First Order Design

Design Overview

Because of the low temperature difference given, power and efficiency will be inherently low. In order to reach the target power of 500 W and keep the engine a reasonable size, the engine should be pressurised. Essentially, pressurising the engine puts more moles of working gas into the same space, meaning higher specific power output per litre. This necessitates a strong and completely sealed engine housing in order to maintain a high working pressure safely. The only connections into and out of the engine are the heating and cooling water and the electrical connections, including the power output from the generator which is sealed inside the engine. A nominal value for engine pressure, po, of 1.0 MPa (10 bar) was chosen as an achievable pressure level for what was predicted to be a large pressure vessel. Because of the large expected size of the engine and high internal pressure, the housing must be very strong. A cylindrical shape is a good choice as it has great strength for a given material type and thickness (which is why almost all pressure vessels are cylinders). A cylinder can be formed easily by putting end caps on a length of pipe, and pipe can be bought that is already manufactured and tested, cutting down on costs and time for custom fabrication of a housing.

Due to this engine being a research prototype it was desired for a range of adjustments to be possible to adjust and test certain parameters of the engine. One such adjustment is the stroke length of the power piston, which in turn affects the compression ratio. The compression ratio, or volume ratio VR, is defined as the gas volume with the piston at top dead centre (TDC) divided by the gas volume with the piston at bottom dead centre (BDC). It is typically very low in LTD engines due to the large overall gas volume that is generally required for these engines to operate. Kolin [9] describes a ‘rule of the thumb’ for approximating the ideal volume ratio which depends on the temperature difference of the engine in question. The formula states:


VR = (1 +—— ) (40)

R ( 1100)

The motion of the displacer in a traditional LTD engine design is limited to sinusoidal motion as it is driven off the same shaft as the crankshaft. In addition, the phase angle is almost always fixed at 90° however this is not necessarily the optimum value, as indicated by the graph in Figure 46 which describes a numerical analysis of a certain Philips Stirling engine. It would be useful to be able to easily alter the phase angle to find the optimum value, and it is partly for this reason that it was decided to operate the displacer in the prototype engine via an electronically controlled motor or actuator. The second major advantage of doing this is the ability to move the displacer in a non-sinusoidal or discontinuous motion, the advantages of which have been discussed in Section It will also be possible to do a direct comparison of sinusoidal vs. discontinuous motion and study the effects on power and efficiency. Further to these benefits, it will allow the losses associated with regenerator and heat exchanger pressure drop and bearing and seal friction to be directly and accurately measured simply by looking at the average power consumed by the motor/actuator.


Power —|— (5) —

1 0


"r . u


! /

————— !—————




50 40 30 — 20 10





150 tf> 180

Figure 46: Graph of power output vs. phase angle for 3 different dead volumes [24]

Engine Configuration and Layout

The design requirements call for a large strong engine housing that can be pressurised, and a displacer format that can be electronically actuated by a motor or solenoid. In addition, the heat exchangers must be fed water externally. While it would have been possible to mimic the traditional ‘pancake’ design of LTD Stirling engines, as shown in Figure 48, on a larger scale, it was believed that there would be several significant issues surrounding this design.

Figure 47 shows two cylinders of identical volume but different dimensions. The cylinder on the left is the sort of shape required for a successful design of a displacer chamber of a
‘pancake’ engine. It is short and has a large flat area on the top and bottom which makes it very difficult to pressurise as the flat end plates will experience a tremendous force acting outwards upon them. The cylinder on the right contains the same volume of gas but can be easily filled to a pressure exceeding that of the left cylinder without requiring prohibitively expensive materials or end plate designs. A driving factor of this project is to arrive at a design for an engine which can be manufactured very cheaply, and the long skinny cylinder can fulfil this requirement. This shape however does not lend itself well to traditional displacer operation, hence why the new concept of the rotary oscillating displacer was conceived.

The final reason this design was unsuitable is the fact that water-to-air heat exchangers would have very difficult to incorporate into this design, especially in an effective manner.

