Applied Mathematical Modelling 40 (2016) 6588–6603
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Two-dimensional formulation for inverse heat conduction problems by the calibration integral equation method (CIEM) Hongchu Chen, Jay I. Frankel∗, Majid Keyhani Mechanical, Aerospace and Biomedical Engineering Department, University of Tennessee, Knoxville, TN 37996-2210, USA
a r t i c l e
i n f o
Article history: Received 6 February 2015 Revised 30 January 2016 Accepted 3 February 2016 Available online 12 February 2016 Keywords: Inverse heat conduction Calibration integral equation method Total surface heat transfer
a b s t r a c t A new calibration integral equation is developed and investigated in the context of twodimensional linear inverse heat conduction. This paper examines a simplified plate geometry possessing three known boundary conditions and one unknown boundary condition. This plate contains of a series of temperature sensors located on a fixed plane below the surface of interest. A total surface heat transfer (W) calibration integral equation is derived based on a fully two-dimensional analysis. In this form, the total surface heat transfer (i.e., the spatially integrated value along the entire surface of interest), is directly derived and implemented bypassing the need to determine the local surface heat flux (W/m2 ). This formulation yields a Volterra integral equation of the first kind that is pertinent to many applications where the total surface heat transfer (W) is required. Numerical results indicate the merit of the approach when examining two dissimilar isotropic materials. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Inverse heat conduction problems (IHCP’s) involve predicting the surface thermal conditions based on in-depth temperature measurements [1]. In conventional heat conduction problems, the interior temperature distribution of a solid body is calculated when the boundary and initial conditions are known. These are direct problems that are well studied and well understood [2]. In many applications, sensors cannot be placed on the boundaries due to harsh surface thermal environment. Thus, a technological void is introduced when a direct measurement is not available. In-depth probe placement is necessary for practical problems associated with numerous aerospace applications involving reentry, combustors, solid rockets, nozzles and fire research [1]. In those cases, the surface boundary condition can be reconstructed when in-depth temperature data are available. This slight variation in probe placement (surface to interior) alters the mathematical framework by producing an ill-posed problem. An ill-posed problem does not satisfy Hadamard’s three conditions for well posedness [1,3]. These three conditions involve existence, uniqueness and stability of the solution. In particular, the discrete noisy temperature data collected at the probe site cause instability in the prediction. Alternatively stated, a small change in the measured data can cause substantial changes in the prediction. This phenomenon can also be explained by the physics of heat diffusion. Heat diffusion damps high frequency oscillations as heat passes through the conducting body. In the opposite direction, high frequency oscillations associated with in-depth measurements are magnified when projected to the surface. Therefore, regularization [1,4] is necessary for all inverse heat conduction problems. A variety of methods have been exploited to resolve the inverse heat conduction problem, among them are Tikhonov regularization [5], function specification [1,6–8], space marching and finite difference [9–12], global time method [13–15], ∗
Corresponding author. Tel.: +1 865 974 5129; fax: +1 865 974 5274. E-mail address:
[email protected] (J.I. Frankel).
http://dx.doi.org/10.1016/j.apm.2016.02.003 S0307-904X(16)30067-1/© 2016 Elsevier Inc. All rights reserved.
H. Chen et al. / Applied Mathematical Modelling 40 (2016) 6588–6603
Nomenclature a A, B b d fsampling F( t) h i, j k Ktc, cal Ktc, run L m M Mf ˆ M n N Nx N(β m ) N0 (η n ) q qcal qrun qs qx qy qˆ Q Qcal Qrun Qrun, exact Qrun, γ Qˆ rj Rγ R¯ γ s t tj tmax T Tcal Trun Ttc, cal Ttc, run T∞ Tˆ u, v x, y, z x xk Greek
