New Design Method For Engine Cooling Fan

Huang Hongbin zheng Shiqin Liu Shuyan Yan Weige

(School of Vehicular Engineering, Beijing Institute if Technology, Beijing 100081)

Abstract Aim To put forward a type of math model for optimizing fan’s twisting law.

Methods This math model was based on turbo-machinery euler equations and calculus of variation, it was conducted for optimizing the aerodynamic parameters along the blade height of the fan and the math method was produced for the optimization of fan’s twisting law.

Results the type 6102Q engine cooling fan was optimized by use of this model,and the calculation data were contrasted with those of iso-reaction coefficiency flow type and free vortex flow type. Some problems existing in long blade can be solved by use of above method.

Conclusion The design paramters needn’t be determined artificially, so calculating results are more rational to a high degree than that from other methods.

Key words: cooling fan, twisting law, optimum design

The design of fan has been a hard work on the orientation of aerodynamics because of the omplicated flow through the blades, so the fan had been designed by use of kaufman theory. This law believes that the flow through the fan blades is of one-dimension , the airflow parameters at the mean blade diameter are taken into account, but the flow through the root and tip is negative. After that, fan was projected according to the simply radial balance equation. Numerical precision was enhanced by use of completely radial equilibrium equation and iso-reaction factor of twisting law to determine the air-flow parameters along the blade radial direction, so the flow losses of tip and root are lessen to certain extent.

In this paper ,the authors put forward a math model for optimizing airflow parameter along blade height by use of euler equations and calculus of variation.

1．MATH MODEL

While the minute matter △G flows around the blade which is formed by two neighboring flow surfaces,according to Euler equation,fan’s power is

△P=? (v2ur2- v1ur1) △G (1)

Where ω is the angular velocity of the fan , v1u is the circumferential speed at the fan inlet , v2u is the circumferential speed at fan outlet , r1 is the fan inlet radius, r2 is the fan outlet radius, For the case of non-guide blade, Eq.(1) becomes

△P=?v2ur2△G (2)

We set up the relations between r1and r2 by use of the flow function based on continuity of flow. The

flow function is constant along the flow surface , and the thoroughfare surface of flow passage region is considered as flow surface. Thus, we have the definition of the flow surface

△G= 2?△? (3)

Substitute Eq.(3) into Eq.(2), and integral Eq.(2),then

P= 2??Where P is the effective power of the fan,?is the flow function of the blade-tip The theoretical power P1 is

?1?0v2ur2d? (4)

Pt= 2???10(v2p/2)d?1 (5)

Where vp is the theoretical speed corresponding to P1 Form Eqs.(4) (5),the fan efficiency is

??1r012?vrq(r0)dr0, (6) 22u2?r02GvpWhere r01 and r02 are internal and out radii respectively at the fan’s inlet stretching region, q(r0) is flow of streams per inlet blade height, G is the flow of matter

On the basis of Euler equations, the fan’s power Ph is

Ph??v2ur2 (7)

Substitute this equation into the first law of thermodynamics

2(H2?H1)?(V2?V1)?2v2uu2?0 (8)

Where v1 is the absolute speed of the fan inlet, v2 is the absolute speed of the fan outlet,H1 is the inlet enthalpy of the fan,H2 is the circumferential speed of the fan outlet.

According to the speed triangle of cascade , substituting the relations between speeds, we can obtain the energy equation of relative motion while static entropy

Keeps constant

2W22t?W12?2(H2t?H1)?u12?u2 (9)

From above equations, actual outlet speed of heat insulation that friction exists Is obtained

21/2W22?(?2(W12?WR2?u12?u2)), (10) Where w2tis relative speed of fan outlet as communal entropy course, w1 and w2are relative speeds of

the fan inlet and outlet respectively, u1 is the circumferential speed of the fan inlet, H2 is the static enthalpy of the fan outlet as communal entropy course, ?2 is the outlet speed parameter of the fan,

WR2?2(H2t?H1).

