#### Механизм работы и самая малая модель мотора стирлинга

Август 17th, 2014

Broadly speaking, a Stirling Engine will fall under one of the four main categories described here, being either alpha, beta, gamma or free piston configurations. There are

Other obscure types of engine such as the thermoacoustic engine and the fluidyne engine which were both discussed briefly in Section 1.4.

Alpha (a) Configuration

The alpha configuration uses no displacer and two power pistons connected in series by a heater, cooler and regenerator. There are two cylinders, the expansion space (hot cylinder) and compression space (cold cylinder). It is a mechanically simple engine and typically produces a high power-to-volume ratio, however there are often problems related to the sealing of the expansion piston under high temperatures. The alpha engine pictured in Figure 29 is a horizontally opposed type, which has the smallest dead space but requires rather length and complicated linkages to join the pistons to the crankshaft. Another variant is the ‘V’ design where both pistons are arranged in a V formation and are attached at a common point on the crankshaft. This means the heater and cooler are separated which reduces thermal shorting losses, however it increases dead space through the need to have an interconnecting passageway, containing the regenerator, between them.

Figure 29: Alpha engine configuration [28] |

This design lends itself well to the double-acting configuration as pictured in Figure 30. In this configuration, there is only one piston per cylinder, but the space both in front and behind it form the expansion and compression spaces respectively. This type of engine was researched a lot by Philips from the 1940’s until the 1960’s and was quite successful, having a good power to weight ratio and capable of very respectable outputs.

Figure 30: 4-cylinder double-acting alpha engine [28] |

Beta (P) Configuration

The beta configuration features both the piston and displacer working inside the same cylinder. This makes it quite compact and there is typically a minimal amount of dead space as there are no interconnecting passageways. It is mechanically simple as both piston and displacer are connected at a common point on the crankshaft, with the only difficulty arising from the fact that the connecting shaft for the displacer must pass through the piston where it must make a pressure-tight seal.

Figure 31: Beta engine configuration [28] |

Gamma (y) Configuration

The gamma configuration is similar to that of the beta type in that it uses a piston and a displacer both directly connected to a common crankshaft. The difference is that they are not in the same cylinder, meaning that the problems of sealing the displacer rod through the piston are avoided. The downside for this is the introduction of a gas flow passageway which increases dead space in the engine. This configuration is probably the easiest to build, especially out of budget materials. It is commonly seen in LTD ‘pancake’ engines (see Section 2.2.5

Figure 32: Gamma engine configuration [28] |

Free Piston Configuration

The invention of the free piston Stirling engine is generally credited to W. T. Beale who first built it in the 1960’s as a solution to overcoming problems with lubricating the crankshaft of a traditional engine [29].

Flibban Condoclof Figure 33: Section view of a Sunpower 100W free-piston Stirling engine [30] |

Free piston Stirling engines are different in that there is no crankshaft and the piston and displacer are not attached to each other (i. e. they are ‘free’). The motion of the piston and displacer is controlled by fluid forces and usually by a spring of some sort. Energy is typically extracted from the engine by means of a linear alternator, though sometimes the piston motion is used directly in pumping applications. The advantages of a free-piston engine are fewer moving parts, meaning greater reliability and simplicity, which also reduces production costs. They can also be compact and lightweight by comparison with more traditional designs. Non-contact gas bearings and planar springs can bring friction down to almost zero in these designs.

Low Temperature Differential (LTD) Engines

The LTD Stirling Engine is not a strict classification of engine type, but since it is of particular relevance to this thesis it is discussed in some detail here. There is no strict definition of what constitutes an LTD engine but it can be taken as being something running on a temperature difference of under 100°C. Anything running at these sorts of temperatures must typically use a heat source other than some sort of combustion, which will typically be at a temperature of several hundred degrees.

Figure 34: Illustration of how temperature difference affects geometry [31] |

Figure 34 provides a useful insight into how the temperature difference affects the geometry and proportions of a Stirling engine. With a high temperature difference it is necessary to maintain a relatively long separation between the hot and cold ends in order to avoid excessive heat loss through short conduction paths, while the heating and cooling surface area is less critical. An LTD engine on the other hand requires a large surface area for heat transfer to allow for adequate heating and cooling of the gas at such low temperatures. There is also less heat conduction from the hot to the cold end so the distance here can be shorter.

