Research Papers

Solution of Inverse Problems in Thermal Systems

[+] Author and Article Information
Yogesh Jaluria

Mechanical Engineering Department,
Rutgers University,
Piscataway, NJ 08854
e-mail: jaluria@jove.rutgers.edu

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Thermal Science and Engineering Applications. Manuscript received October 3, 2018; final manuscript received December 7, 2018; published online June 6, 2019. Assoc. Editor: Ziad Saghir.

J. Thermal Sci. Eng. Appl 12(1), 011005 (Jun 06, 2019) (11 pages) Paper No: TSEA-18-1485; doi: 10.1115/1.4042353 History: Received October 03, 2018

A common occurrence in many practical systems is that the desired result is known or given, but the conditions needed for achieving this result are not known. This situation leads to inverse problems, which are of particular interest in thermal processes. For instance, the temperature cycle to which a component must be subjected in order to obtain desired characteristics in a manufacturing system, such as heat treatment or plastic thermoforming, is prescribed. However, the necessary boundary and initial conditions are not known and must be determined by solving the inverse problem. Similarly, an inverse solution may be needed to complete a given physical problem by determining the unknown boundary conditions. Solutions thus obtained are not unique and optimization is generally needed to obtain results within a small region of uncertainty. This review paper discusses several inverse problems that arise in a variety of practical processes and presents some of the approaches that may be used to solve them and obtain acceptable and realistic results. Optimization methods that may be used for reducing the error are presented. A few examples are given to illustrate the applicability of these methods and the challenges that must be addressed in solving inverse problems. These examples include the heat treatment process, unknown wall temperature distribution in a furnace, and transport in a plume or jet involving the determination of the strength and location of the heat source by employing a few selected data points downstream. Optimization of the positioning of the data points is used to minimize the number of samples needed for accurate predictions.

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Fig. 1

(a) Furnace for optical fiber drawing, indicating the important transport mechanisms and (b) finite zones in the preform and fiber for the calculation of radiative heat transfer in the glass

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Fig. 2

(a) Temperature variation needed for an annealing process, (b) a batch annealing furnace for steel, and (c) modeling of some components of the furnace

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Fig. 3

(a) Comparison between numerical and experimental results on temperatures in the steel coils and (b) comparison between the inverse solution and experimental data on the control sensor temperature

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Fig. 4

(a) Schematic of an experimental system for measuring the temperature distribution in a rod located axially in an optical fiber drawing furnace and (b) computed furnace wall temperature distributions (solid line) from graphite rod data. Experimental points are for a 1.27 cm diameter rod. (c) Resulting furnace wall temperature distribution from the inverse solution and the measured furnace wall control sensor temperature.

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Fig. 5

Two-dimensional buoyant jet in a crossflow, along with locations for measurement and simulation to predict discharge conditions

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Fig. 6

Experimental arrangement for generating a two-dimensional buoyant jet in a free stream flow

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Fig. 7

Results from applying the inverse solution methodology to the flow circumstance of Fig. 5

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Fig. 8

Experimental arrangement and the computational domain for heat sources in a channel flow

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Fig. 9

Forward solution: (a) isotherms due to a finite-sized heat source in channel flow and (b) comparisons between simulation and experimental results on temperature profiles

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Fig. 10

(a, b) Narrowing of the region of uncertainty by optimizing the data locations and (c) actual versus predicted source temperature

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Fig. 11

Errors in the inverse solution using arbitrarily distributed data points and optimized distribution or shape

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Fig. 12

Errors in the prediction of source temperature and velocity in a buoyant jet in crossflow

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Fig. 13

(a) Geometry and coordinate system for a wall plume and (b) comparison of numerical results for natural convection boundary layer flow on an isothermal vertical surface with those in the literature

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Fig. 14

Temperature variation with respect to source location (left) and Ra number (right)

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Fig. 15

Estimation error diagram for location and source strength, given in terms of the Rayleigh number, Ra, for an isothermal heat source

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Fig. 16

Estimation error diagram for location and Ra number for a uniform heat flux input at the source



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