One cylinder-set (or gas path) is considered at a time, as in the coaxial 'beta' and the V-alpha configuration. The Rider concept and the 'gamma' configuration are anomalous and are not covered - the latter due to the adverse influence of the arbitrary additional dead space. Moreover, results may be misleading if applied to one cylinder set (one quarter) of a 4-cylinder 'Rinia' machine, whose first harmonic of the volume phase angle, being 90 degrees, is in conflict with the constraints:

    Temperature ratio NT = TE/TC:    3.3 (e.g. TE = 1020 K, TC = 310 K)

    For V-configuration:

        thermodynamic volume ratio κ:        1.0       

        thermodynamic phase angle α:        120 degrees

    Achieving these values in a coaxial, uniform-bore machine will call for the following values of kinematic   phase angle β and piston/displacer displacement ratio λ:

        β = 60 degrees

        λ = 1.0

    Note that β and λ of a real mechanism (e.g. rhombic drive) inter-convert only approximately with α and κ via the formulae for simple-harmonic volume variations (e.g. those derived by Finkelstein):

Expansion and compression exchangers are each of multi-tube type having tubes of internal diameter dx, lengths Lx and respective numbers nTx.

    Regenerator: stacked from wire screens of hydraulic radius rh given by wire diameter dw and mesh number mw. Volume porosity ¶v is pre-set to 0.75

    Resulting Beale number (given adequate external heating and cooling provision): up to 0.18 depending on mechanical efficiency.

The chart set which follows allows design for either H2 or air/N2. (The different isentropic index γ of He calls for an alternative set of charts scaling from GPU-3 performance on He.)

The worked example proceeds with air or N2 as working fluid - and by addressing the rpm requirement:  If avoidance of gearing is a priority, rpm is the design parameter which is frequently the least flexible, being the value demanded by an electric alternator or by the screw of a boat. This example uses rpm = 1500.

All things being equal, air/N2 tend to demand a larger number of exchanger tubes than H2 (or He). A superfluous tube is one extra potential joint failure at manufacture.  Number of expansion tubes nTxe is read from the middle scale of Fig. 1 at the left-hand graduations. A straight-edge pivoted against acceptable nTxe and intersecting target rpm on the right-hand scale shows the value of power [W] on the left-hand scale which is achievable on selection (to follow) of appropriate length and internal diameter. 1500 rpm and 43 tubes provide the example - largely because they lead to a convenient 1000W (1kW) on the power scale.

Now for the reckoning! On the power scale of Chart 3 connect (1000W) with 1500 rpm on the right and read 67.23 bar on the pressure (pref) scale. This may be the time for a re-think: hydrocarbon lubricant can react explosively with air - and the higher the pressure the greater the hazard. Options include (a) proceeding with N2 (b) pressing ahead in anticipation of completing the mechanical design with the aid of 'solid oil' bearing technology and polymer-alloy rubbing seals throughout, (c) returning to square-one and setting off again with a reduced power target (d) pursuing FastTrack to acquire flow-passage lengths Lxe Lr and Lxc, and then re-designing to an acceptable pref with the aid of ReScale.

Flow passage length Lxe of the individual expansion exchanger tube is read in convenience units (mm) from the air/N2scale of the ‘slide-rule’ of Chart 4: A vertical line (cursor) through 1500 rpm of the bottom (rpm) scale intersects the air/N2 scale at 155 mm.  A value of dxe = 1.91 mm is obtained from Chart 5 in the same fashion, and completes the internal design of the expansion exchanger. (Note the advantageous partial de-coupling from external heat transfer design: internal design is essentially independent of tube spacing, whereas that very spacing determines the hydraulic radius 'seen' by the combustion products flowing between the tubes.)

Resist the temptation to round the dxe value of 1.91 mm to 1.9 (or 2.0 mm) pending consultation of the stock lists of tube suppliers/manufacturers (http://www.welleng.co.uk/tube.html). Sizes which are irrational at first sight may be encountered on stock table as a result of conversion from imperial inch, or as the coincidental combination of a rational outside diameter with a wall thickness manufactured to a US or UK standard gauge specification. Not one of the internal diameters on the Fractional Metric stock list of Coopers Needleworks is a ‘rational’ size! In the 'thin-wall' range, a 14-gauge tube is listed having internal diameter of 1.828 mm.

Time for a check: Whatever happened to Beale number NB in these deliberations?! Using the pref etc. from the charts, and remembering to convert from convenience units to a consistent system:

    NB     =    1000 [W]/{67.23E+05 [Pa] x 32.87E-06[m3] x 1500/60 [-]}

        =     0.181 (cf the 0.18 cited for the GPU-3)

Compression exchanger design proceeds in parallel fashion from a chart-set reduced in number by two, reflecting the fact that values for pref and Vsw do not need to be acquired a second time. The charts follow.

