Transient multimode heat transfer of an industrial automated tape laying process under rapidly changing conditions

Bernhard Rameder; Hubert Gattringer; Andreas Müller; Ronald Naderer

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Transient multimode heat transfer of an industrial automated tape laying process under rapidly changing conditions

Bernhard Rameder, Hubert Gattringer, Andreas Müller, Ronald Naderer

arXiv:2608.25470v1Aug 26, 2026cs.RO

TL;DR

Because nip-point temperature is difficult to measure during rapidly changing ATL conditions, the paper develops a coupled transient thermo-radiative model incorporating the process’s interacting heat-transfer mechanisms. Validated against industrial measurements, the model reproduces transient thermal responses with an overall deviation of only 1.08% (NRMSE).

Problem
Nip-point temperature is difficult to measure directly, creating a need for models that predict it from dynamic tape feed rate and heater current while including all thermal transfer modes.
Method
The model couples transient infrared-heater dynamics, finite-width view factors, mixed convection, and phase-shifted two-layer tape temperatures in a monolithic process simulation.
Results
1.08% (NRMSE) maximum deviation was obtained against industrial ATL measurements.
Takeaways & Limitations
The validated model accurately reproduces transient thermal responses from real-world ATL velocity and current data, supporting thermal-state estimation for the process.
Takeaways & Limitations
The model assumes a diffuse-gray enclosure and lumped optical properties, while emissivity and convection coefficients remain uncertain because of limited measurement equipment.

Abstract

from arXiv · show

This work presents a transient heat-transfer model of an industrial automated tape laying (ATL) process designed to overcome the limitations of conventional thermal models in composite manufacturing. The model solves the heat-conduction equation with coupled advection, conduction, convection, and radiation. A key innovation is the implementation of an analytical view factor approach that accounts for finite emitter and tape widths, thereby correcting systematic overestimations of radiative heat flux inherent in 1.5D simplifications. Furthermore, a local convection assessment incorporates mixed convection effects characterized by the Richardson number, ensuring accuracy across a wide range of process speeds. The ATL system is represented by two interacting subsystems: the moving tape substrate and the infrared heat sources. The tape is discretized using a two-node model that resolves the physical phase shift between the heated and monitored surfaces. Numerical stability under high dynamics is ensured by a monolithic solution strategy using a high-order implicit integration scheme. Model predictions were validated on an industrial ATL line, demonstrating an overall deviation of only 1.08% (NRMSE) under rapid velocity and current modulations. This framework provides a high-fidelity, physics-based foundation for thermal state estimation, supporting consistent in-situ consolidation and improved part quality.

1. Introduction

The study develops a transient ATL thermal model to estimate nip-point temperatures from rapidly changing process inputs when direct measurement is impractical. It combines dynamic infrared-emitter modeling, analytical radiation treatment, local mixed-convection assessment, and a stable two-node solution strategy, achieving 1.08% NRMSE validation error.

Motivation: Continuous fiber-reinforced polymer composites can reduce component weight by up to 70% while maintaining stability, motivating their use in modern manufacturing.The motivation is linked to environmental protection, energy efficiency, and weight reduction.
Motivation: Because direct nip-point temperature measurement is often impractical, a thermodynamic model predicts it from dynamic tape feed rate and infrared heater supply current.Sensors must be placed away from the nip point because of space constraints.
Model contribution: Local convection modeling accounts for boundary-layer development and mixed convection through the Richardson number across industrial start-stop velocities.This replaces the constant heat-transfer-coefficient assumption common in existing ATL models.
Model contribution: The model improves transient fidelity by capturing infrared-emitter inrush-current dynamics and the double-tube geometry of dual emitters.The heater model resolves dynamic thermal processes within the heater geometry, particularly during the inrush current phase.
Validation: 1.08% (NRMSE) overall deviation was obtained when the model was validated against measured temperatures on a self-developed industrial ATL line.The simulated temperature profile showed excellent agreement with the measured value at the defined measuring spot.

