Question

Hot carbon dioxide exhaust gas at \(1 \mathrm{~atm}\) is being cooled by flat plates. The gas at \(220^{\circ} \mathrm{C}\) flows in parallel over the upper and lower surfaces of a \(1.5\)-m-long flat plate at a velocity of $3 \mathrm{~m} / \mathrm{s}$. If the flat plate surface temperature is maintained at \(80^{\circ} \mathrm{C}\), determine \((a)\) the local convection heat transfer coefficient at \(1 \mathrm{~m}\) from the leading edge, \((b)\) the average convection heat transfer coefficient over the entire plate, and (c) the total heat flux transfer to the plate.

Short answer

Question: A flat plate with a length of 1.5 meters and a width of 1 meter is maintained at a surface temperature of 80°C and is exposed to a flow of carbon dioxide at 220°C and 2.5 m/s. Calculate (a) the local convection heat transfer coefficient at 1 meter from the leading edge, (b) the average convection heat transfer coefficient over the entire plate length, and (c) the total heat flux transfer to the plate.

Step-by-step solution

Determine the film temperature

Calculate the film temperature, \(T_f\), as the average of the surface temperature (\(T_s\)) and the gas temperature (\(T_g\)): $$T_f = \frac{T_s + T_g}{2}$$ where \(T_s = 80^{\circ}\mathrm{C}\) and \(T_g = 220^{\circ}\mathrm{C}\). Convert both temperatures to Kelvin by adding 273.15.

Find the Reynolds number

To find the Reynolds number, \(Re\), we will use the following formula: $$Re = \frac{\rho u L}{\mu}$$ Where \(\rho\) is the density of the gas, \(u\) is the velocity, \(L\) is the plate length, and \(\mu\) is the dynamic viscosity. Calculate the Reynolds number at \(1 \mathrm{~m}\) and \(1.5 \mathrm{~m}\). You can find the properties of carbon dioxide at the film temperature in a gas property table, including density and dynamic viscosity.

Determine local and average Nusselt numbers

Now that we have the Reynolds numbers, we need to find the local and average Nusselt numbers, which are dimensionless numbers used to calculate the convection heat transfer coefficient. For flow over a flat plate, we will use the following correlation to find the local Nusselt number (\(Nu_x\)) at \(1 \mathrm{~m}\): $$Nu_x = 0.664 Re_x^{1/2} Pr^{1/3}$$ To find the average Nusselt number (\(Nu_L\)) over the entire plate, we will use the following correlation: $$Nu_L = 0.664 Re_L^{1/2} Pr^{1/3}$$ Please note that \(Re_x\) and \(Re_L\) represent the Reynolds numbers calculated in Step 2 at \(1 \mathrm{~m}\) and \(1.5 \mathrm{~m}\), respectively. In both formulas, \(Pr\) is the Prandtl number, which can be found using gas property tables at the film temperature.

Calculate local and average convection heat transfer coefficients

Now we can calculate the local convection heat transfer coefficient, \(h_x\), at \(1 \mathrm{~m}\) and the average convection heat transfer coefficient, \(h_L\), over the entire plate using the following equations: $$h_x = \frac{Nu_x \cdot k}{x}$$ $$h_L = \frac{Nu_L \cdot k}{L}$$ Where \(k\) is the thermal conductivity of the gas, which can be found using gas property tables at the film temperature.

Determine the total heat flux transfer to the plate

Lastly, we can find the total heat transfer rate, \(q'\), using the following equation: $$q' = h_L A \Delta T$$ Where \(A\) is the surface area of the plate, and \(\Delta T = T_g - T_s\). In this problem, \(A\) can be calculated as the product of the plate's length and width. The plate width is not provided in the exercise, but let's assume it is \(1 \mathrm{~m}\) for simplicity. Now find the total heat flux transfer to the plate following the steps above.