Our quest – to counsel brave knights on the advisability of bringing a shield to a dragon fight – continues.
This is a pretty hairy problem, as it turns out. There are all kinds of crazy things happening when fire heats metal. I dove pretty deep into several rabbit holes this week – from impinging jets on cooled calorimeters and handbooks for estimating the effects of forest fires on power lines to combustion chemistry and a hitherto-unknown-to-me method of heat transfer involving cool metal changing the molecular structure of half combusted gases.
But let’s ignore all that. We’re already dealing with dragon fire and theoretical shields. Let’s c̶h̶a̶n̶g̶e̶ ̶r̶e̶a̶l̶i̶t̶y̶ ̶s̶o̶ ̶i̶t̶s̶ ̶e̶a̶s̶y̶ ̶t̶o̶ ̶a̶n̶a̶l̶y̶z̶e̶ ̶ stick with our totally reasonable simplifications.
Remember from last time that we turned something beautiful, like this:
This is a pretty hairy problem, as it turns out. There are all kinds of crazy things happening when fire heats metal. I dove pretty deep into several rabbit holes this week – from impinging jets on cooled calorimeters and handbooks for estimating the effects of forest fires on power lines to combustion chemistry and a hitherto-unknown-to-me method of heat transfer involving cool metal changing the molecular structure of half combusted gases.
But let’s ignore all that. We’re already dealing with dragon fire and theoretical shields. Let’s c̶h̶a̶n̶g̶e̶ ̶r̶e̶a̶l̶i̶t̶y̶ ̶s̶o̶ ̶i̶t̶s̶ ̶e̶a̶s̶y̶ ̶t̶o̶ ̶a̶n̶a̶l̶y̶z̶e̶ ̶ stick with our totally reasonable simplifications.
Remember from last time that we turned something beautiful, like this:
Into something analyzable, like this:
Today we’ll focus solely on convection. Convection is heat moving between a moving fluid and a solid… in this case from hot dragon’s breath to a metal shield. Usually it looks something like:
That “q” is the heat (in Watts) flowing from the flames to the shield. It depends on the temperature differential and the area of the shield, but it also depends on the convection coefficient “h,” which can be a pesky bugger to find.
“h” depends on lots of things – how quickly the air is moving across the shield, how hot the shield is, how turbulent or laminar the flow is, etc. – and trying to get that all straight in three dimensions and real time can get real nasty real quick. Fortunately, people have created empirical data-fits for a lot of the more common situations.
Shield-in-dragon’s-breath isn’t one of those common situations, but luckily it c̶a̶n̶ ̶b̶e̶ ̶s̶h̶o̶e̶h̶o̶r̶n̶e̶d̶ ̶i̶n̶t̶o̶ very closely matches an equation for vertical plates undergoing forced convection:
“h” depends on lots of things – how quickly the air is moving across the shield, how hot the shield is, how turbulent or laminar the flow is, etc. – and trying to get that all straight in three dimensions and real time can get real nasty real quick. Fortunately, people have created empirical data-fits for a lot of the more common situations.
Shield-in-dragon’s-breath isn’t one of those common situations, but luckily it c̶a̶n̶ ̶b̶e̶ ̶s̶h̶o̶e̶h̶o̶r̶n̶e̶d̶ ̶i̶n̶t̶o̶ very closely matches an equation for vertical plates undergoing forced convection:
But wait! you say. There is nary an “h” to be seen in that equation! Very true, but there is an “Nu,” (the Nusselt number), which is closely related to “h.” With a little mathing, we can use this to estimate h!
First, more assumptions. Let’s say:
First, more assumptions. Let’s say:
- Our knight’s shield is a convenient 1m square (just big enough to crouch behind) at a nice room-temperature 25C.
- The dragon’s breath is moving over the shield at a leisurely 5 m/s (11 MPH)… mostly because that keeps the Reynold’s number in a range where our Nusselt number is still easy to calculate.
- The dragon’s breath is 2150C… because there’s no knowing how dragons make fiery breath, so the approximate flame temperature of a WWII gasoline-burning flame thrower is probably a reasonable approximation.
Unfortunately, it’s hard to find physical properties for dragon’s breath (which is probably a mixture of CO, CO2, H20, and half-digested sheep), so we’ll have to settle for getting our Prandtl number, kinematic viscosity, and thermal conductivity from a table for dry air. Even then, I couldn’t find a table that went as hot as 2150C, so I’m settling for dry air at 2000C. It will have to be close enough.
With all those conditions in hand, it’s now a simple matter to calculate h...
With all those conditions in hand, it’s now a simple matter to calculate h...
…and armed with “h,” we can finally estimate the initial convective heat transfer pouring into the knight’s shield!
That's right! 1124 Watts!
What does that even mean? Tune in next time to find out.
See Part 3 here
Questions? Comments? Suggestions for a new subject? Leave a comment and we'll discuss!
What does that even mean? Tune in next time to find out.
See Part 3 here
Questions? Comments? Suggestions for a new subject? Leave a comment and we'll discuss!