An inverse problem in simultaneous estimating the Biot numbers of heat and moisture transfer for a porous material

A conjugate gradient method based inverse algorithm is applied in the present study in simultaneous determining the unknown time-dependent Biot numbers of heat and moisture transfer for a porous material based on interior measurements of temperature and moisture.It is assumed that no prior information is available on the functional form of the unknown Biot numbers in the present study, thus, it is classified as the function estimation in inverse calculation.The accuracy of this inverse heat and moisture transfer problem is examined by using the simulated exact and inexact temperature and moisture measurements in the numerical experiments. Results show that the estimation on the time-dependent Biot numbers can be obtained with any arbitrary initial guesses on a Pentium IV 1.4 GHz personal computer.     

The BFGS method for estimating the interface temperature and convection coefficient in ultrasonic welding

An inverse heat transfer problem is investigated in the present study by the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method to predict the unknown time-dependent heat generation at the weld interface and convection heat transfer coefficient during an ultrasonic metal welding process based on the knowledge of temperature measurements taken on the horn. With known temperature data at some locations on the horn, the inverse solution was rapidly obtained by solving nonlinear direct problem, Central Finite Difference and Simple Step Method. The proposed method which did not need solving adjoint and sensitivity problem revealed the characteristics of high efficiency, lower iterations for a computational algorithm and high accuracy for estimating values even when measurement error was considered. Besides, a comparison of the BFGS method with some previous methods (i.e. CGM, SCGM) was established. These results show that an excellent estimation on interfacial heat generation (or temperature), as well as a convection heat transfer coefficient, can be simultaneously obtained in this study. The current methodology will provide a useful tool to optimize welding conditions in ultrasonic welding.     

A three-dimensional inverse problem in imaging the local heat transfer coefficients for plate finned-tube heat exchangers

A three-dimensional inverse heat conduction problem in imaging the local heat transfer coefficients for plate finned-tube heat exchangers utilizing the steepest descent method and a general purpose commercial code CFX4.4 is applied successfully in the present study based on the simulated measured temperature distributions on fin surface by infrared thermography.It is assumed that no prior information is available on the functional form of the unknown local heat transfer coefficients in the present study. Thus, it can be classified as function estimation for the inverse calculations.Two different heat transfer coefficients for in-line tube arrangements with different measurement errors are to be estimated. Results show that the present algorithm can obtain the reliable estimated heat transfer coefficients.     

Estimation of the Transient Heat Transfer Rate at the Boundary of an Electronic Chip Packaging

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat transfer rate at the electronic-packaging/heat-sink-assembly interface from the knowledge of temperature measurements taken within the packaging. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat transfer rate can be obtained for the test case considered here.     

An inverse problem in estimating the base heat flux of an annular fin based on the hyperbolic model of heat conduction

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent base heat flux of an annular fin from the knowledge of temperature measurements taken within the fin. The inverse solutions will be justified based on the numerical experiments in which two specific cases to determine the unknown base heat flux are examined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent base heat flux can be obtained for the test cases considered in this study.     

Inverse Thermo-Mechanical Analysis of a Functionally Graded Fin

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent base heat flux of a functionally graded fin from the knowledge of temperature measurements taken within the fin. Subsequently, the distributions of temperature and thermal stresses in the fin can be determined as well. It is assumed that no prior information is available on the functional form of the unknown base heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors and measurement location upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent base heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Estimation of Heat Flux and Thermal Stresses in Functionally Graded Hollow Circular Cylinders

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat flux at the inner surface of a functionally graded hollow circular cylinder from the knowledge of temperature measurements taken within the cylinder. Subsequently, the distributions of temperature and thermal stresses in the cylinder can be determined as well. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Function estimation of laser-induced heat generation in a gas-saturated powder layer heated by a short-pulsed laser

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse heat conduction problem with a dual-phase-lag equation for estimating the unknown space- and time-dependent laser-induced heat generation in a gas-saturated porous medium exposed to short-pulse laser heating from the temperature measurements taken within the medium. Subsequently, the powder particle temperature distributions in the porous medium can be determined as well. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The effect of measurement errors on the estimation accuracy is also investigated. The inverse solutions are justified based on the numerical experiments in which two different forms of heat generation are estimated. Results show that the unknown laser-induced heat generation can be predicted precisely by using the present approach for the test cases considered in this study.     

