analysis of the gpu-3 stirling engine

based on the review paper by timoumi et al,

“design and performance optimization of gpu-3 stirling engines”

energy 33 (2008), page 1100

and the book by dr. israel urieli,

“stirling cycle machine analysis”

dave kopp eee @ nd 2008

(stirling cycle machine analysis)

•these 25kW electric solar stirling engines have been measured at >30% system efficiency•the modular design allows slow, steady construction (partnership, sandia national lab and stirling power systems)

motivation

motivation

• the ideal stirling engine can provide maximum thermal efficiency from a given temperature difference

MAX=T H−T L

T H

withT L=300KT H=500K⇒MAX=40%T H=1000K⇒MAX=70%

http://www.ent.ohiou.edu/~urieli/stirling/isothermal/Schm_summary.html

in a nutshell

energy density: stirling engines operate independent of their heat source, and may be driven by a variety of sources, including solar concentrating (~1kW/m^2), coal (30MJ/kg), etc.

efficiency: 30% is currently achievable at 25kW; much higher efficiencies are predicted in maturing technology or large temperature differences

~ the (theoretical) efficiency of the gpu-3 has been significantly improved in the model discussed in this paper ~

comparison for solar heat power generation:

•stirling engines currently operate at around the 31% theoretically possible efficiency for a silicon solar cell [http://www.sandia.gov/news/resources/releases/2008/solargrid.html]

•stirling engines are planned to replace thermoelectrics by nasa in space applications to cut fuel by a factor of four [http://books.google.com/books?id=V84ZHMmdNmYC&pg=PA509&lpg=PA509&dq=stirling+rtg+nasa&source=web&ots=9ZrTpAXW-Z&sig=SRqq-6XWeSHdgpEVBE8QRDqfAE4&hl=en&sa=X&oi=book_result&resnum=5&ct=result]

how it works: see next slide

(1) heat air, causing gas to expand and push on the piston/flywheel ... doing more work by pressure than consumed in step (3). (2) transfer hot gas

through regenerator to cold space. heat is transferred to the regenerator, precooling the gas. gas pressure drops.

(3) piston/flywheel compresses cooling gas, consuming less work against pressure than produced in step (1).

(4) transfer cold gas through regenerator to hot space. heat is transferred from the regenerator, preheating the gas.gas pressure rises.

stirling cycle

(note: the regenerator is not drawn)

http://www.physics.sfasu.edu/astro/courses/egr112/StirlingEngine/StrilingEngineSpring2004.mpg

IDEAL CYCLE

e: expansion space, heatedc: compression space, cooled

heat in=∮ pdV e=∣Qe∣heat out=∮ pdV c=∣Qc∣work out=∮ pdV e∮ pdV c=∣Qe∣−∣Qc∣

efficiency=∮ pdV e∮ pdV c

∮ pdV e

rough estimate for current engines: power=0.15pavgV swept f

displacer

piston

(wikipedia:stirling engines)

a rhombic drive forces the desired volume variations

Ve = expansion space volumeVc= compression space volume

(stirling cycle machine analysis)

(stirling cycle machine analysis)

model inputs

more model inputs

parameter valueThot (K) 288Tcold (K) 977f (Hz) 41.72mean p (MPa) 4.13gas helium

typical energy

flow diagram

(stirling cycle machine analysis)

energy losses in model(typ % of heat input)

1.viscous drag in heat exchangers2.internal conduction from hot to cold (3%-12%)3.regenerator inefficiency (5%-9%)4.heat absorbed by displacer in hot area is transported to a cold

region, causing convection (“shuttle loss”) (2%-6%)5.irreversible work done to compress gas (0.8%-5%)

1.at high heat transfer coefficients, gas enthalpy transfer past the displacer becomes more important than shuttle loss2.seal leakage can be substantial in the hot areas because good seals can be difficult to make at very high temperatures.

not analyzed:

cutting losses in gpu-3 model ...

theory:

efficiency increased from 39% to 51%!

power rose by 20%!

•decreasing thermal conductivity of regenerator matrix reduces conduction losses•increasing regenerator heat capacity increases regenerator effectiveness•decreasing regenerator porosity to about 65% reduces external conduction losses and improves energy exchange in the regenerator, although continued reduction stops working entirely•increasing the regenerator temperature gradient increases desirable heat exchange E between regenerator and gas faster than it increases loss•optimizing working gas mass. maximum efficiency requires less total working gas mass than does maximum power (in this engine, 0.8g give 40% efficiency; power increases with mass)

paper recommendations: how to improve the GPU-3

efficiency increased from 39% to 51% !power rose by 20% !

item model optimregenerator porosity 65.50%regenerator length (cm) 2.1regenerator diameter (m) 2.4working gas mass (g) 1.15exchanger piston conductivity W/(m K) 1.2exchanger piston area (sq cm) 38.6exchanger piston stroke (cm) 4.7

loss 1. drag in heat exhangerscause of drag: back pressure in exchanger due to friction in narrow tubes.

