THE OVERRIDING OBJECTIVE of these Articles is to turn gas path design into a matter for back-of-the-envelope calculation. If this appears over-ambitious, take the definitions of Reynolds number R_e and Mach number M_a:

    R_e = 4rur_h/m

    M_a = u/ÖgRT

Defining velocity u at a common location allows the ratio:

                            M_a/R_e = ¼m(RT/g)^1/2/r_hp            (1)

(See NOTATION for symbols not defined to this point.)

This may be the most important equation the author has ever keyed into a word-processor! The expression is independent of particle velocity u!  Numerical values of g and μ are fixed when the gas (i.e. gas constant R) is specified. At any given temperature T the relationship between R_e and M_a is therefore determined  by the numerical value of a simple product pr_h - (absolute) pressure p [Pa] with hydraulic radius r_h [m].

Fig. 1    Compressibility vulnerability chart. Applicable to air and N₂ at ambient temperature in square-weave screens close-packed of aperture ratio 0.365. A combination of M_a and R_e above the heavy curve means  fractional pressure drop dp/p enhanced over the incompressible value at the same R_e. Combinations below the line are immune, and dp/p may be predicted using data for incompressible flow, such as the correlations of Kays and London (1964).

A limitation on the rpm of the air- (or N2-) charged engine is compressibility. This causes increased pressure drop relative to the incompressible case at the same Reynolds number R_e. Eqn. 1 forms the basis of a cook-book method of designing out this problem. An example will be given first. Experimental and analytical justification will follow.

M_a^crit is used to denote the threshold of compressibility in terms of Mach number M_a.  This threshold is NOT M_a= 0.3, as for steady pipe flow of the undergraduate engineering syllabus, but is a function of R_e and can be as low as M_a = 0.01.

In Fig. 1 the curve of M_a^crit vs R_e was obtained experimentally for a stack of 8 screens (n_g = 8)close-packed to 0.69 volume porosity (aperture ratio 0.365). For a stack of this number gauzes the choking value of M_a is 0.14. At high R_e the curve is asymptotic to this specific value. Otherwise, it is general, applying to all gases obeying p/ρ = RT and having isentropic index, g = 1.4.

A combination of M_a and R_e which puts the operating point above the curve guarantees that compressibility effects will arise. The straight lines radiating from the origin represent Eqn. 1 (M_a vs R_e) for different values of the parameter pr_h. The parameter takes convenient numerical values: for example, at 1 atm and r_h = 0.01mm, pr_h= 10⁵ [Pa] x 10^-5 [m] = 1.0 Pam.

At  pr_h = 13.5 Pam the parameter line is tangent to the M_a^crit vs R_e curve.  Hydraulic radius r_h of 0.05mm represents values commonly encountered. Corresponding 'critical' p is  p ≥ 13.5[Pam]/0.05x10^-3[m] = 2.7x10⁵ [Pa], or 2.7 atm. Increasing p enlarges the 'window' of R_e (and of corresponding M_a) within which, according to the algebra, operation is free of compressibility. A Philips' MP1002CA air engine in this writer's possession refused to run - even unloaded - unless pressurized well above 1 atmosphere. At rated operating temperature (600C) Ward (1972) coaxed 35.8W from such an engine at 4.14bar  - but with power falling off rapidly with speed at 1800 rpm.

As the product pr_h increases above the critical value of 13.5 Pam, the parameter lines intersect the heavy curve at increasingly high values of M_a - and correspondingly increased R_e - consistent with an extended compressibility-free 'window'. Regardless of the value of pr_h (short of infinity) there is a value of R_e at which the parameter line eventually intersects the choking line (as intuition would insist).

DESIGN RESOURCES CURRENTLY AVAILABLE fall into two categories:

Correlations to C_f vs R_e format as at Fig. 2a, together with the S_tP_r^2/3 vs R_e counterpart (heat transfer). Experimental reduction of source data assumes incompressible flow - as does subsequent usage. Screen stacks assumed.

Correlations to f_m (pressure coefficient) vs M_a format as at Fig. 2b. (There is no known heat transfer counterpart.) f_m is inter-convertible with C_f - see later section. Experimental reduction of source data assumes compressible flow, an isolated screen and independence of R_e.

For the urgent task of exploring and extending the performance envelope of the air- (or N₂-) charged Stirling engine neither resource is suitable. Reasons include the following:

*     C_f vs R_e and S_tP_r^2/3 vs R_e data are fundamentally irrelevant to situations in which approach Mach number, M_a, exceeds a value of about 0.02. The regenerator of the air- (or N2-) charged Stirling engine operates in this range, so use of the data for analysis is potentially misleading. Reliance for first-principles design is hazardous.

