Packed-bed model#
PackedBedModel describes a bed of spherical PCM capsules with the heat transfer fluid
flowing through the voids. At each point along the bed, heat conducts across a capsule
(resolved radially) and into the fluid at its surface. It is built from the same submodels
as the finned-tube models, so it has the same outputs, conserves energy exactly and runs the same experiments.
The default parameters are the experiments of Nallusamy et al. (2007): a 47 l tank of paraffin capsules in water, charged with water at 70 °C from 32 °C.
For the geometry, the equations and the options of the packed-bed model, with sketches, see the Packed bed page of the documentation.
Tutorial 3 introduces this model.
import matplotlib.pyplot as plt
import numpy as np
import pybamm
import pytes
model = pytes.PackedBedModel()
model.print_submodel_info()
fluid: AdvectedFluid (Heat transfer fluid)
requires:
Heat transfer fluid heat gain per unit length [W.m-1]
Initial uniform temperature [K]
provides:
Heat transfer fluid specific enthalpy [J.kg-1]
Inlet specific enthalpy [J.kg-1]
Outlet specific enthalpy [J.kg-1]
Mass flow rate [kg.s-1]
Heat transfer fluid thermal energy [J]
Heat transfer fluid temperature [K]
Inlet temperature [K]
Outlet temperature [K]
Inlet-outlet temperature difference [K]
Heat transfer fluid velocity [m.s-1]
Volume flow rate [m3.s-1]
Pumped volume [m3]
parameters:
Bed cross-sectional area [m2]
Bed length [m]
Bed porosity
Heat transfer fluid density [kg.m-3]
Heat transfer fluid specific heat capacity [J.kg-1.K-1]
Inlet temperature [K]
Mass flow rate [kg.s-1]
PCM melting temperature [K]
Reference temperature [K]
assumptions:
The fluid is incompressible, with constant density.
Axial conduction and dispersion in the fluid are neglected.
The flow is the same in every flow path.
storage: EnthalpyStorage (Phase-change material storage)
requires:
Phase-change material boundary heat flux [W.m-2]
Initial uniform temperature [K]
provides:
Phase-change material enthalpy [J.m-3]
Phase-change material enthalpy rate of change [W.m-3]
Phase-change material conduction heating rate [W.m-3]
Phase-change material thermal energy [J]
Liquid fraction
Phase
Average liquid fraction
Phase-change material temperature [K]
Phase-change material average temperature [K]
Phase-change material cross-section average temperature [K]
Phase-change material surface temperature [K]
Phase-change material thermal conductivity [W.m-1.K-1]
parameters:
Bed cross-sectional area [m2]
Bed porosity
Capsule radius [m]
PCM latent heat [J.kg-1]
PCM liquid phase conductivity [W.m-1.K-1]
PCM liquid phase density [kg.m-3]
PCM liquid phase specific heat capacity [J.kg-1.K-1]
PCM melting range [K]
PCM melting temperature [K]
PCM smoothing width [K]
PCM solid phase conductivity [W.m-1.K-1]
PCM solid phase density [kg.m-3]
PCM solid phase specific heat capacity [J.kg-1.K-1]
Reference temperature [K]
assumptions:
Heat flows along the primary spatial variable only.
Heat flows by conduction only (no convection in the liquid).
The volume change on melting is neglected.
coupling: WallCoupling (Wall coupling)
requires:
Phase-change material conduction heating rate [W.m-3]
Heat transfer fluid specific enthalpy [J.kg-1]
Heat transfer fluid temperature [K]
Phase-change material surface temperature [K]
Mass flow rate [kg.s-1]
provides:
Heat transfer fluid heat gain per unit length [W.m-1]
Phase-change material boundary heat flux [W.m-2]
Heat transfer coefficient [W.m-2.K-1]
Surface Reynolds number
Surface Prandtl number
parameters:
Bed cross-sectional area [m2]
Bed porosity
Capsule radius [m]
Heat transfer coefficient [W.m-2.K-1]
Heat transfer fluid conductivity [W.m-1.K-1]
Heat transfer fluid specific heat capacity [J.kg-1.K-1]
Heat transfer fluid viscosity [Pa.s]
assumptions:
(none)
device energy: DeviceEnergy (Device energy)
requires:
Mass flow rate [kg.s-1]
Outlet specific enthalpy [J.kg-1]
Inlet specific enthalpy [J.kg-1]
Phase-change material thermal energy [J]
Heat transfer fluid thermal energy [J]
Phase-change material average temperature [K]
Phase-change material cross-section average temperature [K]
provides:
Power [W]
Discharged energy [J]
Charged energy [J]
Discharged throughput [J]
Charged throughput [J]
Storage thermal energy [J]
Total thermal energy [J]
Heat loss power [W]
Storage average temperature [K]
Storage cross-section average temperature [K]
Initial uniform temperature [K]
parameters:
Initial temperature [K]
assumptions:
There are no heat losses to the surroundings.
