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The method of lines with numerical differentiation of the sequential temperature–time histories for a facile solution of 1-D inverse heat conduction problems

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Abstract

This work addresses an inverse heat conduction problem (IHCP) in a large planar slab receiving a certain heat flux at one surface while the other surface is thermally insulated. The two different heating conditions to be studied are: (1) constant heat flux and (2) a time-dependent triangular heat flux. For the IHCP, the temperature–time variations at the insulated surface of the large planar slab are obtained with a single temperature sensor. The two temperature–time histories are generated from the solutions of the direct heat conduction problem. The central objective of the paper is to implement the method of lines for the descriptive 1-D heat equation combined with numerical differentiation of: (1) the “measured” temperature–time history at the insulated surface and (2) the temperature–time history at other sub-surface locations. In the end, it is confirmed that excellent predictions of the temperature–time variations at the directly heated surface are obtainable for the two dissimilar heating conditions. This is accomplished with small systems of first-order differential-difference equations, one with two equations and the other with four equations.

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Notes

  1. There are two possible options: one is the forward difference formulation with truncation error of order Δτ continually, and the other is to start with the forward difference formulation with truncation error of order Δτ and immediately switch to the central difference formulation with truncation error of order (Δτ)2 [28].

Abbreviations

c v :

Specific heat at constant volume (J/(kg °C))

k :

Thermal conductivity (W/m °C)

L :

Thickness of large planar slab (m)

q N :

Nominal surface heat flux in Eq. (19) (W/m2)

q R :

Surface heat flux ratio in Eq. (19)

q s :

Surface heat flux (W/m2)

t :

Time (s)

T :

Temperature (°C)

T e :

Equivalent temperature, q s L/k (°C)

T m,i :

“Measured” temperature at the location xm and time ti (°C)

T s :

Surface temperature (°C)

T 0 :

Initial temperature (°C)

x:

Space coordinate (m)

X:

Dimensionless space coordinate, \( \frac{x}{L} \)

α:

Thermal diffusivity, \( \frac{k}{\rho \,c} \) (m2/s)

\( \phi \) :

Dimensionless temperature, \( \frac{{T - T_{0} }}{{q_{s} L/k}} \) and \( \frac{{T - T_{0} }}{{q_{N} L/k}} \)

ρ:

Density (kg/m3)

τ:

Dimensionless time or Fourier number, \( \frac{t}{{L^{2} /\alpha }} \)

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Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Texas at San Antonio, San Antonio, TX, 78249, USA

    Antonio Campo

  2. Department of Mechanical Engineering, Isfahan University of Technology, 84156-83111, Isfahan, Iran

    Mohammad R. Salimpour

  3. Department of Mechanical Engineering, University of Vermont, Burlington, VT, 05405, USA

    John Ho

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  1. Antonio Campo
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  2. Mohammad R. Salimpour
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  3. John Ho
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Correspondence to Antonio Campo.

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Campo, A., Salimpour, M.R. & Ho, J. The method of lines with numerical differentiation of the sequential temperature–time histories for a facile solution of 1-D inverse heat conduction problems. Heat Mass Transfer 49, 369–379 (2013). https://doi.org/10.1007/s00231-012-1088-5

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