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 particles that are spherical in shape 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.

Step by step solution

01

Calculate the mass of the particle

First, we need to find the mass of the alumina particle. To compute the mass, we use the particle density (\(\rho\)) and volume, which is given by the formula for the volume of a sphere: \(V = \frac{4}{3} \pi r^3\) Given the average diameter of the alumina particles is \(60 \mu m\), the radius is \(30 \mu m\). To convert this to meters, we need to divide it by \(10^6\): \(r = \frac{30}{10^6} m\) Now we can calculate the volume: \(V = \frac{4}{3} \pi (\frac{30}{10^6})^3 m^3\) And, finally, we can compute the mass of the particle: \(m = \rho V = (3970 kg/m^3) \frac{4}{3} \pi (\frac{30}{10^6})^3 m^3\)

02

Calculate the heat absorbed by the particle

Next, we will calculate the heat absorbed by the particle as it reaches its melting point. We will use the lumped system analysis equation: \(Q = m c_p (2300 - 20)\) Substitute the specific heat value, \(c_p\), provided and the mass calculated in step 1: \(Q = (m)(800 J/ kg \cdot K)(2280 K)\)

03

Calculate the heating rate

Now we will calculate the heating rate of the alumina particle. We will use the following formula: \(\frac{dQ}{dt} = hA_s\Delta T\) Here, \(h\) is the heat transfer coefficient, \(A_s\) is the surface area of the particle, and \(\Delta T\) is the temperature difference between the plasma jet temperature and the particle temperature. First, we need to find the surface area of the particle, which can be calculated as: \(A_s = 4 \pi r^2\) Substitute the radius in meters: \(A_s = 4 \pi (\frac{30}{10^6})^2 m^2\) \(\Delta T = T_{\infty} - T_m\) Substitute the given values for the plasma jet temperature and the melting point: \(\Delta T = (15000 + 273) K - (2300 + 273) K\) Now we can compute the heating rate: \(\frac{dQ}{dt} = (10000 W/m^2 \cdot K)(A_s)(\Delta T)\)

04

Calculate the time to reach the melting point

Finally, we can calculate the time it takes for the alumina particle to reach its melting point. We can derive the equation for the time by dividing the heat absorbed by the particle by the heating rate: \(t = \frac{Q}{\frac{dQ}{dt}}\) Substitute the values calculated in steps 2 and 3: \(t = \frac{Q}{\frac{dQ}{dt}}\) This will give us the time it takes for the alumina particles to reach their melting point from the moment they are injected into the plasma jet.

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