The conventional multi-scale modelling approach that predicts carbon nanotube (CNT) growth region in heterogeneous flame environment is computationally exhaustive. Thus, the present study is the first attempt to develop a zero-dimensional model based on existing multi-scale model where mixture fraction z and the stoichiometric mixture fraction zst are employed to correlate burner operating conditions and CNT growth region for diffusion flames. Baseline flame models for inverse and normal diffusion flames are first established with satisfactory validation of the flame temperature and growth region prediction at various operating conditions. Prior to developing the correlation, investigation on the effects of zst on CNT growth region is carried out for 17 flame conditions with zst of 0.05 to 0.31. The developed correlation indicates linear ( zlb=1.54zst +0.11) and quadratic ( zhb=zst(7-13zst )) models for the zlb and zhb corresponding to the low and high boundaries of mixture fraction, respectively, where both parameters dictate the range of CNT growth rate (GR) in the mixture fraction space. Based on the developed correlations, the CNT growth in mixture fraction space is optimum in the flame with medium-range zst conditions between 0.15 and 0.25. The stronger relationship between growth-region mixture-fraction (GRMF) and zst at the near field region close to the flame sheet compared to that of the far field region away from the flame sheet is due to the higher temperature gradient at the former region compared to that of the latter region. The developed models also reveal three distinct regions that are early expansion, optimum, and reduction of GRMF at varying zst.

Zero-dimensional model for the prediction of carbon nanotube (CNT) growth region in heterogeneous methane-flame environment
    Abstract
    1 Introduction
    2 Numerical conditions
        2.1 CFD flame model
        2.2 Computational domain
        2.3 Growth rate (GR) model
        2.4 Generation of zero-dimensional model
    3 Results and discussion
        3.1 Multi-scale validation of coflow diffusion flame
        3.2 Development of zero-dimensional model
    4 Conclusion
    Acknowledgements 
    References

• Muhammad Thalhah Zainal(High Speed Reacting Flow Laboratory (HiREF), Universiti Teknologi Malaysia, 81310 Skudai, Malaysia, Malaysia – Japan International Institute of Technology (MJIIT), Universiti Teknologi Malaysia, Jalan Sultan Yahya Petra (Jalan Semarak), 54100 Kuala Lumpur, Malaysia)
• Norikhwan Hamzah(Department of Thermo‑Fluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Malaysia, High Speed Reacting Flow Laboratory (HiREF), Universiti Teknologi Malaysia, 81310 Skudai, Malaysia)
• Mazlan Abdul Wahid(Department of Thermo‑Fluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Malaysia, High Speed Reacting Flow Laboratory (HiREF), Universiti Teknologi Malaysia, 81310 Skudai, Malaysia)
• Natrah Kamaruzaman(Department of Thermo‑Fluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Malaysia, High Speed Reacting Flow Laboratory (HiREF), Universiti Teknologi Malaysia, 81310 Skudai, Malaysia)
• Cheng Tung Chong(China‑UK Low Carbon College, Shanghai Jiao Tong University, Lingang, Shanghai 201306, China)
• Mohd Hanafi Ani(Department of Manufacturing and Materials, Kuliyyah of Engineering, International Islamic University Malaysia, P. O. Box 10, 50728 Kuala Lumpur, Malaysia)
• Shokri Amzin(Department of Mechanical and Marine Engineering, Western Norway University of Applied Sciences (HVL), N5020 Bergen, Norway)
• Tarit Das(Department of Thermo‑Fluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Malaysia)
• Mohd Fairus Mohd Yasin(Department of Thermo‑Fluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Malaysia, High Speed Reacting Flow Laboratory (HiREF), Universiti Teknologi Malaysia, 81310 Skudai, Malaysia)