Question

Hydrogen gas at \(1 \mathrm{~atm}\) is flowing in parallel over the upper and lower surfaces of a 3 -m-long flat plate at a velocity of $2.5 \mathrm{~m} / \mathrm{s}\(. The gas temperature is \)120^{\circ} \mathrm{C}$, and the surface temperature of the plate is maintained at \(30^{\circ} \mathrm{C}\). Using appropriate software, investigate the local convection heat transfer coefficient and the local total convection heat flux along the plate. By varying the location along the plate for \(0.2 \leq x \leq 3 \mathrm{~m}\), plot the local convection heat transfer coefficient and the local total convection heat flux as functions of \(x\). Assume flow is laminar, but make sure to verify this assumption.

Short answer

Based on the given information and our step-by-step solution, we can say that the local convection heat transfer coefficient and the local total convection heat flux can be successfully calculated using the relevant equations and properties of hydrogen gas, providing us with a clearer understanding of the heat transfer behavior along the flat plate. Make sure to verify the laminar flow assumption using the calculated Reynolds number and use the appropriate correlations, equations, and fluid properties to obtain accurate results. Plotting the results will allow for easy visualization of the heat transfer behavior as a function of the position along the flat plate.

Step-by-step solution

Calculate the Reynolds number

First, we need to calculate the Reynolds number, Re, to verify the laminar flow assumption. The Reynolds number is given by: Re = \(\frac{ρ u L}{\mu}\), where \(ρ\) is the density of the fluid, \(u\) is the fluid velocity, \(L\) is the characteristic length, and \(\mu\) is the dynamic viscosity of the fluid. For hydrogen gas at \(120^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\), we can look up the properties in a standard table or use an online fluid properties calculator. In this case, the density of the hydrogen gas is approximately \(ρ = 0.042 \mathrm{~kg/m^3}\) and the dynamic viscosity is approximately \(\mu = 9.74E{-6} \mathrm{~Pa \cdot s}\). Given that the fluid velocity is \(u = 2.5 \mathrm{~m/s}\), we calculate the Reynolds number at various \(x\) locations.

Calculate the local Nusselt number

Given that the flow is assumed to be laminar, we can use an appropriate correlation for the Nusselt number in the form: Nu\(_x\) = \(0.664 \sqrt{\mathrm{Re}_x} \mathrm{Pr}^{1/3}\), where Nu\(_x\) is the local Nusselt number, Re\(_x\) is the local Reynolds number at location \(x\), and Pr is the Prandtl number. For hydrogen gas at \(120^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\), the Prandtl number is approximately Pr = 0.69. We calculate the local Nusselt number at various \(x\) locations using this equation.

Calculate the local convection heat transfer coefficient

The local convection heat transfer coefficient, \(h_x\), can be calculated using the local Nusselt number as: \(h_x\) = \(\frac{\mathrm{Nu}_x \cdot k}{x}\), where \(k\) is the thermal conductivity of the hydrogen gas. For hydrogen gas at \(120^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\), the thermal conductivity is approximately \(k = 0.025\mathrm{~W/(m \cdot K)}\). Calculate \(h_x\) at various \(x\) locations using the calculated Nusselt numbers from the previous step.

Calculate the local total convection heat flux

The local total convection heat flux, \(q_x\), can be calculated using the formula: \(q_x\) = \(h_x \cdot (T_g - T_s)\), where \(T_g\) is the gas temperature (in Kelvin) and \(T_s\) is the surface temperature (in Kelvin). Convert the given temperatures to Kelvin and then calculate \(q_x\) at various \(x\) locations along the plate using the calculated local heat transfer coefficients from the previous step.

Plot the local convection heat transfer coefficient and the local total convection heat flux

Now we have the required data at various \(x\) locations along the plate, it's time to plot the local convection heat transfer coefficient and the local total convection heat flux as functions of \(x\). You can use any appropriate software, like Microsoft Excel or Python libraries like Matplotlib, to plot these graphs.