Disk convector for Stirling engine.

Dehelean, Nicolae ; Ciupe, Valentin ; Maniu, Inocentiu 等

1. INTRODUCTION

The idea of the paper is to improve the thermal transfer from the cylinder head to the working gas. In relation with [alpha]-Stirlling engine principle, the working gas flows from the compression cylinder to the expansion cylinder and passes through the cylinder head. During the gas flow, the cylinder head should be able to transfer a great amount of heat energy. The basic relationship for heat transfer by convection is the well known Newton's Law of Cooling (1).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Where:

h--convection heat transfer coefficient

A--area of the convection surface

[T.sub.w]--wall temperature

[T.sub.f]--fluid temperature

In order to improve the rate of heat transfer, both h and A should be increased. The h coefficient could be improved by the meaning of the basic analysis of heat transfer process relations. The area of the convection surface could be enlarged by a set of constructive attainments.

2. BASIC CONVECTION

Convection thermal transfer is a complex phenomenon. It can be described only by the meaning of the similitude criteria. The best criteria that can be used for this paper seems to be "Nusselt Number" ([N.sub.u]). The [N.sub.u] number gives a relation between fluid thermal gradient and wall thermal gradient (Leca & Pop, 1987).

[N.sub.u] = h 1/[lambda] (2)

Where:

l--flow length

[lambda]--conductivity coefficient

[N.sub.u] = [([R.sub.a]/[R.sub.ac]).sup.1/3] (3)

Where:

[R.sub.a]--Rayleigh Number

[R.sub.ac]--critical Rayleigh Number

[R.sub.a] = [G.sub.r] [P.sub.r] (4)

Where:

[G.sub.r]--Grashof Number

[P.sub.r]--Prandtl Number

[G.sub.r] = g[alpha][DELTA]T[L.sup.3/[v.sup.2] (5)

[P.sub.r] = [eta][c.sub.p]/[lambda] (6)

Where:

[eta]--dynamic viscosity

[c.sub.p]--heat capacity at constant pressure

g--gravity acceleration

[alpha]--thermal expansion coefficient

[DELTA]T--temperature difference

L--length scale

v--kinematic viscosity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Where:

[gamma]--heat capacity ratio

[c.sub.v]--heat capacity at constant volume

[K.sub.T]--isothermal bulk modulus

[K.sub.S]--adiabatic bulk modulus

[K.sub.T] = -V[([partial derivative]P/[partial derivative]V).sub.T] (8)

[K.sub.s] = [c.sub.p]/[c.sub.v] [K.sub.T] (9)

Finally it obtains h = [N.sub.u] [lambda]/l.

To increase the convection heat transfer coefficient h from (2), a good selection of the engine's working gas must be done. The working gas must have a high level of heat capacity ratio [gamma] (7), (5), (4), (2) and heat capacity at constant pressure, [c.sub.p] in (6). The temperature difference [DELTA]T should be as large as possible and L as log as possible as per (5). As Nusselt Number shows the wall material must have a high conductivity coefficient [lambda].

The selection of the wall material must use the A coefficient as parameter. The best metals for wall are copper and aluminum.

3. CONSTRUCTIVE MEASURES TO ENLARGE [??]

The constructive measures could operate on the A parameter of equation (1). In order to enlarge A, a special cylinder head for the engine must be designed, this being the focus of this paper (Figure 1).

The cylinder head is a special disk with a lot of holes. The total areas of the holes represent parameter A in equation (1). The optimization function in the cylinder head design process is to obtain a maximum rate of heat transfer from the cylinder head to the working gas.

An ordinary placement of the holes on the disk is shown in Figure 2. The maximum convection area A that can be obtained in this configuration is given by the equation system (10).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

The second stage of the optimization refers to a set of holes with variable diameter values to obtain a constant heat transfer and a constant flow resistance. This level of optimization requires a disk with variable thickness. This stage will be approached in the future by means of finite element calculus.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

In Figure 3 is shown a possible shape of the disk convector after its first stage of optimization. A special shape surface it is expected to be obtained after a finite element analysis.

4. FUTURE RESEARCH

The proposed solution for the disk convector must be further optimized and tested in real life conditions. This presumes building a working model of the disk convector and testing the distribution of the thermal field by means of a thermographic imaging system.

The further stage of the cylinder head should reveal the best shape of the disk surface, the best distribution of the holes and the variance of the holes diameter. The optimization tasks will consider, in the same time, the flow resistance and the heat transpfer.

5. CONCLUSION

The attempt to improve the performance of the engine implies the increase of the thermal transfer efficiency. In order to improve the rate of heat transfer, both convection heat transfer coefficient h and area of the convection surface A should be increased. In order to enlarge parameter A, a special cylinder head for the engine has to be designed.

The selection of the wall material of the cylinder head needs to consider the conductivity coefficient [lambda]. This parameter must be as high as possible. Therefore it is recomanded to build the cylinder head from copper or aluminum.

6. REFERENCES

Leca, A. & Pop, M. (1987). Indrumar tabele, nomograme si formule termotehnice (Guide-book; tables, nomograms and thermodynamics formulae), Ed. Tehnica, Bucuresti

PVGIS Project, (2008) EC's Joint Research Centre. Available from: http://re.jrc.ec.europa.eu/pvgis/, Accessed: 2009-07-02

Red Rock Energy. (2008) Solar Power Heliostat Arrays. Available from: http://www.redrok.com/main.htm, Accessed: 2009-06-29

*** (2009) http://scienceworld.wolfram.com/physics/ topics/ thermodynamics.html, Accessed: 2009-06-29

*** (2009) http://www.engineeringtoolbox.com/ overall-heat-transfer-coefficient-d_434.html, Accessed: 2009-06-06