1、0_.2;.-,.i, this equation will automatically satisfy thefirst boundary condition of horizontal tangency at x=0. If thed(_x)a = hT2, s is to be satisfied, the conditionsurface condition _kof equation (4) must be satisfiedT2 c (4)M= _+nFrom equations (I), (3), and (4)gmax = T2cTon _ (5) n+l _+nCOC_Pro
2、vided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 2955 7or, if R-n T2, cTO (n+l) equation (5) readily reduces toi i n i*max- + “giEquation (6) suggests that a plot of a.maxstraight line. When values of a*ma x and(6)1versus _ should be afrom figure
3、 2 are used,1 1the plot a. versus _ In figure 4 is obtained. It is seen thatmax_ 0.2 or _ S), the curve deviates some-what from the straight line, curving downward and reaching a limit= 1.0 at i/_ = 0 instead of a value of 1.5 predicted by thestraight line. To make the formula accurate over the enti
4、re range, itis desirable to add a term that will be effective only in the very lowrange of I/# and cause the expression to reach the proper limit ati/_ = O. An exponential term serves this purpose well over the entirerange of _; therefore, the following equation relates p and a*.1 3.25 -lS/ a-T-= 1.
5、5 + o.5 e (8)maxFigure 5 shows the correctness of fit of equation (8) and the exactresults over the range 0 _ 20. For values of _ between 0 and 5,the exponential term is negligible and the fit of the exact resultswith equation (7) is essentially the same as the fit with equation (8).No refined calcu
6、lations have been carried out for values of _ above 20,but a comparison of equation (8) with an asymptotic formula given byCheng (ref. 2) indicates that equation (8) may be in error by as muchas 5 percent at a value of _ = 200. However, since most practicalproblems involve values of _ below 20, equa
7、tion (8) is seen to giveresults of unusually good accuracy over the practical range of B.If, however, greater accuracy is desired in the range of 8 from 5 to20_ the formulaProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACA TN 2955l-L-= 1.o + (8a)
8、*max #2/5can be used in this range together with equation (7) in the range0 B 5. Equation (8) has, however, the advantage of representingthe entire range of 8 with a single formula.In his original paper (ref. 5) Buessem derived a simplifiedformula for this case in another manner. It was assumed that
9、 thestress could be approximated by taking temperature distribution in theplate at the time of maximum stress as the straight line PQ infigure 5. By determining first the surface stress for this temperaturedistribution, and then adjusting the resultant formula so that it wasconsistent with the corre
10、ct surface stress values at two values of_, the following equation was obtained:1 4 (9)a-W-.- = z +maxEquation (9) is very similar in form to equation (7), but it does notfit the correct curve of G* versus 8 quite so well as equation (8)over the entire range of 8. Figure 5 shows the degree of corre-
11、lation between analytical results of figure 2 and the simple formulasgiven in equations (7) and (8). Also shown is the correlation obtainedwith the Buessem formula; which, although very good, is not so close asthat with the formulas presented herein.c_GcThermal shock parameters. - Use can now be mad
12、e of the approximateformulas to correlate the maximum stress developed in a material withthe physical properties of materials. In most cases it is found thatthe value of _ for reasonable heat-transfer coefficients, plate thick-nesses, and conductivities is relatively low so that the term 1.5 inequat
13、ion (7) can be neglected compared to the value 5.25/_ forpractical purposes. In this case, equation (7) becomes equation (10),which can be rewritten as equation (ll).i 5.25a.ma.,. 13 (lO)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2E NACA TN 2933
14、 9orkCmax 3.25 (i-_) (ii)TO = E-_ ahFor the case where failure occurs, Cma x = cb = breaking stress; hence,kCb 3.25 (i-H) (ila)T0, max - E_ ahThis equation states that for the case of a flat plate of thickness aand heat-transfer coefficient h, the maximum shock temperature that canbe withstood by th
15、e plate is proportional to the product kUb/E_. SincePoissons ratio _ is very similar for all materials, it is placed inthe group of terms not involving materials properties. This groupingkUb/E_ is identified as the thermal shock parameter used by Bobrowsky(ref. 4) and by others. Equation (ii) gives
16、a numerical measure ofshock temperature that will cause failure and provides the basis foran index for listing materials in order of merit. Table I shows resultsof tests conducted in reference 4 showing the order of merit of severalmaterials according to the thermal shock parameter kCb/E_. These tes
17、tsconsisted of subjecting a round specimen 2 inches in diameter and1/4 inch thick to thermal shock cycles until failure occurred. In thiscycle the specimen was first heated to furnace temperature and thenquenched in a stream of cold air directed parallel to the faces of thisspecimen. If the specimen
