bluemira.fuel_cycle.tools

Fuel cycle utility objects, including sink algorithms

Classes

`NoiseModeType`	Enumeration of noise search modes.
`FitMethod`	Sink Data fit methods

Functions

`find_noisy_locals`(→ tuple[numpy.ndarray, numpy.ndarray])	Find local minima or maxima in a noisy signal.
`discretise_1d`(→ tuple[numpy.ndarray, numpy.ndarray])	Discretise x and y for a given number of points.
`convert_flux_to_flow`(→ float)	Convert an atomic flux to a flow-rate.
`piecewise_linear_threshold`(→ numpy.ndarray)	Piecewise linear model with initial linear slope, followed by threshold.
`piecewise_sqrt_threshold`(→ numpy.ndarray)	Piecewise square-root model, followed by threshold.
`fit_sink_data`(→ tuple[float, float])	Function used to determine simplified tritium sink model parameters, from
`delay_decay`(→ numpy.ndarray)	Time-shift a tritium flow with a delay and account for radioactive decay.
`fountain`(→ tuple[numpy.ndarray, Ellipsis])	Fountain tritium block. Needs a minimum T inventory to operate.
`_speed_recycle`(→ numpy.ndarray)	The main recycling loop, JIT compiled.
`find_max_load_factor`(→ float)	Finds peak slope in fpy as a function of calendar years
`legal_limit`(→ float)	Calculates the release rate of T from the model TFV cycle in kg/yr.
`_dec_I_mdot`(→ float)	Analytical value of series expansion for an inventory I with a incoming
`_timestep_decay`(→ float)	Analytical value of series expansion for an in-flux of tritium over a time-
`_find_t15`(→ float)	Inter-timestep method solving for dt in the below equality:
`_fountain_linear_sink`(→ tuple[float, float, float, float])	A simple linear fountain tritium retention sink model between a minimum
`_linear_thresh_sink`(→ tuple[float, float, float, float])	A simple linear tritium retention sink model. Used over a time-step.
`_sqrt_thresh_sink`(→ tuple[float, float, float, float])	A simple sqrt tritium retention sink model. Used over a time-step.
`linear_bathtub`(→ tuple[numpy.ndarray, numpy.ndarray, ...)	Bathtub sink model.
`sqrt_bathtub`(→ tuple[numpy.ndarray, numpy.ndarray, ...)	Bathtub sink model with a sqrt inventory retention law.
`fountain_bathtub`(→ tuple[numpy.ndarray, numpy.ndarray, ...)	A fountain and bathtub sink simultaneously.

Module Contents

class bluemira.fuel_cycle.tools.NoiseModeType(*args, **kwds)

Bases: enum.Enum

Enumeration of noise search modes.

MIN

MAX

classmethod _missing_(value: str | NoiseModeType) → NoiseModeType

Parameters:: value (str | NoiseModeType)
Return type:: NoiseModeType

bluemira.fuel_cycle.tools.find_noisy_locals(x: numpy.ndarray, x_bins: int = 50, mode: str = 'min') → tuple[numpy.ndarray, numpy.ndarray]

Find local minima or maxima in a noisy signal.

Parameters:

x (numpy.ndarray) – The noise data to search
x_bins (int) – The number of bins to search with
mode (str) – The search mode [‘min’, ‘max’]

Returns:

local_mid_x – The arguments of the local minima or maxima
local_m – The local minima or maxima

Return type:

tuple[numpy.ndarray, numpy.ndarray]

bluemira.fuel_cycle.tools.discretise_1d(x: numpy.ndarray, y: numpy.ndarray, n: int, method: str = 'linear') → tuple[numpy.ndarray, numpy.ndarray]

Discretise x and y for a given number of points.

Parameters:

x (numpy.ndarray) – The x data
y (numpy.ndarray) – The y data
n (int) – The number of discretisation points
method (str) – The interpolation method

Returns:

x_1d – The discretised x data
y_1d – The discretised y data

Return type:

tuple[numpy.ndarray, numpy.ndarray]

bluemira.fuel_cycle.tools.convert_flux_to_flow(flux: float, area: float) → float

Convert an atomic flux to a flow-rate.

