Question

A stainless steel plate is connected to a copper plate by long ASTM B98 copper-silicon bolts of \(9.5 \mathrm{~mm}\) in diameter. The portion of the bolts exposed to convection heat transfer with air is \(5 \mathrm{~cm}\) long. The air temperature for convection is at \(20^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of $5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. The thermal conductivity of the bolt is known to be \)36 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The copper plate has a uniform temperature of \(70^{\circ} \mathrm{C}\). If each bolt is estimated to have a rate of heat loss to convection of \(5 \mathrm{~W}\), determine the temperature \(T_{b}\) at the upper surface of the stainless steel plate. According to the ASME Code for Process Piping (ASME B31.3-2014, Table A-2M), the maximum use temperature for the ASTM B 98 copper-silicon bolt is \(149^{\circ} \mathrm{C}\). Does the use of the ASTM B 98 bolts in this condition comply with the ASME code?

Short answer

Answer: The temperature at the upper surface of the stainless steel plate is approximately \(21.93^{\circ}\mathrm{C}\).

Step-by-step solution

Find the surface temperature of the bolt

We know the convection heat transfer and the convection heat transfer coefficient. We can use convection heat transfer equation to find the surface temperature of the bolt: $$q = h A \Delta T$$ Where, \(q\) = Heat transfer rate (\(5\mathrm{~W}\)) \(h\) = Convection heat transfer coefficient (\(5\mathrm{~W/m^2\cdot K}\)) \(A\) = Surface area of the bolt in contact with air \(\Delta T\) = Temperature difference between the surface of the bolt and the air (T_surface - \(20^{\circ}\mathrm{C}\)) To find the surface area of the bolt: $$A = \pi dL$$ Where, \(d\) = Diameter of the bolt (\(9.5\mathrm{~mm}\)) \(L\) = Length of the exposed bolt (\(50\mathrm{~mm}\))

Calculate the surface area of the bolt

Using the diameter and length given: $$A = \pi (0.0095\mathrm{~m})(0.05\mathrm{~m}) = 1.497 \times 10^{-3} \mathrm{~m^2}$$

Find the temperature difference between the surface of the bolt and the air

Now we will use the convection heat transfer equation to find the temperature difference between the surface of the bolt and the air: $$\Delta T = \frac{q}{hA}$$ Substituting the known values: $$\Delta T = \frac{5\mathrm{~W}}{5\mathrm{~W/m^2\cdot K} \times 1.497 \times 10^{-3} \mathrm{~m^2}}$$ Calculate the temperature difference: $$\Delta T = 6.68^{\circ}\mathrm{C}$$

Determine the surface temperature of the bolt

Now, we will add the calculated temperature difference to the air temperature to find the surface temperature of the bolt: $$T_{surface} = T_{air} + \Delta T$$ Substituting the known values and calculated temperature difference: $$T_{surface} = 20^{\circ}\mathrm{C} + 6.68^{\circ}\mathrm{C}$$ Calculate the surface temperature of the bolt: $$T_{surface} = 26.68^{\circ}\mathrm{C}$$

Determine the temperature at the upper surface of the stainless steel plate using heat transfer equation

Now, we will use the heat transfer equation to find the temperature \(T_b\) at the upper surface of the stainless steel plate: $$q = kA\frac{\Delta T}{L}$$ Where, \(k\) = Thermal conductivity of the bolt (\(36\mathrm{~W/m\cdot K}\)) Rearranging to find \(\Delta T\): $$\Delta T = \frac{qL}{kA}$$ Substituting the known values: $$\Delta T = \frac{5\mathrm{~W} \times 0.05\mathrm{~m}}{36\mathrm{~W/m\cdot K} \times 1.497 \times 10^{-3} \mathrm{~m^2}}$$ Calculate the temperature difference: $$\Delta T = 48.07^{\circ}\mathrm{C}$$ Now, we will subtract the calculated temperature difference from the copper plate's temperature to find \(T_b\): $$T_b = T_{copper} - \Delta T$$ Substituting the known values: $$T_b = 70^{\circ}\mathrm{C} - 48.07^{\circ}\mathrm{C}$$ Calculate the temperature at the upper surface of the stainless steel plate: $$T_b = 21.93^{\circ}\mathrm{C}$$ Finally, we need to check if the use of ASTM B98 bolts complies with the ASME code. The maximum use temperature for the ASTM B98 copper-silicon bolt is \(149^{\circ}\mathrm{C}\). Since the calculated surface temperature (\(26.68^{\circ}\mathrm{C}\)) and upper surface stainless steel plate temperature (\(21.93^{\circ}\mathrm{C}\)) are both less than the maximum use temperature, the use of the ASTM B98 bolts in this condition complies with the ASME code.