## 710

fluid and the entry of the other. Similar blow and reversal periods exist for the passage of the cold fluid. As lliffe (1948) has pointed out, in practical regenerators the blow period is the same as the reversal period, since the last portion of fluid to enter is driven out by the other fluid through the port by which it came in. fn the hypothetical ideal regenerator. the blow period is always less than the reversal period by the time taken for a gas particle to travel from one end of the regenerator to the other. Therefore, if this effect is ignored, we are assuming that the time for a particle to pass through the regenerator is small compared with the total blow-time.

Other significant simplifying assumptions have been found necessary to render the analysis of operation tractable. Some of these are summarized below.

(a) The thermal conductivity of the matrix must be simple. Nusselt considered four cases:

(i) The thermal conductivity of the matrix is infinitely large. This means there would be no temperature difference in the matrix, and Nusselt's calculation shows that this type would have a poor performance.

(ii) The thermal conductivity of the matrix is infinitely large, parallel lo the fluid flow, and finite, normal to the fluid flow. In practice, this may be approached by a very short regenerator, with a matrix composed of thick walls.

(iii) The thermal conductivity of the matrix is zero, parallel to the fluid flow, and infinitely large, normal to the fluid flow.

(iv) The thermal conductivity of the matrix is zero, parallel to the fluid flow, and finite, normal to the fluid flow.

Cases (iii) and (iv) correspond closest to the practical regenerator, but it is unfortunate that the analyses of these two cases are the most complicated. Schultz (1951), Tipler (1947), and Hahnemann (1948) have examined the effect of longitudinal heat conduction in the walls of regenerator passages, and have demonstrated this to have a negligible effect in certain cases. Saunders and Smoleniec (1948) state that 'for matrices built up in layers, such as gauzes, or matrices made of refractory, the conduction effect is almost certainly negligible".

(b) The specific heats of the fluids and of the matrix material do not change with the temperature.

(c) The fluids flow in opposite directions, and have rn/ei temperatures that are constant both over the flow-section and n'fr/i time.

(d) The heat-transfer coefficients and fluid velocities are constant with time and space. even though they may be different for the two fluids.

(e) Phc rate of mass flow of either fluid is constant during the blow period, even though it may be different for the two fluids, and the blow periods may be different.

Very little theoretical work appears to have been done on regenerators operating under conditions not fulfilling assumptions (b), (c). and (d). and most results are available for operation with equal blow-times and equal mass flow. However. Johnson (1952) and Saunders and Smoleniec (1948) have investigated this latter effect. Saunders and Smoleniec also considered the effect of variation in the specific heats of the fluid and matrix, for a particular case. They found the assumption of constant values, made in (b), resulted in less than one per cent error in the effectiveness.

Another interesting (but impractical) case, considered by Nusselt (1927), was for a regenerator with an infinitely-small reversal-period, and in which the fluids had been switched infinitely often. The theory for the case is simple, and corresponds to that for a 'recuperator', or normal continuous-counterflow heat/exchanger, in which the two fluids flow-continuously, and are separated by metal walls.

Presentation of results

The performance results calculated for regenerators, assumed to be

---—--— >v>n/l!«!Anr itii-micoflrl oKauo h'.uip Iw-cn nr»>«pnli'(l in ¡1

Fig. 7.12. Regenerator effectiveness as a function of the reduced length A and reduced period .t (after Hausen).

Fig. 7.12. Regenerator effectiveness as a function of the reduced length A and reduced period .t (after Hausen).

Hauscn, and reproduced in Fig. 7.12. These have been supplemented by similar curves calculated by Johnson and Saunders and Smoleniec. The curves show that the effect on regenerator effectiveness of variation in two dimensionless parameters called (after Hausen), the 'reduced length' (A) and the 'reduced period' (II). The reduced length (in the flow direction) is defined by

A = HAL/ VCP, where h = heat-transfer coefficient between fluid and matrix, per unit surface area,

A = matrix surface area per unit length, V— fluid-volume flow rate,

The reduced period is defined by n = hAZJMC.

where h and A are as defined above, iV/ = mass of matrix material, C-specific heat of matrix material, Z — blow-time.

