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simplify expression of cooling terms in x-lumped thermal models #3439

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Nov 7, 2023
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simplify expression of cooling terms in x-lumped thermal models #3439

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12 changes: 8 additions & 4 deletions docs/source/examples/notebooks/models/pouch-cell-model.ipynb
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5 changes: 2 additions & 3 deletions pybamm/models/submodels/thermal/lumped.py
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Original file line number Diff line number Diff line change
Expand Up @@ -56,10 +56,9 @@ def set_rhs(self, variables):
# Newton cooling, accounting for surface area to volume ratio
cell_surface_area = self.param.A_cooling
cell_volume = self.param.V_cell
total_cooling_coefficient = (
-self.param.h_total * cell_surface_area / cell_volume
Q_cool_vol_av = (
-self.param.h_total * (T_vol_av - T_amb) * cell_surface_area / cell_volume
)
Q_cool_vol_av = total_cooling_coefficient * (T_vol_av - T_amb)

self.rhs = {
T_vol_av: (Q_vol_av + Q_cool_vol_av) / self.param.rho_c_p_eff(T_vol_av)
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64 changes: 28 additions & 36 deletions pybamm/models/submodels/thermal/pouch_cell/pouch_cell_1D_current_collectors.py
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Original file line number Diff line number Diff line change
Expand Up @@ -58,33 +58,29 @@ def set_rhs(self, variables):
y = pybamm.standard_spatial_vars.y
z = pybamm.standard_spatial_vars.z

# Account for surface area to volume ratio of pouch cell in surface and side
# cooling terms
cell_volume = self.param.L * self.param.L_y * self.param.L_z

# Calculate cooling, accounting for surface area to volume ratio of pouch cell
edge_area = self.param.L_z * self.param.L
yz_surface_area = self.param.L_y * self.param.L_z
yz_surface_cooling_coefficient = (
cell_volume = self.param.L * self.param.L_y * self.param.L_z
Q_yz_surface = (
-(self.param.n.h_cc(y, z) + self.param.p.h_cc(y, z))
* (T_av - T_amb)
* yz_surface_area
/ cell_volume
)

side_edge_area = self.param.L_z * self.param.L
side_edge_cooling_coefficient = (
Q_edge = (
-(self.param.h_edge(0, z) + self.param.h_edge(self.param.L_y, z))
* side_edge_area
* (T_av - T_amb)
* edge_area
/ cell_volume
)

total_cooling_coefficient = (
yz_surface_cooling_coefficient + side_edge_cooling_coefficient
)
Q_cool_total = Q_yz_surface + Q_edge

self.rhs = {
T_av: (
pybamm.div(self.param.lambda_eff(T_av) * pybamm.grad(T_av))
+ Q_av
+ total_cooling_coefficient * (T_av - T_amb)
+ Q_cool_total
)
/ self.param.rho_c_p_eff(T_av)
}
Expand All @@ -94,7 +90,7 @@ def set_boundary_conditions(self, variables):
T_amb = variables["Ambient temperature [K]"]
T_av = variables["X-averaged cell temperature [K]"]

# find tab locations (top vs bottom)
# Find tab locations (top vs bottom)
L_y = param.L_y
L_z = param.L_z
neg_tab_z = param.n.centre_z_tab
Expand All @@ -104,11 +100,10 @@ def set_boundary_conditions(self, variables):
pos_tab_top_bool = pybamm.Equality(pos_tab_z, L_z)
pos_tab_bottom_bool = pybamm.Equality(pos_tab_z, 0)

# calculate tab vs non-tab area on top and bottom
# Calculate tab vs non-tab area on top and bottom
neg_tab_area = param.n.L_tab * param.n.L_cc
pos_tab_area = param.p.L_tab * param.p.L_cc
total_area = param.L * param.L_y

non_tab_top_area = (
total_area
- neg_tab_area * neg_tab_top_bool
Expand All @@ -120,18 +115,22 @@ def set_boundary_conditions(self, variables):
- pos_tab_area * pos_tab_bottom_bool
)

# calculate effective cooling coefficients
# Calculate heat fluxes weighted by area
# Note: can't do y-average of h_edge here since y isn't meshed. Evaluate at
# midpoint.
top_cooling_coefficient = (
param.n.h_tab * neg_tab_area * neg_tab_top_bool
+ param.p.h_tab * pos_tab_area * pos_tab_top_bool
+ param.h_edge(L_y / 2, L_z) * non_tab_top_area
q_tab_n = -param.n.h_tab * (T_av - T_amb)
q_tab_p = -param.p.h_tab * (T_av - T_amb)
q_edge_top = -param.h_edge(L_y / 2, L_z) * (T_av - T_amb)
q_edge_bottom = -param.h_edge(L_y / 2, 0) * (T_av - T_amb)
q_top = (
q_tab_n * neg_tab_area * neg_tab_top_bool
+ q_tab_p * pos_tab_area * pos_tab_top_bool
+ q_edge_top * non_tab_top_area
) / total_area
bottom_cooling_coefficient = (
param.n.h_tab * neg_tab_area * neg_tab_bottom_bool
+ param.p.h_tab * pos_tab_area * pos_tab_bottom_bool
+ param.h_edge(L_y / 2, 0) * non_tab_bottom_area
q_bottom = (
q_tab_n * neg_tab_area * neg_tab_bottom_bool
+ q_tab_p * pos_tab_area * pos_tab_bottom_bool
+ q_edge_bottom * non_tab_bottom_area
) / total_area