подпись: the final reason this design was unsuitable is the fact that water-to-air heat exchangers would have very difficult to incorporate into this design, especially in an effective manner.Another reason for choosing the rotating displacer design is that in existing ‘pancake’ designs the displacer is simply a lightweight space-filler, usually made from polystyrene and supported by one central shaft. In a large scale design the displacer would need additional support mechanisms to avoid it moving in an unintended fashion or from flexing and possibly suffering damage. These extra supports would require extra sliding seals out of the displacer housing, and these seals are quite critical to the engine performance — if they leak or create too much friction it can easily stop the engine from operating. Furthermore the traditional displacer material for these types of engines, polystyrene, would have been unsuitable at the desired temperatures (it tends to soften around 60°C); hence a heavier material would have had to be used. Another factor that led away from this design was the fact that internal pressure would have created huge forces on the large diameter flat end plates that are typical of this engine type due to the short wide nature of the design. This would necessitate very thick material for the end plates as well as very robust flanges and bolts to attach them to the main cylinder, both of which add significantly to cost, weight and difficulty of manufacturing.

First Order Design

Figure 47: Two cylinders of equal volume but differing dimensions

First Order Design

Figure 48: Cutaway view of a typical LTD ‘pancake’ engine [41]

The design concept for the new prototype engine was originally conceived as per the rendering in Figure 49, as a drum with a half-round displacer that would rotate back and forth through almost 180° instead of using a piston type displacer with linear travel. The power piston for this concept is attached to the drum and the heat exchangers are shown in red and blue outside the drum on opposite sides, creating the hot and cold spaces respectively. The regenerator is shown in green as the central element through which the gas flow would be forced by sealing the edges of the displacer to the inside of the drum. This design also offers the advantage of being able to be manufactured largely from relatively cheap materials such as off the shelf steam pipes for the outer housing.

First Order Design

Figure 49: First concept of reciprocating engine design

This ‘reciprocating displacer’ engine concept was decided on as the configuration to be used, and subsequent development then took place. Early on it was realised that the heat exchangers on the outside of the drum would provide insufficient heat transfer, and certainly would not act sufficiently on the gas nearer to the centre of the engine (rather than just the gas near the exchangers at the outside).

The engine was redesigned with a new layout of heat exchangers, and drawn in Solidworks™, with the result shown in Figure 50. The heat exchangers in this configuration are to be wedge shaped elements, which will slot into the large displacer chamber. The regenerator element (green) is flanked by the hot side heat exchanger (red) and cold side heat exchanger (blue), with each element occupying a 40° wedge of the drum volume such that all three elements represent one third of the drum volume. The displacer is a 120° degree wedge of lightweight material such that it occupies one third of the remaining space in the drum. It is connected to a stepper motor which will be housed in the adjacent shorter drum, which will actuate the displacer back and forth through a 120° range of motion such that the gas is forced to flow through the heat exchangers and regenerator. The piston is connected to a crankshaft which is mechanically independent of the displacer. The crankshaft will be connected to an electric generator which will allow the output power to be obtained with only 3 wires protruding from the casing (although there will also be other wires for stepper motors and related control systems).

First Order Design

Figure 50: Early design of prototype engine

Power Output and Efficiency

Power output can be estimated using a variety of methods which take into consideration things like temperature difference, operating speed and pressure, expansion and compression space volumes and regenerator effectiveness. One such estimate is made using the Beale number, an empirically derived number named after William Beale, inventor of the free-piston Stirling engine. He noted that the performance of many Stirling Engines tended to conform to the following simple equation relating indicated power, Po (W), to pressure p (bar), operating frequency f (Hz) and expansion space volume Ve (cm3) with the Beale number Bn.

P0=BnpfVe (41)

Figure 51 shows a graph plotted by measuring data from many Stirling Engines. The solid line in the middle is typical of most Stirling Engines while the upper and lower lines denote unusually high or low performing engines. Based on this graph, with a heater temperature of 90°C (363K), it would be reasonable to expect a Beale number in the range of 0.002 to 0.008. For further calculations a Beale number of 0.005 is used.

