(435g) a Forward Modeling Approach for the Inverse Estimation of Transient Local Heat Fluxes

A forward modeling approach for the inverse estimation of transient
local heat fluxes

Jiu LUO¹,
Ya-Qiao WANG¹, Jia-Li LUO¹, Dong-Chuan MO¹, Yuan-Xiang
FU¹,

Shu-Shen LYU¹,
Hiroyuki OZOE², Yi HENG¹*

1.
School of Chemical Engineering and Technology, Sun Yat-sen University,
Guangzhou, China

2. Kyushu
University, Kasuga Koen 6-1, Kasuga, Fukuoka, 816-8580, Japan

          *
Corresponding author. E-mail address: hengyi@mail.sysu.edu.cn (Yi Heng)

The efficient and robust solution
of
inverse
heat conduction problems (IHCP)
has received
enormous research interest due to its
extensive applications in numerous branches in the last several decades. However, IHCP are often ill-posed
in the sense of Hadamard^[1]. Hence, it is hard to obtain physically stable
solutions. Furthermore,
the solution techniques for IHCP are mostly
limited to the transient solution in one or two space dimensions^[2-5]
or the steady-state solution for regular
computational domains in space of three dimensions (3D)^[6]
due to computational obstacles.

In this work, a benchmark 3D transient IHCP arising in Chemical Engineering is
considered, namely the estimation
of unmeasurable transient
local boiling heat
fluxes on micro/nano porous copper surfaces
from available temperature readings. It is known
that micro/nano surface modification can be used for the enhancement of boiling
heat transfer^[7,8]. An example of modified micro/nano porous copper surface prepared by us and the mechanism for the bubble
departure in a real experiment are shown in Fig. 1 and Fig. 2, respectively. To
the best of our knowledge, few studies addressed the estimation of unknown transient
local boiling heat fluxes. Together with the optical probe information obtained
in the liquid side, the heat-flux distribution and dynamics evaluated by
solving large-scale IHCP is supposed to provide a
data basis for the development of realistic mechanistic heat transfer models
for boiling regimes beyond nucleate boiling. The IHCP solution approach also
has a wide range of applications in other engineering disciplines such as the
estimation of transient heat flux on slab for optimal control of temperatures
in a reheating furnace^[9,10], the
reconstruction of time-and space-dependent boundary heat flux in alloy
solidification problems^[11] and the recovery of high intensity
periodic laser heat flux on the front surface of a target object^[12].

Figure
1 Modified micro/nano porous                Figure 2 Mechanism for the bubble
departure

copper surface (adopted from
[8])                   of modified one (adopted from [8])

A forward modeling approach is
presented in this work for the efficient solution of the abovementioned
benchmark IHCP. As a
first step the temperature field is simulated on the surface boundary by using
a first order approximation derived by a Taylor expansion. The original IHCP
can be transformed into a well-posed forward heat conduction problem in the
second step. The unknown heat flux is then estimated by the solution of the forward problem using the numerical finite-difference
formulae. In contrast to the optimization-based solution method^[13], the proposed forward modeling approach
shows its high computational efficiency and its great superiority to resolve
the heat-flux discontinuity. Besides, it is easy to implement
and it is platform-independent. In this work, Comsol Mutiphysics is used to solve
the arising forward heat conduction problems. Other commercial softwares such
as CFX, Fluent, or open source codes DROPS, NGSolve are also applicable.

A
simulation case study is presented
to demonstrate the efficiency and accuracy of the proposed forward modeling approach. A
thin heater of is considered. Micro/nano porous structure is modeled as
an inverted cone to reflect the sophisticated geometric structure in a real pool
boiling experiment (see Fig. 3).
Inspired by the
microlayer theory^[14] that predicted that heat during boiling is mainly
transferred in the local area of the three-phase contact line by evaporation, a ring-shaped transient
heat flux is used for the simulation (see Fig. 4, bottom). By considering this heat flux as
the Neumann boundary condition of the forward heat
conduction problem, the temperature measurements are generated on a very fine mesh. This finite element space discretization results in
a total number of
54133 tetrahedra. 50 time steps of 0.001s are
considered for the time discretization to simulate a time
period of 0.05s.
Exact and estimated local
transient heat fluxes at a representative time
instant t=0.025s are compared. The
main temporal and spatial characteristics of the simulated heat flux can be
captured (see Fig. 4, top).

Figure 3
Geometrical model for the            Figure 4 Exact and estimated surface heat

micro/nano
structure                        fluxes at time instant t=0.025s

Further
simulations to address the heat-flux discontinuity problems are also considered
in this work. Huang et al.^[15] estimated
the heat flux with leaps by the Conjugate Gradient (CG)
method. The simulated and estimated heat fluxes for noise-free temperature data by Huang et al.^[15]are shown in
Fig. 5. In another case study Cui et al.^[4] tested a
nondifferentiable heat-flux function in a 1D IHCP, namely the heat flux
function in this case study is time dependent and space independent. For comparison, the results produced by our solution approach
for the same problems are shown in Fig. 6 and Fig. 7, respectively.
Together with the results of the abovementioned inverse boiling problem, these
two case studies verified the validity of our approach in
estimating time- and space-dependent heat flux
functions that have leaps.

Figure
5 (1) Exact instantaneous heat flux by Huang et al. (adopted from [15])

    
(2) Estimated instantaneous heat flux by Huang et al. (adopted from [15])

Figure
6 (1) Exact heat flux distribution (t = 12s)               Figure 7 Exact and
estimated heat

 (2) Estimated
heat flux in the present work (t = 12s)             fluxes in the present work

In summary, the
forward modeling approach proposed in this work can transform the IHCP into a
simple-to-solve forward problem. In this way, it is inherently more efficient
than the optimized-based inversion method, which often need to solve dozens or
hundreds of forward problems. The proposed forward modeling approach also shows
its great superiority in estimating the heat-flux functions with leaps. Moreover, due to its efficiency the proposed solution
approach can also be used to yield initial estimates for the optimized-based
inversion method. In this way, high solution accuracy and efficiency can be
ensured simultaneously.

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