length of two-dimensional plate, cm coefficients of Gaussian function, Eq. (26a) height of two-dimensional plate, cm distance between thermocouples and x axis for two-dimensional IHCP sampling frequency, Hz function defined in Eq. (16b) convective heat transfer coefficient, W/(m2 K) x-direction vector and y-direction vector, respectively thermal conductivity, W/(mK) kernel defined in Eq. (14b), °Cm kernel defined in Eq. (14c), °Cm Laplace operator summation index total number of temperature data beyond initial condition multiplying factor transformed function, Eq. (8b) summation index total number of time increments number of thermocouples normalization integrals, m normalization integrals, m heat flux, W/m2 surface heat flux in “calibration” test, W/m2 surface heat flux in “run” test, W/m2 net surface heat flux, W/m2 heat flux in x direction, W/m2 heat flux in y direction, W/m2 Laplace transformed heat flux, W/m2 total heat transfer, W “calibration” test total heat transfer, W exact unknown “run” total heat transfer, W exact unknown “run” total heat transfer in, W predicted unknown “run” total heat transfer in, W Laplace transformed total heat transfer, Ws random number in the interval [–1,1] residual function defined in Eq. (22), J°Cm running average of residual function defined in Eq. (23), J°Cm complex parameter in Laplace transform time, s discrete time, s maximum time range, s temperature, °C “calibration” test positional temperature at thermocouple site, °C “run” test positional temperature at thermocouple site, °C measured thermocouple temperature in “calibration” test, °C measured thermocouple temperature in unknown “run”, °C environmental temperature, °C Laplace transformed temperature, °Cs dummy time variables, s spatial variables, m dummy spatial variable, m kth thermocouple position
α thermal diffusivity, m2 /s β m , ηn Eigenvalues, m−1
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ɛQ ɛT t x
γ γm γ opt σ R, γ σR,∗ γ
H. Chen et al. / Applied Mathematical Modelling 40 (2016) 6588–6603
noise factor of total heat transfer noise factor of temperature time increment, s space increment, m future time parameter, s future time parameter in Eq. (21a), s optimal future time parameter, s square root of time-running variance of residual defined in Eq. (24), J°Cm normalized square root of time-running variance of residual defined in Eq. (25)
non-integer system identification [16–19], exact solution [20], digital filtering [21], conjugate gradient method [22,23], singular-value decomposition (SVD) [24,25], iteration method [26], boundary-element method [27], and neural networks [28,29]. Calibration is a novel approach for resolving the inverse heat conduction problems. The non-integer system identification (NISI) method [16–19] is a calibration method that requires an accurate extraction of the impulse function based on the fractional derivative formulation of the heat equation. This approach has been used in null-point calorimetry. A known net surface source is first used as a calibration source to obtain the relationship between net surface heat flux and temperature response at the sensor site. The sensor characteristics, depth of sensor, and thermophysical properties of the host material are accounted in the calibration coefficients that are determined by a least squares method. For the inverse analysis, the unknown surface heat flux can be estimated using the corresponding sensor data and the calibration coefficients. In all the noted methods, with exception of the NISI method [16–19], thermophysical properties must be known. Probe location must be accurately measured by some means such as x-ray, CT scans, etc. In addition, issues associated with the sensor attachment to the host material must be quantified in order to account for potential delay and attenuation effects. The alternative calibration methodology proposed by Frankel et al. [30], Elkins et al. [31], Frankel and Keyhani [32] inherently contains sensor positioning, sensor characteristics and thermophysical properties of the host material in the final mathematical expression that relates the in-depth measured temperature data to the surface heat flux. The final mathematical expression is presented in terms of a Volterra integral equation of the first kind [33] which is inherently ill-posed. Therefore, regularization is required to stabilize the prediction [34]. The goal of this paper is to develop the calibration method to higher dimensional domains. The format of this paper is as follows. The problem description is given in Section 2. In Section 3, a two-dimensional total surface heat transfer calibration integral equation method is developed and regularization is performed. Numerical results are discussed in Section 4 displaying the merit and novelty of the concept. Numerical simulations produce favorable results for a large set of engineering materials under the assumption of constant thermophysical properties. Finally, Section 5 provides some concluding remarks on the proposed method. 2. Problem description This paper investigates the two-dimensional plate geometry as shown in Fig. 1. Fig. 1 displays a two-dimensional plate of length a, height b and unit depth. It possesses adiabatic conditions on two sides. The top of the plate is exposed to some
Fig. 1. Schematic for two-dimensional plate geometry with time-varying and spatially distributed surface heat flux. Six in-depth thermocouples are indicated as solid circles.