Q= p V

cause of back pressure: gas adhering to, or vibrating against, the tube wall area

wall shear stress1/2 v2 = f fanning

wall shear stresswall areapressure drop pcross sectional area

=1

moody diagram for friction factor f

p= f 1/2 v2 Awall

Across section

Re= v d /=viscosity

(mm

pip

e ro

ughn

ess)

/(mm

pip

e hy

d di

amet

er)

Q= p Vlost power

(http://www.engineeringtoolbox.com/moody-diagram-d_618.html)

loss 2. conduction losses

Q=TRth

thermal resistance Rth=Lk A

cause of loss:metal surfaces, with high thermal conductivity k, allow rapid transport of heat

(1) heat air, causing gas to expand and push on the piston/flywheel ... doing more work by pressure than consumed in step (3). (2) transfer hot gas

through regenerator to cold space. heat is transferred to the regenerator, precooling the gas.gas pressure drops.

(3) piston/flywheel compresses cooling gas, consuming less work against pressure than produced in step (1).

(4) transfer cold gas through regenerator to hot space. heat is transferred from the regenerator, preheating the gas.gas pressure rises.

3. regenerator function

loss 3. regenerator performanceQ loss

Qinto regenerator=1−E design for high effectiveness E

E=1 if:(1) the excess heat of the hot gas that does not do useful work transfers to the regenerator, and(2) no excess heat is lost in the cooling load, and(3) all of the excess heat is transferred back to the gas as it moves through the regenerator to the heater.

E=0 if:(1) all excess heat transferred from the regenerator is lost, or(2) no excess heat is transferred to the regenerator, or(3) there is no regenerator

E≈1 h AwallTc p mT

−1 h, the effective total heat transfer coefficient, is complicated.

4. shuttle loss• cause: displacer picks up heat and

transports it to a cold region, wasting some upon mixing

• phenomenological model:

Q loss=0.4 k pistTd disp sdist

2

g ring Ldisp

thermal conductivity of piston

annular gap between displacer and cylinder

displacer length

displacer strokedisplacer diameterexpansion space

temp – compression space temp

5. gas hysteresis loss• cause: a real gas, when compressed or

expanded, dissipates a small amount of work

derivation: begin with c p∂T∂ t

=k ∂2T

∂ y2∂ p∂ t

, and substitute

p=RT and =c p /cv . assume sinusoidal volume variationsV=V meanV sin t . The heat loss over the cycle is

Q= 12∮−kA[∂T∂ y ]wall dt which is linearized and

simplifed to give the result.

W loss= 132

3−1 pmeanT wall k th VV mean

2

Awall

cp/cv

model

ideal gas T=PV /Rmlinear temperature change in regenerators slope=T / Lengthheat transfer by convection Q=hATregenerators have calculable effectiveness Emass moves a power Q=mc pTthe total gas mass is fixedheat Q from the source either becomes work W, moves to the cold sink, or escapeswork per cycle = (heat in - loss to sink - loss to exterior) power is work times frequencyefficiency is work output over heat input

•decreasing thermal conductivity of regenerator matrix reduces conduction losses•increasing regenerator heat capacity increases regenerator effectiveness•decreasing regenerator porosity to about 65% reduces external conduction losses and improves energy exchange in the regenerator, although continued reduction stops working entirely•increasing the regenerator temperature gradient increases desirable heat exchange E between regenerator and gas faster than it increases loss•optimizing working gas mass. maximum efficiency requires less total working gas mass than does maximum power (in this engine, 0.8g give 40% efficiency; power increases with mass)

paper recommendations: how to improve the GPU-3

efficiency increased from 39% to 51% !power rose by 20% !

item model optimregenerator porosity 65.50%regenerator length (cm) 2.1regenerator diameter (m) 2.4working gas mass (g) 1.15exchanger piston conductivity W/(m K) 1.2exchanger piston area (sq cm) 38.6exchanger piston stroke (cm) 4.7

model outputs: efficiency of the gpu-3

efficiency calculationsexperimental 0.35

0.380.53

timoumi modelurielli model

my model (based on urieli http://www.ent.ohiou.edu/~urieli/)

available at my websitehttp://nd.edu/~dkopp/main/clean/stirling/stirlingMatlab.zip

solar engines: (1) http://www.youtube.com/watch?v=fUrB7KRvxUk (2) http://www.youtube.com/watch?v=tugshxuh-f0

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