Question

The upper surface of an ASTM B152 copper plate is subjected to convection with hot air flowing at \(6 \mathrm{~m} / \mathrm{s}\) parallel over the plate surface. The length of the plate is \(1 \mathrm{~m}\), and the temperature of the hot air is \(400^{\circ} \mathrm{C}\). The maximum use temperature for the ASTM B152 copper plate is \(260^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A-1M). Determine the variation of the local heat flux with the thermal boundary layer thickness on the plate surface for \(0

Short answer

To summarize: In this problem, we aimed to find the variation of the local heat flux with the thermal boundary layer thickness on a copper plate subjected to convection with hot air. We followed these steps: 1. Calculated the Reynolds number and the Prandtl number, which are dimensionless numbers used in fluid dynamics and heat transfer analysis. 2. Found the Nusselt number using appropriate empirical correlations based on the Reynolds and Prandtl numbers. 3. Determined the local heat transfer coefficient and the thermal boundary layer thickness using the Nusselt number, thermal conductivity, and the distance along the plate. 4. Found the local heat flux using the local heat transfer coefficient and the temperature difference between the hot air and the copper plate. 5. Plotted the required graphs (thermal boundary layer thickness vs. distance along the plate and local heat flux vs. thermal boundary layer thickness) to analyze the variation of the local heat flux with thermal boundary layer thickness on the copper plate surface.

Step-by-step solution

Calculate the Reynolds number and the Prandtl number

First, we need to find the Reynolds number (Re) and the Prandtl number (Pr) using the given air properties and flow velocity. The Reynolds number and the Prandtl number are dimensionless numbers used in fluid dynamics and heat transfer analysis. The Reynolds number is given by: \(Re = \frac{\rho V x}{\mu}\) The Prandtl number is given by: \(Pr = \frac{c_p \mu}{k}\)

Find the Nusselt number using appropriate correlations

Next, we will use the Reynolds number (Re) and the Prandtl number (Pr) to find the Nusselt number (Nu) using appropriate empirical correlations. For this problem, we can use the isothermal flat plate correlation, where the flow is considered turbulent, given by: \(Nu_x = 0.664 Re_x^{1/2} Pr^{1/3}\)

Calculate the local heat transfer coefficient and the thermal boundary layer thickness

Now, we calculate the local heat transfer coefficient (h) using the Nusselt number (Nu), the thermal conductivity (k), and the distance (x) along the plate. \(h = \frac{Nu_x \cdot k}{x}\) We can also find the thermal boundary layer thickness (\(\delta_t\)) using the following relation: \(\delta_t = \frac{x}{(0.664\cdot Pr^{1/3})\cdot Re_x^{1/2}}\)

Determine the local heat flux

We can now find the local heat flux (q) using the local heat transfer coefficient (h) and the temperature difference between the hot air and the copper plate. \(q = h \cdot (T_{air} - T_{copper})\)

Plot the required graphs

Finally, using the calculated thermal boundary layer thickness and local heat flux values for various values of x, we can plot the graphs for: (a) Thermal boundary layer thickness vs Distance along the plate (x) (b) Local heat flux vs Thermal boundary layer thickness Upon analyzing the plots, we should be able to understand the variation of the local heat flux with thermal boundary layer thickness on the copper plate surface.