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time varying spatially distributed source, given as q (x, b, t), while the bottom side is exposed to a convective boundary condition with constant heat transfer coefficient, h and ambient temperature, T∞ . Without loss of generality, the initial condition, T(x, y, 0) = 0, and the ambient temperature, T∞ , are set to zero. The plate itself does not have any sources or sinks. As shown, six thermocouples are placed equidistant apart so that common width zones can be defined. The thermocouples are placed at height y = d. The direct analysis would require the de (x, y, t ) = qx (x, y, t )i + qy (x, y, t ) jwhen termination of either or both the temperature field T(x, y, t) and heat flux given by q provided the boundary and initial conditions; and, the appropriate thermophysical properties of the material. In contrast, the inverse problem would require the reconstruction of surface thermal conditions at y = b, x ∈ [0, a], t ≥ 0 using a finite number of in-depth temperature sensors placed at y = d. Accuracy of the estimated surface heat flux depends on many factors such as the number of probes, location and depth of probes, accuracy of both measured data and thermophysical properties, as well as delay and attenuation characteristics of the temperature sensors [1]. The total surface heat transfer is a quantity of significant interest. As such, we revisit Fig. 1 and view the system in a different manner. That is, we derive a total surface heat transfer calibration integral equation that bypasses the need to obtain the local surface heat flux. The calibration integral equation approach reduces systematic errors through careful integration of calibration and analysis. Key to all inverse studies is the ability to extract the optimal regularization parameter. The calibration integral equation method will be shown to possess a fundamental measure that permits its estimation. The proposed methodology is highly robust and requires a simple numerical implementation. 3. Two-dimensional total surface heat transfer calibration integral equation method In this section, a two-dimensional total surface heat transfer calibration equation is derived based on the analytical solution of a two-dimensional transient heat conduction problem. No assumptions are employed in the derivation except for the constant thermophysical properties. Results from numerical simulations will illustrate that the two-dimensional total surface heat transfer calibration integral equation method works well for a large set of engineering materials under the assumption of constant thermophysical properties. 3.1. Derivation of the two-dimensional total surface heat transfer calibration equation The two-dimensional total surface heat transfer calibration equation is derived by placing the analytical solution of the two-dimensional transient heat conduction problem into an input–output relationship. The schematic of the twodimensional transient heat conduction problem to be considered is displayed in Fig. 1. The analytical solution of the proposed two-dimensional linear transient heat conduction problem with the initial condition, T(x, y, 0), and the ambient temperature, T∞ , set to zero is [2]
T (x, y, t ) =
a ∞ ∞ α cos(βm x ) cos ηn (b − y ) t −α (βm2 +ηn2 )(t−u ) e cos(βm x )q (x , b, u )dx du, k N (βm )N0 (ηn ) u=0 x =0 m=0 n=1 x ∈ [0, a], y ∈ [0, b], t ≥ 0,
(1a)
where the x-direction eigenvalues and normalization integrals are defined as
π βm = m , m = 0, 1, 2, 3, ...
(1b)
a
N (βm ) =
a,
m = 0, (1c)
a , m = 1, 2, 3, ... 2
and the y-direction eigenvalues and normalization integrals are defined as
ηn ’ s are positive roots of N0 (ηn ) =
b(ηn2 +
h k
ηn tan (ηn b) = , n = 1, 2, . . . ,
) + hk , n = 1, 2, .... 2 2(η + hk2 ) h2 k2
2 n
(1d)
(1e)
Evaluating Eq. (1a) at y = d yields
T (x, d, t ) =
∞ ∞ α cos(βm x ) cos ηn (b − d ) t −α (βm2 +ηn2 )(t−u ) a e cos(βm x )q (x , b, u )dx du. k N (βm )N0 (ηn ) u=0 x =0 m=0 n=1
Next, formally integrating Eq. (2) over the x-domain produces (allowing the integral to pass through the sums)
(2)
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a
x=0
∞ ∞ α
T (x, d, t )dx =
k
a x=0
m=0 n=1
cos(βm x )dx cos ηn (b − d ) N (βm )N0 (ηn )
t
e−α (βm +ηn )(t−u ) 2
2
u=0
a x =0
cos(βm x )q (x , b, u )dx du, (3a)
and noting
a
x=0
cos(βm x )dx =
a
cos
mπ
x=0
x dx =
a
⎧ ⎨a, m = 0, ⎩
, a a sin( maπ x ) = 0, m = 1, 2, 3, ... mπ x=0
(3b)
there by reduces the double infinite summation of m, n to one infinite summation of n as
a
x=0
T (x, d, t )dx =
∞ α cos ηn (b − d ) k N0 (ηn ) n=1
t
e−αηn (t−u ) 2
a
x =0
u=0
q (x , b, u )dx du.
