def initialise_target(self, c, key):
r"""
Return a starting value for pressure and enthalpy at inlet.
Parameters
----------
c : tespy.connections.connection.Connection
Connection to perform initialisation on.
key : str
Fluid property to retrieve.
Returns
-------
val : float
Starting value for pressure/enthalpy in SI units.
.. math::
val = \begin{cases}
4 \cdot 10^5 & \text{key = 'p'}\\
h\left(p, 300 \text{K} \right) & \text{key = 'h' at inlet 1}\\
h\left(p, 220 \text{K} \right) & \text{key = 'h' at outlet 2}
\end{cases}
"""
if key == 'p':
return 50e5
elif key == 'h':
if c.target_id == 'in1':
T = 300 + 273.15
return h_mix_pT(c.get_flow(), T)
else:
T = 220 + 273.15
return h_mix_pT(c.get_flow(), T)
def energy_balance(self):
r"""
Calculate the residual in energy balance of the water electrolyzer.
The residual is the negative to the necessary power input.
Returns
-------
res : float
Residual value.
.. math::
\begin{split}
res = & \dot{m}_{in,2} \cdot \left( h_{in,2} - h_{in,2,ref}
\right)\\ & - \dot{m}_{out,3} \cdot e_0\\
& -\dot{m}_{in,1} \cdot \left( h_{out,1} - h_{in,1} \right)\\
& - \dot{m}_{out,2} \cdot \left( h_{out,2} - h_{out,2,ref}
\right)\\
& - \dot{m}_{out,3} \cdot \left( h_{out,3} - h_{out,3,ref}
\right)\\
\end{split}
Note
----
The temperature for the reference state is set to 20 °C, thus
the feed water must be liquid as proposed in the calculation of
the minimum specific energy consumption for electrolysis:
:func:`tespy.components.reactors.water_electrolyzer.calc_e0`.
The part of the equation regarding the cooling water is implemented
with negative sign as the energy for cooling is extracted from the
reactor.
- Reference temperature: 293.15 K.
- Reference pressure: 1 bar.
"""
T_ref = 293.15
p_ref = 1e5
# equations to set a reference point for each h2o, h2 and o2
h_refh2o = h_mix_pT([1, p_ref, 0, self.inl[1].fluid.val], T_ref)
h_refh2 = h_mix_pT([1, p_ref, 0, self.outl[2].fluid.val], T_ref)
h_refo2 = h_mix_pT([1, p_ref, 0, self.outl[1].fluid.val], T_ref)
val = (self.inl[1].m.val_SI * (self.inl[1].h.val_SI - h_refh2o) -
self.outl[2].m.val_SI * self.e0 -
self.inl[0].m.val_SI * (self.outl[0].h.val_SI -
self.inl[0].h.val_SI) -
self.outl[1].m.val_SI * (self.outl[1].h.val_SI - h_refo2) -
self.outl[2].m.val_SI * (self.outl[2].h.val_SI - h_refh2))
return val
def initialise_target(self, c, key):
r"""
Return a starting value for pressure and enthalpy at inlet.
Parameters
----------
c : tespy.connections.connection.Connection
Connection to perform initialisation on.
key : str
Fluid property to retrieve.
Returns
-------
val : float
Starting value for pressure/enthalpy in SI units.
.. math::
val = \begin{cases}
5 \cdot 10^5 & \text{key = 'p'}\\
h\left(T=293.15, p=5 \cdot 10^5\right) & \text{key = 'h'}
\end{cases}
"""
if key == 'p':
return 5e5
elif key == 'h':
flow = c.get_flow()
T = 20 + 273.15
return h_mix_pT(flow, T)
def initialise_source(self, c, key):
r"""
Returns a starting value for pressure and enthalpy at component's
outlet.
Parameters
----------
c : tespy.connections.connection
Connection to perform initialisation on.
key : str
Fluid property to retrieve.
Returns
-------
val : float
Starting value for pressure/enthalpy in SI units.