2. Design and instrumentation of an experimental ATL platform

The experimental ATL platform is a modular industrial prototype integrating tape heating, feeding, processing, storage, and controlled consolidation. Instrumentation includes dynamic nip-pressure control and contactless temperature sensing at defined tape and mold locations.

Platform architecture: The self-designed industrial ATL device is organized into modules for heating, tape processing and feeding, consolidation, and tape storage.The prototype design includes infrared heaters, tape feed and guidance, a consolidation unit, and a storage spool.
Consolidation control: A compaction roller mounted on a uniaxial force-controlled Active Contact Flange provides precise, dynamic control of nip-point contact pressure.The Active Contact Flange is supplied by FerRobotics Compliant Robot Technology.
Tape guidance and heating: Guiding sheet-metal parts hold the tape in line during motion, and the process chain also includes an optional inactive heater before the tape cutting unit.The optional heater is located before the tape cutting unit.
Temperature instrumentation: A contactless Optris CS LT infrared sensor measures tape temperature at a defined lower-side location, while a second sensor is aligned with the mold.The corresponding sensor beams are shown as green double-arrows in Fig. 2.

3. Modeling framework … 3.3. Infrared heater source model

The modeling framework describes transient heat transfer for moving tape and interacting infrared heaters in a fixed spatial domain. It couples advection, conduction, convection, radiation, and segmented heater electrical–thermal dynamics to resolve tape and source temperatures.

3. Modeling framework: The fixed Eulerian domain models a tape moving through a heater field, with heat conduction in the tape and boundary exchange through convection and radiation.Enclosure radiation accounts for mutual interactions among the tape, heater tubes, guides, and housing.
3. Modeling framework: The coupled subsystems use tape temperature as the primary state, while heater and radiation models provide boundary fluxes driven by electrical and thermal interactions.Input current i_e maps to filament temperature T_f, which drives the radiative source seen by the tape; quartz temperature T_q depends on neon conduction and radiation.
3.1. Assumptions and reference frame: The reference-frame simplifications neglect tape conduction to guide plates and assume vertical tape motion, while retaining guide plates as radiation shields.These assumptions address the short flyby duration, limited material contact, and geometric simplification of view-factor calculations.
3.2. Governing heat-transfer equations: The tape model includes convection on both exposed surfaces, heater-facing radiation, through-thickness conduction, and predominantly feed-direction advection on a fixed mesh.Lateral temperature variations across the tape width are assumed small under near-uniform radiative exposure.
3.2. Governing heat-transfer equations: ΔT_m,max = 6 K demonstrates that the coupled two-node tape model is needed to resolve distinct heated-side and monitored-side surface temperatures during dynamic transitions.Although Bi ≈ 0.021 supports lumped through-thickness capacitance, asymmetric one-sided radiation violates a uniform-temperature assumption.
3.2. Governing heat-transfer equations: Ri_Lm(L_m, T_s,max, v_T,max) ≈ 6.23 identifies combined natural and forced convection in the modeled operating regime.The convection formulation evaluates forced and natural contributions using Reynolds, Prandtl, Grashof, and Rayleigh numbers.
3.3. Infrared heater source model: The infrared heater source model discretizes each radiator into N_h longitudinal segments to capture localized rapid heating during inrush current.A homogeneous-coil model would heat the full mass simultaneously and underrepresent the actual heater dynamics.
3.3. Infrared heater source model: Each filament segment balances electrical heat generation against radiation, neon conduction, axial wire conduction, and gap heat transfer, while the quartz envelope receives filament radiation and loses heat by radiation, convection, and conduction.Neon transients are neglected as a quasi-stationary radial conduction problem, and the reflective gold layer is assumed equilibrated with the quartz, T_r = T_q.

3.4. Enclosure radiation model · 3.5. Coupling and numerical implementation

The enclosure radiation model computes coupled radiative exchange among tape, heaters, and surroundings using optical properties and geometry-specific view factors. The coupled thermo-radiative equations are reduced to an ODE system and integrated with a stable implicit Radau IIA scheme.

