Estimation of Heat Fluxes From the Internal and External Surfaces of a Band Heater

This report investigates the feasibility of simultaneously estimating heat fluxes from the inner and outer surfaces of a band heater using only temperature measurements on those two surfaces. Knowledge of the energy delivered to the inner surface is needed for a separate study to develop a thermal flow meter. In a previous investigation, the pipe and the heater together were simulated using five different models including different geometries and boundary conditions. The models ranged from simple to complex cases. Direct mathematical models were either found or developed for them and, using the models, five inverse problems were developed and solved with the aim of estimating the heat flux at the inner surface. The investigations demonstrated considerable deviations from the manufacturer band heater data. Some of the energy generated inside the band heater is not delivered to the pipe. This suggests that a significant portion of the power is lost to the surroundings. The current investigation focuses on the band heater only. In the present study, heat transfer from both the inner and outer surfaces of the heater is calculated. Green’s functions form the basis for exact analytical relations for component solutions which are then superimposed to determine the band heater’s response to the two surface heat fluxes and the internal heat generation in the heater. Properties and dimensions of the targeted band heater are used in the model to determine the responses of the surface sensors to the two surface heat fluxes. These responses define the sensitivity of the sensors to the surface heat fluxes. An inverse heat conduction problem is formulated in discrete terms with the goal of estimating the heat fluxes, and Tikhonov regularization is employed to solve the inverse problem. The inverse heat conduction problem solution requires not only the sensitivities of the measurements to the two heat fluxes, but also the sensitivity of the responses to the internal heat generation of the heater. This internal heat generation is presumed known from direct measurement of the heater power consumption. A test problem based on assumed constant heat transfer coefficients at each boundary is employed to generate ideal responses at the sensor locations, and these ideal responses are converted into simulated measurements through addition of Gaussian noise. The generated data are subsequently used in solving the inverse problem to recover the surface heat fluxes. The results demonstrate the heater’s capability to transfer 33.9 W/m2 to the inner surface and 10.2 kW/m2 to the outer surface in 100 seconds. Importantly, this study shows that it is possible to simultaneously estimate the heat fluxes from the two surfaces of the band heater using only measured temperatures from these two surfaces.