Taking the Laplace transform with respect to t [35] produces
L
∞ α cos ηn (b − d ) T (x, d, t )dx = L k N0 (ηn ) x=0 n=1 a
t
e
−αηn2 (t−u )
a x =0
u=0
(4)
q (x , b, u )dx du ,
(5a)
where L is the standard Laplace transform operator given as L [ f (t )] = fˆ(s ) =
∞
t=0
e−st f (t )dt ,
(5b)
where s is a complex number. Upon using the well-known convolution theorem [35], Eq. (5a) reduces to
a
x=0
Tˆ (x, d, s )dx =
∞ α cos ηn (b − d ) −αηn2 t L e k N0 (ηn ) n=1
a x =0
qˆ (x , b, s )dx ,
(6a)
where [36, p. 1022, (29.3.8)]
L e−αηn t = 2
1 . s + αηn2
(6b)
Substituting Eq. (1e) and Eq. (6b) into Eq. (6a) produces
a
x=0
∞ 2(ηn2 + hk2 ) cos ηn (b − d ) α 1 2 + h2 ) + h k s + αηn2 b ( η n n=1 k k2 2
Tˆ (x, d, s )dx =
a x =0
qˆ (x , b, s )dx ,
(7)
allowing Eq. (7) to be written in the input–output format as
a ˆ x=0 T a ˆ x =0 q
(x, d, s )dx ˆ (b, d, α , k, h; s ), =M (x , b, s )dx
(8a)
where ∞ 2(ηn2 + hk2 ) cos ηn (b − d ) α 1 . h2 h 2 k s + αηn2 b( ηn + k 2 ) + k n=1 2
ˆ (b, d, α , k, h; s ) = M
(8b)
Suppose that the total surface heat transfer at the surface y = b is the only quantity of interest. We can define total surface heat transfer per unit depth (z-direction) as
Q (b, t ) =
a
x=0
q (x, b, t )dx, t ≥ 0,
(9a)
or in the frequency domain as
Qˆ (b, s ) =
a x=0
qˆ (x, b, s )dx, Re(s ) ≥ 0.
(9b)
With this definition, we can express Eq. (8a) as
a x=0
Tˆ (x, d, s )dx ˆ (b, d, α , h, k, h; s ). =M Qˆ (b, s )
(10)
ˆ (b, d, α , k, h; s ) do not change during the testing campaign then we can use If the parameters in the transfer function M the concept of calibration reconstruction [30] as
a x=0
Tˆ (x, d, s )dx Qˆ (b, s )
calibration test
=
a x=0
Tˆ (x, d, s )dx Qˆ (b, s )
reconstruction test
.
(11)
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Cross multiplying and employing the convolution theorem [35] yields
t u=0
Qrun (b, u )
a
x =0
Tcal (x , d, t − u )dx du =
t u=0
Qcal (b, u )
a
x =0
Trun (x , d, t − u )dx du, t ≥ 0.
(12)
Here, the subscription “cal” represents data collected during the calibration test while “run” represents measured temperature and unknown total surface heat transfer in the reconstruction run. Eq. (12) can be written in terms of the ideal thermocouple temperature data as
t u=0
Qrun (b, u )
a
x =0
Ttc,cal (x , d, t − u )dx du =
t
u=0
Qcal (b, u )
a x =0
Ttc,run (x , d, t − u )dx du, t ≥ 0,
(13)
which follows the discussion presented in Ref. [30]. Eq. (13) can be written in the compact format as
t u=0
Qrun (b, u )Ktc,cal (t − u )du =
where
Ktc,cal (t − u ) = Ktc,run (t − u ) =
a
x =0
a x =0
t
u=0
Qcal (b, u )Ktc,run (t − u )du,
(14a)
Ttc,cal (x , d, t − u )dx ,
(14b)
Ttc,run (x , d, t − u )dx .
(14c)
Here, Ktc , cal and Ktc , run are obtained by numerical integration. The mid-point rule is applied in this context for discretizing space as
Ktc,cal (t − u ) =
Ktc,run (t − u ) =
a
x =0
a x =0
Ttc,cal (x , d, t − u )dx =
Nx
Ttc,cal (xk , d, t − u )x,
(15a)
k=1
Ttc,run (x , d, t − u )dx =
Nx
Ttc,run (xk , d, t − u )x,
(15b)
k=1
where xk denotes the kth thermocouple position, and x denotes the distance between each thermocouple as indicated in Fig. 1. Here, Nx denotes the total number of thermocouples placed along the x-direction at y = d. 3.2. Regularization-local future time method and the optimality metrics Eq. (14a) is Volterra integral equation of the first kind [33]. Lamm’s local future time method [30,34] is applied to stabilize the equation. The two-dimensional total surface heat transfer calibration integral equation given in Eq. (14a) is alternatively expressed as
t u=0
Qrun (b, u )Ktc,cal (t − u )du = F (t ),
where
F (t ) =
t u=0
Qcal (b, u )Ktc,run (t − u )du,
t ≥ 0,
(16a)
t ≥ 0.
