ReacTran
Reactive Transport Modelling in 1d, 2d and 3d

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tran.cyl_spher: Diffusive Transport in cylindrical (r, theta, z) and...

tran.cyl_spher: Diffusive Transport in cylindrical (r, theta, z) and...
In ReacTran: Reactive Transport Modelling in 1d, 2d and 3d

Description Usage Arguments Details Value See Also Examples

Description

Estimates the transport term (i.e. the rate of change of a concentration due to diffusion) in a cylindrical (r, theta, z) or spherical (r, theta, phi) coordinate system.

Usage

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tran.cylindrical (C, C.r.up = NULL, C.r.down = NULL, 
                  C.theta.up = NULL, C.theta.down = NULL, 
                  C.z.up = NULL, C.z.down = NULL, 
                  flux.r.up = NULL, flux.r.down = NULL, 
                  flux.theta.up = NULL, flux.theta.down = NULL,          
                  flux.z.up = NULL, flux.z.down = NULL, 
                  cyclicBnd = NULL,
                  D.r = NULL, D.theta = D.r, D.z = D.r, 
                  r = NULL, theta = NULL, z = NULL)

tran.spherical (C, C.r.up = NULL, C.r.down = NULL, 
                C.theta.up = NULL, C.theta.down = NULL, 
                C.phi.up = NULL, C.phi.down = NULL, 
                flux.r.up = NULL, flux.r.down = NULL, 
                flux.theta.up = NULL, flux.theta.down = NULL,          
                flux.phi.up = NULL, flux.phi.down = NULL, 
                cyclicBnd = NULL,
                D.r = NULL, D.theta = D.r, D.phi = D.r, 
                r = NULL, theta = NULL, phi = NULL)

Arguments

C

concentration, expressed per unit volume, defined at the centre of each grid cell; Nr*Nteta*Nz (cylindrica) or Nr*Ntheta*Nphi (spherical coordinates) array [M/L3].

C.r.up

concentration at upstream boundary in r(x)-direction; one value [M/L3].

C.r.down

concentration at downstream boundary in r(x)-direction; one value [M/L3].

C.theta.up

concentration at upstream boundary in theta-direction; one value [M/L3].

C.theta.down

concentration at downstream boundary in theta-direction; one value [M/L3].

C.z.up

concentration at upstream boundary in z-direction (cylindrical coordinates); one value [M/L3].

C.z.down

concentration at downstream boundary in z-direction(cylindrical coordinates); one value [M/L3].

C.phi.up

concentration at upstream boundary in phi-direction (spherical coordinates); one value [M/L3].

C.phi.down

concentration at downstream boundary in phi-direction(spherical coordinates); one value [M/L3].

flux.r.up

flux across the upstream boundary in r-direction, positive = INTO model domain; one value [M/L2/T].

flux.r.down

flux across the downstream boundary in r-direction, positive = OUT of model domain; one value [M/L2/T].

flux.theta.up

flux across the upstream boundary in theta-direction, positive = INTO model domain; one value [M/L2/T].

flux.theta.down

flux across the downstream boundary in theta-direction, positive = OUT of model domain; one value [M/L2/T].

flux.z.up

flux across the upstream boundary in z-direction(cylindrical coordinates); positive = INTO model domain; one value [M/L2/T].

flux.z.down

flux across the downstream boundary in z-direction, (cylindrical coordinates); positive = OUT of model domain; one value [M/L2/T].

flux.phi.up

flux across the upstream boundary in phi-direction(spherical coordinates); positive = INTO model domain; one value [M/L2/T].

flux.phi.down

flux across the downstream boundary in phi-direction, (spherical coordinates); positive = OUT of model domain; one value [M/L2/T].

cyclicBnd

If not NULL, the direction in which a cyclic boundary is defined, i.e. cyclicBnd = 1 for the r direction, cyclicBnd = 2 for the theta direction and cyclicBnd = c(1,2) for both the r and theta direction.

D.r

diffusion coefficient in r-direction, defined on grid cell interfaces. One value or a vector of length (Nr+1), [L2/T].

D.theta

diffusion coefficient in theta-direction, defined on grid cell interfaces. One value or or a vector of length (Ntheta+1), [L2/T].

