# 1D Transient Heat Conduction in Biomass Particle
# -----------------------------------------------------------------------------
# number of nodes from center to surface of particle
# if m=5 then nodes=[0, 1, 2, 3, 4] where 0=center and 4=surface
m = 100
# time vector from 0 to max time
tmax = 3.0 # max time, s
nt = 1000 # number of time steps
dt = tmax/nt # time step, s
t = np.arange(0, tmax+dt, dt) # time vector, s
# intraparticle temperature array [T] in Kelvin
# row = time step, column = node point from 0 to m
T = hc(d, rho, cp, k, h, Ti, Tinf, b, m, t)
# Plot Results
# -----------------------------------------------------------------------------
py.close('all')
Tc = T[:, 0] # center temperature profile
Ts = T[:, m-1] # surface temperature profile
Tavg = [np.mean(row) for row in T] # average temperature profile
py.figure(1)
py.plot(t, Tc, 'r--', lw=2, label='center')
py.plot(t, Ts, 'r-', lw=2, label='surface')
py.plot(t, Tavg, 'g-', lw=2, label='average')
py.axhline(Tinf, c='k', ls='--')
cpc = 1003.2 + 2.09 * (T[0] - 273.15) # char heat capacity, J/(kg*K)
kc = 0.08 - (1e-4) * (T[0] - 273.15) # char thermal conductivity, W/(m*K)
cpbar = Yw*cpw + (1-Yw)*cpc # effective heat capacity
kbar = Yw*kw + (1-Yw)*kc # effective thermal conductivity
pbar = pw[0] + pc[0] # effective density
g = np.ones(m)*(1e-10) # assume initial heat generation is negligible
# Solve system of equations [A]{T}={C} where T = A\C for each time step
#------------------------------------------------------------------------------
for i in range(1, nt+1):
# heat conduction
T[i] = hc(m, dr, b, dt, h, Tinf, g, T, i, r, pbar, cpbar, kbar)
# kinetic reactions
B[i], C1[i], C2[i], g = kn(T, B, C1, C2, rhow, dt, i, H)
# update thermal properties
cpw = 1112.0 + 4.85 * (T[i] - 273.15)
kw = 0.13 + (3e-4) * (T[i] - 273.15)
cpc = 1003.2 + 2.09 * (T[i] - 273.15)
kc = 0.08 - (1e-4) * (T[i] - 273.15)
# update wood and char density
pw[i] = B[i]*rhow
pc[i] = (C1[i]+C2[i])*rhow
# update mass fraction vector
dv = (6/np.pi*v)**(1/3) # volume equivalent sphere diameter, m
dsv = (dv**3)/(ds**2) # surface volume equivalent sphere diameter (Sauter), m
dc = v/As # characteristic length, m
# number of nodes from center of particle (m=0) to surface (m)
m = 100
# time vector from 0 to max time
tmax = 3.0 # max time, s
nt = 1000 # number of time steps
dt = tmax/nt # time step, s
t = np.arange(0, tmax+dt, dt) # time vector, s
# intraparticle temperature array [T] in Kelvin
# row = time step, column = node point from 0 to m
T = hc(d, rho, cp, k, h, Ti, Tinf, 1, m, t) # base case, b = 1 for cylinder
Ts = hc(ds, rho, cp, k, h, Ti, Tinf, 2, m, t) # ds case, b = 2 for sphere
Tv = hc(dv, rho, cp, k, h, Ti, Tinf, 2, m, t) # dv case, b = 2 for sphere
Tsv = hc(dsv, rho, cp, k, h, Ti, Tinf, 2, m, t) # dsv case, b = 2 for sphere
Tc = hc(dc, rho, cp, k, h, Ti, Tinf, 2, m, t) # dc case, b = 2 for sphere
# Plot Results
# -----------------------------------------------------------------------------
py.close('all')
T_avg = [np.mean(row) for row in T] # average temperature profile
Ts_avg = [np.mean(row) for row in Ts] # ds average temperature profile
Tv_avg = [np.mean(row) for row in Tv] # dv average temperature profile
Tsv_avg = [np.mean(row) for row in Tsv] # dsv average temperature profile
Tc_avg = [np.mean(row) for row in Tc] # dc average temperature profile
