ABSTRACT
The basic formula governing "flash temperatures", A T, i.e. the average local temperature rise above ambient at contact spots, as given by Jaeger (34) and elaborated in (35,36), is
A T = F(Z, S)AT0 = F(Z, S)7rqr/4XB (20.35)
Here q is the rate of heat evolution at the contact spot in units of wattsec per unit area and time, and A To is the equilibrium temperature rise which would arise on account of q at a circular contact spot at rest, of radius r, between a flat asperity of infinitely extended brush material (with thermal conductivity X) and an infinitely extended insulating substrate. F(Z, S) < 1 is the factor by which the actual temperature rise, AT, is smaller than A To on account of sliding velocity v (expressed through the function Z0) and ellipticity of the spot (expressed through a shape function S). Z0 is a function of the "relative velocity" (Peclet number) vr = v/vo, wherein the characteristic velocity is
vo = K B I r = (X BIDBcB)1 r (20.36) with KB, DB and cB the thermal diffusivity, density and specific heat of the brush material, respectively, (compare Table 20.3), i.e.,
Vr = (r/KB)v = r(DBcB /A.B)v (20.37) Strictly speaking, if the contact spot moves relative to both sides, as it may do in the
case of large monolithic brushes but cannot do with fiber brushes, the relative velocity between contact spots and substrate, vrB, differs from Eq. (20.37) by the amount of the contact spot velocity on the brush, and one must also consider the corresponding relative velocity of the contact spot on the substrate, i.e. ?As. In full generality, then, with the subscripts S and B designating the properties of the brush and substrate materials, and
= Xs/),B, it is (36, Pt.I) F(Z, 8) = [1/Zo(vrB)S(e, VrB)+Ar/SO(v,$))-1
(20.38) where e is the ellipticity of the spot, namely the ratio of the long to the short axis. The functions Zo(vri) and S(e, vri) with subscript i referring to either S or B as appropriate, are not analytical but they have very serviceable approximations as follows:
Zo(vrB :5 2) = 1/[1 + vrB/3] (20.39) Zo(vrB L 2) (9/8)/(v 1/32 + 1/81 /2) (20.40)
both with a maximum error of — 3% in opposite directions at v = 2vrB (36, Pt I, Fig. 2), and
1) = So(e) = (4/3)/[(1 + e314/3)(1 + 3 e3/4))1 /2
(20.41) S(e, yr; > 1) = [el/4 ei/4/ovriv]m e3/4/(8v,01/2] (20.42) The accuracy of Eqs. (20.41) and (20.42) is believed to be similar to that of Eqs.
(20.39) and (20.40), but since complete computations by Jaeger (34) are available for verifying Eqs. (20.39) and (29.40) but only five computed points for Eqs. (20.41) and (20.42), all for vr = 1, one cannot be certain (36, Pt. I, especially Fig. 4). Representative sets of computed functions of F(Z, S) based on Eqs. (20.39) to (20.42) have been published previously (35,36). The relevant curves for fiber brushes, i.e. with stationary, essentially equiaxed contact spots at the fiber ends (4) so that vrB = vr, yrs = 0 and S(e, yr ) = 1, are reproduced in Fig. 20.14. Furthermore, almost univerally actual situations will represent the low-speed case, i.e. v,. < 2, as demonstrated in Fig. 20.15 showing vo(d, )8) for copper fiber brushes, with K = 1.14 x 104[MKS]. Hence with copper brushes sliding on copper substrates, F(Z, R-•-