X. — c ctl endor year Figure 35: Chart of progress in LTD engines from 1980 to 1990 [9] |

Figure 35 charts the progress, in terms of required temperature difference to operate, of the LTD engine through 10 years of development by Ivo Kolin and James Senft, the two pioneers of this type of engine. It shows what a difference 10 years of research and development can do for an idea, going from 44°C to just 0.5°C.

Basics

Gas Laws

As previously mentioned, the underlying principle of the Stirling Engine, or in fact any hotair engine, is that of a gas expanding when heated or contracting when cooled. This principle has been known for thousands of years, but was not really understood until the science of thermodynamics began to be explored around the 17th century. In 1662 Irish physicist Robert Boyle published results of several experiments he had performed on air trapped between mercury in a J-shaped glass tube [32]. Boyle’s Law stated that if the pressure on the gas increased, its volume decreased proportionally and vice-versa; hence the product of pressure and volume remained a constant. Mathematically this can be expressed as the following:

PV = k (20)

In 1834 French physicist and engineer Benoоt Paul Йmile Clapeyron produced the ideal gas law, an equation based on a combination of two other gas laws; Avogadro’s law and the combined gas law.

Avogadro’s law (1811) states that “Equal volumes of ideal or perfect gases, at the same temperature and pressure, contain the same number of particles, or molecules.” The combined gas law is derived from Boyle’s law, as well as two other famous gas laws; Gay — Lussac’s law (1809) and Charles’ law (1802). It states that “The ratio between the pressure — volume constant and the temperature of a system remains constant.”

Using these laws and the gas constant, R, (8.314 J/K. mol) the ideal gas law can be expressed mathematically:

PV = nRT (21)

This equation can be applied to the ideal form of any gas to describe its behaviour under variations in temperature, pressure and volume. An ideal gas is an approximation to a real gas; it is “a model of matter in which the molecules are treated as non-interacting point particles which are engaged in a random motion that obeys conservation of energy. At standard temperature and pressure, most real gases behave qualitatively like an ideal gas” [33]. Under the conditions experienced in a Stirling Engine the ideal gas approximation generally holds true — the deviation from a real gas is expressed using the ‘compressibility factor’. This factor increases at extreme pressure or temperature, however experimental data shows that to achieve only a 1% deviation from the ideal gas approximation requires a pressure of at least 4 MPa at 400 K or 2 MPa at 1000 K [34].

Reynolds Number

The Reynolds number, Nre, is a dimensionless number used to characterize the condition of fluid flow. It is independent of the type of fluid, meaning that a certain Reynolds Number defines a specific velocity profile, whether the fluid be air, helium or nitrogen [14]. The Reynolds Number is used to characterize different flow regimes, such as laminar or turbulent flow: laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion, while turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce random eddies, vortices and other flow fluctuations [35]. Mathematically, Nre is defined using the density of the fluid in question, p (kg/m3), its mean mass velocity, u (= m/pAff) (m/s), hydraulic radius, rh(= Aff/Pw) (m) and coefficient of dynamic viscosity, p (Pa. s):

4p. u. rh 4m. rh

Nre = ———— h = —-h————————— (22)

M M — Aff

Prandtl Number

The Prandtl number Pr is a dimensionless number approximating the ratio of momentum diffusivity (kinematic viscosity) and thermal diffusivity. It is expressed as:

C-pM

(23) |

N =

I Vpr

For gases of interest in regards to a Stirling engine the Prandtl number is near enough to 1 for it be approximated as this in most equations though a value of 0.7 is typically more accurate for air [26].

