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Modeling and Simulation of Solar Dish-Stirling Systems

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Modeling and Simulation of Solar Dish-Stirling Systems
Li Mingzhen
School of Jet Propulsion Beijing University of Aeronautics and Astronautics Beijing 100191, China lmzbuaa@foxmail.com
Dong Jinzhong
School of Jet Propulsion Beijing University of Aeronautics and Astronautics Beijing 100191, China dd309@buaa.edu.cn
Abstract
—A Solar Dish-Stirling System Model is developed for design, optimization, control, and systems development of Dish-Stirling systems. It models the energy transfer in parabolic concentrator and receiver, cycle thermodynamics, heat flow, mechanical dynamics in Stirling engine and the Starter/Generator. The model’s scope extends from the solar energy to thermal, mechanical, and electrical energy out, helping to study complex system interactions among subsystems. It is a non-linear time-domain model, which can be used to simulate transient and dynamic phenomena. And the entire range of system operation can be modeled, from start-up to full power conditions. The model details and simulation results are discussed in this paper.
Keywords-
solar Dish-Stirling system; modeling; simulation
I.
I
NTRODUCTION
Solar energy is one of the most attractive renewable energy sources that can be used as an input energy source for heat engine. Dish-Stirling systems convert the thermal energy in solar radiation to mechanical energy and then to electrical energy. Of all solar technologies, Dish-Stirling systems have demonstrated the highest solar-to-electric conversion efficiency (29.4%) [1], and therefore have the potential to become one of the least expensive sources of renewable energy [2].
Figure 1. Schematic diagram of a Dish-Stirling system
As shown in fig. 1, Dish-Stirling systems produce electricity using concentrated solar thermal energy to drive a Stirling engine. The system utilizes a parabolic mirror to concentrate solar radiation onto a thermal receiver integrated in the Stirling engine. The receiver consists of a heat exchanger designed to transfer the absorbed solar energy to the working fluid. The Stirling engine then converts the absorbed thermal energy to mechanical power by expanding the gas in a piston-cylinder. The linear motion is converted to a rotary motion to turn a generator to produce electricity. One of the key tasks in the research and development work for Dish-Stirling systems is to create computational tools which enable engineers to more accurately mathematically simulate the system’s working process and estimate the system’s steady and dynamic performance. And also computer modeling and analysis of system is crucial to optimization system design and developing the control strategy of the systems. During the last several decades, various models have been developed and used for the simulation of the working process of Stirling engine [3-8]. But most of them based on linear theory are not possible to model transient behavior. Recently, some models have added nonlinearities in the Stirling systems [9][10]. The scope of these models is generally limited to the Stirling engine and not related to the remaining system such as the generator and controller as well as the solar capture in Dish-Stirling systems. This paper will describe in detail the Solar Dish-Stirling system model developed by thermodynamics and dynamics theory. And some simulation results for system’s steady and dynamic working processes are obtained by this model. II.
T
HE
M
ODEL
The Solar Dish-Stirling system model includes the energy transfer in Parabolic concentrator and receiver, cycle thermodynamics, heat flow, mechanical dynamics in Stirling engine and the Starter/Generator. The model’s scope extends from the solar energy to thermal, mechanical, and electrical energy out, helping one to study complex system interactions among subsystems (Fig. 2). Thermal, fluid, mechanical, and electrical aspects can be studied in one model. The model is a non-linear time-domain model, allowing it to simulate transient and dynamic phenomena. The entire range of system operation can be modeled, from start-up to full power conditions.
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Figure 2. Scope of the Dish-Stirling system model
A.
Parabolic Concentrator
The concentrator for the Solar Dish-Stirling systems uses parabolic mirrors mounted on a structure that tracks the sun by pivoting on two axes. The parabolic concentrator reflects direct normal solar radiation into the aperture of the receiver where it is concentrated on the absorber. Fig. 3 is a schematic diagram of dish concentrator.
