Engineering Project Progress Report #1

Jeffrey Chang2/18/09

Proposal

• Investigate different approaches to calculating the radiative heat transfer of a solar collector for a given geometry.

• The Monte Carlo Method can be used calculate the geometric configuration factor

• Compare results to the analytical approach.

Background

• 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: Solel’s Mojave Solar Park (MSP-1) becomes operational in 2011 with 553 MW capacity.

Solar towers absorb energy reflected by mirrors

Solar Energy Generators utilize parabolic collectors to heat pipes

The Parabolic Solar Collector

•Mirrors used to reflect sunlight•Concentrates energy at a focal point•Energy heats a thermal fluid flowing through the pipe•Thermal fluid interfaces with heat exchanger to create high pressure steam•Steam drives turbine generators.

Parabolic mirror

Fluid in pipe

Solar energy

Using the Monte Carlo Method to calculate efficiency

• Assume that solar energy can be modeled as packets of energy or photons.

• Use set of random numbers to represent the number of photons reflecting off the mirror.

• When set becomes large, we are guaranteed a probability distribution.

• Track the probability of various parameters.1) Hitting vs missing the mirror.2) Absorbed vs reflected by the mirror3) Absorbed by the air/gas before hitting the mirror.4) Hitting the focal point (pipe containing thermal fluid)

First Pass at Monte Carlo Analysis(Absorbed by the air)

• Start off simple in 1-D analysis• Use Beer’s Law to calculate the fraction of

transmittance of photons through a gaseous medium

• Track distances of photons traveled.

Beer’s Law – Determine how far photons will fly

x

Photons/energy packets

•Some will be absorbed by the gaseous medium.•Use random number to determine flight distances.

S = -LN(1-Rs)/AKS = Flight distance (dimensionless)Rs = Uniform Random NumberAK= gas absorption coefficient

1 – e^(KS) = % Intensity

ResultsAbsorption coefficient 0.1

# packets 15000Distance (m/m) 1 2 3 4 5 6 7 8 9 10

# packets absorbed 1389 1212 1160 1071 984 905 786 719 677 578Calculated Absorption 9.260% 8.080% 7.733% 7.140% 6.560% 6.033% 5.240% 4.793% 4.513% 3.853%

Exact Absorption 9.516% 8.611% 7.791% 7.050% 6.379% 5.772% 5.223% 4.726% 4.276% 3.869%total Absorption 63.207%

As # of packets increase, absorption % converges to analytical solution

Next Step: Developing Code

• Develop 2-D model for analysis– Set mirror geometry (parabola)

• y=2*C*x^2 • C determines the width of the mirror

– Set target geometry (semicircle)

• x^2+(y-H)^2=R^2• H is the center of target• R is the radius of the target

H

R

Mirror

X-max

YTarget: Half-tube

X-min

Approach

X1,Y1

Target: Half-tube

X3,Y3

X1,Y1

S

Line tangent to starting point 1

Photon Flight Path

X2,Y2

•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 Beer’s Law•Q is selected using RNG•X1 is selected using RNG

L2

L1

L3

•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)

Hit or Miss?

X1,Y1

L3C1