Optimizing Fuel Tank Design for High-Speed

Vehicles

Graduate Student Mathematical Modeling Camp

(GSMMC) 2009

Michael Davis, Claremont Graduate UniversityZhingxing Fu, University of DelawareQunhui Han, University of Delaware

Ivonne Rivas, University of CincinnatiAnna Zemlyanova, Louisiana State University

Faculty Advisor : Thomas Witelski, Duke University

June 9-12, 2009

1 Introduction

The design of fuel tanks for high-speed vehicles can be quite a complex task.There are many aspects of the tank to consider and there are numerousphysical factors that need to be considered when designing an effective fueltank. There are a number of examples of high-speed vehicles that warrantspecial consideration.

Take a high-speed rocket that travels through the air and possibly intoa zero gravity atmosphere. The spinning of the rocket and the zero gravitycan severely effect the shape of the free surface of fuel. In this case, we canask the question, how can we accurately gauge how much fuel is left in ourtank? We cannot use a traditional sensor used in a regular automobile gastank because the free surface is not a flat, uniform shape, so we must look atalternate ways to measure the remaining fuel in the tank without assuminga convenient free surface.

Also, consider a high-speed boat or drag racer. At high speeds, thesloshing that can occur in the tank can drastically affect center of gravityof the vehicle and, depending on the size of the fuel tank and severity of thesloshing, can negatively affect a control system. The forces that act on the

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wall of the tank can also reduce the integrity of the tank over a extendedperiod of time. The questions to ask in this case are: how can we analyzesloshing in a fuel tank and the forces it produces, and how can we design afuel tank that can minimize the amount of sloshing that occurs?

Questions like these can guide engineers in designing the most effectivefuel tank for a given vehicle, but also can lead to some interesting modelingproblems for mathematicians. In this paper, we will review some differentmodels that have been proposed for sloshing in a fuel tank and performsome analysis on the outcomes of sloshing in a rectangular tank. We willalso examine the overall geometry of a fuel tank and how we can accuratelygauge the volume of fuel remaining in a tank given a certain height of fuel.First, we need to consider what reference frame and model would be bestfor our purposes.

2 Reference Frames

2.1 Fixed Frame

In the paper “Resonant sloshing near a critical depth” [4], Waterhouse mod-ified the formulation given by Ockendon [3] of a inviscid, irrotational, two-dimensional fluid motion of mean depth hL in a tank of length πL andfrequency ω in a fixed frame.

Figure 1: Fixed frame depiction: (left) fluid in tank initially at rest at t = 0on 0 ≤ x ≤ π and (right) at t > 0 sloshing fluid volume in the horizontallyoscillating tank on −ε sin(t) ≤ x ≤ π − ε sin(t).

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Waterhouse considering ε as parameter of perturbation that representsthe amplitude of the horizontal forcing on the tank, h the non-dimensionalizedstill water depth, and φ, the velocity of the fluid, to formulate the followingmodel for time-periodic solutions:

∇2φ = 0,

{−ε cos(t) ≤ x ≤ π − ε cos(t)

0 ≤ z ≤ h+ εη(x, t)

φx = ε sin(t),

{x = −ε cos(t),x = π − ε cos(t)

φy = 0, y = 0∇φ(x, y, t) = ∇φ(x, y, t+ 2π)∫ π0 ηdx = 0

(1a)

and the free boundary conditions at y = h+ εη(x, t) are given by:

φy = ηt + εφxηx (1b)

η +Lω2

g

[φt +

12ε(φ2

x + φ2y)]

= 0 (1c)

where η describes the free surface. The linearized solution is given by

φ(x, y) =4

δπ coshhcosx cosh(y + h) sin t. (1d)

Using separation of variables, we discovered that this model would be diffi-cult for us to work with since the boundary solution implies the null solution.Therefore, we consider an alternate model proposed by Frandsen [2], whichis another variation of Ockendon’s model, but in a moving frame.

2.2 Moving Frame

We find that for analysis of sloshing in a rectangular tank, a model associ-ated with a moving frame is a better choice than a fixed frame. We choseFrandsen’s model because it led to better results and easier computation.

Frandsen’s Model:

∇2φ = 0,

φx = 0,

{x = 0x = b

φy = 0, y = −hsφt = −1

2 [(φx)2 + (φy)2]− (g + Y ′′T (t))ζ − xX ′′T (t), y = ζ

ζt = φz − φxζx y = ζ

(2a)

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Figure 2: Moving frame depiction for Frandsen’s model. In the movingreference frame, the tank is fixed on the interval 0 ≤ x ≤ b.

We are looking for the solution in the form:

φ(x, y, t) =∞∑n=0

cosh(nk(y + hs))cosh(nkhs)

cos(nkx)Fn(t) (2b)

ζ(x, t) =∞∑n=0

cos(nkx)Zn(t) (2c)

where x and y are the horizontal and vertical coordinates, t denotes time,hs is the non-dimensionalized still water depth, k = π/b is the wavenumbercorresponding to the first sloshing motion. Using the perturbation approach,we can get

Zn = Z(1)n + εZ(2)

n + . . .

