The differential equation of the electromagnetic injection period.

Blaga, Vasile ; Teusdea, Alin ; Chioreanu, Nicolae 等

1. INTRODUCTION

One writes the Bernoulli relation between sections 1-1 and 2-2 as shown in the figure 1. We obtain a Riccatti-type differential equation of the duration of the injection. The flow passes through the electro-injector and the flow area is also computed.

2. DYNAMIC ANALYSIS OF THE DISCHARGE

We deduce the Bernoulli relation between the sections 1-1 and 2-2 as is shown in figure 1, (Blaga, 2000). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Where we consider the section [S.sub.1] = [S.sub.2] = S (constant section), [v.sub.1] = [v.sub.2] = v the speed through the section, [beta] [congruent to] 1 coefficient of the mean speed, [z.sub.1] = [z.sub.2] position heights, [gamma] specific weight of the gasoline, g gravity acceleration,

[h.sub.r1-2] = ([lambda] 1/D - [k.sup.2] + [summation][xi])[v.sup.2]/2g

drop in energy due to resistances, [lambda] coefficient of the carrier loss, l length of the pipe, D diameter of the pipe, k = [v.sub.c]/v speed ratio, [summation][xi] sum of coefficients of local loads losses, [p.sub.1], [p.sub.2] pressures from the sections 1 and 2 respectively. The relation (1) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Taking in account the relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where m = l x S x [rho] is the gasoline mass accelerated through the pipe and [rho] [member of] [0.682;0.767] is the density of the gasoline.

One obtains the Riccatti-type (Micula & Pavel 1992) differential equation

m dv/dt + [cv.sup.2] = [DELTA]p x S (3)

[FIGURE 1 OMITTED]

where

c = ([lambda]1/D [k.sup.2] + [summation][xi]) [rho] x S/2.

The global resistance coefficient is denoted by [lambda]1/D [k.sup.2] + [summation][xi] = [[xi].sub.g]. Since [[xi].sub.g] = 1/[[alpha].sup.2] where [alpha] is the discharge coefficient ([alpha] = 0,7 ... 0,8) we obtain

c = [rho] x S/2 x 1/[[alpha].sup.2] = [rho] x S/2[[alpha].sup.2] dt = mdv/[DELTA]p x S - [cv.sup.2]

Let [v.sub.0] be the speed from the section y after the stabilization and let be T the time constant (natural time). When the speed is stabilized, from the equation (3) one obtains

dv/dt = 0. Hence [v.sub.0.sup.2] = [DELTA]p x S/c or [v.sub.0] = [square root of [DELTA]p x S/c] and c = [DELTA]p x S/[v.sub.0.sub.2].

We have after some computations

dt = (m x vo/[DELTA]p x S) (d(v/vo)/1-[(v/vo).sup.2]), 1/vo dv = d(v/vo).

The time function is (O'Neill, 1991)

t = [mv.sub.0]/[DELTA]p x S arcth v/[v.sub.0] + C = T x arcth v/[v.sub.0] + C (4)

Where T = m x [v.sub.0]/[DELTA]p x S = m x [square root of [DELTA]p x S/C]/[DELTA]p x S = m/[square root of [DELTA]p x S x C].

From the relation (4) we deduce arcth v/[v.sub.0] = t/T - C, v = [v.sub.0]th(t/T - C). For t=0 one obtains v=0 hence C=0 and v = vo x th t/T. The stationary regime state v = [v.sub.0] is achieved for t = [infinity] therefore

[t.sub.0] = 2,647T = 2,647 m/[square root of [DELTA]p x S x C].

3. THE DISCHARGE THROUGH THE INJECTOR

Q = v x S x y = v x [S.sub.t], where: y = [a.sub.c] x [t.sup.2]/2, t [member of] (0,[t.sub.0])

[a.sub.c] is the lifting acceleration of the needle [S.sub.t] is the plan area (section area) of the needle of the injector, which has a frustum of a cone profile as it can be seen in figure 2. The base of this profile formed by the needle point has the diameter [d.sub.2], and the big base has the diameter of the needle [d.sub.1] = [d.sub.2] + 2e.

