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CHAPTE~ 1 D ESCRIPTION O F T HE I NVERSE H EAT C ONDUCTION P ROBLEM 1.1 I NTRODUCTION I f the heat flux o r t emperature histories at the surface o f a solid a re k nown as functions o f time, then the temperature distribution can be found . This is termed a direct problem. In many dynamic heat transfer situations, the surface heat flux and temperature histories o fa solid must be determined from transient temperature measurements a t o ne o r m ore interior locations; this is a n inverse problem. In particular, during the past two decades the special case o f e stimating a surface condition from interior measurements has come to be known as the inverse heat conduction problem. There a re n umerous o ther inverse problems in transient conduction a nd diffusion. but this particular problem has been so named and is the main subject o f this book. The inverse heat conduction problem is m uch more difficult t o solve analytically than the direct problem. But in the direct problem many experimental impediments may arise in m easuring o r p roducing given b oundary conditions. The physical situation a t the surface may be unsuitable for attaching a sensor, o r the accuracy o f a surface measurement may be seriously impaired by the presence o f the sensor. Although i t is often dilficult t o measure the temperature history o f t he heated surface o f a solid, it is easier to measure accurately the temperature history a t an interior location o r at an insulated surface o f the body. T hus there is a choice between relatively inaccurate measurements o r a difficult analytical problem. An accurate and tractable inverse problem solution would thus minimize both disadvantages a t once. The problems o f d etermining the surface temperature and the surface heat flux histories are equivalent in t he sense that if one is k nown the other can be found in a straightforward fashion. They cannot be independently found since in direct heat conduction problems only one boundary condition can be imposed a t a given time and boundary. Even though this is true, the following 2 CHAP.1 SEC. 1 .2 D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM seemingly contradictory statement can be made: the heat flux is m ore difficult to c alculate accurately than the surface temperature. F or this reason the emphasis in this book is o n the calculation of the surface heat flux history. (The surface temperature is a b y-product o f the heat flux calculations for difference procedures.) F or t he purposes o f this book the inverse heat conduction problem ( IHCP) is defined as follows: T he ! HCP is the estimation o f the surface heat flux history given one o r m ore measured temperature histories inside a heat-conducting body. T he w ord " estimation" is used because in m easuring the internal temperatures, errors are always present t o s ome extent a nd they affect the accuracy o f the heat flux calculation. F urthermore even if discrete d ata a ccurate to a large but finite n umber o f significant figures are used, the heat flux c annot be exactly determined. O ne o f the earliest papers on the ! HCP was published by Stolz l in 1960; it a ddressed calculation of heat transfer rates during quenching o f bodies of simple finite shapes. Stolz I claimed use of his method as early as J une 1957. F or semi-infinite geometries Mirsepassi2 maintained t hat he h ad used the same technique b oth numerically2.3 a nd graphically3 for several years prior to 1960. A Russian p aper by Shumakov 4 on the I HCP was translated in 1957. T he s pace program, starting a bout 1956, gave considerable impetus to the study o f t he inverse heat conduction problem. T he a pplications therein were related t o nose cones o f missiles a nd probes, t o rocket nozzles, and o ther devices. Beck also initiated his work on the ! HCP a bout t hat time a nd developed the basic concepts 5- II t hat p ermitted much smaller time steps than the Stolz method. I O thers whose work h ad a pplication to the space program included Blackwell,12.13 Imber,16-23 Mulholland,24-27 a nd Williams a nd C urry.31 A nother research area t hat extensively required solutions of the I HCP was the testing of nuclear reactor components.J2 - 38 M any of the c omputer p rograms in c urrent use in the United S tates 35 - 38 a ppear t o be based on the method described in a 1970 p aper. 9 O ther a pplications reported for the I HCP included (1) p eriodic heating in combustion chambers o f i nternal combustion engines,39 (2) solidification o f glass,40 (3) indirect calorimetry for laboratory use,41 a nd (4) t ransient boiling curve studies. 42 O ver 300 p apers have been 'w ritten to date on the I HCP o r closely related problems. There have been extremely varied approaches to the inverse heat conduction problem. Thes'e have included the use o f D uhamel's theorem (or convolution integral) which is restricted t o l inear problems. I - 3.5 -7.1 0.28.58 N umerical procedures such as finite differences 8.9. 11 - 13.29 - 31.35 - 38 a nd finite elements 35 . 36 h ave also been employed due to their inherent ability to treat nonlinear problems. Exact solution techniques were proposed by Burggraf,43 Imber a nd Khan,23 L angford,t4 a nd o thers; such techniques have limited use for real,stic p roblems (as discussed in C hapter 2) b ut they can give considerable insight into the I HCP. S ome techniques used Laplace transforms and were also limited t o l inear cases. 41 .44 T he I HCP is o ne o f m any mathematically " ill-posed" p roblems. Such E XAM PLES OF INVERSE P ROBLEMS 3 p roblems are typically inverse problems a nd a re extremely sensitive to measurement errors. (See Section 4.2 for further discussion o f ill-posed problems.) There are a n um?er o f p rocedures that have been advanced for the solution of ill-posed problems m general. O ne o f these was developed by T ikhonov a nd Arsenin in 45 1963. T ikhonov i ntroduced what he called the regularization method t o reduce the sensitivity o f ill-posed problems t o m easurement errors. A modification o f this method is presented herein for m ore efficient solution o f the ! HCP (see Section 4.5). N umerous o ther general procedures for ill-posed problems have been proposed including a technique, well known t o geophysicists called the Bac~us-Gil?:rt technique.28.46.47 T he m athematical techniques for ~olving sets ~f Ill-conditIOned a lgebraic equations called single-value decomposition techmques can also be used for the I HCP. 48 . 57 T he p urposes o f this c hapter a re t o i ntroduce the inverse heat conduction a nd related problems and to provide a general description o f v arious aspects of t~e I HCP. T he c ontents o f the remainder o f this c hapter a re as follows: Secllon 1.2 p.rovides some explicit examples o f t he I HCP a nd related problems. T he.IHCP IS related to function a nd p arameter e stimation in Section 1.3. Sect~on 1.4 gives a description o f t he n ature o f t he measurement errors. In . SectIon. 1. ~ a n answe.r is gi:,en .why the ! HCP is difficult. T he i mportant subject of sensltlVlty coeffiCIents IS dIscussed in Section 1.6. A b rief classification of sol~tion m ethods o f t he I HCP is given in Section 1.7. C riteria for evaluating varIOUS I HCP m ethods are suggested in Section 1.8. T he final section, 1.9, gives the scope o f the book. 1 .2 1.2.1 ;. E XAMPLES O F I NVERSE P ROBLEMS I nverse H eat C onduction P roblem E xamples O ne e xample of t~e . IHCP is t.he e stimation o f the heating history experienced by a s huttle o r mISSIle reenterIng the earth's atmosphere from space. T he h eat flux a t the he~ted s~rfac~ is needed. Figure 1.1 depicts a reentering body a nd a n enlar~ed sectIOn o f ItS s km. T hough the heat flux, d enoted q, may be in general a functIon. o f b oth position y a nd time I, it is a ssumed at present that lateral conduction can be neglected compared to the heat flow n ormal to the surface. Thus the net surface heat flux as a function o f time is e stimated from measurements obtained from an interior temperature sensor at position x as shown in F' I Igure 1.1b. T he measurements are made at discrete times, I I , 12 , • .. o r in gener~,1 ~t tim~, I j a t which the temperature measurement is d enoted Y; . (The word discrete means at several particular times, such as I second, 2 seconds, 3 seconds, etc., but not continuously.) Figure 1.2 is a n illustration of postulated values. An estimated surface heat flux, d enoted qj, is associated with the time /. a t which the corresponding temperature measurement, y;, is made. T he Iru~ value o f t he surface heat flux is simply denoted qj. T he surface heat flux history, 4 CHAP.1 DESCRIPTION OF THE INVERSE HEAT C ONDUCTION P ROBLEM SEC. 1.2 sources include high-temperature fluids that flow in reactor heat exchangers, over reentry vehicle surfaces, o r across turbine blades. The heating can also be by r adiation from any source o r by conduction from an adjacent solid t hat is in thermal contact with the b oundary in question. T o e stimate the surface heat flux history it is necessary to have a m athematical model o f t he heat transfer process. F or example, in the reentry vehicle case shown in Figure 1.1 b, it is assumed that the section of the skin is o f a single material, homogeneous and isotropic, a nd t hat it closely approximates a flat plate. ( A radial segment o f a cylinder o r o ther one dimensional coordinate can be treated in a similar manner.) Then a possible mathematical model for the temperature T in the plate is: Section A (4) (b) !..(k FIGURE 1.1 Example o f reentering vehicle for which the surface heat flux is needed. (a), Reentering vehicle s chematic; (b), section A. ax OT)=PC a T ax at T(x, 0) = To(x) a T = 0 a tx=L ax ....... ~. . :3 ... .. \ . • 5 EXAMPLES OF INVERSE P ROBLEMS Measured temperature. (1.2.1) (1.2.2) (1.2.3a) (1.2.3b) Y i. T he objective is t o estimate the surface heat flux a t discrete times, t;, from at time I i ~ q(t i ) = - k iJT(x, t;)1 ax % =0 Time,t; FIGURE 1 .2 M easured temperatures a t discrete times. •I o• r This problem is q uite different from the direct problem in t hat t he boundary condition is n ot specified a t x =O b ut instead a measured temperature history is given a t o ne o r m ore internal locations. F urther c omplications are that the measured temperatures a re o btained only a t discrete times a nd they inherently have errors in them. Clearly interior measurements contain much less information than given for classical direct problems where the surface conditions are continuous, errorless relations. T he t hermal conductivity, k, density, p, a nd specific heat, c, a re postulated to be known functions of temperature. If a nyone o f these thermal properties varies with temperature, the I HCP becomes nonlinear. T he initial temperature distribution, To(x), is a lso taken as known. The location, X I, o f t he sensor is assumed to be measured a nd t o have negligible error. T he thickness of the plate, L, is also known and considered errorless. The known boundary condition o f perfect insulation given by Eq. (1.2.3a) is only one o f m any t hat c an be prescribed a t x = L. T here can be a convective a nd/or a r adiation condition a t x = L. F or a n I HCP with a single unknown heat flux, it is only necessary t hat t he boundary condition a t x = L be known. F or a t emperature-dependent heat transfer coefficient h a nd also for a radiation condition, the inverse heat conduction problem again becomes nonlinear. F or t he case of a single interior temperature history, the problem can be Calculate.d surface heat flux, q i. at time t; .. .. Time,t; FIGURE 1 .3 - (1.2.4) Representations o f calculated surface heat fluxes. q(t), can be a n a rbitrary single-valued time function; see Figure 1.3, which is a representation o f estimated values o f q(t) a t times ti • In general the heat flux can rise and fall abruptly a nd c an be both positive and negative where negative values indicate heat losses from the surface. The source o f h eating is immaterial to the I HCP procedures. Convective L CHAP.1 6 DESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM subdivided into two separate problems, one o f which is a direct problem as shown in F igure 1.4. T he portion of the body from X=X I t o L, body 2, c an be analyzed as a direct problem because there a re known b oundary c onditions a t b oth boundaries [ T(t)=Y(t) a t X=X I , a T/ax=O a t x =L]. F rom this direct problem the heat flux a t X I can be found from the solution for the temperature distribution in X I ~x~L by using qx,(t) = - k aT\ -a (1.2.5) A x X =Xl T his same heat flux must leave b ody 1 (O~x<xd. Consequently, two c onditions are specified a t X=XI in body 1 a nd none a t x=O. Such a s et o f b oundary conditions for the transient heat conduction equation, Eq. (1.2.1), is related to the mathematical problem being ill-posed. T he I HCP for a single unknown surface heat flux c an be complicated in many ways, some o f which are illustrated in Figure 1.5. T here a re four temperature sensors shown which preclude the simple subdivision shown in Figure 1.4 because the heat flux calculated to leave one subdivision would n ot in general equal the calculated heat flux entering the next subdivision. Another complication is t he composite body o f t hree different materials which may be joined together with either perfect o r imperfect contacts. In addition the plates might q(t) =? xl---1 I- L (0 =1 , Yet) q(t) Inverse ·1 yet) CD Direct f -XI-l FIGURE 1 .4 Subdivision o f a single interior sensor I HCP i nto inverse a nd direct problems. SEC. 1.2 E XAMPLES OF INVERSE P ROBLEMS 7 n ot be flat b ut r ather be parts o f a cylindrical wall. A satisfactory solution o f t he inverse heat conduction problem should permit treatment o f each o f these complicating factors. 