Tuesday, September 15, 2015

Tyrrell 008 and Thunderbird 2

Patrick Depailler at the entry to Mirabeau Bas, Monaco 1978
Perhaps unexpectedly, the 1978 Tyrrell 008 shares an aerodynamic concept with Thunderbird 2: they both employ forward-swept wings.

The conventional wisdom on such wings is that they induce a span-wise component to the wing-flow, directed towards the roots of the wing. This has two consequences:

(i) The strength of the wing-tip vortices is reduced, decreasing vortex-drag.

(ii) Yaw instability is increased. As the vehicle begins to yaw, the effective forward-sweep is increased on the outer-wing, and the effective sweep is reduced on the inner wing. The further reduces the drag on the outer wing, and increases the drag on the inner wing, and this differential drag creates a torque which further increases the yaw angle.

So perhaps the Tyrrell 008 front-wing was designed to improve turn-in response.

Detailed information on the aerodynamic performance of Thunderbird 2 appears to have been lost when Century 21 Productions closed its studio on the Slough Trading Estate in late 1970.

Sunday, September 13, 2015

Tyre friction and self-affine surfaces

The friction generated by an automobile tyre is crucially dependent upon the roughness of the road surface over which the tyre is moving. The theoretical representation of this phenomenon developed by the academic community over the past 20 years has been largely predicated on the assumption that the road can be represented as a statistically self-affine fractal surface. The purpose of this article is to explain what this means, but also to question whether this assumption is in need of some generalisation.

To begin, we need to understand two concepts: the correlation function and the power spectrum of a surface.

Surfaces in the real world are not perfectly smooth, they're rough. Such surfaces are mathematically represented as realisations of a random field. This means that the height of the surface at each point is effectively sampled from a statistical distribution. Each realisation of a random field is unique, but one can classify surface types by the properties of the random field from which their realisations are drawn. For example, each sheet of titanium manufactured by a certain process will share the same statistical properties, even though the precise surface morphology of each particular sheet is unique.

Let us denote the height of a surface at a point x as h(x). The height function will have a mean <h(x)> and a variance. (Here and below, we use angular brackets to denote the mean value of the variable within the brackets). The variance measures the amount of dispersion either side of the mean. Typically, the variance is calculated as:

Var = <h(x)2> − <h(x)>2

Mathematically, the height at any pair of points, x and x+r, could be totally independent. In this event, the following equation would hold:

<h(x)h(x+r)> = <h(x)2>

The magnitude of the difference between <h(x)h(x+r)> and <h(x)2> therefore indicates the level of correlation between the height at points x and x+r. This information is encapsulated in the height auto-correlation function:

ξ(r) = <h(x)h(x+r)> − <h(x)2>

Now the auto-correlation function has an alter-ego called the power spectrum. This is the Fourier transform of the auto-correlation function. It contains the same information as the auto-correlation function, but enables you to view the correlation function as a superposition of waves with different amplitudes and wavelengths. Each of the component waves is called a mode, and if the power spectrum has a peak at a particular mode, it shows that the height of the surface has a degree of correlation at certain regular intervals.

Related to the auto-correlation function is the height-difference correlation function:

C(r) = <(h(x+r)−h(x))2>

This is essentially the variance in height as a function of distance from an arbitrary point x. This is a useful function to plot graphically because it represents the difference between the auto-correlation function and the overall variance, as a function of distance r from an arbitrary point x:

C(r) = 2(Var−ξ(r))

Which brings us to self-affine fractal surfaces. For such a surface, a typical height-difference correlation function is plotted below, (Evaluation of self-affine surfaces and their implications for frictional dynamics as indicated by a Rouse material, G.Heinrich, M.Kluppel, T.A.Vilgis, Computational and Theoretical Polymer Science 10 (2000), pp53-61).

Points only a small distance away from an arbitrary starting point x can be expected to have a height closely correlated with the height at x, hence C(r) is small to begin with. However, as r increases, so C(r) also increases, until at a critical distance ξ||, C(r) equals the variance to be found across the entire surface. Above ξ||, C(r) tends to a constant and ξ(r) tends to zero. ξ|| can be dubbed the lateral correlation length. In road surfaces, it corresponds to the average diameter of the aggregate stones.

