TECHNICAL NOTE 4308 TRANSIENT TEMPEIWTURE DISTNMYTION TWO ...
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--—. -. TECHNICAL NOTE 4308 DISTNMYTION TRANSIENT TEMPEIWTURE TWO-COMPONENT SEMI-INFINITE COMPOSITE SLAB OF ARBITRARY MATERIALS SUBJECTED TO AERODYNAMIC HEATING WITH A DISCONTINUOUS CHANGE IN EQUILIBRIUM TEMPERATURE OR HEAT-TRANSFER COEFFICIENT By Robert L. Trimpi and Robert A. Jones Langley Aeronautical Laboratory Langley Field, Va. Washington September 1958 .— -— ——
Transcript of TECHNICAL NOTE 4308 TRANSIENT TEMPEIWTURE DISTNMYTION TWO ...
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TECHNICAL NOTE 4308
DISTNMYTIONTRANSIENT TEMPEIWTURE
TWO-COMPONENT SEMI-INFINITE COMPOSITE SLAB OF ARBITRARY
MATERIALS SUBJECTED TO AERODYNAMIC HEATING WITH A
DISCONTINUOUS CHANGE IN EQUILIBRIUM TEMPERATURE
OR HEAT-TRANSFER COEFFICIENT
By Robert L. Trimpi and Robert A. Jones
Langley Aeronautical LaboratoryLangley Field, Va.
Washington
September 1958
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TECHLIBRARYKAFB,NM
iB
.NATIONAL ADVISORY COMMITTEE FOR AERONAUTL. ClfJb7)203
TECHNICAL NOTE 4x8.
TRANSIENT TEMPERATURE DISTRIBUTION IN A
TWO-COMPONENT SEMI-INFINITE COMPOSITE SLAB OF ARBITRARY
MATERIALS SUBJECTED TO AERODYNAMIC HEATING WITH A
DISCONTINUOUS CHANGE IN EQUILIBRIUM TEMPERATURE
OR HEAT-TRANSFER COEFFICIENT
By Robert L. Trimpi and Robert A. Jones
suMMARY
A solution is obtained to the transient temperature distributionin a semi-infinite two-component composite slab of arbitrary materialssubjected to an instantaneous application of aerodynamic heating withconstant equilibrium temperature and heat-transfer coefficient. Thenumerical results are tabulated in a form to permit easy computation ofheat-transfer problems typical of aerodynamic testing. ‘Ihesolutionsare valid for finite two-component slabs as long as the times consideredare small compared with the diffusion time of the backing material.
Analytical results obtained from these solutions can be used todetermine (a) the heat-transfer testing time for which the outer skinmay be assumed to act as a calorimeter without exceeding a given erroror (b) correction curves by which the indicated calorimeter heat-transfercoefficient may be multiplied to obtain the true heat-transfer coeffi-cient. For such a correction curve to be valid, the bond between thetwo materials must have negligible thermal resistance, a condition dif-ficult to attain if the slab is not composed of two metals.
Since the differential equation for the temperature distribution islinear, the principle of superposition is valid. Consequently, the wOb-lem of continuously varying equilibrium temperatures and heat-transfercoefficients may be treated by using the tabulated solutions and con-sidering the continuous variation as a series of superimposed stepfunctions.
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INTRODUCTION ‘-
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Themissilesinvolved
increase in flight speeds for both operational and experimentaland airplanes has resulted in renewed interest in the problemsin fluid- and solid-state heat conduction. The two problems of
transient behavior of a two-component slab, consisting of an outer skinwith a backing material, exposed to aerodynamic heating that have becomeof prime importance are: (1) the performance of the surface as measuredby the ability of the material to withstand extreme heating rates on thefull-scale vehicle during the course of its mission and (2) the transienttesting techniques presently in wide use wherein the aerod~amic heatingis varied and the resultant time-wise variation of the temperature of askin, either with or without a backing, is then used to obtain heat-transfer rates and coefficients. For example, these transient testingtechniques are used for shock tubes, free-flight pilotless rocket-propelled models, the sudden insertion of a Dodel into a wind tunnel, orthe rapid variation of the stagnation conditions in wind tunnels.
In the interest of simplicity, these techniques, except that usedfor the shock tube, customarily assume that the outer skin of the slabacts as a calorimeter and absorbs all the aerodynamic heat with negli-gible loss to the backing material. However, even when the backingmaterial has very low conductivity; heat is continually being transferredacross the interface and this heat transfer may introduce large errors inthe answers obtained by the calorimeter assumption.
Two recent papers (refs. 1 and 2) treat the heat transfer to such asemi-infinite composite slab. Reference 1 is applicable to shock-tubetesting wherein a very thin metal plating is formed on an insulator andthe temperature response of the metal is used to determine the heattransfer. This study is applicable to arbitrary heat flux ratesbutassumes the plating to have such negligible thickness that the interface-temperature solution may be found as a perturbation to the surface solu-tion in the absence of the plating. The solutions obtained are valid for
times much greater than the diffusion time in the plating.l Reference 2considers the case of the heating of a composite slab in which the heattransfer is proportional to the difference between wall and equilibrium
1In the discussion of transient heating problems it is convenient to
have reference values of time such as “diffusion time” and “relaxationtime” for comparison purposes. The diffusion time is the quotient of the _ _square of a typical length (for example, the thickness of a slab) dividedby the thermal diffusivity of the material. The relaxation time, appli-cable in exponential decay behavior, is also called the time constant andis that value of time which makes the absolute value of the exponent ofe unity.
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temperatures. The equilibrium temperature is a step function. The outerskin has finite thickness but infinite conductivity; thus, the temperaturein the skin is uniform at any given time. A solution is presented for theheat conduction to the backing n.u%terial,no coupling between this loss ofheat to the backing and the temperature distribution in the outer skinbeing asmuned. This solution is valid for times that are small comparedwith the relaxation time of the outer skin. Reference 2 shows that con-duction even to relatively good insulators introduces appreciable error.
The present investigation was initiated to obtain solutions for theheat transfer to the composite slab without these restrictions. The mag-nitude of the errors arising from the calorimetric assumption and pos-sible correction factors were to be determined. When the thickness of thebacking material is such that the diffusion time for the backing materialis much larger than any of the testing times, the backing material thenacts as if it were of infinite extent during the test. Consequently, solu-tions to the problem of a semi-infinite composite slab would apply underthis condition. The problem considered in this report is the transienttemperature distribution in a semi-infinite two-ccmponent slab subjectedto aerodynamic heating at a rate equal to the product of a constant heat-transfer coefficient and the difference between surface and equilibriumtemperatures. The equilibrium temperature is assumed to be a step functionand the outer skin is assumed to have both finite conductivity and thick-ness. Solutions are found which are applicable for times varying frommuch less than to much greater than the skin diffusion time. The super-position principle permits extension of the results to arbitrary timetisevariation in heat-transfer coefficient and equilibrium temperature. Theinvestigation reported herein was conducted in the Gas Dynamics branch ofthe Langley Aeronautical Laboratory.