### Frequently A and II are combined by the quotient

I I/A - U - {VC\JMC)(Z/L), and called the 'utilization factor', representing the ralio of the sensible-heat capacity of the fluid per blow to the heat-storage capacity of the matrix.

lri practice, regenerators may have different reduced periods and reduced lengths for the hot and cold blows, so that there are four factors to be considered. In these cases, Saunders and Smoleniec recommend that average values be used, suggesting (on the evidence of calculations carried out bv Johnson), that the error is small. This is probably because, even when the actual blow-times are unequal, the reduced periods arc much nearer equality, since a reduction in the actual blow-time 7. is usually accompanied by an increase in the rate of fluid flow V.

The usefulness of the concept of two reduced dimensionless parameters and the curves of regenerator effectiveness is limited by the accuracy of the heat-transfer data. This is generally measured experimentally using the 'single-blow' transient technique, first described by Furnas (1932) and. later, by Saunders and Ford (1940), Johnson (1952), Saunders and Smoleniec (1948), Coppage (1952), Rapley (1960), Vasishta (1969), and Wan (1971). In this technique, the matrix is subjected to a flow of hot fluid, entering with a constant inlet-temperature, and the change in the exit-temperature is measured against time. The theory for 'single-blow' operation was lirst given by Schumann (1929), and may be used to extract, from the measured data, the heat-transfer coefficient relevant to the particular tested matrix. Very careful measurements are required, and there is, in fact, some doubt as to whether this data can be applied to regenerators operating cyclically. A reasonable amount of heat-transfer data is contained in the references given above, but comparison is difficult because several slightly different forms of presentation have been used.

Application of theory to regeneration in Stirling engines

Theories of regenerator operation, discussed above, were developed initially for air-liquefaction and gas-separation plants and for air prehea-ters for boilers. These plants are large and, in general, two regenerators per unit are used, one being heated arid the other being cooled. The blow-times are very long, ranging from ten minutes to several hours.

Later the theory was adapted and extended, during application of regenerative heat exchangers to gas turbines. Here the blow-times are much shorter. Coppage and London (1953) reler to la reversal time of a quarter of a second (two complete cycles per second) which is near the maximum permissible frequency without undue "carry-over loss"' and, again, 'the idealization of no flow-mixing is closely met when the flow-passage length is short, and such shortness of length appears to be good design procedure for the most suitable types of surface*. Most regenerators in gas-turbine engines have a relatively large frontal area and a short flow length, so that, although the blow-time is short, the residence-time of the particle in the matrix is also very short.

The above theory seems applicable, in a reasonably realistic way, to regenerators used in gas-turbine engines and air preheaters, but not applicable to regenerators used in Stirling-cycle engines. The theory is based on assumptions which, clearly, do not apply in the Stirling engine. The most important of these is. perhaps, that the time for a particle to pass through the matrix is small compared to the total blow-time. In a Stirling engine the blow-times are exceedingly short. For example, at the moderate engine speed of 1200 rev/min. or 20c/s, the blow-time is ten times less than the permissible minimum in a gas turbine. We saw earlier (Fig. 7.4) that the blow-times are so short that no particle ever passes right through the matrix. From Fig. 7.5 we saw the actual net flow time through the matrix was about half the complete cycle time, the remaining time being occupied in either filling, or emptying, the dead space. The heat-transfer process that occurs must be very complex, involving a repetitive fluid to matrix, matrix to fluid, fluid to matrix, cyclic relationship, rather like the water bucket passed from hand to hand in a fire-lighting operation. Other important assumptions of the theory are that the inlet conditions, temperature, rates of mass flow, and fluid velocity remain constant with time. Clearly, this is not true in any Stirling-cycle regenerator for the inlet conditions vary constantly, and Fig. 7.5 shows extreme variation in rate of mass flow. The maximum rate of net flow through the matrix is only about half the maximum flow into and out of the expansion space.