First Order Design

0 02

«00 IOOO



Figure 51: Graph of Beale number vs. heater temperature for a range of engines [24]

A second power estimate method is called the West number (after Colin West, inventor of the fluidyne Stirling engine), and is similar to the Beale number except it takes direct account of the temperature difference. The formula is expressed as:

First Order Design

Where Wn is the West number, which has an average value of 0.25 [42] and where a higher number represents a more efficient engine.

Due to the large expected size of the engine and the associated large mass and inertia of all moving components, a slow operating speed is expected. For all calculations a nominal speed of 2 Hz (120 RPM) has been chosen.

The single biggest factor in the engine’s overall efficiency is the temperature difference. In 1824 Sadi Carnot published a work containing a formula which later became quite famous, the formula being the following which states the maximum theoretical efficiency of any heat engine, or the Carnot efficiency ^c, is a function purely of the temperature difference between the hot side temperature Th and the cold side temperature Tc (both temperatures in K).

First Order Design

Based on this the maximum theoretical efficiency of the prototype engine can be calculated to be 19% for a best case scenario of Th = 90°C and Tc = 20°C, or in the worst case scenario of Th = 50°C and Tc = 20°C the efficiency cannot be any higher than 9%. The actual efficiency will be significantly lower than this due to heat losses and stray conduction paths, an imperfect regenerator, dead space and internal friction of bearings and seals. These factors could reduce the actual engine efficiency to around one third of the Carnot efficiency, or somewhere between 3% and 7% overall. It is suggested in literature [24] that a multiplier of 0.58 to the Carnot efficiency would be an optimistic figure; hence a multiplier of 0.33 is considered a modest and conservative estimate. The electric generator will add another loss to this figure so the actual efficiency of heat input to electrical power output will be about 90% of the previous figure. A nominal value of 5% total efficiency is used subsequently for estimations and calculations.

To test the multiplier figure of 0.58 for accuracy it was applied to several engines with known operating temperatures and efficiencies. The results of these calculations are shown in Table 1 below, where it is suggested that the multiplier value is reasonably accurate.

Table 1: Indicated and calculated efficiencies for three engines with published data


Heater temp (K)

Cooler temp (K)





Theoretical efficiency (0.58 x nc)

STM 4-120 RH






Philips 400 HP/CYL






Sunpower EG — 1000






At this point, Equation 41 can be rearranged and solved to find an estimate of the expansion space volume needed to reach the target power output based on the Beale number. Choosing an average Beale number of 0.005 (from Section 3.1.3) and solving for a nominal output of 500 W:

P0 500 3

Ve = =——————— = 5000 cm3 = 51

E Bnpf 0.005 x 10 x 2

Which means the piston must sweep about 5 litres of volume. From Equation 40 the volume ratio VR can be calculated and this will then give an indication of the size of displacer chamber needed.

AT / 70

( AT ( 70

М1+йооН1 + йоо) = 1-06

Vmax Vc + Ve

Vmin Vc

подпись: vmax vc + ve
vmin vc



Ve 5000

Vc =—————————————— — = = 83,000 cm3 = 83 I

C Vn — 1 0.06 ‘

So for a maximum expected temperature difference of 70°C the volume of the displacer chamber (the compression space) should be about 83 litres. The volume of the compression space is fixed once chosen, and since the engine also should run on temperatures less than 70°C, a value of around 130 litres is more appropriate. This allows the engine to run ideally on temperature differences of around 40-50°C, and if the stroke is reduced (thus reducing both VR and Ve) then the temperature difference should be able to drop to around 20°C while still maintaining a close to optimal volume ratio. This does leave the engine somewhat short on compression towards the higher end of its operating temperature range, however in the interests of researching a low temperature differential engine it was considered more beneficial to skew the design in this direction and investigate lower temperature differential operating limits.