.. math::
val = \begin{cases}
5 \cdot 10^5 & \text{key = 'p'}\\
h\left(T=323.15, p=5 \cdot 10^5\right) & \text{key = 'h'}
\end{cases}
"""
if key == 'p':
return 5e5
elif key == 'h':
flow = [c.m.val0, 5e5, c.h.val_SI, c.fluid.val]
T = 50 + 273.15
return h_mix_pT(flow, T)
def desuperheat(ude):
ttd_min = ude.params['ttd_min']
return (
ude.conns[0].h.val_SI - ude.conns[1].h.val_SI -
ude.params['distance'] * (
ude.conns[0].h.val_SI - h_mix_pT(
ude.conns[1].get_flow(),
T_mix_ph(ude.conns[2].get_flow()) + ttd_min)))
def energy_balance_func(self):
r"""
Calculate the energy balance of the diabatic combustion chamber.
Returns
-------
residual : float
Residual value of equation.
.. math::
\begin{split}
0 = & \sum_i \dot{m}_{in,i} \cdot
\left( h_{in,i} - h_{in,i,ref} \right)\\
& -\dot{m}_{out,2}\cdot\left( h_{out,1}-h_{out,1,ref} \right)\\
& + LHV_{fuel} \cdot\left(\sum_i\dot{m}_{in,i}\cdot
x_{fuel,in,i}- \dot{m}_{out,1} \cdot x_{fuel} \right)
\cdot \eta
\end{split}\\
\forall i \in \text{inlets}
Note
----
The temperature for the reference state is set to 25 °C, thus
the water may be liquid. In order to make sure, the state is
referring to the lower heating value, the state of the water in the
flue gas is fored to gaseous.
- Reference temperature: 298.15 K.
- Reference pressure: 1 bar.
"""
T_ref = 298.15
p_ref = 1e5
res = 0
for i in self.inl:
res += i.m.val_SI * (i.h.val_SI - h_mix_pT(
[0, p_ref, 0, i.fluid.val], T_ref, force_gas=True))
for o in self.outl:
res -= o.m.val_SI * (o.h.val_SI - h_mix_pT(
[0, p_ref, 0, o.fluid.val], T_ref, force_gas=True))
res += self.calc_ti() * self.eta.val
return res
def initialise_source(self, c, key):
r"""
Return a starting value for pressure and enthalpy at outlet.
Parameters
----------
c : tespy.connections.connection
Connection to perform initialisation on.
key : str
Fluid property to retrieve.
Returns
-------
val : float
Starting value for pressure/enthalpy in SI units.
.. math::
val = \begin{cases}
10 \cdot 10^5 & \text{key = 'p'}\\
h\left(p, 473.15 \text{K} \right) &
\text{key = 'h' at outlet 1}\\
h\left(p, 473.15 \text{K} \right) &
\text{key = 'h' at outlet 2}\\
h\left(p, 523.15 \text{K} \right) &
\text{key = 'h' at outlet 3}
\end{cases}
"""
if key == 'p':
return 10e5
elif key == 'h':
flow = c.to_flow()
if c.source_id == 'out1':
T = 200 + 273.15
return h_mix_pT(flow, T)
elif c.source_id == 'out2':
T = 200 + 273.15
return h_mix_pT(flow, T)
else:
T = 250 + 273.15
return h_mix_pT(flow, T)
def energy_balance_func(self):
r"""
Calculate the energy balance of the adiabatic combustion chamber.
Returns
-------
residual : float
Residual value of equation.
.. math::
\begin{split}
0 = & \sum_i \dot{m}_{in,i} \cdot
\left( h_{in,i} - h_{in,i,ref} \right)\\
& -\dot{m}_{out,2}\cdot\left( h_{out,1}-h_{out,1,ref} \right)\\
& + LHV_{fuel} \cdot\left(\sum_i\dot{m}_{in,i}\cdot
x_{fuel,in,i}- \dot{m}_{out,1} \cdot x_{fuel} \right)
\end{split}\\
\forall i \in \text{inlets}
Note
----
The temperature for the reference state is set to 100 °C, as the
custom fluid properties are inacurate at the dew-point of water in
the flue gas!
- Reference temperature: 373.15 K.
