Stirling Engine Background

“In all places where there exists a difference of temperature, there can be a production of motive power. ” — Sadi Carnot, 1824

Stirling Engine Components

A Stirling Engine always has 5 main parts to it that are essential to its operation. Figure 17 below shows one of the Stirling brothers’ early engines where the regenerator and cooler are separated from the body of the engine. This makes it simple to see and understand the role of the 5 main parts.

Stirling Engine Background

Figure 17: Stirling brothers’ 4th engine, built in 1840, main parts labelled [9]

The heat exchangers are responsible for transferring all the heat in-to and out of the engine. There are always two heat exchangers, one to heat the working gas and one to cool it. In Figure 17 the hot-side heat exchanger is simply the bottom of the engine, situated over a furnace and the cold-side heat exchanger is a tubular type heat exchanger using water as the cooling fluid. The working gas in the engine flows over and around the tubes containing cold water, and is subsequently cooled. There are many different types of heat exchanger and countless configurations possible, but the basic principle is to have two fluids, one hot and one cold, interacting by some thermally conductive configuration that will cause the outlet temperature of both fluids to approach some point in between both inlet temperatures. Usually there is no direct contact between the fluids; they are separated by some medium, usually a metal of high thermal conductivity. There are also some cases of direct contact heat exchangers such as cooling towers.

Some of the main ways to categorize a heat exchanger are by its design type, its flow arrangement, whether it is single or multiple pass and whether it is regenerative or not.

Common Heat Exchanger Types

Shell and Tube Heat Exchangers

Perhaps the most common type is the shell and tube heat exchanger. They consist of a tube bundle carrying one fluid, enclosed within a shell carrying the other fluid. The tube bundles are connected at the ends, depending on the flow arrangement (see Section 2.1.1.2), to run either back and forth in a series arrangement, in parallel, or a combination of both. Due to the nature of their construction and their shape, shell and tube exchangers are robust and suitable for high pressure and temperature systems [18].

Stirling Engine Background

Figure 18: Typical shell and tube heat exchanger [19]

Tube and Fin Heat Exchangers

Tube and fin heat exchangers are similar in principle to shell and tube types, except that there is no shell covering the tubes so they are more suited to liquid-to-air applications such as car radiators. The tubes are attached to fins which conduct the heat away from (or into) the tubes by effectively increasing their conduction surface area. It is important that the thermal contact between the fins and the tubes is good; hence they are usually bonded in a high thermal conductivity manner (for example brazing).Sometimes the thermal contact is improved by using L-footed joins, where the fin is bent at a right angle where it joins, increasing the area of contact.

Hot Fluid

Stirling Engine Background

Coolant

Figure 19: Tube and fin heat exchanger [20]

Plate and Fin heat Exchangers

In the plate and fin exchanger design, fluid flows through stacked plates, alternating between the hot and cold fluid. Fins sandwiched between the plates increase surface area for heat transfer and space the plates apart. The fins can be made quite dense which gives the plate and fin exchanger type good performance from a small size, however a downside is that it is difficult to make these suited to high pressure differentials between the two fluids.

Stirling Engine Background

— N n N.

**——————- Hot

*2 Flow

Figure 20: Plate and fin heat exchanger [20]

The flow arrangement of a heat exchanger can be put into one of three categories; parallel flow, counter flow and cross flow.

Parallel Flow Arrangement:

9ETF

1

Stirling Engine Background

T 1——————————— 1—

Bl’F INLET OUTLET

Figure 21: Parallel flow heat exchange [19]

Figure 21 depicts a parallel flow arrangement, where both the hot and cold fluids enter from the left and run parallel against each other before exiting from the right hand end. The parallel flow arrangement is ideal for situations when both fluids need to be brought to the same temperature. It has the disadvantages of having high thermal stress at the inlet end where the large temperature difference causes opposing thermal expansion and contraction which can eventually lead to material failure [21]. Also, the cold fluid outlet can never exceed the lowest temperature of the hot fluid, which can be a disadvantage if used as a heater.