1 .2.2 O ther I nverse F unction E stimation P roblems T here are se~eral o ther ~robl.ems related to the inverse heat conduction problem. The IHC~ mv?lves estImatIOn o f t he surface heat-flux time-function utilizing ~easured m tenor t emperature histories. I t is a linear problem (see Section 1.6.1) If th~ ~hermal p roperties a re i ndependent o f t emperature a nd t he boundary c ondltlon a~ t he " known" b oundary is linear. A closely related problem involves the convective b oundary c ondition, aTI -k -a = h[T..,(t)- T(O, t)] X x =o (1.2.6) I f the heat transfer coefficient, h, is k nown either as a constant o r as a function o f time, the estimation o f t he ambient temperature T..,(t) from given internal temperature measurements is a linear, inverse function estimation problem. If h is a k nown function o f T, then the inverse problem becomes nonlinear.49 An~ther import~nt .function estimation problem in connection with Eq. (1.2.6~ IS t he ~etermmatlOn o f h as a function o f time. This is a nonlinear p roblem even If t he dIfferential equation is linear. See Section 1.6 for further discussion o f nonlinearity. T he d etermination o f t he transient heat transfer coefficient is an important technique, for example, for investigating the complete boiling 42 curve. T he estimation o f h(t) is discussed in C hapter 8. An interface contact conductance, he(t), is often used t o model imperfect contact. F or t he interface in Figure 1.5 t he heat flux is related to he by k - UTI UX x =(I. , + L,)- =hc[TI.,=(L, +L,)- - TI.,=(L, + L,).] h T• . ambient temperature ~-----X31--------~ ~-----------~------------~ FIGURE 1 .5 C omposite plate with mUltiple temperature sensors. =_kiJTI a x x=(L, + L,)' (1.2 .7) where the ~ign -+: m eans the material 3 side o f the interface and the sign - means the m atena. 2 side. The problem o f e stimating hc(t) is very similar to that for 1 the convective heat transfer coefficient. Endothermic o r e xothermic chemical reactions can occur inside materials. These ca.n be o~ u nknown magnitudes. Also there can be an energy source due to electnc heatmg, a nuclear sotirce, o r frictional heating. In these cases a n a ppropriate describing equation for one-dimensional plane geometries is U( aT) a x k a x + g(x, t )=pc at aT (1.2.8) 8 C HAP .1 D ESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM where g(x, t) is a volume energy source term. If g is a function of time only, then estimation of g(t) from transient interior temperature measurements is quite similar t o t he one-dimensional I HCP. I fEq . (1.2.8) is linear and the boundary conditions are linear, estimation of g(t) is a linear problem; however, if k = k(T)" t he problem of estimating g(t) becomes nonlinear. When g is a function of both x a nd t, the estimation o f g(x, t) is similar to that of a two-dimensional I HCP. T wo inverse function estimation problems that have received a great deal o f a ttention from mathematicians are called the (improperly posed) Cauchy problem for the two-dimensional Laplace's e quation4s.H-ss a nd the initialboundary value problem for the backward heat equation. S 3.S6 O ne form of the Cauchy problem for the equation, a2 T a2 T ax 2 + ay2 = 0 (1.2.9) is for incomplete specification o f the boundary conditions but some interior measurements o f temperature are given. The objective is to obtain an estimate of T(x, y) for the complete domain including the boundaries. O ne example of a backward heat equation problem is the determination of the initial temperature distribution, To(x), in a finite body given the boundary conditions and some internal transient measurements of temperature. (See Problem 1.26.) T he inverse problems mentioned become more complex as more functions are determined simultaneously. F or example, one might attempt to simultaneously estimate for Figure 1.S t he heat flux q(t) o n the left boundary of the body and TCXl(t) o n the right. This would involve simultaneous estimation of two time functions. If the surface heat flux is a function of position across the surface as shown in Figure 1.6, a n umber of heat flux components would be simultaneously estimated; this is the two-dimensional I HCP a nd is discussed further in C hapter 7. S EC.1.4 F UNCTION E STIMATION V ERSUS P ARAMETER E STIMATION T he words "function estimation" were used in the previous section in connection with the I HCP. In the I HCP, the heat flux is found as an arbitrary, single-valued function of time. The heat flux can be positive o r negative, constant o r a bruptly changing, periodic o r non periodic, and so on. I t may be influenced by h uman decisions. F or example, the pilot o f a shuttle can change the reentry trajectory. In the I HCP problem the surface heat flux is a function of time and may require hundreds of individually estimated heat flux components, qj, t o define it adequately. Related estimation problems are those called " parameter estimation" problems which are also inverse problems but with the emphasis on the estimation o f certain " parameters" o r c onstants o r physical properties. In the context of heat conduction one might be interested in determining the thermal conductivity of a solid given some internal temperature histories and t he surface heat flux a nd other boundary conditions. 49 T he thermal conductivity of A RMCO iron near room temperature, for example, could be a parameter; it is not a function and does not require hundreds o f values of k/ to describe it. The parameter estimation and function estimation problems start to merge if estimates are made o f the thermal conductivity, k, as a function of temperature, T. However, the k(T) function is not arbitrary and is not adjustable by humans. P arameter estimation is a c ompanion subject; a book by Beck a nd Arnold has been written on the subject. 49 A background in parameter estimation is not required to understand this book. The subject o f p arameter estimation has been built on a statistical base but that is not as true for function estimation problems. This book stresses the numerical and mathematical aspects of function estimation rather than the statistical aspects. Certain statistical aspects are included, however, as in Section 1.4 in which measurement errors are discussed. 1.4.1 ~~ • Temperature·sensors FIGURE 1 .6 Surface heat flux as a function of position for a flat plate. 9 1 .3 1 .4 r M EASUREMENTS M EASUREMENTS D escription o f M easurement E rrors In the inverse heat conduction problem there are a number of measured quantities in addition to temperature; such as time, sensor location, and specimen thickness. Each is assumed to be accurately known except the temperature. I f this is n ot true, then it may be necessary, for example, to simultaneously estimate sensor location and the surface heat flux. T he latter problem would involve both the inverse heat conduction and parameter estimation problems and is beyond the scope of this book. If the thermal properties are not accurately known, they should be determined as accurately as possible using parameter estimation techniques. 10 CHAP.1 DESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM T he temperature measurements are assumed to contain the major sources o f e rror o r uncertainty. Any known systematic effects due to calibration errors, presence o f the sensor, conduction and convection losses o r whatever are assumed to be removed to the extent that the remaining errors may be considered to be random. These random errors can then be statistically described. The information provided by the sensors inside the heat-conducting body is incomplete in several respects. First, these measurements are a t discrete locations. There is only a finite number o f sensors, sometimes only one. Hence the spatial variation of temperature is quite incompletely known. Moreover, the measurements obtained from any sensor are available only at discrete times, rather than continuously. Due to the nature o f the measurement errors, a continuous t emperature record might contribute little more information than the discrete values, however. 1 .4.2 S tatistical D escription o f E rrors A set o f eight standard statistical assumptions regarding the temperature measurements is given in this section. These are standard a ssumptions and may not be valid for a particular case. These eight assumptions49 d o provide a yardstick with which to compare the actual conditions. T he r andom errors in the temperature measurements cause random errors in the surface heat flux values. The standard assumptions permit simplifications in the analysis o f r andom errors. T he eight standard assumptions discussed in Beck and Arnold 49 are: 1. T he first standard assumption is t hat the errors are additive o r 11 = 7; + ej (additive errors) (1.4.1) where 11 is the temperature measurement at time tj, 7; is the " true" t emperature at time tj, and ej is the random error at time t j • 2. T he second standard assumption is t hat the temperature errors, ej, have a zero mean (a theoretical quantity), E(ej)=O (zero mean errors) (1 .4.2) where E (') is the "expected value operator."49 A r andom error is o ne that varies as the measurement is repeated b ut the theoretical mean does n ot have to be equal to zero. There can be a bias; t hat is, the error might tend to be positive. I t is frequently possible to calculate and remove the bias. A sample mean is the average t hat is based on actual measurements. T he t rue average Ofej c annot be determined becauseej is u nknown; a typical equation for finding the sample mean o f a r andom variable such as 11 is _ 1 1 ';=J J L j= 1 ljj (1.4.3) SEC.1 .4 M EASUREMENTS 11 where ljj is the j th measurement at time tj a nd there are J measurements at time tj.1f Eq. (1.4.2) is true, the expected value o f ~ is 7;. T he expression given by Eq. (1.4.3) is called the sample mean a nd c an sometimes be used to check the assumption given by Eq. (1.4.2). 3. T he third standard assumption is t hat o f a c onstant variance, V( 11) = 0 '2 (constant variance error) (1.4.4) where V (·) is the "variance operator" a nd is related to the expected value operator by (1.4.5) The symbol 0 '2 does not contain an i subscript, thus Eq. (1.4.4) means that the variance o f 11 is i ndependent o f time tj a nd is a constant. I f the constant variance ass.umption embodied in Eq. (1.4.4) is valid and there is only a single sensor, an estimate o f the "variance o f 1';", d enoted S 2, is (1.4.6) for n measurement times; p is the number o f p arameters being used to estimate 7;, the estimate o f which is denoted 9;, a nd e j is the residual defined by (1.4.7) Ex~r~ssions o f the type giv~n by Eq. (1.4.6) c an be employed to investigate the validity o f the constant variance a ssumption given by Eq. (1.4.4). 4. T he fourth standard assumption relates to the correlations among measurements. F or two measurement errors ej a nd e j where i of j , the two errors are uncorrelated if the covariance o f ej a nd e j is zero o r (1.4.8) The different errors I l j a nd I lj a re uncorrelated if each has no effect on o r relationship to the other. An example o f correlated errors is ej = pe j _ 1 + U j where U j is uncorrelated to the e/S a nd p is a constant. As the sampling rate o f a n automatic data acquisition system increases, the errors tend to become more correlated. High correlation between succeeding temperature measurements indicates that each new measurement is c ontributing much less information than if the correlation were zero. Very high sampling rates (which approach continuous measurements) may contribute little more information than considerably lower rates; that is, larger time steps, 61, between the measurements. A measure o f the correlation between the two succeeding d ata points 1'; a nd 12 C HAP .1 DESCRIPTION OF T HE INVERSE H EAT C ONDUCTION P ROBLEM S EC.1.5 ~ Yi+1 is the sample correlation coefficient, p, defined as : I ejej+1 .:,.j=...;.I:""I-- 13 t hat is, those that vary significantly between successive values, are not permitted, however. • -1 p W HY IS T HE I HCP D IFFICULT? . (1.4.9) I 1 er 1 .5 W HY IS T HE I HCP D IFFICULT? i= (This is a n appropriate estimator for p if Ilj = p llj _ I + Uj which was mentioned previously.) A low correlation is n ear zero and a high correlation is n ear ± 1. ( For further discussion of correlated errors, see Reference 4 9, p p. 301 - 326.) 