18、 survived 25 cycles at one furnace temperature,the furnace temperature was increased 200 F and the tests were repeated.In this way the temperature was raised until failure finally occurred.The table shows that a good correlation was obtained between the maximumtemperature that was achieved and the t
19、hermal shock parameter k_b/E_.When equation (8) is again considered, it is seen that for verylarge values of _ the value 5.25/_ can be neglected compared to theother terms and _*max becomes equal to unity. It is interesting toexamine the meaning of O_ma x = i and to determine under which condi-tions
20、 O_ma x = i is achieved. The condition O_ma x = i means thatE_T 0 (i2)_max - i-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-l0 NACA TN 2933The product _T0 is the contraction in the material that would takeplace if the temperature were reduced by T
21、O and the material allowedto contract freely. If contraction is completely prevented by appli-cation of stress, then _T0 is the elastic strain that must be inducedin the material to prevent this contraction, and this strain multipliedby the elastic modulus becomes the stress that must be applied. Th
22、eterm (i - _) results from the fact that the problem is for an infiniteplate in which equal stresses are applied in two perpendicular directions.In this case E_T0/(I - _) is the stress that must be applied in twoperpendicular directions to completely prevent any contraction in thematerial. Hence, fo
23、r very large values of ah/k, equation (8) statesthat the stress developed is just enough to prevent any thermal expansion.To obtain an index of merit for rating materials under the conditionsof very large _, equation (12) is rewritten as equation (13), whichsuggests that this index is now _b/E_; and
24、 it is seen that the con-ductivity factor has vanished compared with the index kab/E_.Co_bTO ,max - _ (I-_) (13)The implication is that it does not matter what the conductivity of thematerial is_ the temperature that can be withstood is in proportion toOb/Ea. Physically, this result can be understoo
25、d by examining the mean-ing of very large p, which condition can occur either if a is verylarge, if h is very large_ or if k is very small. If a is verylarge, it means that the test body is very large and that the surfacelayers can be brought down to the temperature of the surrounding mediumbefore a
26、ny temperature change occurs in the bulk of the body. The sur-face layers cannot contract because to do so they would have to deformthe remainder of the body, and this cannot be achieved for a very largebody. Henee_ in this case, complete constraint of contraction is imposed,and the stress developed
27、 is EaTo/(I-_ ) irrespective of the actual valueof conductivity. Similarly, for large heat-transfer coefficients h,the same result can be expected. The surface is brought down to the tem-perature of the surrounding medium before the remainder of the body hashad the time to respond to the imposed tem
28、perature difference. Hence,again complete constraint of contraction is imposed, and the stressdeveloped is independent of conductivity. Finally, if the conductivityis very small, again only the surface layers can realize the imposedthermal shock conditions, the remainder of the body remaining essent
29、iallyat the initial temperature. Again, complete constraint against thermalcontraction is imposed and the stress is independent of the precisevalue of k providing it is very small.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 2933 iio!ooGoTh
30、at there are two thermal shock parameters at the two extremes ofthe _ scale, kOb/E_ and Ob/E_, has recently been emphasized byBradshaw (ref. 5). That both thermal shock parameters are necessary todetermine completely the thermal shock resistance of a material has beenemphasized by Buessem. The merit
31、 of equation (8) is that itprovides a simple formula for determining the relative roles of the twoparameters over the complete range of _.Significance of test conditions. The previous result, namely,that the index of merit is proportional to kOb/E_ for low values ofand proportional to ab/E_ for high
32、 values of _, suggests the impor-tance of the test conditions used to evaluate materials. In figure 6there is plotted the temperature that could be withstood in the testspecimens described in reference 4, and shown in table I, for differentvalues of ah, that is, if they had used different specimen t
33、hicknessesor different heat-transfer coefficients instead of the values actuallyused. These curves were obtained from equation (8) in conjunctionwith the material properties given in table I. It is seen thaiat the low values of ah, the test condition actually used being repre-sented at a value of ah