After examining available sizes in steam pipes and other available pipes in large diameters it was found the biggest readily available diameter size was a nominal bore of 800 mm (813 od x 10.0 wt). This spiral pipe was chosen for use as displacer chamber, giving an internal diameter dc of 793 mm. To calculate the required length Lc of the displacer chamber for a swept volume Vc of 130 litres:

1 /^c



3Vc 3 x 0.13

Lc =———- 2 = 2 = 0.79 m ~ 800 mm

C (dc2 (0.7932

Nl) n 2 )

Hence a value of 800 mm was used for the length of the displacer chamber. Note the factor of three used in the equation is because the displacer and heat exchangers take up a third of the volume each, leaving one third of the remaining volume as swept volume.

The cylinder in which the power piston resides needs to be a honed tube, i. e. perfectly round and smooth inside. Available sizes of honed tube are somewhat limited, with 228 mm being the largest readily available size. A cylinder of this diameter has a cross-sectional area of 0.0408 m2, meaning a stroke of only 122 mm is needed to obtain the desired expansion space volume of 5 litres. It was decided to use a maximum possible stroke length of 150 mm to allow some room for adjustment on either side. This would allow the use of higher temperatures through a higher volume ratio.

A stroke length of 150 mm dictates the length of the cylinder (stroke length + height of piston) and the size of the chamber in which the crankshaft resides. This chamber is also made of steam pipe similar to the displacer chamber. A pipe with a nominal bore of 450 mm was chosen for this as it provides adequate clearance for the crank.

Heat Exchangers

Given the cylindrical nature of the displacer chamber and the allowable space for the two heat exchangers, the exchangers are necessarily a wedge shape. Because the engine is pressurised and enclosed the only real option for getting heat in and out is via a pumped liquid. Water was chosen as the fluid as it is cheap and accessible, easy to pump, is suitable at the temperatures chosen and it has a very good heat transfer coefficient, as seen in Table

2. Additives such as antifreeze (ethylene glycol) may be added to the water to raise its boiling point if a heater temperature above 100°C was desired.

Table 2: Estimated heat transfer coefficients of liquids flowing inside tubes (fluid velocity about 1 m/s) [23]


Ai, W/m2 K

Cooling water

2000 — 4000

Paraffins (C6H14 — C11H24)

800 — 1200


1000 — 1500


1400 — 1700


1100 — 1500


600 — 800


400 — 600

At this stage of the design process it is clear what shape the heat exchangers need to be to fit into the engine. An early Solidworks™ rendering of a heat exchanger is pictured in Figure 52, which shows the general shape of the exchanger and its tube and fin arrangement. Air flow, forced by the displacer, will be in parallel with the fins.

The heat exchangers must offer a significant surface area in order to transfer the necessary amount of heat into and out of the engine to produce 500 W at the low predicted efficiency. If efficiency is 5% as estimated in Section 3.1.3, then the heater will need to be able to shift 10 kW of heat into the engine.

Since the shape of the exchanger fins is already determined, the surface area for each fin can be calculated. Ignoring the holes, the wedge shape is 40° of a circle whose diameter is that of the inside of the displacer chamber, namely 793 mm, with the inner circle radius subtracted. This value was not yet known but estimated at 50 mm. This gives an area per fin of:

First Order Design

Figure 52: Initial Solidworks® rendering of heat exchanger

Using the NTU method of finding heat transfer as outlined in Section, and making the following assumptions, the heat transfer for this heat exchanger can be calculated.


• Water flows in the heater pipes, at a rate of 100 ml/s (equivalent to 0.1 kg/s).

• The calculations assume air as the working fluid, pressurised to 1.0 MPa (10 times

Atmospheric pressure)

• The displacer is operating at 2 Hz, effectively forcing the gas volume of 130 litres

Through the exchangers 4 times per second, giving a gas flow rate of 0.52 m3/s and a mass flow rate of 6 kg/s. This also gives a gas velocity of 2.2 m/s.