5. T he fifth standard assumption is t hat the temperature measurement errors have a normal (that is, gaussian) distribution, Ilj has a normal distribution (1.4.10) I f the second, third, a nd fourth standard assumptions are valid, the probability density o f Ilj is given by 1 (-Il~) f (eJ = a .J2rt exp 2a; (1.4.11) T he assumption o f n ormality is frequently valid even if standard assumptions 2 ,3, a nd 4 are n ot; in t hat case a joint p robability density for the errors is needed (Reference 49, p. 230). . 6. T he sixth standard assumption is t hat the statistical parameters such as ( 12 a nd p a re known, Known statistical parameters (1.4.12) 7. T he seventh standard assumption is t hat t he times t l , t 2 , • • • , t ., positions X J> specimen dimensions, a nd t hermal properties are accurately known. X I, X 2,"" Errorless time, dimensions, and configuration o f object in question and thermal property values (1.4.13) In other words, the only source o f e rror is in the measured temperatures. I n statistical terms, the variances o f time, a nd s o on, are zero. 8. T he last standard assumption is t hat there is no prior information regarding the shape of the surface heat flux, N o p rior information regarding the surface heat flux 1.5.1 S ensitivity t o E rrors T he inverse heat conduction problem is difficult because it is extremely sensitive to measurement errors. The difficulties are particularly pronounced as one tries to obtain the maximum a mount o f i nformation from the data. F or t he onedimensional I H C P when discrete values o f t he q c urve a re estimated, maximizing the a mount o f i nformation implies small time steps between qj values (see Figure 1.3). However, the use o f small time steps frequently introduces instabilities in the solution o f the I HCP unless restrictions are employed t hat will be discussed in later chapters. Notice the condition o f small time steps has the opposite effect in the I HCP c ompared to t hat in the numerical solution o f t he heat conduction equation. In the latter, stability problems often can be corrected by reducing the size o f t he time steps. 1 .5.2 E xamples o f D amping a nd L agging; E xact S olutions T he transient temperature response o f a n internal point in an opaque, hcatconducting body is q uite different from that o f a p oint a t t he surface. The internal temperature excursions a re much diminished internally compared to the surface temperature changes. This is a d amping effect. A large time delay o r lag in the internal response can also be noted. These damping a nd lagging effects for the direct problem a re i mportant t o study because they provide engineering insight into the difficulties encountered in the inverse problem. O ne interesting case is t hat o f a semi-infinite body .heated by a sinusoidal surface heat flux o f frequency w, q =qo cos(wt) (1.4.14) " Prior i nformation" m eans information known before any temperature measurements a re m ade for a particular case. I f p rior information exists, then it can be utilized t o o btain better estimates. If, for example, from experience with previous similar tests the heat flux is c onstant over some t.ime perio.d o r .is periodic, this information can be used to improve upon the estimators given I n this book. I t is a ssumed herein t hat little is known a bout t he surface heat flux except t hat it can vary abruptly with time. High-frequency fluctuations o f qj, (1.5.1) where qo is t he maximum value o f t he surface heat flux. After a sufficiently long time, the temperature solution also becomes periodic and is given by T =To+ :0 (~r2 exp [ - x (~r2}os [ wt-x (~r2 - iJ (1.5.2) where a. is the thermal diffusivity, k is t hermal conductivity, and To, a c onstant, is the initial temperature distribution. T he envelope o f Eq. (1.5.2) is ( T-To).nv=qok- I (~r2 ex p [ -x(~r2J (1.5.3) As t he frequency w increases, the envelope decreases. The maximum temperature 14 C HAP.l DESCRIPTION OF THE INVERSE HEAT C ONDUCTION PROBLEM SEC . l.5 rise occurs at x =O a nd is p roportional to w - II2. F or an interior location ( T-To).nv (T-To).nv . x=O = exp [ (W)II2] -x 20: W HY IS THE I HCP DIFFICULT? 1.0 r----~-:------------~_.. x +;;::+ 0 .9 ~ (1.5 .4) 0 .8 which shows that the envelope of interior temperatures sharply decreases for increased x values. The exponential in Eq. (1.5.4) also indicates a large effect as w is increased. T o o btain some insight from this equation, the case is considered wherein the right-hand side is less than om o r x(~r2 > 4.6 15 . 0.7 qc~ f -- L - --l ·I ~ 0.6 ' i. . .... 0.5 II t. 0.4 0.3 F or steel with ex = 1 0- 5 m 2 /s and w = 27t r ad/s= 1 Hz, there is negligible response for x~0.82 cm ( =0.32 in.). This is large damping but ifw were further increased, say by a factor of 100, there would be negligible response for x > 0.08 cm ( = 0.03 in .). T he lagging effect can also be investigated through a n examination of Eq. (1.5.2) . The surface temperature lags 7t/4 radians o r 45 behind the surface heat flux, a nd any interior location lags even more. F or example, for the values of ex = 10 - 5 m 2 /S, w = 27t rad/s, and x = 0.82 cm, there is a lag of 4.6 rad o r 264 which corresponds to a 0.73-s lag of the internal temperature compared t o the surface T. Returning now to the IHCP, consider a transient interior temperature with small fluctuations imposed o n its changing value. These fluctuations can be the result of high-frequency sinusoidal surface heat flux components o r r andom measurement errors. F or a given sensitivity in the temperature sensor, it is possible to specify many different heat flux curves (each having high-frequency components) that will p roduce interior temperatures indistinguishable from one another. This implies that the inverse problem does not have a unique solution. However, the heat flux history t hat caused a thermocouple response can usually be determined to acceptable accuracy for properly designed experiments using the methods to be given. Similar effects to those just described can be found through an examination of the problem of a flat plate exposed to a constant heat flux qc a t x = 0 and insulated at x =L. See Figure 1.7 a nd Table 1.1. T he solution for the temperature distribution for this problem is given by 0 0 (1.5.5) where + _T-To T = qc L /k' + _ ext t = L2 ' (1.5 :6a,b,c) T he dimensionless time defined by Eq. (1.5 .6b) is sometimes called the Fourier number. F or x+ = 1, the insulated surface, the time t+ = 0 .05 can be considered as small and above 0.5 as large since little temperature response occurs before 0.2 0.1 °o~~~~~~~--~--~----~--~----J 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t += ;~ FIGURE 1 .7 T emperalure s inside a plate with a constant h eatllux a t x = 0 a nd insulated at x = L. {+ = 0.05 a nd a "fully-developed"linear-with-time response occurs after t + = 0.5 . F or small times the response at x+ = 0 is quite rapid; in fact, the dimensionless te~perature is expressed by t+)II2 T+(O, { +)=2 ( 1t for t+ < 0 .3 (1.5.7) As t+ -+0 the time derivative of Eq. (1.5.7) goes to infinity, indicating an instantaneous change in the surface temperature when the surface heat flux is applied. F or an interior point the response is slow, being both lagged and damped. As an example, for x + = 1 a nd for small times the T + expression is (see Reference 51, p. 484) (1.5 .8) These expressions yield very small temperatures at early times and the time derivative is zero as t+ -+0. See also Figure 1.7 for x+ = 1 a nd small t + values. Some numerical values for T + a re provided by Table 1.1. F or t+ = 0.05, for example, T +(O, ( +)=0.2523, whereas T+(1, t+)=0.000269, a factor of almost 1000 smaller. This factor increases as t+ becomes smaller. O n the other hand for sufficiently large times, the factor approaches unity. This can be demonstrated as follows : The summation in Eq . (1.5 .5) can be dropped for large times to get (1.5.9) (. SEC. 1 .6 T ABLE 1.1 D imensionless T emperature V alues. T+ ( x+. t +). f or V arious D imensionless T ime a nd D istances f or a F inite P late H eated a t x=O a nd I nsulated a t x =L. T +(x+. t +) = [T(x. t )-To)/q,L/k): x + = x/L: t+ = a.tIP 1+ om 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0 .60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 ........... ' -.J 16 x+ = 0 .0 0.112838 0.159577 0. 195441 0.225676 0.252313 0.276395 0.298541 0.319154 0.338514 0.356826 0.374245 0.390892 0.406863 0.422240 0.437089 0.451466 0.465422 0.479000 0.492236 0.505165 0.566146 0.622842 0.676928 0.729423 0.780946 0.831876 0.882444 0.932790 0.983002 1.033131 1.083210 1.133258 1.183287 1.233305 1.283316 1.333323 x+ = 0 .25 0.004377 0.020235 0.039238 0.058510 0.077297 0.095405 0.112807 0.129537 0.145644 0.161180 0.176198 0.190745 0.204865 0.218598 0.231980 0.245044 0.257820 0.270335 0.282614 0.294679 0.352432 0.407165 0.460054 0.511818 0.562895 0.613553 0.663954 0 .714199 0.764349 0.814440 0.864496 0.914530 0.964551 1.014563 1.064571 1.114576 x+ =0.50 0.000014 0.000802 0.003722 0.008754 0.015366 0.023074 0.031528 0.040486 0.049784 0.059311 0.068992 0.078777 0.088632 0.098535 0.108469 0.118425 0.128395 0.138375 0.148361 0.158352 0.208336 0.258334 0.308333 0.358333 0.408333 0.458333 0.508333 0.558333 0.608333 0.658333 0 .708333 0:758333 0.808333 0.858333 0.908333 0.958333 x+ = 0 .75 0.000000 0.000008 0.000150 0.000702 0.001879 0.003764 0,006360 0.009630 0.013523 0.017986 0.022969 0.028422 0.034302 0.040569 0 .047187 0.054123 0.061347 0.068831 0.076553 0.084488 0. 126735 0.172002 0.219112 0.267348 0.316271 0.365614 0.415212 0.464967 0.514818 0.564726 0.614671 0.664637 0.714616 0.764603 0.814595 0.864591 W HY IS T HE I HCP DIFFICULT? so that T+(O, t +) ~t+ o.()()()()()() +t (1.5.10a) T +(l, t +) ~t+ x+ = 1.0 0.000000 0.()()()()()5 0.000057 0.000269 0.000786 0.001735 0.003207 0.005251 0.007885 0.011104 0 .014887 0.019205 0.024024 0.029306 0.035017 0.041121 0.047584 0.054375 0.061464 0.100516 0.143824 0.189738 0.237244 0.285721 0.334791 0.384223 0.433877 0.483665 0.533536 0.583457 0.633409 0.683379 0 .733361 0.783351 0.833344 17 -i (1.5.10b) Hence, the temperature ratio is T +(O,I+) T +(l, t +) / ++t 1 ~ / + - i ~ 1+ 2t+ for t + ~ 1 (1.5.11) This result is a ppropriate for a " thin" plate which is defined as one with a negligible temperature difference across it compared to the temperature rise. The detailed values for Eq. (1 .5.5) given in T able 1.1 are provided for examples and problems in subsequent chapters. A geometry related to the finite plate is the semi-infinite body. This case with a constant heat flux at x = 0 is discussed further in Section 1.6.2.2 a nd numerical values are given in Table 1.2. T wo other cases o f interest are the solid cylinder and solid sphere, both subjected to a constant heat flux. The center location for each has the greatest lagging and damping; for that reason only T ABLE 1 .2 D imensionless T emperature V alues. f or T+ ( t;). V arious D imensionless T imes f or a S emi-Infinite B ody. T +(t:J = [ T(x. t )-To)/(qcxlk): t ; =a.tlx 2 . r+(t;l t; T +(r;l I.: T +(r;l 0.05 0.06 0.10 0.12 0. 15 0.18 0.20 0.24 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 0.000135 0.000393 0.003943 0.007444 0.014653 0.023792 0.030732 0.046147 0.050254 0.071893 0.094800 0. 118437 0.142456 0.166631 0.190810 0.214891 0.238808 0.262515 0.285982 0.309190 0.332128 0.354791 0.377175 0.399282 1.25 1.50 1.75 2.00 2.50 3.00 3.50 4.00 5.00 6.00 7.00 8.00 9.00 10.00 12.00 15.00 20.00 25.00 30.00 35.00 40.00 50.00 60.00 70.00 0.50579 0.60612 0.70101 . 0.79119 0.95962 1.11505 1.26002 1.39635 1.64825 1.87832 2.09140 2.29076 2.47874 2.65708 2.98997 3.44283 4. 10921 4.69822 5.23182 5.72321 6. 18105 7.01871 7.77678 8.47439 80 .0 90.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 9.1241 9.7345 10.3120 14.9776 18 .5604 21.5817 24.2439 26.6510 28 .8648 30.9254 32.8608 34.6914 t+ 18 CHAP.1 DESCRIPTION OF THE INVERSE HEAT CONDUCTION PROBLEM SEC . 1.6 T ABLE 1 .4 D imensionless T emperatures a t t he C enter o f a S olid S phere the center temperatures are tabulated in Table 1.3 for the solid cylinder a nd in Table 1.4 for the solid sphere. T he e quation for the temperature distribution in the cylinder is (see Reference 51, p. 203) + + +_ + ! T .