Parameters:

flux (float) – The atomic flux [T/m^2/s]
area (float) – The surface area of the flux [m^2]

Return type:

The flow-rate [kg/s]

bluemira.fuel_cycle.tools.piecewise_linear_threshold(x: numpy.ndarray, x0: float, y0: float, m1: float, m2: float) → numpy.ndarray

Piecewise linear model with initial linear slope, followed by threshold.

Parameters:

x (numpy.ndarray) – The vector of x values to calculate the function for
x0 (float) – The x coordinate of the kink point
y0 (float) – The y coordinate of the kink point
m1 (float) – The slope of the first curve
m2 (float) – The threshold value of the function

Return type:

The vector of fitted values

bluemira.fuel_cycle.tools.piecewise_sqrt_threshold(x: numpy.ndarray, factor: float, kink: float, threshold: float) → numpy.ndarray

Piecewise square-root model, followed by threshold.

Parameters:

x (numpy.ndarray) – The vector of x values to calculate the function for
factor (float) – The multiplication factor for the sqrt function
kink (float) – The x value where the behaviour changes from sqrt to constant
threshold (float) – The threshold value of the model

Return type:

The vector of fitted values

class bluemira.fuel_cycle.tools.FitMethod(*args, **kwds)

Bases: enum.Enum

Sink Data fit methods

LINEAR

SQRT

classmethod _missing_(value: str)

Parameters:: value (str)

bluemira.fuel_cycle.tools.fit_sink_data(x: numpy.ndarray, y: numpy.ndarray, method: str | FitMethod = FitMethod.SQRT, *, plot: bool = True) → tuple[float, float]

Function used to determine simplified tritium sink model parameters, from data values.

Parameters:

x (numpy.ndarray) – The vector of x values
y (numpy.ndarray) – The vector of y values
method (str | FitMethod) – The type of fit to use [‘linear’, ‘sqrt’]
plot (bool) – Whether or not to plot the fitting result

Returns:

slope – The slope of the fitted piecewise linear threshold function
threshold – The threshold of the fitted piecewise linear threshold function

Return type:

tuple[float, float]

bluemira.fuel_cycle.tools.delay_decay(t: numpy.ndarray, m_t_flow: numpy.ndarray, tt_delay: float) → numpy.ndarray

Time-shift a tritium flow with a delay and account for radioactive decay.

Parameters:

t (numpy.ndarray) – The time vector [s]
m_t_flow (numpy.ndarray) – The mass flow vector [kg/s] [or any other unit same as return value]
tt_delay (float) – The delay duration (scalar) [yr]

Return type:

The delayed flow vector [kg/s] [or any other unit same as m_t_flow]

bluemira.fuel_cycle.tools.fountain(flow: numpy.ndarray, t: numpy.ndarray, min_inventory: float) → tuple[numpy.ndarray, Ellipsis]

Fountain tritium block. Needs a minimum T inventory to operate. This is a binary description. In reality, the TFV systems modelled here (such as the cryogenic distillation column) can and do operate below I_min.

Parameters:

flow (numpy.ndarray) – \(m_{T_{flow}}\) tritium flow through system [kg/s]
t (numpy.ndarray) – \(t\) time [years]
min_inventory (float) – \(I_{min}\) minimum T inventory for system to operate [kg]

Returns:

m_out – \(m_{T_{flowout}}\) mass flow out [kg/s]
inventory – \(I\) built-up T inventory in system [kg]

Return type:

tuple[numpy.ndarray, Ellipsis]

Notes

\(dt = t[i]-t[i-1]\) n \(I[i] = I[i-1]e^{-ln(2)dt/t_{1/2}}+m_{T_{flow}}dt\) n

if \(I > I_{min}\):: \(I[i] = I_{min}\) [kg]n \(m_{T_{flowout}} = \frac{I_{min}-I[i]}{dt}\) [kg/s] n
if \(I < I_{min}\):: \(I[i] = I[i-1]+m_{T_{flow}}dt\) [kg] n \(m_{T_{flowout}} = 0\) [kg/s]

bluemira.fuel_cycle.tools._speed_recycle(m_start_up: float, t: numpy.ndarray, m_in: numpy.ndarray, m_fuel_injector: numpy.ndarray) → numpy.ndarray

The main recycling loop, JIT compiled.