(16b)
The future time parameter γ is introduced by advancing time through t → t + γ , where γ is the future time parameter. Eq. (16a) becomes
t+γ u=0
Qrun (b, u )Ktc,cal (t + γ − u )du = F (t + γ ),
t ∈ [0, tmax − γ ].
(17)
Observe that the recoverable time domain is reduced by γ . The basic integral definition allows us to express Eq. (17) as
t u=0
Qrun (b, u )Ktc,cal (t + γ − u )du +
t+γ u=t
Qrun (b, u )Ktc,cal (t + γ − u )du = F (t + γ ),
t ∈ [0, tmax − γ ].
(18)
Next, we define v = u – t, and Eq. (18) becomes
t u=0
Qrun (b, u )Ktc,cal (t + γ − u )du +
γ
v=0
Qrun (b, v + t )Ktc,cal (γ − v )dv = F (t + γ ), t ∈ [0, tmax − γ ].
(19)
If γ is small then we can approximate Qrun (b, v + t ) ≈ Qrun (b, t ) since v ∈ [0, γ ]. If this is the case then we can form the approximation
t u=0
Qrun (b, u )Ktc,cal (t + γ − u )du + Qrun (b, t )
γ
v=0
Ktc,cal (γ − v )dv ≈ F (t + γ ), t ∈ [0, tmax − γ ],
(20a)
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or
H. Chen et al. / Applied Mathematical Modelling 40 (2016) 6588–6603
t
u=0
γ
Qrun,γ (b, u )Ktc,cal (t + γ − u )du + Qrun,γ (b, t )
v=0
Ktc,cal (γ − v )dv = F (t + γ ), t ∈ [0, tmax − γ ].
(20b)
Eq. (20b) is a second kind Volterra integral equation for the unknown total surface heat transfer denoted by Qrun,γ (b, t). Here Qrun,γ (b, t) is an approximation to Qrun (b, t) as it depends on γ . Data are collected up to some time defined as tmax . Future information is incorporated into the numerical algorithm. The predicted total surface heat transfer Qrun,γ (b, t) can only be resolved in temporal domain t ∈ [0, tmax − γ ] due to the inclusion of future data to stabilize the numerical implementation. Discretization of Eq. (20b) can be performed using a variety of low-order numerical integration rules, such as the rectangular rule, trapezoidal rule, product integration rules, etc. A right-hand rectangular rule is employed in the present context. The discrete values for Qrun,γ (b, t) can be obtained in the time-marching form where γ → γ m = mMf t, m = 1, 2, 3, …, where t is given as t = tmax /N, N is the number of segments (or samples) in the discretized temporal domain, and Mf is a convenient multiplying factor. Discretizing Eq. (20b) produces i
Qrun,γm (b, t j )
j=1
tj
u=t j−1
Ktc,cal (ti + γm − u )du + Qrun,γm (b, ti )
γm u=0
Ktc,cal (γm − u )du = F (ti + γm ),
i = 1, 2, ...M, (21a)
where M = N – mMf or upon extracting the desired total surface heat transfer at t = ti we obtain
F (ti + γm ) − Qrun,γ (b, ti ) =
ti u=ti −1
i −1 j=1
Qrun,γ (b, t j )
tj u=t j−1
Ktc,cal (ti + γm − u )du +
Ktc,cal (ti + γm − u )du
γm
u=0
Ktc,cal (γm − u )du
, i = 1, 2, ...M,
(21b)
where tj = jt, j = 0, 1, …, M. In this time-marching implementation, the future time parameter γ m must be specified over a range of values and then interrogated until an optimal value is determined based on some measure. The introduced discretization errors are not explicitly indicated but are absorbed in the present notation as the focus of the numerical analysis is on the regularization method. It is understood that discretization errors are present. In the present context, Eq. (16a and b) are the exact formulation of the proposed total surface heat transfer calibration integral equation. Eq. (20a) represents an approximation since a constant total surface heat transfer is assumed in the future time interval [t, t + γ ] in order to convert the first kind Volterra integral equation into a well-posed second kind Volterra integral equation for sufficiently large γ m . If the future time parameter γ is too small then the prediction via Eq. (20b) is unstable. If the future time parameter γ is too large then over smoothing of the prediction occurs as high frequencies are damped. Hence, significant surface physics can be lost. Compromise is required between stability and accuracy. To determine the optimal future time parameter γ , the residual function Rγ (t) is defined as
Rγ (t ) =
t
u=0
Qrun,γ (b, u )Ktc,cal (t − u )du − F (t ),
t ∈ [0, tmax − γ ].
(22)
Next, the running average of residual function is defined as
R¯ γ (t ) =
1 t
t
u=0
Rγ (u )du,
t ∈ [0, tmax − γ ].