(r " .)-2,. + 2(r where P., n = + )2_ ! 4 - 2 ~ e - P:'; J o(r+P.) .~I fI:Jo(P.) ,+ . (1.5.13) a nd where a is the cylinder radius a nd +(+ + ) T lJ r , to [ T(r,I)-To]k qc a + ' al 1. == ' 2' a (1.5.14) The equation for the temperature distribution in the sphere is (see Reference 51, p.242) 2 ~ sin(r+ /3.) _ p',+ + + +_ + I +2 3 T .(r , 1.)=31. + 2 (r) - 10-- + L... p2 • P e • • r n = 1 n sin " (1.5.15) where /3., n = I, 2, . .. , a re the positive roots o f tan fl. = fin (1.5.16) T ABLE 1 .3 D imensionless T emperatures a t t he C enter o f a S olid C ylinder. P (O. t n E [T(O. t )-ToJ/(qca/k). t : E flt/a 2 ,+ . ........... ~ 0.010 0.020 0.030 0 .040 0 .050 0.060 0.070 0.080 0.090 0. 100 0. 110 0. 120 0 .130 0. 140 0. 150 0 .160 0. 170 0 .180 0.190 0.200 T +(O,,:) ,+ . r IO,':) 0.00000 0 .00000 0.00003 0.00028 0.00120 0.00325 0.00676 0.01187 0.01862 0.02692 0.03667 0.04771 0 .05993 0.07317 0.08731 0.10223 0.11784 0.13405 0. 15077 0.16794 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000 0.02692 0.08731 0. 16794 0.25861 0.35413 0.45198 0.55095 0 .65046 0.75022 0.85011 0.95005 1.05002 1.15001 1.25001 1.35000 1.45000 1.55000 1.65000 1.75000 T+(O,,:1 ': T+(O,':1 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.150 0.160 0.170 0.180 0. 190 0.200 (1.5.12) I , 2, . .. , a re the positive roots o f the Bessel function, J dP.)=O 19 SENSITIVITY COEFFICIENTS 0.00000 0.00000 0.00009 0.00088 0.00342 0.00865 0.01699 0.02848 0.04288 0.05988 0.07911 0.10023 0.12293 0.14694 0.1 7203 0.19801 0.22472 0.25203 0.27983 0.30804 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000 0.05988 0.17203 0.30804 0.45293 0.60107 0.75039 0 .90014 1.05005 1.20002 1.35001 1.50000 1.65000 1.80000 1.95000 2.10000 2.25000 2.40000 2.55000 2.70000 the first few of which are PI =4.4934, P2 = 7.7253 a nd P3 = 10.9041. Both Tables 1.3 and 1.4 clearly exhibit the same lagging behavior as the x + = 1 location in Table 1.1. 1.6 S ENSITIVITY COEFFICIENTS 1.6.1 D efinitions o f S ensitivity C oefficients a nd L inearity In function estimation as in parameter estimation a detailed examination of the sensitivity coefficients can provide considerable insight into the estimation problem. These coefficients can show possible areas o f difficulty and also lead to improved experimental design. The sensitivity coetlicient is defined as the first derivative o f a d ependent variable, such as temperature, with respect to an unknown parameter, such as a heat flux component. I f t he sensitivity coeffici~nts are either small o r c orrelated with one another, the estimation problem is difficult and very sensitive to measurement errors. For the inverse heat conduction problem, the sensitivity coefficients o f interest are those o f the first derivatives o f t emperature T a t location X j a nd time with respect to a heat flux c omponent, qM, a nd are defined by 'i (1.6.la) 20 CHAP.1 DESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM for j =I, . .. , J, i =I,2, . .. , n, a nd M =I,2, . .. , n. N ote t hat the number o f times t j equals the number o f h eat flux components. The heat flux component, qM, is s hown in Figures 1.8 a nd 1.9; qM is t he constant heat flux between times t M- 1 a nd t M' I f t here is only one interior location, t hat is, J = I, t he sensitivity coefficient is simply given by aT, X M(t j )=-' (1.6.1 b) aqM F or t he transient problems considered in the I HCP, t he sensitivity coefficients a re z ero for M > i. In other words, the temperature a t time t j is i ndependent of a yet-to-occur future heat flux component of qM, M > i. O ne o f the important characteristics of the I HCP is t hat it is a linear problem if the heat conduction equation is linear and the boundary conditions are lilJear. T he thermal properties (k, p, a nd c), c an be functions o f p osition a nd n ot affect the linearity. They cannot, however, be functions o f t emperature without causing the I HCP to be nonlinear. Linearity, if it exists, is a n important property because it allows superposition in various ways and it generally eliminates the need for iteration in the solution. I f t he linear I HCP is t reated as if it were nonlinear, excessive computer time would be used due to unnecessary iterations. q (/) SEC. 1 .6 S ENSITIVITY COEFFICIENTS 21 O ne way to determine the linearity o f a n estimation problem is t o inspect the sensitivity coefficients. I f t he sensitivity coefficients are not functions o f t he parameters, then the estimation problem is linear. I f they are, then the problem is nonlinear. This can be illustrated using Eq. (1.5.7) and differentiating T with respect to qc. F rom Eq. (1.5.7): aT(O, t) =~ 2 (~)1/2 aqc k nL2 ' + 03 (1.6.2) t<. which is i ndependent o f qt. M ore generally, Eq. (1.5 .5) c an be used to arrive at the same conclusion. An example o f a n onlinear estimation problem is t hat o f e stimating ex. T aking the derivative of T in Eq. (1.5.7) with respect to ex yields (_t_)1/2 ' aT(O, t) = qc L aex k 7texL2 + 03 t<. (1.6.3) The right side o f Eq. (1.6.3) is a function o f ex; t hus the estimation o f ex from transient temperature measurements is a nonlinear problem. ( If t he m ore lengthy Eq. (1.5.5) is used, the same conclusion is reached.) This principle o f the sensitivity coefficients being independent o f t he parameter to be estimated can be employed in cases when the solution is n ot explicitly known. The equations for a flat plate with temperature-independent properties are given as a n example: aa -a [k(x)T ] = pc(x) - T ax qM I FIGURE 1 .8 Hea~ flux history with a conSlant heat flux q. . and arbitrary elsewhere. I 0 I M-I a TI -k- ax (1.6.4) at - k a TI = {qM=constant, t M a x x =O q(t), t >tM I I I ax = qloss - 1 < t<t M (1.6.5) (1.6.6) x =L 1M (1.6.7) where TM - I (x) d enotes the temperature distribution a t time tM _ 1 a nd qto,. is a heat flux history due to losses that are independent o f qM' T he heat flux, q(t), for t > tM is a n arbitrary function o f time. The thermal properties can be functions of x. And the parameter o f interest is the heat flux, qM, which is a c onstant between times tM - I a nd tM' See Figure 1.9. T he temperature distribution at time t M - 1 is k nown and is given by Eq. (1.6.7). I t is desired to find the differential equation and boundary conditions for the qAt sensitivity coefficient defined by X M(x, t )=aT(x, t) DqAt FIGURE 1 .9 H eat flux components. (1.6.8) F or t < tM - I , t he solution is X M(X, t) = 0; t hat is, the body has not yet been 22 C HAP. 1 DESCRIPTION OF T HE INVERSE H EAT C ONDUCTION P ROBLEM exposed t o qM' F or times greater t han tM - I , Eqs. (1.6.4)-(1.6.7) are differentiated with respect to qM t o o btain, aXM] = p c(x)aXM -a [ k(x) ax ax at _{I, - k aXMI ax x =O - 0, S EC.1.6 23 S ENSITIVITY COEFFICIENTS and thus XM(t)=O, for then t~IM-I' T he initial condition for XM(t) for l >t M_ 1 is (1.6.15) (1.6.9) The complete solution for X ,\ / is XM(t)=O, (1.6. 10) t~IM-I (1.6 .16a) (1.6.l6b) a XMI -0 ax x =L- (1.6. 11) (1.6.16c) (1.6.12) Equations (1.6.9)- (1.6.12) describe the mathematical problem for the sensitivity coefficient X M which can b e explicitly found if the functions k(x) a nd pc(x) a re known. Notice t hat it is n ot necessary t o k now qM, q(t), o r even TM_1(x) to obtain a solution for the sensitivity coefficient X M(X, t) b ecause Eqs. (1.6.9)(1.6.12) are n ot functions o f qM, q(t), o r TM_1(x). An i mportant c onclusion t hat c an be drawn from Eqs. (1.6.9)- (1.6.12) is t hat the estimation problem for qM is linear, a consequence o f X M(X, t) n ot being a function o f qM' T his means t hat t he (unknown) value of qM is n ot needed to find its sensitivity coefficient. I t is a lso significant t hat t he same differential equation is given for X M(X, t), Eq. (1.6.9), as for T(x, t), Eq. (1.6.4). Also the boundary conditions are o f t he same type; t hat is, the gradient conditions given by Eqs. (1.6.5) a nd (1.6.6) for T(x, t) a re still gradient conditions, Eqs. (1.6.10) a nd (1.6.11) for X M(x,t). T he m ain differences are t hat t he XM(x, t) b oundary c onditions a re s impler a nd t he initial condition is zero. D ue t o t his similarity between the T(x, t) a nd XM(x, t) problems, the same solution procedure o r c omputer p rogram c an be used for the XM(x, t) s olution as for the T(x, t), which can result in considerable programming a nd c omputational efficiency. which is s hown in Figure 1.10a. T he sensitivity coefficients X I , . .. , X 4 a re shown in F igure 1.10b . In general, well-designed experiments for estimating the surface heat flux component qM w ould have large values of the sensitivity coefficient X M(I). Hence, Eq. (1.6.16b) indicates t hat the r atio o f t he surface area A to the thermal capacity (pc V) s hould be as large as possible. T hat is, a t hin foil is deSIred. Several observations can be made regard ing the I HCP t hrough the examination of Figure 1.10. First, for the heat flux c omponent qM, m easurements before time 1M - I yield n o i nformation regarding qM because Eq. (1.6.l6a) is t rue. Consequently measurements after time 1M - I a re needed t o e stimate qM' Because X M r emains greater than zero for t > 1M _ I ' t here is a n "infinite memory" o f qM' In o ther words, the temperature a t a ny time subsequent t o 1M _ I is affected by qM' A second observation concerns the case of estimating m ore t han o ne component o f qM, such as q l' q2, a nd q3' T o e stimate several parameters simultaneously, it is necessary t hat t he sensitivity coefficients be linearly independent. 49 T his means t hat a t least one C;,/=O a nd t hat c onstants C 1 , C 2 , a nd C 3 1 .6.2 O ne-Dimensional S ensitivity C oefficient E xamples 1 .6.2.1 l umped B ody C ase T he simplest transient heat conduction sensitivity coefficient is for a lumped body described by the differential equation a nd initial condition, q (t)A=pcV dT d r' T(O) = To ( 1.6.l3a,b) where pc, A, a nd V a re all constants. T he q(t) function can be such as those given in Figures 1.8 o r 1.9. T he sensitivity coefficient for qM is o btained by differentiating Eq. (1.6.13a) with respect t o qM t o get ( X M == a T/aqM), C~dXM={l, t M-1<t<t M pA dt 0, o therwise (1.6.14) °o~~~--L---~­ ' 101 + 1 (a) F IGURE 1.10 (b) Sensili vil Y coefficients for the l umped body case . 24 CHAP. 1 DESC-RIPTION OF T HE INVERSE H EAT C ONDUCTION P ROBLEM c annot be found such that (1.6.17a) over the domain o f t he measurements. Referring to Figure 1.10b, if measurements o f T a re m ade a t t l , t 2, a nd t 3, t here would be no set o f c onstants that would m ake Eq. (1.6.17a) true. Hence, q l' q2, a nd q3 can be estimated if Y1, Y2, a nd Y3 a re known along with the temperature a t time zero. Another way to look a t t he linear dependence of the sensitivity coefficients X I' X 2, a nd X 3 is t o write Eq. (1.6.17a) in matrix form for the three different times, t i> tj ' a nd t b ... SEC. 1 .6 25 S ENSITIVITY COEFFICIENTS ing q /s as revealed by the proportionality o f t he sensitivity coefficients to each other over large time periods as shown in Figure 1.10b. 1 .6.2.2 S emi-Infinite B ody. T he s olution for the temperature in a semi- infinite p lanar body (defined by q " is x :;': 0) subjected to a constant surface heat flux, i T(x, 1 )= To+2 q (exl)112 ierfc [ (4cct)-1/2] 7 (1.6.18a) which can be written in dimensionless form as T+(t;)=2(t;)112 i erfc[(4t;)-1/2] X I(ti) X1(t j ) [ X I (tt) (1.6.17b) T+ F or linear dependence this equation must be true for arbitrary C i values, but not all Ci c an equal zero. Equation (1.6.17b) is t rue if and only if the determinant o f the square matrix o n t he left is e qual to zero. O ne such example is for the matrix associated with t2, t2,5, a nd t 3, [~ .1 ~.5] k [T(x, t )- To] qc x (1.6.18c, d) This dimensionless temperature is t abulated for some values o f 1.2. The function, ierfc(z), is ierfc(z) = 1t- 1/ 2 exp( - Z2)_ Z (1.6.18b) erfc(z) t; in Table (1.6.19) At t he heated surface the temperature is (1.6. 