Counter Flow Arrangement:

Figure 22 depicts a counter flow arrangement where the cold fluid enters left and exits right, and the hot fluid enters right and exits left. This type is generally better at heat exchange than the parallel flow arrangement with all other things being equal. It is possible for the cold fluid to approach the inlet temperature of the hot fluid unlike the parallel flow and there is also less thermal stress due to more uniform temperature distribution.

BZT

Stirling Engine Background

9Cf F

Figure 22: Counter flow heat exchange [19] 22

Figure 23 depicts a cross flow arrangement, where the fluids flow perpendicular to one another. The performance of this type of exchanger is typically in between that of a parallel type and a counter flow type, and they are often used in applications where one of the fluids changes state (2-phase flow) [22].

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Figure 23: Cross flow heat exchange [19]

Heat Exchanger Performance Analysis

The rate of heat transfer, Q (W), by a heat exchanger is a valuable quantity to know for design and analysis. It is given by the following equation:

Q = U0A0ATlm (1)

Where Uo is the overall heat transfer coefficient (W/m2K), Ao is the effective cross-sectional heat transfer area (m2) and ATim is the log mean temperature difference (K). In order to find the value of Q, the other values must be known. Finding Uo can be done by solving the following equation:

1

Uo=T7±J^ (2)

Ht + h2 + k

Where h] is the convective heat transfer coefficient between the exchanger wall and the hot fluid (W/m2Kj, h2 is the convective heat transfer coefficient between the exchanger wall and the cold fluid (W/m2Kj, tw is the exchanger wall thickness separating the hot and cold fluids (m) and k is the thermal conductivity of the exchanger wall material, (W/m. K) which

Can be looked up in a table (see Appendix B Appendix A). The values for h depend on the

Fluid properties such as viscosity and flow velocity and whether the convection type is natural or forced. Typical values for h are 10-100 W/m2K for air and 500-10000 W/m2K for water. Typical values for Uo can be found in Appendix B.

If insufficient data is available to solve the previous equation for Uo, it is possible to estimate this value graphically. Figure 24 shows such a graph, based on the heat transfer of a single fiinned tube of the given dimensions carrying a hot liquid, being cooled by forced air flow. The value of Uo can be estimated knowing only the velocity of the cooling air, u (m/s), and the value of heat transfer coefficient for the hot fluid, ai (W/m2K), which is found by looking up the value on a table (see Table 2, Section 3.1.5).

Firrhjbe characteristic t

Stirling Engine Background

U [m/п]

Figure 24: Overall heat transfer coefficient for a given size of single tube and fin [23]

The temperature across the heat exchanger profile is not linear, but rather depends on a logarithmic relationship known as the log mean temperature difference or LMTD. If the inlet and outlet temperatures are known for both fluids, then the LMTD can be calculated using the following equation for counter flow exchanger designs:

AT,

Lm

In

(T — t2) (T2 — t1)

(T -12) — (T2 — to

 

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(3)

 

Stirling Engine Background

Or for parallel flow designs:

AT _(J1-t1)-(T2-t2)

AT Im —

Lm ^-ti) (4)

(T2 t2)

Where T is the hot fluid inlet temperature, T2 is the hot fluid outlet temperature, 11 is the cold fluid inlet temperature and t2 is the cold fluid outlet temperature.

In the case of a cross flow exchanger design a correction factor dependant on design factors is applied to the LMTD formula to obtain the correct numerical solution.

Another method for looking at the rate of heat transfer in a heat exchanger is to use the NTU (number of transfer units) method which can be used when insufficient data is available for the LMTD method. In order to use the NTU method, the values of heat capacity rates (product of specific heat capacity C (J/kgK) and mass flow rate in kg/s), Ch and Cc, for both the hot and cold fluids must be known. The smaller one of the two values is denoted as Cmin and the bigger as Cmax. Using the methods mentioned previously for finding Uo and ^o, the value for NTU can then be found:

U0A0

NTU — (5)

Cmin

The value of NTU represents the ability of a given heat exchanger to exchange heat between fluids. Formally, it is the ratio of a given heat exchanger’s capacity for heat exchange by convection per unit temperature difference to the capacity of the flowing fluid to carry that heat away per unit temperature rise [14].

