[R-sig-dyn-mod] Radial Diffusion with prescribed flux

[cfriedalek .]

Thanks Karline. So I think I need to consider the heated central zone as an
infinitely long cylinder of finite radius rather than a line of zero
radius. This would mimic the real situation of a heated slender wire in a
domain.

I understand how I can specify a two zone grid where, say, the inner 1% of
the total radius is the "wire" as follows:

xgrid <- setup.grid.1D(x.up = 0, x.down = c(0.01, 1), L=c(0.01, 0.99), N =
c(1, 99))

Then setup the properties as before.

For constant diffusion coefficient of value 1:

r1 <- setup.prop.1D(grid = xgrid, func = function(r) r)

For different diffusion coefficients for wire and domain, say, 10 and 1
respectively:

D.coeffs <- c(10, 1)
wire_radius <- 0.01
r1 <- setup.prop.1D(grid = xgrid, func = function(r) r*D.coeffs[[(r >
wire_radius)+1]])


However now I am unclear how to proceed.

How do I specify a heat source that applies only to the inner zone of the
grid? The heat source is a electrical resistance wire so the power
generation is Volt^2/Resistance (Watts). I guess that must be done in the
transport function but I'm not sure how. Also, do I need to specify the D
parameter since it appears in the properties?

radial_diffusion <- function (t, Y, parms, A){
  tran.1D(C = Y, flux.up = flux.up, flux.down = flux.down, D = D.coeff, A =
A, dx = xgrid)
# Add heat generation term for the wire zone
}

I hope you can help.

Regards

Chris

PS: how to a achieve nice code layout in these posts. Compared to other
posts it looks like I'm missing some part of CR/LF. I'm using R on Windows.

On Tue, Feb 10, 2015 at 5:50 AM, Karline Soetaert <Karline.Soetaert at nioz.nl>
wrote:

> Chris, the problem is that it multiplies the flux with 'A', the surface,
> and in your case, A at the origin = 0, so the flux vanishes (there is no
> surface to cross).
> Karline
>
> -----Original Message-----
> From: R-sig-dynamic-models [mailto:
> r-sig-dynamic-models-bounces at r-project.org] On Behalf Of cfriedalek .
> Sent: zaterdag 7 februari 2015 9:43
> To: r-sig-dynamic-models at r-project.org
> Subject: [R-sig-dyn-mod] Radial Diffusion with prescribed flux
>
> Hi
>
> Thanks for ReacTran and deSolve. To help my understanding I'm trying to
> simulate a line source heating problem in 1-D. Heat is generated along the
> central axis of an infinitely long cylinder of finite radius and diffuses
> radially to the exterior wall of the cylinder of radius R. I'm modelling
> this as a 1-D line along the radial (diffusion) direction with a
> prescribed, positive heat flux at the origin and an insulated exterior wall
> (zero heat flux). My code is shown below.
>
> The problem I'm having is that when I set a positive heat flux at r=0 and
> zero heat flux at r=R the solver returns zero for the temperature (C) field
> at all times.
> However if I set a negative heat flux at r=R (heat flow into the cylinder)
> and zero at r=0 I get a sensible result (the wall temperature rises in time
> and eventually so too does the center temperature). I don't understand why
> this doesn't work for the case with heating in the center.
>
> Have I set this up properly?
>
> I'm really enjoying working with R and R-studio so even though I have
> solved this problem with other programming languages, I would really like
> to migrate to working with R. Getting my head around this is one step of
> the way. Ultimately I want to extend this to modelling a real line-source
> device with zones of different materials and electrical heating in the
> core, so any help is greatly appreciated.
>
> regards
>
> Chris
>
> # 1-D radial diffusion
> library(ReacTran)
> library(deSolve)
> N <- 10
> xgrid <- setup.grid.1D(N = N, L = 1)
> r1 <- setup.prop.1D(grid = xgrid, func = function(r) r) Yini <- rep(0, N)
> times <- seq(from = 0, to = 10, length.out = 101) D.coeff <- 0.05
> radial_diffusion <- function (t, Y, parms, A){
>   tran.1D(C = Y, flux.up = flux.up, flux.down = flux.down, D = D.coeff, A
> = A, dx = xgrid) } # heating from outside - THIS WORKS flux.up <- 0
> flux.down <- -1 # # heating from inside - THIS FAILS # flux.up <- 1 #
> flux.down <- 0 # solve
> out1 <- ode.1D(y = Yini, times = times, func = radial_diffusion, parms =
> NULL, nspec = 1, A = r1) # Plot Results par (mfrow=c(1, 2)) # time
> evolution at center "1" and wall "N"
> plot(out1[1:nrow(out1), "time"], out1[1:nrow(out1), as.character(N)], type
> = "l", lwd = 2,
>      xlab = "time", ylab = "C", main = "Time Evolution")
> lines(out1[1:nrow(out1), "time"], out1[1:nrow(out1), as.character(1)])
>
> # radial distribution every 10 time steps plot(xgrid$x.mid, out1[1,
> 2:(N+1)], type = "l", lwd = 2, xlab = "r/R", ylab = "C",
>      main = "Radial Distribution", ylim=c(0, max(out1[1:nrow(out1),
> as.character(N)])))
> time_indicies <- seq(1, length(times), by=10) for (i in time_indicies)
> lines(xgrid$x.mid, out1[i, 2:(N+1)]) par (mfrow=c(1, 1))
>
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