r"""Refrigerators and heat pumps: coefficients of performance and their Carnot limits.
Perkins's vapour-compression machine pumps heat from a cold space; Kelvin
saw that the same machine run against the weather heats a building with
less work than the heat it delivers. Both are bounded by the reversed
Carnot cycle, whose two COPs differ by exactly one. The vapour-compression
cycle itself is run on a Peng-Robinson refrigerant.
"""
from __future__ import annotations
import numpy as np
from engineeringkit._validation import finite as _finite
from engineeringkit._validation import interval as _interval
from engineeringkit._validation import nonnegative as _nonnegative
from engineeringkit._validation import positive as _positive
from engineeringkit.thermodynamics.core.base import VaporCompressionResult
from engineeringkit.thermodynamics.systems.peng_robinson import R134A, refrigerant_properties
from engineeringkit.thermodynamics.systems.steady_flow import compression_outlet_enthalpy
__all__ = ["carnot_cop_refrigerator", "carnot_cop_heat_pump", "refrigeration_cop", "heat_pump_cop", "vapor_compression_cycle"]
[docs]
def carnot_cop_refrigerator(T_hot, T_cold):
r"""Coefficient of performance of a reversible refrigerator, :math:`T_c/(T_h - T_c)`.
Parameters
----------
T_hot, T_cold : float
Absolute temperatures of the heat sink and the cooled space [K], with
:math:`T_h > T_c > 0`.
Returns
-------
float
Maximum heat removed per unit work [-].
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/thermodynamics`.
"""
_finite(T_hot=T_hot, T_cold=T_cold)
_positive(T_hot=T_hot, T_cold=T_cold)
if np.any(np.asarray(T_hot) <= T_cold):
raise ValueError("hot reservoir must be hotter")
return T_cold / (T_hot - T_cold)
[docs]
def carnot_cop_heat_pump(T_hot, T_cold):
r"""Coefficient of performance of a reversible heat pump, :math:`T_h/(T_h - T_c)`.
Parameters
----------
T_hot, T_cold : float
Absolute temperatures of the heated space and the heat source [K],
with :math:`T_h > T_c > 0`.
Returns
-------
float
Maximum heat delivered per unit work [-], one more than the
refrigerator COP between the same temperatures.
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/thermodynamics`.
References
----------
W. Thomson, "On the economy of the heating or cooling of buildings by
means of currents of air," *Proc. Phil. Soc. Glasgow* 3 (1852) 269-272.
"""
_finite(T_hot=T_hot, T_cold=T_cold)
_positive(T_hot=T_hot, T_cold=T_cold)
if np.any(np.asarray(T_hot) <= T_cold):
raise ValueError("hot reservoir must be hotter")
return T_hot / (T_hot - T_cold)
[docs]
def refrigeration_cop(q_cold, work_input):
r"""Refrigerator coefficient of performance, :math:`Q_c/W`.
Parameters
----------
q_cold : float
Heat removed from the cold space [J or W], nonnegative.
work_input : float
Work (or power) supplied [J or W, as ``q_cold``], positive.
Returns
-------
float
COP [-].
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/thermodynamics`.
References
----------
J. Perkins, "Apparatus and means for producing ice, and in cooling
fluids," British Patent 6662 (1834).
"""
_finite(q_cold=q_cold, work_input=work_input)
_nonnegative(q_cold=q_cold)
_positive(work_input=work_input)
return q_cold / work_input
[docs]
def heat_pump_cop(q_hot, work_input):
r"""Heat-pump coefficient of performance, :math:`Q_h/W`.
Parameters
----------
q_hot : float
Heat delivered to the warm space [J or W], at least ``work_input``.
work_input : float
Work (or power) supplied [J or W, as ``q_hot``], positive.
Returns
-------
float
COP [-], at least one.
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/thermodynamics`.
"""
_finite(q_hot=q_hot, work_input=work_input)
_positive(work_input=work_input)
if np.any(np.asarray(q_hot) < work_input):
raise ValueError("heat-pump heat output must be >= work input")
return q_hot / work_input
[docs]
def vapor_compression_cycle(p_evaporator, p_condenser, *, fluid=R134A, superheat=0.0, subcooling=0.0, compressor_efficiency=1.0):
r"""Vapour-compression refrigeration cycle on a Peng-Robinson refrigerant.
Parameters
----------
p_evaporator : float
Evaporator pressure [Pa], positive and below ``p_condenser``.
p_condenser : float
Condenser pressure [Pa], below the fluid's critical pressure.
fluid : CubicFluid, optional
Refrigerant; :data:`R134A` by default.
superheat : float, optional
Superheat of the vapour leaving the evaporator [K], nonnegative.
subcooling : float, optional
Subcooling of the liquid leaving the condenser [K], nonnegative.
compressor_efficiency : float, optional
Isentropic efficiency of the compressor [-] in (0, 1].
Returns
-------
VaporCompressionResult
The four states (compressor inlet and outlet, condenser outlet,
evaporator inlet), the heat absorbed and rejected and the
compressor work per kilogram, and both COPs.
Notes
-----
Perkins's cycle: the vapour is compressed, condensed, throttled through
an expansion valve at constant enthalpy and evaporated. Heat exchangers
have no pressure drop. Properties come from
:func:`refrigerant_properties`. Inputs are raw SI numbers with no
dimensional metadata; see :doc:`/guides/thermodynamics`.
References
----------
Y. A. Çengel and M. A. Boles, *Thermodynamics: An Engineering
Approach*, 9th ed., New York: McGraw-Hill (2019), Ch. 11.
Examples
--------
>>> res = vapor_compression_cycle(0.14e6, 0.8e6)
>>> round(res.cop_refrigerator, 2)
3.94
"""
_positive(p_evaporator=p_evaporator, p_condenser=p_condenser)
if p_evaporator >= p_condenser:
raise ValueError("p_evaporator must be below p_condenser")
_nonnegative(superheat=superheat, subcooling=subcooling)
_interval("compressor_efficiency", compressor_efficiency, 0, 1, closed="right")
suction = refrigerant_properties(fluid, p_evaporator, x=1.0)
if superheat:
suction = refrigerant_properties(fluid, p_evaporator, suction.T + superheat)
h_s = refrigerant_properties(fluid, p_condenser, s=suction.s).h
discharge = refrigerant_properties(fluid, p_condenser, h=compression_outlet_enthalpy(suction.h, h_s, compressor_efficiency))
liquid = refrigerant_properties(fluid, p_condenser, x=0.0)
if subcooling:
liquid = refrigerant_properties(fluid, p_condenser, liquid.T - subcooling)
valve_exit = refrigerant_properties(fluid, p_evaporator, h=liquid.h)
return VaporCompressionResult(
fluid=fluid,
labels=("compressor inlet", "compressor outlet", "condenser outlet", "evaporator inlet"),
states=(suction, discharge, liquid, valve_exit),
q_evaporator=suction.h - valve_exit.h,
q_condenser=discharge.h - liquid.h,
compressor_work=discharge.h - suction.h,
)