Fn = F (1)n + εF (2)

n + . . .

solving these equations yield

Z(1)′′n + ω2

n(1 + Z ′′T )Z(1)n = −bnω2

nX′′T (2d)

with ZT and XT representing the vertical and horizontal excitation, respec-tively.

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3 Horizontal Sloshing in a Fuel Tank

φ(x, z, t) and the free surface elevation ζ can be obtained in the followingform: n(t), hs is a still water depth, k = π/b is the wavenumber correspond-ing to the first sloshing motion. Solving equation (2d) with the horizontalresonant frequencies

XT (t) = aH cos(ωHt) and Z ′′T = 0,

we get the new equation

Z(1)′′n + ω2

n(1 + Y ′′T )Z(1)n = bnω

2naHω

2H cos(ωHt).

Its solution is

Z(1)n (t) = −

aHbnω2Hω

2n cos(t ωH)

ω2H − ω2

n

.

Figure 3: Horizontal frequencies vs. maximum of the wave at six differentfuel level heights.

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From figure 3, we can see that there is only one frequency carrying thesystem to an unstable term at each height (i.e. when the amplitude increasedrastically).

3.1 Associated forces on the wall of the gas tank

Consider the classical perturbation approach. Represent the functions Fnand Zn as the asymptotic expansions with respect to the small parameter ε

Fn = F (1)n + εF (2)

n + . . . , Zn = Z(1)n + εZ(2)

n + . . . .

In the main approximation O(ε0) we obtain the following equations de-scribing the linear sloshing under the horizontal excitation XT (t):

Z(1)′′n + ω2

nZ(1)n = −bnω2

nX′′T ,

Z(1)′n − ω2

nF(1)n = 0,

where ωn =√kn tanh(knhs), kn = nk, b0 = b/2, b2n = 0, b2n+1 =

−4b/((2n+ 1)π)2, n ≥ 1.By solving these equations in the case of harmonic horizontal excitation

XT (t) = ah cos(ωht) (assuming we are far away from the initial momentt = 0) we obtain in the case of non-resonant frequency ωh 6= ωn:

Z(1)n =

bnahω2nω

2h

ω2n − ω2

h

cos(ωht),

F (1)n = −

bnahω3h

ω2n − ω2

h

sin(ωht);

and in the case of resonant frequency ωh = ωn:

Z(1)n =

12bnahω

3ht sin(ωht),

F (1)n =

12bnahωh(t cos(ωht) + sin(ωht)).

One of the important aspects in the design of the fuel tanks is identifyingthe maximum force acting on the tank wall. This force can be computed bythe integration of the pressure along the tank wall:

F =∫ 0

−hs

p

ρ

∣∣∣∣x=0

dy.

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Figure 4: Dependence of the maximal force acting on the tank wall on thefrequency of the horizontal oscillation.

Using linearized Bernoulli’s equation [1]:

∂φ

∂t+p

ρ+ gz = 0,

we obtain for the non-resonant case

Fmax = |ahω4h

∞∑n=0

bnω2n − ω2

h

1nk

tanh(nkhs)|+gh2

s

2.

It is easy to see that for the resonant frequency the force will tend to ∞ astime increases.

The dependence of the maximal force Fmax on the frequency of the hor-izontal excitation can be seen in fig. 4. Notice that the force changes littlefor non-resonant frequencies, but increases steeply for the frequencies closeto the resonant frequencies.

Similarly, we can find the maximal free surface elevation for the non-resonant case:

ζmax = maxx

{ahω

2h

∞∑n=0

bnω2n

ω2n − ω2

h

cos(nkx)

}.

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Again for the resonant case this value infinitely increases as t→∞.Assume that the level of fuel is changing according to the law h = hin−

kfut, where hin is initial level of fuel and kfu is the coefficient of the fueluse. As the level of fuel changes the frequency of the horizontal excitationmay become a resonant frequency. The graph of the maximal free surfaceelevation for this case is shown in fig. 5.

Note: There exists an unresolved issue here because there is not justone resonant frequency, but this figure illustrates how one of those resonantfrequencies can be reached by only discreasing the level of fluid.

Figure 5: Dependence of the maximal oscillation amplitude on the time asthe height of fluid is lowered over time.

4 Geometry of the Fuel Tank

The shape of the tank is one of the basic elements of design. We start froma vertical cylinder case. The equilibrium free surface of a liquid inside avertical cylinder of circular cross-section is given by the following equations

1r

d

dr

{rdf/dr

[1 + (df/dr)2]12

}−Bf(r)− λ = 0 (0 < r < 1)

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f ′(0) = 0 f ′(1) = cot θ

The quantities f , r, B and λ are dimensionless, where B = ραa2/σ is theBond number, ρ is the liquid density, α is the vertical acceleration field, a isthe radius of the cylinder (which is fixed in our cylinder case), and σ is thesurface tension. λ is selected by a compatibility condition to make f(0) = 0,af(r) is the height of the surface at a point r, and θ is the given contactangle between the surface and the wall. B = ρga2/σ in the outer space withstrong surface tension will form a meniscus.

