The effect of transient heat flux on heat flux sensor response and calibration is analyzed. A one dimensional case was studied in order to elucidate the key parameters and trends for the problem. It has the added advantage that the solutions to the governing equations can be obtained by analytic means. The analytical results obtained to date indicate that the transient response of a heat flux sensor depends on the thermal boundary conditions, the geometry and the thermal properties of the sensor. In particular it was shown that if the thermal diffusivity of the sensor is small, then the transient behavior must be taken into account

Dybbs, A.

Ling, J. X.

English

NASA Technical Reports Server

N91-24323
:\,_alysis of tlw, Transient Calibrat ion of l|eat FJux
Sens(-)rs - one 1)imensional Case
A. Dyhbs and .l.X. Ling
Case C,enter for Complex Flow Measurements, C3FM
Department of Mectianit:at Engineering
Case Western Reserve University
Cleveland, Ohio 1,1106
ABSTRACT
The purpose of this analysis is to determitte the effect of transient heat flux on heat
Il_tx sensor response and ('alii)ration.
We have begun this work by studying a, one dimensional ease. This was done to
elucidate the key liaral_eters and trends for the lmfl)lem. II has lhe added adva.ntage thai.
ihe solutions to the governing eqlia/i¢ms ('ai! lie ()htained by altalylical inoai-is. The
analylical results ol_l,ained i() date indi(:al,(: til_ll, the transient resl)onse ()1"a I_eal flux s(,nsor
(lepelMs eli 1,he I,ll(;i'illal hol!ndary eOlidilJOliS, l lif_ ge,llllll_,l,l'y and Ih(, lli4!rnial lil'Olierl,ies o1'
i,ho $P1"1801", Ill t)a,rl, icul,_r it was <_hown lhai, if i,lte llierlnal di[filsivily of i,ho SOllS()l" is ,_inall,
say for a, nl_terial slloh as Nichrollie, then the I,railsielil, hehavior mllSI, be, i,aken ii_to
aeeOUll t.
!13
https://ntrs.nasa.gov/search.jsp?R=19910015010 2020-03-19T18:12:25+00:00Z
INTIit)I)I.!C, TION
The purpose el II,is analysis was to determine tile effect of transient heat flux on
sensor response and calilmttion.
Consider the case of a. heat flux sensor momited in a wall subjected to both radiative
a,d convective he_t transfer on the top side and purely convective heat transfer on the
bet/ore side as shown in Fig. 1, where x and r are the axial and radial coordinates,
respectively; L and C I) and d are tlie ttiicl<nessies and dlaiiieters of the two layers; 1<is the
therJnal conductivity, a is the diffusivity, T is temperature, lh and h.z are the convective
I_eal transfer coefficients, qo is I.he radiant incident heat flux, and Too is the a._nbient
lemperature. The subscril_ts i and a refer to the different materials of the sensor.