Rationale behind FastTrack.

Like every other feature of this Site, FastTrack exploits Similarity principles. This is not the same thing as insisting that all Stirling engines are similar (although unifying features reassuringly emerge). Rather, it is a recognition of the benefits of studying, say, the buckling characteristics of structural columns in terms of the dimensionless parameter slenderness ratio d/L, rather than dealing separately with each d and each L.

The basic similarity parameters of the Stirling engine are:

    Temperature ratio NT = TE/TC

    Volume ratio κ = VC/VE

    thermodynamic phase angle α (or, for practical purposes, the first harmonic thereof)

    Dead space ratios δ = Vd/Vw: δxe = Vdxe/Vsw, δr = Vdr/Vsw etc.

    Specific heat ratio or isentropic index γ.

The gas processes of two engines charged to difference pressures pref and/or running at different rpm can be arranged to be similar only if the engines share identical numerical values of these basic parameters.

Approximating to that similarity in practice calls for a further constraint:

Geometric similarity of the flow passages: prototype and derivative must both have cylindrical tubes, or must both have slots of the same depth/width ratio; both must have regenerators of geometrically-similar matrix material, e.g. square-weave wire gauze of the same volume porosity ¶v.

It is now possible (in principle) to constrain the operating conditions of the two machines (working gas, reference pressure pref, rpm) and the finer detail of the gas path (internal diameters dx, lengths Lx and respective numbers nTx of tubes) so as to ensure that (variable) temperature distributions and pressure drops follow identical histories over the entire 360 degree cycle. This is bound to result if Reynolds number Re and Mach number Ma at any given location can be arranged to undergo the same respective variations. If the cycle histories of local Re are the same, then so are the cycle histories of Stanton number St and of friction factor Cf at comparable locations. If, simultaneously with this, the Ma histories are the same, then local cycle histories of fractional pressure drop dp/p may be arranged to be identical also.

Analytically complex? With the basic similarity conditions already satisfied, less daunting than might be feared!

Reynolds number is defined Re = 4ρurh/μ. It may be re-expressed in terms of mass rate m' = dm/dt on noting that m' = ρuAff, in which Aff is free-flow area. For the multi-tube exchanger Aff is ¼nTxπdx2.

The Site page Without the computer introduced specific mass σ = mRTC/prefVsw and specific mass rate σ' = m'RTC/ωprefVsw. Substituting the expression for Aff and the definition of σ' into Re:

                                                            

    4Re/π    =    σ'/{(dx/Vsw1/3)nTx}  pref /{ωμ0f(T)}  ω2Vsw2/3/RTC    

The term f(T) corrects a datum value μ0 of coefficient of dynamic viscosity μ to the value at the nominal temperature(TE or TC) of the exchanger, e.g., μTE = μ0f(TE).

Site page Without the computer demonstrates that the cycle history of specific mass rate σ' computed from the ideal adiabatic reference cycle at any reproducible location varies little between the benchmark engines GPU-3, P-40, V-160 etc. Where hypothetic engines share identical numerical values of the basic similarity parameters, an infinite range of such engines has identical cycle history of σ'.

Term ω2Vsw2/3/RTC is equivalent to the square of Mach number, Ma, and requires to be independently similar, as σ' already is. For the cylindrical duct rh = dx/4. Angular speed ω can be re-written ω = 2πrpm/60. Similarity of Re (and thus of St and Cf) is now achieved by ensuring commonality of the numerical value of the dimensionless group DGRe:

    DGRe = pref/{rpmμ0f(T)(dx/Vsw1/3)nTx}

Pressure drop dp in steady flow in a parallel duct of hydraulic radius rh and length Lx is: dp = ½ρu2CfLx/rh. Using rh = dx/4 again, fractional pressure drop dp/p over length Lx of the exchanger is:

    dp/p =2γ [u2/(γRT)]Cf(Re) Lx/dx

The term u2/(γRT) is the square of Mach number Ma. With similarity of Cf taken care of (by similarity of Re), similarity of dp/p is ensured by arranging similarity of the product γMa2Lx/dx. The easiest way of achieving this is to impose similarity of the 3 groups independently. γ is already subject to a basic similarity condition. Ma is dealt with by making the same substitutions as for Re. There are now the similarity groups DGMa and DGLd:

    DGMa    = rpmVsw/{(√RTC)dx2nTx}

    DGLd = Lx/dx

A plausible design requirement is a specified value of power Pwr [W] at specified rpm. This calls for the designer to establish the Vsw, pref and, for each exchanger, the Lx, dx and nTx - 5 numerical values - which will cause the engine to achieve the specified performance. So far there are three similarity conditions DGRe, DGMa and DGLd, so two more are required to make the number of equations equal to the number of unknowns.