The Stirling Cycle

Thermodynamically speaking, heat is a raw material which has to be transformed into the final product, mechanical work, by means of a power cycle. The Stirling cycle is actually only one of many different possible power cycles that any kind of engine, theoretical or real, can possess. These cycles are all named after their author, and some of the better known cycles are listed here:

• 1690 — Papin — 2 isobaric and 2 isochoric processes

• 1807 — Cayley — 2 isobaric and 2 polytropic processes

• 1816 — Stirling — 2 isothermal and 2 isochoric processes

• 1824 — Carnot — 2 isothermal and 2 adiabatic processes

• 1852 — Joule — 2 isobaric and 2 adiabatic processes

• 1853 — Ericsson — 2 isobaric and 2 isothermal processes

• 1867 — Otto — 2 adiabatic and 2 isochoric processes

• 1873 — Reitlinger — 2 isothermal and 2 polytropic processes

• 1893 — Diesel — 2 adiabatic, 1 isobaric, 1 isometric process

• 1894 — Lorenz — 2 polytropic and 2 adiabatic processes

• 1896 — Crossley — 2 polytropic and 2 isochoric processes

Each cycle has its own unique combination of isobaric (constant pressure), isochoric (constant volume), isothermal (constant temperature), polytropic (pressure and volume obey a curve relation) or adiabatic (heating or cooling with no change in entropy) processes to form a cycle. The Stirling cycle contains two isothermal and two isochoric processes, which can be illustrated by thep-Vand T-s diagrams in Figure 36.

The p-V diagram is an indication of how much pressure the working gas in under at any given moment in the cycle, relative to the total engine volume. As the changes in pressure cause the power piston to move in or out, the total engine volume is altered.

The T-s diagram plots gas temperature against the gas entropy, s. Entropy, measured in joules per Kelvin (J/K), is a measure of the unavailability of a system’s energy to do work [36]. In other words, under maximum entropy, there is a minimum of energy available for doing work.

The Ideal Stirling Cycle

In the following analysis of the ideal Stirling cycle, the following assumptions are made: [11]

1) The working substance is a perfect gas.

2) Flow resistance everywhere is zero: pressure always uniform throughout engine.

3) Regenerator loss zero: the gas enters the hot end at exactly the heater temperature and enters the cold end at exactly the cooler temperature.

4) Zero heat loss by conduction, etc.; all heat added to engine passes to the gas.

5) Isothermal expansion and compression; zero temperature drop across heat-exchange surfaces. At each point in the engine the temperature has some constant value.

6) Volumes of expansion space and of compression space vary in the ideal discontinuous (non-sinusoidal) manner such as illustrated in Figure 43, Section 2.3.2.2.

7) Mechanical friction assumed zero.

8) Dead space is assumed to be zero

Figure 36: Idealized Stirling Cycle p-V and T-s diagrams [37] |

Figure 37: P-type engine shown at the four points in the Stirling cycle [11] |

I. (1 ^ 2) Isothermal compression: This stage occurs as the power piston is travelling inwards, compressing the gas and reducing the overall gas volume which in turn raises its pressure. The amount of work done compressing the gas, Wc, is equal to the area under the p-V diagram between points 1 and 2. This work is extracted from the flywheel in a real engine. During this part of the cycle, heat is removed by the cooler to the outside environment. The amount of heat removed, Qc, is equal to the area under the T-s diagram between points 1 and 2.

Figure 38: Work done compressing gas, Wc, and heat removed Qc |

II. (2 ^ 3) Isochoric heating: At this point, the piston is at TDC (i. e. at its most inwards point) and remains approximately still, keeping the volume constant. Heat is added to the gas from the regenerator as the displacer moves through its range of motion, bringing the gas temperature up from Tc to Th and the gas pressure up to pmax. At the start of this part of the cycle, entropy is at a minimum value meaning that there is a maximum amount of energy available to do work. This makes sense given that the gas is compressed and ready to do work on the piston when the available energy stored in the regenerator is released. The amount of heat added from the regenerator, Qr1, is equal to the area under the T-s diagram between points 2 and 3. No work is done during this part of the cycle.

Figure 39: Heat added to regenerator, Qrl |

The amount of heat added, Qe, is equal to the area under the T-s diagram between points 3 and 4.

Figure 40: Work performed by expanding gas, We, and heat added Qe |

IV. (4 ^ 1) Isochoric cooling: During this part of the cycle, the piston is at BDC (i. e. at its outermost point) and remains approximately still, keeping the volume constant. Heat is absorbed from the gas by the regenerator as the displacer moves through its range of motion, bringing the gas temperature down from Th to Tc and the gas pressure down to pmin. The amount of heat added to the regenerator, Qr2, is equal to the area under the T-s diagram between points 4 and 1 and should be equal to Qri, the amount of heat removed from the regenerator earlier in the cycle. No work is done during this part of the cycle.

