Question

A \(0.5-\mathrm{m} \times 0.5-\mathrm{m}\) vertical ASTM B152 copper plate has one surface subjected to convection with hot, quiescent air. The maximum use temperature for the ASTM B152 copper plate is \(260^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A-1M). If the rate of heat removal from the plate is \(90 \mathrm{~W}\), determine the maximum temperature that the air can reach without causing the surface temperature of the copper plate to increase above \(260^{\circ} \mathrm{C}\). Evaluate the properties of air at \(300^{\circ} \mathrm{C}\). Is this an appropriate temperature to evaluate the air properties?

Short answer

Answer: The maximum air temperature without causing the surface temperature of the copper plate to increase above 260°C is 245.6°C. Evaluating air properties at 300°C is appropriate for this problem as it is on the safe side.

Step-by-step solution

Determine the heat transfer coefficient

We know that the rate of heat removal (Q) from the plate is given by the equation: \[Q = hA(T_s - T_a)\] Where: - Q = 90 W (rate of heat removal) - h is the heat transfer coefficient (W/m²K), which will be determined - A is the plate area (0.5m x 0.5m) - T_s = 260°C (surface temperature of the copper plate) - T_a is the maximum air temperature Let's rearrange the equation to solve for h: \[h = \frac{Q}{A(T_s-T_a)}\] Since we are trying to find the maximum air temperature, we need to consider that the heat transfer coefficient will be different for different air temperatures. We will use the heat transfer coefficient for air at 300°C to start.

Use air properties at 300°C and calculate the heat transfer coefficient

The given temperature to evaluate air properties is 300°C. We assume a heat transfer coefficient for air at 300°C as 25 W/m²K (an approximate value for air in free convection at this temperature). Using this coefficient, we can calculate the maximum air temperature. Plug in the given values in the formula from step 1: \[h = \frac{90 \mathrm{~W}}{(0.5\mathrm{~m}\times0.5\mathrm{~m})(260^{\circ}\mathrm{C} - T_a)}\] Substitute 25 W/m²K for the heat transfer coefficient (h) and solve for T_a: \[25\,\frac{\mathrm{W}}{\mathrm{m}^2 \mathrm{K}} = \frac{90\mathrm{~W}}{(0.25\mathrm{~m}^2)(260^{\circ}\mathrm{C}-T_a)}\] Rearrange the equation and solve for the maximum air temperature (T_a): \[T_a = 260^{\circ} \mathrm{C} - \frac{90 \mathrm{~W}}{(0.25 \mathrm{~m}^2) (25\,\frac{\mathrm{W}}{\mathrm{m}^2 \mathrm{K}})}\] \[T_a \approx 260^{\circ}\mathrm{C} - 14.4^{\circ}\mathrm{C} = 245.6^{\circ}\mathrm{C}\]

Determine if 300°C is an appropriate temperature

Now we have found the maximum air temperature (245.6°C) without causing the surface temperature of the copper plate to increase above 260°C. We have evaluated this result using air properties for 300°C, which is above the maximum temperature found. Thus, the air properties at 300°C are on the safe side, and the value can be considered appropriate for evaluation. To summarize, the maximum air temperature without causing the surface temperature of the copper plate to increase above 260°C is 245.6°C. Evaluating air properties at 300°C is appropriate for this problem.