*    The f_m vs M_a format discounts viscous effects and the influence of proximity of the adjacent screen.  f_m increases with increasing M_a, eventually tending to infinity.

Fig. 2a. Specimen C_f vs R_e correlation. Valid for incompressible flow. Presentation stylized, but to the format of Kays and London (1964). The traditional geometric parameter, volume porosity ¶_v is represented here by d_wm_w (algebraic conversion dealt with later).

Fig. 2b. Specimen f_m vs M_a correlation. Valid for compressible flow. Presentation stylized, but to the format of Pinker and Herbert (1967). The traditional geometric parameter, aperture ratio a, is represented here by d_wm_w (algebraic conversion dealt with later).

*    neither friction factor C_f  nor pressure coefficient f_m is a logical parameter by which to characterize or compute pressure drop.

*    Re-acquiring pressure loss correlations in terms of Δp/p (or the 'per screen' value, Δp/p|_screen) instead of C_f or f_m brings 3 benefits: (a) the hazard of a misleading result is eliminated (d) greater insight, and (c) the design process becomes more intuitive.

*    Displaying the re-acquired data to the new format reveals crucial sensitivity of Δp/p and of Stanton number S_t to isentropic index, g.

(Steady vs oscillatory environment: Some argue that steady flow data can never properly serve the oscillating flow case. This writer disagrees: local acceleration intensity through the interstices far exceeds that due to the cyclic nature of the flow.) 

Eqn. 1 may now be used to illustrate the potential pitfalls embodied in conventional C_f vs R_e correlations. The concern which will come to light extends to the counterpart heat transfer correlations S_tP_r^2/3 vs R_e.

A hazard exposed

Take r_h = 0.05mm (0.05 x 10^-3 m) and  p = 10 atm (10⁶ Pa) and use Fig. 3a to read off M_a corresponding to the peak (10⁵) R_e of the correlation (Fig. 2a). Result: M_a/R_e = 2.11 x 10^-5, so that for  R_e = 10⁵, Mach number M_a = 2.11 - or more than twice the speed of sound!!

Explanation: A  high value of u, or of r_h (or of both) yields high R_e. The C_f - R_e correlation does not distinguish.

Eqn. 1 promises further short-cuts in the 'cook-book' approach to regenerator design. Figs. 3 apply to air (and N2) at (a) ambient temperature and (b) at T = 600^oC. The plotting arithmetic allows M_a/R_e to be read directly in terms of p in atm (or bar) and r_h in mm.  In engines charged with H₂ or He the rpm at which compressibility phenomena are expected probably lie beyond limits set by mechanical inertia.

Fig. 3a    Graphical representation of Eqn. 1 for air at ambient temperature.

APPLICABILITY:    The issues identified apply to all matrices whose passages afford significant contraction and expansion of the flow.  The problem can be addressed – and rectified – in those cases for which it is possible to quantify - or at least to categorize - flow passage geometry. Square-weave gauzes made from round wire are a case in point, since all sizes (ideally) which share a common value of the product d_wm_w have geometrically-similar flow passages - regardless of the absolute magnitudes of d_w and m_w individually.

Fig. 3b    Graphical equivalent of Eqn. 1 for air at 600 deg C. Figs 3a and b serve for nitrogen, N₂. R_e and M_a are based on particle speed u at a common location - e.g. the point of minimum flow area.

FLOW PASSAGE GEOMETRY

Let wire diameter and mesh number (wires/unit distance) be denoted by d_w and m_w respectively. For the square-weave gauze  the (dimensionless) product d_wm_w defines volume porosity ¶_v, aperture ratio a and hydraulic radius ratio r_h/d_w.

    volume porosity ¶_v = 1 – ¼pd_wm_wÖ{1 + (d_wm_w)²}

                                    » 1 – ¼pd_wm_w

    aperture ratio a = (1 - d_wm_w)²

(The expression for a employed by Pinker and Herbert (1967) is more sophisticated, but remains a function of d_wm_w only.)

    hydraulic radius ratio r_h/d_w = ¼¶_v/(1 - ¶_v)  (valid for any uniformly-distributed wire matrix)

Reciprocal formulae are useful when checking commercial availability of gauzes against design requirements:

a = [1 - (1 - ¶_v)4/p]²

    ¶_v = 1 – ¼p(1 - √a)

     d_wm_w = 1 - √a

     » 1 - 4¶_v/p

High aperture ratio a accompanies high volume porosity ¶_v. Returning to Fig. 3, a small aperture ratio is associated with greatest vulnerability to compressibility effects. This is consistent with a view of the individual gauze aperture as a convergent-divergent nozzle.

