One-dimensional Heat Conduction Domain Research Articles

Inverse determination of thermal conductivity using semi-discretization method

In this work a semi-discretization method is presented for the inverse determination of spatially- and temperature-dependent thermal conductivity in a one-dimensional heat conduction domain without internal temperature measurements. The temperature distribution is approximated as a polynomial function of position using boundary data. The derivatives of temperature in the differential heat conduction equation are taken derivative of the approximated temperature function, and the derivative of thermal conductivity is obtained by finite difference technique. The heat conduction equation is then converted into a system of discretized linear equations. The unknown thermal conductivity is estimated by directly solving the linear equations. The numerical procedures do not require prior information of functional form of thermal conductivity. The close agreement between estimated results and exact solutions of the illustrated examples shows the applicability of the proposed method in estimating spatially- and temperature-dependent thermal conductivity in inverse heat conduction problem.

ESTIMATION OF TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY AND HEAT CAPACITY PER UNIT VOLUME WITH A SIMPLE INTEGRAL APPROACH

We present an integral approach to estimate temperature-dependent thermal conductivity and heat capacity per unit volume simultaneously in a transient non-linear heat conduction medium. It is assumed that the thermal property is a monotonic function of the temperature and it can be expressed as a piecewise linear function. Hence, the present inverse heat conduction problem is converted to a parameter identification problem determining the unknown coefficients of the model functions of the thermal conductivity and heat capacity per unit volume. The spatial temperature distribution required for the integral approach is modeled as a third-order function of position in a one-dimensional heat conduction domain with heated and insulated walls. Four coefficients of the approximated temperature distribution are determined from the heat fluxes imposed and the temperatures measured at both ends at each measurement. Hence, the proposed algorithm does not require any internal measurements

Direct estimation of time-dependent boundary heat flux

We present a non-iterative direct method to estimate time-varying boundary heat flux in a one-dimensional heat conduction domain with heated and insulated walls. The proposed method relates the temporal variation of the surface heat flux to the temperature variation at the heated surface based on the analytical temperature distribution. Several illustrative examples are examined to verify the performance of the proposed method. The uncertainty analysis of the estimated heat fluxes is discussed. The proposed method may also be useful to make sufficiently accurate initial guesses for more sophisticated method which usually adopt the iterative scheme

Inverse estimation of time-dependent boundary heat flux with an adaptive input estimator

We present preliminary results from the adaptive input estimator developed to estimate time-dependent boundary heat flux in a one-dimensional heat conduction domain with heated and insulated walls. For the estimation, the algorithm requires only the temperatures measured at the insulated wall. The estimator is so composed of Kalman filters connected in parallel as to reconstruct the unknown input effectively. Computer simulation results reveal that the proposed adaptive input estimator has improved estimation performance even for step-changing heat flux

Estimation of temperature-dependent thermal conductivity with a simple integral approach

We present an integral approach to estimate temperature-dependent thermal conductivity in a transient non-linear heat conduction medium. It is assumed that the thermal conductivity is a monotonic function of the temperature and it can be expressed as a piecewise linear function. Hence, the present inverse heat conduction problem is converted to a parameter identification problem determining the unknown coefficients of the thermal conductivity function that is approximated to be piecewise linear. The spatial temperature distribution required for the integral approach is modeled as a third-order function of position in a one-dimensional heat conduction domain with heated and insulated walls. Four coefficients of the approximated temperature distribution are determined from the heat fluxes imposed and the temperatures measured at both ends at each measurement. Hence, the proposed algorithm does not require any internal measurements. Numerical experiments are successfully introduced to verify the present approach. The proposed method may also be useful to make sufficiently accurate initial guesses for the inverse heat conduction problem to determine the thermal conductivity having an arbitrary functional form

Non-iterative estimation of temperature-dependent thermal conductivity without internal measurements

In this work a direct integration method is proposed to estimate temperature-dependent thermal conductivity in a one-dimensional heat conduction domain without internal measurements. By approximating the spatial temperature distribution in the domain as a third-order polynomial of position and by integrating the heat conduction equation over the spatial and temporal domain, the present method estimates the thermal conductivity directly. Also, this method does not require any prior information on the functional form of the thermal conductivity. Some illustrative examples are examined to verify the proposed approach. The proposed approach may also be useful to make sufficiently accurate initial guesses for sophisticated algorithms usually based on iterative refinement scheme.

