TS-134
An Inverse Geometry Problem in Estimating Frost Growth on An Evaporating Tube
Cheng-Ching HUANG and Cheng-Hung HUANG
Department of Naval Architecture and Marine Engineering
National Cheng Kung University
Tainan, Taiwan, 701, R. O. C.
ABSTRACT
When humid air comes into contact with a surface whose temperature is below the dew point of water vapor in air and also below the freezing point frost deposition takes place over the surface. The phenomena of the frost growth is very complicated and therefore it is very difficult to model mathematically the behavior of frost growth and predict it. In the present study a transient inverse geometry heat conduction problem (shape identification problem) is solved using the Conjugate Gradient Method (CGM) and Boundary Element Method (BEM)-based inverse algorithm to estimate the unknown irregular frost thickness and shape.
Results obtained by using the conjugate gradient method to estimate the frost growth are justifieded on the numerical experiments. It is concluded that the accurate frost shape can be estimated by the conjugate gradient method except for the initial and final time The reason and improvement of this singularity are addressed, Finally the effects of reducing the number of sensors and increasing the measurenment errors on the inverse solutions are discussed.
Key Words: Inverse problem, Moving boundary problem, Frost estimation.
I. INTRODUCTION
Frost formation phenomena are encountered in numerous fields of industry dealing with low temperatures, including gas coolers, refrigerators, purification of gases, cryogenics, aeronautics.
The frosting process involves simultaneous heat and mass transfer under unsteady state condition. Several attempts have been made to study the frost growth process on both analytical (or numerical) and experirmental analysis. For example, the analytical (or numerical) prediction can be found in references [1-3], semi-empirical approaches in [4-5] and experimental observation in [6-7]. Recently. LeGall, et al [8] and Lee, et al [9] proposed respectively the analytical model for one-dimensional frost growth on a flat plate with many assumptions and the results are in good agreement with the experimental data. However, the prediction of two-dimensional frost shape is never found in the literature.
The objective of the present study is to propose a new approach using the technique of Inverse Geometry Problem (IGP) and measurement temperatures on the evaporating tube to estimate two-dimensional frost growth (i.e, the thickness of frost is function of positions as well as time on the tube surface).
For the conventional frost estimation problems [1-9], the enthalpy formulation, mass transfer relations and many other empirical and semi-empirical formulas are needed to perform the calculation. Since many assumptions were used, the solution must exist many uncertainties.
However, one may look at the problem from the other viewpoint i.e. from the viewpoint of inverse problem. The idea is that the thickness and shape of the formative frost must effect the measured temperatures on the surface of evaporating tube. By using those measured temperatures (which reflect the influence of the real frost thickness and shape) the shape and thickness of the existence frost can be estimated based on the technique of Inverse Geometry Problems (IGP). Under this consideration the mechanism of frost formation and how the frost growth become totally not relevant to our problem formulation and therefore the enthalpy as well as mass transfer equations are not needed in our mathematical formulations. Only the conduction equation is needed.
The applications of Inverse Geometry Problems can be found in several engineering fields, such as the determination of cooling passages in turbine blades [10], the estimation of optimal hull form in shipbuilding industries [11] and nondestructive evaluation (NDE) [12], etc.
In the recent work by Huang, et al [13], a steady shape identification problem in estimating the shape and locations of multiple cavities has been solved successfully by using the steepest descent method (SDM). The present work extended the previous work [13] to a transient shape identification problem and used conjugate gradient method (CGM) to fasten the rate of convergence. Moreover in the present study, the thermal resistance of evaporating tube is rather small in comparison with the frost, therefore the tube region is neglected.
