Computational psychrometric analysis as a control problem: case of cooling and dehumidification systems

Abstract

Psychrometric chart is a basic tool for analysis of processes in air-conditioning systems. While psychrometric calculators and numerical psychrometric charts are widely available, there is a lack of numerical algorithms for solving the sizing problem by using psychrometric analysis. After introducing a classification of modelling problems in direct and inverse (based on the relation between physical and computational causality), this paper defines the design problem as a set of inverse problems of control and parameter optimization. A non-linear model is obtained by assembling the models of the elementary processes. The paper proposes to solve the direct and control problems by using a method similar to Newton-Raphson’s, and the parameter optimization problem by using least-squares. Open-source implementation, published on Zenodo repository, and interactive, reproducible environment, available on Binder web service, accompany this paper (Ghiaus [2021]. PsychroAn_cool: Psychrometric analysis of cooling systems as a control problem. Zenodo.).

KEYWORDS:

• Inverse problem
• HVAC
• design
• sizing
• steady-state
• Python

Disclosure statement

No potential conflict of interest was reported by the author(s).

Nomenclature

Latin letters	=
$A$	=	surface area, m2
AHU	=	air handling unit
$C$	=	constant
CAV	=	constant air volume
$K$	=	controller gain
${\dot{H}}_i$	=	enthalpy rate of the air in state $i$, $\text{W}$
$M$	=	molar mass, kg/kmol
$\dot{Q}$	=	heat flow rate, W
$R$	=	universal gas constant, $\text{J}/\text{K kmol}$
SHR	=	sensible heat rate, -
$T$	=	temperature, K
$U$	=	overall heat transfer coefficient, W/m2 K
VAV	=	variable air volume
$c$	=	specific heat capacity of the dry air, $\text{J}/\text{kg}\text{K}$
$c_w$	=	specific heat capacity of the water vapour, $\text{J}/\text{kg K}$
$d$	=	derivative
$f$	=	function representing the saturation curve
$h_i$	=	specific enthalpy of dry air in state $i$, $\text{J}/\text{kg}$
$l$	=	specific latent heat for water vaporization, $\text{J}/\text{kg}$
$l_k$	=	length of thermal bridge $k$, m
$\dot{m}$	=	mass flow rate of dry air, $\text{kg}/\text{s}$
$n$	=	molar fraction
$p$	=	pressure, Pa
$v$	=	unspecified variable
$w_i$	=	humidity ratio of moist air in the state $i$, $\text{k}{\text{g}}_{\text{vapor}}/\text{k}{\text{g}}_{\text{dry air}}$
Greek letters	=
$\mathrm{\Delta }$	=	difference operator
$\alpha$	=	mixing ratio
$\text{β}$	=	bypass factor
$ϵ$	=	arbitrarily small number
${\theta }_i$	=	temperature of the air in state $i$, ${}^{\circ }\text{C}$
$\phi$	=	relative humidity, %
$\psi$	=	overall heat transfer coefficient of a thermal bridge, W/m K
Bold letters	=
$\mathbf{A}$	=	matrix of coefficients
$\mathbf{b}$	=	vector of constant terms
$\mathbf{x}$	=	vector of unknowns
Subscripts	=
$BL$	=	building
$CC$	=	cooling coil
$HC$	=	heating coil
$TZ$	=	thermal zone
$da$	=	dry air
$i$	=	air infiltration
$1\dots 5$	=	states in the psychrometric chart
$l$	=	latent
$ls$	=	least squares
$o$	=	outdoor
$s$	=	sensible or saturation
$sp$	=	set-point
$t$	=	total
$vs$	=	vapour at saturation
$w$	=	humidity ratio
$\theta$	=	temperature
${\theta }_s^0$	=	initial guess of the temperature corresponding to the saturation point
Superscripts	=
${}^{\mathrm{\prime }}$	=	derivative of a function
$0$	=	initial value