The first is acquired by re-expressing - in terms of Lx, dx and nTx - similarity of exchanger fractional dead space δxe = Vdxe/Vsw = ¼πdx2LxnTx/Vsw. The constants may be dropped, yielding dead-space criterion DGδx:

    DGδx = dx2LxnTx/Vsw

Finally, look no further than the original similarity parameter NB variously attributed to Beale and Finkelstein. The constant which converts rpm to frequency f is dropped, and convenience units used for pref and Vsw. The symbol NB is replaced by DGNB (because the numerical value will no longer be ≈ 0.15):

     DGNB = power [W]/{pref [bar]Vsw[cc] rpm}

The (unknown) parameter values for the engine to be designed (the derivative) can now be equated to corresponding values for an engine of known specification and (preferably exemplary) performance - the prototype. The process transfers to the right-hand side values already prescribed for the derivative - power Pwr[W] and rpm. Variables for the derivative are on the left:

pref/{(dx/Vsw1/3)nTx}        =    DGRe rpm μ0f(T)

Vsw/(dx2nTx)        =    DGMa/rpm√RTC

Lx/dx                    =     DGLd

dx2LxnTx/Vsw                =    DGδx

pref Vsw                =    Pwr/(DGNB rpm)

The equations are non-linear in the unknowns. An apt succession of row divisions and substitutions leads to explicit expressions for individual solutions. On the other hand, a solution which serves as a stencil for a more comprehensive treatment is probably a worthwhile investment. Accordingly, the equations are linearized by taking logs. The resulting array of coefficients is:

pref         Vsw        Lx         dx         nTx        RHS

1        1/3          0        -1        -1         log{DGRe rpm μ0f(T)}

0        1              0        -2        -1        log{√RTC DGMa/rpm}

0        0              1        -1          0        log(DGLd)

0        -1             1         2          1        log(DGδx)

1        1              0         0          0         log{Pwr/(DGNB rpm)}

This is easily coded for numerical solution (by a library routine such as SIMQX).  The five DG are substituted by respective numerical values from a suitable prototype gas path specification. One solution run generates the design charts for all derivative gas paths.

For reasons which a competent analyst might have anticipated, solutions for pref, Vsw and nTx emerge in the form pref = C1Pwra1rpmb1, Vsw = C2Pwra2rpmb2, nTx = C3Pwra3rpmb3. Both Lx and dx are unexpectedly independent of Pwr, and so can be represented Lx =  C4rpmb4 and dx =  C5rpmb5. This allows the former three solutions to be graphed against Pwr with rpm as parameter. Each of the curves for Lx and dx is evidently a unique line plotted against rpm. The parallel-scale alignment nomogram is equivalent to the conventional x-y-parameter presentation. Its superior resolution explains its choice as the format for the design charts of this section.

Regenerator design is completed as for the tubular exchangers - except for a final step to convert hydraulic radius rh via volume porosity ¶v to wire diameter dw and mesh number mw which specify the required grade of wire gauze.

Design charts follow for hydraulic radius rh and overall stack height Lr

Chart 9.    Virtual slide rule for regenerator stack height Lr [mm]. Place cursor at derivative rpm on bottom-most horizontal scale. Read Lr from uppermost graduations when scaling from H2 to Air/N2. Note the low value (14.32 mm). Read from intermediate graduations when scaling to H2.

Chart 10    Virtual slide rule for regenerator hydraulic radius rh [mm]. Place cursor at derivative rpm on bottom-most horizontal scale. Read rh from uppermost graduations when scaling from H2 to Air/N2. Read from intermediate graduations when scaling to H2.

A simple exercise for the hand calculator converts hydraulic radius rh to wire diameter dw [mm] mesh number mw [wires/mm or wires/inch]. In the equation, symbol ¶v represents volume porosity, which has been pre-set for this example to 0.75 [-].

    dw    =    4rh [mm] (1 - ¶v)/¶v    [mm]    =    0.0253 mm, or about 0.001 inch.

    mw    =    ¶v/πrh [wires/mm]   =    12.49 wires/mm, or 317 wires/inch (≈ 320 mesh).

A manufacturer of precision wire gauzes holding a comprehensive stock range is G Bopp and Co.