Question

Polyvinylchloride automotive body panels \((k=0.092\) $\left.\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, c_{p}=1.05 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, \rho=1714 \mathrm{~kg} / \mathrm{m}^{3}\right), 1 \mathrm{~mm}$ thick, emerge from an injection molder at \(120^{\circ} \mathrm{C}\). They need to be cooled to \(40^{\circ} \mathrm{C}\) by exposing both sides of the panels to \(20^{\circ} \mathrm{C}\) air before they can be handled. If the convective heat transfer coefficient is $15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and radiation is not considered, the time that the panels must be exposed to air before they can be handled is (a) \(0.8 \mathrm{~min}\) (b) \(1.6 \mathrm{~min}\) (c) \(2.4 \mathrm{~min}\) (d) \(3.1 \mathrm{~min}\) (e) \(5.6 \mathrm{~min}\)

Short answer

A) 3000 s B) 1800 s C) 2700 s D) 4500 s

Step-by-step solution

Calculate the initial temperature difference

First, we need to find the initial temperature difference between the PVC panels and the air. This can be calculated using the following formula: $$\Delta T_initial = T_{panel} - T_{air}$$ Where, \(T_{panel} = 120 °C\) is the initial temperature of the panels \(T_{air} = 20 °C\) is the temperature of the air

Calculate the heat transfer surface area

Next, let's find the heat transfer surface area (\(A\)) of the PVC panels. Since the panels are 1mm thick and both sides are exposed to air, the surface area is given by: $$A = 2 \times 1 \times 10^{-3} m²$$

Calculate the heat transfer coefficient for PVC panels

Now, we need to find the overall heat transfer coefficient for the PVC panels. We can use the following formula: $$\frac{1}{h_{o}} = \frac{1}{h_{conv}} + \frac{\delta}{k}$$ Where, \(h_{o}\) is the overall heat transfer coefficient \(\delta = 1 \times 10^{-3} m\) is the thickness of the panel \(h_{conv} = 15 \ \frac{W}{m².K}\) is the convective heat transfer coefficient \(k = 0.092 \ \frac{W}{m.K}\) is the thermal conductivity of the panels Solving for \(h_{o}\): $$h_{o} = \frac{1}{\frac{1}{15}+\frac{1*10^{-3}}{0.092}} $$

Calculate the total heat loss

Now, we can calculate the total heat loss from the panels during cooling. We can use the following energy balance equation: $$Q = m_{pvc} \cdot c_{p} \cdot \Delta T$$ Where, \(Q\) is the total heat loss \(m_{pvc}\) is the mass of the PVC panels \(c_{p} = 1.05 \ \frac{kJ}{kg.K} = 1050 \ \frac{J}{kg.K}\) is the specific heat capacity of the panels \(\Delta T = 120°C - 40°C = 80 °C\) is the temperature difference during cooling

Calculate the mass of the PVC panels

To find the mass of the PVC panels, we need to know their volume. Since we are not given the shape or dimensions of the panels, we assume they are cooled uniformly and are in perfect contact with air. So we can write: $$m_{pvc} = \rho \cdot V$$ Where, \(\rho = 1714 \ \frac{kg}{m³}\) is the density of the PVC \(V\) is the volume of the panel According to our assumptions, the volume \(V\) is not affecting the result, hence the mass \(m_{pvc}\) can be treated as a constant factor.

Calculate the time needed for the panels to cool

Finally, we can calculate the required time (\(t\)) for the panels to cool down using the following formula: $$Q = h_{o} \cdot A \cdot \Delta T \cdot t$$ Solving for \(t\): $$t= \frac{Q}{h_{o}A\Delta T}$$ Choose the correct answer from the given multiple choice options. By following these steps, the student should be able to calculate the time it takes for the PVC automotive body panels to cool down from 120°C to 40°C by exposing them to 20°C air.