To understand what a self-affine fractal surface is, first recall that a self-similar fractal surface is a surface which is invariant under magnification. In other words, the application of a scale factor x → a⋅x leaves the surface unchanged.

In contrast, a self-affine surface is invariant if a separate scale factor is applied to the horizontal and vertical directions. Specifically, the scale factor applied in the vertical direction must be suppressed by a power between 0 and 1. If x represents the horizontal components of a point in 3-dimensional space, and z represents the vertical component, then it is mapped by a self-affine transformation to x → a⋅x and z → aH⋅z, where H is the Hurst exponent. In the height-difference correlation function plotted above, the initial slope is equal to 2H, twice the value of the Hurst exponent.

Note, however, that road surfaces are considered to be statistically self-affine surfaces, which is not the same thing as being exactly self-affine. If you zoomed in on such a surface with the specified horizontal and vertical scale-factors, the magnified subset would not coincide exactly with the parent surface. It would, however, be drawn from a random field possessing the same properties as the parent surface, hence such a surface is said to be statistically self-affine.

A yet further adaptation is necessary to make the self-affine model applicable to road surfaces. Roads are known to be characterised by two distinct length-scales: the macroscopic one determined by the size of aggregate stones, and the microscopic one determined by the surface properties of those stones, (see diagram below).

One attempt to adapt the self-affine model to road surfaces introduces two distinct Hurst exponents, one for the micro-roughness and one (purportedly) for the macro-roughness, as shown below, (Investigation and modelling of rubber stationary friction on rough surfaces, A.Le Gal and M.Kluppel, Journal of Physics: Condensed Matter 20 (2008)):

This, however, doesn't seem quite right. The macro-roughness of a road surface is defined by the morphology of the largest asperities in the road, the stone aggregate. Yet, as Le Gal and Kluppel state, a road surface only displays self-affine behaviour "within a defined wave length interval. The upper cut-off length is identified with the largest surface corrugations: for road surfaces, this corresponds to the limit of macrotexture, e.g. the aggregate size."

It's not totally clear, then, whether the macro-roughness of a road surface falls within the limits of self-affine behaviour, or whether it actually defines the upper limit of this behaviour.

So whilst the notion that a road surface is statistically self-affine appears, at first sight, to have been empirically verified by the correlation functions and power spectra taken of road surfaces, perhaps there's still some elbow-room to suggest a generalisation of this concept.

For example, consider mounded surfaces. These are surfaces in which there are asperities at fairly regular intervals. In the case of road surfaces, this corresponds to the presence of aggregate stones at regular intervals. Such as surface resembles a self-affine surface in the sense that it has a lateral correlation length ξ||. However, there is an additional length-scale λ defining the typical spacing between the asperities, as represented in the diagram below, (Evolution of thin film morphology: Modelling and Simulations, M.Pelliccione and T-M.Lu, 2008, p50).

In terms of a road surface, whilst ξ|| characterizes the average size of the aggregate stones, λ characterizes the average distance between the stones.

In terms of the height-difference correlation function C(r), a mounded surface resembles a self-affine surface below the lateral correlation length, r < ξ||. However, above ξ||, where the self-affine surface has a constant profile for C(r), the profile for a mounded surface is oscillatory (see example plot below, ibid. p51). Correspondingly, the power spectrum for a mounded surface has a peak at wavelength λ, where no peak exists for a self-affine surface.

The difference between a mounded surface and a genuinely self-affine surface is something which will only manifest itself empirically by taking multiple samples from the surface. Individual samples from a self-affine surface will show oscillations in the height-difference correlation function above the lateral correlation length, but the oscillations will randomly vary from one sample to another. In contrast, the samples from a mounded surface will have oscillations of a similar wavelength, (see plots below, from Characterization of crystalline and amorphous rough surface, Y.Zhao, G.C.Wang, T.M.Lu, Academic Press, 2000, p101).