Figure 3. Schematic diagram of dish concentrator
As shown in fig. 3, the diameter D
coll
and the focal length of the concentrator f are two important structural parameters. They have the relations as follows:
( )
2tan41Df
rimcoll
ϕ=
(1)
θ+=
cos1f 2R
P
(2) Where
φ
rim
is the rim angle, R
p
is the distance from the
concentrator to focal point, θ
is the polar axis angle. There are several imperfections in the concentrator system that contributes to the spread of the beam. Leonard D. Jaffe [11][12] gives an empirical model to estimate the concentrator imperfections. The energy losses can be described by intercept efficiency
φ
is given:
σ−=φ
2f
C21exp1
(3)
Where, C is the concentration ratio, C=A
coll
/A
rec
, A
coll
is the projected area of concentrator, A
rec
is the aperture area of receiver.
The heat flux distribution error in focal plane σ
f
:
rimrimrimrim222f
sincos3cos21
ϕϕϕϕ+δ=σ
(4) The total error in a Dish-Stirling concentrator
δ
:
2sun2w2spec2slp2
4
σ+σ+σ+σ=δ
(5) Where,
σ
slp
is the slope error,
σ
spec
is the specular reflectance error,
σ
w
is the tracking error,
σ
sun
is the sun’s width error.
B.
Receiver
The Dish-Stirling receiver absorbs thermal energy from the parabolic concentrator and transfers the absorbed thermal energy to the working fluid in the Stirling engine. The receiver is responsible for the majority of the thermal losses that occur before the energy is converted into electricity in the Stirling engine [13]. Mechanisms that contribute to the total receiver thermal loss include conduction through the receiver housing, convection from the cavity, and radiation through the aperture opening to the ambient environment. A receiver energy balance with the loss mechanisms can be viewed in fig. 4.
Figure 4. Receiver energy balance for a Dish-Stirling system
1)
Radiation Losses:
The radiation losses in the receiver contribute to a significant fraction of the total losses in the receiver. There are two ways solar radiation contributes to losses from the receiver. The first results from thermal radiation being emitted from the aperture due to the large temperature difference between the cavity walls and the parabolic mirror. The second results from solar radiation being reflected off from the cavity walls and back through the aperture.
a)
Radiation due to emission:
The general equation for net radiation exchange due to emission is given by:
)TT(Aq
4amb4recrecemirad
−⋅⋅ε⋅σ=
(6) Where,
ε
emi
is the effective emissivity of the cavity aperture,
σ is S
tefan Boltzmann’s constant, T
rec
is the cavity interior temperature, and T
amb
is the ambient temperature.
b)
Radiation due to emission:
To determine the radiation losses due to reflection off of the cavity surfaces, the effective absorptance of a cavity rece
iver α
abs
is required.
2)
Convection Losses
The convective losses in the receiver represent a significant fraction of the total losses in a Dish-Stirling system [13]. A model for predicting convection losses was developed by Stine and McDonald [14] to predict convective losses in receivers. This correlation is given in (7).
( ) ( ) ( )
scavrec47.218.0
ambwall31
d d cosTTGr 088.0 Nu
ϕ=
(7)
)d d (982.012.1s
cavrec
⋅−=
(8) Where, Gr is the Grashof number based on the average internal width of the cavity length d
cav
, Nu is the Nusselt number based on the length d
cav
T
wall
is the average internal wall temperature, and
φ
is the tilt angle of the cavity. The convective losses is given by (9):
)TT(Ahq
ambrecreccon
−⋅⋅=
(9)
3)
Conduction Losses
Conduction losses are minimal in the receiver since they can be easily controlled by adding insulation without appreciable losses in other components in the Stirling dish system. So, it’s neglected in this paper.
4)
Total Power Intercepted by Receiver
The total power intercepted by the aperture in the receiver can be approximated expressed using (10).
( )
( )
[ ]
ambrec4amb4recemirecabsref collrad tot
TThTTA
EGAIq
−⋅+−⋅σ⋅ε⋅−
α⋅φ⋅⋅⋅ρ⋅⋅=
(10) Where, I
rad
is the direct normal insolation,
ρ
ref
is the mirror specular reflectance, G is the cover rate of the concentrator, E is the transmittance.
