In this paper, an algorithm will be presented that enables solving the two-phase inverse Stefan problem, where the additional information consists of temperature measurements in selected points of the solid phase. The problem consists in the reconstruction of the function describing the heat transfer coefficient, so that the temperature in the given points of the solid phase would differ as little as possible from the predefined values. The featured examples of calculations show a very good approximation of the exact solution and stability of the algorithm.
The paper presents the results of calculations related to determination of temperature distributions in a steel pipe of a heat exchanger taking into account inner mineral deposits. Calculations have been carried out for silicate-based scale being characterized by a low heat transfer coefficient. Deposits of the lowest values of heat conduction coefficient are particularly impactful on the strength of thermally loaded elements. In the analysis the location of the thermocouple and the imperfection of its installation were taken into account. The paper presents the influence of determination accuracy of the heat flux on the pipe external wall on temperature distribution. The influence of the heat flux disturbance value on the thickness of deposit has also been analyzed.
A direct problem and an inverse problem for the Laplace’s equation was solved in this paper. Solution to the direct problem in a rectangle was sought in a form of finite linear combinations of Chebyshev polynomials. Calculations were made for a grid consisting of Chebyshev nodes, what allows us to use orthogonal properties of Chebyshev polynomials. Temperature distributions on the boundary for the inverse problem were determined using minimization of the functional being the measure of the difference between the measured and calculated values of temperature (boundary inverse problem). For the quasi-Cauchy problem, the distance between set values of temperature and heat flux on the boundary was minimized using the least square method. Influence of the value of random disturbance to the temperature measurement, of measurement points (distance from the boundary, where the temperature is not known) arrangement as well as of the thermocouple installation error on the stability of the inverse problem was analyzed.
Inverse boundary problem for cylindrical geometry and unsteady heat conduction equation was solved in this paper. This solution was presented in a convolution form. Integration of the convolution was made assuming the distribution of temperature T on the integration interval (ti, ti+ Δt) in the form T (x, t) = T (x, ti) Θ + T (z, ti+ Δt) (1 - Θ), where Θ ϵ (0,1). The influence of value of the parameter Θ on the sensitivity of the solution to the inverse problem was analysed. The sensitivity of the solution was examined using the SVD decomposition of the matrix A of the inverse problem and by analysing its singular values. An influence of the thermocouple installation error and stochastic error of temperature measurement as well as the parameter Θ on the error of temperature distribution on the edge of the cylinder was examined.
Direct and inverse problems for unsteady heat conduction equation for a cylinder were solved in this paper. Changes of heat conduction coefficient and specific heat depending on the temperature were taken into consideration. To solve the non-linear problem, the Kirchhoff’s substitution was applied. Solution was written as a linear combination of Chebyshev polynomials. Sensitivity of the solution to the inverse problem with respect to the error in temperature measurement and thermocouple installation error was analysed. Temperature distribution on the boundary of the cylinder, being the numerical example presented in the paper, is similar to that obtained during heating in the nitrification process.
The article presents the prototype of a measurement system with a hot probe, designed for testing thermal parameters of heat insulation materials. The idea is to determine parameters of thermal insulation materials using a hot probe with an auxiliary thermometer and a trained artificial neural network. The network is trained on data extracted from a nonstationary two-dimensional model of heat conduction inside a sample of material with the hot probe and the auxiliary thermometer. The significant heat capacity of the probe handle is taken into account in the model. The finite element method (FEM) is applied to solve the system of partial differential equations describing the model. An artificial neural network (ANN) is used to estimate coefficients of the inverse heat conduction problem for a solid. The network determines values of the effective thermal conductivity and effective thermal diffusivity on the basis of temperature responses of the hot probe and the auxiliary thermometer. All calculations, like FEM, training and testing processes, were conducted in the MATLAB environment. Experimental results are also presented. The proposed measurement system for parameter testing is suitable for temporary measurements in a building site or factory.
The paper presents investigations related to solving of a direct and inverse problem of a non-stationary heat conduction equation for a cylinder. The solution of the inverse problem in the form of temperature distributions has been obtained through minimization of a functional being the measure of the difference between the values of measured and calculated temperatures in M points of the heated cylinder. The solution of the conduction equation was presented in the convolutional form and then numerically integrated approximating one of the integrand with a step function described with parameter Θ ∈ (0, 1]. The influence of the integration parameter Θ on the obtained solution of the inverse problem (including a number of temperature measurement points inside the heated body) has been analyzed. The influence of the parameter Θ on the sensitivity of the obtained temperature distributions has been investigated.
The paper presents the solution to a problem of determining the heat flux density and the heat transfer coefficient, on the basis of temperature measurement at three locations in the flat sensor, with the assumption that the heat conductivity of the sensor material is temperature dependent. Three different methods for determining the heat flux and heat transfer coefficient, with their practical applications, are presented. The uncertainties in the determined values are also estimated.