per-area totals: PerAreaTotals (Per-area totals)
requires:
Power [W]
Discharged energy [J]
Storage thermal energy [J]
Total thermal energy [J]
provides:
Power per unit area [W.m-2]
Discharged energy per unit area [J.m-2]
Storage thermal energy per unit area [J.m-2]
Total thermal energy per unit area [J.m-2]
parameters:
Bed cross-sectional area [m2]
assumptions:
(none)
geometry parameters:
Bed length [m]
Capsule radius [m]
t_end = 160 * 60
solution = pybamm.Simulation(model).solve(
[0, t_end], t_interp=np.linspace(0, t_end, 321)
)
E = solution["Total thermal energy [J]"].entries
discharged = solution["Discharged energy [J]"].entries
print("max conservation error [J]:", np.max(np.abs(E - E[0] + discharged)))
print("stored energy [kWh]:", E[-1] / 3.6e6)
print(
"energy between 32 and 70 degC [kWh]:",
pytes.energy_between(model, 273.15 + 32, 273.15 + 70) / 3.6e6,
)
print(
"stored energy per unit area [MJ.m-2]:",
solution["Total thermal energy per unit area [J.m-2]"].entries[-1] / 1e6,
)
max conservation error [J]: 2.60770320892334e-08
stored energy [kWh]: 2.3031771680909383
energy between 32 and 70 degC [kWh]: 2.635389843901675
stored energy per unit area [MJ.m-2]: 81.45822913734806
The measured fluid temperatures at a quarter, half, three quarters and the full height of the bed (points) against the model (lines).
Z = model.default_parameter_values["Bed length [m]"]
fig, ax = plt.subplots(figsize=(6, 4))
for position, fraction in [("0.25L", 0.25), ("0.5L", 0.5), ("0.75L", 0.75), ("1L", 1)]:
t_data, T_data = np.loadtxt(
f"data/Nallusamy2007_HTF_{position}.csv", delimiter=",", skiprows=1, unpack=True
)
t = solution["Time [s]"].entries
T = solution["Heat transfer fluid temperature [degC]"](t=t, z=fraction * Z)
(line,) = ax.plot(t / 60, T, label=f"{fraction:g} L")
ax.plot(t_data, T_data, "o", color=line.get_color(), markersize=3)
ax.set_xlabel("Time [min]")
ax.set_ylabel("Fluid temperature [°C]")
ax.legend()
plt.show()
The fluid temperatures agree with the measurements to 1 K to 2 K (root mean square). Inside the capsules the agreement is looser, about 5 K at 0.8 of the capsule radius, which depends on where the thermocouples sat and on the capsule shell, which the model neglects.
With options={"storage along flow path": "averaged"}, one capsule stands for the average along the bed and
exchanges heat with the fluid averaged along it: much faster, but it cannot capture the
thermal front moving through the bed. Every model has this option.
Note the dip of the averaged outlet below the initial 32 °C early on: the heat the capsule takes from the averaged fluid is taken uniformly along the bed, including near the outlet, where the fluid is still cold. This is a known artefact of this formulation, the published reduced model; energy is still conserved exactly.
averaged = pytes.PackedBedModel(options={"storage along flow path": "averaged"})
averaged_solution = pybamm.Simulation(averaged).solve(
[0, t_end], t_interp=np.linspace(0, t_end, 321)
)
print(
f"solve time: resolved {solution.solve_time}, averaged {averaged_solution.solve_time}"
)
t_data, T_data = np.loadtxt(
"data/Nallusamy2007_HTF_1L.csv", delimiter=",", skiprows=1, unpack=True
)
fig, ax = plt.subplots(figsize=(6, 4))
for sol, label in [(solution, "resolved"), (averaged_solution, "averaged")]:
ax.plot(
sol["Time [min]"].entries, sol["Outlet temperature [degC]"].entries, label=label
)
ax.plot(t_data, T_data, "ko", markersize=3, label="measured")
ax.set_xlabel("Time [min]")
ax.set_ylabel("Outlet temperature [°C]")
ax.legend()
plt.show()
solve time: resolved 1.532 s, averaged 75.950 ms