An inverse problem of parameter estimation for heat and mass transfer in capillary porous media

This work deals with the solution of an inverse problem of parameter estimation involving heat and mass transfer in capillary porous media, as described by the dimensionless linear Luikov’s equations. The physical problem under picture involves the drying of a moist porous one-dimensional medium. The main objective of this paper is to simultaneously estimate the dimensionless parameters appearing in the formulation of the physical problem by using transient temperature and moisture content measurements taken inside the medium. The inverse problem is solved by using the Levenberg-Marquardt method of minimization of the least-squares norm with simulated measurements.     

Estimation of heat generation at the interface of cylindrical bars during friction process

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat generation at the interface of cylindrical bars during friction process from the knowledge of temperature measurements taken within the bar. It is assumed that no prior information is available on the functional form of the unknown heat generation; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat generation can be obtained for the test case considered in this study. The current methodology can be applied to the prediction of heat generation in continuous-drive friction welding or in breaking systems.     

An iterative regularization method in estimating the transient heat-transfer rate on the surface of the insulation layer of a double circular pipe

In this study, a conjugate gradient method based inverse algorithm is applied to estimate the unknown space- and time-dependent heat-transfer rate on the surface of the insulation layer of a double circular pipe heat exchanger using temperature measurements. It is assumed that no prior information is available on the functional form of the unknown heat-transfer rate; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The accuracy of the inverse analysis is examined by using simulated exact and inexact temperature measurements. Results show that an excellent estimation on the space- and time-dependent heat-transfer rate can be obtained for the test case considered in this study.     

Coupled heat and mass transfer in a counter flow hollow fiber membrane module for air humidification

Hollow fiber membrane based air humidification offers great advantages over the traditional methods because the liquid water droplets are prevented from mixing with the process air, while water vapor can permeate through the membranes effectively. The novelty in this research is that the coupled heat and moisture transport in a hollow fiber membrane module for air humidification is investigated, both numerically and experimentally. The air stream and the water stream flow in a counter flow arrangement. It is found that the membranes play a key role in humidification performances. For sensible heat transfer, both the liquid side and the membrane side resistance can be neglected, while the total heat transfer coefficients are determined by the air side heat transfer coefficients. In contrast, in mass transfer, only the liquid side resistance can be neglected, while the total mass transfer coefficients are co-determined by membrane properties and the air side convective mass transfer coefficients.     

Estimation of heat flux on the surface of an initially hot cylinder cooled by a laminar confined impinging jet

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown space-and time-dependent heat flux at the surface of an initially hot cylinder cooled by a laminar confined slot impinging jet from the knowledge of temperature measurements taken on the cylinder’s surface. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. The results show that an excellent estimation on the space-and time-dependent heat flux can be obtained even the distributions of thermal properties inside the cylinder is unknown.     

Optimal Heat Input for Estimating Luikov's Parameters in a Heat and Mass Transfer Problem

In this article, the problem of computing an optimal heat input in Luikov's heat and mass transfer problem is detailed and analyzed. The main objective is the establishment of an optimal time-dependent heat flux profile with the goal of maximizing the temperature and moisture sensitivities of some parameters to this excitation in a drying process. Such maximization makes the estimation of the desired parameters possible, easier, and with limited uncertainty intervals. It also helps to reduce the linearity dependence between the parameters of interest and the number of temperature and moisture sensors used. The estimation of the optimal heat input is obtained with Uzawa's algorithm, while the estimation of parameters is performed with Levenberg-Marquardt's method of minimization of the ordinary least-square criterion. The six dimensionless parameters characterizing Luikov's equations are estimated successfully with this optimal heat flux profile, which also helps to reduce the number of both temperature and moisture sensors needed in the estimation procedure. By doing so, the objective of estimating simultaneously the six parameters which appear in the formulation of Luikov's physical problem is reached by using a limited transient temperature and/or moisture measurements taken anywhere in the drying medium.     

