Coagulation Property Package

This package implements property relationships for water density as a function of temperature, pressure, and mass fraction of suspended/dissolved solids from Engineering Toolbox. (2003) and water viscosity as a function of temperature from D.S. Viswananth, G. Natarajan. (1989).

Note: TDS = Total Dissolved Solids and TSS = Total Suspended Solids

This coagulation property package:

• supports only ‘H2O’, ‘TDS’, ‘TSS’, and ‘Sludge’ as Components

• supports only liquid phase

• is formulated on a mass basis

• does NOT support formulations on a molar basis

• includes mass density correction for fraction of suspended/dissolved solids

Sets

Description	Symbol	Indices
Components	\(j\)	[‘H2O’, ‘TDS’, ‘TSS’, ‘Sludge’]
Phases	\(p\)	[‘Liq’]

State variables

Description	Symbol	Variable	Index	Units
Component mass flowrate	\(M_j\)	flow_mass_phase_comp	[p, j]	\(\text{kg/s}\)
Temperature	\(T\)	temperature	None	\(\text{K}\)
Pressure	\(P\)	pressure	None	\(\text{Pa}\)

Parameters

Description	Symbol	Parameter	Index	Units
Fluid specific heat capacity	\(c_p\)	cp	None	\(\text{J/kg/K}\)
Reference density (at 273 K)	\(\rho_{ref}\)	ref_dens_liq	None	\(\text{kg/}\text{m}^3\)
Relative density contribution from salt mass fraction	\(\rho_s\)	dens_slope	None	\(\text{kg/}\text{m}^3\)
First density temperature coefficient	\(A\)	dens_param_A	None	\(\text{K}^{-2}\)
Second density temperature coefficient	\(B\)	dens_param_B	None	\(\text{K}^{-1}\)
Third density temperature coefficient	\(C\)	dens_param_C	None	\(\text{dimensionless}\)
First pressure coefficient	\(\alpha\)	ref_pressure_correction	None	\(\text{dimensionless}\)
Second pressure coefficient	\(\beta\)	ref_pressure_slope	None	\(\text{Pa}^{-1}\)
Reference viscosity (at 273 K)	\(\mu_{ref}\)	mu_A	None	\(\text{kg/}\text{m/}\text{s}\)
First viscosity temperature coefficient	\(\mu_B\)	mu_B	None	\(\text{K}\)
Second viscosity temperature coefficient	\(\mu_C\)	mu_C	None	\(\text{K}\)

The parameters provided are valid between 0 and 350 °C and up to 600 bar.

Properties

Description	Symbol	Variable	Index	Units
Component mass fraction	\(x_j\)	mass_frac_phase_comp	[p, j]	\(\text{dimensionless}\)
Mass density of fluid	\(\rho\)	dens_mass_phase	[p]	\(\text{kg/}\text{m}^3\)
Phase volumetric flowrate	\(Q_p\)	flow_vol_phase	[p]	\(\text{m}^3\text{/s}\)
Mass concentration	\(C_j\)	conc_mass_phase_comp	[p, j]	\(\text{kg/}\text{m}^3\)
Dynamic viscosity	\(\mu\)	visc_d_phase	[p]	\(\text{Pa}\cdotp\text{s}\)
Enthalpy flow	\(H\)	enth_flow	None	\(\text{J/s}\)

Relationships

Description	Equation
Component mass fraction	\(x_j = \frac{M_j}{\sum_{j} M_j}\)
Mass density	\(\rho = (\rho_{ref} + \rho_s \cdotp \sum_{j} x_j) \cdotp (A \cdotp T^2 + B \cdotp T + C) \cdotp (\alpha + \beta \cdotp P)\)
Volumetric flowrate	\(Q = \frac{\sum_{j} M_j}{\rho}\)
Mass concentration	\(C_j = x_j \cdotp \rho\)
Dynamic viscosity	\(\mu = \mu_{ref} \cdotp exp( \frac{\mu_B}{T - \mu_C} )\)
Enthalpy flow	\(H = c_p \cdotp \sum_{j} M_j \cdotp (T - 273)\)

Scaling

This coagulation property package includes support for scaling, such as providing default or calculating scaling factors for almost all variables. The only variables that do not have scaling factors are the component mass flowrate and the user will receive a warning if these are not set.

The user can specify the scaling factors for component mass flowrates with the following:

# relevant imports
import watertap.property_models.unit_specific.coagulation_prop_pack as props
from idaes.core.util.scaling import calculate_scaling_factors

# relevant assignments
m = ConcreteModel()
m.fs = FlowsheetBlock(dynamic=False)
m.fs.properties = props.CoagulationParameterBlock()

# set scaling for component mass flowrate
m.fs.properties.set_default_scaling('flow_mass_phase_comp', 1, index=('Liq','H2O'))
m.fs.properties.set_default_scaling('flow_mass_phase_comp', 1e2, index=('Liq','TDS'))
m.fs.properties.set_default_scaling('flow_mass_phase_comp', 1e2, index=('Liq','TSS'))
m.fs.properties.set_default_scaling('flow_mass_phase_comp', 1e3, index=('Liq','Sludge'))

# calculate scaling factors
calculate_scaling_factors(m.fs)

The default scaling factors are as follows:

• 1e-2 for temperature

• 1e-6 for pressure

• 1e-3 for mass density

• 1e3 for dynamic viscosity

Scaling factors for other variables can be calculated based on their relationships with the user-supplied or default scaling factors.

Reference

Engineering Toolbox. Water - Density, Specific Weight, and Thermal Expansion Coefficients. (2003) https://www.engineeringtoolbox.com/water-density-specific-weight-d_595.html [Accessed 02-01-2022]

D.S. Viswananth, G. Natarajan. Data Book on the Viscosity of Liquids. Hemisphere Publishing Corp. (1989)