After defining NTU it is possible to find the heat exchanger effectiveness, s, a dimensionless measure of how much heat the exchanger can actually exchange compared to the theoretical maximum value. It is found by the following equation:

1 — p(-NTU(1-C))

TOC o "1-5" h z S — —————————————————- (6)

-C e(~NTU(1~C)) V ‘

Where C=Cmin/Cmax. Next, the maximum theoretical heat transfer amount possible, Qmax, is found by the following equation:

Qmax — Cmin(Th, i — Tc, i) (7)

Where Th, i is the initial (inlet) hot fluid temperature and Tci is the initial cold fluid temperature, both in K. Finally, the total overall heat transfer rate in Watts is found by multiplying the maximum possible heat transfer by the effectiveness of the heat exchanger:

(8)

The heat exchanger design criteria must also take into consideration a number of other factors, such as allowable size and weight, cost, required efficiency, types of fluids to be used and materials available, operating pressures and temperatures.

Displacer

The purpose of the displacer, or displacer piston, is to simply move the working gas around inside the engine. The heat exchangers create two distinct regions inside the engine; a hot region and a cold region. These regions occupy the majority of the volume inside the engine. It is the role of the displacer to shuttle the working gas alternately between these regions in order to alternately heat and cool the gas, giving rise to the necessary expansion and contraction that facilitates the operation of the engine.

Ideal characteristics of a displacer are for it to be lightweight so that it can accelerate and decelerate quickly without drawing too much power from the engine, it should be thermally non-conductive to avoid unwanted heat transfer, it should be rigid and should offer minimal flow resistance while keeping dead-space around its edges to a minimum.

Most existing Stirling engines use a displacer that looks not unlike a piston. The main difference usually between a displacer and a piston is that the displacer is a loose fit inside the cylinder which allows gas to flow past it through the annulus when it moves.

The layout of the displacer in relation to the power piston is often used to classify the Stirling engine into one of three main categories, as discussed in Section 2.2. One of these categories, the alpha type, does not use a displacer but rather two power pistons which are connected in a similar manner as a piston and displacer.

Power piston

The power piston is similar to that of a piston found in an internal combustion engine. Its job is to transmit power created by pressure acting on the piston face to the crankshaft of the engine. The piston slides within a cylinder and is tightly sealed against the cylinder walls by the piston rings in order to maintain the necessary pressure differential across the piston for motive power.

Design criteria for the piston are that it is light weight and perfectly balanced (this is actually achieved by counterbalancing the crankshaft), as well as being made of a material suitable for use at the design temperature. In some cases thermal expansion must be considered where it can mean that the piston expands to the point of seizing in the cylinder.

The piston rings can be made of metal, rubber or other suitable materials. They need to be capable of sealing against the design pressure difference, which is typically quite low for a Stirling engine. In most cases some form of lubricant is used to prevent excessive friction between the cylinder wall and the piston rings, though care must be taken to ensure that the

Lubricant will not vaporize and subsequently condense inside the regenerator, causing it to become blocked and lose effectiveness.

Regenerator

The regenerator is probably the single most studied and talked about feature of a Stirling engine. As mentioned in section 1.1, the regenerator (then referred to as the ‘economiser’) was the main point of Robert Stirling’s 1816 patent. The point of the regenerator is to act as a temporary heat storage element that is able to quickly absorb heat from the hot working fluid and transfer it back again into the cold working fluid. This greatly reduces the amount of thermodynamic work needed to be performed by the heat exchangers and in turn greatly increases the overall efficiency of the engine. John Ericsson stressed the importance of the regenerator when in 1855 he famously said “…. we will show practically that bundles of wire are capable of exerting more force than shiploads of coal… " [14] The graph of Figure 25 illustrates this point to some degree, showing a typical relationship between the effectiveness of a regenerator (with 0 being no regenerator and 1.0 being the perfect regenerator) versus the overall efficiency of a Stirling engine.

