I N a recent Technical Note, Hall l considered free molecule heating of micron size particles at hypersonic speeds. In the course of the analysis he made use of the drag and heating expressions for spheres in free molecule flow taken from Ref.2. The purpose of this Comment is to point out that the expression quoted by Hall from Ref. 2 for the recovery factor is erroneous. The error is only important for low values of the speed ratio (Mach number) so it probably does not affect Hall's results, which are for hypersonic flow. However, the error appears in so many standard works on free molecule flow that it seems worthwhile to point it out.In fact, it seems that the formulas for drag and heating of spheres in free molecule flow have suffered more than the usual number of misprintings. Not only is the heating formula wrong in Ref. 2, it is also wrong in the related books, Refs. 3 and 4. A different incorrect version appears in Ref. 5. Furthermore, the drag formula is in error in Refs. 3 and 5, although Hall has used the correct version in Ref. 1.On the other hand, some of the earlier papers in the field presented the correct formulas. As examples one can cite the papers of Sauer 6 and Oppenheim 7 for heating rate, and Stalder and Zurick 8 for drag. From these works, or by following the derivation in Ref. 3 for example, one can obtain the heating rate to a sphere of radius a and surface temperature T wt in a stream of density p, temperature T, and speed £7, consisting of a gas with gas constant R and specific heat ratio 7. The expression iswhere a is the thermal accommodation coefficient, T r the recovery temperature, and St f the modified Stanton number. The latter two quantities are expressed in terms of the speed ratio 5, which is related to the Mach number by S=(y/2) 1/2 M=U/(2RT) 1/2 (2)It is the ratio of flow speed to the most probable random speed in the flow. St' and T r are correctly expressed aswhere r' is the modified recovery factor '-y ierfc(S)]+(7-S-2 /2)erf(S) r = -S 2 + Sierfc(S)+0.5erf(S)The error that appears in Ref. 2, Ref. 3 [Eq. (6-5)], Ref. 4 [Eq. (36)], and is repeated by Hall ! in his Eq. (3), is in the last term in the denominator of /*', Eq. (5), where they have a factor of 1/S 2 . This factor magnifies the importance of that term at low values of S, and causes r' -»0 as S->0, instead of the correct limit r' -8/3. The error in Ref. 5 [Eq. (10. 5. 23)] is in St', where the factor S has been omitted from the first term in the numerator. The expressions given here are in agreement with the spherical results in Table 1 of Ref. 7, and also with the monatomic (7 = 5/3) and diatomic (7 = 7/5) results of Ref. 6. Hall and others frequently use more conventional expressions for Stanton number and recovery factor, which do not have the primes on the symbols, and are related to the modified expressions defined above by = r'y/(y + (6) The advantage of the modified quantities is that they depend only on the speed ratio S, not on 7 or a.For completeness, the correct expression for a sphere drag coefficien...