Conceptually, what's particularly interesting about mounded surfaces is that they're generalisations of the self-affine surfaces normally assumed in tyre friction studies. Below the lateral correlation length-scale ξ||, a mounded surface is self-affine (M.Pelliccione and T-M.Lu, p52). One can say that a mounded surface is locally self-affine, but not globally self-affine. Note that whilst every globally-affine surface is locally self-affine, not every locally self-affine surface is globally self-affine.

A self-affine road surface will have aggregate stones of various sizes and separations, whilst a mounded road surface will have aggregate stones of similar size and regular separation.

In fact, one might hypothesise that many actual road surfaces in the world are indeed locally self-affine, but not globally self-affine. For this to be true, it is merely necessary for there to be some regularity in the separation of aggregate within the asphalt. If the distance between aggregate stones is random, then a road surface can indeed be represented as globally self-affine. However, if there is any regularity to the separation of aggregate, then the surface will merely be locally self-affine. If true, then existing academic studies of tyre friction have fixated on a special case which is a good first approximation, but which does not in general obtain.

Thursday, September 03, 2015

BMW's F1 'rocket fuel' and aromatic hydrocarbons

The story of BMW's turbo 'rocket fuel' has long since passed into Formula 1 legend, but there's a longer and deeper story here, involving the German war effort, some organic chemistry, and the history of oil refining techniques. But let's begin with the legend, and the breakthrough which enabled the Brabham-BMW of Nelson Piquet to win the 1983 Drivers' Championship:

[BMW motorsport technical director, Paul] Rosche telephoned a contact at chemicals giant BASF and asked if a different fuel formulation might do the trick. After a little research, a fuel mix was unearthed that had been developed for Luftwaffe fighters during World War II, when Germany had been short of lead. Rosche asked for a 200-litre drum of the fuel for testing and, when it arrived, he took it straight to the dyno.

"Suddenly the detonation was gone. We could increase the boost pressure, and the power, without problems. The maximum boost pressure we saw on the dyno was 5.6 bar absolute, at which the engine was developing more than 1400 horsepower. It was maybe 1420 or 1450 horsepower, we really don't know because we couldn't measure it — our dyno only went up to 1400." ('Generating the Power', MotorsportMagazine, January 2001, p37).

An aromatic hydrocarbon called toluene is commonly held to have been the magic compound in this fuel brew, but erstwhile Brabham chief mechanic Charlie Whiting goes further:

"There were some interesting ingredients in it, and toluene has been mentioned. But it would have had far more exciting things in it, I think, than toluene. I suspect – well, I know – that it was something the BMW engineers had dug out of the cupboard from the Second World War. Almost literally rocket fuel," ('Poacher Turned Gamekeeper', MotorsportMagazine, December 2013, p74).

Before we delve into the chemistry of fuels, let's establish some context here. The current F1 turbo engine regulations require detonation-resistant fuels with a high calorific value per unit mass. Detonation resistance enables one to increase the compression ratio, and thereby increase the work done on each piston-stroke, while the limits on total fuel mass and fuel mass-flow rate require fuel with a high energy content per unit mass.

In contrast, in the 1980s the regulations required detonation-resistant fuels with a high calorific value per unit volume. From 1984, the amount of fuel permitted was limited, but the limitation was defined in terms of fuel volume rather than mass, hence fuel with a high mass-density became advantageous. By this time, the teams had already followed BMW's lead and settled upon fuels with a high proportion of aromatic hydrocarbons.

To understand the significance of this, we need to start with the fact that there are four types of hydrocarbon:

(i) Paraffins (sometimes called alkanes)
(ii) Naphthenes (sometimes called cycloalkanes)
(iii) Aromatics (sometimes called arenes)
(iv) Olefins (sometimes called alkenes)

Methane, ethane and propane. Each larger disk represents a carbon atom; each white disk represents a hydrogen atom; and each black disk represents a covalent bond.
Each hydrocarbon molecule contains hydrogen and carbon atoms, bound together by covalent bonds. The hydrocarbon types differ from each other by the number of bonds between adjacent atoms, and by the overall topology by which the atoms are connected together. So let's briefly digress to consider the nature of covalent bonding.