D.z

diffusion coefficient in z-direction, defined on grid cell interfaces for cylindrical coordinates. One value or a vector of length (Nz+1) [L2/T].

D.phi

diffusion coefficient in phi-direction, defined on grid cell interfaces for cylindrical coordinates. One value or a vector of length (Nphi+1) [L2/T].

r

position of adjacent cell interfaces in the r-direction. A vector of length Nr+1 [L].

theta

position of adjacent cell interfaces in the theta-direction. A vector of length Ntheta+1 [L]. Theta should be within [0,2 pi]

z

position of adjacent cell interfaces in the z-direction (cylindrical coordinates). A vector of length Nz+1 [L].

phi

position of adjacent cell interfaces in the phi-direction (spherical coordinates). A vector of length Nphi+1 [L]. Phi should be within [0,2 pi]

Details

tran.cylindrical performs (diffusive) transport in cylindrical coordinates

tran.spherical performs (diffusive) transport in spherical coordinates

The boundary conditions are either

This is also the order of priority. The cyclic boundary overrules the other. If fixed concentration, fixed flux, and cyclicBnd are NULL then the boundary is zero-gradient

A cyclic boundary condition has concentration and flux at upstream and downstream boundary the same. It is useful mainly for the theta and phi direction.

** Do not expect too much of this equation: do not try to use it with many boxes **

Value

a list containing:

dC

the rate of change of the concentration C due to transport, defined in the centre of each grid cell, a Nr*Nteta*Nz (cylindrical) or Nr*Ntheta*Nphi (spherical coordinates) array. [M/L3/T].

flux.r.up

flux across the upstream boundary in r-direction, positive = INTO model domain. A matrix of dimension Nteta*Nz (cylindrical) or Ntheta*Nphi (spherical) [M/L2/T].

flux.r.down

flux across the downstream boundary in r-direction, positive = OUT of model domain. A matrix of dimension Nteta*Nz (cylindrical) or Ntheta*Nphi (spherical) [M/L2/T].

flux.theta.up

flux across the upstream boundary in theta-direction, positive = INTO model domain. A matrix of dimension Nr*Nz (cylindrical) or or Nr*Nphi (spherical) [M/L2/T].

flux.theta.down

flux across the downstream boundary in theta-direction, positive = OUT of model domain. A matrix of dimension Nr*Nz (cylindrical) or Nr*Nphi (spherical) [M/L2/T].

flux.z.up

flux across the upstream boundary in z-direction, for cylindrical coordinates; positive = OUT of model domain. A matrix of dimension Nr*Nteta [M/L2/T].

flux.z.down

flux across the downstream boundary in z-direction for cylindrical coordinates; positive = OUT of model domain. A matrix of dimension Nr*Nteta [M/L2/T].

flux.phi.up

flux across the upstream boundary in phi-direction, for spherical coordinates; positive = OUT of model domain. A matrix of dimension Nr*Nteta [M/L2/T].

flux.phi.down

flux across the downstream boundary in phi-direction, for spherical coordinates; positive = OUT of model domain. A matrix of dimension Nr*Nteta [M/L2/T].

See Also

tran.polar for a discretisation of 2-D transport equations in polar coordinates

tran.1D, tran.2D, tran.3D

Examples

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## =============================================================================
## Testing the functions
## =============================================================================
# Grid definition
r.N     <- 4   # number of cells in r-direction
theta.N <- 6   # number of cells in theta-direction
z.N     <- 3   # number of cells in z-direction

D       <- 100 # diffusion coefficient
 
r      <- seq(0,   8, len = r.N+1)       # cell size r-direction [cm]
theta  <- seq(0,2*pi, len = theta.N+1)   # theta-direction - theta: from 0, 2pi
phi    <- seq(0,2*pi, len = z.N+1)       # phi-direction (0,2pi)
z      <- seq(0,5, len = z.N+1)          # cell size z-direction [cm]
 
