def delta_temperature_underwood_callback(b):
r"""
This is a callback for a temperature difference expression to calculate
:math:`\Delta T` in the heat exchanger model using log-mean temperature
difference (LMTD) approximation given by Underwood (1970). It can be
supplied to "delta_temperature_callback" HeatExchanger configuration option.
This uses a cube root function that works with negative numbers returning
the real negative root. This should always evaluate successfully. This form
is
.. math::
\Delta T = \left(\frac{
\Delta T_1^\frac{1}{3} + \Delta T_2^\frac{1}{3}}{2}\right)^3
where :math:`\Delta T_1` is the temperature difference at the hot inlet end
and :math:`\Delta T_2` is the temperature difference at the hot outlet end.
"""
dT1 = b.delta_temperature_in
dT2 = b.delta_temperature_out
temp_units = pyunits.get_units(dT1[dT1.index_set().first()])
# external function that ruturns the real root, for the cuberoot of negitive
# numbers, so it will return without error for positive and negitive dT.
b.cbrt = ExternalFunction(library=functions_lib(),
function="cbrt",
arg_units=[temp_units])
@b.Expression(b.flowsheet().time)
def delta_temperature(b, t):
return ((b.cbrt(dT1[t]) + b.cbrt(dT2[t])) / 2.0)**3 * temp_units
def test_cbrt_values():
m = pyo.ConcreteModel()
flib = functions_lib()
m.cbrt = pyo.ExternalFunction(library=flib, function="cbrt")
assert (abs(pyo.value(m.cbrt(-27.0)) + 3.0) < 0.00001)
assert (abs(pyo.value(m.cbrt(0.0))) < 0.00001)
assert (abs(pyo.value(m.cbrt(27.0)) - 3.0) < 0.00001)
def eq_lmtd(b, t):
dT_in = b.delta_temperature_in
dT_out = b.delta_temperature_out
temp_units = pyunits.get_units(dT_in)
dT_avg = (dT_in + dT_out) / 2
# external function that ruturns the real root, for the cuberoot of negitive
# numbers, so it will return without error for positive and negitive dT.
b.cbrt = ExternalFunction(library=functions_lib(),
function="cbrt",
arg_units=[temp_units**3])
return b.lmtd == b.cbrt((dT_in * dT_out * dT_avg)) * temp_units
def test_cbrt_hes():
m = pyo.ConcreteModel()
flib = functions_lib()
m.cbrt = pyo.ExternalFunction(library=flib, function="cbrt")
h = 1e-6
tol = 1e-5
for v in [-27.0, -0.5, 0.5, 27]:
f1, g1, h1 = m.cbrt.evaluate_fgh(args=(v, ))
f2, g2, h2 = m.cbrt.evaluate_fgh(args=(v + h, ))
hfd = (g2[0] - g1[0]) / h
assert (abs(h1[0] - hfd) < tol)
def delta_temperature_underwood_callback(b):
"""
This is a callback for a temperature difference expression to calculate
:math:`\Delta T` in the heat exchanger model using log-mean temperature
difference (LMTD) approximation given by Underwood (1970). It can be
supplied to "delta_temperature_callback" HeatExchanger configuration option.
This uses a cube root function that works with negative numbers returning
the real negative root. This should always evaluate successfully.
"""
# external function that ruturns the real root, for the cuberoot of negitive
# numbers, so it will return without error for positive and negitive dT.
b.cbrt = ExternalFunction(library=functions_lib(), function="cbrt")
dT1 = b.delta_temperature_in
dT2 = b.delta_temperature_out
@b.Expression(b.flowsheet().config.time)
def delta_temperature(b, t):
return ((b.cbrt(dT1[t]) + b.cbrt(dT2[t])) / 2.0)**3
def delta_temperature_chen_callback(b):
r"""
This is a callback for a temperature difference expression to calculate
:math:`\Delta T` in the heat exchanger model using log-mean temperature
difference (LMTD) approximation given by Chen (1987). It can be
supplied to "delta_temperature_callback" HeatExchanger configuration option.
This uses a cube root function that works with negative numbers returning
the real negative root. This should always evaluate successfully.
"""
dT1 = b.delta_temperature_in
dT2 = b.delta_temperature_out
temp_units = pyunits.get_units(dT1)
# external function that ruturns the real root, for the cuberoot of negitive
# numbers, so it will return without error for positive and negitive dT.
b.cbrt = ExternalFunction(library=functions_lib(),
function="cbrt",
arg_units=[temp_units**3])
@b.Expression(b.flowsheet().time)
def delta_temperature(b, t):
return b.cbrt(dT1[t] * dT2[t] * 0.5 * (dT1[t] + dT2[t])) * temp_units