• The overall heat transfer coefficient from liquid to gas is 30 W/m2K, as read off the

Graph in Figure 24 for a velocity of 2.2 m/s and ai of 3000.

• There are 150 fins per heat exchanger giving a total surface area for heat transfer, Ao,

Of 8.1 m2.

• The inlet temperature of the water to the heater is 90°C.

• The inlet temperature of the cold air to the heater is 20°C.

Specific heats: cair = 1000 J/kgK and cwater = 4200 J/kgK

Densities: pair = 12 kg/m3 (at 10 times atmospheric pressure) and pwater = 1000 kg/m3 Volumetric flow rates: Vair = 0.5 m3/s and Vwater = 0.001 m3/s Calculating mass flow rates:

TOC o "1-5" h z ^air Vair X Pair 6 ^3/s (46)

^water Vwater X pwater 0^ kg/s (47)

And calculating the heat capacity rates:

Cair ^air X ^air 6000 J/Ks Cmax (48)

^water ^water x ^water 420J/Ks Cmin


And combining 48 and 49:

C = Bmin = 0.07 (50)



Calculating NTU from Equation 5:

NTtf = -^ = 0.58

C ■


Theoretical maximum heat transfer from Equation 7:

Qmax = Cmin(^h, i _ ^c. i) = 29.4 fcW

Exchanger effectiveness from Equation 6:

1 _ p(-NTO(l-C))

Ј =———————— = 0 43

Ј 1 _ Ce(-WTU(l-C)) 0.43

And finally, total heat transfer from Equation 8:

Q = Qmax x Ј = 12.6 fcW

So with the previous assumptions, 12.6 kW of heat can be transferred from the water into the air. Given the accuracy of these calculations this number is acceptable. Further calculations show that increasing the flow rate of water proportionally increases the theoretical maximum heat transfer but reduces the effectiveness, leading only to a marginal increase in overall heat transfer.


The regenerator for this design is a modular element, meaning it can be pulled out and changed easily to a different regenerator to enable experimentation with different regenerator types. The shape and size of the regenerator is exactly the same as that of the heat exchangers, and it is situated in between the heater and cooler.

Since the size and shape of the regenerator is predetermined, the fill material and its packing density is the only variable to adjust. Table 3 and Table 4 list experimental results for six different regenerator packing materials. The best results in terms of both regenerator effectiveness and indicated engine efficiency are obtained from mild steel wool. There appears to be some discrepancy in the results in terms of regenerator effectiveness not correlating with overall engine efficiency. This must be attributed to dead space or pressure drop changes that are not immediately obvious. For instance, kao wool (a product used as high temperature insulation, often as a replacement for asbestos) is listed as having the highest heat capacity and good regenerator effectiveness, however it yields low engine efficiency, whereas stainless steel wool has a relatively low heat capacity and regenerator effectiveness but it still yields reasonable engine efficiency.

Regenerator material

Fibre Diameter

(M *)

Denei ty (g*/ec)

Matrix Weight (g*)



Thermal Maaa of matrix (3/*C)

NtId Steel Wool






Stslnlin Sttil Wool






Stainleee Steel Math






Silica wool






Glaae fibre






Као wool






Table 4: Measured effectiveness of 6 regenerator materials in a small air charged test engine [25]

Regenerator Material

Th ( к)

T*h < K>

Тгк< K)

Tk < K>

Regenerator E f fee tivaneaa(?C)



Mild Steel Wool







Stainlea» Steal Wool







Stainleea Steal Aaah







Silica Waal







Glaee fibre







Kao Wool







Based on these tables, it seems that mild steel wool is a good place to start for testing of the regenerator. The regenerator will be constructed as a framework with some mesh walls to enclose the steel wool within, and ideally some sort of thermal insulation layer to prevent direct heat conduction through the regenerator (this is a direct path for heat loss from heater to cooler).