- Reference pressure: 1 bar.
"""
T_ref = 373.15
p_ref = 1e5
res = 0
for i in self.inl:
res += i.m.val_SI * (
i.h.val_SI - h_mix_pT([0, p_ref, 0, i.fluid.val], T_ref))
for o in self.outl:
res -= o.m.val_SI * (
o.h.val_SI - h_mix_pT([0, p_ref, 0, o.fluid.val], T_ref))
return res + self.calc_ti()
def energy_balance_deriv(self, increment_filter, k):
r"""
Partial derivatives for reactor energy balance.
Parameters
----------
increment_filter : ndarray
Matrix for filtering non-changing variables.
k : int
Position of derivatives in Jacobian matrix (k-th equation).
"""
# derivatives determined from calc_P function
T_ref = 298.15
p_ref = 1e5
h_refh2o = h_mix_pT([1, p_ref, 0, self.inl[1].fluid.val], T_ref)
h_refh2 = h_mix_pT([1, p_ref, 0, self.outl[2].fluid.val], T_ref)
h_refo2 = h_mix_pT([1, p_ref, 0, self.outl[1].fluid.val], T_ref)
# derivatives cooling water inlet
self.jacobian[k, 0, 0] = self.inl[0].h.val_SI - self.outl[0].h.val_SI
self.jacobian[k, 0, 2] = self.inl[0].m.val_SI
# derivatives feed water inlet
self.jacobian[k, 1, 0] = (self.inl[1].h.val_SI - h_refh2o)
self.jacobian[k, 1, 2] = self.inl[1].m.val_SI
# derivative cooling water outlet
self.jacobian[k, 2, 2] = -self.inl[0].m.val_SI
# derivatives oxygen outlet
self.jacobian[k, 3, 0] = -(self.outl[1].h.val_SI - h_refo2)
self.jacobian[k, 3, 2] = -self.outl[1].m.val_SI
# derivatives hydrogen outlet
self.jacobian[k, 4, 0] = -(self.outl[2].h.val_SI - h_refh2 + self.e0)
self.jacobian[k, 4, 2] = -self.outl[2].m.val_SI
# derivatives for variable P
if self.P.is_var:
self.jacobian[k, 5 + self.P.var_pos, 0] = 1
def calc_parameters(self):
r"""Postprocessing parameter calculation."""
CombustionChamber.calc_parameters(self)
T_ref = 298.15
p_ref = 1e5
res = 0
for i in self.inl:
res += i.m.val_SI * (i.h.val_SI - h_mix_pT(
[0, p_ref, 0, i.fluid.val], T_ref, force_gas=True))
for o in self.outl:
res -= o.m.val_SI * (o.h.val_SI - h_mix_pT(
[0, p_ref, 0, o.fluid.val], T_ref, force_gas=True))
self.eta.val = -res / self.ti.val
self.Q_loss.val = -(1 - self.eta.val) * self.ti.val
if self.inl[1].p.val < self.inl[0].p.val:
msg = (
"The pressure at inlet 2 is lower than the pressure at inlet 1 "
"at component " + self.label + ".")
logging.warning(msg)
def derivatives(self, vec_z):
r"""
Calculate partial derivatives for given equations.
Returns
-------
mat_deriv : ndarray
Matrix of partial derivatives.
