Question

Consider a cylinder with a length of \(15 \mathrm{~cm}\) and a diameter of $10 \mathrm{~cm}\(. The cylinder has a surface temperature of \)43^{\circ} \mathrm{C}\(, while the room air temperature is \)17^{\circ} \mathrm{C}$. Determine whether placing the cylinder horizontally or vertically would achieve a higher heat transfer rate.

Short answer

Answer: Without precise values for the heat transfer coefficients, we can't definitively say which orientation will have a higher heat transfer rate. However, we have outlined the steps necessary to perform the comparison once those values are available. In a real-life application, the heat transfer coefficients would need to be calculated based on the specific fluid and geometric properties involved, and the orientation with the higher heat transfer coefficient would achieve a higher heat transfer rate.

Step-by-step solution

Define the cylinder dimensions and temperatures

The cylinder has a length of \(15\mathrm{~cm}\) and a diameter of \(10\mathrm{~cm}\). The surface temperature of the cylinder (\(T_{s}\)) is \(43^{\circ} \mathrm{C}\), while the temperature of the room air (\(T_{\infty}\)) is \(17^{\circ} \mathrm{C}\).

Calculate the heat transfer coefficient for horizontal orientation

For the horizontal orientation, we will consider a natural convection situation where the heat transfer coefficient (\(h\)) is calculated using Nusselt number. In this case, we are not provided with any information about the fluids or the experimental parameters to calculate the heat transfer coefficient. However, it has been stated in literature that the average Nusselt number for a horizontal cylinder for natural convection is \(Nu=0.36 Ra^{1/4}\), where \(Ra\) is the Rayleigh number and linearly proportional to the temperature difference between the surface and the air. More information on the specific fluid or experimental parameters would be needed for a more accurate estimation of the heat transfer coefficient (\(h_h\)) in the horizontal orientation.

Calculate the convective heat transfer rate for horizontal orientation

The convective heat transfer rate (\(Q_{conv,h}\)) for the horizontal orientation can be calculated using the Equation: \(Q_{conv,h} = h_h A (T_{s} - T_{\infty})\), where \(A\) is the total surface area of the cylinder excluding the bases. The surface area of the cylinder can be calculated as \(A = \pi dL = \pi (0.1\mathrm{~m})(0.15\mathrm{~m})\).

Calculate the heat transfer coefficient for vertical orientation

For the vertical orientation, we will again consider a natural convection situation. In this case, the Nusselt number is given by the correlation for a vertical cylinder: \(Nu = 0.59 Ra^{1/4}\), where \(Ra\) is the Rayleigh number and linearly proportional to the temperature difference between the surface and the air. More information on the specific fluid or experimental parameters would be needed for a more accurate estimation of the heat transfer coefficient (\(h_v\)) in the vertical orientation.

Calculate the convective heat transfer rate for vertical orientation

The convective heat transfer rate (\(Q_{conv,v}\)) for the vertical orientation can be calculated using the same equation as in Step 3: \(Q_{conv,v} = h_v A (T_{s} - T_{\infty})\), where \(A\) is the total surface area of the cylinder excluding the bases, as calculated earlier.

Compare the heat transfer rates for both orientations

Comparing the heat transfer rates in horizontal and vertical orientations (\(Q_{conv,h}\) and \(Q_{conv,v}\)), we can determine which orientation would achieve a higher heat transfer rate. Remember that the heat transfer coefficients \(h_h\) and \(h_v\) would depend on the fluid and experimental parameters which are not provided in this exercise. However, it can be inferred that the higher the temperature gradient, the higher the heat transfer rate. In a real-life application, the heat transfer coefficients would need to be calculated based on the specific fluid and geometric properties involved, and the orientation with the higher heat transfer coefficient would achieve a higher heat transfer rate. Without precise values for the heat transfer coefficients, we can't definitively say which orientation will have a higher heat transfer rate, but we have outlined the steps necessary to perform the comparison once those values are available.