# just use left and right for clarity
Expand All @@ -141,21 +140,14 @@ def set_boundary_conditions(self, variables):
self.boundary_conditions = {
T_av: {
"left": (
pybamm.boundary_value(
bottom_cooling_coefficient * (T_av - T_amb),
"left",
)
/ pybamm.boundary_value(lambda_eff, "left"),
pybamm.boundary_value(-q_bottom / lambda_eff, "left"),
"Neumann",
),
"right": (
pybamm.boundary_value(
-top_cooling_coefficient * (T_av - T_amb), "right"
)
/ pybamm.boundary_value(lambda_eff, "right"),
pybamm.boundary_value(q_top / lambda_eff, "right"),
"Neumann",
),
}
},
}

def set_initial_conditions(self, variables):
Expand Down
47 changes: 22 additions & 25 deletions pybamm/models/submodels/thermal/pouch_cell/pouch_cell_2D_current_collectors.py
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Original file line number Diff line number Diff line change
Expand Up @@ -58,20 +58,22 @@ def set_rhs(self, variables):
y = pybamm.standard_spatial_vars.y
z = pybamm.standard_spatial_vars.z

# Calculate cooling
Q_yz_surface_W_per_m2 = -(self.param.n.h_cc(y, z) + self.param.p.h_cc(y, z)) * (
T_av - T_amb
)
Q_edge_W_per_m2 = -self.param.h_edge(y, z) * (T_av - T_amb)

# Account for surface area to volume ratio of pouch cell in surface cooling
# term
cell_volume = self.param.L * self.param.L_y * self.param.L_z

yz_surface_area = self.param.L_y * self.param.L_z
yz_surface_cooling_coefficient = (
-(self.param.n.h_cc(y, z) + self.param.p.h_cc(y, z))
* yz_surface_area
/ cell_volume
cell_volume = self.param.L * self.param.L_y * self.param.L_z
Q_yz_surface = pybamm.source(
Q_yz_surface_W_per_m2 * yz_surface_area / cell_volume, T_av
)

# Edge cooling appears as a boundary term, so no need to account for surface
# area to volume ratio
edge_cooling_coefficient = -self.param.h_edge(y, z)
Q_edge = pybamm.source(Q_edge_W_per_m2, T_av, boundary=True)

# Governing equations contain:
# - source term for y-z surface cooling
Expand All @@ -88,10 +90,8 @@ def set_rhs(self, variables):
T_av: (
self.param.lambda_eff(T_av) * pybamm.laplacian(T_av)
+ pybamm.source(Q_av, T_av)
+ pybamm.source(yz_surface_cooling_coefficient * (T_av - T_amb), T_av)
+ pybamm.source(
edge_cooling_coefficient * (T_av - T_amb), T_av, boundary=True
)
+ Q_yz_surface
+ Q_edge
)
/ self.param.rho_c_p_eff(T_av)
}
Expand All @@ -102,24 +102,21 @@ def set_boundary_conditions(self, variables):
y = pybamm.standard_spatial_vars.y
z = pybamm.standard_spatial_vars.z

# Calculate heat fluxes
q_tab_n = -self.param.n.h_tab * (T_av - T_amb)
q_tab_p = -self.param.p.h_tab * (T_av - T_amb)
q_edge = -self.param.h_edge(y, z) * (T_av - T_amb)

# Subtract the edge cooling from the tab portion so as to not double count
# Note: tab cooling is also only applied on the current collector hence
# the (l_cn / l) and (l_cp / l) prefactors. We also still have edge cooling
# the (l_cn / l) and (l_cp / l) prefactors. We still have edge cooling
# in the region: x in (0, 1)
h_tab_n_corrected = (self.param.n.L_cc / self.param.L) * (
self.param.n.h_tab - self.param.h_edge(y, z)
)
h_tab_p_corrected = (self.param.p.L_cc / self.param.L) * (
self.param.p.h_tab - self.param.h_edge(y, z)
)

negative_tab_bc = pybamm.boundary_value(
-h_tab_n_corrected * (T_av - T_amb) / self.param.n.lambda_cc(T_av),
negative_tab_bc = (self.param.n.L_cc / self.param.L) * pybamm.boundary_value(
(q_tab_n - q_edge) / self.param.n.lambda_cc(T_av),
"negative tab",
)
positive_tab_bc = pybamm.boundary_value(
-h_tab_p_corrected * (T_av - T_amb) / self.param.p.lambda_cc(T_av),
"positive tab",
positive_tab_bc = (self.param.p.L_cc / self.param.L) * pybamm.boundary_value(
(q_tab_p - q_edge) / self.param.p.lambda_cc(T_av), "positive tab"
)

self.boundary_conditions = {
Expand Down
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