Question

A 150 -mm-diameter and 1 -cm-long vertical rod has water flowing across its outer surface at a velocity of \(0.5 \mathrm{~m} / \mathrm{s}\). The water temperature is uniform at \(40^{\circ} \mathrm{C}\), and the rod surface temperature is maintained at \(120^{\circ} \mathrm{C}\). Under these conditions, are the natural convection effects important to the heat transfer process?

Short answer

Answer: No, natural convection effects are not important to the heat transfer process in this case, as the forced convection is the dominant heat transfer mechanism.

Step-by-step solution

Calculate Grashof Number (Gr)

Grashof Number is a dimensionless number that represents the ratio of buoyancy force to viscous force acting on a fluid. It can be used to determine the importance of natural convection. The formula for Grashof Number for a vertical rod is: \(Gr = \frac{g \beta (T_s - T_{\infty}) d^3}{\nu^2}\) where \(g\) is the gravitational acceleration (9.81 m/s²), \(\beta\) is the coefficient of thermal expansion (for water, \(\beta \approx 3 \times 10^{-4} K^{-1}\)), \(T_s\) is the surface temperature of the rod (120°C), \(T_{\infty}\) is the water temperature (40°C), \(d\) is the diameter of the rod (150 mm = 0.15 m), and \(\nu\) is the kinematic viscosity of the fluid (for water at 40°C, \(\nu \approx 6.4 \times 10^{-7} m^2/s\)).

Compute Reynolds Number (Re)

Reynolds Number is a dimensionless number that represents the ratio of inertial forces to viscous forces within a fluid. It can be used to determine the flow regime and the importance of forced convection. The formula for Reynolds Number for a vertical rod is: \(Re = \frac{\rho U d}{\mu}\) where \(\rho\) is the fluid density (for water at 40°C, \(\rho \approx 992 kg/m³\)), \(U\) is the flow velocity (0.5 m/s), \(d\) is the diameter of the rod (0.15 m), and \(\mu\) is the dynamic viscosity of the fluid (for water at 40°C, \(\mu \approx 6.37 \times 10^{-4} kg/ms\)).

Compare Gr/Re²

To determine the importance of natural convection, we will compare the Grashof Number to the square of the Reynolds Number. If Gr/Re² is significantly larger than 1, the natural convection effects are expected to be important. If it is significantly smaller than 1, the forced convection would be the dominant heat transfer mechanism. Thus: \(Gr/Re² = \frac{g \beta (T_s - T_{\infty}) d^3}{\nu^2} \times \frac{\mu^2}{\rho^2 U^2 d^2}\)

Calculate and compare Gr/Re²

Plugging in the values obtained from Step 1 and Step 2, we get: \(Gr/Re² = \frac{9.81 \times 3 \times 10^{-4} K^{-1} \times (120 - 40) K \times (0.15)^3 m^3}{(6.4 \times 10^{-7} m^2/s)^2} \times \frac{(6.37 \times 10^{-4} kg/ms)^2}{(992 kg/m^3)^2 \times (0.5)^2 m^2/s^2}\) After calculating, we get \(Gr/Re² \approx 0.0076\) Since Gr/Re² is significantly smaller than 1, the forced convection is the dominant heat transfer mechanism in this situation, and the natural convection effects can be neglected. Therefore, the conclusion is that natural convection effects are not important to the heat transfer process in this case.