CHALLENGE QUESTION: (See the Answer Below!)
How long could a knight survive against a fire-breathing dragon?
As a knight you do this all the time, fight dragons that is. But finding yourself engulfed in a dragons fiery breath, you are beginning to question that career move. If it weren't for your trusty shield you would have been burnt to a crisp by now. This dragon's breath seems to be particularly long lasting however...
How long will it take for the dragon's 1000 K breath to melt through your shield?
You recall the master shield crafter telling you about the thermophysical properties of your shield:
- 2 cm thick, cp = 903 J/Kg K roh = 2702 Kg/m3 k = 237 W/m K melting point = 933 K.
- Dragon breath is 1000 K
- Your finely honed knightly senses tell you the Biot number for this situation is around 0.01.
Let's Congratulate Nooruddin Jamali of Middle East Technical University!
Here's the Correct Answer:
The first thing to notice is that this a transient heat transfer problem.
Since the Biot number is <1 we can use the lumped capacitance method to solve the problem.
This means that the temperature of the knight's shield is uniform at any time.
Also assume the shield is initially at room temperature.
We then calculate the time taken for only one surface to reach the melting point.
h is calculated from the Biot number: Bi=h*L/k where L=0.02m, k=237W/m*K and Bi=0.01
Energy in -Energy out = Energy accumulated
Energy out=0 [Assuming that the body if the knight doesn't absorb any heat.]
Rearranging we have,
dT/[(Tflame-T)] = [h/(rho)*L*Cp]*dt
Taking the integral with t from 0 to t and T from 298K to 933K.
It takes approximately 16 minutes for the shield to start melting. This gives the knight enough time to escape to another position and make an attack.
Here is Al Rolston's Response:
The knight could use a shield made from the ablative properties of Refrasil (silica fibers) and phenolic resin. This is the same construction as the shield used on re-entry space vehicles. Now, the knight can get close enough to slay the dragon with a stiletto. The shield will be one-fourth the weight of the old, metal shield, and withstands over 2,000 degrees F.
Challenge author, Kendall Fosse:
Nooruddin Jamali answered the problem correctly using the lumped capacitance method and reverse engineering an h value to use in a simple heat balance. Nooruddin's solution is the same as the one I prepared, and much more elegant. Al Rolston didn't give an answer to the question but instead suggested changing the material of the shield to be more heat resistant. This is a delightful contribution, as it demonstrates the engineering thought process.