Development of the method of approximate solution of the nonstationary problem of thermal transmission through flat wall

Abstract

In the present work, we propose a method for approximate analytical solution to the nonstationary problem of heat transfer through a flat wall in the concentrated statement. In the course of the study, three issues were consistently addressed: 1. symmetrical heating of a body, 2. asymmetrical heating, 3. nonstationary heat transfer. At the first stage, we solved in approximate analytical statement the problem on symmetrical heating of a plate. The solution obtained has an error. The availability in the scientific literature of exact analytical solution, in a distributed (one-dimensional) statement, allowed us to assess the accuracy of the obtained approximate solution. It does not exceed the limits permissible for engineering calculations. A special feature of the developed method is the possibility of its application as an integral part in solving the problems on nonstationary heat transfer. At the second stage, we solved a problem on asymmetrical heating of a plate. By using numerical study, the character of displacement of the minimum in temperature profile for thickness of the plate was identified. This made it possible, when applying the method developed at the previous stage, to obtain a solution to the problem on asymmetrical heating of the body. A special feature of the solutions is the developed approach to determining position of the temperature minimum for thickness of the plate. Such an approach was employed as another constituent part for solving a problem on nonstationary heat transfer. At the third stage, numerical study allowed us to identify a characteristic point of varying temperature profile and the trajectory of its motion in the process of nonstationary heat transfer. Based on these data and applying the developed method, we demonstrated the possibility of approximate analytical solution to the problem on nonstationary heat transfer through a flat wall. In all cases, when the exact solutions were lacking, assessment of error in approximate representation was conducted by comparing with the results of numerical calculations. The error did not exceed 6 %

Для дослідження процесу нестаціонарної теплопередачі через плоску стінку був розроблений наближений аналітичний метод рішення задачі в зосередже-ній постановці. Працездатність методу продемонстрована на прикладі рішення задач, що послідовно ускладнюються, для плоскої пластини: симетричного, несиметричного нагріву і нестаціонарної теплопередачі. Показана адекватність отриманих рішень в рамках точності інженерних розрахунків

Для исследования процесса нестационарной теплопередачи через плоскую стенку разработан приближенный аналитический метод решения задачи в сосредоточенной постановке. Работоспособность метода продемонстрирована на примере решения последовательно усложняющихся задач для плоской пластины: симметричного, несимметричного нагрева и нестационарной теплопередачи. Показана адекватность полученных решений в рамках точности инженерных расчетов