A hybrid analytical heat conduction model was developed to predict the transient thermal evolution at the particle–substrate interface during the deposition of cold spray process. First, three-dimensional heat conduction model based on classical diffusion approach was developed to predict the transient surface temperature of the substrate. To that end, the traveling wave solution technique was utilized in order to take into account the effect of the movement of the cold spray nozzle. The results of the analytical model were validated with the experimentally measured surface temperature which was obtained by employing a low-pressure cold spray unit to generate the impingement of a compressed air jet on a flat substrate. The analytical model was further utilized to investigate the effect of the non-dimensional characteristic velocity of the traveling heat source on the surface temperature profile of the substrate. It was found that both the maximum surface temperature and the spatial variation of surface temperature profile of the substrate decreased as the non-dimensional characteristic velocity increased. The mathematical model was further extended by developing a one-dimensional hyperbolic (non-Fourier) heat conduction model to predict the temperature rise at the particle–substrate interface during the cold spray deposition process. In order to validate the results of the hybrid model, a three-dimensional finite element model was developed in ABAQUS to simulate the thermal and dynamic behavior of a single particle upon impact. The results of the hybrid analytical model for the temperature at the particle–substrate interface were compared to the results of the numerical model, and good agreement was found. It was concluded that by coupling the classical diffusion theory and hyperbolic heat conduction approach, the proposed hybrid analytical model can be utilized to predict the transient temperature of the particle–substrate interface during the cold spray deposition process a priori before experimentation.

中文翻译：

---

冷喷涂沉积过程中的颗粒-基材瞬态热演化：混合热传导分析

开发了一种混合分析热传导模型来预测冷喷涂过程沉积过程中颗粒-基材界面处的瞬态热演化。首先，开发了基于经典扩散方法的三维热传导模型来预测基板的瞬态表面温度。为此，为了考虑冷喷嘴运动的影响，使用了行波求解技术。分析模型的结果通过实验测量的表面温度进行了验证，该表面温度是通过使用低压冷喷涂装置在平坦基材上产生压缩空气射流的冲击而获得的。进一步利用分析模型来研究行进热源的无量纲特征速度对基板表面温度分布的影响。结果表明，随着无量纲特征速度的增加，基体表面最高温度和表面温度分布的空间变化均减小。通过开发一维双曲线（非傅立叶）热传导模型来进一步扩展数学模型，以预测冷喷涂沉积过程中颗粒 - 基材界面的温升。为了验证混合模型的结果，在 ABAQUS 中开发了一个三维有限元模型来模拟单个粒子在撞击时的热和动态行为。将粒子-基材界面温度的混合分析模型的结果与数值模型的结果进行了比较，发现了良好的一致性。得出的结论是，通过耦合经典扩散理论和双曲线热传导方法，所提出的混合分析模型可用于在实验前先验地预测冷喷涂沉积过程中颗粒 - 基材界面的瞬态温度。