added; equation (10) should then assume the form already complicated investigation. The effect of dry friction as well as thin-layer lubrication is currently being investigated by the second author. Fs = F„ cos(y - Cs) - Fr sin (y - Cs) The reviewers were hasty to conclude that the second term in 2 equation (15) is an error. Vis the total volume of oil as defined in ± "vV, + {Fq s~in (y - C3) + Fr cos (y - Cs)f the nomenclature and not half volume as used in current litera ture. The coefficient (y ) is correct. This, however, would not affect the method of analysis. The wri 4 Regarding the final remark, the authors are grateful to be ters also question whether, in equation (15), the number % in the shown another way of achieving the same results. second term is accurate. The stability analysis boils down to examining the eigenvalues of the approximate fundamental solution Z (2ir) of the matricial equation Y' = AY with initial values given by a unit matrix. Slightly reworded, we want to find the state transition matrix of Investigation of Pulse Tube 1 the state equation y' = Ay. A simpler approach to the numerical Refrigeration solution would be to obtain Z (2TT) using a Picard iteration solu Downloaded from http://asmedigitalcollection.asme.org/manufacturingscience/article-pdf/96/2/703/6501783/703_2.pdf by guest on 27 September 2021 tion: R. C. Longsworth.2 The theoretical relation for no load cold end temperature based on an isothermal process in the hot end
Z(2TT)= Dzk(2it) predicts temperatures that are slightly warmer than those pro fc=0 posed previously based on the assumption of an isentropic process where in the hot end. Most of the data reported in [12] confirms the Z0=I isentropic assumption; however, the spread in data is biased towards the isothermal assumption. It would be most interesting Z„(2TT) = jA{r)Zn.x{T)d7 for n = 2,.. . k o to know the no load cold end temperatures based on the present data and whether or not they agree with the present theory. The advantage of this method is that the solution is obtained di The proposed relation for heat pumping rates is oversimplified rectly from the state formulation equation (21) instead of solving in that the proportionality constant K is not really a constant but the static differential equation (18). The algorithm of the new is dependent on pulse rate, tube geometry, pressure ratio, etc. In method depends mainly on one recursive formula. However, the developing the empirical relation for heat pumping rate given in main advantage lies in the fact that sufficient components Zn (4) based on over 250 tests with helium it was found that the data (2?r) can be used to reduce the error in the solution to any re correlated poorly based on the assumption (equation 19) that the quired degree. A constraint equation can also be included in the effective mass of gas is proportional to PH ~ PL- Better correlation digital solution to check the eigenvalues. Having tr(A) = —air + was achieved assuming the effective mass of gas to be proportional a T = 1. = | Y | exp {-« ~ i( + 2ir) + a cos T} 0 2 2 The present data are shown plotted in the figure using the corre lation given in [4] along with some limited data for air given in [12]. thus, \G\ = exp (-277a!) The correlating equation from [4] is = ^1^2^-3 in nm tr/Lor* (Tc - Tco) LJtD? (PHf' , ^ _ , qm CpP U 1} j2 = 5J9 TH-TC0 —R — " 345 where \m = e ™", the eigenvalues of G. The equation for the curve drawn through the data points in Accordingly, a-y + 02 + 03 = —jcii- This constraint condition can this figure is be used to check the accuracy of the eigenvalues or, at least, to -26.7 NF estimate the third value. Qout =1 = 5.6<7m(l -e ) Finally, the writers would like to point out that they enjoyed in which the Fourier number, Afc, is evaluated on the basis of reading and discussing the analysis of a fine applied metal cut properties at TH and PH. ting problem. From this figure it is seen that reasonably good correlation is obtained with the preceding relations and also the present data Author's Closure seem to be consistent with previous data. The authors wish to thank Professors Massoud and Morcos for In error: their comments. It is obvious that ay is a function of y, but it is 1 Equation 9 should be: Lc = (Lt-Lo)/ir1/2 + Lo/rc also obvious that the stability analysis is based on a linearized 2 Page 3, right column, paragraph 1: If Lt, Lo, Th, and IT are model with small variations from the mean as given by equation known .... (18). Under these assumptions it is quite legitimate to visualize uy as a parameter reflecting the effect of n only. 1 The authors were also aware of the possibility of including dry By K. G. Narayankhedkar and V. D. Mane, published in the February 1973 issue of the JOURNAL OF ENGINEERING FOR INDUSTRY, friction as well as other recent refinements to the mathematical TRANS. ASME, Series B, Vol. 95, No. 1, pp. 373-378. modeling of the cutting forces. However, it was decided that their 2 Manager Development Engineer, Air Products and Chemical, Inc., inclusion in the analysis at this stage might further complicate an Mem. ASME. Journal of Engineering for Industry MAY 1974 / 703 Copyright © 1974 by ASME 3 Fig. 5: a) Omit Lo from title block since it is a variable, data were not reported. However, experimentation is under way b) Longsworth relation per reference [4] lies parallel to the curve for the same. shown but passes through the first data point, 9.9 watts at 31.5 The proposed relation for heat pumping rate is, no doubt, a mm. simplified relation. However, it provides a reasonably good corre 4 Fig. 9: a) Omit Lt from title block since it is a variable. lation, when proportionality constant K is taken equal to 1, com b) Move data point for 400 mm tube from 40 to 45 mm. bined with the assumption that the effective mass of gas is pro portional to (PH - PL)- Discussion about the correlation from [4] and its extension for Additional References the present data forms a valuable supplement to our paper. [12] Longsworth, R. C, "An Analytical and Experimental Investigation We may mention here that the correlating equation .from [4] does of Pulse Tube Refrigeration," Ph.D. thesis, Syracuse University, Syracuse, not hold good for pulse rates above optimum pulse rate. As per this New York, 1967. correlation, the heat pumping rate increases with increase in the pulse rate, whereas the heat pumping rate should decrease at Author's Closure pulse rates above optimum pulse rate as per experimental inves We thank Dr. Longsworth for his discussion on this paper. It is tigation. Downloaded from http://asmedigitalcollection.asme.org/manufacturingscience/article-pdf/96/2/703/6501783/703_2.pdf by guest on 27 September 2021 quite interesting to note that the spread of the data reported in Corrections in equation (9) and Fig. 9 may be made as suggest [12] is biased towards the isothermal assumption. This itself ed by Dr. Longsworth. However, there are no mistakes in page 3, justifies the assumption of isothermal process in the hot end. As right column, paragraph 1. the heat pumping rates were determined only at cold temperature In Fig. 5, Dr. Longsworth's relation does not pass through the Tc = Tn, the no load cold end temperatures based on present first data point. NOTICE All manuscripts submitted to ASME Headquarters after July 1, 1974 shall use STANDARD INTERNATIONAL (SI) units. Other units may be included parenthetically. Please be advised that your work must conform to the above. By action of the ASME Council, Item 286, June 13-14, 1973. A handbook, ASME Orientation and Guide for Use of Metric Units, is avail able at a cost of $1.00 to members and $2.00 for non-members. It may be obtained from: ASME Order Department United Engineering Center 345 East 47th Street New York. N. Y. 10017 Telephone: 212-752-6800, Ext. 255/256 704 / MAY 1974 Transactions of the ASME