SYMBOLS
a,b constants
A,B,C,D constants
c specific heat
Gn functions definedby equation (23)
H= h/k
h
G
A‘o
heat-transfer coefficient
indicated heat-transfer coefficient
indicated heat-transfer coefficient at E = O; PCZ(dT/dt)o
Te - To
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iil indicated heat-transfer coefficient at E =1
J coefficient of ~ in expression for P and equal to +1or -1
k thermal conductivity
L=~=2E7‘1
h
1
m,n
P
P
Q
r~=P~
s = p/4a2
T
T=
To
Trelsx
trelax
t,t’,u
x
a
13=~
-L
Laguerre polynomials (see appendix A)—
distance from surface to interface
integers
parameter in inverse Nlace transform
variable of the Laplace transform
heat flux to surface per unit area
temperature
equilibrium temperature
temperature at ~ = O
relaxation temperature
relaxation time
time variables
per unit time
coordinate measured normal to surface exposed to heating
thermal diffusivity, k/pc
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r~ solutions to r~ tan rs = L
P density of material
q,x,T dummy variables
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-1-r inverse Laplace transform
A bar over a symbol denotes the Laplace transform. Except as desig-nated in the preceding list, the subscripts 1 or 2 refer to evaluation inregion 1 or 2.
TEE(3RY
tureface
General Solution
The problem to be solved is the one-dimensional transient tempera-distribution in a semi-infinite two-component slab which has onesuddenly exposed to a fluid. This fluid till transfer heat to the
exposed face at a rate proportional to the difference between the constantequilibrium temperature of the fluid and the outer smface temperature ofthe slab. In the interval O < x< Z, the slab is composed of a materialwith properties P1~ cl} kl, and ~; and, in the interval Z < x< co,
it is composed of a material with pr&perties P2, C2, ~, and ~. NO
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thermal resistance is assumed at the interface in the slab. The mathe-matical formulation of this problem is given in terms of the followinggoverning differential eqyations and boundex’yconditions (see fig. 1):
b2T2 1 ~T2 o—-. —=
ax2 a2 aT
‘1 =T2=0
bT1kl —
ax ( )+hTe-T1 =0
‘1 = T2
(t >0, Z<x<m)
(t <0, all x)
(t> o, X+m)
(X=o, t>o)
(x=2, t>o)
(x=2, t>o)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
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/ ‘ptT (t)dt and the definitionsThe MPlace transforms ?(p) = ‘--eJ()
of q= np CL are then employed to reduce equations (1) to (7) to the fol- :“lowing forms:
g2 - @2= o
(O< X<2) (la)
(2< x < I=) (2a)
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I&o (x +=)
d?l hTe—.
‘1 & h?l+Y=O (x = o)
51 = ?2 (x = 2)
kl ~’%> (x = z)
The general solutions to equations l(a) and 2(a) are:
T1 = Ae-qlx + Beqlx
?2 = Cc-%? + Deqa
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(4a)
(5a)
(6a)
(7a)
(8)
(9)
The value of D must be identically zero to satisfy equation h(a).Application of the boundary conditions (eqs. 6(a) and T(a)) yields “
(lo)
C = (1 + ~)Ae-q12+qJ
(11)
The remaining constant A is then found from equation 5(a) to be
(-1
A=Hfiql - *1 e-2q1z
PA1‘Pql+H1
)(12)
The solutions to the differential equations in the transformed planebecome
fil H1 1 1
[
e-lq~x—.= — 1+~e-ql(=-x)Te
(o<x<z)n.l+*ll&l-*l e-2qlz
ql + H1 (13)
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[- 1
kl 1-B ~-~)-ql z+~+
e (x> z)
—(i4)
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-1 ~1Z()The inverse LaPlace transform~
of equation (13) is known
for the special case a = 0, ~ = 1 which corresponds to a perfect insu-lator as the backing material. If the expc.mentialsof equation (13) areexpressed in hyperbolic form, the inversion on page 259 of reference 3applies.
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/.\
.=
(15) - -
where 17s are the roots of the equation
rs tan rs =
If the slab is composed entirely of
‘1 = k2)~ equations (13) and (14) become
transform
L (16) “-
the same material(P
= 0; .
identical and have as their
(17).-
For the general case of u # O, f3# 1, the inverse transform ofi \Te as expressed in the form of equations (13)and (14) is neither
known nor easily determined. Recourse is then made to expansion of theright-hand side of these equations. (Note that I~I~ 1.) An e~ansionof this ty-pegenerally results in solutions which will converge morerapidly for small values of time.
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substitution of eqyation (18) into eqyations (13) and (14) yields
Z1I f (q-l- %)n-Q(a+g)+ ~n+l (ql -Hl)n -q1Z[2(n+l)-~]
—=Te ‘1
;0 p (qI+ H1~+lsT P’ ‘ ‘l)n+’
.
(19)
If the orders of summation and integration of the inversion processare assumed to be interchangeable, then
-1 -
Z( )‘2~
=Hl(l+ ~)
Pn+l
(1 Hl)n e-qlz[2(n+l)-~]
y (q: +-%)n+’ tJ
(ql - H1 )n
n+l(% + ‘1)
(a)
(22)
Let the function Gn(P,u,L) be defined as follows:.
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@@J)
In appendix Athe series has the
it is shown that the inversion of the general term offOrm .-.
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(24)~-l~:+-H~jle-qlzp] =Gn(p,u,L) ,
Substitution of equation (24) into
m
J
equations (21) end (22) produces —
~n+1C#n[2(n+l)-~,m,L] (25)
n=O
(26) .
The formal integrations required for the evaluation of Gn for a4
range of values of n from O to 5 are @ven in appendix B. The
(&n ahassociated ttiewise and spacewise derivatives
)~ and — are also
at
presented. These derivatives permit the evaluation of both the correc-tion curves to the indicated calorimetric heat-transfer coefficient andthe amount of heat-crossing a given station, in order that estimate ofthe flux through the backing material may be obtained.
Significance of parameters L, h, and B
The physical significance of the nondimensional parameters L, 1,
and j3 wilJ.be briefly discussed at this point. The param?ter L = ~kl
is proportional to the ratio of the temperature difference required totransport a given amount-of heat per unit area across the slab to thetemperature difference required to transport the same amount of heat per,.
