Question

Air is flowing in parallel over the upper surface of a flat plate with a length of \(4 \mathrm{~m}\). The first half of the plate length, from the leading edge, has a constant surface temperature of \(50^{\circ} \mathrm{C}\). The second half of the plate length is subjected to a uniform heat flux of $86 \mathrm{~W} / \mathrm{m}^{2}$. The air has a free-stream velocity and temperature of \(2 \mathrm{~m} / \mathrm{s}\) and \(10^{\circ} \mathrm{C}\), respectively. Determine the local convection heat transfer coefficients at $1 \mathrm{~m}\( and \)3 \mathrm{~m}$ from the leading edge. As a first approximation, assume the boundary layer over the second portion of the plate with uniform heat flux has not been affected by the first half of the plate with constant surface temperature. Evaluate the air properties at a film temperature of \(30^{\circ} \mathrm{C}\). Is the film temperature \(T_{f}=30^{\circ} \mathrm{C}\) applicable at \(x=3 \mathrm{~m}\) ?

Short answer

Based on the given data, the local convection heat transfer coefficients at 1m and 3m from the leading edge of the flat plate are approximately 74.52 W/m²·K and 36.53 W/m²·K, respectively. However, after checking the applicability of the initial film temperature assumption of 30°C, it is found that it is not applicable at x=3m as the calculated film temperature is significantly different (11.15°C). Further adjustments to the film temperature should be made for more accurate results at x=3m.

Step-by-step solution

Calculate the Reynolds number at x=1m and x=3m

At the given film temperature, we can look up the properties of air: kinematic viscosity (v), thermal conductivity (k), and Prandtl number (Pr). For a film temperature of \(30^{\circ} \mathrm{C}\) (use tables or engineering software to find these values): - Kinematic viscosity, \(\nu = 15.68 * 10^{-6} \thinspace \mathrm{m^2/s}\) - Thermal conductivity, \(k = 0.0262 \thinspace \mathrm{W/m \cdot K}\) - Prandtl number, \(Pr = 0.71\) Now, let's calculate the Reynolds number (Re) at x=1m and x=3m. The Reynolds number is given by \(Re_x = \frac{U_\infty x}{\nu}\). For x=1m: \(Re_1 = \frac{2 \thinspace \mathrm{m/s} \cdot 1 \thinspace \mathrm{m}}{15.68 * 10^{-6} \thinspace \mathrm{m^2/s}} = 127536\) For x=3m: \(Re_3 = \frac{2 \thinspace \mathrm{m/s} \cdot 3 \thinspace \mathrm{m}}{15.68 * 10^{-6} \thinspace \mathrm{m^2/s}} = 382608\)

Calculate the Nusselt numbers in both constant surface temperature and constant heat flux parts of the plate

Now, we will use the relationships for the Nusselt numbers for both regions of the plate and the corresponding Reynolds and Prandtl numbers. For the constant surface temperature part of the plate (first half), we have: \(Nu_x = 0.332 \thinspace Re_x^{1/2} \thinspace Pr^{1/3}\) For constant heat flux part of the plate (second half), we have: \(Nu_x = 0.453 \thinspace Re_x^{1/2} \thinspace Pr^{1/3}\) Since x=1m lies in the constant surface temperature part, for x=1m: \(Nu_1 = 0.332 \thinspace (127536)^{1/2} \thinspace (0.71)^{1/3} = 74.52\) Since x=3m lies in the constant heat flux part, for x=3m: \(Nu_3 = 0.453 \thinspace (382608)^{1/2} \thinspace (0.71)^{1/3} \approx 109.58\)

Compute the local convection heat transfer coefficients

The heat transfer coefficients can be calculated using the formula: \(h_x = \frac{Nu_x \cdot k}{x}\) For x=1m: \(h_1 = \frac{74.52 \thinspace \mathrm{(W/m \cdot K)}}{1 \thinspace \mathrm{m}} = 74.52 \thinspace \mathrm{W/m^2 \cdot K}\) For x=3m: \(h_3 = \frac{109.58 \thinspace \mathrm{(W/m \cdot K)}}{3 \thinspace \mathrm{m}} \approx 36.53 \thinspace \mathrm{W/m^2 \cdot K}\) So the local convection heat transfer coefficients at x=1m and x=3m from the leading edge are 74.52 W/m²·K and 36.53 W/m²·K, respectively.

Check the applicability of the film temperature Tf=30°C at x=3m

Now let's check if the film temperature of 30°C is applicable at x=3m. To do this, we need to find the local wall temperature at x=3m. Since the heat flux is constant, we have: \(T_w = T_{\infty} + \frac{q''x}{h_3}\) where \(T_{\infty}\) is the free-stream temperature (10°C) and \(q'' = 86 \thinspace \mathrm{W/m^2}\) is the constant heat flux. \(T_w = 10^{\circ} \mathrm{C} + \frac{86 \thinspace \mathrm{W/m^2} \cdot (3\thinspace \mathrm{m} - 2\thinspace \mathrm{m})}{36.53 \thinspace \mathrm{W/m^2 \cdot K}} \approx 12.3^{\circ} \mathrm{C}\) Now, to check the applicability of the film temperature of 30°C at x=3m, we need to verify if the average of the wall temperature and the free stream temperature is close to 30°C. \(T_f = \frac{T_w+T_\infty}{2} = \frac{12.3^{\circ} \mathrm{C} + 10^{\circ} \mathrm{C}}{2} = 11.15^{\circ} \mathrm{C}\) Since the film temperature at x=3m (11.15°C) is significantly different from the initial assumption of 30°C, we can conclude that the film temperature Tf=30°C is not applicable at x=3m.