(23)
Both Rγ (t) and R¯ γ (t ), from Eqs. (22) and (23), respectively can be discretized by a simple and consistent integration rule. The right-hand rectangular rule is employed herein as with Eq. (21). The time-running variance of residual function is defined through
σR,2 γ (ti ) =
2 1 Rγ (t j ) − R¯ γ (ti ) , i i
i = 1, 2, . . . , M.
(24)
j=1
A second measure based on the normalized version of Eq. (24) can be defined to emphasize or enhance other effects. Following this logic, we define the normalized square root of the time-running variance of the residual as
σR,∗ γ (ti ) =
σR,γ (ti ) , i = 1, 2, . . . , M, ||σR,γ (ti )||∞
(25)
where ||σ R, γ (ti )||∞ is the maximum norm given as max |σR,γ (ti )|. These four metrics assist in estimating the optimal future 1≤i≤M
time parameter γ . Eq. (24) is a novel function in the context of inverse analysis. 4. Numerical results Two contrasting isotropic materials are considered in this section for understanding the basic formulation and numerical implementation of Eq. (21b).
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Fig. 2. The surface heat flux in the “run” test, qrun (x, b, t ) (copper).
4.1. Numerical simulation using copper as the host material In this sub-section, the host material of the slab in the simulation for the two-dimensional total surface heat transfer calibration integral equation method is copper with thermal conductivity k = 401 W/(mK) and thermal diffusivity α = 117 × 10−6 m2 /s. The length of slab is a = 12 cm, the height of slab is b = 1 cm with unit depth (1 cm). The convective heat transfer coefficient used in the simulation is h = 100 W/(m2 K). Six thermocouples are placed equidistant apart at y = d = 5 mm with x1 = 1 cm and x = 2 cm. The sampling frequency fsamplin g used in the simulation is 200 Hz while the total simulation time is tmax = 10 s. These are reasonable parameter choices for a laboratory experiment. The “calibration” surface heat flux is uniform in the spatial domain and constant in the temporal domain. That is, we assign the net calibration surface heat flux as qcal (x, b, t ) = 100 W/cm2 . The “calibration” total surface heat transfer is the spatially integrated value of the “calibration” surface heat flux producing Qcal (b, t) = 1200 W (for surface area of 12 cm2 ). A uniform calibration source offers significant simplicity and minimizes integration errors during the calibration test. In contrast, the “reconstruction” test surface heat flux is given as a power function in the spatial domain and Gaussian function in the temporal domain and mathematically expressed as
qrun (x, b, t ) = 10 0 0 x0.5 e−
(t−A )2 B2
W/cm2 ,
(26a)
where
A = tmax /2,
(26b)
B = tmax /8,
(26c)
as shown in Fig. 2. The exact total surface heat transfer in the “reconstruction” test is the spatially integrated value along the entire surface. For simplicity, the “reconstruction” heat flux is now referred to as “run”. Performing this integration yields
Qrun,exact (b, t ) =
a x=0
q run (x, b, t )dx =
20 0 0 3/2 − a e 3
(t−A )2 B
( W ).
(26d)
Thermocouple data are generated by adding artificial noise to the positional temperatures (ignoring signal delay and attenuation effects) in accordance to
Ttc (xk , d, t j ) = T (xk , d, t j ) + εT r j ||T (xk , d, t j )||∞ , k = 1, 2, ...Nx , j = 1, 2, ...N,
(27)
for both the “calibration” and “run” tests. Here ε T denotes a noise factor, rj denotes the jth random numbers chosen in the interval [–1,1] which are obtained by the MATLAB command “2∗(rand-0.5)”. Noise is also added to the total calibration surface heat transfer per
Qcal (b, t j ) = Qcal, exact (b, t j ) + εQ r j ||Qcal, exact (b, t j )||∞ ,
j = 1, 2, ...N.
(28)
Table 1 presents the parameters and properties for the two-dimensional total surface heat transfer analysis for the copper test campaign. Fig. 3 presents the exact total surface heat transfer Qcal , exact (b, t) and noisy total surface heat transfer Qcal (b, tj ) data for the “calibration” test. Fig. 4 displays the surface temperature for the “calibration” denoted as Tcal (x, b, t). The positional temperature at the thermocouple site, Tcal (x, d, t), and the noisy thermocouple data, Ttc,cal (x, d, tj ), for the “calibration” test are shown in Fig. 5. Temperature for the “calibration” test is uniform in the spatial domain due to the uniform surface heat
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H. Chen et al. / Applied Mathematical Modelling 40 (2016) 6588–6603 Table 1 Properties and parameters for two-dimensional total surface heat transfer calibration integral equation method simulation using copper as the host material. Property / Parameter
Value
k
401 W/(mK) 117 × 10−6 m2 /s 100 W/(m2 K) 12 cm 1 cm 0.5 cm 0.01 0.01 200 Hz 20 0 0 10 s 0.005 s 10 6
α h a b d ɛT ɛQ fsampling N tmax t Mf Nx
Fig. 3. The exact total surface heat transfer Qcal , exact (b,t) and the noise added total surface heat transfer Qcal (b,tj ) (copper).