17c) T(O, t )=To+2 1 which has a determinant o f zero. Note that the first two columns o f Eq. (1.6.17c) are equal and thus linear dependence is seen by simple inspection. Equation (1.6.17a) is satisfied by setting C 1 = - C 2 a nd C 3 = 0. T he heat flux components ql ' q2, a nd q3 c annot be simultaneously a nd uniquely estimated by using Y;, l j, a nd Y,. if Eq. (1.6.17b) is valid. The final observation regarding the sensitivities shown in Figure 1.10 is that, though there is a n infinite memory o f q M, t here is a perfect correlation between qM a nd qM + 1 for times greater than tM + I ' T his is because the sensitivity coefficients for X M a nd X M + 1 a re equal for t >tM+1 as shown in Figure 1.10b . Expressed another way, the measurement Y c ontains information regarding 100 ql b ut it may not be efficient t o use this information; Y also contains informa100 tion a bout q2," " qloo a nd is affected in exactly t he same m anner by changes in each o f these q c omponents. O n t he o ther h and Y1 is n ot affected by q2, . .. , q 1 00 a nd neither is Y by q 1 0 1 , • • •• Consequently an estimation scheme that 100 simultaneously estimates q l,"" qloo by using Y1, . .. , Y may n ot be much 100 better than one that estimates the qi values in a sequential m anner t hat depends mainly o n previous times. (By sequential, it is m eant that q M-1 is estimated before a nd is completely independent of q M') Even though a temperature a t a large time contains information a bout a h eat flux component a t a n early time, it may not be advisable to use t hat i nformation to estimate the early q component. This conclusion is a result o f t he high correlation between the interven- (~)(~r2 (1.6.20) which is similar to Eq. (1.5.7) for early times in a finite plate. Notice that for Eqs. (1.6.19) a nd (1.6.20) t he qc sensitivity coefficient, X (x, t )= aT(x, t) aqc (1.6 .21) is e qual t o t he temperature rise for qc= I . I t is c onvenient to examine the case o f x =O a nd x fO separately. F or x=O, 1 (t 2X(O, t )= - (ext)1 2 = 2 -)112 k 1t 1tkpc (1.6.22a) a nd for x fO (1.6.22b) T he dimensionless sensitivity coefficient, X+=~aT=T+ x aqc (1.6.23) is t abulated in Table 1.2 a nd plotted in Figure 1.11. T he p lot for x = 0 is included for comparison ; for the x =O case, the x in Eq. (1.6.18d) and in Eq. (1.6.23) should be interpreted as the x o f t he interior location. [This is confusing b ut it permits the x =O a nd x fO cases to be plotted in the same graph. N ote t hat 26 , CHAP,1 DESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM 100~------~------~~------~------~--------, SEC. 1 .6 S ENSITIVITY COEFFICIENTS 27 In order to demonstrate this difference, the x = 0 a nd x + 0 cases are treated separately. The x =O case is considered first. F or t <t M- 1, the sensitivity coefficient at x = 0, x M(O, t):; BT(O, t) 10 =~ {[CX(/-~M - t)r2 _[CX(t~tM>J/} 0 .1 Interior location ( x" 0) 0.01 0.001 = Note: For the surface, the x in X+ a nd t+ is arbitrary a nd may b e s et equal to unity. (See Equations (1 .6.1&, d) and (1.6.220.).) S ensitivity coefficients for semi-infinite b ody h eated w ith a c onstant h eat flux. Eq. (1.6.22a) can be normalized by multiplying by k /x only if x + 0.] F or a single constant heat flux, the sensitivity coefficient behaves in the same manner as the temperature. The temperature rise for a semi-infinite body is similar to that o f a finite body at early times. T he surface temperatures (and sensitivity coefficients) are numerically equal for the finite and semi-infinite bodies at small times and differ by only 1%a t t + = I; = 0.3 (see Problem 1.20). Because o f similarities between the temperature rise a nd the sensitivity coefficient and also between the finite plate and semi-infinite body, many o f the same comments made in Section 1.5 regarding the lagging and damping effects can be applied to a semi-infinite body as well as to a finite plate. In the I HCP t he complete heat flux history is sought. Hence, it is necessary to 'examine the sensitivity coefficients for a general heat flux component, qM, which is depicted in Figure 1.9. As c an be seen from Figure 1.11, the behavior o f the surface temperature is quite different from that o f an interior temperature. t >I M (1.6.25) The. ~econd form of Eq. (1.6.25) is constructed using the principle o f superpOSItIOn . See Problems 1.21 a nd 1.22. These expressions given by Eq. (1.6 .25) can be plotted on a single curve if one lets I lt=t M-t M_ t and rewrites Eq. (1.6.25) as 1l/)t/2 2 ( t-IM_1)1/2 kXM(O , t) ( -;;= 1t 1/2 - -Il-I, 0 .0001 L -______.....L.._ ___- -L._ _I ..-______....L..._ _____- -J_ _____- - ' 0 .001 0.01 0.1 10 100 FIGURE 1.11 (1.6.24) BqM is zero. F or I> 1M - 1 there are two nonzero expressions, 2 [CX(/-t M_ d]1 /2 XM(O, t)=1< 1t ' t M- 1< t<tM x + = 2(t+ /".)1/2 (See Equation (1.6.22a).) IM- 1< t<tM 1t~/ 2 [ C-;;-J/2 -C-~~-1 -1Y'], t >I M (1.6.26) Th ~ s expression i.s plotted in Figure 1. 12 for M = 1, 2, 3, and 4. T he graph is vahd for any chOice o f M provided I II is a positive constant. T he sensitivity coefficients plotted in Figure 1.12 are for the surface o f a semi-infinite body and are similar in some respects to those shown in Figure 1.10 which are for a lumped body. The similarities include the instantaneous response to changes in the surface heat flux . A difference is t hat the semi -i nfinite body responds more rapidly and the effect o f heating over a time interval gradually Semi·infinite body; x =0 1.2 1.0 ~ ;;1· 0.8 ~ 0 .6 "" 0 .2 °0~---~----~2---~3L---4~~ t /dt FIGURE1 ,12 S ensitivily coefficients for hea t flux c omponents for surface o f a sem i- infinite b ody . . _._-_._----_ .. _ --_.- - - - - - - - 28 C HAP. 1 D ESCRIPTION O F T HE I NVERSE HEAT C ONDUCTION P ROBLEM S EC. 1 .6 diminishes. There is an "infinite memory" in the sense that the sensitivity coefficient given for t > tM by Eq. (1.6.26) only approaches zero for t - tw-+ 0 0. T he decision regarding the times at which to make measurements to estimate certain parameters is called experimental design. As noted previously, because XM(O, t)=O for t <t M- 1 t emperature measurements prior to t M- 1 c annot be used to estimate qM' O n the other hand, if only measurement times much larger than tM are used to estimate qM, difficulties are encountered for two reasons. First, the X M value is relatively small, indicating little sensitivity regarding qM a nd hence little information. Second, the X M , X M + I , . •• functions tend to become correlated, t hat is, have nearly the same shape and thus approach linear dependence. Consequently the best choice of measurement times for estimating qM from surface temperature measurements of a semi-infinite body must include tM, t M+ I , a nd slightly larger t values. The case o f a n interior temperature measurement history has substantially different characteristics than that for x = 0 j ust discussed. The dimensionless sensitivities for q l' q2, a nd q3 are shown in Figure 1.13. Unlike the x = 0 curve, the results for x =1=0 c annot be combined into a single curve. The dimensionless sensitivity curves for q l' q2, a nd q3 are denoted X i, X t, and X j ; the curves have exactly the same shape but they are shifted to the right for I'lt; from X ;' t o X ;'+I' T here are several important observations t hat can be made in regard to Figure 1.13. First, the X;"s display a lag which is most clearly shown in Figure 1.13a by X i. Even though ql starts at t=O as shown by Figure 1.9, X i in Figure >0.05. This is quite unlike the x =O 1.13a does n ot appreciably rise until behavior for a semi-infinite body for which Figure 1.12 shows the greatest gradient of X i as t-+O. T he second observation is t hat the magnitudes of the increase with I'lt;. Small values o f I'lt; give small values of causing the estimation o f the surface heat flux t o become inaccurate. x~ 2.5 t; (b) xt, xt, ... 0.4 0.3 X~ t; 0.2 xt's xt 0.03 r --r--,.---,---r---;r--.--r---r-,.---,---, 29 S ENSITIVITY C OEFFICIENTS t .t;= 1 0.1 ~ 0 ! 0 2 3 4 5 6 7 8 9 10 t; (e) ff F IGURE 1 .13 Dimensionless sensitivity coefficients for q" q2 and qJ for an interior point in a semi-infinite body. I 'll: =0.05, 0.25, a nd I . xt t .t; = 0.05 0.7 t; ( a) 0.8 0.9 l .0 Third, each curve has a maximum which occurs farther from the time o f the end o f the heating pulse as I'lt; becomes small. F or example, for I'lt; = 1, X i has a maximum at t ; = 1.3. Also the value o f X i a t t ; equal to one I'lt; is a bout 87% o f the maximum X i for I'lt; = 1. This is in contrast to the I'lt; =0.05 case where the maximum X i is a t t ; = 0.45 which is 8 I'lt; values after the heating ends. Moreover, X i a t t ; =L'lt; is only 0.7% o f the maximum X i. T he fourth and final observation is t hat all the X ;"s tend to become correlated increases. as t; 30 C HAP. 1 DESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM These characteristics of sensitivity coefficients provide some insight into the design of an estimator for the I HCP utilizing an interior sensor. O ne o f the most obvious points is t hat temperature measurements at times greater than tM are needed t o estimate qM, particularly as IJ.t; becomes small. Because X i for IJ.t; =O.OS is nearly zero a t time t; = O.OS {the end of nonzero q d, Y 1 would yield very little information regarding q I . T he Y2 measurement, however, is significantly affected by q l, b ut insignificantly by q2. Hence, a more effective strategy for estimating qM for the case of IJ.t; =O.OS would use "f. ~ 1 YM and YM + 1 rather than omitting YM + I . Even a better procedure might be to utilize more future (relative to tM) temperature measurements than YM+I . If, however, a large number of future temperature measurements is used, for example, Y2 , • •• , Y to estimate q l then the procedure might not be efficient due to 100 the high correlation among the sensitivity coefficients. When r future temperatures such as YM, YM+I , • .. , YM+r - 1 are used to estimate qM, there is extra information for estimating qM which manifests itself as extra algebraic equations involving qM . This information is needed, but due to measurement errors, it is not completely consistent. O ne way to use all this information is to employ the method of least squares which is discussed in C hapter 4. Also see Problems 1.9 - 1.14. 1 . 6.2.3 P late I nsulated o n O ne S ide T he heat flux sensitivity coefficients for a flat plate heated on one side and insulated on the other are discussed in this section. This is the same geometry as shown in t he inset of Figure 1.7. F or a heat flux o f infinite duration the temperature distribution is given in equation form by Eq. (1.S.5), in t abular form by Table 1.2, a nd in graphical form by Figure 1.7. These results can also be interpreted as being equal to the dimensionless heat flux sensitivity coefficient o f SEC. 1 .6 S ENSITIVITY COEFFICIENTS 31 and IJ.t is the duration of heating. This result is even more complicated than that for a semi-infinite body because X i depends on t+ a nd 1J.t+ a nd also x+. F or the large dimensionless times o f t+ -lJ.t+ > O.S (1.6.31) the second expression of Eq. (l.6.29) goes to the time- and space-independent value of (l.6.32) This expression shows the dependence of X; on the duration of heating; hence, as more components of q are estimated over a fixed time period, the sensitivity coefficients become smaller and t hus· the q{t) curve becomes more difficult to estimate. Notice that as t + increases, the temperatures increase as shown by Figure 1.7 b ut the sensitivity coefficients d o not. The sensitivity coefficient X i normalized with respect to xtmax is plotted in Figure 1.14. I n order to present results compactly, the time has been made dimensionless by normalization with respect to the heating duration, 1J.t. 1 .6.3 T wo-Dimensional S ensitivity C oefficient E xample In this section the estimation of two-dimensional heat flux histories is investigated. The heat flux is a function of time as previously discussed but also is a function of position over the surface. Further, the heat flux is subdivided into a (1.6.27) 0.8 % +=xIL=l T he sensitivity coefficient for q l where q= { ql' 0, O <t<tl=lJ.t t >t l ( l.6.28) can be readily found using Eq. (l.5.S). Because the other coefficients for q2, . . . have exactly the same shape but are displaced IJ.t a part, it is only necessary to examine X i which is given by k aT X i=--=T+{x+ t+) L aq< " ~ >< 0.4 0.2 O<t+<lJ.t+ (1.6.29) where I loll A ut ~. + _a.lJ.t = -2- L (1.6.30) FIGURE 1 .14 H eal Hux sensilivilY coefficienls al insulaled surface o f Hal pl ale. 32 C HAP. 1 DESCRIPTION OF THE INVERSE HEAT C ONDUCTION PROBLEM n umber o f s imple building blocks of s hort d uration o ver small regions. As before, each h eat flux c omponent is a ssumed c onstant o ver position as well as over time. See F igure 1.6 for the spatial variation a nd F igure 1.9 for the time variation. O ther functional variations such as linear, parabolic, sinusoidal, a nd so on, a re p ossible b ut t he basic ideas can be m ore easily presented for the c onstant a pproximation. T he t wo-dimensional a nd t ime-dependent h eat flux, q(y, t) is a pproximated by (see P roblem 1.23) q(y, t) = L L jj;(J, t) i (1.6.33) j a nd Yj-I/2~y<Yj+I / 2 (6) 1. .34 for constant h eat flux " building blocks." T he sensitivity coefficient for qji is t hen _ aT(x, y, t) X ji (x, y, t ) = a qji m athematically by (1.6.36) , , (1.6.35) ~T =qo, -k i ir x =O At y <o (X aT (1.6.37) { ax = 0, y>O I aT t JI 33 S ENSITIVITY COEFFICIENTS f; w here i refers t o t ime a nd j t o position. T he function j ji(y, t) is given by t;_I~t<t; J:.. ( y, t )={qjj, otherwise 0, SEC. 1 .6 ay ->0 I ! An example o f a t wo-dimensional body is a semi-infinite b ody h eated with a space-variable flux as shown in Figure 1.15. T he c oordinate x goes into the body s tarting a t t he surface; y is p arallel t o t he surface b ut t here is n o n atural s tarting point. T he h eat flux q(y, t) is usually a c ontinuous f unction in b oth t ime a nd p osition b ut it is a pproximated b y a series o f u niform pulses over s hort d istances a nd t ime periods, as discussed previously. T he sensitivity coefficients for this problem can be developed from the T(x, y, t) s olution for a semi-infinite body heated continuously over negative y values; this problem can be described as y-> ± 0 0 - T - > To (1.6.38a) for x-> 0 0 (1.6.38b) T(x, y, 0) = To (1.6.38c) T he s olution for all x's is given in Reference 50, b ut t o simplify the t reatment only the solution for x =o is used, which is given in C arslaw a nd J aeger [ Reference 51, p. 264]. T he s olutio.n 51 for Egs. (1.6.36)- (1.6.38) for x =o a nd y=/=O is T+( +) ,y T(O , y, t )- To qo(at/n)1/2/k ~ ( y+) er c T- y+ 2nl/2 E I [ (y+)2] - 4- , T +( Y+ )=2-erfc(!iJ)+ ly+1 E [ (y+)2] Y 2 21[1/2 1 4 , Y< ° y >o (1.6.39a) (1.6.39b) where y+ a nd E1(z) (see Reference 52, p. 227) a re defined by +_ y = Y (at)1/2' _ra) u J -I-u E 1(z)= (1.6.40a,b) e du T he e xpression given by Eg. (1.6.39) can be plotted as a function o f y+ with as a attention to whether y is positive o r negative. Figure 1.16 gives a plot o f function o f Iy+ 1/2 for positive a nd n egative y values. F or t he special locations corresponding to y--> - 00 , y=O, a nd y-> 0 0, t he surface temperatures are : T; t T = To+2qo ( - kn pc " Semi·infinite body )1/2 ,y-> - 00 (1.6.41a) (1.6.41 b) (1.6.41c) T he sensitivity coefficients for the various terms qji c an be constructed using the principle o f s uperposition with Eg. (1.6.39). T o b e morl" specific, the surface FIGURE 1 .15 Semi-infinite body with a space- and time-variable heat flux t hat is a pproximated by constant elements. L 34 C HAP. 1 D EStRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM S EC.1.6 S ENSITIVITY COEFFICIENTS 35 T: The notation [(2s - I)/(t: )1/2] means t hat y+ in Eq. (1.6.39) is t o be replaced by ( 2s-1)/(£:)1/2 . F or positive values of 2 s-1 (or 2 s+ I). Eq. (1.6 .39a) is used. whereas Eq. (1.6.39b) is used for negative values. F or the time o f n 6 t a nd n = 2 .3 • .. . • the sensitivity coefficient for q OI is +_ T+ y T'1 - Xcii , sn = 1-(n;)"2 { T: T(O , y, t ) - T. qo(atl,,)"' 11t y +a (n_I)I+)1/2{ ~ 'l ( (~i2 [(I~;S:~1~2]_ T:L~::~1~2]} 7t. T: [ 2 s-1 ] [2S+I]} ((11_1)1:)1/ 2 - T: ( (n-l)t:)112 (1.6.44) 0.3 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.9 1.0 IY+112 • n JJ ... -...:~ FIGURE 1 .16 Dimensionless surface temperature for a semi-infinite body heated uniformly over one-half the surface. I\ 0.6 I shown in Figure 1.15 which is exposed to a number o f equally spaced heat flux c omponents for time t j is considered. Let the heated regions be equal to 6 y= 2a. A p rototype sensitivity coefficient is the one for q OI which is for heating over the region - a< y <a a nd for a heating duration of 0 < t<6t. T he dimensionless sensitivity coefficient for the location. x =O. y =2sa. (s=O. I. ...) a nd time t =n6t(n=I.2• ... ) is needed. The notation is 0.5 : ,. n - k aT(O. y. . 6 t) . ( t:)1/2 { + [ 2S-I] + [ 2S+ I ] } Ol,$l=~ aqol I, Ty (t:)1/2 - Ty (£:)1 1 2 (1.6.42) Semi ·infinite • +0 >.; .I 02 t,; + --+ = 0.05 Use right axis I I I 0 .1 0 .1 / / . - -e \ ' -.. t: = 1 Use left axis ·- ;-. ~'.·IX~1'2. 1 ................. ~.""'""- -----::::::. _e--e 4 -e--e __ : 5 Time index, (1.6.43) J2 body 3 where 0 .2 ::~ +0 0.3 X+ II I I J2 >.; where the first two subscripts refer to the location of the application of the heat flux c omponent (i.e .• j = 0) a nd to the time o f the applied heat flux component (i = I). and the last two subscripts are for the location and time o f evaluation. The first two subscripts are for the cause (i.e .• the impulse) and the last two are for the effect (i.e .• response). The sensitivity coefficient a t time t = 6 t. (n = I). a t location y .=2sa is o btained by superimposing two solutions taken from Eq. (1.6.39). ~ o 6 7 8 9 10 n FIGURE 1 .17 Sensitivity coellicients for heat Hux components along surface of a semi-infinite body. 36 C HAP. 1 D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM T he general sensitivity coefficient for q OI by + X ji • s .= {O, q ji a t Ys a nd In is related t o the one for n <i X + I.ls-jl . n O i+l, (1.6.45) >-. n,....1 Hence if the Xcii sensitivity coefficients are known for s = 0, 1,2, . .. and 1, a t the ';~rface of a semi-infinite body, the sensitivity coefficients a t the surface can be found for all other F igure 1.17 gives plots o f XciI. O n' Xcii. I n, a nd Xci1.2n for = I . These curves are those with dots and dashed lines. Use the left axis for these curves. T he Xcil ,on curve is the largest since it is for the response directly below the heat pulse; it is a maximum at n = 1, the end o f the heat pulse, a nd decays thereafter. The Xcil.2n curve is much smaller in magnitude than the Xcil.o. curve and its maximum is displaced in time. Also shown in Figure 1.17 are two curves for = 0 .05, which is a relatively small value. These are the continuous curves that have crosses for which the right axis is used. In this case the Xcii . o. curve is similar to that for = 1 b ut the Xcil . l. curve is very small in magnitude and there is a very large lag in response. This means that the heat impulse for the small time steps o f = 0.05 affects mainly the temperature response at the location o f the pulse. There is little correlation (or cross-coupling) of the individual heat pulses. T he uncorrelated nature o f the spatial variation o f the sensitivity coefficients for small time steps and with surface temperature measurements has experimental design implications. If, for example, the heat transfer coefficients (or equivalently heat fluxes if the ambient temperatures are known) are needed a round a j et engine turbine blade, a transient experiment with surface thermocouples could be used. T he calculational time steps should be small t.o reduce spatial correlation between the components o f the heat transfer coeffiCient. 1 1= i, ..., X;;.•• . t: I: t: t: 1 .7 C LASSIFICATION OF M ETHODS T he methods for solving the inverse heat conduction problem can be classified in several ways, some o f which are discussed in this section. . O ne classification relates to the ability o f a m ethod to treat nonhnear as well as linear IHCP's. This book emphasizes basic algorithms t hat c an be employed for both linear a nd n onlinear problems. The. tW? basic procedures given herein are the function specification and r:gulanzatlOn .methods. Both o f these can be used for nonlinear problems prOVided the nonhnear heat conduction equation is solved. Some methods o f solution of the I HCP are inherently linear such as t hose based o n the Laplace transform; such methods are not considered because the nonlinear case is m ore important for industrial applications. S EC.1 .7 C LASSIFICATION OF M ETHODS 37 T he m ethod o f solution o f the heat conduction equation is a nother way to classify the I HCP . Methods o f solution include the use of Duhamel's theorem, finite differences, finite elements and finite control volumes. The use o f D uhamel's theorem restricts the I HCP algorithm to the linear case, whereas the other procedures can treat the nonlinear problem. Duhamel's theorem is used frequently in this book because the basic I HCP algorithms are easier to use and program for simple calculations than the finite difference and other methods. Moreover for linear problems, the answers for the surface heat flux are nearly identical for all of the methods mentioned provided the heat conduction equation is solved accurately. Consequently, experience acquired using Duhamel's theorem incorporated in a basic I HCP a lgorithm is also relevant to the other methods when used for linear I HCP 's. The time domain utilized in the I HCP can also be used to classify the method of solution. Three time domains have been proposed: (1) only to the present time, (2) t o the present time plus a few time stops, a nd (3) the complete time domain. The use o f measurements only t o the present time with a single temperature sensor allows the calculated temperatures to match the corresponding measured temperatures in an exact manner; that is, the calculated temperatures equal the measured values. This is called the Stolz method. I Such exact matching is intuitively appealing but the algorithms based on it frequently are extremely sensitive to measurcmenl errors. In the second method, a few future temperatures (associated with future times) are used ; the associated algorithms are called "sequential." Greatly improved algorithms are obtained compared with exact matching. T he improvements are noted in the considerably reduced sensitivity to measurement errors and in the much smaller time steps that are possible. Small time steps permit more detailed information regarding the time variation o f the surface heat flux t o be found. T he whole domain estimation procedure is also very powerful because very small time steps can be taken but it is n ot as computationally efficient as the use o f only a few future temperatures. Both the function specification and regularization methods can be employed in sequential a nd whole domain estimation forms. The last classification to be mentioned is relative to the dimensionality of the IHCP. If a single heat flux history is t o be determined, the I HCP c an be considered as one-dimensional. In the use o f D uhamel's theorem, the physical dimensions o f the problem are not o f concern ; that is , the same procedure is used for physically one-, two-, o r three-dimension bodies provided a single heat flux history is to be estimated. I f two o r more heat flux histories are estimated and Duhamel's theorem is used, the problem is multidimensional. When the finite difference o r the other methods for nonlinear problems are employed, the dimensionality o f the problem depends on the number o f space coordinates needed to describe the heat-conducting body; one coordinate would give a one-dimensional problem, two-coordinates a two-dimensional problem, and so on. 38 CHAP. 1 1 .8 DESCRIPTION OF THE INVERSE HEAT CONDUCTION PROBLEM CRITERIA FOR E VALUATION OF IHCP M ETHODS In o rder t o evaluate the several I HCP procedures, various criteria are needed. The criteria proposed in Reference 10 a re given in this section and are as follows: SEC.1.9 5 . T he m ethod should n ot require continuous first-time derivatives o f the surface ~eat flux. Furthermore, step changes o r even m ore a brupt changes In the surface heat fluxes should be permitted. 6 . Knowledge o f t he precise starting time o f the application o f the surface heat flux should n ot be required. The start o f h eating is frequently n ot synchronized with the discrete times t hat t emperatures are measured. R~so~s for this might be that the starting time is n ot accurately known o r IS dIfficult t o measure. Precise times a t which a brupt changes in the heat flux OCCur may also be unknown. 7 . T he method should n ot be restricted to any fixed number o f observations. 8 . C omposite solids should be permitted. 9. 1 0. 11 . 1 2. T emperature-variable properties should be permitted. C ontact conductances should n ot be excluded. T he m ethod should be easy t o p rogram. T he c omputer cost should be moderate. 1 3. T he user should n ot have to be highly skilled in mathematics in o rder t o use the method o r t o a dapt it to o ther geometries. 1 4. T he method should be capable o f t reating various one-dimensional coordinate systems. 1 5. T he m ethod should permit extension to more than one heating surface. 1 6. T he method should have a statistical basis a nd p ermit various statistical assumptions for the measurement errors. . T he functi~n s~cific~tion a nd regularization methods are capable o f satisfyIng all th~se c ntena prOVIded t he nonlinear heat conduction equation is a pproximated USIng m ethods such as the finite difference, finite element, o r finite control volume methods. If t he heat conduction equation is solved by Duhamel's theorem a nd the 39 function specification o r regularization method is used, all the criteria can be satisfied except t hat o f t reating the nonlinear problem (criterion number 9). 1.9 1 . T he predicted temperatures and heat fluxes should be accurate if the measured d ata a re o f high accuracy. 2 . T he m ethod should be insensitive t o measurement errors. 3 . T he method should be stable for small time steps o r intervals. This permits the extraction o f m ore information regarding the time variation o f surface conditions than is p ermitted by large time steps. 