# Intial conditions 
C <- array(dim = c(r.N, theta.N, z.N), data = 0)

# Concentration boundary conditions
tran.cylindrical (C = C, D.r = D, D.theta = D, 
  C.r.up = 1, C.r.down = 1,
  C.theta.up = 1, C.theta.down = 1, 
  C.z.up = 1, C.z.down = 1,
  r = r, theta = theta, z = z )

tran.spherical (C = C, D.r = D, D.theta = D, 
  C.r.up = 1, C.r.down = 1, C.theta.up = 1, C.theta.down = 1, 
  C.phi.up = 1, C.phi.down = 1,
  r = r, theta = theta, phi = phi)

# Flux boundary conditions
tran.cylindrical(C = C, D.r = D, r = r, theta = theta, z = z,
  flux.r.up = 10, flux.r.down = 10,
  flux.theta.up = 10, flux.theta.down = 10,
  flux.z.up = 10, flux.z.down = 10)

tran.spherical(C = C, D.r = D, r = r, theta = theta, phi = phi,
  flux.r.up = 10, flux.r.down = 10,
  flux.theta.up = 10, flux.theta.down = 10,
  flux.phi.up = 10, flux.phi.down = 10)

# cyclic boundary conditions
tran.cylindrical(C = C, D.r = D, r = r, theta = theta, z = z,
  cyclicBnd = 1:3)
tran.spherical(C = C, D.r = D, r = r, theta = theta, phi = phi,
  cyclicBnd = 1:3)

# zero-gradient boundary conditions
tran.cylindrical(C = C, D.r = D, r = r, theta = theta, z = z)
tran.spherical(C = C, D.r = D, r = r, theta = theta, phi = phi)

## =============================================================================
## A model with diffusion and first-order consumption
## =============================================================================

N     <- 10          # number of grid cells
rr    <- 0.005       # consumption rate
D     <- 400

r       <- seq (2, 4, len = N+1)
theta   <- seq (0, 2*pi, len = N+1)
z       <- seq (0, 3, len = N+1)
phi     <- seq (0, 2*pi, len = N+1)

# The model equations
Diffcylin <- function (t, y, parms)  {
  CONC  <- array(dim = c(N, N, N), data = y)
  tran  <- tran.cylindrical(CONC, 
        D.r = D, D.theta = D, D.z = D,
        r = r, theta = theta, z = z,
        C.r.up = 0,  C.r.down = 1,
        cyclicBnd = 2)
  dCONC <- tran$dC  - rr * CONC
  return (list(dCONC))
}

Diffspher <- function (t, y, parms)  {
  CONC  <- array(dim = c(N, N, N), data = y)
  tran  <- tran.spherical (CONC, 
        D.r = D, D.theta = D, D.phi = D,
        r = r, theta = theta, phi = phi,
        C.r.up = 0,  C.r.down = 1,
        cyclicBnd = 2:3)
  dCONC <- tran$dC  - rr * CONC
  return (list(dCONC))
}

# initial condition: 0 everywhere, except in central point
y   <- array(dim = c(N, N, N), data = 0)
N2  <- ceiling(N/2)

y[N2, N2, N2] <- 100  # initial concentration in the central point...

# solve to steady-state; cyclicBnd = 2, 
outcyl <- steady.3D (y = y, func = Diffcylin, parms = NULL,
                  dim = c(N, N, N), lrw = 1e6, cyclicBnd = 2)

STDcyl <- array(dim = c(N, N, N), data = outcyl$y)
image(STDcyl[,,1])

# For spherical coordinates, cyclic Bnd = 2, 3
outspher <- steady.3D (y = y, func = Diffspher, parms = NULL, pos=TRUE,
                  dim = c(N, N, N), lrw = 1e6, cyclicBnd = 2:3)

#STDspher <- array(dim = c(N, N, N), data = outspher$y)
#image(STDspher[,,1])

## Not run: 
  image(outspher)

## End(Not run)

Example output

Loading required package: rootSolve
Loading required package: deSolve
Loading required package: shape
$dC
, , 1

          [,1]     [,2]     [,3]     [,4]     [,5]      [,6]
[1,] 254.37813  72.0000  72.0000  72.0000  72.0000 254.37813
[2,]  92.26424  72.0000  72.0000  72.0000  72.0000  92.26424
[3,]  79.29513  72.0000  72.0000  72.0000  72.0000  79.29513
[4,] 132.86486 129.1429 129.1429 129.1429 129.1429 132.86486