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unit area from the fluid to the slab. This may be illustrated for thesteady-flow case as follows:
Q=h~e -T(o)] =~T(0) jT(l)
Therefore,
T(0) - T(1) =hz _ ~ ‘
Te - T(0) 5
Now if L is small, a small temperature difference across the outerskin is indicated; thus, the surface temperature and heat transfer whenT(0) +Te will be sensitive to changes in the temperature at the inter-
face. Since the interface temperature is in turn influenced stronglybythe backing material, the relative influence of the backing materialincreases with a decrease in L“ for times of the order of the relaxationtime. (For very large times the backing material must always be thedominating factor.) The significance of L may be realized in an alter-nate way by considering that, when L is small owing to a thin skin, highthermal conductivity, or a low heat-transfer coefficient, a greater por-tion of the heat transferred to the surface is available for transfer tothe backing material and the backing rmterial can exert a larger influ-ence on the temperature time history of the skin for finite times. Aplot of the parameter L obtained for various skin thiclmess and materialcombinations with h = 0.1 Btu/sq ft-sec-% is shown h figure 2. A rangeof L from 0,001 to 0.5 appears to cover practical outer skins whichmight be used for high-speed vehicles.
The significance of the parameter A =r
fi.=w is most easilypcl
described from consideration of the aerodynamic heating of a slab ofthickness 1 having itiinite conductivity (~T/& . O). The slab acts
as a calorimeter absorbing all the heat input. This slab is initiallyat T = O and is subjected to a step input in equilibrium temperature att = o. The temperature of
T—=Te
the slab at time t &y be expressed as
. !z&1 -ez2=l-
Thus, the value A = 1 corresponds to theslab calorimeter.
For a perfectly insulated skin havingness, the temperature distribution departs
-12e
relaxation time in such a
finite conductivity and thick-from that of the calorimeter
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as L increases from
ture is approximately
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.zero. However, at the value X = 1, the tempera.
-,
the relaxation value given by()Trela=Tel-~*
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Since t a A2, the ratio of any time t to the relaxation time trehx
is simply 72.—
The parameter ~ denotes the relative insulating perfection of thebacking material. A value of ~ = 1 applies to a perfect insulator;~ = O describes the homogeneouscomposite slab constructed of the samematerial throughout; and ~ = -1 describes a perfect conductor whichmaintains a constant temperature at the interface at ~ = 1.
—Values of
~ appropriate to various combinations of outer skin and backing materialare shown in figure 3. The outer skin materials have common abscissas —and the intersection of these vertical lines with the curves for thebacking materials determine the appropriate value of ~. If the materialsfor skin and backing are interchanged, the sign of ~ is reversed.
Particular Solutions .—
Since many aerodynamic heat-transfer experiments are performed witha model which has a skin of high conductivity and density compared withthat of the backing material, it is desirable to have a form of solutionapplicable to these circumstances. The solution for a perfect insulatorbacking material is either the series given in equation (15) or that .
given in equation (25) when a value of f3= 1 is used. These alternatesolutions converge most rapidly at opposite lilmes, The number of terms .-required to express the timewise derivative for equation (15) is large
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as t +0 and decreases rapidly with an increase in time; thus for values “-of t greater than about one-half the remtion time, only one or twoterms are required to give the value of ‘1lTe and its timewise and
spacewise derivatives with acceptable accu&cy. Conversely, althoughone term is sufficient for equation (25) when t +0, the number of termsreqyired increases with time. A series thalrwill converge rapidly forthe times of interest for aerodynamic testing purpose’swhen p $=1 may
—
be obtained by applying the solution of equation (25) as a perturbationtO the solution of equation (15). If T1/Te for ~ = 1.0 is added and .—
.—subtracted to the right-hand side of equation (25), the resulting equa-tion is
?=(3),=J$-‘n)GJa+E20)L)+! - ‘n+l)’nE(n+’)-~~Ll)=(27)
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()
T1where
~ ~=1is evaluated from equation (15) for the appropriate
.values of L, m, and ~.
Of particular interest is the temperature history on the surfaceexposed to the transferring fluid (~between the two different materials.tion (27) produces the following set
.
= O) and thatSubstitutionof equations:
m
on the interface (~ = 1)of k = O into equa-
1
A correspondingby substituting ~ =
L
set of eq.tions1 into equation
(28)
(29)
(30)
may be obtained for the interface(27):
- 13n(l+ B)]Gn(2n+@L) (31)
The evaluation of the terms of equations (25) and (26) becomesprogressively more difficult as n increases. Consequently, if the
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temperature at the interface (~ = 1) has to be evaluated
NACA TN 4308
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or if the values tabulated in-this report are to be used, then the tem-perature distribution in the backing material can advantageously be .obtained by Duhamel’s method.
Since the temperature distribution in a semi-infinite slab (2 < x< w)when one surface at x = 2 undergoes a ugit-s@p-function increase In tem~ -perature is (ref. 3)
Duhamd’s
*(xJt) =e
T2(x)t) = erfc ~
r (34)2 a2t
technique yields
~~:j$)[~BnGn[~+l,~,$ld. (35’](X-t) (l+p) ‘t
F2 a2
RESULTS AND DISCUSSION
General Results
The values of Gn and its derivatives for n = O to 5 and
E = O and 1 were computed on a card-progamed calculator. Details ofthis procedure are discussed in appendix C. The results of these cmn-putations are given in table I for values of L between 0.001 and 0.5with 0.02~ As 8. Table I is arranged for ,futurecomputing convenience.Values on the left for E = O are alined so that pairs in G or itsderivatives on the ssme horizontal line are operated on by the sam? powerof ~ as prescribed by equations (28)to (,30).
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As discussed in appendix C the computing procedure at times involvedthe subtraction of two large and nearly equal numbers obtained by approxi-mations. Certain voids appear in table I where an inspection of themachine results indicated insufficient accuracy. There was no rigorousrule for determining which answers to discard, but all answers with lessthan foux significant figures were usually eliminated. Those with lessthan three significant figures were always eliminated.
Typical Values of Gn and its associated derivatives which are
required at ~ =The values shownin the braces of
O are plotted against -L for A = 1.0. (See fig. 4.)are those needed in.the evaluation of the first seriesequations (28) to (3o). It is immediately obvious that .
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.the convergence of the series is much more rapid for large values of Lthan for small values of L. For example, at L = 0.5 the ratio of ~to Go is of the order of 10-6 whereas at L = 0.001 the ratio is of
the order of one.
An indication of the accuracy which results from consideration of afinite number of terms to represent the infinite series of eqyation (25)may be obtained frcm figure 5. In this figuxe are plotted the values ofthe ratio of T1/Te for j3= 1 when the upper limit of sunmation in
equation (~) is reduced from infinity to OJ 1, 2, 3, 4, or 5 to thatobtained from table II. (Tkble II contains the values of T1/Te as
determined from equation
)
15) by using a sufficient nuniberof terms toget the desired accuracy. The ratio for values of L = 0.001, 0.1,and 0.5 is plotted as a function of X. The number of terms requiredto reduce the error to a given limit is seen to increase with an increasein A at a constant L and to increase with a decrease in L at a con-stant h. For a constant L, the behavior of TllTe for any finite num-
ber of terms becomes osci~atory as A increases: The errors forOs~<l at a given A and L shouldbe less than those indicated in.figure 5 for j3= 1 since the terms neglected contain ~n or ~*l.In other words, the neglected terms for 0~~<1 haveasmallercon-tribution than those for P = 1.0.