3.4.1. Modeling basis: The radiation enclosure includes tape segments, infrared heater surfaces, and black-body surroundings, with rectangular heater surrogates simplifying the double-tube geometry.Small emitter-to-tape distances justify the surrogate representation.
3.4.2. Surface discretization and view factors: View factors are assigned across heater, tape, and surrounding surfaces while obeying A_θF_θ−ψ = A_ψF_ψ−θ to ensure cavity energy conservation.The surface sequence contains four heater surfaces, N_x tape surfaces, and the surroundings.
3.4.2. Surface discretization and view factors: 159% is the overestimation of real radiation exchange area introduced by the 1.5D simplification, motivating the full 3D view-factor formulation.The comparison uses real geometric dimensions with l_q≈2.5w_m.
3.4.3. Spectral emission and gray approximation: Planck’s law and Stefan-Boltzmann radiation provide the spectral basis, while a first-10-term infinite-series approximation reduces calculation time for band radiation.The gray approximation is appropriate when emissivity varies slowly over dominant emission bands and spectral filtering is weak.
3.4.4. Optical properties: Temperature-dependent emissivity, reflectivity, and transmissivity for tape, emitters, surroundings, and quartz determine Planck-weighted irradiation, radiosity, and net radiative heat flux.SWIR-band emissivity and transmissivity are critical for modeling energy absorption and gold-reflector effectiveness.
3.4.5. Radiosity, irradiation and net radiative heat flux: Under the diffuse-gray approximation, radiosity and irradiation are solved across all surfaces using optical properties and geometric view factors, yielding surface-specific net radiative heat flux.The resulting heat flow is counted positive when leaving the corresponding surface.
3.5. Coupling and numerical implementation: The coupled thermo-radiative formulation is naturally a differential-algebraic system, but analytical elimination of radiosity and irradiation reduces it to an ODE system whose radiation flux depends nonlinearly on the thermal state.The reduced state contains nodal temperatures and their associated thermal differences.
3.5.1. Time integration scheme: A 2-stage Radau IIA method of order 3 integrates the stiff transient ODE system using implicit Runge-Kutta time integration.The method has general order 2s−1 and was selected for stability properties in stiff differential equations.

4. Experimental setup and evaluation

The thermo-radiative model was evaluated on an industrial ATL test rig under dynamically varying tape velocity and heater current, with temperature measurements used for validation. Results showed consistent transient tracking, while grid-refinement tests supported retaining the baseline discretization as an accuracy–efficiency trade-off.

Experimental setup: The test rig combined a spatially fixed ATL unit, an industrial handling robot, and a plane test mold.The robot handled the mold and placed tape during the process.
Experimental setup: The validation trajectory simultaneously varied tape motion and infrared-heater supply current while recording tape-side temperature at a measuring point.The trajectory was generated through optimization to minimize process time using the maximum available heating power.
Model evaluation: The model consistently captured measured temperatures during transient tape-velocity changes and strongly fluctuating current modulations, including nonlinear radiation effects without compromising stability.The temperature field was evaluated at both the measuring point and nip point, resolving heat flow through the tape thickness.
Model evaluation: Inrush current concentrated heat generation and produced high heating rates in specific heater segments, especially near the free filament coil at the heater center.After the heater approached a quasi-stationary state, the heat terms increasingly equalized across segments.
Grid independence: 0.17 % NRMSE (0.61 K RMSE) was obtained for the refined tape grid with Nx = 60 and Nh = 15, confirming independence from tape discretization.The baseline configuration used Nx = 30 and Nh = 15.
Grid independence: 1.86 % NRMSE (6.93 K RMSE) resulted from the refined heater grid with Nx = 30 and Nh = 31, while simulation time increased approximately 148 % to 862 s.The baseline simulation required 347 s, so the baseline grid was retained for macro-level accuracy and computational efficiency.