Parameters:

m_start_up (float) – An initial guess for the start-up inventory [kg]
t (numpy.ndarray) – The time vector [years]
m_in (numpy.ndarray) – The array of tritium flow-rates required for fusion [kg/s]
m_fuel_injector (numpy.ndarray) – The array of tritium flow-rates fuelling the plasma [kg/s]

Returns:

The tritium in the stores

Return type:

numpy.ndarray

bluemira.fuel_cycle.tools.find_max_load_factor(time_years: numpy.ndarray, time_fpy: numpy.ndarray) → float

Finds peak slope in fpy as a function of calendar years Divides implicitly by slightly less than a year

Parameters:

time_years (numpy.ndarray) – The time signal [calendar years]
time_fpy (numpy.ndarray) – The time signal [fpy]

Returns:

The maximum load factor in the time signal (over a one year period)

Return type:

float

bluemira.fuel_cycle.tools.legal_limit(max_load_factor: float, fb: float, m_gas: float, eta_f: float, eta_fuel_pump: float, f_dir: float, f_exh_split: float, f_detrit_split: float, f_terscwps: float, TBR: float, mb: float | None = None, p_fus: float | None = None) → float

Calculates the release rate of T from the model TFV cycle in kg/yr.

\(A_{max}\Bigg[\Big[\dot{m_{b}}\Big((\frac{1}{f_{b}}-1)+ (1-{\eta}_{f_{pump}})(1-{\eta}_{f})\frac{1}{f_{b}{\eta}_{f}}\Big)+ \dot{m_{gas}}\Big](1-f_{DIR})(1-f_{tfv})(1-f_{detrit})+\dot{m_{b}} \Lambda f_{TERSCWPS}\Bigg]\times365\times24\times3600\)

Where:

\(\dot{m_{b}} = \frac{P_{fus}[MW]M_{T}[g/mol]} {17.58 [MeV]eV[J]N_{A}[1/mol]} [g/s]\)

Parameters:

m_gas (float) – mass of gas flow [kg/s]
mb (float | None) – tritium inventory gross burn rate [kg/s]
p_fus (float | None) – fusion power [W]
dimensionless (All other parameters are)
max_load_factor (float)
fb (float)
eta_f (float)
eta_fuel_pump (float)
f_dir (float)
f_exh_split (float)
f_detrit_split (float)
f_terscwps (float)
TBR (float)

Returns:

release rate of T [kg/yr]

Return type:

legal_limit

Raises:

FuelCycleError – Fusion power or burn rate must be specified

bluemira.fuel_cycle.tools._dec_I_mdot(inventory: float, eta: float, m_dot: float, t_in: float, t_out: float) → float

Analytical value of series expansion for an inventory I with a incoming flux of tritium (kg/yr).

\(I_{end} = Ie^{-{\lambda}{\Delta}t}+{\eta}\dot{m}\sum_{t=0}^{{\Delta}t}e^{-\lambda(T-t)}\)

\(I_{end} = Ie^{-{\lambda}{\Delta}t}+{\eta}\dot{m}\dfrac{e^{-{\lambda}T}\big(e^{{\lambda}({\Delta}t+1/2)}-1\big)}{e^{\lambda}-1}\)

Returns:

output inventory

Raises:

ValueError – Output inventory < 0

Parameters:

inventory (float)
eta (float)
m_dot (float)
t_in (float)
t_out (float)

Return type:

float

bluemira.fuel_cycle.tools._timestep_decay(flux: float, dt: float) → float

Analytical value of series expansion for an in-flux of tritium over a time- step. Accounts for decay during the timestep only.

\(I_{end} = I\dfrac{e^{-{\lambda}T}\big(e^{{\lambda}({\Delta}t+1)}-1\big)}{e^{\lambda}-1}\)

Parameters:

flux (float) – The total inventory flowing through on a given time-step [kg]
dt (float) – The time-step [years]

Returns:

The value of the total inventory which decayed over the time-step.

