N. S. Heaps

English

MAGiC ARC Repository

C.P. No. 299 (16,932) A&C. Technical Report C.P. $Jti 299 A&C. Technkal RepOK MINISTRY OF SUPPLY AERONAUTICAL RESEARCH COUNCIL CURRENT PAPERS Transient Thermal Stress in a Flat Plate Due to Non-Uniform Heat Transfer Across One Surface N. S. Heaps, E.Sc. LONDON: HER MAJESTY’S STATIONERY OFFICE 19S6 FOUR SHILLINGS NET C.P. NO* 299 U.C.L. i43. 533.319 : 5356.Cll.6 i1 Xii Iv -2- 1 hL2 O.Ok2 - * -- = --2 - + 7.306 . 133 ks G I Introduction The problem solved in this paper is that of a long rectangular plate at uniforn temperature which 1 2 suddenly exposed ta a higher temperature in the immediate proximity of its top surface.* On the exposecl surfaoe s heat transfer coeffioient i.5 postulated whioh varies inversely as the square of the distsnoe f'ron one of the longitudinal edges. A state of no heat transfer is postulated for the edges of the plate. W&r these ooii- dltions there is a non-uniform transferenoe of heat to the plate, giving rise to transient temperature gradients and thermal stresses within the plate. The temperatures snd stresses are derived theoretically with certain dmplifyjng assumptions. The solution in terms of Bessel &notions is then used to estinate transient therms1 stresses educed in sn akcrsft w5ng when suddenZ,y acquiring a high supersonio velooity. Hofd cal*,ulated the stzesses in a spar web by assdng a unifarm heat trender coefficient on the dng surfaces, and thiq work was later extended by Parkes2. The stresses new considered are those in a wing skin due to chordwise variation of heat transfer coeffioient on the wing surfaoes clesortiea by we3. Th? law of variation of heat trsnsfer coefficient postulated was chosen to give a convenient analytioal solution of the heat corduotion equation> ana TV be at the sezne t%re a reasonable approximation to the actual variation of heat transfer ooef'floient in f'Eght. 2 Statement of the prcblem Diagrams of the plate under consideration are sho~m in ?Ji&l[(a)~(b)]~ 0 is a paid on o%e of the longitdnal edges end x, y are distances measured from 0 in lateral and longituclinal directions~ respectively. The width of the plate is denoted by L, and its thickness by s. The them1 conductivity of the material of %e $ate is den&d. by k Initially the piate and its surroundings are at a uniform temperature e The teuperature in the kdide proxizity of the top surfers 'tkez JLges 5nstantaneously to a constant value 8,. &at is SUbS~qL13il%7iJ transferred across the surface at a rate govern& by a heat transfer ooefflcient, h(x), whioh is postulatea in the form: where p end q sre srbitrsry non-&mensionsl constants such that p > 4, q > 0. A state of no heat transfer is postulated across the other surfaces of the plate. Under these conditions trsnsient temperatme gradients ed hence transient the-1 stresses develop in the plate, heat conduction l,&d.ng place until the temperature throughout the plate is 8,. Since the thictiess of the plate is srkLl in ~~~pezison tith its lateral end longi- tdinal dimensions the temperature at any psition on the pl.ate is assumed oonstant through the thickness ti 3s aended by f3(x,t). On this assump tion it is require& to solve the problem of lineer heat oondudion parallel to ox, thus to determ5ne the temperature distribution 0(x,t) ad. l&nce the thernd stresses in the piate. In determining the thermal stresses it is pdzulated that the plate is free from applied forces* Other stresses due to applied forces WY be determined separately, +he total strssses being obtainedby superpk.tion~ * I’M theory is equally applicable to a code6 @tee -l&w -5- -6- lb3 constant coeffx5c~d5s An are deterrcine~, in 2Pp12ndxx I, to wtx3Yy tnc titial condition, e = e. at tzo. It is subscqwmtly found that a - % e. - 6, ’ determined from equation (I.