34、 = approximately i0, the order of merit of thematerials, that is, the temperatures which could be withstood withoutfailure, is in agreement with the experimental observations. For highervalues of ah the index of merit can be reversed. For example, at anah of 80, zircon becomes better than beryllium
35、oxide, and for even highervalues of ah beryllium oxide, which was quite good at the low valuesof ah, becomes the poorest of all materials. This reversal is due tothe fact that beryllium oxide has outstandingly good thermal conductivityand that at the low values of ah, the index of merit takes advant
36、ageof this good conductivity. At the higher values of ah, the effect ofthe good conductivity grmdually diminishes until at very high valuesthe high thermal conductivity has no beneficial effect at allThe importance of the possibility of reversal of merit index shouldbe emphasized because it strongly
37、 suggests that test conditions shouldsimulate as closely as possible the intended use of the material. If,in order to obtain more rapid failure and thereby to expedite the test-ing procedure, more drastic conditions are imposed than the true appli-cation warrants, the order of merit of materials can
38、 be reversed, andthe results rendered meaningless. The results of simple tests recentlyconducted at the Lewis laboratory will serve as experimental verification.The tests were conducted on specimens of beryllium oxide Be0 andal_ninum oxide AI203 under two conditions of quench severity. All speci-men
39、s were disks 2 inches in diameter and 1/4 inch thick and were quenchedon their outer periphery while the sides were insulated. Air and waterProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12 NACATN 2933sprays were used as the quenching media. Figure
40、7 shows the analyticaland experimental results. The curves show the variation of thermalshock resistance with quench severity for the two materials. Thesecurves were obtained from equation (8) in conjunction with mate-rial properties listed by Bradshaw (ref. 5). For BeO these proper-ties were apprec
41、iably different from those given in reference 4; hence,the curves for BeO in figures 6 and 7 are different. However, Bradshawpresents data for both BeO and A1203, and these data seem better toillustrate the experimental results. It is seen from figure 7 that forlow values of ah, BeO is superior to A
42、I203, but that for severe quenchesBeO becomes distinctly inferior. This behavior is due, as previouslymentioned, to the high conductivity of Be0, which is of value in improv-ing thermal shock resistance primarily for mild quenches. At the severequenches AI203 assumes superiority owing primarily to i
43、ts better relativebreaking strength. The experimental results are shown in the insert inthe upper right section of figure 7. In the air quench the superiorityof Be0 is evidenced by the fact that it withstood any temperature lessthan 1425 F, while the AI203, failed at i000 F. In the water quenchBe0 b
44、ecame inferior to AI203, failing when quenched from 800 F, whileAI203 withstood quenching until the temperature was 950 F. Because theactual air and water temperatures just before impingement on the speci-mens were not known, the true quench temperatures are not determinable,but qualitatively these
45、tests certainly indicate the importance ofquenching conditions on the determination of the relative merit of mate-rials in their resistance to thermal shock.Another Snteresting aspect of these tests is that the cross-overpoint P between the two curves occurs very near the flat part of theAI203 curve
46、, at which point the BeO curve is still fairly steep. Inthese tests, the quenching conditions must have been in the region ofthe crossover point, because actual reversal of index of merit wasobserved. This may explain why the failure temperature did not changemuch in the two types of quench for the
47、AI203, but appreciable changewas observed for the Be0. Again, however, this conclusion must be con-sidered very qualitative because the temperatures of the air and thewater were not known.Stress at center of plate. - Thus far only the maximum stressdeveloped at the surface of the brittle plate has b
48、een discussed, andalso it has been tacitly assumed that the duration of the thermal shockwas sufficient to allow the maximum stress to be developed. For quench-ing from a high temperature, the surface stress is tensile, and ingeneral failure occurs at the surface. In the case of rapid heating,rather
49、 than rapid cooling, the surface stress is compressive, and surfacefailure may occur as a result of spalling, or as a result of the shearstress induced by the compression. Failure may, however, occur firstnot at the surface, but at the center of the plate where the largesttensile stress is developed. An important case is that in which theCOProvided by IHSNot for ResaleNo reproduction or networki
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