C.
Stirling Engine Cycle Thermodynamics
The Stirling cycle third-order analysis which is a nonlinear model uses control volumes to directly solve one-dimensional governing equations. The model represents the working space that consists of the expansion space, the heater, the regenerator, the cooler and the compression space by a series of control volumes [5]. The control volume model is shown in fig. 5.
Figure 5. Control volume model
In the third-order model the heat transfer and gas dynamical processes occurs in the internal gas circuit of the engine are described by the set of partial differential equations of mass, momentum, energy conservation and gas state equation. Mass equation:
oi
mmdtdm
′−′=
(11) Momentum equation:
( )
( ) ( )
0xPuD2f uxut
2h2
=∂∂+ρ+ρ
∂∂+ρ∂∂
(12) Energy equation:
dt)mT(d
CdtdV pmTCmTC
dtdQ
voo pii p
+=′−′+
(13) Gas state equation: PV=mRT (14) Several assumptions are inherent in the use of these equations: (1) Flow is one dimensional. (2) Kinetic energy can
be neglected in the energy equation. (3) The pressure is uniform throughout the working space at a given time in applying these equations. (4) The time derivative term in the momentum equation is neglected. By solving above differential equations with the piston moving equation, the heat transfer and friction coefficient correlations and the heat boundary conditions, the thermodynamic cycle parameters in control volumes can be obtained [6]. The indicated work including pressure-drop losses is calculated according to:
( )
∫
+=
cceei
dVPdVPW
(15)
Where, V
e
, V
c
and P
e
, P
c
are the expansion-space and compression-space volume and pressure. The indicated power is just the indicated work per cycle times the engine frequency. The conduction and shuttle transfer losses increase the heat input and heat output by the same amount. Since it is assumed that they do not interact with the working space, they affect the efficiency but not the power calculations. The net heat into the engine per cycle is defined to be the basic heat input plus conduction and shuttle losses minus one-half the total pressure-drop loss.
2WQQQQ
f shconih
−++=
(16) Where, Q
h
is the net heat input, Q
i
is the basic heat input, Q
con
is the conduction loss, Q
sh
is the shuttle loss, W
f
is the pressure-drop loss. The detail analysis methods of these losses were developed by Martini [15]. So, engine efficiency is:
hii
QW
=η
(17) In Dish-Stirling systems, engine absorbed heat is the same as receiver intercepted heat. Under certain direct normal insolation and engine mean pressure, the energy balance can be established through the hot end temperature of the engine, which linked the Dish concentrator-receiver model and the Stirling engine thermodynamics model together.
D.
Stirling Engine Crank-Piston Mechanism Dynamics
Assuming the engine crankshaft and connecting rod are rigid, and the deformation does not occur, the following second-order differential equations of motion could be established.
lvslf iP
s
IIITTTT
++−−−=θ
(18) Where,
θ
s
is the crank angle, T
p
is indicated torque, T
i
is the inertia torque, T
f
is the friction torque, T
l
is the load torque, I
v
is the variable inertia, I
s
is the engine inertia, I
l
is the load inertia.
1)
Inticated Torque
Figure 6. Four-cylinder double-acting engine arrangement
Figure 6 shows a four-cylinder double-acting Stirling engine arrangement. The gas pressure in each cylinder applies a force, PA
p
, to the piston which is converted to a torque, T
p
, on the crankshaft by the crank-piston mechanism. Considering four cylinders each with a double-acting piston 90 degrees out of phase with the next cylinder:
)270(LA)PP(T
)180(LA)PP(T
)90(LA)PP(T
)(LA)PP(T
p p344 p
p p233 p
p p122 p
p p411 p
+θ−=
+θ−=
+θ−=
θ−=
(19) The leverage of the crank-piston mechanism varies with crank angle,
)sinr 2l1(sinr l)]2cosr 4l(cosLr 4lr [L
222c2 p
θ+θθ+θ+−=
(20) Where, r is the crank radius, l is the length of connecting rod, A
p
is the piston cross-section area. The value of cylinder pressure torque is determined by crank angle and cylinder transient pressure which can be computed by Stirling engine thermodynamics in section C.