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Illustrative Example.
The effects of the backing material in a composite slab under typicaltransient test conditions will be illustratedby the following study. Thenumerical results obtained are found by using the values of table I forOS n= 5 and those of table 110 The mathematical assumption of an infi-nite extent of backing material to represent the finite backing of the testis valid for values of ~ near unity as long as the times considered aremuch less than the diffusion time of the backing. The”diffusion times,which are equal to the square of the thickness of the backing materialdivided by the diffusivity of the backing material, for l/&-inch-thickbalsa and mahogany are approximately 120 and 170 seconds, respectively.Consider two stainless-steel outer skins (k = 0.0028 Btu/sec-ft-OF,a = 5.2x 10-5 sqft/see) with thicknesses of 0.060 inch and 0.030 inchbacked by either a perfect insulator (~ =1.(XO), balsa (~s O.975), ormahogany (~ = O.~0). These composite slabs are subjected to a stepinput in equilibrium temperature (for example, by sudden exposure to theairstream of a wind tunnel) and the heat-transfer coefficient h is0.056 Btu/sq ft-sec-9F. The values of L me then 0.10 and O.~ for the0.060- and O.On-inch thicknesses, respectively.
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The resulting behavior of the outer skin is illustrated in figure 6which shows the temperature distribution at the outer face (g = O) as afunction of time. The effect of the heat lost to the backing material isevident in a lower temperature in the outer.skin at a given time> a largereffect being observed for the thinner skin.
An indicated heat-transfer coefficient is often obtained from thetemperature-time history of such an experiment. This heat-transfer coef-ficient is determined by assuming that the slab acts as a calorimeter withan infinite value of k; that is,
fhPc,*Te-T
Since the value of ~ depends on the value of 5 at which T is meas-
ured for finite k, the term to or ;I will be used to designate the
values of ; which would be found when T and dT/dt are evaluated at
k =Oorl.
The ratio of the indicated heat-transfer coefficient to the trueheat-transfer coefficient ~/h is plotted as a function of time in fig-ures 7 and 8. Figure 7 shows the ratio at ~ = O and 1.0 for the0.030-inch-thick skin. Except for the initial discrepancy at early times
the values of fio and ~1 are nearly equal. At t = O, to iS infinite
while ~1 is zero, but after about 0.1 second the difference is minor.
Figure 8 shows the ratio ~olh as a function of time for both the
0.030-inch and O.~0-inch s~n. It is evident that ho = h on@ at a
single discrete point for each combinationof skin thickness and backingmaterial. Furthermore, even the perfect insulator does not give a ratio
of unity for ~/h except at one value of time. It may be shown for the
perfect insulator that the asymptotic value of ;/h iS
~ r12Mm -=—t+mh L
~+$L1~=1--3
+...
1+ 2 L2~L+y
(L<< 1)
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Thus even the perfect insulator gives errors of 5/3 and 10/3 percentfor the cases considered (0.030-inch and O.@O-inch-thick skins). Thelarge increase with time of the deviation of fi/h from unity for theimperfect insulators is striking. This error increases with time because,as the heat transferred into the skin at ~ = O decreases monotonicallywith time, the heat transferred out at ~ = 1 first rises with time to amaximum (at approximately h = 1.0) and then subsequently decreases butat a rate slower than that at ~ = O. An approximate method for esti-mating the amount of heat conducted across the interface for the condi-tions of A<< 1 and k + m was also presented in reference 2.
If an error of 10 percent were set on the deviation of indicated totrue heat-transfer coefficient, the time limit for useful data with therespective backings composed of mahogany and balsa would be about 1 sec-ond and 2.5 seconds When Z = 0.030 inch and 2.4 seconds and 5.7 secondswhen Z = 0.C60 inch. If, instead of using a time limit, a temperaturerise limit for a 10-percent error were desired, it may be obtained froma plot similar to that of figure 9. For 2 = 0.030 inch, temperaturerises greater than 30 percent of Te for a mahogany backing or 65 percent
of Te for a balsa backing would give”values of ~/h deviating from
unity by more than 10 percent. Approximately the same limits also applyfor 2 = 0.060 inch.
CONCLUDING RIMARKS
A solution has been obtained to the transient temperature distribu-tion in a semi-infinite two-component composite slab of arbitrary mate-rials subjected to an instantaneous application of aerod-ic heating.with constant equilibrium temperature and heat-transfer coefficient. Thenumerical results are tabulated in a form to permit easy computation ofheat-transfer problems typical of aerodynamic testing. The solutions mevalid for finite two-c~onent slabs as long as the times considered aresmll compared with the diffusion the of the backing material.
Analytical results obtained from these solutions can be used todetermine (a) the heat-transfer testing time for which the outer skin maybe assumed to act as a calorimeter without exceeding a given error or(b) correction curves by which the indicated calorimeter heat-transfercoefficient may be multiplied to obtain the true heat-transfer coefficient.For such a correction curve to be valid, the bond between the two mate-rials must have negligible thermal resistance, a condition difficult toattain if the slab is not composed of two metals.
Since the differential equation for the temperature distribution islinear, the principle of superposition”is valid. Consequently, the problem
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of continuously varying equilibrium temperatures and heat-transfer coef-ficients may be treated by using the tabulated solutions and consideringthe continuous variation as a series of superposed step functions.
Langley Aeronautical.Laboratory,National Advisory Committee for Aeronautics,
Langley Field, Vs., June 4, 1958.
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APPENDIX A
INVERSION OF ‘I!KEIAPJACE TRANSFORM
The steps reqyiredequations (21) and (22)sion of the expression
.-
to invert the general terms of the series ofwill be deteti-ned in this section. The inver-
1- - 7
will first be found and then the constants a and b will he assignedpertinent values.