In this analysis we make the following simplifying assumptions
colistaJlt tliermal prolmrties;
constant, uniform ambient temperal.ures;
• constant, uniform convection coefficients;
constant, uniform radiant, in(hle_t, heat. flux;
(il perlLx:t thermal col,l.act I)etween tl.. regions of different I<;
(7 the equation_ are one-dimensional;
S.teady - State Sol,_ti<m To determine a base line case we first solve for the
steady-state case.
For this case, the governing equations and corresponding boundary condil.ion,q are,
reader the above assumpiions.
(l'-Tl
7Vd_- = 0 {}<x>f
- - (i)
{-N_.-= 0 (_<x ?_ 1, (2)
d T t T
-kl x(TT-- +hi (Tl- ml)-q0 =0 at x= 0 (:1)
i14
"l'j = T,,
d Ta = k., dT:_
<IT2
- h._, (T_ -'l' ..,)
at x = g
at, x -- L
(i)
(:3) au<l (5) are rea'rangecl as:
-kn -_._-----t- li, t*, = I_llj
trf_
k._,a-x-- -t- h..,T., = h..,f..,
where t't =Tnoo
f., = T._
qo
a_ x = l) Cl-a)
a.t x = L (5-a)
The solutions to equs. (I) an<l (2) are
'I'1 = At x + lIT
T2=+\,2x+B_
_l'he coefficienl_s At. Bi, .'\_, I]2 can lw. d_terllfine<l iu t;ellllS cff II,_ t)otmda,Lv {'OIl<]ili_)llS
(:I-a), (1), and (:_-a).
The sensor out,lint is based oil 1.1_, I.elnperal, ure dil'lk_rence bel,ween two fixe_l I>oi_t'_
where l,l_e two a lmne.l wires are a.t,1;a<:htxl. Assllllle I}lal, lira co_n<liimt,es _l" t,hese two points
are xi an<l x_ resl)e<:tivo, ly (ll_ey are iis_tally at <li['fere_t layers), t.llen
Tt (xl) = AIx, + PJl
T._, (x_) = .'\ex_ + I_,,
l lence,
AT s = % (xl) T.; (x._>) ((;)
It5
ORIGINAL PAGE IS
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Wlt('F('
•'\I=_A::, l_l=(k)/\j + h,f))lht
- l<dljh,) _--
.A., = Iqk,,h_ + kthih21., + k_hlh.2t +
NoL(' 1.ha.I L_STs is indcpendc, nt of time.
l!_s(('ady - State Solu(,h)n
For u _stea(Iy st, aLe ca,so, we have
a.j _,_-- = _-77-- O_<x_< t, t>O
i? T l = c)T.,
a_)-_-- _- O_<x_<L, t>O
with
(TF, Tz hll'j a,t x 0,
-lq _Kx--- + hj = =
T =Tm = '2 at, x /_, I, > 0
_)Tl -k., 8T._
lq-_l __ a,t;x=C, t>O
k,) _T2
__+ h,2T,2= h,2f'2 a.t.x= L,t > 0
lqk2h2 - kllhh2/'
1,>0
(7)
(s)
(9)
and
'rj (x,o) = T.: (x,O) = T() 10)
lhc, solu(,ions to ecluS. (7)and (8)with )lonhomogeneous boundary condil, ions (9)and (10)
(:a_l 1)e expressed in t,l_c fOl'lll
T i (×,1,) = 0i (×,t) + _i (×) t't + ¢'i (×) 1".2 (11)
i= I,'2
where 1,he funcl, ions :;i(x), t/,i(x), and Oi(x,l ) _lre Lhe soluLi(ms ()1"l,he t'ollmvil_g s i )l)r() )Ictus:
116
ORIGINAL PAGE IS
OF.. POOR QU._JTY
Therefore,
(h)
)
(_1- ,_l 0
(r_- 2 - -__-
(j 2 _72
- + )--ax- = 0
wi(Ik -kl _ + hl:i
(IX
91 =_>_'
kl (dx,_ = k2 d-_ ",CIx
ka _ + h2T2 = 0
= hill
_, (x) C'= ..Ix + 131
_>,(x) = C,,x + D_
d x 2=0
(ix -_ = 0
with -lq (If't
+ hi _"l = 0