, , 2

           [,1]     [,2]     [,3]     [,4]     [,5]       [,6]
[1,] 182.378131  0.00000  0.00000  0.00000  0.00000 182.378131
[2,]  20.264237  0.00000  0.00000  0.00000  0.00000  20.264237
[3,]   7.295125  0.00000  0.00000  0.00000  0.00000   7.295125
[4,]  60.864860 57.14286 57.14286 57.14286 57.14286  60.864860

, , 3

          [,1]     [,2]     [,3]     [,4]     [,5]      [,6]
[1,] 254.37813  72.0000  72.0000  72.0000  72.0000 254.37813
[2,]  92.26424  72.0000  72.0000  72.0000  72.0000  92.26424
[3,]  79.29513  72.0000  72.0000  72.0000  72.0000  79.29513
[4,] 132.86486 129.1429 129.1429 129.1429 129.1429 132.86486


$flux.r.up
     [,1] [,2] [,3]
[1,]  100  100  100
[2,]  100  100  100
[3,]  100  100  100
[4,]  100  100  100
[5,]  100  100  100
[6,]  100  100  100

$flux.r.down
     [,1] [,2] [,3]
[1,] -100 -100 -100
[2,] -100 -100 -100
[3,] -100 -100 -100
[4,] -100 -100 -100
[5,] -100 -100 -100
[6,] -100 -100 -100

$flux.theta.up
          [,1]      [,2]      [,3]
[1,] 190.98593 190.98593 190.98593
[2,]  63.66198  63.66198  63.66198
[3,]  38.19719  38.19719  38.19719
[4,]  27.28370  27.28370  27.28370

$flux.theta.down
           [,1]       [,2]       [,3]
[1,] -190.98593 -190.98593 -190.98593
[2,]  -63.66198  -63.66198  -63.66198
[3,]  -38.19719  -38.19719  -38.19719
[4,]  -27.28370  -27.28370  -27.28370

$flux.z.up
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]  120  120  120  120  120  120
[2,]  120  120  120  120  120  120
[3,]  120  120  120  120  120  120
[4,]  120  120  120  120  120  120

$flux.z.down
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,] -120 -120 -120 -120 -120 -120
[2,] -120 -120 -120 -120 -120 -120
[3,] -120 -120 -120 -120 -120 -120
[4,] -120 -120 -120 -120 -120 -120

$dC
, , 1

         [,1]      [,2]       [,3]          [,4]      [,5]       [,6]
[1,]  0.00000 52.648031 105.296063 -7.446152e+17 52.648031 105.296063
[2,]  0.00000  5.849781  11.699563 -8.273503e+16  5.849781  11.699563
[3,]  0.00000  2.105921   4.211843 -2.978461e+16  2.105921   4.211843
[4,] 65.30612 66.380572  67.455022 -1.519623e+16 66.380572  67.455022

, , 2

         [,1]     [,2]     [,3]     [,4]     [,5]         [,6]
[1,]  0.00000  0.00000  0.00000  0.00000  0.00000 8.933952e-14
[2,]  0.00000  0.00000  0.00000  0.00000  0.00000 9.926613e-15
[3,]  0.00000  0.00000  0.00000  0.00000  0.00000 3.573581e-15
[4,] 65.30612 65.30612 65.30612 65.30612 65.30612 6.530612e+01

, , 3

         [,1]      [,2]       [,3]          [,4]      [,5]       [,6]
[1,]  0.00000 52.648031 105.296063 -7.446152e+17 52.648031 105.296063
[2,]  0.00000  5.849781  11.699563 -8.273503e+16  5.849781  11.699563
[3,]  0.00000  2.105921   4.211843 -2.978461e+16  2.105921   4.211843
[4,] 65.30612 66.380572  67.455022 -1.519623e+16 66.380572  67.455022


$flux.r.up
     [,1] [,2] [,3]
[1,]  100  100  100
[2,]  100  100  100
[3,]  100  100  100
[4,]  100  100  100
[5,]  100  100  100
[6,]  100  100  100

$flux.r.down
     [,1] [,2] [,3]
[1,] -100 -100 -100
[2,] -100 -100 -100
[3,] -100 -100 -100
[4,] -100 -100 -100
[5,] -100 -100 -100
[6,] -100 -100 -100

