Question

A \(0.5-\mathrm{m} \times 0.5-\mathrm{m}\) vertical ASTM A240 410S stainless steel plate has one surface subjected to convection with a cold, quiescent gas at \(-70^{\circ} \mathrm{C}\). The type of cold gas that the plate surface is exposed to alternates between carbon dioxide and hydrogen. The minimum temperature suitable for the stainless steel plate is \(-30^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A-1M). Determine the heat addition rate necessary for keeping the plate surface temperature from dropping below the minimum suitable temperature, such that it is applicable to both carbon dioxide gas and hydrogen gas.

Short answer

Answer: The main goal is to ensure that the surface temperature of the steel plate does not drop below -30°C for both carbon dioxide and hydrogen gases.

Step-by-step solution

State the convection heat transfer equation

The convection heat transfer equation is given by: \(q = hA(T_s - T_\infty)\) where \(q\) is the heat transfer rate, \(h\) is the convection heat transfer coefficient, \(A\) is the surface area, \(T_s\) is the surface temperature, and \(T_\infty\) is the fluid temperature.

Calculate the convective heat transfer coefficient for carbon dioxide and hydrogen

Before we calculate the needed heat addition rate, we would need the heat transfer coefficients for carbon dioxide and hydrogen. However, this exercise does not provide sufficient information to directly compute these coefficients. You must be provided with additional values, such as fluid properties (thermal conductivity, viscosity, etc.) or relationships that allow calculating the heat transfer coefficients. Assuming that we have these heat transfer coefficients for carbon dioxide and hydrogen (\(h_{CO_2}\), and \(h_{H_2}\)), we can proceed to the next step.

Calculate the surface area of the steel plate

Given that the steel plate dimensions are 0.5 m x 0.5 m, we can calculate its surface area as follows: \(A = 0.5\,\text{m} \times 0.5\,\text{m} = 0.25\,\text{m}^2\)

Determine the convective heat transfer rate for carbon dioxide and hydrogen

Now, we can determine the convective heat transfer rate for both gases using the convection equation: 1. Carbon dioxide: \(q_{CO_2} = h_{CO_2} A (T_s - T_\infty)\) 2. Hydrogen: \(q_{H_2} = h_{H_2} A (T_s - T_\infty)\)

Determine the heat addition rate required for both gases

As we need to determine the heat addition rate such that it is applicable to both gases, we would choose the maximum heat transfer rate found in step 4: \(q_{required} = \max\{q_{CO_2},\,q_{H_2}\}\) The heat addition rate required to keep the plate surface temperature from dropping below -30°C for both carbon dioxide gas and hydrogen gas will be \(q_{required}\).