kl _-- = k., d <',>
fiX " (-T.x-
(] ('I
k,., c.-1"7-_ + h2x2 = IL.,I2
al x = 0
al.x = _
atx=t
atx=L
a tx=0
aL._ = #
a.l.x = (
alx=l,
we ha.ve
f'i (×) = l';, x + F,
fL' (x) = E., x + i:.,
Ill lifts case, the sensor ¢)_iI l>i_l will l)e d(,l)(u)(hu)l, on l i)))e. 'I'o deI(_r)))il)e If)(; s(ms()r
OULl)ut we calcula(,e !,he l,e npera(tlre d f(uen< e l() l)e
ORIGINAL PAGE IS
OF POOR QUALITY
117
_X'l'u (1,) = % (x,t)- T._,(x2,1_)
- 0, (.,:,,_.)- a( x.,,t)+ [ _,(x,)- _:(x.,)lr,
+ [¢,(_,)- ¢_(x._,)la
Comput.ational results show that
[_,(xj) - V,_,(x=,)lf, + (¢_(x,)- ¢_(x2)1t'2 = AT s
"l'l,creIbre,
,',T'_(t)= O,(x_,t)- O.,(x2t)+ aT s
(c) Ol(x,t) and 02(x,t) are determined from the following homogeneous
problem:
0 _"Ol OOj
aJ Ox"- = <)t
02 02 00.2
a..., Ox"- = Ot
00j
$1"_' =ll)l_Cl, to -kl _ + hlO_ 0
01 0:_ at x = /'
00_ 002
1<1_ = l,:,, _ at, x = f
30,,
k., i-aT-x + h.,O., = O at x = L
a.t. x = 0
(12)
]el,
=>
and Oi(x,O ) = T ° -9i(x)lt- Oi(x)l )
Oi(x,i,)= xi{×)I'(i,}
_,]2 1
l_(I,)=e 'n
118
wh4'l'e/h_ _|1"() d_J.,_l_'l]Vall|[ (_'S whi('la will I)e determined tal,er.
= _.t/--- x)Xll I (X) All ] Si I' a'l_'_ X) + BI I _ (i"_*S ( _
I I
/4]ll _11
X.,.. (x) = "2,_'\Si,l (_ x) + B.,n" C,.s (_x)
\Vii.hour. loss of _;eneralit ,,, ,,{'_ ohm.bose A,u = I, lllll, '_"211' ;lllll B?I I
the corresponding I)our, lary conditiunS(alJd sevrrai pa,_es of a lgel._ra).
l?inally,
o0
ATU(t) -
li=!
C. [X,n(x)- X (×2)]e-lS_n"+ &Ts
eali I)e dele, rrililled wiih
(i3)
whore C
n
is (letermined hy the inilial i:_indil.i(ilis. The e_lilathnl for Lhe del, ei'nlinal, ion of
die eigeiivalues 17n is
sin ? + Ill')sin ")
K cos ?- Klli')sin ")
0
--Sill I1
,ros l I
I • I,
c,ls(-Tq)-ll-,qsui(Tq)
=(I
where I<1I' = /7'i f, _1= ;3h t, I li : 17-/7'
d-<,
I '2
k i n'i I!., k.,
Sonic addil, i(iual ah_elira will lead t,,i tllt_ i'(lllowin_ li'ans(:(,n(ll al (,quiii, iOli I'()r tile
(l(,I,Crllliiial loll of I.ho ¢''-l_(li'valilf,s 3 l:
I19
di-i-
I
li., t . ,l,-f iJ,,) (i,i)
_('wi'ilot'qllnli()li(13) as:
whoro
oo e_ltli.2 t
'_.Vl'n(t) = E C'*.,n + ATS
I1 --_ |
(15)
c ,1= c [ x,!, (x,)- x.2.(x)l
Numeri('al exnu_plc; shows i,hal. t.li-) << 17,.)-"<< ,J':l'-)<< ..... , so we can just 1,ake l,he
t'ir,_l ll'rlll t(I _/i)l)roxinia.l.e Lhe resull.
A,l-,U 3 7=c,e-J' t+AT s
lii ol.li(,r words, wo lla.VO:
1)ul. AT u = 0 a.l. I, = 0
Therel'ore, cl =-A"r s
,VI,n
,__VI,S= I -o
I
(i_)
WIi(H'0 :]1 C{tll lic deternd ,led t'rl)lii Eqlia .gill
16' o×a, lnining Eq,_. (14) a,ii(l (16):
(l) ATu(t) = r'X''I'S _'S I,-. + ;
II). The I'olh)wing ()l)selVal io is
(7) The transielil. 1)rocess (li,l)(,ii(ls ()ii I ll(, I)onndary c()iidil.il),i,_ (I,>li:_),
l.]te _OOlil(!l, ry ([,I,)_!nd ill(, l)i'Ol)Oll ip._ (kl.l%,,nl,ri.x)t)l" lilt, SOliS()r.