Fig. 4. The surface temperature in the “calibration” test, Tcal (x, b, t) (copper).
flux in the “calibration” test. Fig. 6 displays the exact total surface heat transfer Qrun,exact (b, t) for the “run” test calculated according to Eq. (26d). Fig. 7 presents the surface temperature in the “run” test which is denoted as Trun (x, b, t). Positional temperatures and the noisy thermocouple data at the probe sites x1 –x6 for the “run” test are also shown in Fig. 7. The residual-based metrics defined in Section 3.2 are employed here for determining the optimal future time parameter γ opt . Fig. 8 displays the residual function, Rγ (t), defined in Eq. (22) over time for increasing future time parameter γ based on the “calibration” and “run” thermocouple data as well as the “calibration” total surface heat transfer and the predicted
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Fig. 5. The positional temperature at thermocouple site Tcal (x, d, t) and the thermocouple temperature Ttc,cal (x, d, tj ), respectively (copper).
Fig. 6. The exact “run” total surface heat transfer Qrun , exact (b, t) to be reconstructed (copper).
“run” total surface heat transfer. As the future time parameter increases, the residual’s magnitude increases. Fig. 9 shows the corresponding running average of the residual defined in Eq. (23) as R¯ γ (t ). The smoothing operation of integration is clearly observed in Fig. 9. Fig. 10 displays the square root of time-running variance of the residual function defined in Eq. 2 (t ) while Fig. 11 displays the normalized version of σ 2 (t ) per Eq. (25) as σ ∗ (t ). In this form (Fig. 11), both (24) as σR, γ i R,γ i R,γ i instability and over-smoothing effects can be quickly identified. Small γ ’s produce unstable predictions (see for example γ = 0.10 s) while large γ ’s produce over-smoothed predictions (see for example γ = 0.50 s) of the total surface heat transfer. The normalized plot possesses key physical effects. In fact, a bundling effect occurs for γ = 0.20–0.30 s for times, t > 7.5 s. This range of γ -values offer the best prediction based on the balance between bias (data) and variance (model). This γ -range of the spectrum is where the optimal prediction for the total surface heat transfer will lie. In fact, it is desired to retain as much of the high frequency content in the signal for recovering actual physics. Hence, the optimal estimator is chosen as γ opt = 0.20 s. The highly favorable predicted time history of the “run” total surface heat transfer based on γ opt = 0.20 s is presented in Fig. 12.
4.2. Numerical simulation using stainless steel as the host material In this section, the host material of the slab in the simulation is changed from pure copper to stainless steel with thermal conductivity k = 14.7 W/(mK), and thermal diffusivity α = 3.75 × 10−6 m2 /s. The sampling frequency fsampling used in the simulation is 50 Hz while the total simulation time is tmax = 40 s. All other parameters are the same as the copper case. The relevant parameters for this analysis are presented in Table 2. For this case, the functional forms of the “calibration” surface heat flux and the “run” surface heat flux are identical to the copper case. However, their magnitudes are reduced by a factor of 10 (qcal (x, b, t ) = 10 W/cm2 , Qcal (b, t ) = 120 W, etc.). Figs. 13 and 14 present the “run” surface heat flux and the “run” total surface heat transfer, respectively. The objective is to reconstruct the total surface heat transfer displayed in Fig. 14.
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Fig. 7. The positional temperature at various thermocouple sites Trun (xk , d, t) (k = 1, 2, …, 6) and the thermocouple temperature Ttc,run (xk , d, tj ), respectively (copper).
Fig. 8. The local residual function, Rγ (t) defined in Eq. (22) over the discrete γ -spectrum (copper).
Fig. 9. The running average of residual function, R¯ γ (t ) defined in Eq. (23) over the discrete γ -spectrum (copper).
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Fig. 10. The square root of time-running variance of residual, σ γ (ti ) defined in Eq. (24) over the discrete γ -spectrum (copper).
Fig. 11. The normalized square root of time-running variance of residual, σγ∗ (ti ) defined in Eq. (25) over the discrete γ -spectrum (copper).
