# 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.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)