(3) The I.)'ailSielll. l('!rlil (e -/¢i'-'1') iil_iy 1)o very inil)Orl.nlll, if 31 is Siiial].
f:?lllI)e liiadO
120
ORIGINALPAGE IS o
oF POmlQ,.JAu-ry
'l'(_ i ltl._tratu, t.l_e impact of these results consider a typical example. Assume that D
- 0.S (.in, f = 0.02 era, I, = I1.11 era, choose Ii.z/h= = 1.0. See Fig. I. The material of the
snl)l,yer is alumcl whose i)t'operties _le k._ = .z37 w/re.l,:, n_ = 9.7 x 10-5 m-/s. If the
i mie'i_ds (>f the foil ;1re pm'e _zic,kot _tll<l ni<'lLr_m_e (_ll(,_ Ni, 20% Cl), resl)ectivel_,'. The
reslfivs are shown in l,l_l, t
l'ahl_I
I:(>il Xlalev'ial n, (In_/s) f1, e-/t '(at t = 0.05 second)
Nickel 2.3 x 10-5 " _- _ 0-716._,1.) 8.15 x I
(I.,1"e)
Ni('h rome
(SOt/, Ni, 20% Cr)
:].4 x 10-6 6.4:]_ 1.2.6 x 10-1
,,\s can be seen ft'f)ln TaMe t, if we _:h(x)se Ni-Cr a ll()y as the fiill vtmL(,rial, the
2I
transient l,_rn_ "e -/_1 " wottld !)e important.
SI ; _I _1A 1-IY
\Ve have, inilinte(t a study of the lrnnsienl ho;iI flux respmlse ()1"hva.v Iltlx sevls()vs.
his lmper present,,_ a one dill_onsiotml _naly_is ot the l)rohlem. This was done I,o elttcifh_le
1_('key parametev's and t,r(;n(l t'¢)r Illo I)r()l)te_n. The t'osHII.s of a ho;tl tl,vx seltsor _lelx'nds _ll
t1/(' thermal I)om_dary (:'()n(IJtion.% l,he g(x)nletry and th(, thernml I)rOt_er/i_,._ o1 i ho sons_)r.
In I>a.rti(:_fiar, it. w;_s shown lhat Jl" lhe I/mrm;ll dill'tt._ivilv o1"t;1_('sen,_or is ._niall, ._ay [or a
l_ldll(__rJallike NJ{;hr(-)mo, t.llell the LlitllSJ(llil. Ix,havJor linl._l, be l ilk(_ti illl(i it(:COtllil.
ORIGINAL PAGE IS
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IZl
1_EFER ENCES
I. Atl<inson, W.II. and Slxange, R.R.: "Developnlent of Advanced High
Temp6rature Ileal, Flux Sensors," NASA CR-1656181 NASA Lewis Research
Center, September, 1982.
"2. Alkinson, W.II., Cyr, M.A., and Strange, R.R.: "Turbine Blade
al_d Vane IIeat Flux Sensor Development Phase I - Final tleport,"
NASA CR-168297, NASA Lewis Resom'ch Center, August, 1984.
:_. llolman, J.P.: Heal, IransIer, Mc-Graw-lfill Book Company, 19_.
.I. Atl<inson, W.It., Cyr, M.A., and Strange, R.R.: "Turbine Blade
and Vane I!eat Flux Sensor Development Phase II - Final Report,,"
NASA CR-17=1995, NAS,,\ l,ewis Research Center, October, 198.5.
Wood_'uff, L.W., Iiea'ne, L.F., and Keliher, T.J.: "lnterpre-
t.ation of Asymptoti( Caorimeter _leasurements," AIAA Journal,
Vol. 5, No. 4, April, 1967, pp. 795-797.
,
Schematic of Transient Analysis
qO
TB I, h I
Fig. i
E
Tin2, h2
122


Analysis of the transient calibration of heat flux sensors: One dimensional case

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