Consultation of manufacturers' literature suggests that gauzes of low d_wm_w giving high aperture ratio are not available off-the-shelf. Some manufacturers offer to weave to customers' specifications.

The experimental work of Su (1986) may be unique in documenting tests on stacks under conditions of compressible flow - tests which embodied recognition of dynamic similarity principles. Aperture ratio a, Reynolds number R_e and number n_g of screens were held constant while Mach number M_a was varied. Fig. 4 shows specimen results for n_g = 8.

Fig. 4. Su (1986) plotted pressure coefficient f_m against M_a with R_e as parameter taking the following values: 1: R_e =  380; 2: R_e =  570; 3: R_e =  760; 4: R_e =  930; 5: R_e =  1140; 6: R_e =  1240;

Undergraduate engineering courses teach the criterion that compressibility becomes an issue only for M_a ≥ 0.3.  But that is for pipe flow - and flow through a porous medium is not pipe flow. (A rather silly referee for Cryogenics, relying on a text-book  half-a-century behind the times (Oswatitsch 1952), challenged Fig. 4 with 'what experimental evidence do we have that friction factor at given Reynolds number (and porosity) really depends on the Mach number?' Well, only an entire PhD-worth of experimental and simulated results for stacks of different n_g - all leading to results consistent with Fig. 4.))

The lower the Reynolds number R_e, the lower is the value of Mach number M_a at which pressure drop is influenced by compressibility. In pores of the size encountered in air engines,  and at R_e = 120, values of M_a no greater than 0.01 make pressure drop  a function of M_a!

Limiting rpm of an air- (or N₂-) charged Stirling engine coincides (Organ 2005d) with the threshold of the compressibility effects already identified.

On this basis, extending the performance envelope calls for flow - and heat transfer - data acquired as a function of prevailing R_e and M_a (and g) - and collated for use accordingly.

Many argue that the definitive tool for ab initio gas path design is the computer simulation. Provided the computer is fed with appropriate flow and heat transfer correlations, and provided those correlations are unravelled in a manner consistent with the way in which they were originally condensed, one would not argue with the protagonists.

On the other hand, there is nothing to beat a rule-of-thumb - particularly if it is in tune with the intuition. This author has a vision of a repertoire of 'conditional design guidelines' which, satisfied simultaneously, bracket a viable design. Such a 'conditional guideline' emerges from consideration of Fig 4:

A guideline for first-principles design

     Minimize the ratio M_a/R_e (consistent with other design constraints) to minimize the adverse influence of compressibility.

Fig. 3a allows this to be related to practice: For given working fluid M_a/R_e is minimized by maximizing the value of the product r_hp. This is so regardless of temperature (see Fig. 3b). In other words:

    design around the highest practicable charge pressure p_ref. Work the design of the (wire mesh) regenerator around the largest hydraulic radius r_h permitted by heat transfer considerations.

Because Eqn. 1 (and thus Fig. 3) is independent of u, the favourable condition is achieved over the whole cycle.

A tentative design guideline emerges from consideration of Fig. 3:

    design around the lowest d_wm_w permitted by heat transfer considerations (i.e. highest volume porosity ¶_v and/or highest numerical value of aperture ratio a) .

THE NEW CORRELATIONS

Putting the present exercise to effect involves first of all re-examining the concept of friction factor. This will reveal that the C_f vs R_e concept has been adopted after inadequate thought for its applicability.

What's wrong with friction factor (and pressure coefficient)?

In their pioneering work of the 1950s, Kays and London chose the C_f vs R_e correlation to characterise flow in all the channel geometries they tested, from parallel ducts to wire screen stacks.  The choice is at odds with the fact that pressure gradient in an irregular flow passage cannot be equated to any definable wall shear stress (as it can for pipe flow). The inherent assumption of incompressibility is a separate concern.

To see this more clearly, return to the straightforward case of steady, incompressible flow in the parallel cylindrical tube, and focus on the laminar case. Take a flow passage element of length dx, free-flow area A_ff and wetted perimeter p_w. Pressure drop (minus) dp times A_ff is balanced by shear stress τ_w at the wall times area p_wdx wetted at the perimeter. The force balance algebra yields (minus) dp/dx = τ_w/r_h, where hydraulic radius r_h denotes the ratio A_ff/p_w.