$.), is a non-d3mcnsiond parameter 4.3 ZheriiT stresses - It 1s a&zawd that the stress systeln u the plate 2s entirely lon~itu&na~. It nay, therefore, be deterrancd. on the basis that lateral cross-sections remaxn PLane, frw which it follows t&t f = E(- cd3 t e, * a,x) (161 vrhere E denotea 'Yang's rx&&s, a denotes the coefficxnt of linear expansz.on, ad 99 ?j are constants. The plate ie considered to be fret fran apPl~d forces and, therefax, 1 L fax = 0, -i 0 L ; Cq7) / xfax zz 0, I * j -7- -8- -9- 16%, 211°F, 367*F, 502*F, eter 25 sec.5; c3f%x jo sets; after 100 sees; af%c?r 200 scc?3; &Yxx J+oo sax3. L s k P c t h z h(x) Q Q x 9 ft fY B Th u -2 0 ft ( 3vft) set lb -7 f-t- B 'T'h U I% SW2 s-y&cl - a = a(x,t) a0 % f = f(x,t) E x = x(e) Lid of Syticls (ccrhl) &scr33tion -- = texperature m the plzte (cm&at thro~ugh the thduxss) at a pcsltion x, at tme t. z initiLt tempcral,ure of the pkte and 1% l.lcLlem surro~s. = temperature 21 the ixmdia~e prcxm~ty of the top surface for t > 0 (the fin.51 tmpemture oZ the plde). = lcng~tudiml stress in tiie p%ate = Young's oKv.l~Lls. z coeffxknt of lxiear ezqxnsion. = arbitrmy non-dimmsiond constants such i&t P>&, q>o, introduced in equation (I). invclvtng = non-d~neneional vzmable cf equation (12). zz .lth a~ nti roots (in aecc+in~ ctier cf ~+gade) 0f equation (12j. = function of C satwfying equation (9). z mkpen3ent vanable. = l3essel &notion of the First Kind of order P. - 12 - E. 1 2 3 4 5 6 7 a 3 IO II - 14 - 'The constant coeffxients Aa in eqmtmn (13) may now 3e determined. Equation (13) mst mtxdy the init3.d condition that 9 = go at t = OS and, therefore, ; ~KiLt.i.&&g both sides of equation (23) by <- J&3.&) and thw wkgm- ting beixecn 0 and I2 using eque.tion.5 (22), gives - 15 - . - 16 - as fol~om: - 17- I(n, F9 i, , ’ -1 p = y (J = I, 2, . . . . 6) , - ,i8 - - 1y - f’ = 0.~1169 =z, .f tJ - 2C - L. - -- -~~---- -- l - LONG LONG EDGE tzDGE --------- ----- %- 0 (0) PLAN 4 TOPSURFACE 8 1 p3td +i,p>c. x X+dx (b) LATERAL CROSS -SECTION. FIG. i(a8bI THE PLATE, SHOWING NOTATION. -v I+, AIR STREAM. 07 -v LEADING EDGE. ------v (Cl) PLAN 0~X PLATE I 09 LATERAL CROSS-SECTION . FIG. 2(agb) AERODYNAMIC HEATING OF THE PLATE, SHOWING THE DIRECTION OF THE SUPERSONIC AIR STREAM OVER THE TOP SURFACE. 7 6 4 1 I I I I I 1 5 t- - - I I I \ \ 1 \ ' )I \ \ \ \ \. -. , I I I ( I \ \ \ \ 0 43.2 3 2 I 9 t POSTULATCD COEFFICIENT FOU p = 6.5, ( 7 = 2.3s~ IO! ---- ----- ACRODYNAMIC HEATING:- PREDICTED COEFFICIENT FOR A LAMINAR OUNDARY LAYER ON A FLAT PLATE OF THE AIR EVALUATED AT CONDITIONS ATAN ALTITUDE OF 50,000 FT> POSTULATEQ COEFFKIENTFOR p = 3.5, FIG. 3. HEAT TRANSFER COE FFICENTS. 03 0.4 .- 0.5 < .kx F 0.0005 4 .o I 0.00 375 - 000875 - - ; 0.7 oe 03 IO (a) TEMPERATURE DlSTlilBuTlON. 0.2 @I STRESS DlSTRl6UTlON. FlG.4(aab) TEMPERATURE AND STRES: DISTRIBUTIONS IN THE PLATE AT VARIOUS TIMES FOR +j3’ g =v + 0.1774 @=3*5, q,= O.l774~clO’).THE NUMBERS ON THE CURVES ARE VALUES OF ct/Lz (CORRESPONDING TO t = 200,500, l,500,3,500 SECS. FOR THE PLATE TAKEN AS EXAblPLE).THE RESULTS ARE ASSOCIATED WITH AERODYNAMIC HEATING WITH A LAMINAR BOUNDARY LAYER. 06 04 02 OOOCO625 ,/@- -- - , ,‘I I I iO.000125 1 (Cl) TEMPERATURE DISTRIBUTION. 0~0000625 -0.6 (b) STRESS DlsTRlBUTiON. FIG. 5 (at b) TEMPERATURE AND STRESS DISTRIBUTIONS IN THE PLATE AT VARIOUS TIMES FOR -& * = O-04’ + 2.306 (9 = $5, ‘J:2*306X ld).THE NUMBERS C%i Tl x”ES ARE VALUES -0~ E*/L~ (CORRESPONDING TO t = 2 FOR THE PLATE TAKEN AS SE&. ASSOCIATED WITH AERODYNAMIC HE-ATING WITH A TURBULENT BOUNDARY LAYER. IN AERODYNAMIC . Q-Q 0.001 0 002 0.003 0.004 o-005 oax lx I I -F I I I I I 400 600 lit00 WQ 2,000 q+cQ t sacs., FOR THE PLATE TAKEN ~5 EKAMPLL. - . FIG. 6 MAXIMUM STRESS IN THE PLATE AT VARIOUS TIMES. THE CURVES ARE BASED ON VALUES OBTAINED FROM FlG.4.AND FIG. 5,AS SHOWN. C.P. No. 299 (16,932) A.R.C. Technical Reprt To be purchased from York House, Kmgsway, London xv c 2 +q Oxford Street, London w I P 0 Box sb9v London s E I 13~ Castle Street, Edmburgh 2 109 St Mary Street, Cardd? 39 Kmg Street, Manchester 2 Tower Lane, Bristol I 2 Edmund Street, Bmmngham 3 So ChIchester Street, Belfast or through any bookseller S.O. Code No. 23-9009-99 C.P. No. 299 

Transient thermal stress in a flat plate due to non-uniform heat transfer across one surface

https://reports.aerade.cranfield.ac.uk/bitstream/1826.2/312/1/arc-cp-0299.pdf

Transient thermal stress in a flat plate due to non-uniform heat transfer across one surface

Abstract

Similar works

Full text

Available Versions

MAGiC ARC Repository