Mathematical analysis of capillary thermal mass flow sensors for gas and liquid flow measurement

Abstract

An analytical temperature solution for a capillary thermal mass flow sensor (CTMFS) was derived and validated against a detailed thermofluidic numerical model. Approximate formulas for predicting mass flow rates in gases and liquids were developed using a Taylor series expansion of the temperature difference equation between the exit and entrance of the heater zone. Interestingly, while the gas flow rate was linearly proportional to the measured temperature difference, the liquid flow rate was inversely proportional to it. This divergence arises from the differing dominance of heat transfer mechanisms: conduction through the capillary wall in gases and convection within the flow in liquids. In gases, mass flow measurement occurs under conduction-dominant conditions, where convection-induced temperature differences remain within a linear range. In liquids, the high flow heat capacity rate renders wall conduction negligible, resulting in an inverse linear relationship between flow rate and temperature difference. While gas flow measurements exhibit increased nonlinearity at higher flow rates, liquid flow measurements encounter nonlinearity challenges at low flow rates. To quantify these effects, degrees of nonlinearity for gas and liquid flows were defined, providing criteria for establishing upper and lower flow rate limits, respectively. Design parameters for gas flow sensors involve a trade-off between linearity and sensitivity, whereas liquid flow sensitivity is primarily governed by heater power. This study presents novel insights into the measurement principles and design guidelines for CTMFS, contributing to improved accuracy in both gas and liquid flow measurements.