Question

In cryogenic equipment, cold air flows in parallel over the surface of a \(2-\mathrm{m} \times 2-\mathrm{m}\) ASTM A240 \(410 \mathrm{~S}\) stainless steel plate. The air velocity is \(5 \mathrm{~m} / \mathrm{s}\) at a temperature of \(-70^{\circ} \mathrm{C}\). 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). The plate is heated to keep its surface temperature from going below \(-30^{\circ} \mathrm{C}\). Determine the average heat transfer rate required to keep the plate surface from getting below the minimum suitable temperature.

Short answer

Based on the given conditions, we have determined the average heat transfer rate needed to maintain the plate's surface temperature above -30°C is 13,795.2 Watts. This is achieved through a convective heat transfer process, with the calculated convective heat transfer coefficient being 86.22 W/(m²K).

Step-by-step solution

Calculate surface area of the plate

First, calculate the surface area of the plate. A = Length × Width = 2m × 2m = 4 m²

Calculate the convective heat transfer coefficient (h)

We can calculate the convective heat transfer coefficient using empirical correlations for flow over a flat plate. A commonly used correlation for flow over a flat plate with constant temperature is Churchill and Bernstein correlation (for air). The correlation is given by: \(h = 0.037 \times Re^{0.8} \times Pr^{1/3} \times \frac{(μ/μ_s)^{0.14}}{k/L}\) where, Re = Reynolds number = (density × velocity × Length) / dynamic viscosity Pr = Prandtl number = dynamic viscosity × specific heat / thermal conductivity μ = dynamic viscosity of air μ_s = dynamic viscosity of air at the plate's surface k = thermal conductivity of air L = length of the plate To calculate h, we first need to find Reynolds number (Re) and Prandtl number (Pr). From the given temperature -70°C and using air property tables, we can approximate the values of air properties: Density (ρ) = 1.12 kg/m³ Dynamic viscosity (μ) = 1.77e-5 kg/(ms) Specific heat (cp) = 1.009 kJ/(kgK) Thermal conductivity (k) = 2.99e-2 W/(mK) Calculate Re: Re = (ρ × v × L) / μ = (1.12 kg/m³ × 5 m/s × 2 m) / 1.77e-5 kg/(ms) = 63417 Calculate Pr: Pr = (μ × cp) / k = (1.77e-5 kg/(ms) × 1.009 kJ/(kgK)) / 2.99e-2 W/(mK) = 0.72 Assuming the air properties do not change significantly at surface temperature, we can approximate μ_s as μ (1.77e-5 kg/(ms)). Now, calculate convective heat transfer coefficient (h): h = 0.037 × 63417^0.8 × 0.72^(1/3) × (1.77e-5/1.77e-5)^0.14 / (2.99e-2 W/(mK)/2m) h = 86.22 W/(m²K)

Calculate heat transfer rate to maintain the plate's surface temperature

Now, we can use the convective heat transfer formula to determine the average heat transfer rate required to maintain the plate's surface temperature above -30°C: Q = h * A * (Ts - T∞) = 86.22 W/(m²K) × 4 m² × (-30 + 70) Q = 13795.2 W The average heat transfer rate required to keep the plate's surface temperature from going below the minimum suitable temperature is 13,795.2 Watts.