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A New Efficient Formula for the Thermal Diffusivity Estimation from the Flash Method Taking into Account Heat Losses in Rear and Front Faces

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Abstract

The flash method is a commonly used technique for estimating the thermal diffusivity of solid materials. In this work, we develop a new efficient formula for the thermal diffusivity calculation to process the flash method data in the case of heat losses at the rear and front faces. First, the transient heat equation is analytically solved using the Green functions under time-dependent convection boundary conditions. The property of the descending part of the analytical solution, containing series which converges very quickly for long times, allows to express the thermal diffusivity as a function of the slope of the natural logarithm of the temperature rise and the Biot numbers related to heat losses at the rear and front faces. These Biot numbers are calculated from the ratio between the integrals of the complete temperature trends of the two faces. For this purpose, we developed a novel way to evaluate these integrals using the analytical solutions in Laplace domain. To verify the accuracy of these formulas, we simulated experimental data adding a Gaussian noise to the theoretical temperature results of the flash method data.

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Abbreviations

a:

Thermal diffusivity (m2·s−1)

Cp :

Specific heat (J·k−1·kg−1)

h1 :

Heat transfer coefficient at front face (W·m−2·k−1)

h2 :

Heat transfer coefficient at rear face (W·m−2·k−1)

λ:

Thermal conductivity (W·m−1·k−1)

e:

Length of the sample (m)

Q0 :

Amount of heat absorbed through the front surface of the sample per unit area (J·m−2)

Bi1 :

\(\frac{{eh_{1} }}{\lambda }\) the Biot number at front face

Bi2 :

\(\frac{{eh_{2} }}{\lambda }\) the Biot number at rear face

ρ:

Density of the sample (kg·m−3)

Tma :

\(\frac{{Q_{0} }}{{e \cdot \rho c_{p} }}\) adiabatic limit temperature (°C)

\(\tilde{T}\) :

Simulated front-surface temperature rise data (°C)

T:

Simulated rear-surface temperature rise data (°C)

t:

Time (s)

td :

Pulse duration (s)

ti :

ith sampled time (s)

\(\tilde{t}_{N}\) :

Final sampled time at front face (s)

t Z :

Final sampled time at rear face (s)

x:

Spatial coordinate (m)

r:

Normally distributed random number (°C)

σ:

Standard deviation

N:

Number of temperature samples recorded at front face (excluding initial temperature)

Z:

Number of temperature samples recorded at rear face (excluding initial temperature)

\(\varepsilon\) :

Relative error (%)

T(0,t):

The theoretical rise temperature at front face (°C)

T(e,t):

The theoretical rise temperature at rear face (°C)

S:

\(\int\limits_{0}^{ + \infty } {T\left( {x,t} \right)} dt\) (°C·s)

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Appendix A. Analytical Expression of the Temperature Rise

Appendix A. Analytical Expression of the Temperature Rise

The system (A1) can be reduced to a problem with homogeneous boundary conditions.

$$\left\{ \begin{aligned}& \frac{{\partial^{2} T\left( {x,t} \right) \, }}{{\partial x^{2} }} = \frac{1}{a}\frac{\partial T}{\partial t}. \hfill \\& - \lambda \frac{{\partial T\left( {x,t} \right) \, }}{\partial x}\left| {_{x = 0} } \right. = - h_{1} \cdot T\left( {0,t} \right) + Q_{0} \times f\left( t \right) \hfill \\& - \lambda \frac{{\partial T\left( {x,t} \right) \, }}{\partial x}\left| {_{x = e} } \right. = h_{2} \cdot T\left( {e,t} \right) \hfill \\& T\left( {x,0} \right) = 0 \hfill \\ \end{aligned} \right.$$
(A1)

We suggest that

$$T\left( {x,t} \right) = K\left( {x,t} \right) + F\left( {x,t} \right)$$
(A2)

where K(x,t) is a function satisfying the boundary conditions (Eq. A3)

$$\left\{ \begin{aligned} &\frac{{\partial K\left( {x,t} \right) \, }}{\partial x}\left| {_{x = 0} } \right. = \frac{{h_{1} }}{\lambda } \cdot K\left( {0,t} \right) - g\left( t \right). \hfill \\ &\frac{{\partial K\left( {x,t} \right) \, }}{\partial x}\left| {_{x = e} } \right. = - \frac{{h_{2} }}{\lambda } \cdot K\left( {e,t} \right). \hfill \\ \end{aligned} \right.$$
(A3)