Attempts to analyse the regenerators of Stirling engines by any of the recommended procedures require the adoption of 'average' conditions for the flow. Such gross approximation is required to determine these 'average' values that the value of the ultimate result is thought to be highly questionable. No recommendation can be made, at this stage, for the application of any theories of regenerator operation as aids to regenerator design.

Although the situation is unsatisfactory at present, there is reason to hope for improvements. Smith and co-workers ai the Massachusetts Institute of Technology have made a promising start (Ovale and Smith.

1968, 1969). They discuss an approximate solution for the thermal performance of a Stirling-cycle regenerator, in which ihere is provision for non-steady pressure (and mass-flow) conditions, including the possibility of sinusoidal variation, with a phase difference in the peak values. By assuming a second-order polynomial form for the temperature field in the regenerator, a closed solution was obtained for the net enthalpy flux. The theory remains highly idealized, with the assumption that the gas temperature and matrix temperature, at one location, are practically constant with lime, and that there are no wall (or fluid-friction) effects. However, at present, the theory does not appear to be sufficiently well developed to be of direct use in regenerator design. Köhler and coworkers al the Philips Laboratories. Eindhoven, have done more research on regenerators in Stirling engines than anybody else, but. unfortunately, little of this work has been published. Although Dr. Köhler presented a series of lectures on regenerators at the Technische Hochschule, Delft, in

1969. these were never published in the open literature.

Nodal analysis of Stirling engines, exemplified by Schock (1978b), Urielli (1977), Finkelstein (1975a), and Tew (1978)t. attempts to simulate exactly the operation of all components in a Stirling system including the regenerator. This is accomplished by dividing the system into a number of cells and applying rigorous thcrmofluid analysis to each node in turn.

Nodal analysis requires access to large high-speed computers and so is not suitable for general use. However there is little doubt that the time is near at hand when sufficient experience will be accumulated for the preparation of a series of consolidated design charts for general use in the selection of an optimum regenerator configuration

### Experimental performance

Little appears to have been published about the effect of imperfect regeneration on the performance of Stirling-cycle machines, or about experimental work on regenerators, tested under conditions approximating to those present in a Stirling engine.

Davies and Singham (1951) carried out some experiments on a small thermal regenerator, composed of brass and copper wire gauzes, subjected to the oscillating flow of a constant volume of air, at atmospheric pressure and at a frequency of five cyclcs per second. The air was heated on one side ol the regenerator, and cooled on the other. Continuous records of the temperature of the air were taken on both sides of the matrix. It was concluded from these experiments that:

(1) for a given gauze matrix, the regenerator efficiency increases with the matrix weight, but the improvement takes place at a progressively diminishing rate,

(2) for a given matrix weight, the regenerator efficiency increases with decreases in the diameter of the gau/.e wire.

Tests, with equal weights of brass and copper gauze, gave approximately the same values for regenerator efficiency. Thus, although the copper had a thermal conductivity about three times greater than that of the brass, this appears to have had little effect. It was concluded that, with line wires of these materials, the conductivity lag is extremely small. In these tests, the regenerator efficiency was obtained by analysis of the continuous fluid-temperature records, measured at each end of the regenerator matrix.

Experiments by Walker (1961a), with a scries of different regenerators on the Philips gas refrigerating machine, have conlirmcd the second conclusion reached by Davies and Singham. namely, that reduction in wire-diameter increases the effectiveness of the regenerator. The criterion of performance was taken to be the quantity of liquid air produced by the machine, operating at a constant speed and the mean pressure of the working fluid. A reduction iri the wire-diameter, with approximately constant matrix weight and porosity, resulted in an increase in the surface area for heat transfer.

Work by Murray, Martin, Baylcy and Rapley (1961) has shed some light on the performance of regenerators under sinusoidal flow conditions. It was found that frequency appeared to have little effect on the heat-transfer process, but the shape of the wave has a significant effect. With pulsating flow, the effectiveness of the tested gauze matrices was appreciably below that obtained under steady flow conditions. With flame-trap matrices, an improvement in the heat-transfer rate, in unsteady flow, was noticed.

