[Paper Review] Modeling of a multiple source heating plate

S. Scholz, Lothar Berger|arXiv (Cornell University)|Nov 30, 2020

Stability and Controllability of Differential Equations15 references4 citations

TL;DR

This paper proposes a quasi-linear, distributed parameter model of a 2D multiple-source heating plate using a finite volume method to simulate thermal dynamics with spatially characterized actuators and sensors. The key finding is that realistic actuator spatial distribution causes oscillatory temperature profiles uncorrectable by diffusion alone, necessitating precise modeling and advanced control for uniform heating.

ABSTRACT

Heating plates describe the transfer of heat from actuators to a target object. In other words, they separate the heat sources and heated object and can be further used to apply a specific heat distribution on this object. Therefore, an exact description of their thermal dynamics and an efficient coordination of their actuators is necessary to achieve a desired time-dependent temperature profile accurately. In this contribution, the thermal dynamics of a multiple source heating plate is modeled as a quasi-linear heat equation and the configuration of the spatially distributed actuators and sensors are discussed. Furthermore, the distributed parameter system is approximated using a Finite Volume scheme, and the influence of the actuators' spatial characterization on the plate's thermal dynamics is studied with the resulting high-dimensional system.

Motivation & Objective

• To develop a physically accurate, distributed parameter model of a heating plate with multiple spatially distributed heat sources.
• To analyze how the spatial characterization of actuators and sensors influences thermal dynamics and temperature uniformity.
• To demonstrate that idealized models fail to capture real-world thermal behavior due to non-uniform actuator effects.
• To establish a foundation for future control design that accounts for distributed actuators and sensors in 2D and 3D thermal systems.
• To provide a numerical framework for simulating and controlling complex thermal processes with high spatial and temporal resolution.

Proposed method

• Model the heating plate as a 3D quasi-linear heat equation with temperature-dependent thermal properties.
• Apply a finite volume spatial discretization to approximate the distributed parameter system, transforming it into a high-dimensional ODE system.
• Incorporate spatially distributed actuators and sensors with realistic spatial characterization (e.g., non-uniform heating zones).
• Use the forward Euler method for time integration in a 2D simulation environment implemented in Julia with DifferentialEquations.jl.
• Validate results through visualization of temperature distributions and comparison of input/output signals across idealized vs. realistic actuator scenarios.
• Compute average input and output signals to assess controller performance despite divergent thermal behaviors.

Experimental results

Research questions

• RQ1 How does the spatial distribution of heating actuators affect the resulting temperature profile on the heating plate?
• RQ2 To what extent does the diffusive nature of the heat equation compensate for non-uniform actuator characteristics?
• RQ3 Can a naive controller achieve uniform temperature distribution when actuators have realistic spatial characteristics?
• RQ4 What is the impact of sensor and actuator spatial resolution on the observability and controllability of the thermal system?
• RQ5 How do temperature-dependent material properties (thermal conductivity, emissivity) influence the system's dynamic response?

Key findings

• The nominal actuator scenario achieved a final temperature on the topside close to the reference (400 K) with an error of approximately 1 K, indicating good performance under idealized assumptions.
• In contrast, the realistic actuator scenario produced a strongly oscillatory temperature distribution along the horizontal axis, demonstrating significant non-uniformity.
• Despite nearly identical input signals between scenarios, the thermal behavior differed drastically, highlighting that controller signals alone cannot indicate system performance.
• The finite volume discretization successfully captured the influence of actuator spatial characterization on the thermal dynamics, validating its use in high-dimensional systems.
• The results show that the diffusive nature of the heat equation cannot fully compensate for non-uniform actuator distributions, necessitating precise modeling of spatial characteristics.
• The study establishes that modeling must include realistic actuator and sensor spatial distributions to achieve accurate thermal control in industrial applications.