Question

After heat treatment, the \(2-\mathrm{cm}\)-thick metal \({ }_{\mathrm{a}}\) length of \(10 \mathrm{~m}\). The plates enter the cooling chamber at an initial temperature of \(500^{\circ} \mathrm{C}\). The cooling chamber maintains temperature of \(10^{\circ} \mathrm{C}\), and the convection heat transfer coefficient is given as a function of the air velocity blowing over the plates \(h=33 V^{0.8}\), where \(h\) is in \(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(V\) is in \(\mathrm{m} / \mathrm{s}\). To prevent any incident of thermal burn, it is necessary for the plates to exit the cooling chamber at a temperature below \(50^{\circ} \mathrm{C}\). In designing the cooling process to meet this safety criterion, use appropriate software to investigate the effect of the air velocity on the temperature of the plates at the exit of the cooling chamber. Let the air velocity vary from 0 to $40 \mathrm{~m} / \mathrm{s}$, and plot the temperatures of the plates exiting the cooling chamber as a function of air velocity at the moving plate speed of 2,5 , and \(8 \mathrm{~cm} / \mathrm{s}\).

Short answer

Answer: The temperature of the metal plates at the exit of the cooling chamber decreases with increasing air velocity. This is because the convection heat transfer coefficient increases with increasing air velocity, leading to a faster cooling rate for the metal plates. This can be determined by analyzing the cooling process using Newton's Law of Cooling, calculating the heat transfer rate as a function of air velocity, and plotting the exit temperatures for different plate speeds.

Step-by-step solution

Understand the problem variables

We are given the following variables: - Initial temperature of metal plates, \(T_i = 500^\circ\text{C}\). - Cooling chamber temperature, \(T_c = 10^\circ\text{C}\). - Convection heat transfer coefficient, \(h = 33 V^{0.8}\). - Air velocity, \(V\) (varying from \(0\) to \(40\,\text{m/s}\)). - Plate speeds: \(2\,\text{cm/s}\), \(5\,\text{cm/s}\) and \(8\,\text{cm/s}\) (different scenarios).

Apply Newton's Law of Cooling

We will use Newton's Law of Cooling to model the cooling process, which states: $$\dfrac{dT}{dt} = k (T-T_c)$$ where \(T\) is the temperature of the metal plate, \(T_c\) is the constant temperature of the cooling chamber, \(t\) is time, and \(k\) is a constant.

Calculate heat transfer rate

The heat transfer rate, \(Q\), can be calculated using the formula: $$Q = hA(T - T_c)$$ Where \(h\) is the convection heat transfer coefficient, \(A\) is the surface area of the metal plate, and \((T - T_c)\) is the temperature difference between the plate and the cooling chamber.

Relate heat transfer rate and air velocity

We can write the heat transfer rate as a function of air velocity using the given formula for \(h\): $$Q(V) = (33V^{0.8})A(T - T_c)$$

Solve the cooling rate equation

Now, we will solve the cooling rate equation for \(k\) and obtain an equation for the temperature \(T\) as a function of time \(t\) and air velocity \(V\). $$k = \dfrac{hA}{C}$$ where \(C\) is the heat capacity of the metal plate. Integrating the Newton's cooling law equation with this value of \(k\) will give us the temperature \(T\) as a function of time \(t\) and air velocity \(V\).

Determine the exit temperature

After obtaining the equation for \(T\), we can calculate the exit temperature of the metal plates for different air velocities and plate speeds. The exit temperature is the temperature of the plates after a certain amount of time spent in the cooling chamber, depending on their speed. Create a function that calculates the exit temperature as a function of air velocity for each of the given plate speeds and model the cooling process.

Plot the results

Now that we have the exit temperature data, we can plot the temperatures of the plates exiting the cooling chamber as a function of the air velocity. In this plot, we will have three curves representing the different plate speeds of \(2\,\text{cm/s}\), \(5\,\text{cm/s}\), and \(8\,\text{cm/s}\). To summarize, we used Newton's Law of Cooling to model the cooling process and calculated the heat transfer rate for different air velocities and plate speeds. We then plotted the exit temperature as a function of air velocity for each plate speed to investigate the effect of air velocity on the temperature of the plates at the exit of the cooling chamber.