Question

Plasma spraying is a process used for coating a material surface with a protective layer to prevent the material from degradation. In a plasma spraying process, the protective layer in powder form is injected into a plasma jet. The powder is then heated to molten droplets and propelled onto the material surface. Once deposited on the material surface, the molten droplets solidify and form a layer of protective coating. Consider a plasma spraying process using alumina $(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, \)\rho=3970 \mathrm{~kg} / \mathrm{m}^{3}\(, and \)\left.c_{p}=800 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ powder that is injected into a plasma jet at \(T_{\infty}=15,000^{\circ} \mathrm{C}\) and $h=10,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The alumina powder is made of spherical particles with an average diameter of \(60 \mu \mathrm{m}\) and a melting point at \(2300^{\circ} \mathrm{C}\). Determine the amount of time it would take for the particles, with an initial temperature of $20^{\circ} \mathrm{C}$, to reach their melting point from the moment they are injected into the plasma jet.

Short answer

Answer: It takes approximately 275 microseconds for the alumina particles to reach their melting point when injected into a plasma jet.

Step-by-step solution

Calculate the temperature difference

First, let's find the temperature difference between the initial temperature of the particle \(T_i\) and the melting point of the alumina \(T_m\): \(\Delta T = T_m - T_i\) Where, \(T_i = 20^{\circ}\mathrm{C}\) (initial temperature) \(T_m = 2300^{\circ}\mathrm{C}\) (melting point) \(\Delta T = 2300 - 20 = 2280\mathrm{~K}\)

Calculate the Biot number (Bi)

Biot number (Bi) is a dimensionless value that represents the ratio of internal thermal resistance to the convective heat transfer coefficient. Calculate Bi using the following formula: \(\text{Bi} = \frac{h L_c}{k}\) Where, \(h = 10,000 \mathrm{~W/m^2\cdot K}\) (convective heat transfer coefficient) \(L_c = \frac{d_p}{6}\) (characteristic length, d_p is the diameter of the particle) \(d_p = 60 \times 10^{-6} \mathrm{~m}\) (particle diameter) \(k = 30 \mathrm{~W/m\cdot K}\) (thermal conductivity of alumina) \(L_c = \frac{60 \times 10^{-6}}{6} = 10 \times 10^{-6}\mathrm{~m}\) \(\text{Bi} = \frac{10,000 \times 10 \times 10^{-6}}{30} = 0.003333\)

Calculate the Fourier number (Fo)

The Fourier number (Fo) is a dimensionless value that represents the ratio of the rate of heat conduction to the rate of heat energy storage. Calculate Fo using the following formula: \(\text{Fo} = \frac{\alpha t}{L_c^2}\) Where, \(\alpha = \frac{k}{\rho c_p}\) (thermal diffusivity) \(\rho = 3970 \mathrm{~kg/m^3}\) (density of alumina) \(c_p = 800 \mathrm{~J/kg\cdot K}\) (specific heat capacity of alumina) \(t\) is the time we want to find. \(\alpha = \frac{30}{3970 \times 800} = 9.424 \times 10^{-6} \mathrm{~m^2/s}\) Rewrite the equation for time: \(t = \frac{\text{Fo} L_c^2}{\alpha} \)

Determine the Fourier number (Fo) using the dominant thermal energy storage process assumption

Let's assume the convective heat transfer is dominant and the process can be analyzed in the regime of \(0.1 \leq \text{Bi} \leq 100\). It has been shown that \(\text{Bi} \leq 0.1\) corresponds to the situation when the temperature varies significantly along the way from the center to the surface of the sphere. In this case, the average temperature of the sphere can be evaluated as follows: \(\frac{T - T_i}{T_{\infty} - T_i} = 1 - e^{-3\text{Fo}}\) Plug in the values: \(\frac{2280}{14,980} = 1 - e^{-3\text{Fo}}\) Solve for Fo: \(\text{Fo} = -\frac{1}{3} \ln (1 - \frac{2280}{14,980}) = 0.245\)

Calculate time required for the particles to reach their melting point

Now, use the Fo value found in step 4 and substitute it into the equation found in step 3 to find the time it takes for the particles to reach their melting point: \(t = \frac{0.245 \times (10 \times 10^{-6})^2}{9.424 \times 10^{-6}}\) \(t = 0.000275\mathrm{~s}\) So, it would take approximately \(0.000275\mathrm{~s}\) or about \(275\mathrm{~\mu s}\) for the alumina particles to reach their melting point from the moment they are injected into the plasma jet.