Regenerator design—a practical guide

In the absence of adequate theoretical assistance in regenerator design, a few helpful suggestions are offered below. They are not intended to be fundamental rules.

The regenerator designer must attempt to solve the problem of satisfying a number of conflicting requirements. To minimize the temperature excursion of the matrix, and thus improve the overall effectiveness of the regenerator, the ratio of the heat capacity of the matrix to thai of the gas tM C IhA r \ I/-I K/» -------:------ — •

On I he other hand, the Huid-friction loss must be limited. We saw, in Chapter 3. that the effect of the pressure drop across the matrix is to reduce the range of the pressure excursion in the expansion space, thereby adversely affecting the area of the expansion-space P-V diagram. This reduces the net work-output and thermal efficiency of a prime mover, and the amount of heat lifted and coefficient of performance of a cooling engine. The fluid-friction loss is minimized by a small, highly porous matrix.

A third, and most important, consideration is that of dead space. The size of the dead space influences the ratio of maximum to minimum volume of working space, and this directly affects the ratio of maximum to minimum pressure. For maximum specific output, both ratios should be as high as possible, and, for this to be achieved, the dead space should be made as small as possible. This can be achieved by a small, dense matrix.

To improve the heat-transfer performance, and establish the minimum temperature difference between the matrix and the fluid, it is necessary to expose the maximum surface area for heat transfer between the fluid and matrix. Therefore, the matrix should be finely divided, with preferential thermal conduction at a maximum normal to the flow, and minimum in the direction of the flow.

Finally, it is important to appreciate that the regenerator acts as an exceedingly effective filter of the working fluid, so that any oil, or grease, particles arc retained in the tine flow-passages. In the case of a cooling engine, any impurities in the working fluid that condense in the low-temperature region of the expansion space will accumulate in the regenerator. This build-up is cumulative, and has the effect of increasing the fluid-friction losses, so that the pressure excursion in the expansion space is decreased, and the performance of the cooling engine progressively diminishes. In the case of the prime mover, any accumulation of oil particles in the regenerator inhibits the flow of working fluid, and increases the pressure loss. The temperature in the expansion space thereby increases, and may be even further increased, because more fuel is supplied in an attempt to restore the lost power. This increase in temperature carbonizes the fuel, thereby further blocking the flow passage, and the process continues in cumulative fashion, until catastrophic overheating of the engine occurs. From this aspect, the regenerator should olfer minimal obstruction to the flow.

Thus, we have the following desirable characteristics for a regenerative matrix:

tor maximum heat capacity—a large, solid matrix, for minimum flow losses -a small, highly porous matrix, for minimum dead space—a small, dense matrix.

for maximum heal transfer—a large, finely-divided matrix, for minimum contamination—a matrix with no obstruction.

Clearly, it is impossible to satisfy all these conflicting requirements. With our present understanding of the cycle, it is not possible to quantify the relative significance of the various aspects.

### Prime movers

In most engine designs, considerable attention is given to the regenerator, and comparatively little to the problem of the heater and cooler. As a consequence, heal transfer to and from the engine is poor, and the engine fails to operate satisfactorily. This stimulates yet further interest in the regenerator, with the investment of much experimental effort in trying different regenerator arrangements. Frequently, surprise is expressed when this produces absolutely no effect on engine performance except that a reduction in size results in improved operation. Surprise becomes confusion when the experiments are extended to the point of diminishing the regenerator to such an extent that it has, in effect, been completely removed from the engine. It is a matter of experience that, in small low-pressure engines, removal of the regenerator nearly always results in improved performance. This is because the gains due to a reduction in the dead space, ami. to a lesser extent, a reduction in the conducting path oí the regenerator enclosure and in fluid-friction losses, more than offset the loss of thermal capacity and area for heat transfer of the regenerative matrix.

In most small, low-speed machines (up to, say, 5 cm bore, with less than 5-6 atm pressure and operating at below 1000 rev/min). it is adequate (for a start, at least), not to incorporate a formal regenerator in the engine design, but. rather, to depend on the action of a regenerative annulus around the displacer.