"""
######################################################################
# derivatives fluid, mass flow and reactor pressure are static
k = self.num_nw_fluids * 4 + 5
######################################################################
# derivatives for energy balance equations
T_ref = 293.15
p_ref = 1e5
h_refh2o = h_mix_pT([1, p_ref, 0, self.inl[1].fluid.val], T_ref)
h_refh2 = h_mix_pT([1, p_ref, 0, self.outl[2].fluid.val], T_ref)
h_refo2 = h_mix_pT([1, p_ref, 0, self.outl[1].fluid.val], T_ref)
# derivatives cooling water inlet
self.mat_deriv[k, 0, 0] = -(
self.outl[0].h.val_SI - self.inl[0].h.val_SI)
self.mat_deriv[k, 0, 2] = self.inl[0].m.val_SI
# derivatives feed water inlet
self.mat_deriv[k, 1, 0] = (self.inl[1].h.val_SI - h_refh2o)
self.mat_deriv[k, 1, 2] = self.inl[1].m.val_SI
# derivative cooling water outlet
self.mat_deriv[k, 2, 2] = - self.inl[0].m.val_SI
# derivatives oxygen outlet
self.mat_deriv[k, 3, 0] = - (self.outl[1].h.val_SI - h_refo2)
self.mat_deriv[k, 3, 2] = - self.outl[1].m.val_SI
# derivatives hydrogen outlet
self.mat_deriv[k, 4, 0] = - self.e0 - (self.outl[2].h.val_SI - h_refh2)
self.mat_deriv[k, 4, 2] = - self.outl[2].m.val_SI
# derivatives for variable P
if self.P.is_var:
self.mat_deriv[k, 5 + self.P.var_pos, 0] = 1
k += 1
######################################################################
# derivatives for temperature at gas outlets
# derivatives for outlet 1
if not vec_z[3, 1]:
self.mat_deriv[k, 3, 1] = dT_mix_dph(self.outl[1].to_flow())
if not vec_z[3, 2]:
self.mat_deriv[k, 3, 2] = dT_mix_pdh(self.outl[1].to_flow())
# derivatives for outlet 2
if not vec_z[4, 1]:
self.mat_deriv[k, 4, 1] = - dT_mix_dph(self.outl[2].to_flow())
if not vec_z[4, 2]:
self.mat_deriv[k, 4, 2] = - dT_mix_pdh(self.outl[2].to_flow())
k += 1
######################################################################
# derivatives for power vs. hydrogen production
if self.e.is_set:
self.mat_deriv[k, 4, 0] = - self.e.val
# derivatives for variable P
if self.P.is_var:
self.mat_deriv[k, 5 + self.P.var_pos, 0] = 1
# derivatives for variable e
if self.e.is_var:
self.mat_deriv[k, 5 + self.e.var_pos, 0] = -(
self.outl[2].m.val_SI)
k += 1
######################################################################
# derivatives for pressure ratio
if self.pr_c.is_set:
self.mat_deriv[k, 0, 1] = self.pr_c.val
self.mat_deriv[k, 2, 1] = - 1
k += 1
######################################################################
# derivatives for zeta value
if self.zeta.is_set:
f = self.zeta_func
if not vec_z[0, 0]:
self.mat_deriv[k, 0, 0] = self.numeric_deriv(
f, 'm', 0, zeta='zeta')
if not vec_z[0, 1]:
self.mat_deriv[k, 0, 1] = self.numeric_deriv(
f, 'p', 0, zeta='zeta')
if not vec_z[0, 2]:
self.mat_deriv[k, 0, 2] = self.numeric_deriv(
f, 'h', 0, zeta='zeta')
if not vec_z[2, 1]:
self.mat_deriv[k, 2, 1] = self.numeric_deriv(
f, 'p', 2, zeta='zeta')
if not vec_z[2, 2]:
self.mat_deriv[k, 2, 2] = self.numeric_deriv(
f, 'h', 2, zeta='zeta')
# derivatives for variable zeta
if self.zeta.is_var:
self.mat_deriv[k, 5 + self.zeta.var_pos, 0] = (
self.numeric_deriv(
f, 'zeta', 5, zeta='zeta'))
k += 1
######################################################################
# derivatives for heat flow
if self.Q.is_set:
self.mat_deriv[k, 0, 0] = -(
self.inl[0].h.val_SI - self.outl[0].h.val_SI)
self.mat_deriv[k, 0, 2] = - self.inl[0].m.val_SI
self.mat_deriv[k, 2, 2] = self.inl[0].m.val_SI
k += 1
######################################################################
# specified efficiency (efficiency definition: e0 / e)
if self.eta.is_set:
self.mat_deriv[k, 4, 0] = - self.e0 / self.eta.val
# derivatives for variable P
if self.P.is_var:
self.mat_deriv[k, 5 + self.P.var_pos, 0] = 1
k += 1
######################################################################
# specified characteristic line for efficiency
if self.eta_char.is_set:
self.mat_deriv[k, 4, 0] = self.numeric_deriv(
self.eta_char_func, 'm', 4)
# derivatives for variable P
if self.P.is_var:
self.mat_deriv[k, 5 + self.P.var_pos, 0] = 1
k += 1
def derivatives(self):
r"""
Calculates matrix of partial derivatives for given equations.