We assume that K(x, t) is written as follows:

$$K\left( {x,t} \right) = C_{1} \left( t \right)x + C_{2} \left( t \right).$$
(A4)

The coefficients A1 and A2 are calculated using the boundary conditions (Eq. A3):for x = 0,

$$A_{1} \left( t \right) = \frac{{h_{1} }}{\lambda } \cdot A_{2} \left( t \right) - g\left( t \right).$$
(A5)

for x = e,

$$\begin{aligned} &C_{1} \left( t \right)\left( {1 + \frac{{e \cdot h_{2} }}{\lambda }} \right) = - \frac{{h_{2} }}{\lambda } \cdot C_{2} \left( t \right), \hfill \\ &\Rightarrow C_{1} \left( t \right) = \frac{{ - g\left( t \right)\frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}\quad C_{2} \left( t \right) = \frac{g\left( t \right)}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}. \hfill \\ \end{aligned}$$
(A6)

K(x,t) can be written as follows:

$$K\left( {x,t} \right) = \frac{{ - g\left( t \right)\frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}x + \frac{g\left( t \right)}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}.$$
(A7)

Then, the function F(x,t) must be the solution of the problem:

$$a \cdot \frac{{\partial^{2} F\left( {x,t} \right) \, }}{{\partial x^{2} }} + p\left( {x,t} \right) = \frac{{\partial F\left( {x,t} \right)}}{\partial t}.$$
(A8)

And

$$\left\{ \begin{aligned} &\frac{{\partial F\left( {x,t} \right) \, }}{\partial x}\left| {_{x = 0} } \right. = \frac{{h_{1} }}{\lambda } \cdot F\left( {0,t} \right). \hfill \\ &\frac{{\partial F\left( {x,t} \right) \, }}{\partial x}\left| {_{x = e} } \right. = - \frac{{h_{2} }}{\lambda } \cdot F\left( {e,t} \right). \hfill \\ & F\left( {x,0} \right) = T\left( {x,0} \right) - K\left( {x,0} \right) = \frac{g\left( 0 \right)}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}\frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}x + \frac{ - g\left( 0 \right)}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}} = 0. \hfill \\ \end{aligned} \right.$$
(A9)

With

$$p\left( {x,t} \right) = - \frac{{\partial K\left( {x,t} \right)}}{\partial t}.$$
(A10)

The first step in resolving the system (Eqs. A8 and A9) is to consider the PDE (Eq. A8) homogenous:

$$p(x,t) = 0.$$

Using the method of separating the variables, we obtain

$$F\left( {x,t} \right) = A\left( x \right) \times B\left( t \right).$$
(A11)

The eigenvalues fn are the solutions of the following transcendent equation:

$$tan\left( {f_{n} \cdot e} \right) = \frac{{f_{n} \left[ {\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{\lambda }} \right]}}{{f_{n}^{2} - \left( {\frac{{h_{1} }}{\lambda }} \right) \times \left( {\frac{{h_{2} }}{\lambda }} \right)}}.$$
(A12)

For this problem, the normalized eigenfunctions are

$$A_{n} \left( x \right) = a_{n} \left[ {cos\left( {f_{n} \cdot x} \right) + \frac{{h_{1} }}{{\lambda \cdot f_{n} }} \cdot sin\left( {f_{n} \cdot x} \right)} \right].$$
(A13)

The orthogonality properties of the normalized eigenfunctions An (x):

$$\int\limits_{0}^{e} {A_{n} \left( x \right)} \times A_{m} \left( x \right)dx\left\{ \begin{aligned} = 0\quad if\;n \ne m. \hfill \\ = 1\quad if\;n = m. \hfill \\ \end{aligned} \right.$$
(A14)
$$\begin{aligned} &\int\limits_{0}^{e} {A_{n} \left( x \right)} \times A_{n} \left( x \right)dx = 1 \hfill \\& \Rightarrow \int\limits_{0}^{e} {a_{n}^{2} \left[ {cos\left( {f_{n} \cdot x} \right) + \frac{{h_{1} }}{{\lambda \cdot f_{n} }} \cdot sin\left( {f_{n} .x} \right)} \right]}^{2} dx = a_{n}^{2} \cdot \int\limits_{0}^{e} {\left[ {cos\left( {f_{n} \cdot x} \right) + \frac{{h_{1} }}{{\lambda \cdot f_{n} }} \cdot sin\left( {f_{n} \cdot x} \right)} \right]}^{2} dx = 1 \hfill \\& \Rightarrow a_{n} = \frac{1}{{\sqrt {\int\limits_{0}^{e} {\left[ {cos\left( {f_{n} \cdot x} \right) + \frac{{h_{1} }}{{\lambda \cdot f_{n} }} \cdot sin\left( {f_{n} \cdot x} \right)} \right]}^{2} dx} }}. \hfill \\ \end{aligned}$$