The length of the generating line is h = y cos [alpha] and the area of the passing section is [S.sub.t] = [pi] [d.sub.1] + [d.sub.2]/2 h = [pi] [d.sub.1] + [d.sub.2]/2 y cos [alpha], hence Q = [pi]([d.sub.1] + [d.sub.2]/2)cos [alpha] x [a.sub.c] x [t.sup.2]/2 x [v.sub.0]th t/T.

We denote by A = [pi]([d.sub.1] + [d.sub.2]/2)cos [alpha] x [a.sub.c]/2 x [v.sub.0], t [greater than or equal to] [t.sub.0], t [member of] ([t.sub.0], [t.sub.1])

Where [t.sub.1] is the time up to the end of the stroke of needle and [t.sub.0] is the stabilization time of the liquid and denote by [Q.sub.1] = [v.sub.0] x S x y . With the notation A' = [pi]([d.sub.1] + [d.sub.2]/2)cos [alpha] x [a.sub.c]/2 [v.sub.0] = A we obtain [Q.sub.1] = A' x [t.sup.2] or [Q.sub.1] = A x [t.sup.2], t [greater than or equal to] [t.sub.0]; t [member of] ([t.sub.0], [t.sub.1]). The gasoline volume which passes over the injector for t [member of] (0, [t.sub.0]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where th t/T = v/[v.sub.0]. In order to integrate in (5) we will proceed to the linearization of the function (Chioreanu, 2006)

tg[alpha] = 0,99v/[t.sub.0] = v/t with [t.sub.0] = T x arcth0,99 = 2,647T.

After linearization we obtain :

v = 0,99[v.sub.0]/2,647T x t = 0,374 [v.sub.0]/T x t.

Finally, one obtains

Q = [pi] x ([d.sub.1] + [d.sub.2]/2)cos [a.sub.c] x [t.sup.2]/2 x 0,375 [v.sub.0]/T t = A 0,375/T [t.sup.3] = B[t.sup.3], B = 0,375 A/T. After a short computation, the gasoline volume which passes over the injector it can be determined from the formula dV = B[t.sup.3]dt. One obtains

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [t.sub.1] = [square root of 2[y.sub.max]/[a.sub.c]]. The gasoline overall volume [V.sub.t] is

[V.sub.t] = [V.sub.0] + [V.sub.1] = B/4 [t.sup.4] + A/3([t.sub.1.sup.3] - [t.sub.0.sup.3]).

[FIGURE 2 OMITTED]

We are presented the gasoline discharge dependent on lifting time t and the gasoline discharge dependent on the lifting height of the needle of the electro-injector, respectively, for the following values: [d.sub.1] = 3mm; [d.sub.2] = 1mm; y = 0.15mm; [rho] = 0.7kg/[dm.sup.3]; [DELTA]p = 1.1daN/[cm.sup.2]; t = 9/[micro]s.

4. CONCLUSIONS

The quality of the pulverization of the fuel depends on a several factors related to the construction conditions (e.g. injection system type and also injector system type, the number, shape and the sizes of structural dimensions of pulverization orifices) and working conditions (e.g. injection pressure, backpressure and air density from the combustion chamber, flow rate of the jet). Very important factors are also the physic-chemical properties of the fuel: combustion value, fractional composition, octane number, gasoline purity and the running conditions (Blaga, 2007).

5. REFERENCES

Blaga, V. (2000). The modeling injection from gasoline at engines with spark lightning through sparking, The University from Oradea Publishing House, ISBN 973-8083-41-9, Oradea.

Blaga, V. (2007). Engines for motor vehicles and tractors, The University from Oradea Publishing House, ISBN 978-973-759 408-2, Oradea.

Chioreanu, N. & Chioreanu, S. (2006). Heat engine for nonconventional motor vehicles, The University from Oradea Publishing House, ISBN 973-759--065-1, Oradea.

Micula, Gh. & Pavel, P. (1992). Differential and Integral Equations through Practical Problems and Exercises, Kluwer Academic Publishers, ISBN 0-7923-1890-0, U.K.

O'Neill, P.V. (1991). Advanced Engineering Mathematics, Wadsworth Eds, ISBN-10: 0534552080, U.K.