Question

A long cylindrical stainless steel rod \(\left(c_{p}=\right.\) \(450 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \rho=7900 \mathrm{~kg} / \mathrm{m}^{3}, \varepsilon=0.30\) ) with mechanically polished surface is being conveyed through a water bath to be quenched. The \(25-\mathrm{mm}\)-diameter stainless steel rod has a temperature of \(700^{\circ} \mathrm{C}\) as it enters the water bath. \(\mathrm{A}\) length of \(3 \mathrm{~m}\) of the rod is submerged in water as it is conveyed through the water bath during the quenching process. As the stainless steel rod enters the water bath, boiling would occur at \(1 \mathrm{~atm}\). In order to prevent thermal burn on people handling the rod, it must exit the water bath at a temperature below \(45^{\circ} \mathrm{C}\). Determine the speed of the rod being conveyed through the water bath so that it leaves the water bath without the risk of thermal burn hazard.

Short answer

What is the speed of the stainless steel rod as it moves through the water bath for quenching to ensure that the rod exits the water bath with a temperature below \(45^{\circ} \mathrm{C}\), given the following information: - The stainless steel rod has a diameter of \(25\,\mathrm{mm}\), a length of \(3\,\mathrm{m}\), and a density of \(7900\,\mathrm{kg/m^3}\) - The initial temperature of the rod is \(800^{\circ} \mathrm{C}\) and the required final temperature is \(45^{\circ} \mathrm{C}\) - The heat transfer coefficient is given by the formula: \(h = 6.124 \times εv^{0.805}\), where ε is the emissivity and has a value of 0.30

Step-by-step solution

Calculate the mass of the stainless steel rod

Before we calculate the speed, let's first calculate the mass of the stainless steel rod. Since we know its density (\(\rho = 7900\,\mathrm{kg/m^3}\)), diameter (\(D = 25\,\mathrm{mm}\)), and length (\(L = 3\,\mathrm{m}\)), we can determine the mass using the formula: Mass = Density × Volume $$m = \rho \times V$$ Where \(V\) is the volume, which can be calculated as: $$V = \pi \times r^2 \times L$$ Now, let's find the mass (m).

Calculate the heat transfer rate required

Now we will calculate the heat transfer rate (Q) necessary to bring the temperature of the stainless steel rod down below \(45^{\circ} \mathrm{C}\) using the formula from the analysis: $$Q = mc_p(T_{initial} - T_{final}) $$ We plug in the values given and the mass that we calculated in step 1.

Determine the speed of the rod

Given the heat transfer rate (Q) and the length of the submerged rod, we can now calculate the speed (v) at which the rod must be conveyed through the water bath so that it leaves the water bath without the risk of thermal burn hazard. To do this, we'll use the formula: $$v = \frac{Q}{hA(T_{initial} - T_{final})L} $$ Where: - h is the heat transfer coefficient [W/m²K] - A is the surface area of the rod [m²] - L is the length of the rod submerged in the water bath [m] We already have Q, now let's calculate the surface area A, knowing the diameter D and length L: $$ A = πDL $$ Since the given ε (emissivity) is 0.30, we assume that steel rod is not a black body. For black bodies, the heat transfer coefficient (h) between the steel rod and water bath can be calculated by the equation: $$ h = 6.124 \times εv^{0.805}$$ Now that we have the required information, we can determine the speed of the stainless steel rod by plugging in the values in the formula for v. After finding the speed of the stainless steel rod, we can conclude that conveying it at this speed through the water bath will allow it to leave the bath without posing a risk of thermal burn hazard.

Key concepts

Thermal Burn Hazard

The risk of a thermal burn hazard arises when handling materials that are too hot, leading to potential injuries. In scenarios involving industrial processes, especially with metals, it's crucial to manage their temperatures as they are handled.

• Thermal burns occur when skin comes into contact with hot surfaces, causing damage to tissue.
• Safety standards require that materials should cool down to a safe temperature to prevent burns.
• In this exercise, the stainless steel rod must be under 45°C when it exits the water bath to prevent burns.

Understanding and preventing thermal burns is a critical safety aspect of industrial operations, ensuring that both the process and personnel remain secure.

Stainless Steel

Stainless steel is a durable, corrosion-resistant alloy commonly used in various applications, including industrial tools and components. This material has properties that make it suitable for processes requiring high temperatures and potential exposure to moisture.

• Its high density and specific heat capacity mean it can store significant amounts of thermal energy.
• It's often mechanically polished for improved surface finish, which affects its heat transfer characteristics.
• Stainless steel's thermal properties make it a popular choice for heat-intensive manufacturing processes.

Understanding these characteristics helps in designing processes that efficiently manage temperature changes, like the quenching process detailed in this exercise.

Quenching Process

The quenching process is a rapid cooling method that changes the properties of metals. This technique is frequently used to improve hardness or toughness, but it also plays an essential role in safely reducing temperatures.

• The process involves immersing the metal in a liquid, commonly water or oil, which absorbs heat quickly.
• This exercise specifically looks at using a water bath at atmospheric pressure to cool down a stainless steel rod from 700°C.
• Quenching requires precise control to avoid warping or cracking due to thermal shock.

The success of quenching comes down to accurate calculations of time and speed to ensure metals cool to safe temperatures without defects.

Heat Transfer Coefficient

The heat transfer coefficient is a parameter that quantifies the rate of heat exchange between surfaces and fluids. It plays a crucial role in determining how quickly a metal can be cooled in a quenching process.

• The coefficient depends on factors such as material surface properties, fluid type, and flow conditions.
• In this solution, it helps calculate the speed needed for the rod to travel through the water bath to achieve the desired cooling.
• Using the given emissivity, this value was derived from the properties of the steel rod and the behavior of water in the bath.

Calculating the heat transfer coefficient precisely is key for ensuring that safety temperatures are met and material properties are maintained post-quench.