Thermally Controlled
2009
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Art Meets Industry: Helicopter Aerial Photography is made for Industrial Photography Demands
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When we think of the concept of aerial photography we envision some guy taking a Nikon up in an aeroplane and shooting pastoral photos of the earth below, in "fly-by" fashion. But when it comes to aerial photography today, this is in reality the least you can do. Advancements in technology over the past couple of decades have led to wireless systems and gyroscopic stabilization, giving the chance to take focused, perfect quality images via radio controlled helicopter aerial photography. In fact, remote controlled helicopters are the ideal platform for aerial photography, and the applications are often for commercial and industrial photography purposes.
Not only for taking images of the decorative spires of buildings (though that is actually one thing that could be done with a camera mounted on a radio-controlled helicopter), mounting a camera on an unmanned aerial vehicle allows for aerial photography to better serve in commercial and industrial photography capacities. When the helicopter is fitted out with an infrared sensor, it can sense the temperature difference between itself and its surrounding environment. An exceedingly important feature if, for instance, the camera is required to take pictures of a petrochemical stack presently out of control and aflame, 500 feet up in the air, and burning at, say, 2000 degrees Celsius. This suggests the helicopter and camera can be maintained at a thermally safe distance, and it implies that high resolution photographs can be taken of what is going wrong at the source of the malfunction. It makes possible for crews to find out how to respond to the emergency, saving time and cash, not to mention how much safer this approach is in comparison to sending some poor guy up there in a space-age industrial heat protection suit to check it out.
The remote controlled helicopter is rigged to permit the operator to see what the camera sees. Once the helicopter is in the right position, a switch on the transmitter can trigger the camera. The result is extremely high quality, detailed aerial photography, and inspectors can inspect from the security of the ground.
Gaining experience on how to fly radio-controlled helicopters well takes years. For industrial purposes, a professional who specializes in elevated equipment inspections is positively required. Flares and Stacks Incorporated is one company that specializes in serving the industrial community with professional radio controlled helicopter aerial photography. Appreciating that industrial photography is a robust and growing niche of the photography professional's universe means that photography is hardly just for local picture studios these days.
Katherine Parker blogs on aerial photography and its useful applications to industrial photography needs.
Separation of Variables / Boundary Conditions?
The edges of a square sheet of thermally conducting material are at x=0, x=L, y= -L/2 and y=L/2
The temperature of these edges are controlled to be:
T = T0 at x = 0 and x = L
T = T0 + T1sin(pi*x/L) at y = -L/2 and y = L/2
where T0 and T1 are constants. The temperature obeys Laplace's equation grad² T (x,y) = 0
(a) Find the general solution to the equation d²X(x) / dx²
(b) Use the method of separation of variables to find the solution to Laplace's equation that objeys the boundary conditions. You'll need to consider a superposition of two solutions: one with a separation constant equal to 0 and a second for which the separation constant is nonzero.
Apologies, part (a) should read
Find the general solution to the equation d²X(x) / dx² = 0
Using separation of variables, write
T = X(x)Y(y)
Then from Laplace's equation
X"Y + XY"= 0
which can be written as
X"/X = -Y"/Y = k^2 or -k^2
where k is a constant since this is the only way to satisfy this equation. The general solution for X is
X = (A0 + A1*x) and Y = (A3 + A4*y) for k = 0
X = C1*sin(kx) + C2*cos(kx) for k^2.
X = C3*sinh(kx) + C4*cosh(kx) for -k^2
From the form of the edge condition at y = -L/2 and y = L/2 we see that X must have the form
X = C1*sin(pi*x/L) + A0*A3
and the other solutions for X(x) do not apply here. Thus
k = pi/L
and from the Y equation
Y = B1*cosh(pi*y/L) + B2*sinh(pi*y/L)
The edge conditions give
C1*B1*cosh(pi/2) = T1, B2 = 0, and A0*A3 = T0
Thus the solution is
T = T0 + T1*sin(pi*x/L)*cosh(pi*y/L)/cosh(pi/2)
Once you see how this method works the steps in the solution can be shortened by looking at the edge conditions and writing
T = T0 + Y(y)*sin(pi*x/L)
Separation of variables is a powerful method for linear pde's with constant coefficients
Finding out the Maximum Speed of Thermally Controlled Fans