Returns
-------
mat_deriv : ndarray
Matrix of partial derivatives.
"""
mat_deriv = []
######################################################################
# derivatives for cooling liquid composition
deriv = np.zeros((self.num_fl, 5 + self.num_vars, self.num_fl + 3))
j = 0
for fluid, x in self.inl[0].fluid.val.items():
deriv[j, 0, 3 + j] = 1
deriv[j, 2, 3 + j] = -1
j += 1
mat_deriv += deriv.tolist()
# derivatives to constrain fluids to inlets/outlets
deriv = np.zeros((3, 5 + self.num_vars, self.num_fl + 3))
i = 0
for fluid in self.inl[0].fluid.val.keys():
if fluid == self.h2o:
deriv[0, 1, 3 + i] = -1
elif fluid == self.o2:
deriv[1, 3, 3 + i] = -1
elif fluid == self.h2:
deriv[2, 4, 3 + i] = -1
i += 1
mat_deriv += deriv.tolist()
# derivatives to ban fluids off inlets/outlets
deriv = np.zeros((3 * len(self.inl[1].fluid.val.keys()) - 3,
5 + self.num_vars, self.num_fl + 3))
i = 0
j = 0
for fluid in self.inl[1].fluid.val.keys():
if fluid != self.h2o:
deriv[j, 1, 3 + i] = -1
j += 1
if fluid != self.o2:
deriv[j, 3, 3 + i] = -1
j += 1
if fluid != self.h2:
deriv[j, 4, 3 + i] = -1
j += 1
i += 1
mat_deriv += deriv.tolist()
######################################################################
# derivatives for mass balance equations
# deritatives for mass flow balance in the heat exchanger
deriv = np.zeros((3, 5 + self.num_vars, self.num_fl + 3))
deriv[0, 0, 0] = 1
deriv[0, 2, 0] = -1
# derivatives for mass flow balance for oxygen output
o2 = molar_masses[self.o2] / (molar_masses[self.o2] +
2 * molar_masses[self.h2])
deriv[1, 1, 0] = o2
deriv[1, 3, 0] = -1
# derivatives for mass flow balance for hydrogen output
deriv[2, 1, 0] = (1 - o2)
deriv[2, 4, 0] = -1
mat_deriv += deriv.tolist()
######################################################################
# derivatives for pressure equations
# derivatives for pressure oxygen outlet
deriv = np.zeros((2, 5 + self.num_vars, self.num_fl + 3))
deriv[0, 1, 1] = 1
deriv[0, 3, 1] = -1
# derivatives for pressure hydrogen outlet
deriv[1, 1, 1] = 1
deriv[1, 4, 1] = -1
mat_deriv += deriv.tolist()
######################################################################
# derivatives for energy balance equations
deriv = np.zeros((1, 5 + self.num_vars, self.num_fl + 3))
T_ref = 293.15
p_ref = 1e5
h_refh2o = h_mix_pT([1, p_ref, 0, self.inl[1].fluid.val], T_ref)
h_refh2 = h_mix_pT([1, p_ref, 0, self.outl[2].fluid.val], T_ref)
h_refo2 = h_mix_pT([1, p_ref, 0, self.outl[1].fluid.val], T_ref)
# derivatives cooling water inlet
deriv[0, 0, 0] = - (self.outl[0].h.val_SI - self.inl[0].h.val_SI)
deriv[0, 0, 2] = self.inl[0].m.val_SI
# derivatives feed water inlet
deriv[0, 1, 0] = (self.inl[1].h.val_SI - h_refh2o)
deriv[0, 1, 2] = self.inl[1].m.val_SI
# derivative cooling water outlet
deriv[0, 2, 2] = - self.inl[0].m.val_SI
# derivatives oxygen outlet
deriv[0, 3, 0] = - (self.outl[1].h.val_SI - h_refo2)
deriv[0, 3, 2] = - self.outl[1].m.val_SI
# derivatives hydrogen outlet
deriv[0, 4, 0] = - self.e0 - (self.outl[2].h.val_SI - h_refh2)