The formal definition of friction factor C_f = |τ_w|/1/2ρu². Substituting into the expression just acquired:

    C_f = |dp/dx|r_h/1/2ρu²

For the ideal gas ρ = p/RT. Substituting and recalling the definition of Mach number M_a:

    C_f = |dp/dx|r_h/1/2gpM_a²                 (2) 

The definition is at its most perverse and uneconomical in the context of a Newtonian fluid. There is an analytical solution for shear stress at the wall, namely, τ_w = 2μu/r_h (Rouse 1946) so that dp/dx is (minus)2μu/r_h². Substituting into Eqn. 2 gives C_f  = 16/R_e - despite the fact that flow in this regime is independent of R_e - indeed of M_a! White (1974) considers the use of C_f for calculations in this flow regime an 'outrageous' idea (his p 123).

But tradition is tradition. Experimental determination of C_f involves measuring pressure gradient dp/dx at given flow rate, dividing by M_a and plotting in function of Reynolds number R_e. Subsequent use of the information to calculate dp/dx in a some other flow situation calls for evaluating R_e, reading off corresponding C_f and multiplying the value up by the local M_a.

Over and above this perversion, does a numerical value for C_f of, say 0.18 at R_e = 675 convey any design information at all? The focus of design interest is pressure drop. Even more pertinent is  fractional pressure drop, Δp/p, because a Δp/p value of, say, 1% is of wider relevance than a Δp value of, say, 1.1 atm.

The parameter f_m, favoured in the context of compressible flow through the single screen is as un-revealing as C_f, as may be seen by recalling what they have in common:

Relationship between friction factor C_f and pressure coefficient f_m

    C_f = (dp/dx)r_h/½ru² = dp/p|_dx(r_h/dx)/½gM_a²

     f_m = dp|_screen/½ru² = dp/p|_screen/½gM_a²

Stack length element dx = 2d_wn_g. For the square-weave screen  r_h/d_w = 1/4¶_v/(1 - ¶_v), so that:

                  C_f =  1/8f_m¶_v/(1 - ¶_v) 

Abandoning the perverse tradition of C_f and f_m and working instead of Δp/p|_screen allows fractional pressure drop to be read off directly from numerical values of any two of M_a, R_e and S_g (= Stirling number, pr_h/ωμ)? Wouldn't that be a step in the direction of making Stirling engine design a less inscrutable business? There is the bonus that the new format reflects compressibility effects of which the traditional C_f vs R_e correlations are ignorant.

The new format

Pressure drop correlations are sought in the form: Δp/p = Δp/p(viscosity, compressibility, flow passage geometry, gas properties). For rectangular-weave wire screens, geometry may be characterized by the dimensionless parameter d_wm_w (see later section). The relevant gas property is specific heat ratio g, so that we anticipate:

    Δp/p = Δp/p(viscosity, compressibility, d_wm_w, g)

Provisionally return to laminar Newtonian flow in the the parallel duct element. Re-write the expression for dp/dx as dp/p = -2μu(dx/r_h)/pr_h. Now define Stirling number S_g:

                        S_g =  pr_h/μu                             (3)                           

Substituting gives dp/p = -2S_g^-1dx/r_h, which should keep White happy.

Now return to the gauze stack. At very low-Reynolds-number-flow C_f = const/R_e (see Fig. 2a) - in other words, as for the parallel duct - but with a different constant. Express dp/p = (minus)½ru²C_fdx/pr_h for the gauze stack by substituting  R_e = 4rur_h/m. The resulting dp/p = (minus)½constS_g^-1dx/r_h . We are now justified in writing:

    Δp/p = Δp/p(S_g, compressibility, d_wm_w, g)dx/r_h

By simple substitution, 4gS_gM_a² = R_e, so that inclusion of M_a in the parameter list accounts for compressibility and for that part of the flow regime characterized by Reynolds number R_e:

     Δp/p = Δp/p(S_g, M_a, d_wm_w, g)dx/r_h

For the individual square-weave wire screen dx = 2d_w, so that dx/r_h is a dimensionless function of (dimensionless) volume porosity ¶_v - itself a function only of d_wm_w.

It is finally clear that experimental measurements of Δp across a small number n_g of screens may be converted into correlations of the form:

    Δp/p½_screen = Δp/p{S_g, M_a, d_wm_w, g}    (4)                                    

M_a remains an algebraic device, but gives continuity of Eqn. 5 between compressible and incompressible regimes. S_g and M_a are evaluated in terms of u at a common location. The point of minimum free-flow area is appropriate.

Over a gauze stack of finite length L, p and r are lowest, and u and M_a highest at the downstream end. If any gauze chokes, it will be the downstream gauze. This means that dp/dx may not be evaluated as (p_out – p_in)/L and must be measured - and eventually applied - locally.

