Question

Hot water vapor flows in parallel over the upper surface of a 1-m-long plate. The velocity of the water vapor is \(10 \mathrm{~m} / \mathrm{s}\) at a temperature of \(450^{\circ} \mathrm{C}\). A coppersilicon (ASTM B98) bolt is embedded in the plate at midlength. The maximum use temperature for the ASTM B98 copper-silicon bolt is \(149^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A-2M). To devise a cooling mechanism to keep the bolt from getting above the maximum use temperature, it becomes necessary to determine the local heat flux at the location where the bolt is embedded. If the plate surface is kept at the maximum use temperature of the bolt, what is the local heat flux from the hot water vapor at the location of the bolt?

Short answer

In order to calculate the local heat flux at the bolt's location we need to perform the following steps: 1. Calculate the Reynolds and Prandtl numbers using the given velocity, temperature, and properties of water vapor. 2. Evaluate the Nusselt number based on the Reynolds and Prandtl numbers. 3. Determine the convective heat transfer coefficient using the Nusselt number, thermal conductivity, and characteristic length. 4. Apply the convective heat transfer equation to find the local heat flux with the calculated convective heat transfer coefficient, the difference in temperature between the plate and water vapor, and the surface area. Follow these steps and provide the calculated local heat flux.

Step-by-step solution

Calculate the Reynolds number and Prandtl number

First, we need to find the Reynolds and Prandtl numbers using the given velocity and temperature values. For water vapor at \(450^{\circ} \mathrm{C}\), the dynamic viscosity \(\mu = 2.61 \times 10^{-5} \mathrm{~Pa} \cdot \mathrm{s}\) and thermal conductivity \(k = 0.071 \mathrm{~W} / (\mathrm{m} \cdot \mathrm{K})\). From this, we can derive the Prandtl number, Pr: Pr = \(\frac{c_p \mu}{k}\), where \(c_p\) is the specific heat at constant pressure, which for water vapor is \(c_p = 2.1 \mathrm{~kJ} / (\mathrm{kg} \cdot \mathrm{K})\) at 450 °C. For Reynolds number Re, we have: Re = \(\frac{VD}{\nu}\), where \(V\) is the velocity, \(D\) is the characteristic length (1 meter in this case), and \(\nu\) is kinematic viscosity, which can be calculated using \(\frac{\mu}{\rho}\), with \(\rho\) being the mass density of water vapor at the given temperature. After calculating the Reynolds and Prandtl numbers, proceed to the next step.

Calculate the Nusselt number

Now that we have the Reynolds and Prandtl numbers, we can evaluate the Nusselt number. In this scenario, we'll assume that the flow is turbulent and apply the Dittus-Boelter equation for the Nusselt number: Nu = \(0.023 \times Re^{0.8} \times Pr^{0.3}\). Using the Reynolds and Prandtl numbers from Step 1, calculate the Nusselt number.

Determine the convective heat transfer coefficient

Convert the Nusselt number to the convective heat transfer coefficient \(h_c\): \(h_c = \frac{Nu \times k}{D}\) Use the calculated Nusselt number (Nu), the thermal conductivity (k), and the characteristic length (D = 1 meter) to calculate the convective heat transfer coefficient.

Calculate the local heat flux

Finally, apply the convective heat transfer equation to find the local heat flux at the location of the bolt: \(q = h_c \times A \times (T_S - T_R)\) where \(T_S = 149^{\circ} \mathrm{C}\), the plate surface temperature, and \(T_R = 450^{\circ} \mathrm{C}\), the temperature of the water vapor. Use the calculated convective heat transfer coefficient (\(h_c\)) and an area \(A=1\ \mathrm{m}^2\) (since the entire length of the plate is in contact with the vapor) to obtain the local heat flux from the hot water vapor at the bolt's location.