Particle–Substrate Transient Thermal Evolution During Cold Spray Deposition Process: A Hybrid Heat Conduction Analysis

Abstract

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.

Appendix

A boundary condition of the third kind (Robin condition), including the effects of radiation from the substrate, was supposed to be applied at the front surface of the substrate (at z = 0) to model the heat flux due to impinging jet (Eq 6 in the manuscript).

$$- k_{\text{s}} \left[ {\frac{{\partial T\left( {x,y,z = 0,t} \right)}}{\partial z}} \right] = h\left( {x,y} \right)\,\left[ {\,T_{\text{AW}} \left( {x,y} \right) - T\left( {x,y,z = 0,t} \right)\,} \right]\, + \,\varepsilon \sigma \left[ {\,T^{4} \left( {x,y,z = 0,t} \right)\, - \,T_{\infty }^{4} \,} \right].$$

(34)

The second term on the right-hand side of abovementioned equation represents the heat loss from the front surface by radiation to the ambient at temperature, T_∞. It has been reported that if the relative temperature difference between the substrate and the ambient, $\frac{{\left| {T - T_{\infty } } \right|}}{{T{}_{\infty }}}$, is on the order of one or less, the radiation term in Eq 2-4 can be linearized and reformulated as (Ref 34)

$$\varepsilon \sigma \left[ {\,T^{4} \left( {x,y,z = 0,t} \right)\, - \,T_{\infty }^{4} \,} \right]\, \approx \,h_{\text{rad}} \left[ {\,T\left( {x,y,z = 0,t} \right)\, - \,T_{\infty } \,} \right],$$

(35)

where h_rad is the radiative heat transfer coefficient and is defined as (Ref 34)

$$h_{\text{rad}} \,= \,\varepsilon \sigma \left[ {\,T^{3} \left( {x,y,z = 0,t} \right)\, + \,T\left( {x,y,z = 0,t} \right)\,T_{\infty }^{2} + \,T^{2} \left( {x,y,z = 0,t} \right)T{}_{\infty }\, + \,T_{\infty }^{3} \,} \right]\, \approx \,4\varepsilon \sigma T_{\infty }^{3} .$$

(36)

where ε is the emissivity of the substrate and it was assumed to be 0.96, and σ is the Stefan–Boltzmann constant (σ = 5.67 × 10⁻⁸ W/m² K⁴). According to the experimental results previous studies study (Ref 9, 10), the relative temperature difference between the substrate and the ambient was measured to be on the order of one. Therefore, the aforementioned assumption of linearization of the radiation term was valid and it can be expressed as

$$- k_{\text{s}} \left[ {\frac{{\partial T\left( {x,y,z = 0,t} \right)}}{\partial z}} \right] = h\left( {x,y} \right)\,\left[ {\,T_{\text{AW}} \left( {x,y} \right) - T\left( {x,y,z = 0,t} \right)\,} \right]\, + \,h_{\text{rad}} \,\left[ {\,T\left( {x,y,z = 0,t} \right)\, - \,T_{\infty } \,} \right].$$

(37)

By applying a scale analysis on the radiation term in the above-mentioned equation, it can be deduced that the radiation effects on the temperature distribution within the substrate were negligible. In other words, by incorporating the approximate values of each term in this equation, and considering T_∞ = 21 °C (294 K), the radiative heat transfer coefficient will be on the order of one, while the convective heat transfer coefficient of an impinging air jet is expected to be on the order of 10² (Ref 9). In light of the aforementioned scale analysis and in order to simplify the problem, the effects of radiation were not considered in the mathematical modeling. Therefore, the boundary condition was reformulated as

$$- k_{s} \left[ {\frac{{\partial T\left( {x,y,z = 0,t} \right)}}{\partial z}} \right] = h\left( {x,y} \right)\,\left[ {\,T_{\text{AW}} \left( {x,y} \right) - T\left( {x,y,z = 0,t} \right)\,} \right].$$

(38)

Regarding the accuracy of the FLIR measurements, a detailed error analysis was conducted in a previous study (Ref 9) in order to predict the uncertainty propagation in the results of the developed semi-empirical analytical model. Please note that the emissivity of substrate which was used in the scale analysis was identical to that in the experimental measurements.