We achieve

$$a_{n} = \frac{{\left( {\frac{{\lambda \cdot f_{n} }}{{h_{1} }}} \right)}}{{\sqrt {\left( {\frac{e}{2} - \frac{{sin\left( {2f_{n} \cdot e} \right)}}{{4f_{n} }}} \right) + \left( {\frac{\lambda }{{h_{1} }}} \right) \times sin\left( {f_{n} \cdot e} \right) + \left( {\frac{{\lambda \cdot f_{n} }}{{h_{1} }}} \right)_{{}}^{2} \times \left( {\frac{e}{2} + \frac{{sin\left( {2f_{n} \cdot e} \right)}}{{4f_{n} }}} \right)} }}$$
(A15)
$$A_{n} \left( x \right) = \frac{{cos\left( {f_{n} \cdot x} \right) + \frac{{h_{1} }}{{\lambda \cdot f_{n} }}sin\left( {f_{n} \cdot x} \right)}}{{\sqrt {\int\limits_{0}^{e} {\left[ {cos\left( {f_{n} \cdot x} \right) + \frac{{h_{1} }}{{\lambda \cdot f_{n} }} \cdot sin\left( {f_{n} \cdot x} \right)} \right]}^{2} dx} }}$$
(A16)

We express the unknown solution F(x, t) as a generalized eigenfunctions series of Fourier with time-dependent coefficients:

$$F\left( {x,t} \right) = \sum\limits_{n = 1}^{\infty } {B_{n} \left( t \right)} \cdot A_{n} \left( x \right).$$
(A17)

After replacement Eq. A17 in Eq. A8 we obtain

$$a \cdot \sum\limits_{n = 1}^{\infty } {\left( {B_{n} \left( t \right)\frac{{\partial^{2} A_{n} \left( x \right)}}{{\partial x^{2} }}} \right)} + p\left( {x,t} \right) = \sum\limits_{n = 1}^{\infty } {A_{n} \left( x \right)\frac{{\partial B_{n} \left( t \right)}}{\partial t}} .$$
(A18)

We have

$$\left\{ \begin{aligned} &\frac{{\partial A_{n} \left( x \right)}}{\partial x} = a_{n} \cdot \left( { - f_{n} \cdot sin\left( {f_{n} \cdot x} \right) + \frac{{h_{1} }}{\lambda }cos\left( {f_{n} \cdot x} \right)} \right) \hfill \\& \frac{{\partial^{2} A_{n} \left( x \right)}}{{\partial x^{2} }} = \frac{{\partial \left( { - a_{n} \cdot f_{n} \cdot sin\left( {f_{n} \cdot x} \right) + a_{n} \frac{{h_{1} }}{\lambda }cos\left( {f_{n} \cdot x} \right)} \right)}}{\partial x} = - f_{n}^{2} \cdot a_{n} \cdot \left( {cos\left( {f_{n} \cdot x} \right) + \frac{{h_{1} }}{{\lambda \cdot f_{n} }}sin\left( {f_{n} \cdot x} \right)} \right) \hfill \\ \end{aligned} \right.$$

\(\Rightarrow\)

$$\frac{{\partial^{2} A_{n} \left( x \right)}}{{\partial x^{2} }} = - f_{n}^{2} \cdot A_{n} \left( x \right).$$
(A19)

Replacement of Eq. A19 in Eq. A18:

$$p\left( {x,t} \right) = \sum\limits_{n = 1}^{\infty } {A_{n} \left( x \right)\left[ {\frac{{\partial B_{n} \left( t \right)}}{\partial t} + a \cdot f_{n}^{2} \cdot B_{n} \left( t \right)} \right]} .$$
(A20)

Then we also develop p(x, t) as a generalized eigenfunctions series of Fourier with a time-dependent coefficient.

$$p\left( {x,t} \right) = \sum\limits_{n = 1}^{\infty } {P_{n} \left( t \right)} \cdot A_{n} \left( x \right)$$
(A21)

We multiply Eq. A21 by An(x):

$$A_{n} \left( x \right) \times p\left( {x,t} \right) = A_{n} \left( {x,\mu } \right) \times \sum\limits_{n = 1}^{\infty } {P_{n} \left( t \right)} \times A_{n} \left( x \right).$$
(A22)