One type of displacer system with a regenerative annular duct, used with success by William Beale in small free-piston Stirling engines and also by the writer, is shown in Fig. 7.13 The displacer is made of a thin-walled, low conductivity stainless-steel tube, closed at the hot end by an inverted 'top-hat' section which is machined from a solid bar so as to be a close fit in the tube. After assembly, the seam may be gas-welded, and the joint section trimmed and trued by grinding. Inside the displacer, a series of radiation shields may be provided, as shown, either cut from solid material or fabricated. The lower end of the displacer is closed by another closely fitting plate. Since this end operates in the cooled zone, the end plate can be of light alloy or stainless steel. An epoxy-cement joint has been found adequate for fixing. Good results have been obtained with disolacers about three rliamt»**»™ Ir»n« Thn rUc«l»nv»r nn»™!»

Fig. 7.13. Detail cif regenerative anttulus.

A. regenerative annular gap 0.015-0.030 in.

Ii. thin-\v;iU sections to minimize thermal conduction.

C, epoxy joint.

D, rulon guide-ring.

E, radiation shield

F, welded scams.

G, stiffening rings.

H, cooling jacket.

Fig. 7.13. Detail cif regenerative anttulus.

A. regenerative annular gap 0.015-0.030 in.

Ii. thin-\v;iU sections to minimize thermal conduction.

C, epoxy joint.

D, rulon guide-ring.

E, radiation shield

F, welded scams.

G, stiffening rings.

H, cooling jacket.

in a cylinder, also of low-conductivity stainless steel and having a thin cross-section, except for occasional circumferential stiffening rings, left during manufacture. The top of the cylinder is closed by another inverted Top-hat' section with an external welded joint. The lower end of the cylinder is, of course, attached by a flange to the cooled compression-space cylinder. The cylinder of the displacer may actually be shorter than the displacer, so that the bottom cooled end of the displacer operates within the compression-space cylinder. This makes it possible to mount a guide ring of P.T.F.E.-based material around the lower end of the displacer. and have it operate on a cooled wall-section.

The annulus formed between the displacer and the cylinder is then the flow passage, connecting the expansion and compression spaces. It acts as a regenerator, since the top end is always in the heated section, and the bottom end always in the cooled section. It is a simple device, but remarkably effective if the displacer and cylinder wall are reduced to very thin sections, to minimize thermal conduction losses. The gap between the displacer and the cylinder wall is a critical dimension with regard to heat transfer, and should be between <1.015 and 0.030 inches. It is important, also, that a regular annulus be established with a uniform circumferential gap. to equalize heat-transfer and fluid-flow effects. The problem of heat transfer in an annular duct, with an axial temperature gradient and a reciprocating internal member, does not appear to have been studied, and might be a topic of considerable appeal for a university research program.

The limits of applicability of the regenerative annular duct are not known, but it is likely that the system would become less and less effective as the cylinder bore, cylinder pressure, or engine speed were increased. The initial inadequacy would appear, perhaps, in the heater section, and some improvement might be gained by providing an extended surface for heat transfer by using internal finning: but this would be difficult to accomplish without substantially increasing the dead space. Eventually, it would become necessary to resort to increasingly complicated heaters, probably of external-tubular form, and it is at this point that a regenerative matrix becomes worthwhile. By this time, however, one is developing an engine of advanced form that would probably evolve with close similarities to machines of the Philips type.