5. Conclusions

The validated thermo-radiative ATL model reproduces transient thermal responses under real-world operating data, with a maximum deviation of 1.08% (NRMSE). Its modeling contributions improve transient-state representation and radiative and convective heat-transfer assessment, while limitations remain in optical properties, runtime, and parameter uncertainty.

Validation: The model accurately reproduces transient thermal responses driven by synchronized industrial measurements of velocity, supply current, and surface temperature.It captures sharp heating onset at motion start and the subsequent transition to steady-state conditions within sensor margins.
Modeling contributions: Heater segmentation captures highly dynamic internal thermal behavior, while redefined view factors prevent overestimation of net incident radiative flux.The local convection assessment additionally accounts for boundary-layer development and transition to mixed convection.
Validation: 1.08 % (NRMSE) was the maximum deviation compared to industrial ATL measurements.This quantitative result summarizes the model’s agreement with the industrial ATL line.
Limitations: Limitations include diffuse-gray enclosure and lumped optical-property assumptions, non-targeted real-time execution, and uncertainty in emissivity and convection coefficients.These limitations reflect underrepresented band-specific effects, high-fidelity computational coupling, and limited measurement equipment.
Future work: Future investigations will use the validated model for trajectory optimization and AI-assisted process design under robot, process, workspace, and surface-temperature constraints.The planned optimization targets robot paths and speed profiles that exploit available heating power while respecting kinematic and process limits.

Supplementary material: Transient multimode heat transfer of an industrial automated tape laying process under rapidly changing conditions

The supplementary material documents the datasets, analyses, formulations, and calibration protocols supporting the numerical modeling and experimental investigations of transient multimode heat transfer in industrial automated tape laying. It also covers automated tape laying, thermoplastic composites, optical properties, view factor formulation, and parameter tuning.

Supporting data: The document provides thermal and optical property datasets supporting the ATL thermal model and experiments.These datasets are identified as comprehensive and material-specific.
Model formulations: It includes extreme value analyses and detailed formulations for internal heat transport mechanisms.These materials support the numerical modeling presented in the main manuscript.
Calibration: Calibration protocols are provided to support the numerical modeling and experimental investigations.The supplementary document explicitly identifies calibration as part of its supporting material.
Scope: The stated keywords identify automated tape laying, thermoplastic composites, optical properties, view factor formulation, and parameter tuning.These keywords summarize the supplementary material’s topical scope.

S1. Dimensionless numbers

The model determines convective heat-transfer coefficients from dimensionless-number analysis that identifies flow regimes and transport mechanisms. Comparing Nusselt, Prandtl, Richardson, Grashof, and Rayleigh numbers provides the basis for selecting appropriate Nusselt correlations.

Dimensionless numbers: Nusselt-number correlations determine the convective heat-transfer coefficients at the model boundaries.The coefficients are denoted h_m,c and h_m,s.
Dimensionless numbers: The Prandtl number describes the ratio between momentum and thermal transport.It relates velocity-boundary-layer thickness to thermal-boundary-layer thickness.
Dimensionless numbers: The Richardson number evaluates the relative contributions of free and forced convection.It is used when temperature-induced density gradients occur alongside forced flow.
Dimensionless numbers: The Grashof number quantifies buoyancy-driven free-convection intensity relative to viscous effects.It represents the ratio of buoyancy forces to viscous forces within the fluid.
Dimensionless numbers: The Rayleigh number scales the resulting convective heat-transport capability.Systematic comparison of the dimensionless numbers defines flow characteristics and supports appropriate Nusselt-equation selection.

S2. Fluid flow regime analysis and extreme values

The analysis shows that transient ATL convection depends nonlinearly on temperature, position, velocity, and material, requiring a local mixed-convection model rather than a constant forced-convection approximation. During deceleration and cooling, natural convection increasingly dominates and causes divergence from the simplified baseline.