The electrons in an atom are stacked in so-called 'shells', each of which can contain a maximum number of members. The first shell can contain only two electrons, while the second can contain eight. If the outermost electron shell possessed by an atom is incomplete, then the atom will be disposed to interact or bond with other atoms.

A neutral hydrogen atom has one electron, so its one and only shell needs one further electron to complete it. A neutral carbon atom has six electrons, two of which fill the lowermost shell, leaving only four in the next shell. Hence, another four electrons are required to complete the second shell of the carbon atom.

In covalent bonding, an electron from one atom is shared with an adjacent atom, and the adjacent atom reciprocates by sharing one of its electrons. This sharing of electron pairs enables groups of atoms to complete their electron shells, and thereby reside in a more stable configuration. In particular, a carbon atom, lacking four electrons in its outermost shell, has a propensity to covalently bind with four other neighbours, while a hydrogen atom has a propensity to bind with just one neighbour. By this means, chains of hydrocarbons are built.

Methane, for example, (see diagram above) consists of a single carbon atom, bound to four hydrogen atoms. The four shared electrons from the hydrogen atoms complete the outermost shell around the carbon atom, and each hydrogen atom has its one and only shell completed by virtue of sharing one of the carbon atom's electrons.

If there is a single covalent bond between each pair of carbon atoms, then the hydrocarbon is said to be saturated. In contrast, if there are more than one covalent bond between a pair carbon atoms, the molecule is said to be unsaturated.

Saturated ethane in a state of unconcealed glee compared to the glum unsaturated ethylene, and the vexatious triple-bonded acetylene, (this and the above taken image from 'BP - Our Industry', 1958, p69).
Now, to return to our classification scheme, paraffins are non-cyclic saturated chains, (there is a sub-type called iso-paraffins in which the chain contains branching points); naphthenes are cyclic saturated chains; aromatics are cyclic (semi-)unsaturated chains; and olefins are non-cyclic unsaturated chains, (with a sub-type of iso-olefins in which the chains have branching points).

Aromatic compounds possess a higher carbon-to-hydrogen ratio than paraffinic compounds, and because the carbon atom is of greater mass than a hydrogen atom, this entails that aromatic compounds permit a greater mass density. This characteristic was perfect for the turbo engine regulations in the 1980s, and toluene was the most popular aromatic hydrocarbon which combined detonation-resistance and high mass density.

To put toluene into context, we need to begin with the best-known aromatic hydrocarbon, benzene. This is a hexagonal ring of six carbon atoms, each one of which is bound to a single hydrogen atom. Toluene is a variant of this configuration in which one of those hydrogen atoms is replaced by a methyl group. The latter is one of the primary building blocks of hydrocarbon chemistry, a single carbon atom bound to three hydrogen atoms. The carbon atom in a methyl group naturally binds to another carbon atom, in this case one of the carbon atoms in the hexagonal ring. Hence toluene is also called methyl-benzene.

Closely related to toluene is xylene, another variant of benzene, but one in which two of the hydrogen atoms are replaced by methyl groups. (Hence xylene is also called dimethyl-benzene). If the two methyl groups are bound to adjacent carbon atoms in the ring, the compound is dubbed o-xylene; if the docking sites of the two methyl groups are separated from each other by two steps, then the result is dubbed m-xylene; and if the docking sites are on opposite sides of the ring, the compound is called p-xylene.

Most teams seem to have settled on the use of toluene and xylene. By mid-season 1987, for example, Honda "reached an 84% level of toluene," (Ian Bamsey, McLaren Honda Turbo - A Technical Appraisal, p32).

With respect to the Cosworth turbo used by Benetton in 1987, Pat Symonds recalls that "the problem was the engine had been developed around BP fuel, and we had a Mobil contract. Fuels then weren’t petrol, they were a chemical mix of benzene, toluene and xylene. We kept detonating pistons, and it wasn’t until mid-season that we got it right," (Lunch with Pat Symonds, MotorsportMagazine, September 2012). In fact, Pat attests that the Cosworth fuel was an equal mix of benzene, toluene and xylene, (private communication).