Let s =+. It is then necessary to find the inversion for4-at
-1
e2-[-%b F (2@-l)n 14a2s(2G+l)n+1
(2&- l)n1(26+1)*‘A2)The square-root image rehtion (see page 123 of ref. 4) can be used toevaluate eqyation (A2). Thus if
f(t’) =~-l[.(s)]
then
coU2
1
-— -1
4t’ f(u)du=&oe z *
g(w)
and the evaluation of the following expression is reqtired:
(A3)
-1
Z[ e-2abs (2s - l)n
s 1(2s +“l)n+l
20
The term
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and
-1
Z[1
(2s - l)n =
(2s + l)n+l
(seepage 298 of ref. 5 and pwe 1-29of
o
1
(o< T< @b)
1(A4)
ref. 4.) The
designates Laguerre polynomi~ls of the following form:
(2ab < T)
J
nn (.’r)rn
Ln(T) =
u )n-m ~:
m=o
n
Ln(T) =I
1
(-T)m
(n ~!’m)! (m!)2m=O
.-
(A5)
symbol %(T)
—
(A6)
Application of the convolution theorem to equations (Ah) and (A5) resultsin:
Z’-’[--]=*L“-%b“’’~(’)‘l)dT+
J /1 u ~-z %(T) (0)dT
u-2ab
J’u-2ab
~ e-T/2 L (T)dT=-2 ,~ n (A7)
.
.
NACA TN 4308 21
Combination of
.-1 -2ab&
z{ [$’6
L
For S = Cp,
equation (A7) and the image formula (A3) yields
(2G - l)n
1}(2G + l)n+l
wU2
J
:=
I
& 12u-2ab -T
‘& 2ab 2Ln(T)dT du
o
(A8)
Ct ‘ = t, it may be shown that, if
z-~(sj=f(t’)then
Consequently,
-1
zor
since c 1= ~, equation (A8) may be operated on to yield:
(A9)
4ac
(A1O)
Ifar
=H a and b=~,&
equation (A1O) becomes
22 NACA TN 4308.
. .
and
are substituted inequations result:
xu=— --
r4?X ~
X2
‘T%
the right-hand side of equation (All), the following
- Gn(P,u,L) (A12) .
.
.
.
.
NACA TN 4308
APPENDIX B
23
.
.
ZIG aGnEXPRESSIONS FOR THE VARIOUS Gn, ~, AND —
&2 ae
FOR VALUES OF n FROM O TO 5
Formal integration of the eqyations for the Gn for values of n
from O to 5 and subsequent partial differentiation with respect to 02and ~ produced the following expressions:
‘o= erfc * - e“erfc @
G2 = erfc ~ - (1+ 8.2@2 )+ 4A2L enerfc $ + ($ A%@)&-$
*2
)8J ~3/2(1 + A@)e-*2—=
ae-Lj(l+ 8A@ + 8A%@2+ 4A*L e“erfc $+z
G4 = erfc $ - e~erfc
36~L512 + 40A2L3@ + 32T
NACA TN 4308.
~ A3L3/2@3 +
)~3L3/2 e -@
36AL5/2$ + 4012L3@2 + ‘20A2L3+ ~ A3L7~2#3 +
.
!( 6433/2@3 +324241 + 16A2L#2 + 8A2L; ~ h L ~AL@ +
.
3 3/2@)+ ;F~48fL@ + 64A3L3/2@2 + 64A3L3/2 ,32A4L2$2+ 8h4L2 + 32A L
423
)32A L@ + 80A4L2@ e
-F—
~G4
(
‘+2 j 4.8hL3/2~=e + 144A2L2@ + U8A3L5/2 + 128X3L5/2@2 + 32A4L3@3 +
36
)80A4L3# -
(
3/2enerfc $ j + 16AL $ + 48A2L2(? i-24h2L2 + ~ A3L5/2@3 +
3 3/2 3243#
432 4364AL @+@L +32AL@+8hL+L
).-
.
/’,-
.
NACA ‘INL308 25
( 3/2$+ 240~2L2fj2“+-e~erfc @ $ 60AL
480h3fi/2@ + 1601.4L3~ + 480X4L3@2
35~2$3+120A2L2 + 320A L
+ 120A4L3 + :8 ~5L7/2@ +/
)( J ~U3/2 + 2401%2$ + 320A3$/2 +128A5L7/2ff3+ 96A5L7/2@+ 3L + —3G
320h3L5’2$ -i-160A4L3$3+ kOOA4L3#-i-~A5L7/2$4 + ~A5L7/2@2 +
)256 ~5L7/2e-@T
26
%=(
-eQerfc @ llL2 +z
320A3L7/2@+ ~
.
.
.
.
.#f
NACA TN 4308 27
APPENDIX C
& aGnNUMERICAL COMPUTATIONS FOR TKE VARIOUS Gn, ~, ANDj —
&2 ak
The computations necessary to evaluate the eqpatiortsof appendix Bwere carried out on a card-programed calculator. On this machine, thenumber of significant figures in any computational step was limited toeight, a limitation which resulted in a pr~essive d~inishing of thesignificant figures in the answers as the value of n increased at agiven A and L. The reason for the loss of significant figures is the
fact that each answer(except for a
*)Eis the difference between two
numbers; thus, when th~se numbers be~ornelarge and near~ equa~~ theresultant difference has few significant figures.
It was also found that the complimentary error functions had to becomputed to within a very small fraction of a percent of the correctvalue in order to obtain usable accuracy. The above-mentioned subtractionof two large but nearly equal numbers, one of which contains the compli-mentary error function.as a multiplier, is the reason fm the particularaccuracy required. The following expressions were used to compute thecomplimentary error function of @ and V in the ranges of the argu-ments listed (~ is taken as the illustrative argument):
. erfc#=~e -P(alq + a2q2 + a3q3 + ahqk + a~rj3fi )
where
1n =,
1+ 0.3275911@
al = 0.22583685
a2 = -0.25212867
a3 = 1.2596951
a~ = -1.2878225
= 0.94064607a5
28 NACA TN 43o8
(erfc @=o. oo@069z02-o:oo21782842E 1-2.5E+3.8333333E2-3.9583333E3+
2.8083333E4-1.2847222E5+ 0.24w07$Jw6+ 0.v66766G7 -
)O.U4g0024e8+ 0.03617587U9 (2S @ ~ 2.8) (C2)
where e =@ -2.5 and
76.331
T + 10;~18 )]
where X = 2#2.
(2.8 < @< M) (C3)
..”
the approximation found on page 169 of reference 6.by this approximation became unacceptable in this .
Eqwtion (Cl) isThe errors Introducedapplication above a value of the argument of 2. Equation (C!3)was orig-inally thought to be useful in the range of arguments from 2 to co since
1 ed2the product of the first bracketed expression and ——Gfl
represent the
first eight terms of the semiconvergent series for erfc @ valid forlarge @ and the constants of the second bracketed expression are chosento give true answers at @ = 2, 2.5, and 3. However, it was discoveredthat this eqpation was also unacceptable in.the range. 2<~< 2.8. Ausable equation (eq. (C2)) for this range was found by employing the first10 terms of the Taylor expansion of erfc @ about the point @ = 2.5.
--
,/
NACA TN 43o8 29
REFERENCES
1. Vidal, R. J.: Model Instrumentation Techniques for Heat Transferand Force Measurements in a Hypersonic Shock Tunnel. Rep.No. AD-917-A-1 (Contract No. AF33(616)-2387), Cornell AeronauticalLab., kc., Feb. 1%6.