Fig. 12. Comparison between the predicted total “run” surface heat transfer Qrun , γ (b, t) using γ opt = 0.20 s and the exact total “run” surface heat transfer Qrun,exact (b, t) (copper).
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H. Chen et al. / Applied Mathematical Modelling 40 (2016) 6588–6603 Table 2 Properties and parameters for two-dimensional total surface heat transfer calibration integral equation method simulation using stainless steel as the host material. Property/Parameter
Value
k
14.7 W/(mK) 3.75 × 10−6 m2 /s 100 W/(m2 K) 12 cm 1 cm 0.5 cm 0.01 0.01 50 Hz 20 0 0 40 s 0.02 s 10 6
α h a B D ɛT ɛQ fsampling N tmax t Mf Nx
Fig. 13. The surface heat flux in the “run” test, qrun (x, b, t ) (stainless steel).
As in Section 4.1, noise is added to the positional temperatures in the “calibration” and “run” tests for generating thermocouple data. Similar to the copper case, noise is added to the total surface heat transfer in the “calibration” test data. The positional temperatures and the noisy thermocouple data at the six probe sites (Nx = 6) for the “calibration” and “run” tests are presented in Figs. 15 and 16, respectively. Again, idealized thermocouples are assumed (no delay and attenuation). Following the procedure outlined in Section 4.1, the optimal regularization parameter γ opt is estimated but now in the context of stainless steel as the host material. In the normalized square root of time-running variance of residual plot (Fig. 17), both instability and over-smoothing effects are readily identified. Small γ ’s produce unstable predictions (see for example γ = 1.0 s) while large γ ’s produce over-smoothed predictions (see for example γ = 4.0 s) of the total surface heat transfer. A bundling effect occurs as labeled in Fig. 17. Here, γ = 2.0 s is extracted as the value for optimal future time parameter. Comparison between the predicted total “run” surface heat transfer Qrun , γ (b, t) using γ = 2.0 s and the exact total ∗ (t ) from Figs. 11 and 17 preserve “run” surface heat transfer Qrun,exact (b, t) is displayed in Fig. 18. Also, observe that σR, γ i the similar characteristic between the copper and stainless steel materials. Again, a highly favorable estimation of the total surface heat transfer is recovered as indicated in Fig. 18. The amount of total surface energy input (J) can be calculated by integrating the total surface heat transfer (W) with respect to time. The comparisons of exact and predicted total surface energy inputs using optimal future time parameters for both copper and stainless steel are shown in Table 3. The percent differences between exact and predicted total surface energy input are smaller than 0.03% for both copper and stainless steel.
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Fig. 14. The exact “run” total surface heat transfer Qrun , exact (b, t) to be reconstructed (stainless steel).
Fig. 15. The positional temperature at thermocouple site Tcal (x, d, t) and the thermocouple temperature Ttc,cal (x, d, tj ), respectively (stainless steel).
Fig. 16. The positional temperature at various thermocouple sites Trun (xk , d, t) (k = 1, 2, …, 6) and the thermocouple temperature Ttc,runl (xk , d, tj ), respectively (stainless steel). Table 3 Comparison of exact total surface energy input and predicted total surface energy input using the optimal future time parameter γ opt . Material
Exact total surface energy input (J)
Predicted total surface energy input (J)
Percent difference
Copper Stainless steel
6164.8 2465.9
6164.3 2466.5
0.0074% 0.0239%
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Fig. 17. The normalized square root of time-running variance of residual, σγ∗ (ti ), defined in Eq. (25) over the discrete γ -spectrum (stainless steel).
Fig. 18. Comparison between the predicted total “run” surface heat transfer Qrun , γ (b, t) using γ = 2.0 s and the exact total “run” surface heat transfer Qrun,exact (b, t) (stainless steel).
5. Conclusions A transformative calibration methodology for resolving the total surface heat transfer for two dimensional linear inverse heat conduction problems has been investigated. The calibration methodology is derived based on mathematical reasoning infused with experimental insight. The primary advantage of the calibration methodology over the other inverse technologies lies in the explicit removal of sensor positioning, sensor characteristics and thermophysical properties. These parameters are implicitly contained in the final mathematical expression that relates the in-depth measured temperature data to the surface boundary condition. The numerical results illustrate excellent robustness and accuracy of this approach for predicting total surface heat transfer for both copper and stainless steel samples. Acknowledgments The results presented in this article were supported under the NASA Cooperative Agreement NNX10AN35A. References [1] [2] [3] [4] [5] [6] [7]
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