PRNG) to
produce uniform distribution of inputs for any
single parameter. - Use the Inverse Transform method to transform
random numbers into parameter inputs for
analysis.
8Applying Monte Carlo Method to calculate
efficiency
- Assume that solar energy can be modeled as
packets of energy or photons. - Use ITM to transform pseudo-random numbers to
represent the photon emission locations
reflecting off the mirror. - Track the probability of various parameters.
- Hitting or missing the target
- Absorbed by gas before hitting the target.
-
9Validating MC for example of two perpendicular
infinitely long plates
Schematic of two infinitely long plates of
unequal lengths
Target
Emission surface
Analytical equation
(where Hh/w)
10Validating MC for example of photon absorption
and attenuation
where I0 is initial photon intensity I is
attenuated intensity of photon S is the finite
flight distance of the photon K is the gas
absorption coefficient
- Use Beers Law to calculate the fraction of
transmittance of photons through a gaseous medium - Use ITM to convert random number generated to a
usable parameter (attenuated photon flight
distance S). - Track distances of photons traveled.
11MC approximation is close to analytical results
Study to calculate photon absorption at 1-ft.
intervals
exact solution for photon traveling at interval
photons N15,000
12Next Step Validating MC for Semicircular Geometry
Schematic of infinitely long concentric
semicylinder and cylinder
Analytical equations
- Set target geometry (semicircle)
- x2(y-H)2R22
- H is the center of target.
- R1 is the radius of the target.
- R2 is radius of the collector.
- P is angle of tangent line wrt to x-axis.
Use ITM to convert random numbers generated for
following parameters - q (photon flight
angle) - X1 (photon emission location)
13MC approximation for configuration factor is
close to analytical results
MC approximation must be modified to EFFECTIVE
EFFICIENCY for blockage factor
Max efficiency is only about 25
14Next Step Applying MC for Parabolic Geometry
Configuration factor 5 more efficient than
semicircular geometry
Collector
- Develop 2-D model for analysis
- Set mirror geometry (parabola)
- y2Cx2
- C determines the width of the mirror
15Results show that both C1 and C2 are acceptable
shapes for effective efficiency
16Attenuated efficiencies reduced by 10 (C1 and
C2 still acceptable cases)
17Summary
- MC method successfully validates exact solutions
for 2 examples. Very useful for approximating
solutions for simple models. - Parametric optimization analysis concludes that
narrower parabolas are more efficient
(non-attenuated and attenuated) - More detailed models can be examined (i.e.,
additional parabolic geometries, secondary
reflections, focal point validation)
18Y
Target Half-tube
Semi circular Collector
19 7 The reciprocity rule requires that
8 Solving for F2-1
9 The areas of the target and collector are
10a,b Substituting 6 and 9a,b into 8
results in
11
12
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21Approach
X3,Y3
Target Half-tube
Photon Flight Path
L2
S
L3
q
X2,Y2
L1
X1,Y1
X1,Y1
- Point 1 (X1,Y1) Starting point of photon
(emitting point). - Point 2 (X2,Y2) Projected point of photon onto
tangent line - Point 3 (X3,Y3) End point of photon.
- S calculated using Beers Law
- Q is selected using RNG
- X1 is selected using RNG
Line tangent to starting point 1
22Hit or Miss?
C1
L3
X1,Y1
- Conditions for Hitting the Target
- If point 3 (X3,Y3) remains on the edge or inside
the target. - If line equation L3 intercepts semicircle
equation C1 - And if point 3 lies above the mirror
- And if point 3 is in left quadrant of the mirror
(given point is on the right side)
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