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# time_integral

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# time_integral

{ 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