4 . T emperature m easurements from one o r m ore sensors should be permitted. SCOPE OF BOOK SCOPE OF BOOK The scope o f t he book is mainly limited t o the inverse heat conduction problem. I t is o ne o f a class o f ill-posed problems. However, many o f t he techniques given herein apply t o a wide variety o f ill-posed problems. A number o f s olution methods are presented a nd t he emphasis is o n general methods t hat c an meet the criteria in Section 1.8. C hapter 1 h as given a n i ntroduction t o the subject. Chapter 2 presents some exact solutions, one o f which is due to Burggraf. 43 T hough this exact solution is restricted in its application, the insight gained from it is very important. C hapter 3 presents two different basic procedures for solving direct transient heat conduction problems. T he first is based o n a numerical form o f D uhamel's integral. T he second method approximates the partial differential equation for transient heat conduction by a set o f algebraic equations, t hat is, difference equations. T he s econd method is more powerful in t hat it can treat the nonlinear problems. C hapter 4 o pens with a discussion o f ill-posed problems a nd t hen presents a number o f m ethods for the I HCP. M any researchers have contributed t o these methods. T he t wo basic classes are the function specification a nd regularization methods. A procedure called the trial function method unifies these approaches. Two ways t hat the function specification a nd regularization methods can be used are called sequential a nd whole domain. Various modifications o f t hem a re discussed. Each o f these methods can utilize the Duhamel a nd difference equation procedures a nd t hus is applicable for both linear a nd n onlinear problems. An important p art o f this chapter is a discussion o f a digital filter form o fthe I HCP. Such a form can be used for the very efficient implementation o f the linear I HCP algorithms. T he final section o f C hapter 4 discusses some criteria for comparing estimation procedures. C hapter 5 p resents a n umber o f test cases. Utilizing a numerical approximation o f D uhamel's integral for the solution o f t he heat conduction equation, these test cases are used t o investigate a number o f t he I HCP algorithms. T he s tudy o f I HCP a lgorithms is facilitated by using a numerical form for Duhamel's integral because only a single equation is needed a t each time rather t han a set o f difference equations. C hapter 5 e nds with a discussion o f o ptimal choices o f p arameters in the function specification a nd regularization methods. C hapter 6 uses the finite control volume method t o a pproximate the heat conduction equation for the one-dimensional inverse heat conduction problem. T he sequential function specification a nd regularization methods developed in C hapter 4 a re used. In addition, some space marching techniques are discussed; these methods are unique t o t he difference equation approach since analogous 40 CHAP.1 DESCRIPTION OF THE INVERSE HEAT C ONDUCTION PROBLEM REFERENCES e~uatio~s b ased o n D uhamel's integral a re n ot available. T he c hapter concludes With a list o f c omputer p rograms available for the I HCP. C hapter 7 is for multiple heat flux estimation. Two o r m ore h eat flux histories c an be estimated a t t he same time. This case is commonly encountered in twoor t hree-dimensional inverse heat conduction problems. C hapter 8, t he last chapter, discusses methods for estimating the heat transfer coefficient. O ne way is t o use the I HCP m ethods to calculate the surface heat fl~x history. An alternate procedure calculates the heat transfer coefficient directly. 16. Imber, M., A Temperature Extrapolation Method for Hollow Cylinders, , 1/,1,1 J. I I, 117 - 118 (1973). 17. Imber, M., Comments on " On Transient Cylindrical Surface Heat Flux Predicted from Interior Temperature Responses," , 1/,1,1 J. 14,542 - 543 (1975). 18. Imber, M., Inverse Problem for a Solid Cylinder, , 1/,1,1 J. 17,91 94 (1979). 19. Imber, M., Nonlinear Heat Transfer in P lanar Solids: Direct and Inverse Applications, A lAA J. 17,204 212 (1979). 20. Imber, M., A Temperature Extrapolation Mechanism for Two-Dimensional Heat Flow, A lA A J. 12, 1087 - 1093 (1974). 21. Imber, M., The Two Dimensional Inverse Problem in Heat Conduction, Fifth International Heat Transfer Conference, Tokyo, Japan (1974). 22. Imber, M., Two-Dimensional Inverse Conduction Problem- Further Observations," A IAA J. 13,114- 115 (1975). 23. Imber, M. and Khan, J., Prediction of Transient Temperature Distributions with Embedded Thermocouples, A IAA J. 10,784- 789 (1972). 24. Mulholland, G. P., Gupta, B. P., and San Martin, R. L., Inverse Problem o f Heat Conduction in Composite Media, ASME Paper, 75-WA/HT-83 (1975). 25. Mulholland, G. P. and San Martin, R. L., Inverse Problem of Heat Conduction in Composite Media, Third Canadian Congress of Applied Mechanics, Calgary, Alberta, Canada (May 1971). 26. Mulholland, G. P. and Cobble, M. H., Diffusion Through Composite Media, Int. J. Heal Mass Transfer 15,147 - 152 (1972). 27. Mulholland, G. P. and San Martin, R. L., Indirect Thermal Sensing In Composite Media, Int. J. Heal Mass Transfer 16, 1056 - 1060 (1973) . 28. Hills, R. G. and Mulholland, G. P. ; Accuracy and Resolving Power of One-Dimensional Transient Inverse Heat Conduction Theory as Applied to Discrete and Inaccurate Measurements, Int. J. Heal Mass Transfer 22,1221 - 1229 (1979). 29. Randall, J. D., Embedding Multidimensional Ablation Problems in Inverse Heat Conduction Problems Using Finite Differences, 6th Int. Heat Transfer Conf., Toronto, Ont., Aug. 7 - 11, 1978. Publ. by Nail. Res. Council of Can., Toronto, Ont., 1978. Available from Hemisphere Publ. Corp, Washington, D.C., Vol. 3, 129 - 134. 30. Randall, J. D., Finite Difference Solution of:the Inverse Heat Conduction Problem and Ablation, Technical Report, Johns Hopkins University, Laurel, Maryland (1976), Proceedings o f the 25th Heat Transfer and Fluid Mechanics Institute, Univ. o f California, Davis (1976). 31. Williams, S. D. and Curry, D. M., An Analytical Experimental Study for Surface Heat Flux Determination, J. Spacecrafl 14,632- 637 (1977). 32. France, D. M., Chiang, T., Carlson, R. D., and Minkowycz, W. J., Measurements and Correlation of Critical Heat Flux in a Sodium Heated Steam G enerator Tube, Technical Memorandum, ANL-CT-78-15, Argonne National Laboratory, Argonne, IL (1978). 33. France, D. M., Carlson, R. D., Chiang, T., and Priemer, R., CHF-Induced Thermal Oscillations Measured in a n LMFBR Steam Generated Tube Wall, Technical Report, ANL-CT-78-I, Argonne National Laboratory Argonne, IL (1977). 34. France, D. M. and Chiang, T., Analytical Solution to Inverse Heat Conduction Problems with Periodicity, J. Heat Transfer 102, 579 - 581 (1980). 35. Bass, B. R., Applications of the Finite Element to the Inverse Heat Conduction Problem Using Beck's Second Method, J. Eng. Ind. 102, 168 - 176 (1980). 36. Bass, B. R .,lncap: A Finite Element Program for One-Dimensional Nonlinear Inverse Heat Conduction Analysis, Technical Report NRCjNUREG/CSD/TM-8, O ak Ridge National Laboratory (1979). 37. Muzzy, R. J., Avila, J. H. and Root, R. E., Topical Report: Determination of Transient Heat Transfer Coefficients and the Resultant Surface Heat Flux from Internal Temperature Measurements, General Electric, GEAP-20731 (1975). REFERENCES I. Stolz, G., Jr., Numerical Solutions to an Inverse Problem of Heat Conduction for Simple Shapes, J. Heat Transfer 8 2,20-26 (1960). 2. Mirsepassi, T. J., Heat-Transfer Charts for Time-Variable Boundary Conditions, Br. Chem. Eng. 4, 130 - 136 (1959). 3. Mirsep~si, T. J., Graphical Evaluation of a Convolution Integral, Mathematical Tables and Olher Aides to Computallon 13, 202 - 212 (1959). 4. Shumakov, N. V., A Method for the Experimental Study o f the Process o f Heating a Solid Body, Soviet Physics- Technical PhYSics (Translated by American Institute o f Physics) 2 771 . (1957). ' 5. ~ ' -\:) Beck, J. V., Correction of Transient Thermocouple Temperature Measurements in HeatConducti?g Solids, Part II, The Calculation of Transient Heat Fluxes Using The Inverse Convolution, AVCO Corp., Res. a nd Adv. Dev. Div., Wilmington, MA., Tech. Report RADTR-7-6O-38 ( Part II), March 30,1961. 6. !J1 Beck, J. V., Calculation of Surface Heat Flux From an Internal Temperature History ASME Paper 62-HT-46 (1962). ' 7. Beck, J. V., "Surface Heat Flux Determination Using an Integral M ethod" Nuc/. Eng Des 7 170 - 178 (1968). ' .. , 8. Beck, J. V. a nd Wolf, H., The Nonlinear Inverse Heat Conduction Problem ASME Paper 65-HT-40 (1965). ' , 9. Beck, J. V., Nonlinear Estimation Applied to the Nonlinear Heat Conduction Problem Int. J. H eat Mass Transfer 13, 703 716 (1970). ' 10. Beck, J. V., Criteria for Comparison o f Methods of Solution o f the Inverse Heat Conduction Problem, Nuc/. Eng. Des. 5 3,11-22 (1979). I I. Beck, J. V., Litkouhi, B., a nd St. Clair, C. R., Jr., Efficient Sequential Solution o f the Nonlinear Inverse Heat Conduction Problem, Numer. Heat Transfer 5 ,275- 286 (1982). 12. ~Iackwell, B. F., A New iterative Technique for Solving the Implicit Finite-Difference Equalions for t.he Inverse Problem of Heat Conduction, unpublished technical report, Sandia Laboratones, Albuquerque, NM (1968). 13. Blackwell, B. F., An Efficient Technique for the Numerical Solution o f the One-Dimensional Inverse Problem of Heat Cond\lction, Numer. Heat Transfer 4 ,229-239 (1981). 14. Langford, D., New Analytic Solutions of the One-Dimensional Heat Equation for Temperature and Heat Flow Rate Both Prescribed at the Same Fixed Boundary (with Applications to the Phase Change Problem), Q. Appl. Math. 2 4,315-322 (1967). 15. Woo, K. C. a nd Chow, L. C .,lnverse Heat Conduction by Direct Inverse Laplace Transform Numer. Heal Transfer 4, 499-504 (1981). ' 2 2 42 CHAP.1 DESCRIPTION OF THE INVERSE HEAT CONDUCTION PROBLEM 38. Snider, D. M., INVERT 1.0- A Program for Solving the Nonlinear Inverse Heat Conduction Problem for One-Dimensional Solids. E G&G Idaho, Inc., Idaho Falls, Idaho, EGG-2068 (1981). 39. 40 . 41 . 42 43 . 44. 45 . 46. 47 . 48 . 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. Alkidas, A. L., H eat Transfer Characteristics o f a Spark-Ignition Engine," J . Heat Transfer 102, 189- 193 (1980). Howse, T. K. J., Kent, R., a nd Rawson, H., The Determination o f G lass-Mould Heat Fluxes from Mould Temperature Measurements, Glass Technol. 12,91 - 93 (1971). Sparrow, E. M~ Haji-Sheikh, A., a nd Lundgren, T. S., T he Inverse Problem in Transient Heat Conduction, J . Appl. Mech., Trans. A SME, Series E, 86, 369 - 375 (1964). Lin, D. Y. T~ a nd Westwater, J. W., Effect o f Metal Thermal Properties o n Boiling Curves O btained by the Quenching Method, Heat Transfer 1982- Munchen Conference Proceedings, Hemisphere Publ. Corp., New York, 1982, p p. 155 - 160, Vol. 4. Burggraf, O. R., An Exact Solution o f the Inverse Problem in Heat Conduction Theory and Applications, J . Heat Transfer 116C, 373 - 382 (1964). Grysa, K., Cialkowski, M. J. a nd Kaminski, H ~ An Inverse Temperature Field Problem o f the Theory o f T hermal Stresses, Nucl. Eng. Des. 64, 169 184 (1981). Tikhonov, A. N. a nd Arsenin, V. Y., Solutions o f III-Posed Problems, V. H . Winston & Sons, Washington, D.C., 1977. Backus, G. a nd Gilbert, F., Uniqueness in the Inversion o f I naccurate Gross Earth D ata Phil. Trans. R . Soc. London Ser. A 266, 123 - 192 (1970). ' Nolet, G., Simultaneous Inversion o f Seismic Data, Geophys. J . R. Astr. Soc. 55, 679 - 691 (1978). Mandrel, J., Use o f t he Singular Value Decomposition in Regression Analysis, Am. Stat . 36, 15 - 24 (1982). Beck, J . V. a nd Arnold, K. J ., Parameter Estimation in Engineering and Science, Wiley, New York,1977. Litkouhi, B. a nd Beck, J. V., T emperatures in Semi-Infinite Body Heated by Constant Heat Flux O ver H alf Space, Heat Transfer 1982- Munchen Conference Proceedings, Hemisphere Publ. Corp., New York, 1982, pp. 21 - 27, Vol. 2. Carslaw, H. S. a nd Jaeger, J. c., Conduction o f Heat in Solids, 2nd ed., Oxford Univ. Press, L ondon, 1959. Abramowitz, M. a nd Stegun, I. A., Handbook o f Mathematical Functions with Formulas, Graphs and Mathematical Tables, N ational Bureau o f S tandards, Applied Mathematics Series, Vol. 55, 1964. Payne, L. E., Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, PA., 1975 . H adamard; J., Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, New Haven, CT, 1923. Cannon, J. R. a nd Douglas, J., The Cauchy Problem for the Heat Equation, S IAM J . Numer. Anal. 3, 317- 336 (1967). John, F., Numerical Solution o f t he Heat Equation for Preceeding Times, Ann. Mat . Para. Appl.40, 129- 142 (1955). Lawson, C. L. a nd Hanson, R. J ., Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, NJ, 1974. Murio, D. A., T he Mollification Method a nd the Numerical Solution o f a n Inverse Heat Conduction Problem, S IAM J . Sci. Stat . Comput. 