We integrate Eq. A22 from 0 to e:

$$\int\limits_{0}^{e} {A_{n} \left( x \right) \times p\left( {x,t} \right)} dx = \int\limits_{0}^{e} {\left[ {A_{n} \left( x \right) \times \sum\limits_{n = 1}^{\infty } {P_{n} \left( t \right)} \times A_{n} \left( x \right)} \right]} dx.$$
(A23)

Assuming the uniform convergence:

$$\int\limits_{0}^{e} {A_{m} \left( x \right) \times p\left( {x,t} \right)} dx = \sum\limits_{n = 1}^{\infty } {P_{n} \left( t \right)} \int\limits_{0}^{e} {A_{n} \left( x \right)} \times A_{m} \left( x \right)dx.$$
(A24)

The orthogonality relation of the normalized eigenfunctions leads to:

$$\int\limits_{0}^{e} {A_{m} \left( x \right) \times p\left( {x,t} \right)} dx = P_{n} \left( t \right) \cdot \int\limits_{0}^{e} {A_{n}^{2} \left( x \right)} dx, ,$$
(A25)
$$P_{n} \left( t \right) = \int_{0}^{e} {p\left( {x,t} \right).A_{n} \left( x \right)dx} .$$
(A26)

Replacement of Eq. A21 in Eq. A18 gives

$$\sum\limits_{n = 1}^{\infty } {\left( {P_{n} \left( t \right) \cdot A_{n} \left( x \right)} \right)} = \sum\limits_{n = 1}^{\infty } {A_{n} \left( x \right)\left[ {\frac{{\partial B_{n} \left( t \right)}}{\partial t} + a \cdot f_{n}^{2} \cdot B_{n} \left( t \right)} \right]} .$$
(A27)

This leads to

$$\frac{{\partial B_{n} \left( t \right)}}{\partial t} + a \cdot f_{n}^{2} \cdot B_{n} \left( t \right) = P_{n} \left( t \right).$$
(A28)

The time-dependent ODE solution mentioned above (Eq. A28) is presented as follows:

$$B_{n} \left( t \right) = B_{n} \left( 0 \right) \cdot e^{{ - a \cdot f_{n}^{2} \cdot t}} + \int\limits_{0}^{t} {G_{1} \left( {t,\mu } \right)} \cdot P_{n} \left( \mu \right)d\mu .$$
(A29)

In the equation (Eq. A29), τ is the auxiliary time and G1(t, μ) is the first-order Green function defined as follows:

$$G_{1} \left( {t,\mu } \right) = e^{{ - a \cdot f_{n}^{2} \cdot \left( {t - \mu } \right)}} .$$
(A30)

We use the initial condition to obtain the initial coefficient Bn(0):

$$F\left( {x,0} \right) = \sum\limits_{n = 1}^{\infty } {B_{n} \left( 0 \right)} \cdot A_{n} \left( x \right) = 0.$$
(A31)

Assuming uniform convergence of the series and the orthogonality property of the eigenfunctions An(x), we obtain

$$B_{n} \left( 0 \right) = \int_{0}^{e} {F\left( {x,0} \right) \cdot A_{n} \left( x \right)dx} = 0.$$
(A32)

The equation (Eq. A29) becomes

$$B_{n} \left( t \right) = e^{{ - a \cdot f_{n}^{2} \cdot t}} \cdot \int\limits_{0}^{t} {e^{{a \cdot f_{n}^{2} \cdot \mu }} } \cdot P_{n} \left( \mu \right)d\mu .$$
(A33)

Finally, the solution of F(x, t) can be written as follows:

$$F\left( {x,t} \right) = \sum\limits_{n = 1}^{\infty } {A_{n} \left( x \right)} \cdot e^{{ - a \cdot f_{n}^{2} \cdot t}} \cdot \int\limits_{0}^{t} {e^{{a \cdot f_{n}^{2} \cdot \mu }} } \cdot P_{n} \left( \mu \right)d\mu .$$
(A34)

The temperature T(x,t) can be written as follows:

$$\begin{aligned} T\left( {x,t} \right) & = \sum\limits_{n = 1}^{\infty } {a_{n} \left[ {cos\left( {f_{n} \cdot x} \right) + \frac{{h_{1} }}{{\lambda \cdot f_{n} }} \cdot sin\left( {f_{n} \cdot x} \right)} \right]} \cdot e^{{ - a \cdot f_{n}^{2} \cdot t}} \cdot \int\limits_{0}^{t} {e^{{a \cdot f_{n}^{2} \cdot \mu }} } \cdot P_{n} \left( \mu \right)d\mu \\ & \quad + \frac{{ - g\left( t \right)\frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}x + \frac{g\left( t \right)}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}} \\ \end{aligned}$$
(A35)

