Heat Transfer Design 1 — Plasma Actuator Cable, Turbojet Combustor Fins, and Heat Treatment  
Instructions
$\xi$ 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.
05.05.14
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.
Design Problem #4
Consider a combustor of a turbojet engine made of a 1 m long hollow steel cylinder, with the cylinder outer radius being of $r_{\rm o}=0.3$ m and the cylinder inner radius being of $r_{\rm i}=0.25$ m. Gases flow within the combustor at a temperature of $2000^\circ$C with a convective heat transfer coefficient between the cylinder and the gases of $h_{\rm in}=4$ W/m$^2$$^\circ$C (including radiation). On the outside of the combustor, some cool air is flowing at a temperature of 10$^\circ$C and a convective heat transfer of $h_{\rm out}=5$ W/m$^2$$^\circ$C (including radiation). You weld 8 steel fins on the outside of the combustor to cool it, as depicted below.
combustor-and-fins.png  ./download/file.php?id=5090&sid=aa688c9c1e4179c044a5db201988b6e8  ./download/file.php?id=5090&t=1&sid=aa688c9c1e4179c044a5db201988b6e8
Each fin is 3 mm thick with a width of 1 m (spanning the length of the cylinder). You wish to minimize the length of the fins $L_{\rm fin}$ as much as possible to keep the weight of the combustor down while resulting in sufficient cooling so that the combustor temperature anywhere (i.e., anywhere within the steel) doesn't exceed 800$^\circ$C. What would be the optimal fin length that would accomplish this? Specifically, given the following air, gases, and steel properties:
Matter$\rho,~{\rm kg/m}^3$$c,~{\rm J/kg^\circ C}$$k,~{\rm W/m^\circ C}$
Gases29000.1
Steel780048550
Air110000.03
Do the following:
(a)  Find the heat transfer at the base of one fin $q_{\rm fin}$ that will result in the combustor steel temperature not exceeding $800^\circ$C anywhere.
(b)  Find the fin length $L_{\rm fin}$ that yields the heat transfer found in part (a).
Hint: You can assume that the outer surface of the combustor is insulated except for the fins.
04.13.19
Design Problem #5
Quenching is a heat treatment process in which a metal is first heated to a high temperature and rapidly cooled in a liquid bath. The rapid cooling changes the properties of the metal and makes it more stiff and more brittle. We here consider the quenching of an infinitely-long and infinitely-wide plate of steel with a thickness $D$, which is first heated to a temperature of $1000^\circ$C and then immersed in a bath of water for an amount of time $\Delta t$. The convective heat transfer coefficient within the bath can be taken as $h=2500~\rm W/{m^2}^\circ C$ and the water temperature far from the steel plate can be taken as $30^\circ$C. For the quenching to be successful, the steel plate should be removed out of the water when its minimum temperature reaches $100^\circ$C. Further, for the quenching not to introduce excessive stresses, the steel plate shouldn't be too thick so that the rate of change of temperature within the steel doesn't vary by more than 25% (that is, the change in temperature of the steel at one location shouldn't exceed the change in temperature of the steel at another location by more than 25%). Put in another way, we wish that the steel plate thickness $D$ is low enough so that the following constraint is satisfied within the steel through the cooling process: $$ \frac{(T_{x,t=0}-T_{x,t=\Delta t})_{\rm min}}{(T_{x,t=0}-T_{x,t=\Delta t})_{\rm max}}\ge 0.75 $$ Given the latter, determine:
(a)  The surface temperature of the steel plate after the cooling process is completed.
(b)  The temperature at the center of the steel plate after the cooling process is completed for a steel plate with the maximum possible thickness $D_{\rm max}$ that prevents excessive stresses from forming.
(c)  The maximum allowable thickness $D_{\rm max}$ of the steel plate that would prevent excessive stresses from forming.
(d)  The amount of time $\Delta t$ in seconds that the steel plate with the thickness $D_{\rm max}$ needs to be immersed in the bath of water.
You can use the following steel properties: $$ \rho=7800~{\rm kg/m^3},~~~~c=465~{\rm J/kg^\circ C},~~~~k=50~{\rm W/m^\circ C} $$ Hint: You don't need to know the value of $D_{\rm max}$ to determine the temperature at the center in part (b).
Answers
1.  $742^\circ {\rm C} - \frac{22^\circ{\rm C}}{3600~{\rm s}}t$.
2.  7.9 cm.
3.  0.0034 m.
4.  0.274 m.
5.  0.22 m, 732 s.
Due on Wednesday April 24th at 9:00. Do Problems #3, #4, and #5 only. The last 2 problems are challenging — you should start early.
For problem 4, it's fine to assume negligible heat transfer from the fin tip. This simplifies the analysis and I will not take away points if you do so.
04.24.19
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