A direct integration approach for the estimation of time-dependent boundary heat flux

In a one-dimensional heat conduction domain with heated and insulated walls, an integral approach is proposed to estimate time-dependent boundary heat flux without internal measurements. It is assumed that the expression of the heat flux is not known a priori. Hence, the present inverse heat conduction problem is classified as a function estimation problem. The spatial temperature distribution is approximated as a third-order polynomial of position, whose four coefficients are determined from the heat fluxes and the temperatures at both ends at each measurement. After integrating the heat conduction equation over spatial and time domain, respectively, a simple and non-iterative recursive equation to estimate the time-dependent boundary heat flux is derived. Several examples are introduced to show the effectiveness of the present approach.

An integral approach to the inverse estimation of temperature-dependent thermal conductivity without internal measurements

An integral approach is developed to estimate temperature-dependent thermal conductivity without internal measurements. It is assumed that the thermal conductivity can be expressed as a linear function. Hence, the present inverse heat conduction problem is converted to a parameter estimation problem that determines the unknown coefficients of the thermal conductivity function. In a one-dimensional heat conduction domain with heated and insulated walls, the spatial temperature distribution is modeled as a third-order function of position, whose four coefficients are determined from the heat fluxes and the temperatures at both ends at each measurement. Several examples are successfully introduced to verify the present approach. The proposed approach may also be useful to make sufficiently accurate initial guesses for the inverse heat conduction problem to determine the thermal conductivity having an arbitrary function form.

Second-order finite difference approximation for inverse determination of thermal conductivity

A second-order finite difference procedure is presented for the inverse determination of the thermal conductivity in a one-dimensional heat conduction domain. In this approach, the thermal conductivity of the material is reconstructed by using the available temperature data at discrete grid points. The accuracy of the computational algorithm is investigated. To confirm the validity of the numerical procedure, various comparative examples are presented. The close agreement between the current results and the exact solutions confirms that the proposed finite difference procedure is an effective technique for the inverse determination of thermal conductivity. The method is applicable for linear and nonlinear spatially dependent, as well as temperature dependent, thermal conductivities. In addition, a key feature of the present technique is that a priori knowledge of the functional form for the thermal conductivity is not required.

Inverse determination of thermal conductivity for one-dimensional problems

Two finite difference procedures are presented for the inverse determination of the thermal conductivity in a one-dimensional heat conduction domain. The thermal conductivity is reconstructed from the inverse analysis based on the assumption that the temperature measurements are either available continuously over the entire domain or at discrete grid points. The convergence and stability of the computational algorithms are investigated. It is concluded that both procedures are first-order accurate methods. A comparison of the exact thermal conductivity with the one estimated was made to confirm the validity of the numerical procedures. The close agreement between the two results confirms that the proposed finite difference techniques are effective procedures for the inverse determination of thermal conductivity in a one-dimensional heat conduction domain. The methods are applicable for linear and nonlinear spatially- as well as temperature-d ependent thermal conductivities. Additionally, the special feature of the present techniques is that a priori knowledge of the functional form for the thermal conductivity is not mandatory. Nomenclature C = arbitrary constant F = function defined by Eq. (10) / = function defined at the boundary G = function defined by Eq. (10) g — heat generation, W/nr k = thermal conductivity, W/m-°C kM = initial thermal conductivity calculated at dT/dx — 0 in the discrete formulation, W/m-°C k(} = initial thermal conductivity calculated at dT/dx = 0 in the continuous formulation, W/m-°C q = heat flux, W/m2 T = temperature, °C t = time, s / = selected time in the continuous formulation, s /,- — selected time in the discrete formulation, s A' = spatial coordinate, m XM = spatial coordinate where dT/dx = 0 in the discrete formulation, m A',, = spatial coordinate where dT/dx = 0 in the continuous formulation, m Subscripts /, /, k = indices Superscript = approximated value