Question

A \(0.6-\mathrm{m} \times 0.6-\mathrm{m}\) horizontal ASTM A203 B steel plate has its lower surface subjected to convection with cold, quiescent hydrogen gas at \(-70^{\circ} \mathrm{C}\). The minimum temperature suitable for the steel plate is \(-30^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A-1M). The lower plate surface has an emissivity of \(0.3\), and thermal radiation exchange occurs between the lower plate surface and the surroundings at \(-70^{\circ} \mathrm{C}\). Determine the heat addition rate necessary for keeping the lower plate surface temperature from dropping below the minimum suitable temperature.

Short answer

Now, plug in the values into the formula: Q_conv = 25 * 0.36 * (-30 - (-70)) Q_conv = 25 * 0.36 * 40 Q_conv = 360 W So, the heat loss through convection is 360 W. #tag_title#Step 2: Calculate heat loss through radiation #tag_content#To calculate the heat loss through radiation, we first need to find the emissivity (ε) of the steel plate. Given that an uncoated steel surface has an emissivity of about 0.8, we can use this value. Next, we need to calculate the surface temperature and gas temperature in Kelvin. T_surface_K = T_surface + 273.15 = -30 + 273.15 = 243.15 K T_gas_K = T_gas + 273.15 = -70 + 273.15 = 203.15 K Now, we can use the following formula to calculate the heat loss through radiation: Q_rad = ε * σ * A * (T_surface_K^4 - T_gas_K^4) where Q_rad = Heat loss through radiation (W) ε = Emissivity (unitless) σ = Stefan-Boltzmann constant (5.67 × 10^-8 W/(m^2 K^4)) T_surface_K = Surface temperature in Kelvin (K) T_gas_K = Gas temperature in Kelvin (K) Substitute the values and constants in the formula: Q_rad = 0.8 * 5.67 × 10^-8 * 0.36 * (243.15^4 - 203.15^4) Q_rad ≈ 63.17 W So, the heat loss through radiation is approximately 63.17 W. #tag_title#Step 3: Calculate the total heat addition rate required #tag_content#Now, we can add the heat loss rates through convection and radiation to find the total heat addition rate required to maintain the lower plate surface temperature at its minimum suitable temperature. Q_total = Q_conv + Q_rad Q_total = 360 W + 63.17 W Q_total = 423.17 W Hence, the heat addition rate required to maintain the lower plate surface temperature at its minimum suitable temperature is approximately 423.17 W.

Step-by-step solution

Calculate heat loss through convection

First of all, we need to determine the convective heat transfer coefficient (h) for hydrogen gas in order to calculate the heat loss through convection. Unfortunately, this information is not provided in the problem statement; therefore, we will assume a representative value for h. Let's take h = 25 W/(m^2 K) for the convective heat transfer coefficient, which is a reasonable value for quiet gases. Now, we can calculate the heat loss through convection using the following formula: Q_conv = h * A * (T_surface - T_gas) where Q_conv = Heat loss through convection (W) A = Surface area (m^2) T_surface = Surface temperature (in this case -30°C) (°C) T_gas = Gas temperature (in this case -70°C) (°C) h = Convective heat transfer coefficient (W/(m^2 K)) Substitute the given values and constants in the formula: A = 0.6 * 0.6 = 0.36 m^2 T_surface = -30°C T_gas = -70°C