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.
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 pressure of wet water vapor inside a condenser has a great impact on the efficiency of thermal cycle. The value of this pressure depends on the mass share of inert gases (air). The knowledge of the spots where the air accumulates allows its effective extraction from the condenser, thus improving the conditions of condensation. The condensation of water vapor with the share of inert gas in a model tube bank of a condenser has been analyzed in this paper. The models include a static pressure loss of the water vapor/air mixture and the resultant changes in the water vapor parameters. The mass share of air in water vapor was calculated using the Dalton’s law. The model includes changes of flow and thermodynamic parameters based on the partial pressure of water vapor utilizing programmed water vapor tables. In the description of the conditions of condensation the Nusselts theory was applied. The model allows for a deterioration of the heat flow conditions resulting from the presence of air. The paper contains calculations of the water vapor flow with the initial mass share of air in the range 0.2 to 1%. The results of calculations clearly show a great impact of the share of air on the flow conditions and the deterioration of the conditions of condensation. The data obtained through the model for a given air/water vapor mixture velocity upstream of the tube bank allow for identification of the spots where the air accumulates.
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.
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.