At Ferrari, AGIP later recalled that their toluene and xylene based fuel reached density values of up to 0.85, in some contrast with the paraffinic fuels of the subsequent normally-aspirated era, with density values of 0.71 or 0.73. "Given the ignition delays of heavy products, we had to add more volatile components that would facilitate that ignition," (Luciano Nicastro, Head of R&D at AGIP Petroli, 'Ferrari Formula 1 Annual 1990', Enrico Benzing, p185).

Renault, in contrast, claim to have used mesitylene, as Elf's Jean-Claude Fayard explains:

"We found a new family of hydrocarbons which...contained a strong proportion of mesitylene [trimethyl-benzene] and they had a boiling point of 150C, but with a combustion capability even higher than that of toluene," (Alpine and Renault, Roy Smith, p142).

Mesitylene is a variant of benzene in which three methyl groups are docked at equal intervals around the hexagonal carbon ring, (naturally, mesitylene is also called trimethyl-benzene).

Now, the fact that Paul Rosche grabbed a barrel of aviation fuel used by the Luftwaffe is significant because German WWII aviation fuel differed substantially from that used by the allies. Faced with limited access to crude oil, and a poorly developed refining industry, the Germans developed war-time aviation fuels with a high aromatic content.

Courtesy of the alkylation process, the original version of which was developed by BP in 1936, the allies could synthesise iso-octane from a reaction involving shorter-chain paraffins, such as iso-butane, and olefins such as butene or iso-butene. By definition, iso-octane has an octane rating of 100, defining the standard for detonation-resistance. Using 100-octane fuel synthesised by the alkylation process, the British were able to defeat the Luftwaffe in the 1940 Battle of Britain.

In contrast, German aviation fuel was largely obtained from coal by applying hydrogenation processes. With limited capacity to produce paraffinic components, the initial B-4 grade of aviation fuel used by the Germans had an octane range of only 87-89, a level which itself was only obtained with the addition of the anti-detonation agent, Lead Tetra-Ethyl. A superior C-3 specification of aviation fuel was subsequently produced, with an octane rating of 95-97, but only by substantially increasing the proportion of aromatic hydrocarbons:

"The B-4 grade...contained normally 10 to 15 percent volume aromatics, 45 percent volume naphthenes, and the remainder paraffins...The C-3 grade was a mixture of 10 to 15 percent volume of synthetic isoparaffins (alkylates and isooctanes)...[and] not more than 45 percent volume aromatics," (US Navy, Technical Report No. 145-45. Manufacture of Aviation Gasoline in Germany, Composition and Specifications).

The Germans, however, also included some interesting additives:

"The Bf 109E-8's DB601N engine used the GM-1 nitrous oxide injection system...Injected into the supercharger inlet, the gas provided additional oxygen for combustion at high altitude and acted as an anti-detonant, cooling the air-fuel mixture," ('The Decisive Duel: Spitfire vs 109', David Isby).

"Additional power came from water-methanol and nitrous-oxide injection," ('To Command the Sky: The Battle for Air Superiority over Germany, 1942-44', Stephen L.McFarland and Wesley Phillips, p58).

At which point, one might recall Charlie Whiting's suggestion that the 1983 BMW fuel brew "had far more exciting things in it" than toluene. This, despite regulations which explicitly stated that fuel should be 97% hydrocarbons, and should not contain "alcohols, nitrocompounds or other power boosting additives." Still, there's breaking the rules, and then there's getting caught breaking the rules. Perhaps BMW were a little naughty in 1983, before settling down with an 80% toluene brew.

The current turbo regulations, however, require a much lower aromatic content, stipulating the following maxima:

Aromatics wt% 40
Olefins wt% 17
Total di-olefins wt% 1.0
Total styrene and alkyl derivatives wt% 1.0

Which entails, in a curious twist, that the current maximum aromatic content almost matches that of the C-3 aviation fuel developed in war-time Germany...