$flux.theta.up
          [,1]      [,2]      [,3]
[1,] 190.98593 190.98593 190.98593
[2,]  63.66198  63.66198  63.66198
[3,]  38.19719  38.19719  38.19719
[4,]  27.28370  27.28370  27.28370

$flux.theta.down
           [,1]       [,2]       [,3]
[1,] -190.98593 -190.98593 -190.98593
[2,]  -63.66198  -63.66198  -63.66198
[3,]  -38.19719  -38.19719  -38.19719
[4,]  -27.28370  -27.28370  -27.28370

$flux.phi.up
     [,1]      [,2]      [,3]         [,4]       [,5]       [,6]
[1,]    0 110.26578 110.26578 7.797592e+17 -110.26578 -110.26578
[2,]    0  36.75526  36.75526 2.599197e+17  -36.75526  -36.75526
[3,]    0  22.05316  22.05316 1.559518e+17  -22.05316  -22.05316
[4,]    0  15.75225  15.75225 1.113942e+17  -15.75225  -15.75225

$flux.phi.down
     [,1]       [,2]       [,3]          [,4]      [,5]      [,6]
[1,]    0 -110.26578 -110.26578 -7.797592e+17 110.26578 110.26578
[2,]    0  -36.75526  -36.75526 -2.599197e+17  36.75526  36.75526
[3,]    0  -22.05316  -22.05316 -1.559518e+17  22.05316  22.05316
[4,]    0  -15.75225  -15.75225 -1.113942e+17  15.75225  15.75225

$dC
, , 1

          [,1]      [,2]      [,3]      [,4]      [,5]      [,6]
[1,] 15.549297 6.0000000 6.0000000 6.0000000 6.0000000 -3.549297
[2,]  9.183099 6.0000000 6.0000000 6.0000000 6.0000000  2.816901
[3,]  7.909859 6.0000000 6.0000000 6.0000000 6.0000000  4.090141
[4,]  1.649900 0.2857143 0.2857143 0.2857143 0.2857143 -1.078471

, , 2

          [,1]      [,2]      [,3]      [,4]      [,5]      [,6]
[1,]  9.549297  0.000000  0.000000  0.000000  0.000000 -9.549297
[2,]  3.183099  0.000000  0.000000  0.000000  0.000000 -3.183099
[3,]  1.909859  0.000000  0.000000  0.000000  0.000000 -1.909859
[4,] -4.350100 -5.714286 -5.714286 -5.714286 -5.714286 -7.078471

, , 3

           [,1]      [,2]      [,3]      [,4]      [,5]       [,6]
[1,]   3.549297  -6.00000  -6.00000  -6.00000  -6.00000 -15.549297
[2,]  -2.816901  -6.00000  -6.00000  -6.00000  -6.00000  -9.183099
[3,]  -4.090141  -6.00000  -6.00000  -6.00000  -6.00000  -7.909859
[4,] -10.350100 -11.71429 -11.71429 -11.71429 -11.71429 -13.078471


$flux.r.up
     [,1] [,2] [,3]
[1,]   10   10   10
[2,]   10   10   10
[3,]   10   10   10
[4,]   10   10   10
[5,]   10   10   10
[6,]   10   10   10

$flux.r.down
     [,1] [,2] [,3]
[1,]   10   10   10
[2,]   10   10   10
[3,]   10   10   10
[4,]   10   10   10
[5,]   10   10   10
[6,]   10   10   10

$flux.theta.up
     [,1] [,2] [,3]
[1,]   10   10   10
[2,]   10   10   10
[3,]   10   10   10
[4,]   10   10   10

$flux.theta.down
     [,1] [,2] [,3]
[1,]   10   10   10
[2,]   10   10   10
[3,]   10   10   10
[4,]   10   10   10

$flux.z.up
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]   10   10   10   10   10   10
[2,]   10   10   10   10   10   10
[3,]   10   10   10   10   10   10
[4,]   10   10   10   10   10   10

$flux.z.down
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]   10   10   10   10   10   10
[2,]   10   10   10   10   10   10
[3,]   10   10   10   10   10   10
[4,]   10   10   10   10   10   10