Although thorough analysis of a regenerator is very complicated, it is possible to fairly easily calculate its basic performance parameters such as effectiveness and heat

подпись: although thorough analysis of a regenerator is very complicated, it is possible to fairly easily calculate its basic performance parameters such as effectiveness and heatRegenerator design is so complex and involved that it would take many pages of text and equations to begin to describe their thermodynamic operation. It is a balancing act of several factors, namely reducing dead space and flow restriction while maintaining a high level of effectiveness. These criteria are by their very nature contradicting — a high value of effectiveness necessarily means that the regenerative matrix is able to store a lot of heat in it, meaning that it must have a significant amount of material within it. To have a significant amount of material the regenerator must either be large and sparsely packed, meaning a large dead volume is inherent, or else the regenerator must be small and densely packed, meaning a high flow resistance is unavoidable.

Stirling Engine Background

REGENERATOR EFFECTIVENESS Figure 25: Effect of regenerator effectiveness on engine efficiency [24]

Regeneration per cycle, as per the methods outlined in Mansoor [25]. To begin an analysis it is necessary to know either the size and shape of the regenerator matrix and the matrix material parameters, or the amount of heat regeneration per cycle needed Qr and the temperature swing ATS. Assuming the regenerator size, shape and material parameters are known, a quantity called the matrix porosity, e, can be defined. It is the ratio of free space to total space in the regenerator, and is calculated as:

Mm

E = 1 PmVreg (9)

Where Vreg is the regenerator volume, mm is the mass of the matrix and pm is the matrix density. Using the result for e, the free-flow area is found by multiplying with the total cross-sectional area of the matrix ^m.

(10)

подпись: (10)Aff = eAr

Then using the free-flow area, the mass flow rate per unit area, m0, can be calculated:

M

Mo=-r~ (11)

Aff

Where m is the mass flow rate per second, which is given by:

(12)

подпись: (12)

Mf

подпись: mfM = 2xn^x


Where mf is the mass of gas that flows per half cycle and ns is the revolutions per second, equivalent to ns/60. Using m0 the Reynolds number can be calculated as follows:

Modh

Nre =^LjL (13)

Where u is the dynamic viscosity of the fluid and dh is the hydraulic diameter of the matrix, given by:

4V

Dh=-zres — (14)

Oreg

Where Sreg is the surface area for heat transfer of the regenerator matrix. This can be calculated using the porosity, regenerator volume and the diameter of the wire (or fibre), dw, that comprises the matrix:

(15)

подпись: (15)

Jreg

подпись: jreg^reg


From finding the Reynolds number, the value of convective heat transfer coefficient in the matrix, h, can be looked up from a table such as Table 5. This gives an intermediate value, Y3, which is then used in the following equation to find h:

‘2l3

подпись: '2l3

Pr

подпись: pr(16)

Where Npr is the Prandtl number (see Section 2.3.1.2). It is often assumed to be 1 for all gases of interest in respect to a Stirling engine [14] however a value of 0.7 is applicable for air [26].

At this point it is possible to define the effectiveness of the regenerator, qreg. It is given by:

Vreg = A +2 (17)

Where A is the ‘reduced length’ of the regenerator, which is the same as NTU. It is a dimensionless value that represents how many Watts per square meter per unit temperature difference. NTU can be found by following equation:

A = NTU = h’Sreg (18)

M. cp

It should follow from these calculations that nreg and A will fit to Hausen’s temperature recovery ratio curves on one of the lines represented by the parameter n, or ‘reduced period’.

Stirling Engine Background

Figure 26: Graph of regenerator effectiveness vs. NTU for different values of n [14]

The value of reduced period can be found using the following equation:

(19)

подпись: (19)H. Sreg. t

N =

Where c is the specific heat capacity of the matrix and t is the duration of the period, i. e. the time it takes for the gas to blow through the matrix.

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