deriv[0, 4, 2] = - self.outl[2].m.val_SI
# derivatives for variable P
if self.P.is_var:
deriv[0, 5 + self.P.var_pos, 0] = 1
mat_deriv += deriv.tolist()
######################################################################
# derivatives for temperature at gas outlets
deriv = np.zeros((1, 5 + self.num_vars, self.num_fl + 3))
# derivatives for outlet 1
deriv[0, 3, 1] = dT_mix_dph(self.outl[1].to_flow())
deriv[0, 3, 2] = dT_mix_pdh(self.outl[1].to_flow())
# derivatives for outlet 2
deriv[0, 4, 1] = - dT_mix_dph(self.outl[2].to_flow())
deriv[0, 4, 2] = - dT_mix_pdh(self.outl[2].to_flow())
mat_deriv += deriv.tolist()
######################################################################
# derivatives for power vs. hydrogen production
if self.e.is_set:
deriv = np.zeros((1, 5 + self.num_vars, self.num_fl + 3))
deriv[0, 4, 0] = - self.e.val
# derivatives for variable P
if self.P.is_var:
deriv[0, 5 + self.P.var_pos, 0] = 1
# derivatives for variable e
if self.e.is_var:
deriv[0, 5 + self.e.var_pos, 0] = - self.outl[2].m.val_SI
mat_deriv += deriv.tolist()
######################################################################
# derivatives for pressure ratio
if self.pr_c.is_set:
deriv = np.zeros((1, 5 + self.num_vars, self.num_fl + 3))
deriv[0, 0, 1] = self.pr_c.val
deriv[0, 2, 1] = - 1
mat_deriv += deriv.tolist()
######################################################################
#pr_c.val = pressure ratio Druckverlust (als Faktor vorgegeben)
# derivatives for zeta value
if self.zeta.is_set:
deriv = np.zeros((1, 5 + self.num_vars, self.num_fl + 3))
deriv[0, 0, 0] = self.numeric_deriv(self.zeta_func, 'm', 0)
deriv[0, 0, 1] = self.numeric_deriv(self.zeta_func, 'p', 0)
deriv[0, 0, 2] = self.numeric_deriv(self.zeta_func, 'h', 0)
deriv[0, 2, 1] = self.numeric_deriv(self.zeta_func, 'p', 2)
deriv[0, 2, 2] = self.numeric_deriv(self.zeta_func, 'h', 2)
# derivatives for variable zeta
if self.zeta.is_var:
deriv[0, 5 + self.zeta.var_pos, 0] = (
self.numeric_deriv(self.zeta_func, 'zeta', 5))
mat_deriv += deriv.tolist()
######################################################################
# derivatives for heat flow
if self.Q.is_set:
deriv = np.zeros((1, 5 + self.num_vars, self.num_fl + 3))
deriv[0, 0, 0] = - (self.inl[0].h.val_SI - self.outl[0].h.val_SI)
deriv[0, 0, 2] = - self.inl[0].m.val_SI
deriv[0, 2, 2] = self.inl[0].m.val_SI
mat_deriv += deriv.tolist()
######################################################################
# specified efficiency (efficiency definition: e0 / e)
if self.eta.is_set:
deriv = np.zeros((1, 5 + self.num_vars, self.num_fl + 3))
deriv[0, 4, 0] = - self.e0 / self.eta.val
# derivatives for variable P
if self.P.is_var:
deriv[0, 5 + self.P.var_pos, 0] = 1
mat_deriv += deriv.tolist()
######################################################################
# specified characteristic line for efficiency
if self.eta_char.is_set:
mat_deriv += self.eta_char_deriv()
######################################################################
return np.asarray(mat_deriv)