Temperature dependency: Mixed convection is necessary because temperature changes alter buoyancy and air viscosity, producing a self-regulating thermal damping effect ignored by constant coefficients.At zero velocity, natural convection couples nonlinearly with temperature.
Position dependency: Local convection coefficients spike at the leading edge and decay along the path, so spatial averaging can conceal severe localized thermal shocks.The leading-edge spike occurs as x→0 because the thermal boundary layer thickness is zero.
Velocity dependency: Natural convection dominates at low speeds, while forced Sakiadis flow takes over at higher velocities; forced-only models therefore miss stationary-phase thermal losses.The mixed model retains a non-zero convection baseline at v_T = 0 m s−1.
Temperature and material impact: High-temperature polymers such as PEEK exhibit substantial core-temperature deviations over time when the convection model is changed.Convective losses increase strongly with the temperature difference from the surroundings.
Regime transition during deceleration and cooling: As velocity decreases and heater current falls, natural convection becomes more dominant, shifting the Richardson number and amplifying divergence from the static forced-only baseline.The final transient temperature plot cannot isolate velocity, position, and temperature effects because they change simultaneously.

S3. Detailed formulations for filament and envelope heat transport mechanisms

This section formulates conductive and radiative heat transport in the tungsten filament coil and quartz glass envelope. It also accounts for gas conduction, surface radiation, contact-wire losses, geometric view factors, and multiple reflections between coil windings.

Overview: The emitter formulation explicitly defines conductive and radiative transport equations for both the tungsten filament coil and quartz glass envelope.These formulations remain consistent with the simplified representations in the main text.
Filament conduction: Filament conduction is computed segmentwise from tungsten conductivity, wire cross-sectional area, and unwound segment length, including losses to thicker contact wires.The contact wires are located at the filament coil’s start and end.
Gap heat transfer: The net heat flux across the filament gap combines gas conduction through neon with surface-to-surface radiation between adjacent windings.Neon conductivity is temperature-dependent and evaluated at the approximated mean gap temperature.
Coil radiation: Radiative exchange between adjacent coil windings uses gray, diffuse surfaces, an effective segment area, and a network resistance model with geometric view factors.The formulation captures multiple reflections between curved surfaces even when coil spacing makes the view factor less than unity.
Envelope conduction: Heat conduction along the quartz glass cylinder is modeled using thermal conductance based on quartz conductivity, hollow-cylinder area, and segment length.The resulting heat flow is evaluated along the corresponding cylinder axis.

S4. Material properties and optical characterization

The section establishes temperature-dependent thermo-physical and optical material inputs for transient thermal modeling of the ATL system. It combines literature-based property models with experimental emissivity characterization for the composite tape.

Thermo-physical properties: Temperature-dependent constituent properties are required to model transient infrared-emitter behavior and the surrounding domain accurately.The framework accounts for material-property variations rather than assuming constant parameters across the operating thermal window.
Thermo-physical properties: Tungsten specific heat capacity and electrical resistivity are mapped across their operational temperature ranges to represent filament energy balance, inrush currents, and thermal inertia.The cited ranges are 298 K < T_f < 1900 K for specific heat and 300 K ≤ T_f ≤ 3655 K for resistivity.
Thermo-physical properties: Neon thermal conductivity is implemented as a temperature-dependent function over 100 K ≤ T_n ≤ 2500 K, while fused-quartz specific heat defines the protective casing behavior.The neon conductivity expression is k_n(T_n) = −3.601 802 1 × 10−3 + 1.429 062 3 × 10−3T^0.63362378 W m−1 K−1.
Optical properties: Optical characterization uses temperature-dependent tungsten emissivity models, literature gold emissivity, and Fresnel- and reflection-based calculations for quartz transmissivity and emissivity.The tungsten approach compares Fresnel and temperature-dependent Lorentz–Drude formulations, while quartz properties use reference material data and spectral integration.
Tape emissivity: Both tape-emissivity procedures yielded consistent values, but Method A was adopted because camera radiometric corrections improved robustness to reflected and transmitted radiation.The carbon-fiber-reinforced HDPE tape was measured using a contact thermometer and thermal infrared camera after heating to a steady, uniform temperature.

