Materials and fluids#

Materials#

Storage materials: the enthalpy-temperature relation, the conductivity and the liquid fraction of the material a storage submodel holds.

A material maps its enthalpy per unit volume H and its internal states to a temperature. The materials here have no internal states (internal_states is empty), so the temperature is a function of H alone. Materials with kinetics or hysteresis will add internal states without changing the storage submodels (DESIGN D7).

Enthalpies are per unit volume and relative to 0 K. Energy variables subtract the enthalpy at the reference temperature, so this reference never shows in the outputs.

pytes.materials.smooth_min(a, b, width)[source]#

Smooth approximation of min(a, b), (a + b - sqrt((a - b)**2 + width**2)) / 2.

It rounds the corner at a = b over a distance of order width, is never larger than min(a, b), differs from it by at most width / 2 (at the corner), and is exactly min(a, b) when width is 0.

Parameters:
Returns:

pybamm.Symbol – The smoothed minimum.

pytes.materials.smooth_max(a, b, width)[source]#

Smooth approximation of max(a, b), (a + b + sqrt((a - b)**2 + width**2)) / 2. Never smaller than max(a, b); see smooth_min().

class pytes.materials.SensibleMaterial(prefix='Metal')[source]#

Bases: object

A material that does not change phase, such as the metal of the fins.

Parameters:

prefix (str, optional) – Prefix of the parameter names (default “Metal”): “<prefix> conductivity [W.m-1.K-1]”, “<prefix> density [kg.m-3]” and “<prefix> specific heat capacity [J.kg-1.K-1]”.

Variables:

  • c_p (k_param, rho,) – Conductivity [W.m-1.K-1], density [kg.m-3] and specific heat capacity [J.kg-1.K-1].

  • rho_c_p (pybamm.Symbol) – Volumetric heat capacity [J.m-3.K-1].

  • internal_states (tuple) – Empty: the temperature is a function of the enthalpy alone.

  • changes_phase (bool) – False: the material has no liquid fraction.

  • H_reference (H_scale,) – Typical enthalpy scale and reference [J.m-3], those of 100 K and 300 K. A storage submodel may override them (e.g. to scale like the model’s temperatures).

H2T(H, states=())[source]#

Temperature [K] as a function of the enthalpy per unit volume [J.m-3]

T2H(T)[source]#

Enthalpy per unit volume [J.m-3], relative to 0 K, at temperature T [K]

k(H, states=())[source]#

Conductivity [W.m-1.K-1] (constant)

class pytes.materials.PhaseChangeMaterial[source]#

Bases: object

Thermal properties of a material that melts, and the enthalpy-temperature relations derived from them. The enthalpy is per unit volume and relative to 0 K.

Subclasses set the solid and liquid volumetric heat capacities rho_c_p_s and rho_c_p_l [J.m-3.K-1], the volumetric latent heat rho_L [J.m-3], the solid and liquid conductivities k_s and k_l [W.m-1.K-1], the melting temperature T_m [K], the melting range melting_range [K] and the smoothing width smoothing_width [K], then call _set_derived().

The material melts between the solidus T_m - melting_range / 2 and the liquidus T_m + melting_range / 2 (DESIGN D7). In between (the mushy zone) the enthalpy is linear in temperature: the latent heat spread evenly, plus the mean of the solid and liquid heat capacities. A melting range of 0 gives a sharp melting point, where the temperature stays at T_m while the material melts.

Separately, the corners of the relations at the solidus and liquidus are smoothed over smoothing_width, a numerical device that makes the models faster. A width of 0 gives exact corners. Both may be 0.

Variables:

  • internal_states (tuple) – Empty: the temperature is a function of the enthalpy alone.

  • changes_phase (bool) – True: the material has a liquid fraction.

  • T_liquidus (T_solidus,) – Solidus and liquidus temperatures [K].

  • rho_c_p_mushy (pybamm.Symbol) – Sensible heat capacity in the mushy zone [J.m-3.K-1], the mean of the solid and liquid ones.

  • H_liquidus (H_solidus,) – Enthalpy at the solidus (fully solid) and at the liquidus (fully liquid) [J.m-3].

  • dT_smooth (pybamm.Symbol) – The smoothing width, floored at 1e-9 K so that derivatives stay defined where the arguments of smooth_min() and smooth_max() are equal (its effect is at most 5e-10 K).

  • H_reference (H_scale,) – Scale and reference for the enthalpy as a solver variable: the enthalpy of the mushy zone and its middle. The scaled enthalpy is then the liquid fraction minus 1/2 while melting, and of order one over the operating range, as the sensible heat is comparable to the latent heat.

H2T(H, states=())[source]#

Temperature as a function of the enthalpy and the internal states (none here).

Below the solidus the temperature is H / rho_c_p_s; in the mushy zone it rises linearly from the solidus to the liquidus temperature with the liquid fraction; above the liquidus it is T_liquidus + (H - H_liquidus) / rho_c_p_l. The corners are smoothed over dT_smooth.

Parameters:

  • H (pybamm.Symbol) – Enthalpy per unit volume, relative to 0 K [J.m-3].

  • states (tuple, optional) – Internal states, in the order of internal_states (none for this material).

Returns:

pybamm.Symbol – Temperature [K].

T2H(T)[source]#

Enthalpy as a function of the temperature.

This is the exact (unsmoothed) relation. Away from the solidus and the liquidus it inverts H2T() up to about dT_smooth**2 / (4 d), with d the distance to the nearer of them. For a sharp melting point (melting range 0), the material at exactly T = T_m is taken to be fully solid.