2. Cooper, Morton, and Mayo, Edward E.: Normal Conduction Effects onHeat-Transfer Data During Transient Heating of !Ihti-SkinModels.Jour. of the Aero. Sci. (Readers’ Forum), vol. 24, no. 6, June 1957,pp. 461-462.
3. Carslaw, H. S., and Jaeger, J. C.: Conduction of Heat in Solids.The Clarendon Press (Herd), 1947.
4. Magnus, Wilhelm, and Oberhettinger, Fritz: Formulas and Theoremsfor the Special Functions of Mathematical Physics. Chelsea Pub.Co. (New York), 1949.
5. Churchill, RuelV.: Modern Operational hkthematics in Engineering.McGraw-Hill Book Co., Inc., 1X.
6. Hastings, Cecil, Jr.: Approximations for Digital Computers.
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1.0 .632KI -3 .67sQ 3.6776 .63192 3.6794
1.2 .76302 -2.3698 2.3690 . 762sQ 2.3702
1.4 .85% -1.4092 1.4086 . yJ91 1.40~
1.6 .9226!3 -7.7330 x 10+ 7.- x 10+ .-1 7.7360 x 10+
1.8 .%080 -3.9198 3.9184 .@78 3.9204
2.0 .98L66 a. 8337 1.8330 .98163 1.8yio
2.5 .99W7 -1.9343 X 10J5 1.9336 x LO-6 .gg8c6 1.9347 x ld
3.0 .W88 -1.2378 X 10-7 1.2374 x U-7 .99876 l.qti x 10-7
3.5 1.0 -4. ed76 x U-9 4. 80* x 10-9 1.0 k.8oe2 x IO-9
4.0 1.0 -1.1317 x 1O-1Q 1.1313 x K@ 1.0 1.1319 x 10-10
L = O.~
0.1 0.OI.U544.9418 X 10-3 4.9337 x 10-3 0.00919 4.9459 x“10-3
.2 .04076 -4.7s62 4.78@ .03s38 4.8002
.J+ .14s07 -4 .25b6 k.2477 < J46gb 4.25&
.6 .30308 -3.4&.6 3.4789 .30135 3.4875
.8 .47304 -2.6348 2.6305 1 .k~~ 2.6370
1.0 .6z4 -1.8393 L 8363 .63u23 1.8408
1.2 .76290 -LU355 L 1836 .76231 1.1855
1.4 .85893 -7.0535 x 1o-4 7.0419 x lo~ .83er8 7.059k x 10-4
1.6 .s=50 -3. m 3.87X .52231 3.8782
1.8 .*9 -1.9653 1.g621I
.s6%0 L9669
2.0 .98+5$” -9.20W X 10-5 9.M79 x 10+ .98155 9.2K9 x 10-5
2.5 . 9m5 -9.7345 x d 9.7185 x 10-6 .9305 9.7430 x 10-6
3.0 .99987 -6.2520 x 10-7 6.2417 X 10-7 .99387 6.2372 x M-7
3.5 1.0 -2.4568 X Ii@ 2.4328X 104 1.0 2.4* X 104
4.0 1.0 ..5.7665 x 10-10 5.7570 x 1O-1Q 1.0 5.7710 x 1O-1Q
.
,
.
NACA TN 4308 .
IMUMTED-ATVARIW9VMIUQS W L AND A-Continued
E-O I ~ = 1.0
0.1
..2
A
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
2.5
3.0
3.5
4.0
0.1
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
2.5
3.0
3-5
4.0—
0.03472
.06!%%
.11CL42
.31720
.47924
.63242
.75986
.W
.91875
.957%
.97983
.597P
.mEJk
.5%99
1.0
0.04852
.09356
.19k28
.33192
.4%38
.634r2
.75708
.85a76
.91493
.~~l
.97792
.59732
.ma3
.%599
1.0
-9.6528 X 10-2
-9.3094-8.2858
-6.8280
-5.2076
-3.6758
-2.4014
-1.4519
-8.u50 X 10-3
-4.2078
-2.0170
-2.2868 x 10A
-1.5983 x 10-5
-6.8846 X 10-7
-1.82e6 x 10-8
-0.19030
-. lm?g
-. 161J.4
-.13362
-.10278
-7.3176 X 104
-4.fyseA
-2.gag
.1.7o14
-o. gga x 1o-3
4.4156
-5.3660 x lc+
-4.0820 x 10-5
-1.9445 x 1o-6
-P.~gy6 X loa
L = 0.1
0.16877
9.3568 x 10-2
8.0172
6.6066
5.038$3
3.5%62.3236
1.4048
7.i%16 X LO-3
k.opk
1.%M5
2.2@ x lo~
1.5465 x 10-5
6.6614 x 10-7
1.7693 x 10-8
L = 0.2
0.465s
.21g21
.15106
.12516
9.6284 x lo-
6.8548
4.55X4
2.7%11.5938
8.4292 x 10-3
4.1364
5.0269 x lo~
3.8237 x 10-s
1.an5 x IO-6
5.4292 x 10-8
0.cQ078
.rE285
.12964
.28278
.45300
.61388
.74~6
.8k75a
.91465
.95*
.g@31
.99760
.S%@
.99%9
1.0
0.CXXQ4
.0U72
.1.E44
.26408
.43*
.595s2
.W40
.63%0
.9c&8
.95044
.97568
.gg-@
.99978
.%s99
1.0
2.1252x ID-2
9.lo5k
8.421.4
6.9396
5.2926
3.-cm2.44C6
1.4756
8.25E!0X 10-3
4.2768
2.0502
2.3242X 104
1.6244 x 10-5
6.9YE X 10-7
1.&l& x 10-8
0.W667
+3594
.16&J7
.13788
● .KEQ7
7.5704 x m-p
5.0U6
3.0831
1.7558
9.2860 x 10-3
4.5%4
5.5372 x ld
4.2120 x 10-5
2.a%5 x 10-6
5.9808 x 10JJ
67
68 NACA TN 4308
0.1
,2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
2.5
3.0
3.5,
4.0
0.1
.2
.4
.6
,8
1.0
1.2
1.4
1.6
1.8
2.0
2.5
3.0
3.5
4.0
3.01326
.04234
.l~p
.30388
.47%4
.63218
.76278
. %&s
.92232
.96056
.9@54
.BE@!