* Figure 41: Heat absorbed from regenerator, Qr2 |

Comparing Figure 38 and Figure 40 it can be seen that the amount of work spent compressing the gas at a low temperature, Wc, is less than the amount of work performed by the hot expanding gas on the piston, We, meaning that there is net work per cycle transferred to the crankshaft. This amount of net work, Wnet, is shown in Figure 42 as the area inside the p-V diagram. The first law of thermodynamics states that the amount of net

Heat added to the cycle must be equal to this value. The amount of net heat is shown on the T-s diagram in Figure 42 as the enclosed area, Qnet.

Figure 42: Net work per cycle, Wnet, and net heat input per cycle, Qnet |

The amount of net work per cycle can be calculated by the following method. Firstly, the work done on the gas during the expansion and compression stages is calculated:

TOC o "1-5" h z r2 v ■

Wc = I pdv = nRTc InV^ (24)

Jl vmax

-I |

4 Vmax (25)

We = I pdv = nRTh In

Vmin

Also, because there are no losses to consider, the work done for each part of the cycle is the same as the heat added:

Wc = Qc (26)

And similarly:

We = Qe (27)

And then the net work can be calculated as the sum of the two. The value of Wc will be negative due to the sign convention.

Wnet = Wc + We=nR(Th — Tc)n]^ (28)

*min

Now, the total cycle efficiency is defined as the ratio between the total net work output and the total heat input:

(29) |

Wnet

‘ Qe

Substituting (28), (27) and (25) into (29):

NR{Th-Tc)r^ (Th-T) v = ^^ = ((ill-bd = Vc (30)

I’m

NRT^m^

H Vmin

Which shows that an ideal Stirling cycle with a perfect regenerator is equal to the maximum theoretical Carnot efficiency, nc-

The Non-Ideal Stirling Cycle

For the reasons mentioned at the start of Section 2.3.2.1, the real Stirling cycle will fall well short of the expectations of the ideal cycle. Of these eight factors, the three main non-ideal contributors — imperfect regeneration, non-sinusoidal piston motion and dead space volume — will be analysed in more detail. The effect of pressure drop across the regenerator and heat exchangers was discussed in Section 2.1.4 and the effects of heat loss and mechanical friction are mentioned in Section 2.3.3.

Referring to Figure 41, the heat absorbed by the regenerator can be expressed as a product of the temperature difference, gas mass and specific heat at constant volume, cv:

Qr2=mcv(Th-Tc) (31)

If the effectiveness of the regenerator, qreg, is considered, then the values of Qr1 and Qr2 in Figure 39 and Figure 41 are no longer equal. The shortfall in heat able to be supplied back from the regenerator, Qo, must be made up for by the heater.

Qo = Qrl — VregQr2 = (1 — tfreg^ mcv(Th — Tc) (32)

The overall efficiency must now allow for the extra heat input due to the imperfect regenerator:

W t nR(Th-Tc)n^

TOC o "1-5" h z „ net __________________________ vmin___________ /"jox

= n + n = v (33)

Ve n° nRTh lnVm^Ј+ (1 — Vreg) rncv{Th — TO

• m in

Which can be rearranged to give the following:

1 ^ (1 Vreg^mcv(Th Tc) (34)

NRTuln"^

Vmin

Which, given that Th > Tc and Vmax > Vmin, shows that the overall efficiency is less than the Carnot efficiency if the regenerator effectiveness is less than 1, or to put it another way:

V <Vc (for Vreg < 1) and ri = ric (for Vreg = 1) (35)

Kolin [38] states that: “Due to the influences of dead space and sinusoidal piston motion, the actual indicator [Stirling cycle p-V and T-s] diagram is rounded and smaller than it should be” This means that less net work is produced per cycle as the area enclosed is smaller. Figure 43 shows what the ideal piston and displacer motion would look like. The ideal piston motion is quasi-sinusoidal, being a symmetrical waveform with equal time spent idle at the top and bottom of the range of motion. The displacer motion is somewhat different, and hard to quantify. When the piston is at its outmost point of travel, or BDC, the displacer should move through its range of motion to shift all the gas to the compression space in the time that the piston is idle at BDC. It should then remain still whilst the piston compresses the gas, before moving the gas back into the expansion space in the time it takes for the piston to pause at TDC then move back to BDC.