The temperature rise on the front face(x = 0) is written as

$$T\left( {0,t} \right) = \frac{g\left( t \right)}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}} + \sum\limits_{n = 1}^{\infty } {a_{n} } \cdot e^{{ - a \cdot f_{n}^{2} \cdot t}} \cdot \int\limits_{0}^{t} {e^{{a \cdot f_{n}^{2} \cdot \mu }} } \cdot P_{n} \left( \mu \right)d\mu$$
(A36)
$$T\left( {0,t} \right) = \sum\limits_{n = 1}^{\infty } {\frac{{ - 2\left( {\frac{{e^{2} \cdot T_{ma} }}{{t_{d} \cdot a}}} \right) \cdot \left( {\left( {Bi_{2} } \right)^{2} + \left( {e \cdot f_{n} } \right)^{2} } \right) \cdot \left[ {e^{{ - f_{n}^{2} a \cdot t}} - e^{{f_{n}^{2} a \cdot \left( {t_{d} - t} \right)}} } \right]}}{{\left( {e \cdot f_{n} } \right)^{4} + \left( {e \cdot f_{n} } \right)^{2} \left[ {\left( {Bi_{1} } \right)^{2} + \left( {Bi_{2} } \right)^{2} + Bi_{1} + Bi_{2} } \right] + \left[ {\left( {Bi_{1} } \right)^{2} \times \left( {Bi_{2} } \right)^{2} + \left( {Bi_{2} } \right) \times \left( {Bi_{1} } \right)^{2} + \left( {Bi_{1} } \right) \times \left( {Bi_{2} } \right)^{2} } \right]}}}$$
(A37)

The rise temperature on the rear face (x = e) is written in the following form:

$$\begin{aligned} T\left( {e,t} \right) & = \sum\limits_{n = 1}^{\infty } {a_{n} \left[ {cos\left( {f_{n} \cdot e} \right) + \frac{{h_{1} }}{{\lambda \cdot f_{n} }} \cdot sin\left( {f_{n} \cdot e} \right)} \right]} \cdot e^{{ - a \cdot f_{n}^{2} \cdot t}} \cdot \int\limits_{0}^{t} {e^{{a \cdot f_{n}^{2} \cdot \mu }} } \cdot P_{n} \left( \mu \right)d\mu \\ & \quad + \frac{{ - g\left( t \right)\frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}}e + \frac{g\left( t \right)}{{\frac{{h_{1} }}{\lambda } + \frac{{h_{2} }}{{\lambda \left( {1 + \frac{{h_{2} }}{\lambda } \cdot e} \right)}}}} \\ \end{aligned}$$
(A38)
$$T\left( {e,t} \right) = \sum\limits_{n = 1}^{\infty } {\frac{{ - 2\left( {\frac{{e^{2} \cdot T_{ma} }}{{t_{d} \cdot a}}} \right) \cdot \left( {\left( {Bi_{2} } \right)^{2} + \left( {e \cdot f_{n} } \right)^{2} } \right) \cdot \left[ {\cos \left( {f_{n} \cdot e} \right) + \left( {\frac{{Bi_{1} }}{{e \cdot f_{n} }}} \right) \cdot sin\left( {f_{n} \cdot e} \right)} \right] \cdot \left[ {e^{{ - f_{n}^{2} a \cdot t}} - e^{{f_{n}^{2} a \cdot \left( {t_{d} - t} \right)}} } \right]}}{{\left( {f_{n} \cdot e} \right)^{4} + \left( {f_{n} \cdot e} \right)^{2} \left[ {\left( {Bi_{1} } \right)^{2} + \left( {Bi_{2} } \right)^{2} + Bi_{1} + Bi_{2} } \right] + \left[ {\left( {Bi_{1} } \right)^{2} \times \left( {Bi_{2} } \right)^{2} + \left( {Bi_{2} } \right) \times \left( {Bi_{1} } \right)^{2} + \left( {Bi_{1} } \right) \times \left( {Bi_{2} } \right)^{2} } \right]}}}$$
(A39)

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Chihab, Y., Garoum, M. & Laaroussi, N. A New Efficient Formula for the Thermal Diffusivity Estimation from the Flash Method Taking into Account Heat Losses in Rear and Front Faces. Int J Thermophys 41, 118 (2020). https://doi.org/10.1007/s10765-020-02704-w

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