Parameters:

T (pybamm.Symbol) – Temperature [K].

Returns:

pybamm.Symbol – Enthalpy per unit volume, relative to 0 K [J.m-3].

k(H, states=())[source]#

Conductivity [W.m-1.K-1] as a function of the enthalpy [J.m-3], interpolated linearly in the liquid fraction between k_s and k_l.

liquid_fraction(H, states=())[source]#

Liquid fraction as a function of the enthalpy [J.m-3]: the fraction of the mushy zone’s enthalpy, (H - H_solidus) / (H_liquidus - H_solidus), clipped to [0, 1] with the corners smoothed over the enthalpy widths matching dT_smooth (rho_c_p_s * dT_smooth and rho_c_p_l * dT_smooth). In the mushy zone it is also (T - T_solidus) / melting_range. It is between 0 and 1 up to an overshoot of order (rho_c_p * dT_smooth / rho_L)**2 when smoothed.

class pytes.materials.PureMaterial(prefix='PCM')[source]#

Bases: PhaseChangeMaterial

A phase-change material on its own: rho_c_p_s = rho_s c_p_s, rho_c_p_l = rho_l c_p_l, rho_L = rho_s L (the latent heat per unit volume uses the solid density, neglecting the volume change on melting).

Parameters:

prefix (str, optional) – Prefix of the parameter names (default “PCM”), e.g. “<prefix> latent heat [J.kg-1]”, “<prefix> melting range [K]” and “<prefix> smoothing width [K]”.

Variables:

smoothing_width (k_s, k_l, rho_s, rho_l, c_p_s, c_p_l, L, T_m, melting_range,) – “<prefix> solid phase conductivity [W.m-1.K-1]”, “<prefix> liquid phase conductivity [W.m-1.K-1]”, “<prefix> solid phase density [kg.m-3]”, “<prefix> liquid phase density [kg.m-3]”, “<prefix> solid phase specific heat capacity [J.kg-1.K-1]”, “<prefix> liquid phase specific heat capacity [J.kg-1.K-1]”, “<prefix> latent heat [J.kg-1]”, “<prefix> melting temperature [K]” (the middle of the melting range), “<prefix> melting range [K]” (from solidus to liquidus; 0 for a sharp melting point) and “<prefix> smoothing width [K]” (numerical; 0 for exact corners).

class pytes.materials.CompositeMaterial(pcm, filler, volume_fraction)[source]#

Bases: PhaseChangeMaterial

A phase-change material and a sensible filler (such as metal fins) homogenised into one material, with the filler taking up a volume fraction delta:

  • heat capacities: volume averages, rho_c_p_s = delta rho_c_p_filler + (1 - delta) rho_c_p_s,PCM (and similarly for the liquid);

  • latent heat: rho_L = (1 - delta) rho_L,PCM, as only the PCM melts;

  • conductivities: volume averages, k_s = (1 - delta) k_s,PCM + delta k_filler (and similarly for the liquid), i.e. the parallel arrangement, for heat flowing along the filler.

Parameters:

Fluids#

Heat transfer fluid (HTF) properties.

The fluid-property object is the only way a model obtains fluid properties (DESIGN D8). Its properties are functions of the fluid state, the specific enthalpy h and the pressure p, the natural inputs of property tables, so that they stay valid for a fluid whose temperature is flat across a phase change (a refrigerant, later). h(T, p) gives the enthalpy for the inlet and initial conditions. Power is defined from the specific enthalpy, mdot (h_out - h_in).

The heat transfer coefficient is not a fluid property: it depends on the geometry and the flow too, and belongs to the surface flux law of a coupling (DESIGN D6, D8, see pytes.submodels.flux_laws).

class pytes.fluids.ConstantPropertyLiquid(prefix='Heat transfer fluid', T_ref=0)[source]#

Bases: object

A liquid with constant density and specific heat capacity, so that its specific enthalpy is linear in temperature: h = c_p (T - T_ref). The pressure is ignored.

The enthalpy reference T_ref only shifts h; energy variables and power use differences of h, so it never shows in the outputs.

Parameters:

  • prefix (str, optional) – Prefix of the parameter names (default “Heat transfer fluid”).

  • T_ref (pybamm.Symbol or float, optional) – Temperature at which h = 0 [K] (default 0 K).

Variables:

mu_param (rho_param, c_p_param, k_param,) – “<prefix> density [kg.m-3]”, “<prefix> specific heat capacity [J.kg-1.K-1]”, “<prefix> conductivity [W.m-1.K-1]” and “<prefix> viscosity [Pa.s]”. The conductivity and the viscosity are used by the flow-dependent heat transfer coefficients (pytes.submodels.NusseltCorrelation); the conductivity is also the hook for axial diffusion.

h(T, p=None)[source]#

Specific enthalpy [J.kg-1] at temperature T [K] (p is ignored)

T(h, p=None)[source]#

Temperature [K] at specific enthalpy h [J.kg-1] (p is ignored)

c_p(h, p=None)[source]#

Specific heat capacity dh/dT [J.kg-1.K-1] at specific enthalpy h

rho(h, p=None)[source]#

Density [kg.m-3] at specific enthalpy h [J.kg-1]

k(h, p=None)[source]#

Conductivity [W.m-1.K-1] at specific enthalpy h [J.kg-1]

mu(h, p=None)[source]#

Dynamic viscosity [Pa.s] at specific enthalpy h [J.kg-1]