Dead space volume, Vs, is defined as the volume of gas that does not take part in the cycle. It is the volume of all the ‘free space’ in the regenerator, heat exchangers and clearances and interconnecting ducts. The effect of dead space volume is to decrease the work done per cycle in almost linear proportion to the percentage of dead volume in the cycle [24]. To this end, it is useful to define a dead space ratio, 5:

(36) |

6 = V±

Vsw

Where Vd is the total dead space volume in the engine and Vsw is the total swept volume of the engine — that is, the volume of gas swept by the displacer. The dead space ratio can be

used to find the Schmidt Factor, Fs, which predicts the reduction in power compared to the ideal situation of no dead space. The Schmidt Factor is equivalent to the ratio of the actual area enclosed by the p-V diagram to the area of the ideal diagram, in other words the area of the oval in the p-V diagram in Figure 43 divided by the area of the trapezium. The Schmidt Factor is given by the following empirical formula:

(37) |

Fs = 0.74 — 0.68 S

Multiplying the Schmidt Factor with calculations of power under the ideal situation will give a good estimate of actual power produced [9]. The factor 0.74 comes from the reduction in area associated with sinusoidal motion. If discontinuous motion is employed this factor can be increased towards unity as appropriate by the degree of discontinuity introduced.

Figure 43: Piston and displacer motion, discontinuous (left) and sinusoidal (right) and resulting p-V diagrams [11] |

Due to losses associated with real world conditions compared with ideal situations used for analytical purposes, it is impossible for a practical

Flow Losses

Flow losses, or pumping losses, are caused by the resistance of air flow through the heater, cooler and regenerator. This can be found using the pressure drop across the regenerator, Ap, [24], which is calculated from:

W,0 CwLreq /oo

Ap =———————————————— — (38)

2 rhp

Where Cw is a friction factor which is a function of the Reynolds number (see Section 2.3.1.2). Its value can be estimated using the chart of experimental values in Figure 44, once the Reynolds number and porosity are found. Lreg is the length of the regenerator in the flow direction and rh is the hydraulic radius, equal to the hydraulic diameter dh/4. The definitions of hydraulic diameter, mass flow rate per unit area (m0)and porosity are found in Section 2.1.4.

After finding the pressure drop, power loss is given by:

Pioss = 2nsApVe (39)

Where ns is the engine speed in Hz and Ve is the expansion space volume. 1 >4 *(10 i 34 5ilo‘ * S 4 ( I l(f t 14 ( 1 ! 34 6S.0* Figure 44: Experimental data of friction factor vs. Reynolds number for wire screen matrices [24] |

It is more difficult to predict the losses in the heat exchangers as each different exchanger design will have a different method of finding the pressure drop. The pressure drop across heat exchangers is usually small in comparison with the regenerator [24] and in numerical examples is typically around 10% of the value of regenerator loss [39].

Friction Losses

Friction losses are caused by all contact points within the engine, that being bearings, seals and piston rings. Calculating friction losses is very difficult and the only really reliable method is to measure them [24].

Heat Losses

Heat losses are losses that will result in additional heat needing to be added to the input to maintain the same output. They fall under one of the following categories:

Shuttle Loss:

Shuttle loss is caused by the displacer moving through a temperature gradient, such that it absorbs heat when it is occupying the hot space and loses the heat into the cold space at the other end of its cycle. It is difficult to calculate the actual loss, but logic and intuition will serve to understand the concept; losses will be reduced through an increase in clearance between displacer and cylinder or a longer displacer (resulting in a less steep temperature gradient), or through a reduction of the cylinder diameter or using a less thermally conductive gas.

Heat Conduction:

This includes all heat conduction paths in the engine between the hot and cold spaces. Heat is mostly conducted through solid, thermally conductive material (i. e. metal) but also through gas conduction. These paths differ between engine designs, and are typically greater in smaller engines where conduction paths are shorter. Heat is also conducted away from the hot space not just to the cold space but to the ambient sink (outside environment of the engine). Again, logic will show that heat conduction is proportional to temperature difference and conductivity, and inversely proportional to conduction path length.

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