(230i) A Time-Space Adaptive Mesh Refinement Strategy for the Inverse Estimation of Transient Local Heat Flux

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

Inverse heat conduction problems (IHCP) have been studied extensively over the last
decades in different engineering fields such as steel quenching [1], an
intumescent fire retardant paint [2], pool boiling [3], falling film [4] and reheating furnace [5]. Among all these studies, the efficient
solution of three dimensional (3D) IHCP in pool boiling has attracted more and
more attention. It is well known that boiling heat transfer is an
efficient and stable method of heat transfer due to its
large latent heat change during phase transition. It is widely
used to many engineering applications [6]. However,
IHCP in pool boiling are typically ill-posed [7] and
computationally expensive, especially if nonlinear, three-dimensional models with high-density mesh are considered. Although
there exist efficient iterative regularization methods [3,8], it is currently
still not possible to reconstruct the boiling heat flux (a typical IHCP in pool
boiling) at a practically reasonable cost because of their computational
bottleneck.

In this work, we perform research towards the goal to
provide effective and near-real-time
evaluation of a special type of benchmark problem, which is a crucial task for
better understanding of the local boiling heat transfer mechanism and the
development of heat-flux soft sensors in practice. This goal can be achieved by
using a forward modeling approach [9] (our recent work), a time-space
adaptive mesh refinement (TSAMR) strategy that extends the space-only AMR
technique [10] (present work) and a high performance computing platform over
multiple nodes (work in progress). Mathematically,
the benchmark 3D transient IHCP arising in pool boiling considered in this work
is as follows:

where
the 3D transient nonlinear forward heat conduction model lacks the information
about unknown boiling heat flux q on the boiling boundary.
This quantity needs to be reconstructed from temperature data on the back sideof the heating foil by using our forward modeling approach [9]. The new
time-space adaptive mesh refinement (TSAMR) strategy proposed in this work is given below.

Step
2, which intensifies regions with
large heat-flux changes and guides the
grid refinement at a time instant t_i, is implemented by

where
a, b are parameters used to control trade-off between peaks and average values.
In this study a, b are chosen to be 0.9 and 3, respectively.

A case study is presented below to demonstrate the efficiency
of the new TSAMR strategy. Inspired by the micro-layer theory [11],
we simulate three local ring-shaped heat flux peaks at the fluid-heater
interface. A thin heating foil of, which corresponds to the 3D computational
domain, is considered for the numerical computation. The simulation time is 0.1 s, and the numerical
time step is 0.001s. The simulated three heat-flux peaks in time are:

and

and

The
simulated heat flux on the boiling surface is mathematically equivalent to q= q₁+ q₂+ q₃. Its spatial and temporal components are shown in Figure 1. The
solution of the forward problem (1) with the well-defined heat flux q on a
sufficiently fine mesh leads to the temperature data that is used for the IHCP
solution by means of the forward modeling approach [9].

Figure
1: (1) Locations of the three heat-flux peaks; (2) Temporal dynamics of the heat-flux peaks.

Figure 2: From left to right: gradually refined meshes used for the
inverse solution at t₁=
0.015 s (a), t₂= 0.035 s (b).

The
multi-level refined meshes used at selected time instants are shown in Figure 2.
The error estimator of heat flux using different refined meshes is defined by

where the weighting factor w is chosen to be 0.2 in this study. The mean error is
mathematically calculated by, where n is the total number of available discrete
points. If E<=3%, the iterative procedure terminates and
we obtain the estimated heat flux for this time instant by using the locally
refined mesh selected by the TSAMR strategy.

Figure 3:
A comparison of the exact heat flux (top row), the inverse estimated
heat flux with a single fine mesh (second row), the estimated heat-flux
distribution using the TSAMR strategy.

The simulation results of the heat-flux estimation are shown
in Figure 3. The estimated
heat-flux peaks at t₁= 0.015s, t₂= 0.035s and t₃= 0.075s
are 0.035MW/m², 0.139MW/m²and 0.104MW/m², respectively
(expected heat-flux peaks are 0.035 MW/m², 0.141MW/m² and 0.106MW/m²). While the heat flux estimated by the new TSAMR approach is very close to the exact
one in shape, scale and position, the computational effort are significantly
reduced (mesh size used at t₁, t₂ and t₃ are ca. 15%,
6% and 2.1%, respectively).
Moreover, the TSAMR-based IHCP solution procedure was also tested for simulated
noisy measurements and results of satisfactory quality were obtained.

Table 1: Computational cost from
a viewpoint of mesh size: a comparison of the use of very fine mesh and TSAMR
strategy.

In summary, the new TSAMR strategy proposed in this work can be used for the efficient heat-flux estimation at multiple bubble
nucleation sites during boiling. On one hand, the size of computational model is much smaller without
loss of solution quality; on the
other hand, together with the forward modeling approach [9], the new TSAMR approach
is valuable for the near-real-time estimation of heat-flux at multiple bubble
nucleation sites. Besides the
application in boiling heat transfer, the method can also be
applied in other fields that have similar heat-flux estimation tasks. In the future work, the new TSAMR strategy will
be further analyzed mathematically, especially in the refinement strategy (2)
and the error estimator (6). Future work will also include the multi-nodes
scalability analysis for the IHCP and TSAMR codes and we aim to further improve
the computational efficiency of the heat flux estimation procedure, and our
long-term goal is to develop an efficient heat-flux soft sensor that can
satisfy the near-real-time estimation tasks arising in practice.

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