Question

A cryogenic gas flows at \(5 \mathrm{~m} / \mathrm{s}\) in parallel over the plate. The temperature of the cold gas is \(-50^{\circ} \mathrm{C}\), and the plate length is \(1 \mathrm{~m}\). The minimum temperature suitable for the ASTM A240 \(410 \mathrm{~S}\) plate is \(-30^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A-1M). In the interest of designing a heater, determine the total heat flux on the plate surface necessary to maintain the surface temperature at \(-30^{\circ} \mathrm{C}\). Use the following gas properties for the analysis: $c_{p}=1.002 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.02057 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \mu=1.527 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\(, and \)\rho=1.514 \mathrm{~kg} / \mathrm{m}^{3}$.

Short answer

Answer: The total heat flux required to maintain the surface temperature of the plate at -30°C is 2341.3 W/m².

Step-by-step solution

Convert temperatures to Kelvin

Let's start by converting the given temperatures to Kelvin so that our calculations are consistent. To convert from Celsius to Kelvin, add 273.15. \(T_{gas} = -50°C + 273.15 = 223.15 K\) \(T_{plate} = -30°C + 273.15 = 243.15 K\)

Calculate the Reynolds number

To determine the flow regime and heat transfer situation, we need to calculate the Reynolds number for the gas flowing over the plate. The formula for the Reynolds number is: \(Re = \frac{\rho * v * L}{\mu}\) Given values: - \(v = 5~m/s\) (gas velocity) - \(L = 1~m\) (plate length) - \(\rho = 1.514~kg/m^3\) (density of the gas) - \(\mu = 1.527 × 10^{-5}~kg/(m·s)\) (dynamic viscosity of the gas) \(Re = \frac{(1.514~kg/m^3) * (5~m/s) * (1~m)}{1.527 × 10^{-5}~kg/(m·s)} = 496397\)

Calculate the Prandtl number

We need the Prandtl number to find the average heat transfer coefficient. The Prandtl number is the ratio of momentum diffusivity to thermal diffusivity: \(Pr = \frac{c_p * \mu}{k}\) Given values: - \(c_p = 1.002~kJ/(kg·K)\) (specific heat at constant pressure) - \(k = 0.02057~W/(m·K)\) (thermal conductivity) First, let's convert the specific heat from \(kJ/(kg·K)\) to \(W/(kg·K)\), multiplying by 1000: \(c_p = 1.002~kJ/(kg·K) * 1000 = 1002~W/(kg·K)\) Now, calculate the Prandtl number: \(Pr = \frac{(1002~W/(kg·K)) * (1.527 × 10^{-5}~kg/(m·s))}{0.02057~W/(m·K)} = 0.743\)

Calculate the average heat transfer coefficient

The average heat transfer coefficient (\(h_{avg}\)) can be calculated using the correlations for flow over an isothermal flat plate. Since we have a Reynolds number of 496397, it's a turbulent flow. For turbulent flow over a flat plate, we use the following relation: \(h_{avg} = 0.036 * Re^{0.8} * Pr^{1/3} \frac{k}{L}\) Now, calculate the average heat transfer coefficient: \(h_{avg} = 0.036 * (496397)^{0.8} * (0.743)^{1/3} * \frac{0.02057~W/(m·K)}{1~m} = 117.04~W/(m^2·K)\)

Calculate the total heat flux

Now we have all the values required to calculate the heat flux on the plate surface. Use the popular formula for heat transfer by convection: \(q = h_{avg} * A * (T_{plate} - T_{gas})\) In our case, the area of the plate is not given, but we are looking for the heat flux per unit area, so we can ignore the area in the calculation. \(q = (117.04~W/(m^2·K)) * (243.15 K - 223.15 K) = 2341.3~W/m^2\) The total heat flux on the plate surface necessary to maintain the surface temperature at -30°C is 2341.3 W/m².