Identification of the flowrate based upon a solution of an inverse heat transfer problem

Abstract

In the considerations given in this paper, a steady momentum and heat exchange in the inlet area of the tube of a thermal flowmeter is regarded. The process of the momentum and heat exchange in the region of a thermal flowmeter will be described by means of a system of differential equations, comprising the continuity equations, the motion equation, the energy equation for the fluid, and the energy equation for the tube wall material (solid). The mentioned system of differential equations with proper boundary conditions constitutes a boundary problem called “the direct problem”. By introducing an internal temperature response to the considerations, the response in this case being the temperature value measured at a given point in the tube wall, we get an additional condition by comparing that value with the temperature calculated numerically. The mentioned direct boundary problem and the recently formulated condition of temperatures equality constitute an overdetermined system of equations that describe a non-equivocal determination of the value of the Reynolds number as a parameter of those equations. Mathematically, this leads to a form of inverse problem in which a coefficient in the differential equations is to be determined so that the solution satisfies an overdetermined set of boundary conditions.