Return type:

float

bluemira.fuel_cycle.tools._find_t15(inventory: float, eta: float, m_flow: float, t_in: float, t_out: float, inventory_limit: float) → float

Inter-timestep method solving for dt in the below equality:

\(Ie^{\lambda{\Delta}t}+{\eta}\dot{m}\dfrac{e^{-{\lambda}{\Delta}t} \big(e^{{\lambda}({\Delta}t+1/2)}-1\big)}{e^{\lambda}-1}=I_{lim}\)

\({\Delta}t=\dfrac{ln\bigg(\dfrac{Ie^{\lambda}-I-{\eta}\dot{m}} {I_{lim}e^{{\lambda}}-I_{lim}-{\eta}\dot{m}e^{\lambda/2}}\bigg)}{\lambda}\)

Returns:

dt relative to t_in of crossing point

Parameters:

inventory (float)
eta (float)
m_flow (float)
t_in (float)
t_out (float)
inventory_limit (float)

Return type:

float

bluemira.fuel_cycle.tools._fountain_linear_sink(m_flow: float, t_in: float, t_out: float, inventory: float, fs: float, max_inventory: float, min_inventory: float, sum_in: float, decayed: float) → tuple[float, float, float, float]

A simple linear fountain tritium retention sink model between a minimum and a maximum. Used over a time-step.

Parameters:

m_flow (float) – The in-flow of tritium [kg/s]
t_in (float) – The first point in the time-step [years]
t_out (float) – The second point in the time-step [years]
inventory (float) – The inventory of tritium already in the sink [kg]
fs (float) – The tritium release rate of the sink (1-absorbtion rate)
max_inventory (float) – The threshold inventory of the sink at which point it saturates
min_inventory (float) – The minimum inventory required for the system to release tritium
sum_in (float) – Accountancy parameter to calculate the total value lost to a sink
decayed (float) – Accountancy parameter to calculate the total value of decayed T in a sink

Returns:

m_out – The out-flow of tritium [kg/s]
inventory – The amount of tritium in the sink [kg]
sum_in – Accountancy parameter to calculate the total value lost to a sink
decayed – Accountancy parameter to calculate the total value of decayed T in a sink

Raises:

ValueError – Undefined behaviour for inventory > maximum Outflow greater than inflow Negative inventory Negative outflow

Return type:

tuple[float, float, float, float]

bluemira.fuel_cycle.tools._linear_thresh_sink(m_flow: float, t_in: float, t_out: float, inventory: float, fs: float, max_inventory: float, sum_in: float, decayed: float) → tuple[float, float, float, float]

A simple linear tritium retention sink model. Used over a time-step.

Parameters:

m_flow (float) – The in-flow of tritium [kg/s]
t_in (float) – The first point in the time-step [years]
t_out (float) – The second point in the time-step [years]
inventory (float) – The inventory of tritium already in the sink [kg]
fs (float) – The tritium release rate of the sink (1-absorbtion rate)
max_inventory (float) – The threshold inventory of the sink at which point it saturates
sum_in (float) – Accountancy parameter to calculate the total value lost to a sink
decayed (float) – Accountancy parameter to calculate the total value of decayed T in a sink

Returns:

m_out – The out-flow of tritium [kg/s]
inventory – The amount of tritium in the sink [kg]
sum_in – Accountancy parameter to calculate the total value lost to a sink
decayed – Accountancy parameter to calculate the total value of decayed T in a sink

Return type:

tuple[float, float, float, float]

bluemira.fuel_cycle.tools._sqrt_thresh_sink(m_flow: float, t_in: float, t_out: float, inventory: float, factor: float, max_inventory: float, sum_in: float, decayed: float, *, _testing: bool) → tuple[float, float, float, float]

A simple sqrt tritium retention sink model. Used over a time-step.

Parameters:

m_flow (float) – The in-flow of tritium [kg/s]
t_in (float) – The first point in the time-step [years]
t_out (float) – The second point in the time-step [years]
inventory (float) – The inventory of tritium already in the sink [kg]
factor (float) – The multiplication factor of the sqrt function
max_inventory (float) – The threshold inventory of the sink at which point it saturates
sum_in (float) – Accountancy parameter to calculate the total value lost to a sink
decayed (float) – Accountancy parameter to calculate the total value of decayed T in a sink
_testing (bool)

Returns:

m_out – The out-flow of tritium [kg/s]
inventory – The amount of tritium in the sink [kg]
sum_in – Accountancy parameter to calculate the total value lost to a sink
decayed – Accountancy parameter to calculate the total value of decayed T in a sink

Return type:

tuple[float, float, float, float]

Notes

\(I_{sequestered} = factor \times \sqrt{ t_{fpy}}\)

The time in the equation is sub-planted for the inventory, to make the retention model independent of time.