2)
Inertia Torque and Variable Inertia
The inertia torque is equal to the product of leverage of crank-piston mechanism Lp and an inertia force. The inertia torque from a single crank-piston mechanism is :
)]2cosr l(cosMl[)]sinr 2l1(sinr l)][2cosr 4l(coslr 4lr [T
22222i
θ+θθ−
⋅θ+θθ+θ+−=
(21) The mass moments of inertia of the engine crankshaft consist of a constant part and a variable part. The constant part of the engine inertia is the sum of the inertias of the engine masses. The variable part of engine inertias is the effective inertias of the engine masses that have reciprocating linear motion like the pistons,
)]2sinr l(sinMl[)]sinr 2l1(sinr l)][2cosr 4l(coslr 4lr [I
2222v
θ+θ⋅θ+θθ+θ+−=
(22)
3)
Friction Torque
Friction torque is crucial to determining effective engine output torque. However, the friction of the motion mechanism in the Stirling engine is still unknown, making it difficult to accurately calculate. As the friction loss associated with the engine speed, the empirical formula is used in this paper [8].
n30PnnaT
mmazf
⋅π=
(23) Where, n is the engine speed, n
max
is the maximum speed (RPM), P
m
is the cycle mean pressure (MPa), the coefficiency a is determined by the test data.
4)
Load
The Dish-Stirling system needs to start the engine to synchronous speed firstly, then the motor changes into a generator state. A induction generator is often used as a starter/generator to convert the mechanical power into electric power. As the starter, it provides a start torque for system starting. As the generator, it converts the mechanical shaft power into electricity which is supplied to the grid, and induction generator can have efficiencies over 94% [16].
E.
Mean Pressure Control
Control systems are necessary to regulate the torque, power output and speed of the Stirling engine. In Dish-Stirling systems, engine speed is held constant with varying load condition in case of stationary-constant speed, fixed frequency electric power generators. The power output of Stirling engine is directly proportional to the mean cyclic pressure of the working fluid. Thus, the simplest method would be to vent the engine for a power reduction and to supply working fluid for increasing power. III.
M
ODEL VALIDATION
A.
Steady Analysis
Figure 7. Specifications of the SES Solar Dish-Stirling System
U.S. company SES in cooperation with Boeing developed a 25kW Dish-Stirling system DECC in 1998 [17].The main parameters are shown in fig. 7. Table 1 shows the predicted results of DECC using the Dish-Stirling system model. The model predicted results compared with the performance data in fig. 7 shows good accuracy.
TABLE 1. Simulation Results Direct normal insolation (W/m2) 1000 Sun Power by Concentrator (W) 91608 Total Power Intercepted by Receiver (W) 64646 Concentrator-receiver efficiency (%) 70.57 Receiver or engine hot end temperature (K) 1013 Engine cycle mean pressure (MPa) 14 Engine cold end temperature (K) 303 Engine speed (RPM) 1800 Generator efficiency (%) 94 Output Power (W) 24344 Stirling engine efficiency (%) 40.06 Solar Dish-Stirling system efficiency (%) 26.57
B.
Dynamic Analysis
Fig. 8 is the simulation results and test data of output power for a Stirling engine in feeding process. It is found that the simulation results agree well with the test data.
Figure 8. Output power during feeding
IV.
S
IMULATION EXAMPLE
A.
System Start-up
To overcome the friction and all kinds of heat losses, the system must meet certain conditions for starting, that is the receiver cavity temperature and absorbed heat determined by the direct normal insolation as well as the engine cycle mean pressure. Fig. 9 shows the direct normal solar insolation data collected on March 4, 2008 in Beijing [18]. By the model we

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