### Cooling engines

The regenerator in a cooling engine appears to be much more important than in a prime mover, but by happy coincidence, the materials problem is less severe.

low-conductivity compressed-paper sleeve, and made up from the random packing of short lengths of copper wire 0.001 inches in diameter, mounted in annular form around the displacer. The author has found woven wire mesh of copper and phosphor bronze to be effective packing for regenerative matrices. These can be had in a wide variety of mesh densities and wire sizes. As the mesh density increases and the wire diameter decreases, the price per unit area increases very steeply to the point where it is doubtful that the material could be used for production machines. An annular regenerator is very expensive, because the centre section, punched from the screen, is 'wasted'. Wire screens can be 'sintered' easily, to form a stable semi-rigid block. One way is to pack the screen in some form that can be loaded with a weight. Then, the wire screen is cleaned by immersion in nitric (or hydrochloric) acid, and the loaded assembly is heated for a short period in a furnace with a reducing atmosphere. On removal, it will be found that the screen has 'sintered' to a solid assembly that can be lightly machined. It is important to arrange the screen so that the wires are normal to the axes of flow, otherwise the axial conduction may be too high. Sintering with light loading does not appear to significantly increase the axial conduction of the screens, and because of a considerably reduced porosity, it does improve the pack.

It is not possible to make specific recommendations for design, although the following points merit consideration. The wire used should be fine (0.001 to 0.002 inches in diameter), closely packed, and compressed, to minimize voids. A dead-space ratio of one is a good target, but difficult to achieve, and at least half the dead space should be regenerator void-volume. As a rule, the regenerator arrangement should be of such proportions that the total cross-section of the duct is equal to that of a right normal cylinder, having a diameter equivalent to its length.

Heat-transfer and Huid-friction characteristics of dense-mesh wire screens

The heat-transfer and fluid-friction characteristics of a variety of porous media were given by Coppage and London (1956) and have been supplemented by later data. However, little data on the flow in dense wire screens has been published for the size ranges of interest when considering the regenerators of Stirling-cycle cooling engines. Values measured al the University of Calgary by Vasishta (1969), and by Wan (1971), are included here, but no other values are known with which these results may be compared. To validate the experimental apparatus, Vasishta did obtain some results for stainless-steel mesh, in sizes comparable with those studied by Coppage. and found the results to be in close agreement.

The heat-transfer and fluid-friction data for two sizes of screen are given in Figs. 7.14 and 7.15 respectively, both sizes of screen were woven

0 20 40 60

Reynolds No. .VRr

0 20 40 60

### Reynolds No. .VRr

Ftp. 7.1-1. Heat-transfer characteristics of dense-mesh wire screen (after Walker 1972), (a) 400 x 400 strands per inch, 0.001 inch wire diameter. |b) 200x 200 strands per inch, 0.002 inch wire diameter.

from phosphor-bronze wire, having the following composition:

Tin 3.5-3.8 per cent; Phosphorous 0.3-0.35 per cent; Iron 0.1 per cent; Lead 0.05 per cent: Zinc 0.3 per cent: Copper rem. Density 8875 kg m J (554 lb ft n): Thermal conductivity 81.3 Wm"' K '1 (47 Btu h"1 ft-1 °F~l)

Specific heat 0.435 kJ kg-1 K (0.11)4 Btu lb 1 °F ') The two screen-sizes investigated were

(a) 200x200 strands per inch. 0.0021 in wire diameter,

(b) 400x400 strands per inch, 0.001 in wire diameter.

The heat-transfer characteristics are presented as the Nusselt number Nnu. as a function of the Reynolds number /srRc, defined as follows:

iVNu = (4rh/k)(fi//), N„c = p,VdJn, = (4rJ p.,p)(\Vff A,), where rh calculated hydraulic radius of the screen, h ~ heat-transfer coefficient. /(, = thermal conductivity of the fluid, p, = density of fluid, V = volume flow rate of fluid in matrix. Wt = mass flow rate of fluid in

0 20 40

Reynolds No .VK>J

Fio. 7.15. Fluid-friction characteristics of dense-mesh wire screen (after Walker 1972).

(a) 400x400 strands pei inch, 0.001 inch wire diameter.

(b) 200x200 strands per inch. 0.002 inch wire diameter.

matrix. A, = frontal area, p = calculated porosity, /x, = dynamic viscosity of fluid, where p = (volume of matrix-volume of metali/volume of matrix and rh= total volume of connected void spaces/total surface area = volume of matrixx porosity/total surface area.

The fluid-friction characteristics are presented as the Fanning friction factor N|, as a function of the Reynolds number defined as follows:

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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