Heat and mass transfer in plate-fin sinusoidal passages with vapor-permeable wall materials

Laminar forced flow and heat mass transfer in sinusoidal plate-fin small passages encountered in compact heat mass exchangers are investigated. The duct is similar to a traditional plate-fin heat exchanger, but vapor-permeable materials like polymer membranes, paper, and ceramics can be used as the duct materials so both sensible heat and moisture can be exchanged simultaneously. Heat conduction and mass diffusion in the fins and heat and moisture convection in the fluid are analyzed simultaneously as a conjugate problem. Their fully developed Nusselt and Sherwood numbers under various aspect ratios and fin conductance parameters are calculated. The results found that though fins extend the heat transfer area, they are less effective compared to a traditional compact heat exchanger with metal foils. Most unfortunately, fin efficiencies for moisture transfer are even much smaller than those for heat transfer due to the low fin mass conductance parameters. For such heat mass exchangers, the use of fins can be regarded mostly as supporting materials, rather than as mass intensification techniques.     

Three dimensional numerical modeling of simultaneous heat and moisture transfer in a moist object subjected to convective drying

The drying behavior of a moist object subjected to convective drying is analyzed numerically by solving heat and moisture transfer equations. A 3-D numerical model is developed for the prediction of transient temperature and moisture distribution in a rectangular shaped moist object during the convective drying process. The heat transfer coefficients at the surfaces of the moist object are calculated with an in-house computational fluid dynamics (CFD) code. The mass transfer coefficients are then obtained from the analogy between the thermal and concentration boundary layer. Both these transfer coefficients are used for the convective boundary conditions while solving the simultaneous heat and mass transfer governing equations for the moist object. The finite volume method (FVM) with fully implicit scheme is used for discretization of the transient heat and moisture transfer governing equations. The coupling between the CFD and simultaneous heat and moisture transfer model is assumed to be one way. The effect of velocity and temperature of the drying air on the moist object are analyzed. The optimized drying time is predicted for different air inlet velocity, temperature and moisture content. The drying rate can be increased by increasing the air flow velocity. Approximately, 40% of drying time is saved while increasing the air temperature from 313 to 353 K. The importance of the inclusion of variable surface transfer coefficients with the heat and mass transfer model is justified.     

Identification of wall heat flux for turbulent forced convection by inverse analysis

An inverse problem for turbulent forced convection between parallel flat plates is investigated. The space- and time-dependent heat flux at the upper wall is estimated from the temperature measurements taken inside the flow. In the present study, the conjugate gradient method is adopted for the estimation of the unknown wall heat flux. No prior information is needed for the functional form of the wall heat flux in the inverse analysis. The effects of the measurement errors, the functional form of the wall heat flux, and the location of the sensors on the accuracy of the estimation are investigated. The reconstruction of the wall heat flux is satisfactory when simulated exact or noisy data are input to the inverse analysis. The sensitivity coefficients are discussed in this paper. As expected, it is shown that the accuracy of the estimation can be improved when the sensors are located closer to the upper wall.     

Numerically solving twofold ill-posed inverse problems of heat equation by the adjoint Trefftz method

The inverse problem endowing with multiple unknown functions gradually becomes an important topic in the field of numerical heat transfer, and one fundamental problem is how to use limited minimal data to solve the inverse problem. With this in mind, in the present article we search the solution of a general inverse heat conduction problem when two boundary data on the space-time boundary are missing and recover two unknown temperature functions with the help of a few extra measurements of temperature data polluted by random noise. This twofold ill-posed inverse heat conduction problem is more difficult than the backward heat conduction problem and the sideways heat conduction problem, both with one unknown function to be recovered. Based on a stable adjoint Trefftz method, we develop a global boundary integral equation method, which together with the compatibility conditions and some measured data can be used to retrieve two unknown temperature functions. Several numerical examples demonstrate that the present method is effective and stable, even for those of strongly ill-posed ones under quite large noises.     

Recovering both the space-dependent heat source and the initial temperature by using a fast convergent iterative method

In this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green’s second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome the highly ill-posed behavior of the second inverse problem, of which the convergence rate and stability are examined. This results in quite accurate numerical results against large noise.     

Estimation of heat flux and temperature distributions in a composite strip and homogeneous foundation

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat flux and temperature distributions for the system composed of a multi-layer composite strip and semi-infinite foundation, from the knowledge of temperature measurements taken within the strip. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. Results show that an excellent estimation on the time-dependent heat flux can be obtained for the test case considered in this study.