Question

Engine valves $\left(c_{p}=440 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.\( and \)\left.\rho=7840 \mathrm{~kg} / \mathrm{m}^{3}\right)$ are to be heated from \(40^{\circ} \mathrm{C}\) to \(800^{\circ} \mathrm{C}\) in 5 min in the heat treatment section of a valve manufacturing facility. The valves have a cylindrical stem with a diameter of \(8 \mathrm{~mm}\) and a length of \(10 \mathrm{~cm}\). The valve head and the stem may be assumed to be of equal surface area, with a total mass of \(0.0788 \mathrm{~kg}\). For a single valve, determine \((a)\) the amount of heat transfer, \((b)\) the average rate of heat transfer, \((c)\) the average heat flux, and \((d)\) the number of valves that can be heat treated per day if the heating section can hold 25 valves and it is used \(10 \mathrm{~h}\) per day.

Short answer

(a) Calculate the amount of heat transfer: $$ Q = (0.0788\, kg)(440\, \frac{J}{kg\cdot K})(800\, ^{\circ}C - 40\, ^{\circ}C) = 26309.6\, J $$ (b) Calculate the average rate of heat transfer: $$ \dot{Q} = \frac{26309.6\, J}{5\, min} = \frac{26309.6\, J}{300\, s} = 87.7\, \frac{J}{s} $$ (c) Calculate the average heat flux: Based on the given mass, length, and density of the valve, we can calculate the total surface area: $$ A = \frac{m}{h\rho} = \frac{0.0788\, kg}{(0.05\, m)(7800\, \frac{kg}{m^3})} = 3.208 \times 10^{-5}\, m^2 $$ Now, we can calculate the average heat flux: $$ q'' = \frac{\dot{Q}}{A} = \frac{87.7\, \frac{J}{s}}{3.208 \times 10^{-5}\, m^2} = 2.733 \times 10^6\, \frac{J}{m^2\cdot s} $$ (d) Calculate the number of valves heat treated per day: First, we find the number of cycles per day: $$ \text{Number of cycles} = \frac{10\, hr}{5\, min} = \frac{10\, hr}{\frac{1}{12}\, hr} = 120 $$ Now, we can calculate the total number of valves that can be heat treated per day: $$ \text{Number of valves per day} = (25\, \text{valves})(120) = 3000\, \text{valves} $$

Step-by-step solution

(a) Calculate the amount of heat transfer

To calculate the amount of heat transfer (Q) required to heat a single valve from 40°C to 800°C, we will use the formula: $$ Q = mc_p\Delta T $$ where: - \(m\) is the mass of the valve (0.0788 kg) - \(c_p\) is the specific heat of the valve (440 J/kg·K) - \(\Delta T\) is the change in temperature (800°C - 40°C) We can plug in the given values to find the amount of heat transfer: $$ Q = (0.0788\, kg)(440\, \frac{J}{kg\cdot K})(800\, ^{\circ}C - 40\, ^{\circ}C) $$

(b) Calculate the average rate of heat transfer

To calculate the average rate of heat transfer (\(\dot{Q}\)), we will use the formula: $$ \dot{Q} = \frac{Q}{t} $$ where: - \(Q\) is the amount of heat transfer calculated in step (a) - \(t\) is the time for the heat treatment (5 min) We can plug in the calculated heat transfer value and the given time to find the average rate of heat transfer: $$ \dot{Q} = \frac{Q}{5\, min} $$

(c) Calculate the average heat flux

To calculate the average heat flux (\(q''\)), we will use the formula: $$ q'' = \frac{\dot{Q}}{A} $$ where: - \(\dot{Q}\) is the average rate of heat transfer calculated in step (b) - \(A\) is the surface area of the valve Since the valve head and stem are assumed to be of equal surface area, we can say the total surface area of the valve is \(2A\) (considering both head and stem area): $$ m = \rho V = \rho (A_{stem}h + A_{head}h) $$ Given that \(A_{stem} = A_{head}\), we can divide this equation by the density (\(\rho\)) to obtain the total surface area (\(A\)) in terms of the mass (\(m\)): $$ A = \frac{m}{h\rho} $$ We can plug in the values provided for mass, length, and density of the valve, and then calculate the average heat flux using the equation above.

(d) Calculate the number of valves heat treated per day

To calculate the number of valves that can be heat treated per day, we will use the information provided about the heating section capacity and its daily usage time. We will multiply the number of valves that can be held in the heating section (25 valves) by the number of cycles the heating section can run per day. First, we will find the number of cycles that can be run per day: $$ \text{Number of cycles} = \frac{\text{Total daily usage time}}{\text{Time per cycle}} $$ Using the given values for total daily usage time (10 hours) and time per cycle (5 min), we can calculate the number of cycles per day. Finally, we can multiply this value by the number of valves that can be held in the heating section (25 valves) to find the total number of valves that can be heat treated per day.