The values for the threshold and factor must be obtained from detailed T retention modelling.

Here, we’re tacking the growth of the inventory to a function, but decay is not accounted for in this function. We have to add decay in the sink and ensure this is handled when calculation the absorbtion and out-flow.

bluemira.fuel_cycle.tools.linear_bathtub(flow: numpy.ndarray, t: numpy.ndarray, eta: float, bci: int, max_inventory: float) → tuple[numpy.ndarray, numpy.ndarray, float, float]

Bathtub sink model.

Parameters:

flow (numpy.ndarray) – The vector of flow-rates [kg/s]
t (numpy.ndarray) – The time vector [years]
eta (float) – The bathtub tritium release fraction
bci (int) – The blanket change index. Used if a component is replaced to reset the inventory to 0.
max_inventory (float) – The threshold inventory for the bathtub.

Returns:

m_out – The out-flow of tritium [kg/s]
inventory – The amount of tritium in the sink [kg]
sum_in – Accountancy parameter to calculate the total value lost to a sink
decayed – Accountancy parameter to calculate the total value of decayed T in a sink

Return type:

tuple[numpy.ndarray, numpy.ndarray, float, float]

bluemira.fuel_cycle.tools.sqrt_bathtub(flow: numpy.ndarray, t: numpy.ndarray, factor: float, bci: int, max_inventory: float, *, _testing: bool = False) → tuple[numpy.ndarray, numpy.ndarray, float, float]

Bathtub sink model with a sqrt inventory retention law.

Parameters:

flow (numpy.ndarray) – The vector of flow-rates [kg/s]
bci (int) – The blanket change index. Used if a component is replaced to reset the inventory to 0.
t (numpy.ndarray) – The time vector [years]
factor (float) – The sqrt model multiplication factor
max_inventory (float) – The threshold inventory for the bathtub.
_testing (bool) – Used for testing purposes only (switches off decay).

Returns:

m_out – The out-flow of tritium [kg/s]
inventory – The amount of tritium in the sink [kg]
sum_in – Accountancy parameter to calculate the total value lost to a sink
decayed – Accountancy parameter to calculate the total value of decayed T in a sink

Return type:

tuple[numpy.ndarray, numpy.ndarray, float, float]

bluemira.fuel_cycle.tools.fountain_bathtub(flow: numpy.ndarray, t: numpy.ndarray, fs: float, max_inventory: float, min_inventory: float) → tuple[numpy.ndarray, numpy.ndarray, float, float]

A fountain and bathtub sink simultaneously.

Parameters:

flow (numpy.ndarray) – Tritium flow through system [kg/s]
t (numpy.ndarray) – Time [years]
fs (float) – efficiency of bathtub
min_inventory (float) – Minimum T inventory for system to operate [kg]
max_inventory (float) – Maximum T inventory for system to operate [kg]

Returns:

m_out – The out-flow of tritium [kg/s]
inventory – The amount of tritium in the sink [kg]
sum_in – Accountancy parameter to calculate the total value lost to a sink
decayed – Accountancy parameter to calculate the total value of decayed T in a sink

\(dt = t[i]-t[i-1]\)

\(I[i] = I[i-1]e^{-ln(2)dt/t_{1/2}}\)

if \(I < I_{min}\):
\(I[i] += m_{T_{flow}}dt\)

\(m_{T_{flowout}} = 0\)

if \(I >= I_{max}\):
\(I[i] = I_{max}\)

\(m_{T_{flowout}} = m_{T_{flow}}\)

if \(I < I_{max}\):
\(m_{T_{flowout}} = {\eta}m_{T_{flow}}\)

\(I += (1-{\eta})m_{T_{flow}}dt\)

Return type:

tuple[numpy.ndarray, numpy.ndarray, float, float]