The tubular type instrument (flux tube) was developed to identify boundary conditions in water wall tubes of steam boilers. The meter is constructed from a short length of eccentric tube containing four thermocouples on the fire side below the inner and outer surfaces of the tube. The fifth thermocouple is located at the rear of the tube on the casing side of the water-wall tube. The boundary conditions on the outer and inner surfaces of the water flux-tube are determined based on temperature measurements at the interior locations. Four K-type sheathed thermocouples of 1 mm in diameter, are inserted into holes, which are parallel to the tube axis. The non-linear least squares problem is solved numerically using the Levenberg-Marquardt method. The heat transfer conditions in adjacent boiler tubes have no impact on the temperature distribution in the flux tubes.
The aim of this paper is analysis of the possibility of determining the internal structure of the fibrous composite material by estimating its thermal diffusivity. A thermal diffusivity of the composite material was determined by applying inverse heat conduction method and measurement data. The idea of the proposed method depends on measuring the timedependent temperature distribution at selected points of the sample and identification of the thermal diffusivity by solving a transient inverse heat conduction problem. The investigated system which was used for the identification of thermal parameters consists of two cylindrical samples, in which transient temperature field is forced by the electric heater located between them. The temperature response of the system is measured in the chosen point of sample. One dimensional discrete mathematical model of the transient heat conduction within the investigated sample has been formulated based on the control volume method. The optimal dynamic filtration method as solution of the inverse problem has been applied to identify unknown diffusivity of multi-layered fibrous composite material. Next using this thermal diffusivity of the composite material its internal structure was determined. The chosen results have been presented in the paper.
This article presents a new efficient method of determining values of gas flow parameters (e.g. axial dispersion coefficient, DL and Pèclet number, Pe). A simple and very fast technique based on the pulse tracer response is proposed. It is a method which combines the benefits of a transfer function, numerical inversion of the Laplace transform and optimization allows estimation of missing coefficients. The study focuses on the simplicity and flexibility of the method. Calculations were performed with the use of the CAS-type program (Maple®). The correctness of the results obtained is confirmed by good agreement between the theory and experimental data for different pressures and temperature. The CAS-type program is very helpful both for mathematical manipulations as a symbolic computing environment (mathematical formulas of Laplace-domain model are rather sophisticated) and for numerical calculations. The method of investigations of gas flow motion is original. The method is competitive with earlier methods.
A new method for measurement of local heat flux to water-walls of steam boilers was developed. A flux meter tube was made from an eccentric tube of short length to which two longitudinal fins were attached. These two fins prevent the boiler setting from heating by a thermal radiation from the combustion chamber. The fins are not welded to the adjacent water-wall tubes, so that the temperature distribution in the heat flux meter is not influenced by neighbouring water-wall tubes. The thickness of the heat flux tube wall is larger on the fireside to obtain a greater distance between the thermocouples located inside the wall which increases the accuracy of heat flux determination. Based on the temperature measurements at selected points inside the heat flux meter, the heat flux absorbed by the water-wall, heat transfer coefficient on the inner tube surface and temperature of the water-steam mixture was determined.
A method for determining time-optimum medium temperature changes is presented. The heating of the pressure elements will be conducted so that the circumferential stress caused by pressure and fluid temperature variations at the edge of the opening at the point of stress concentration, do not exceed the allowable value. In contrast to present standards, two points at the edge of the opening are taken into consideration. The first point, P1, is located at the cross section and the second, P2, at the longitudinal section of the vessel. It will be shown that the optimum temperature courses should be determined with respect to the total circumferential stress at the point P2, and not, as in the existing standards due to the stress at the point P1. Optimum fluid temperature changes are assumed in the form of simple time functions. For practical reasons the optimum temperature in the ramp form is preferred. It is possible to increase the fluid temperature stepwise at the beginning of the heating process and then increase the fluid temperature with the constant rate. Allowing stepwise fluid temperature increase at the beginning of heating ensures that the heating time of a thick-walled component is shorter than heating time resulting from the calculations according to EN 12952-3 European Standard.
The aim of the paper is a steady-state inverse heat transfer problem for plate-fin and tube heat exchangers. The objective of the process control is to adjust the number of fan revolutions per minute so that the water temperature at the heat exchanger outlet is equal to a preset value. Two control techniques were developed. The first is based on the presented mathematical model of the heat exchanger while the second is a digital proportional-integral-derivative (PID) control. The first procedure is very stable. The digital PID controller becomes unstable if the water volumetric flow rate changes significantly. The developed techniques were implemented in digital control system of the water exit temperature in a plate fin and tube heat exchanger. The measured exit temperature of the water was very close to the set value of the temperature if the first method was used. The experiments showed that the PID controller works also well but becomes frequently unstable.
Identification of coefficients determining flow resistance, in particular Manning’s roughness coefficients, is one of the possible inverse problems of mathematical modeling of flow distribution in looped river networks. The paper presents the solution of this problem for the lower Oder River network consisting of 78 branches connected by 62 nodes. Using results of six sets of flow measurements at particular network branches it was demonstrated that the application of iterative algorithm for roughness coefficients identification on the basis of the sensitivity-equation method leads to the explicit solution for all network branches, independent from initial values of identified coefficients.