According to the triangle of speed in the three-dimension space, we have

22221/2v2u?u2?(?2(w12?wR?u12?u2)?v2 (11) z?v2t) Huang Hongbin et al./ New Design Merhod for Engine Cooling Fan

Substituting Eq.(11) into Eq.(6) yields

1r022?222222220.5(12) ???{?rf2?r{?[?v?(1??)???r]?v?vf22pRf222r}}q0dr0 22frG01vpFor the fan of non-guide blade,v1f =v1r , v1r=0 According to flow continuity

qdr0 =

((??1v1rrf1rf'1/r1)dr0 (13)

qdr0 =((??2v2rrf2rf'2/r2)dr0 (14)

where ?1??1/?2, ?2??1/?2, ?1 is the inlet speed factor , ?2 is the outlet speed factor , ?1 is inlet flow matter factors, ?2, is outlet flow matter factors,rf1 is inlet streamline radius, rf2 is outlet streamline radius.

Eq.(12) belongs to the extreme value problem with qualifications, it can be solved by use of Lagrangian multiplier, the Lagrangian function is

2222220.5 F?2?{{2?/VP2[?rf22?rf2{?2[?2v2p?(1??)?R??rf2]?v2z?v2r}??1(r0)??2(r0)}q0/2???1(r0)?1v1zrr1rr,1/r1??2(r0)?2v2zrr2rr,2/r2 (15) Where ?1(r0) and ?2(r0) are lagrangian multipliers

According to the relation of aerodynamics, the relationship of densities between inlet and outlet are

221/(k?1)?2= ?1[1?(k?1)?R/(2va (16) 2)]222Thus va2?(k?1)?R?va1 (17)

Where va1 is inlet sound speed, va2 is outlet sound speed.

For the extreme value problem of Eq.(12),we make use of the Euler-lagrangian equations

v2r?F??rq0?0,2f2v2u?v2rGvpv2rq0?F2?1' (18) ?r??(r)?rr202r2r2?0 2f2v2u2?r2?v2wGvp2q0?F2??vf212''??1?1??w??(r)vrr?0, (19) R101rf1f1?wRGv2v2?r?w2u1Rp???Fd?Fd1?()???1(r0)?1v1r?[??1(r0)?1v1rrf1]?0,?rf1dr0?rf1dr0r122?rf2q0?Fd?F2?12?()?(2?r?v??)??(r)?rr2rr'2f22u2022?rf2dr0?rf2v2w2?r2Gvp?d1[??2(r0)?2v2rrf2]?0dr0r2 (20)

Where

2220.5v2u?{?2[?2v2p?(1??2)wR??2rf22]?v2z?v2r} (21)

From Eq.(18) we have v2r= 0, Integrating Eq.(8) and Eq.(19)

2Gvvp22u?(r)v2= (22) 20rrf22?2Substituting Eq.(13) into Eq.(21)

d?1(r0)1d?11dr11dv1??（r)[???0 (23) 20dr0?1dr0r1dr0v1zdr0Substituting Eq.(14) into Eq.(22),we get

?2v2r1rf2 r222?rf2drf2d?01dv2z1dr22?2??(r)[?]}?0 (24) (2?r?v??){+ 20f22udrvdrrdrvdrGv201z0102w0pSo we obtain the extreme equations corresponding to the efficiency,i.eEqs.(13)(14)(16)(17)(20)(23)-(26).To sum up, we can obtain a conclusion that the streamline dip of the fan outlet section ought to keep zero,it is calculated by use of radial balance equation.

2 OPTIMUM DESIGN

2.1 Variables, Objective Function and Restraints

The reaction parameters along radial direction were taken for design variables, so objective function is

???minf[Ω(j)], (25) Where Ω（j）is reaction parameters, j is the number of streamlines along radial direction of blade .The equation about ? determined by Eq.(12).

Some restraints should be taken into account from designing and experimental courses of fan:

That the reaction parameters must keep positive along the radial direction (i.e, Ω>0) would protect separated flow at the root, and the reaction parameters must also be larger than 0.50 for relative speed to keep slow at the root. At the tip, these parameters must be smaller than 0.75,for the sake of little leakage. The geometry expanding degree of the fan passageway along the radial direction must keep larger than 1.0,that is sin?2/sin?1>1, where ?1and?2are respectively the flow angles of fan inlet and outlet. Relative inlet and outlet maches must be restrained because they influence fan sound i.e M?1<0.3 and M?2<0.3.

Axial part of absolute fan outlet speed v2z must be positive along radial direction, otherwise the