S5. Analytical formulations and references for view factor calculation

This section compiles the analytical view-factor formulations, geometric definitions, and original references needed to model radiative heat transfer in the geometrically complex setup. It covers rectangle-to-rectangle configurations with parallel and perpendicular emitter–tape alignment.

Analytical formulations and references for view factor calculation: The section collects extensive analytical view-factor derivations, geometric definitions, and literature references to preserve the main manuscript’s conciseness and readability.These formulations support precise determination of view factors for radiative heat transfer in the numerical model.
Parallel rectangles: For parallel alignment, the view factor uses formulations for rectangles in parallel planes from references.This case assumes the first heat source and material strip are fully parallel.
Perpendicular rectangles: For perpendicular alignment, the view factor applies rectangle-to-rectangle formulations in perpendicular planes from references.The corresponding view factor is computed using Eq. S38 by exchanging 𝐺∥ with the relevant perpendicular configuration.
Geometric definitions: Figures S7 and S8 depict the geometric setups for calculating the parallel-rectangle and perpendicular-rectangle cases, respectively.The formulations include geometric terms such as (𝑥2 + 𝜉2).

S6. System parameter tuning and calibration

The section consolidates material, heater, geometrical, thermal, and spectral parameters before systematically tuning uncertain values against experimental temperature profiles. Composite-tape heat capacity is analytically estimated by a linear rule of mixtures, while tuned parameters are iteratively varied within plausible physical ranges to improve agreement with measurements.

Parameter consolidation: Tables S2 and S3 compile the baseline material-specific and heater-specific parameters used by the numerical solver.Table S3 distinguishes nominal untuned values from actual calculation values and identifies uncertain parameters subsequently tuned.
Parameter consolidation: Composite-tape specific heat capacity c_p,m is analytically determined with a linear rule of mixtures because manufacturer specifications were unavailable.The constituent mass fractions are derived from their corresponding volume fractions.
Calibration: The uncalibrated model uses estimated or comparable nominal values from Table S3 as a reference for assessing calibration necessity and efficiency.Figures S9 and S10 show simulated temperature profiles at the nip point and measuring point, including comparison with measured measuring-point temperature.
Calibration: Tuning iteratively modifies uncertain Table S3 parameters within plausible physical ranges until simulated results closely align with experimental data.The tuned parameter set corresponds to the actual values in Table S3 and improves model accuracy as reported in the main manuscript.

Nomenclature

The nomenclature defines thermal, geometric, material, temperature, radiation, and process-model symbols used throughout the automated tape laying analysis. It also lists abbreviations for methods, materials, control concepts, and error metrics.

Geometry and radiation: Geometric and radiative symbols describe emitter segments, exchange areas, heater positions, emissive powers, emissivity, transmissivity, and radiation-cavity surfaces.The definitions include finite emitter geometry, adjacent-winding exchange areas, infrared-heater coordinates, spectral emission, and effective transmissivity.
Thermal properties: Thermal quantities include areal heat capacity, specific heat capacities, thermal conductances, heat-transfer coefficients, and volumetric heat-generation rates.These symbols cover tape layers, tungsten filaments, composite material, neon gaps, mounting wires, fluids, and radiation cavities.
Heat-transfer terms: Heat-transfer terms distinguish conduction, radiation, convection, and combined gap heat between adjacent filament windings and tape surfaces.The nomenclature separately identifies heat fluxes acting on filaments, tape, and quartz glass, as well as line-source heat densities.
Temperature variables: Temperature symbols distinguish two-node tape-surface temperatures, sensor-captured segment means, surrounding-side measurements, and emissivity-corrected quartz-glass measurements.The nomenclature also defines the tape surface facing the radiation cavity and simulated quartz-glass temperatures captured by the sensor.
Abbreviations: Abbreviations cover automated tape laying, differential-algebraic and ordinary differential equations, finite element methods, model predictive control, materials, and error metrics.Listed metrics include normalized root mean squared error and root mean squared error, while materials include HDPE and PEEK.