2, 17 - 34(1981). r , 43 PROBLEMS PROBLEMS 1.1. Give a mathematical description of the I Hep for a solid, homogeneous sphere when the temperature is measured at the center point. The sphere radius is denoted a. Make a sketch of a sphere showing the various quantities. Form a graph that shows typical discrete temperatures measured at time steps of fl.1 with an initial temperature of To = constant. Also show an associated surface heat flux history. 1 2. A certain automobile brake is composed of a cast iron drum and two brake shoes. The rotating drum has an inner radius of 10 cm and an outer radius 10.7 cm. The heat transfer in the drum is assumed to be only in the radial direction and there is a convective boundary condition at the outer drum radius. The brake shoes are 0.5 cm thick, are inside the drum, and are stationary. They cover only 75% of the angular area of the drum. The inner surface of the brake shoes (the surface not in contact with the drum) can also be considered to have a convective boundary condition. Describe the inverse heat conduction problem(s) for determining the surface heat flux based on the drum surface area. Describe the problem mathematically and through the use of any needed sketches. 1.3. The temperature distribution in a semi-infinite body (x ~O) is given by T- To=~T,~ To) erfc [(4<%:)1 /2] where T, is the surface temperature and To is the initial temperature. a. Derive an expression for the heat flux for any x. Also give the expression for x=O. Answer: k (T,- To)(1tCXl)-1/2 for x = 0. b. Plot the heat flux versus CXI/E2 for x = E and also show on the same plot the x = 0 curve. 1 .4. F or a semi-infinite body with a surface temperature given by T(O,t)=O, 1 <0 T(0,1)=Ct,/2, n = 1, 2, 3, . .. , I~O the temperature distribution is given by (Reference 51, p. 63) T = cr (n; 2 ) (4/)"/2i" erfc [ (:r)1/2] where (Reference 51, p. 483) ~ i" erfc(z)= 1"0 i ,-I erfc(u)du, L n = 1,2, . .. , jO erfc(z)=erfc(z) 44 CHAP.1 D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM r Time(s) Derive the expression for the surface heat flux, 3 4 5 6 7 Find a n expression for T(O , t)/q(O, t) a nd c omment o n t he result. Does this provide a relation t hat c an be used for the I HCP? G ive reasons. 1824 1920 2016 2112 2208 Time(s) lj(O F) 136.04 132.94 130.Q7 127.39 124.88 F ind t he average o f t he temperatures given in Problem 1.6. Calculate a nd p lot the residuals defined by for the ext/x values o f om, 0.1, 1, to, a nd 100. C ompare with the exact result o f P roblem 1.3a a nd give some conclusions. 1 .6. A solid copper billet 1.82 in. long a nd 1.00 in. in diameter is heated in a furnace a nd t hen removed. T he density-specific heat p roduct o f t he copper is 51 Btu/ft3-F. Some o f t he temperature measurements a re given below. T he h eat transfer model is dT e j= l j- Y; p cV Y t =qA where Y; is t he average temperature. W hat is t he main reason for the lack o f r andomness in the residuals? where V is t he volume a nd A is the heated surface area (the cylindrical sides a nd t he flat ends). a. Using the forward difference approximation, b. c. i =l with respect t o t he parameters. F or t he model where - 1 2 lj(O F) 1632 1728 142.93 139.34 Time(s) 8 9 1• t =- L tj n j=1 l j+l- lj - I 2 ~t derive the estimators for /11 a nd /12 which a re respectively denoted a nd P2' Which approximation (forward, backward, o r central) gives the best results? Time(s) W '- I, T he m ethod o f o rdinary least squares involves minimizing the s um o f s quares function S, ~t c alculate the heat flux for M = 9 6 s using all the given data. P lot t he results versus time. P lot results o n t he same figure for the backward difference approximation. R epeat p art (a) using the central difference approximation, dt d. 1 .9. l j+l-lj I, dTI 118.04 115 .97 114.13 112.35 1.8. 1'.- T(x, t) x de 2496 2592 2688 2784 to 11 12 13 Use t he forward difference method t o a pproximate the first derivative o f e - I a t t = 2. Use ~t = 0.001, om, 0.1, a nd 1. Use t he exact values o f e - I first a nd t hen repeat the solution by adding the r andom e rrors o f 0.000464, 0.000137,0.002455, - 0.000323 a nd - 0.000068 t o e - I for t =2, 2.001, 2.01,2.1, a nd 3, respectively. F or example, Y = e- 2 + 0 .000464=0.135799. I (These r andom values have a s tandard d eviation o f 0 .001 a nd a re t aken from the first row o f T able 5.2.) W hat c onclusions c an be d rawn from the approximations as ~t-+O'fol' t he exact values a nd from those with the r andom e rrors? 2 dTI lj(O F) 1.7. 1 .5. Use t he relation in Problem 1.3 for T a t x n ot equal to zero t o e valuate q(O, t) ~k 45 P ROBLEMS 2304 2400 A-I • lj(O F) 122.46 120.18 f 31=Y= - L l j n j=1 P2 L L (ej-t)lj L (t j -t)2 PI 46 CHAP.1 DESCRIPTION OF THE INVERSE HEAT CONDUCTION PROBLEM 1 .10. Using the linear model in Problem 1.9, estimate PI and P2 for the data o f P roblem 1.6. Also calculate and plot the residuals. Comment on the time variation o f the residuals. r 47 PROBLEMS 1.11 for the d ata o f Problem 1.6. Calculate the average estimated standard deviation a nd sample correlation coefficient. Discuss your results. 1.14. Derive estimators for PI a nd P2 in the model 1 .11. F or measurements equally spaced in time, orthogonal polynomials can be used in least squares estimation. (See Reference 49, p. 248.) A polynomial model is T j=Po+P l t j+P2t f+ . .. +Pr1i T j=PIXjl +/12 X j2 using the ordinary least square method which requires that s = L (1';~ Tj)2 (a) where t j+ I = tj+i\t. This model can be rewritten as Tj=cxoPo(tj)+CXIPI(tj)+CX2P2(tj)+ . .. +cxrPr(t;) (b) be minimized with respect to PI a nd P2' Answer: where Po(t;), PI(t j), a nd P2(t j) a re Po(t j) = 1 t ·-t n+ 1 PI(t j)= 'i\t = i- -2-' ( ft ft tj_~2 P2(t·)= , i \t _1 t= c.,= L Xj.X ft L n j =1 i =-1 tj n 2_1 --12 d .= il , (c) L ; =1 1';X jl 1.15. With ordinary least squares, show that the estimator, p, o f Pin the model T =/1is Verify that these Pj(t j ) functions for r =2 a re orthogonal; that is, show that I• P=- L 1'; n ft L Pj(tj)P.(t;)=O for j fk fO for j =k j =1 j =1 a nd for the standard statistical assumptions being valid, show that the variance o f Pis a ndj, k=O, 1 ,2. Derive (d) for m=O, I , . .. , r. 1.12. F or the polynomial and orthogonal polynomial models of Problem 1.11, prove for r = I t hat 1 .16. Show that for the random variables 1'; a nd 1), a. V (1)=E(YJ>-E 2(1) b. cov( Y;, 1) = E( 1'; 1) - E( 1';)E( 1) 1.17. W hat is the expected value for the random variable I l t hat has the probability density o f I 1.( y) = { 0 and for r = 2 t hat 1 [(1)2 - -1_1] ni \t 2 a~ PO=CXO-CXI - + CX 2 i\t A P (A I A 2t\ 2 2 A I 1 = CX -CX i \c) i \t' A a2 /12 = (i\t)2 1.13. Using the r =2 o rthogonal polynomial model given in Problem 1.11, estimate the parameters using ordinary least squares in Eq. (b) o f Problem f orO<y<1 otherwise What is t he variance o f e? 1 .18. W hat are the mean and the variance of a random variable I l t hat has the probability density J .(y)=aexp[ - (y-WJ, - oo<y<oo W hat is the value o f a ? 1 .19. Using the first row o f the random numbers o f Table 5.2, generate a set r 48 CHAP.1 D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM r of 10 correlated random numbers using I:j+I=0.5I:j+Uj+l, 49 P ROBLEMS problems 1 and 2: i =I,2, . .. , 9 ar;1 - k - '= qj(t) a x x=o where the Uj are the entries in Table 5.2, with 1:1 =U I the first entry. Calculate the sample average and the associated residuals. Next calculate the sample correlation coefficient. Repeat for the next two rows of Table 5.2. Give some conclusions. a1x'; Ix =L-0 a 1';(x, 0 )=0 1 .20. a. Plot the ratio of the temperature rise at x = 0 o f a finite plate with a constant heat flux a t x = 0 and insulated at x = L t o the temperature rise at x = 0 of a semi-infinite body heated at x = 0 by the same value of constant heat flux. Evaluate the ratio (at least) for I1.t/ L2 = 0.05, 0.1,0.15,0.5,1, and 5. P lot on a semi-logarithmic scale. b. Repeat part (a) for the temperature rise at x = L in both bodies. c. Repeat part (a) for the temperature rise a t x = L /2 in both bodies. d. Compare the results for the different cases particularly for small I1.t/L2 values. e. F or the same additive measurement errors in both geometries, over what regions of x a nd t would an I HCP procedure give the same results? Different results? for i = 1 for problem 1 and i = 2 for problem 2. 1 .22. Use the summation relation in Problem 1.21 t o verify Eq. (1.6.25). What are q l(t) and q2(t) for Eq. (1.6.25)? 1 .23. Make a three-dimensional plot of q (y, t) versus y (horizontal axis) and t (drawn as an axis 30° t o the y-axis) for q ll = 3, q12 = 5, ql3 = 7 q21 = 2, 11. q 32=2, q 33=3 1 .24. Derive Eq. (1.6.44). 1 .25. Derive Eq. (1.6.45). 11. = constant - k aaTI = q l(t)+q2(t), x x=o 1 .26. The solution for the temperature in an infinite body is a- x T I-To [ J a +x] T =To + - 2 ert.: 2(l1.t)I /2+ erh 2(l1.t)I/2 k =constant where - 00 < x < 0 0 a nd the initial temperature distribution is (Reference 51, pp. 54, 55) a TI -0 a x x =L - T =To show that the solution T (x, t) is equal to the solution ofthe three problems, where the three problems are described by T =TI 2 a x2'=Tr' Ixl<a Using the principle of superposition give the temperature distribution in a A . infinite body for the initial temperature distribution of T (x, t )= To(x, t )+ TI(x, t )+ T2(x, t) a r; ar; Ixl>a T =TI T(x,O)=F(x) 11. q 23=4 q31=1, 1 .21. F or the problem a 2T a T a x 2 = a t' q 22=3, Ixl<a T =T2 a <x<3a i=O, 1 ,2 T = To problem 0: otherwise Plot the sensitivity coefficients, aTo - =0 ax a t x =O a nd X = aT(O, t) loTI L and X 2= oTtO, t) ---ar;- versus t for I1.t/a 2 =0 to 5. (See Reference 51, p. 55.) The backward heat T o(x,O)=F(x) I f", 50 C HAP.1 D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION PROBLEM c.onduction problem is the estimation o f the initial temperature distributIOn from temperatures measured at later times. Comment on the relative difficulty o f recovering I i and T2 from measurements o f T a t x = 0 and 2a for the two cases of measurements of (1.t/a 2 equal to (a) 0.2, 0.4, 0.6, 0.8, and I ; and (b) 2, 4, 6, 8, a nd 10. r I I , I I i 1 .27. a. F or Eqs. (1.6.4), (1.6.5), and (1.6.7) a nd the convective boundary condition at x = L o f -k Ox L = h[T(L, t) oT/ T",,(t)] wher~ .h is constant, derive the differential equation and boundary b. condltl?ns for the sensitivity coefficient defined by Eq. (1.6.8). ~elate m words the results of part (a) to the problem of a unit step IDcrease o f surface heat flux. c. F or the case of x = L , t M _ I = 0, t < t M , k (x) = constant, c (x) = constant and hL/k = I, find numerical values for k X M/L for t + = 0.25, 0.5, a nd 1. C ompare the values with those for x /L in Table 1.1. CHAPTER 2 EXACT S OLUTIONS OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM 2.1 I NTRODUCTION Exact solutions of the inverse heat conduction problem are very important because ( I) they provide closed form expressions for the heat flux in terms of temperature measurements, (2) they give considerable insight into the characteristics of inverse problems, and (3) they provide standards of comparison for approximate methods. Inverse heat conduction problems can be divided into steady-state and transient problems. The steady-state inverse problem is simpler in that the only necessary thermal property is the thermal conductivity k, and a temperature history is not required. The transient inverse heat conduction problem can be divided into two categories: lumped thermal capacitance and distributed thermal capacitance. The transient case requires many discrete temperature measurements. F or lumped thermal capacitance, the important thermal property is the volumetric heat capacity p c. If the thermal capacitance is distributed, then the thermal conductivity k must be known in addition to the volumetric heat capacity. Throughout Chapter 2, the thermal properties are assumed to be independent of temperature ; this assumption is one of the weaknesses of exact solutions. Section 2.2 considers one-dimensional steady-state problems in which the temperature is known at two or more locations. Section 2.3 examines the lumped thermal capacitance case and some numerical approximations to the exact solution. Section 2.4 considers a planar semi-infinite body for which the surface temperature history is known; an approximate technique for numerically evaluating the resulting integral and an example problem are presented. Section 2.5 presents the development of an exact solution for a one-dimensional planar body with a temperature sensor at an arbitrary depth E below the heated 51 -