Question

Air at \(20^{\circ} \mathrm{C}\) flows over a 4-m-long and 3 -m-wide surface of a plate whose temperature is \(80^{\circ} \mathrm{C}\) with a velocity of $7 \mathrm{~m} / \mathrm{s}$. The length of the surface for which the flow remains laminar is (a) \(0.9 \mathrm{~m}\) (b) \(1.3 \mathrm{~m}\) (c) \(1.8 \mathrm{~m}\) (d) \(2.2 \mathrm{~m}\) (e) \(3.7 \mathrm{~m}\) (For air, use $k=0.02735 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=0.7228, \nu=1.798 \times\( \)10^{-5} \mathrm{~m}^{2} / \mathrm{s}$ )

Short answer

(a) 1.0 m (b) 1.3 m (c) 1.5 m (d) 1.7 m #Answer# (b) 1.3 m

Step-by-step solution

Determine the Reynolds number for the transition

The critical Reynolds number for transition from laminar to turbulent boundary layer flow is approximately 5 x 10^5. So, we have: \(Re_{crit} = 5 \times 10^5\)

Use the Reynolds number to find the length for the transition

For the boundary layer flow, Reynolds number can be given by the equation: \(Re_x = \frac{\rho U_inf x}{\mu} = \frac{U_inf x}{\nu}\) Where \(Re_x\) is the Reynolds number, \(\rho\) is the density of the air, \(U_inf\) is the freestream velocity, \(x\) is the length for the transition point, \(\mu\) is the dynamic viscosity, and \(\nu\) is the kinematic viscosity. Use the given values for the freestream velocity and kinematic viscosity along with the critical Reynolds number to solve for 'x': \(5 \times 10^5 = \frac{7 x}{1.798 \times 10^{-5}}\)

Solve for the transition length 'x'

Re-arrange the equation from the previous step to solve for 'x' and plug in the given values to calculate the transition length: \(x = \frac{5 \times 10^5 \times 1.798 \times 10^{-5}}{7}\) \(x = 1.284 \mathrm{~m} \approx 1.3 \mathrm{~m}\) Therefore, the length of the surface for which the flow remains laminar is: Answer: (b) \(1.3 \mathrm{~m}\)