Fig. 5    Specimen Δp/p|_gauze chart constructed by re-working Su's experimental data from the 1986 dissertation. Su covered a limited range of R_e. Stirling engine design calls for wider wider coverage. The value of d_wm_w was 0.396. A significantly different value of d_wm_w leads to a different chart.  

HEAT TRANSFER CORRELATIONS

Correlations available for wire gauzes are of the form S_tP_r^2/3 vs R_e with ¶_v as parameter. The same range of R_e is covered as for the C_f vs R_e counterpart, so that at low R_e where M_a > 0.01 the relevance of the data is unknown. S_t is Stanton number h/r½u½c_p and P_r is Prandtl number, μc_p/k (of which more below). 

An additional concern arises The juxtaposition of 'weighting factor' P_r^2/3 with S_t is by analogy with the exact solution of the boundary-layer equation for the flat plate (Chapman 1967 - p 281 of 4th printing 1969). It embodies a presumption that,  for P_r = unity, Reynolds' analogy applies.

If it has been established that Reynolds' analogy applies to flow within the interstices of the regenerator, the fact has yet to come to the attention of this writer - who would not, however, dispute the relevance of physical properties, μ, c_p, k which constitute P_r. Provisionally it appears prudent to include P_r with the parameter list in a way which avoids the Reynolds' analogy assumption:

        S_t = S_t{(P_r), g, S_g, M_a, d_wm_w}        (5)

Something of the character of the eventual S_t map may be inferred by re-working a specimen Kays and London correlation. At any given volumetric porosity ¶_v, S_tP_r^2/3 = aR_e^b is a close fit to their curve (NB incompressible case).  R_e may be re-written 4gS_gM_a², so that (algebraically) S_tP_r^2/3 = f(S_g, M_a, g).

For ¶_v = 0.725, coefficient a = 0.589 and b = -0.385. Fig. 6 is a transformation of the fitted Kays and London curve over the range of R_e and M_a used to generate a map of dp/p|_gauze vs  S_g (Fig. 5). S_tP_r^2/3 is chosen for the display, being the dependent variable of the K & L data.  Horizontal lines are curves of constant R_e. A different value of M_a is constant along each of the inclined lines. S_tP_r^2/3 is everywhere independent of M_a, readily verified by entering the map with trial values.

Fig. 6    Re-working Kays' and London's data on the basis that R_e = 4gS_gM_a² gives an impression of the likely form of a S_tP_r^2/3 chart based on dynamic similarity principles. Experimental re-acquisition the data over the appropriate range of M_a may be expected to distort the map. (The range of R_e and M_a plotted is that covered by Su's curves of Fig. 4.)

Re-acquiring the S_t values experimentally with dynamic similarity in mind (i.e., in function of S_g with M_a as parameter) may be expected lead to a distorted version of Fig. 6. When nature and extent of the distortion are known, any guidelines for first-principles design which emerge will be based on the following: S_t is the parameter of the solution of the equation for steady-flow convective heat exchange: (T_g - T_w)/(T_g0 - T_w) = exp(-S_tx/r_h). Distance x which the fluid has to travel before attaining a given temperature T_g is minimised by maximizing S_t.

From the provisional Fig. 6 (incompressible case transformed to new coordinates) S_t for a given gas (given P_r) is independent of S_g. Provisionally, therefore, the guideline deduced earlier from consideration of dp/p stands: operate at maximum charge pressure and maximum hydraulic radius.

Progress with extending the operational envelope of the air engine calls flow and heat transfer data to be re-acquired over a wide range of S_g, M_a, for both values of g (1.4, 1.66) and for all candidate matrix materials.

There is no reason to believe that a Stirling engine cannot function with the flow processes subject to a degree of compressiblity - if designed to do so. Indeed, it remains to be demonstrated that operation in this regime is without advantage. Availability of correlations to the new format would allow first-principles exploration of operation beyond established performance envelopes.

There is no overwhelming technical challenge to acquiring data in this form, except for the range of candidate materials to be covered. This suggests the need for a coordinated programme based on test apparatus of agreed specification and on common test procedures and data reduction.

Individuals and organizations interested in contributing to a data base to supersede the classic compendium of Kays and London (1964, 1994) may wish to contact allan.j.o@btinternet.com

Apparatus, instrumentation, data capture and interpretation for re-acquisition of heat transfer data to the new format are opportunities for greater sophistication than has been applied to date.

BIBLIOGRAPHY With the exception of starred* items, the titles in the bibliography pre-date recognition of the problem now addressed. On the other hand, look out for The Air Engine Project, Allan J Organ, Woodhead Publishing (date to be announced).

Home