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{ TIME_INTEGRAL.PDE  

 This example illustrates use of the TIME_INTEGRAL function in time-dependent problems.

}  

 

title  

"Float Zone"  

 

coordinates  

 xcylinder('Z','R')  

 

variables  

 temp (threshold=100)  

 

definitions  

 k = 0.85                           {thermal conductivity}  

 cp = 1                             { heat capacity }  

 long = 18  

 H = 0.4                             {free convection boundary coupling}  

 Ta = 25                             {ambient temperature}  

 A = 4500                           {amplitude}  

 

 source = A*exp(-((z-1*t)/.5)^2)*(200/(t+199))  

 

 tsource = time_integral(vol_integral(source))  

 

initial value  

 temp = Ta  

 

equations  

 temp:  div(k*grad(temp)) + source = cp*dt(temp)  

 

boundaries  

region 1  

  start(0,0)  

  natural(temp) = 0 line to (long,0)  

  value(temp) = Ta line to (long,1)  

  natural(temp) = -H*(temp - Ta) line to (0,1)  

  value(temp) = Ta line to close  

feature  

  start(0.01*long,0) line to (0.01*long,1)  

 

time -0.5 to 19  

 

monitors  

for t = -0.5 by 0.5 to (long + 1)  

elevation(temp) from (0,1) to (long,1) range=(0,1800) as "Surface Temp"  

contour(temp)  

contour(dt(temp))  

 

plots  

for t = -0.5 by 0.5 to (long + 1)  

elevation(temp) from (0,0) to (long,0) range=(0,1800) as "Axis Temp"  

 

histories  

history(temp,dt(temp)) at (0,0) (1,0) (2,0) (3,0) (4,0) (5,0) (6,0) (7,0) (8,0)  

                  (9,0) (10,0) (11,0) (12,0) (13,0) (14,0) (15,0) (16,0)  

                  (17,0) (18,0)  

history(tsource) as "Total Source"  

 

end