Question

A 12-cm-diameter and 15-m-long cylinder with a surface temperature of \(10^{\circ} \mathrm{C}\) is placed horizontally in air at $40^{\circ} \mathrm{C}$. Calculate the steady rate of heat transfer for the cases of (a) free-stream air velocity of \(10 \mathrm{~m} / \mathrm{s}\) due to normal winds and (b) no winds and thus a free-stream velocity of zero.

Short answer

Question: Calculate the steady rate of heat transfer for a horizontal cylinder in two different cases: (a) free-stream air velocity of 10 m/s and (b) free-stream velocity of zero. Given the diameter of the cylinder is 12 cm, the length of the cylinder is 15 m, the surface temperature of the cylinder is 10°C, and the ambient air temperature is 40°C.

Step-by-step solution

Given data

First, write down the given data: - Diameter of cylinder: \(D = 12 \mathrm{~cm} = 0.12 \mathrm{~m}\) - Length of cylinder: \(L = 15 \mathrm{~m}\) - Surface temperature of cylinder: \(T_s = 10^{\circ} \mathrm{C}\) - Ambient air temperature: \(T_\infty = 40^{\circ} \mathrm{C}\) - Free-stream air velocity: \(V_\infty\) (Different for each case)

Calculate the Reynolds number and Prandtl number

We need to determine the Reynolds number, \(Re_D\), and the Prandtl number, \(Pr\), in both cases. The Reynolds number can be calculated as: \(Re_D = \frac{VDρ}{μ}\) The Prandtl number can be calculated as: \(Pr = \frac{c_pμ}{k}\) Assuming air properties: - Density, \(ρ = 1.204 \mathrm{~kg/m^3}\), - Dynamic viscosity, \(μ = 1.8 \times 10^{-5} \mathrm{~kg/m s}\), - Specific heat, \(c_p = 1005 \mathrm{~J/kg K}\), - Thermal conductivity, \(k = 0.0255 \mathrm{~W/m K}\). Calculate \(Re_D\) and \(Pr\) for each case by substituting the values of \(V_\infty\) in the first case, \(V_\infty = 10 \mathrm{~m/s}\), and in the second case, \(V_\infty = 0\).

Calculate the Nusselt number

For both cases, we need to calculate the Nusselt number using the appropriate correlations for a cylinder in a cross-flow. For case (a) with a free-stream air velocity of 10 m/s, we can use the Churchill-Bernstein correlation (which is valid for \(0.4 < Pr < 4000\) and \(Re_D > 0\)): \(Nu_D = 0.3 + \frac{0.62Re_D^{1/2}Pr^{1/3}}{\left(1 + \frac{(0.4/Pr)^{2/3}}{Re_D}\right)^{1/4}}\) For case (b) with no winds, we can use the correlation for natural convection on a horizontal cylinder (which is valid for \(0.6 < Pr < 2000\) and \(10^4 < Gr_D < 10^{12}\), where \(Gr_D\) is the Grashof number): \(Nu_D = 0.6 + 0.387Ra_D^{1/6}\left(1 + \left(\frac{0.559}{Pr}\right)^{9/16}\right)^{8/27}\) First, calculate the Grashof number only for case (b): \(Gr_D = \frac{gL^3ρ^2\beta(T_s - T_\infty)}{μ^2}\) Assuming the coefficient of volume expansion, \(β = \frac{1}{T_\infty + 273.15 \mathrm{~K}}\) Then calculate \(Nu_D\) for both cases using the appropriate correlations.

Calculate the heat transfer coefficient

The heat transfer coefficient, \(h\), can be calculated as, \(h = \frac{Nu_Dk}{D}\) Calculate \(h\) for both cases using the calculated values of \(Nu_D\).

Calculate the steady rate of heat transfer

Finally, calculate the steady rate of heat transfer, \(Q\), using the convective heat transfer equation: \(Q = hA(T_s - T_\infty)\) The surface area of the cylinder, \(A = πDL\). Substitute the known values of \(h\), \(A\), \(T_s\), and \(T_\infty\) to find the steady rate of heat transfer in both cases.