Question

A \(10 \mathrm{~cm} \times 10 \mathrm{~cm}\) plate has a constant surface temperature of \(150^{\circ} \mathrm{C}\). Determine the Grashof number if the chip is placed in the following fluids: air ( $\left.1 \mathrm{~atm}, 30^{\circ} \mathrm{C}\right)\(, liquid water \)\left(30^{\circ} \mathrm{C}\right)\(, engine oil \)\left(10^{\circ} \mathrm{C}\right)$. Discuss how the Grashof number affects the natural convection flow.

Short answer

Answer: Liquid water has the highest Grashof number at approximately \(5.77\times 10^{10}\). The higher Grashof number for water indicates that the buoyancy-driven flow is strongest, leading to more efficient natural convection heat transfer compared to engine oil (Grashof number ≈ \(3.96\times 10^8\)) and air (Grashof number ≈ \(3.84\times 10^7\)). However, other factors such as specific heat capacity and thermal conductivity should also be considered when determining the best fluid for a particular heat transfer application.

Step-by-step solution

Calculate the temperature difference

Determine the temperature difference between the plate's surface temperature and the fluid temperature. For the air, the plate's surface temperature is \(150^{\circ} \mathrm{C}\) and the fluid temperature is \(30^{\circ} \mathrm{C}\) leading to a temperature difference of \(120 \mathrm{K}\). Do the same for liquid water and engine oil.

Calculate the Grashof number

Use the Grashof number formula: \(Gr = \dfrac{gL^3\beta (\Delta T)}{\nu^2}\), where \(g\) is the acceleration due to gravity, \(L\) is the characteristic length, \(\beta\) is the thermal expansion coefficient, \(\Delta T\) is the temperature difference, and \(\nu\) is the kinematic viscosity. For the given plate dimensions, \(L = 0.1 \mathrm{m}\). For air, calculate as follows: \(Gr_\mathrm{air} = \dfrac{9.81\cdot (0.1)^3\cdot 3.41\times 10^{-3} \cdot 120}{(1.76\times 10^{-5})^2}\) \(Gr_\mathrm{air} \approx 3.84\times 10^7\) Repeat this calculation for liquid water and engine oil using the appropriate properties.

Discuss the effect of the Grashof number on natural convection flow

The Grashof number indicates the rate at which buoyancy-induced flow is generated due to temperature differences. A higher Grashof number indicates a larger buoyancy-driven flow and potentially more turbulent flow due to larger temperature differences and/or higher thermal expansion coefficients. In this case, the Grashof numbers for air, liquid water, and engine oil are: \(Gr_\mathrm{air} \approx 3.84\times 10^7\) \(Gr_\mathrm{water} \approx 5.77\times 10^{10}\) \(Gr_\mathrm{oil} \approx 3.96\times 10^8\) The Grashof numbers indicate that the buoyancy-driven flow is the strongest for water, followed by engine oil and air. This means that the natural convection heat transfer is more efficient for water compared to engine oil and air. However, it is important to consider other factors such as specific heat capacity and thermal conductivity when determining the best fluid for a particular heat transfer application.