$dC
, , 1

          [,1]       [,2]      [,3]      [,4]       [,5]      [,6]
[1,]  9.549297  4.7746483  9.549297 -9.549297 -4.7746483 -9.549297
[2,]  3.183099  1.5915494  3.183099 -3.183099 -1.5915494 -3.183099
[3,]  1.909859  0.9549297  1.909859 -1.909859 -0.9549297 -1.909859
[4,] -5.166427 -5.8485196 -5.166427 -7.894797 -7.2127049 -7.894797

, , 2

          [,1]      [,2]      [,3]      [,4]      [,5]          [,6]
[1,]  0.000000  0.000000  0.000000  0.000000  0.000000 -4.677806e-15
[2,]  0.000000  0.000000  0.000000  0.000000  0.000000 -1.559269e-15
[3,]  0.000000  0.000000  0.000000  0.000000  0.000000 -9.355612e-16
[4,] -6.530612 -6.530612 -6.530612 -6.530612 -6.530612 -6.530612e+00

, , 3

          [,1]       [,2]      [,3]      [,4]       [,5]      [,6]
[1,] -9.549297 -4.7746483 -9.549297  9.549297  4.7746483  9.549297
[2,] -3.183099 -1.5915494 -3.183099  3.183099  1.5915494  3.183099
[3,] -1.909859 -0.9549297 -1.909859  1.909859  0.9549297  1.909859
[4,] -7.894797 -7.2127049 -7.894797 -5.166427 -5.8485196 -5.166427


$flux.r.up
     [,1] [,2] [,3]
[1,]   10   10   10
[2,]   10   10   10
[3,]   10   10   10
[4,]   10   10   10
[5,]   10   10   10
[6,]   10   10   10

$flux.r.down
     [,1] [,2] [,3]
[1,]   10   10   10
[2,]   10   10   10
[3,]   10   10   10
[4,]   10   10   10
[5,]   10   10   10
[6,]   10   10   10

$flux.theta.up
     [,1] [,2] [,3]
[1,]   10   10   10
[2,]   10   10   10
[3,]   10   10   10
[4,]   10   10   10

$flux.theta.down
     [,1] [,2] [,3]
[1,]   10   10   10
[2,]   10   10   10
[3,]   10   10   10
[4,]   10   10   10

$flux.phi.up
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]   10   10   10   10   10   10
[2,]   10   10   10   10   10   10
[3,]   10   10   10   10   10   10
[4,]   10   10   10   10   10   10

$flux.phi.down
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]   10   10   10   10   10   10
[2,]   10   10   10   10   10   10
[3,]   10   10   10   10   10   10
[4,]   10   10   10   10   10   10

$dC
, , 1

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

, , 2

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

, , 3

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0


$flux.r.up
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0
[5,]    0    0    0
[6,]    0    0    0

$flux.r.down
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0
[5,]    0    0    0
[6,]    0    0    0

$flux.theta.up
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0

$flux.theta.down
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0

$flux.z.up
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

$flux.z.down
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

$dC
, , 1

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

, , 2

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

, , 3

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0


$flux.r.up
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0
[5,]    0    0    0
[6,]    0    0    0

$flux.r.down
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0
[5,]    0    0    0
[6,]    0    0    0

$flux.theta.up
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0

$flux.theta.down
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0

$flux.phi.up
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

$flux.phi.down
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

$dC
, , 1

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

, , 2

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

, , 3

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0


$flux.r.up
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0
[5,]    0    0    0
[6,]    0    0    0

$flux.r.down
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0
[5,]    0    0    0
[6,]    0    0    0

$flux.theta.up
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0

$flux.theta.down
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0

$flux.z.up
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

$flux.z.down
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

$dC
, , 1

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

, , 2

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

, , 3

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0


$flux.r.up
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0
[5,]    0    0    0
[6,]    0    0    0

$flux.r.down
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0
[5,]    0    0    0
[6,]    0    0    0

$flux.theta.up
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0

$flux.theta.down
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0
[4,]    0    0    0

$flux.phi.up
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

$flux.phi.down
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0

Warning message:
In stodes(y = y, times = times, func = func, parms = parms, nnz = c(nspec,  :
  steady-state not reached

ReacTran documentation built on May 2, 2019, 9:38 a.m.

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