The volume of the regenerator matrix to be packed is about 40 litres or 40,000 cm3. The density of steel wool is typically around 0.1 g/cm3 when unpacked [43] as opposed to the density of mild steel itself at 7.8 g/cm3 (as according to Table 2). This will give a total mass for the regenerator matrix of 4 kg.

Using the method outlined in Section 2.1.4 the thermal properties of this matrix can be calculated. Firstly, the porosity and free-flow area are calculated:

Mm 4

E = 1——— — = 1———————- = 0.987

PmYreg 7800 x 0.04

Aff = eAm = 0.987 x 0.24 = 0.237 m2 Then the mass flow rate and mass flow rate per unit area are calculated: m = 2 x ns x = 2 x 2 x 1.56 = 6.24 fcg/s

M 6.24 2

M»=^=0:237 =[1] ‘3/m s

Then calculating heat transfer surface area, hydraulic diameter and Reynolds number:

/1_eKreg /1_ 0.987 0.04 2

5reo =41————————————————————- )-^ = 41 1 x -= 70 m2

Re5 V e / dw V 0.98M 30 x 10-6

4Kreo 4 x 0.04

Dh =———— =———— = 2.86 mm

5 70

‘-’reg ‘ u

M0dh 26.3 x 0.00286

Nre = —^ = —77^———r— = 4,066 re Mair 1.85 x 10-5

Once the Reynolds number is known, the value for convective heat transfer coefficient can be found. The value of Y3 is first looked up on a table such as Table 5, and then the value of h is found by solving the equation. For the value of porosity of 0.987 the value of 0.95 in Table 5 is accurate enough to use, with the resulting Y3 value being 2.75 Nre’0’48, which is calculated as being 0.051. Then h is found:

H = r3.m. cp. Wpr-2/3 = 0.051 x 6.24 x 1000 x 1.27 = 404^/m2^

Table 5: Table of selected Y3 values for known values of porosity and Reynolds number [25]

E Nre < 105

0.95 Y3 = 2.75 N/0’48

0.60 Y3 = 0.5 Nre-0396

A 4.53 ^ = A + 2 = 4.53 + 2 = °’69

For confirmation, it is possible to check the result graphically using the computed value of reduced period.

H. Sreo. t 4°4×7°x°.25

N =—— =——————————- = 4.°4

Rnm. c 4×437

Using the reduced period line on Hausen’s curves that most closely represents 4.04 (i. e. 0) it is shown that the values of A and nreg correspond closely, with the value for efficiency read off the graph of 0.67 being close to 0.69.

First Order Design

Figure 53: Hausen’s curves with the figures for the regenerator design marked in

In addition to these calculations it is also possible to predict the power loss caused by the pressure drop across the regenerator by following the method in Section

Firstly, by referring to the graph in Figure 44 the friction factor Cw is found using the Reynolds number and porosity calculated for the regenerator in the previous section. The value obtained is 0.4. One other value must be found, the length of the regenerator in the flow direction, Lreg. This is a somewhat difficult value to find due to the regenerator not having a constant ‘length’ owing to its wedge shape. The value has been estimated at 10 cm for this calculation, which is an intermediate value so should be a reasonable approximation. Now Equation 38 can be solved using the values already found:

Ap =—— ——— =——- 72rH—————————— ф = 1630 Pa

2rhp 2 x (2^) x 10-3 x 12

And from Equation 39, power loss is:

Pfoss = 2nsAp!4 = 40 Watts

This figure is acceptable for a first design iteration. It indicates that about 40 Watts of power is required by the stepper motor(s) just to overcome the fluid friction involved in pushing the gas through the regenerator matrix. Other factors contributing to this power requirement will be the fluid friction associated with forcing the gas through the heat exchangers, and the dynamic friction of the sliding seal of the displacer against the chamber wall. Power loss through fluid friction in the heat exchangers will be of an order of magnitude lower than that calculated for the regenerator due to the relatively large spacing between fins (which gives it a very large effective hydraulic radius, leading to a low pressure drop as per Equation 38.

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