Question

Two metal plates are connected by a long ASTM B 98 copper-silicon bolt. A hot gas at \(200^{\circ} \mathrm{C}\) flows between the plates and across the cylindrical bolt. The diameter of the bolt is \(9.5 \mathrm{~mm}\), and the length of the bolt exposed to the hot gas is \(10 \mathrm{~cm}\). The average convection heat transfer coefficient for the bolt in crossflow is correlated with the gas velocity as \(h=24.6 \mathrm{~V}^{0.62}\), where \(h\) and \(V\) have the units \(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and $\mathrm{m} / \mathrm{s}$, respectively. The maximum use temperature for the ASTM B98 bolt is \(149^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A-2M). If the gas velocity is \(10.4 \mathrm{~m} / \mathrm{s}\), determine the minimum heat removal rate required to keep the bolt surface from going above the maximum use temperature.

Short answer

When a hot gas at 200 C flows between two metal plates connected by a bolt with a diameter of 9.5 mm, and the length of the bolt exposed to the hot gas is 10 cm; calculate the minimum heat removal rate required to keep the bolt surface temperature below the maximum use temperature of 149 C, given the average convection heat transfer coefficient correlation, h=24.6 V^0.62, and the gas velocity V=10.4 m/s. To solve this problem, follow these steps: Step 1: Calculate the convection heat transfer coefficient (h) using the given correlation and gas velocity. h = 24.6 * (10.4)^0.62 h ≈ 93.27 W/m²·K Step 2: Determine the bolt surface area exposed to the hot gas. d = 0.0095 m L = 0.1 m A = 2 * pi * (0.0095 / 2) * 0.1 A ≈ 0.00299 m² Step 3: Calculate the required heat removal rate (Q) using the heat transfer formula. Q = h * A * (T_gas - T_max) Q = 93.27 W/m²·K * 0.00299 m² * (200 C - 149 C) Q ≈ 44.50 W The minimum heat removal rate needed to keep the bolt surface temperature below 149 C is approximately 44.50 W.

Step-by-step solution

Find the Convection Heat Transfer Coefficient (h)

Using the given correlation for the average convection heat transfer coefficient, h = 24.6 V^0.62, and the gas velocity V = 10.4 m/s, we can calculate h as follows: h = 24.6 * (10.4)^0.62 Plug in the values, and solve the equation for h.

Determine Bolt Surface Area Exposed to Hot Gas

We are given the diameter of the bolt, d = 9.5 mm, and the length of the bolt exposed to the hot gas, L = 10 cm. First, convert these values to meters for consistent units: d = 0.0095 m L = 0.1 m Now, calculate the surface area of the cylindrical part of the bolt exposed to the hot gas using the formula A = 2 * pi * r * L, where r is the radius of the bolt (r = d/2): A = 2 * pi * (0.0095 / 2) * 0.1 Calculate A using the values given.

Calculate the Required Heat Removal Rate

Now that we have the convection heat transfer coefficient (h) and the surface area exposed to the hot gas (A), we can use the heat transfer formula to determine the required heat removal rate, Q: Q = h * A * (T_gas - T_max), where T_gas is the temperature of the gas and T_max is the maximum use temperature. We are given T_gas = 200 C and T_max = 149 C. Plug in the values for h, A, T_gas, and T_max and solve the equation for Q. Once we find the value of Q, we will have the minimum heat removal rate needed to keep the bolt surface from going above its maximum use temperature.