Figure 6: The physical configuration of the tank demonstrating the menis-cus.

Since λ is unknown, one classical way is to use the “shooting method”and find the approximate value of λ. But here, we take the derivative atboth sides of the ODE to eliminate λ

d

dr

{1r

d

dr

{rdf/dr

[1 + (df/dr)2]1/2

}−Bf(r)

}= 0 (0 < r < 1) (3)

wheredf(r)/dr = 0 r = 0, df(r)/dr = cot θ r = 1.

Define y1 = f(r)

y2 =rf ′√

1 + f ′2

y3 = y′2

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We change the differential equation (3) into a first order ordinary differentialequation system as follows:

y′1 =y2√r2 − y2

2

y′2 = y3

y′3 =y3

r+

By2r√r2 − y2

2

with the boundary conditions:y1(0) = 0y′1(0) = 0y2(1) = cos θ

Solving this system with Matlab, we have a simulation of static menisciin a vertical circular cylinder. Notice, from figure 6, that the menisci issymmetric about z = f(r), f indicates the distance between the equilibriumposition and the fuel. Actually, this graph shows the curve of the liquid inreality. We can see that the tangent of the curve is significantly bigger nearthe wall of the tank.

Next, we consider the sphere case, which is used in spacecraft. Weknow that B = ραa2/σ, where a varies directly with the height of fluid.To simplify and approximate B, we set B = a2. The angle φ between thesurface and the sphere wall changes as well. Here φ = β + θ. We will try tofigure out the relationship between β and a from figure 7.

By definition, ϕ is the angle between the surface and the radius of thesphere, β is the angle between the wall of sphere and the perpendicular wall,θ is the angle between the perpendicular wall and the surface. From Figure7 (above), we can figure out that ϕ+ β + θ = π/2, cos(β) = a (with R = 1),thus we have β = arccos(a), φ = θ+β. And the boundary condition changesto

∂f(r)∂r

= cot(θ + β) = cot(θ + arccos(a))

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Figure 7: The relationship of angles β, ϕ, and θ.

For the sphere case,

1r

d

dr

{rdf/dr

[1 + (df/dr)2]12

}−Bf(r)− λ = 0 (0 ≤ r ≤ 1)

f(0) = 0f ′(0) = 0f ′(1) = cot(θ + arccos(a))

We take the following steps:

1. Set r̂ = ar, where 0 ≤ a ≤ 1.

2. Pick a such that 0 ≤ r̂ ≤ a.

3. Define r = r̂a , B = a2.

4. Solve BVP for f(r) by Matlab.

5. Then we find the relation between H and R, where H is the height ofthe fluid in the sphere, R is the radius of the spherical tank.

Figure 8 (below) shows that, for different level of fluid, how the curve ofthe fluid at the center of the tank looks like, and we can find the volume ofthe fluid beneath.

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Figure 8: Different levels of fluid in the spherical tank.

Finally, we also consider finding the volume of fluid in the tank by theMonte Carlo integration method. First, we can define a domain of possibleinputs for either a 2D or 3D model for the tank. In our case, we can use asphere with a certain volume of fluid at hand. Next, we can uniformly scat-ter random inputs throughout the sphere. Counting the number of inputsthat fall in the constraints of the liquid and dividing the total number ofinputs in the sphere will yield a ratio of the volume of liquid and the volumeof the sphere, hence giving the amount of fuel in the tank at any given time.

5 Conclusion and Future Work

Although we did not have the time to completely consider all aspects offuel tank design, we were able to analyze specific pieces that affect overalltank design. In the sloshing section, we were able to look at the resonancefrequencies that result in maximal force on the tank wall of a rectangulargas tank during horizontal excitation. We also examined how the resonantfrequencies change as the height of the fuel diminishes. Some questions totake this further would be, how can baffles reduce the amount of sloshing,and how does the overall tank geometry affect the amount of sloshing?

Looking at the overall geometry of the gas tank, we were able to find a

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relationship between the height of the fluid and the overall volume of fluidremaining given several different shapes of fuel tanks. We found that theMonte Carlo integration can be a very useful tool to find volume regardlessof shape, but it is not practical when it comes to application in an actualfuel tank. Further work in the geometry of the fuel tank could includelooking at specific types of fuel level sensors presently used and formingrecommendations for improvements or new designs.

References

[1] Dodge, F.T., The new “dynamic behavior of liquids in moving con-tainers”, Southwest Research Institute, San Antonio, Texas, 2000,195 p.

[2] Frandsen, J.B., Sloshing motions in excited fuel tanks, Journal ofComputational Physics, 196 (2004) pp. 53-87.

[3] Ockendon, J.R. and Ockendon, H. Resonant surface waves, Jour-nal of Fluid Mechanics, 59 (1973), pp. 397-413.

[4] Waterhouse, D.D., Resonant sloshing near a critical depth, Journalof Fluid Mechanics, 281 (1994), pp. 313-318.

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