Heat Transfer Design 1 — Altimeter Fins, Plasma Actuator Cable, and Annealing Oven  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Design Problem #1
After obtaining a Masters degree from Pusan National University, you are hired soon afterwards by the Pohang Iron and Steel Company (POSCO). Your first project consists of designing an oven to anneal steel. Annealing is a form of heat treatment which causes changes in the strength, hardness, and other properties of the material. The annealing process that POSCO wishes to perform consists of first heating the steel to a temperature of 780$^\circ$C and then to cool the material slowly no faster than 22$^\circ$C per hour. This rate of cooling must be maintained for 5 hours. To prevent the steel from cooling too rapidly, the temperature inside the oven must be carefully adjusted as a function of time. Knowing that the effective convective heat transfer coefficient (including radiation) inside the oven corresponds to $h=20$ W/m$^2\cdot^\circ$C, that the object to be annealed is a cube with each side measuring $20$ cm, determine quantitately how the temperature of the air inside the oven should be varied as a function of time in order to anneal the material properly. Then, compare graphically the temperature of the air within the oven to the average temperature of the steel for the first five hours of the annealing process.
Design Problem #2
You are working for KAI (Korea Aerospace Industries) and are in charge of the design of the cooling system of the altimeter installed in the cockpit of the A50 fighter jet. The altimeter requires 50 Watts of power to operate and has dimensions of 10 cm$~\times~$10 cm$~\times~$10 cm. The design of the cooling system should be such that it keeps the back surface of the altimeter below 60$^\circ$C while minimizing additional weight. Recalling the theory learned in your Heat Transfer course that you took several years ago at PNU, you decide to cool the altimeter by installing on its backside 10 aluminum fins with a thickness of 2 mm. The fins are rectangular, have a width equal to the one of the altimeter, and are long enough that the tips can be considered insulated. Knowing that the air behind the instrument panel is at a temperature of $20^\circ$C with an associated convective heat transfer coefficient of $h=12~$W/m$^2\cdot^\circ$C, find the value of the fin length that matches the design constraints. Take into consideration the fact that the convective heat transfer coefficient is not known accurately and may vary by as much as 30%.
Design Problem #3
The aircraft company you are working for is considering the use of plasma actuators to delay stall beyond the critical angle of attack. Plasma actuators can delay stall by injecting heat and applying electromagnetic forces on a region of the airflow that has been ionized. The heat injected and the applied forces alter the turbulent eddies within the boundary layer, and this can result in the flow remaining attached to the airfoil even when the angle of attack is increased beyond the critical point. In order to operate, the plasma actuators must be fed a power of 50 KiloWatts with a voltage difference of $200$ Volts. You are assigned the task of designing the polyethylene-covered copper cable linking the power supply to the plasma actuators. Noting that the power supply is located $10$ m away from the plasma actuators, it is desired to find the optimal cable design which minimizes weight while keeping the temperature of the polyethylene insulator below melting point. The cable is located inside the wing, where the air temperature is of $-5^\circ$C and the convective heat transfer coefficient is known to be equal to $h={\rm 10~W/m^2\cdot^\circ}$C. For safe operation the polyethylene layer is given a thickness of 0.5 cm. The electrical resistivity of copper at 20$^\circ$C can be taken as 16.8 n$\Omega\cdot$m. The melting point and the thermal conductivity of polyethylene can be taken as $120^\circ$C and 0.5 W/m$^\circ$C, respectively. Design the cable with a safety margin: take into consideration that the convective heat transfer coefficient may have an error of $30$% and do not let the maximum temperature within the polyethylene approach its melting point by less than 40$^\circ$C.
1.  $742^\circ {\rm C} - \frac{22^\circ{\rm C}}{3600~{\rm s}}t$.
2.  7.9 cm.
3.  0.0034 m.
Due on Wednesday April 25th at 15:00. Do Problems #1 and #2 only.
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