Question

A 150-mm-diameter and 1-m-long rod is positioned horizontally and has water flowing across its outer surface at a velocity of $0.2 \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, the natural convection effects are not significant to the heat transfer process for the given horizontal rod with water flowing across its surface, as the ratio of Grashof number to Reynolds number squared is much smaller than 1.

Step-by-step solution

Calculate Reynolds number (Re)

The Reynolds number allows us to characterize the flow around the rod as laminar or turbulent. It is calculated by: \(Re = \frac{D\rho V}{\mu}\) where D is the diameter of the rod, ρ is the density of the water, V is the flow velocity, and μ is the dynamic viscosity. We need to look up the values of density and viscosity of water at 40°C to calculate Re properly. Density of water at 40°C = \(\rho = 992 \mathrm{~kg/m^3}\) Dynamic viscosity of water at 40°C = \(\mu = 6.52 \times 10^{-4} \mathrm{~Pa \cdot s}\)

Calculate the Reynolds number

Using the given values: \(Re = \frac{0.15 \mathrm{~m} \cdot 992 \mathrm{~kg/m^3} \cdot 0.2\mathrm{~m/s}}{6.52 \times 10^{-4} \mathrm{~Pa \cdot s}}\) \(Re \approx 4.56 \times 10^4\) The flow is turbulent since the Reynolds number is greater than \(4 \times 10^4\).

Calculate Grashof number (Gr)

The Grashof number allows us to characterize the importance of natural convection. It is calculated by: \(Gr = \frac{g \beta (T_s - T_\infty)D^3}{\nu^2}\) where g is the acceleration due to gravity, β is the coefficient of thermal expansion, \(T_s\) is the surface temperature, \(T_\infty\) is the fluid temperature far from the surface, D is the diameter of the rod, and ν is the kinematic viscosity. We need to look up the coefficient of thermal expansion and kinematic viscosity of water at 40°C to calculate Gr properly. Coefficient of thermal expansion at 40°C = \(\beta = 3.43 \times 10^{-4} \mathrm{K^{-1}}\) Kinematic viscosity of water at 40°C = \(\nu = 6.57 \times 10^{-7} \mathrm{m^2/s}\)

Calculate the Grashof number

Using the given values: \(Gr = \frac{9.81 \mathrm{m/s^2} \cdot 3.43 \times 10^{-4} \mathrm{K^{-1}} \cdot (120 - 40) \cdot (0.15 \mathrm{~m})^3}{(6.57 \times 10^{-7} \mathrm{m^2/s})^2}\) \(Gr \approx 3.62 \times 10^4\)

Calculate the ratio of Grashof number to Reynolds number squared (Gr/Re^2)

We can determine the importance of natural convection relative to forced convection by comparing the ratio of Gr/Re^2. Gr/Re^2 = \(\frac{3.62 \times 10^4}{(4.56 \times 10^4)^2}\) Gr/Re^2 \(\approx 1.74 \times 10^{-6}\) Since Gr/Re^2 is much smaller than 1, the forced convection is significantly more important than natural convection to the heat transfer process. Therefore, under these conditions, the natural convection effects are not important to the heat transfer process.