.999m
L.o
1.0
0.02484
.05446
.15972
.30976
.47593
.63219
.76140
.85692
.g2070
.95937
.9&76
.9979
.9%X%
.WEJ9
1.0
-9. Wp! x 10-3
-9.5766-8.4970
-6.S6U2
+.2136
-3.6782
-2.3722
-1. 4Y27
-7.7678 X I.C@
-3.?4W-1. %59
-1.9628 X 10-5
-1.2659 x 10-6
-4.95&5 x lo~
-1.lea5 x 10-9
-4.8738 x 10=’
-4.7277
-4.2o14
-3.4512
-2.6204
-1 .Q$w
-1.1930
-7.1540 x 10-3
-3.9548
-2. G313
.9.6193 x II@
-Lop
-7.0375 x 10-6
-2.8798 x 1o-7
-7.2015 x 10-9
L = O.01”
9.8358 X 10-3
9.54608.4698
6.9390
5.1970
3.6664
2.3664
1. 4oee
7.7430 x lo~
3.933-4l.&cO
1.9566 X 10-5
1.2Q8 x M+
4.9428 x m-a
1.1767 X 10-9
L = O.m
6.u36 x 10-2
4.6533
4.1322
3.3944
2.5772
1.&88
1.1734
7.0362x 10-3
3. 89s76
1.9979
9.4607 x 10-4
1.0349
6.9a7 X 104
2.8324 X 10-7
7.0830 x 10-9
0.00834
.03756
.I-46c%
.30040
.476434
.6x34
.76163
. em
.ZJ-93
.sk4336
.9~2
.99803
.99987
1.0
1.0
0. W308
.03070
. 13%0
.2w41
.46276
.62294
.75540
.5333
.91871
.55@5
.98328
.99784
.9%E6
.59399
1.0
9.8248 X 10-3
9.59368.5120
6.9736
5.2228
3.6848
2.3764
1.4152
7.7816 X 10A
3.9510
1.-1:9633 x 10-5
1.26& x 1o-6
4.9674 x lo~
1.IIQ6 X 10-9
3. 5%!4 x 1O-Q
4.7633
4.2361
3.4794
2.6420
1.8343
1.2029
7.2u28X 10-3
3.9975
2.0481
9.6982X 10-4
1.0608
7.0953x I.&6
2.%36 x 10-7
7.2610 x 10-9
.
—.
,
“
NAW TN k308 69
INSULATEDPLATEAT VAKKXJSVALUE2Cl?’L AND ?,.Continued
3.0 g = 1.0
A T1 + + T1 &
~>
<~
s #
L = 0.3
0.1 O.@ago -0.2%233 0.84238 0 0.(X)103,
.2 .u48 -.26526 .3m4 .W3% .13318
.4 .2M28 -.23zi2 .21551 .0%34 .2k388
.6 .3469 -.1%04 .1~1 .24622 .ZOw
.8 .49315 -.15205 .13799 .41534 .15918
1.0 .63441 -.10968 9.%s4 x 10-2 .57829 .Klkee
1.2 .75476 -7.3572x 1o-2 6.6768 .71712 7.7016 X 10*
l.k . 8%703 -4.5891 4.1648 .82355 4.&41
~.6 .9u26 -2.6622 2.416iY .89763 2.78P
1.8 .95212 -1.4363 1.3035 .94477 1.5036
2.0 .$JT5g8 -7.2066 x 10-3 6.51MxIx 10-3 .97=9 7.5444x 10-3
2.5 .9ma -9.3516 X 104 8. U.870 x lo~ .99fM 9.7% x lo~3.0 .95974 -7.7094 x 10-5 6.9966 X 10-5 .599P 8.07c% X 10-5
3.5 .99939 -4.0385 x 10-6 3.6650 x 1o-6- .99398 k.gn x 10-6
k.o 1.0 -1.3423 X 10-7 1.2182x 10-7 1.0 l.t!oyx 10-7
L -0.4
0.1 0.06732 -0.372s9 U@l o 0. cm17
.2 .12&8 -.34877 .57422 . OCQ88 .10955
.4 .23690 -.30524 .2-7893 .081iJ :3r266
.6 .36087 -.25$5 .22498 .z?gl~ .27U4
.8 .m3 -. lgg& .l~& .39748 .2E335
1.0 .63@5 -. I.4558 . lag .5a05 ‘.15448
1.2 .m87 -9.88M x 10-2 8.6968 x 1o-2 .70194 .U14su
1.4 . 8b362 -6.2554 5.5036 .8u38 6.63& x m-
1.6 .90775”-3.@9 3.2k65 .88874 3.9154
1.8 .94929 -2.(X2% 1.7848 .g38a3 2.1527
2.0 .Sm@ -1.0394 g.lwlax 10-3 .968% 1.I.OX
2.5 .93541 -U1357 x 10-3 1.2631 .99567 1.5233 x 10-3
3.0 .99968 -1.~ x lo~ Ll@ x ld .*1 1.3554 x K@
3.5 .93998 -7.3174x 1o-6 6.43& X 10A .ggm3 7.7650 X 10&
4.0 1.0 -2.7017 x 10-7 2.3770 x IO-7 1.0 2. %70 x 10-7
70 NACA TN 4308.
+ +
TABLE II.- VALUES OF ~, ~, AND ~AT~a~
= O AND 1.0 FOR PERFECTLYe &2
INSULATED PLATE N VARIOUS VALUES OF L AND A - Concluded
Eo= k = 1.0
A ‘1 + + T1 q
~$
~s s
L = 0.5
0“1 o ●07504 -0.46248 1.7633 0.000030
.2 .14151 -.42$z4 .78278 .00152 .08179
.4 .25600 -.37200 .3455i .06922 .36890
.6 .37499 -.31250 .26710 .21305 *33547
.8 .50787 -.24606 .21002 .38028 .26447
1.0 .63796 -.18102 .15450 ●54409 .lg4~
1.2 .75139 -.12430 .10610 .68693 .13361
1.4 .&&g -7*9755x 10-2 6.~72 x 10-2 .799148.5719X 10-2
1.6 .90442 -4.7792 4.0791 .87964 5.1365
1.8 .94650 -2.6748 2.2830 .93264 2.8748
2.0 .97204 -1.3982 1.1933 .96479 1.5027
2.5● 99510 -2.0490 x 10-3 1.7488 X 10-3 .99484 2.2022 X1O-3
3.0 ● 99961 -1.9596 “x 10-4 1.6725 x 10-4 .99951 2.1061 x 10-4,
3*5 .99998 -1.2226 X10-5 1.0436 X 10-5 ● 99997 1.3141 x 10-5
4.0 1.0 -4.9836x 10-7 4.2536 X 10-7 1.0 5.3563 X 10-7
NACA TN 4308
.
71
TX*C
.
“
Qih~-[ui,ar)
Boundary condition
t<0:Tl=T2=Te=0
f >0: Te=constant
h =constant
At X=o, (=o●
Q=- + ~
axAt interface, x=J,[<
TI=T2 - -
Figure 1.- Sketch of the composite slab showing pertinent boundaryconditions.
72 NACA TN 4308
I0°
I6’—
-1.
:+aEo
I 64L.001
\
\
\
\
\
\
uo
:
I I I
\
\
\
\
\
\
E
c..+c
i
Ml.01
Skinhickness,
i n.
\
0.30
\
ao
\
.10
\
.05
\
01
I
\
.00 [“i.<
I
\
\
\
\
\
:aa
:
\
I
Thermal conductivity k, Btu/sec ft “F
—
\
\
\
\
\
.a)>=0
h
Figure 2.- Plot of the nondimensional parameter L as a functionskin material and thickness for h = O.lBtu/sq ft-~-sec.
.
.of
, ● t ,
II I I I I I I r
II 1
I :;
I I II Balsa, (pck~~O.00002,II1.0 ,
I 1 Spruce, mahogany, etC., (p Ck)2” o.000157 II A _ ~ ~
1I
+ ~Glriss, (pck)2m 0.0056
.8:; :1
1’ 1 :1(
I /N= 01
1. ml o~I
I ;1 -1 ~1I
:1 .~ I— —
q ,6I I > ~ —
I
L“ ,? I gl I
: 21 al I ImEo
~ — I~ -
/ Ia. I
II I
: 11: I I– Ilro I
I
.2 ,, tI I‘Backing material I
I I‘---Outer skin I
I“ i
II 1
I I I I
Y I I I I Io A .8 1.2 L6 2,0 2.4 2.8 3,2
( ““+ec”’parameter (pck)l, sq ft.
Figure 3.- Variation of the paramter ~ with the parameter (pck) ~ for wious outer sktis
and backing materials. The appropriate value of $ is determined by the intersection of thebacking-material curve with the absclsaa for the outer skin. If the outer skin and backingmaterial. are interchanged, the sign of ~ is reversed.
4w
74
00 ,.
I 0-?
Gn
,0-3
I 0-4
Id
I 0-6.001
—
—
//+——
—
—
—
—
.
.
5
Figure 4.- Values of
against
\
s2 .0I
(a)
2Gn~ and
L for
—
/
\
—
-
—
L
Gn .
J~ applicable at ~ = O plotted
.
.
.
.
.
.
.
,..
.001
—
—
005 .01L
(b) ~.
.05 .1
.
.
Figure 4.- Continued.
NACA TN 4308
.
.
,00
n
0/ ‘/ ‘
10-t
/ I-
/ /~O-2 — — — 2
0
10-3 \\
,0-4\
1(3-5 \
10-6.001 .005 .01
L.05 .1 .5
.
-.
(c) j *.
Figure 4.- Concluded.
.
.
1 , . ,
I
,0 .1
Figure ~.-‘tio ‘f ‘l/Te at g
from equation (15)
Value of upper sutnmat on
limit in equation(2 4
I 0: I
.2 .6 I 2 4
A
(a) L= O.001.
(’
P/
(/
10 20
!ssH
i
= o for a finite number of term in equation (25) to T1/Te ~
plotted against the parameter A. $ = 1. -a
— .
0 Volue of upper summationLal
limit in equation(251
;
:80;
uA3
lL4
B5
“?01 .1 .2
A
—
—
~
\
\
—
.4 .6 .8 I 2
(b) L = 0.1.
Figure 5.- Continued.
46
)
I
, . .
1,4
L 20.-+0L
0
; 1.0-1-0Lwn
E
? .8
.6~.01
,
R
Value of upper summation
limit in equation (25)
.1 ‘ .2 .4 ,6 .8 I 2 46
A
(c) L = 0.5.
Figure ~.- Concluded.
$ 1.0
2nE
5/
.8‘Perfect insulator (/3=1.000)
.6
.4f
!u
Z, in.
0.060
I ––––– .030
.2#
2 4 6 8 10 12 14 [6 18 2
Time, sec
F@ure 6.- l’enrperature distribution at the outer face plottdd against tires for composite slabs , L‘having sta~lew-steel skins suddenly exposed to ae%dynmic-heating at t = O.- The skin ‘Q
thicknesses are O.C@ and O.0~ inch and the insulati~ backing mterials are mhogany, balsa,and a perfect insubtor. h = 0.056Btu/sqft-’%-sec. ~
, . ,,,
, ,
. . , 1
L 1.3 ‘oac
a
E 1.2 ‘
o (
+-C 0
.% E ––––– 1.0
.:\lcl~u+.o+-OcOal.-
::1.0I
cm00 Perfect insulator (~. 1.000)I-u -- .2. .-
:: ;
**
:’% .9 ‘j ---& --Csc
w? I Balso (@=.975) ‘-a - %~++ Io0 I
.--clc
.8 J a.- Mahogany (/3=.950)
% ;
o:-+ I
z.70
.4 .8 1.2 1.6 2,0 2.4 2.8 3.2 3:6
Figure 7.- Ratio of Indicated heat-tranafer
against time for a composite slab having
insulating backing materials. h = 0.056
Time, sec
coefficient to true heat-transfer coefficient plotteda 0.030-inch-thick stainless-steeL skin and various
Btu/aq ft-%%ec.
.-
J 1,40
z
a
z1, in.
- 1.2 0.0600 ————— .030
0,,
: 140 -. --- .- ---- .–– ___ _T. —- —.- ----- ----- --- --- --- --- .-
:-J k& > ~ - ‘Perfect insulator (~= 1.000)
.- . \u \
‘.8\ \
EC \-
; ,aJ \ ‘BSSISSJ (8=.975)\00v .- \ \
w- \ i>
:= \ Y ‘
~:.6\ \
Wb. \ i,: u,
u- \ \
;:! \
\x
~ :, \ \ \G+, . 4 \ \u2
~,< \
:\
‘ \’1 Mahogany (/3=.950/~
G .2c \ ,\
\ \.- ---%. . / 1 \..0 < ‘ -- -_
~.
0’.-
&cl 4 8 12 16 20 24 28 32 36Time, sec
Figure 8.- Ratio of imdicated heat-transfer coefficient at E = O to true heat-transfer coeff%cient plotted against the for camposlti slabs bavi.ng stainless-steel skins of thichess z
and various insulating bac.king materials. h = O. 0% Btu/sq ft-%-sec.
. . . .
CDIII
43o8 83
I.4
1.2
1.0
.8
.6
.4
.2
(
\Perfect insulator (~= 1.000)
/ ‘mBalsa (6=.975)
(
–––--– too
.
.2 .4 .6 .8 1.0
Ratio of local temperature to equilibrium temperature, _T,
Tc
Figure 9.- Ratio of indicated heat-transfer coefficient to true heat-transfer coefficient plotted against ratio of local temperature toequilibrium temperature for composite slab having a 0.030-inch-thickstainless-steel.outer skin and various insulating backing materials.h = 0.056 Btu/sq ft-%-sec.
NACA -Langley FteId, V&
