# Potash Species Model

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Related Links: Potash Solubility, Potash Evaporator, Potash Properties Utility

## General Description

The Potash species model may be used to calculate certain properties of aqueous liquids within a MOP (Muriate of Potash) project. Density, Heat Capacity (and hence Enthalpy) and Viscosity may be calculated using the equations defined by in papers by Laliberte et al1,2,3. The Boiling Point Elevation (BPE) may be calculated using the Fabuss Korosi equation.

The Laliberte and Fabuss Korosi equations used in the species model are given in the Model Theory section below.

Notes:

1. The Laliberte and/or Fabuss Korosi equations may be selected in the Plant Model, as described below.
2. If the Laliberte species model is selected, only AQUEOUS species are considered when calculating liquid properties. If any of the following species are present they will be used in the calculations:
• KCl(aq), NaCl(aq), MgCl2(aq), CaCl2(aq), LiCl(aq), CaSO4(aq), NaBr(aq) and KBr(aq).
• All other aqueous species are IGNORED.
3. All of the properties that are not explicitly calculated by this model are calculated using the Standard Species Model.
4. The Four Stage Crystallisation Example and Three Stage Evaporator Example, which are distributed with SysCAD in the Examples Folder, demonstrate the use of the Potash species model in SysCAD projects.

## Model Theory

### Liquid Density Calculations

The Laliberte solution density is calculated using the water density, pw and the solutes apparent density using following equation:

$\mathbf{\mathit{p_m=\frac{1}{\frac{m_w}{p_w}+\sum{\frac{m_i}{p_{app,i}}}}}}$

The Apparent density of each solute in solution is calculated from:

$\mathbf{\mathit{p_{app,i}=\frac{(c_0(1-m_w)+c_1)*e^{(0.000001(T+c_4)^2)}}{(1-m_w)+c_2+c_3*T}}}$

Where:

 mw = mass fraction of water mi = mass fraction of solute species i pw = density of water (at stream temperature and pressure), kg/m3 papp,i = solute i apparent density, kg/m3 pm = solution density, kg/m3 T = Temperature in °C c0 to c4 = dimensionless empirical constants for each solute species.

Notes:

1. The constants for most of the aqueous species are valid for temperatures between 0 and approximately 100°C.
2. If the unit temperature is outside of the species temperature range, then SysCAD will use the values at the temperature limit.
3. If the species Mass Fraction in a unit exceeds the maximum mass fraction, then SysCAD will continue to use the Laliberte equation, but will log a warning that the values are questionable.
4. The constants for CaSO4(aq) are only valid for a single temperature, 25°C, and hence these values are suspect.
5. Water density is calculated in SysCAD as described here: Water and Steam Properties.
6. Solid density is calculated using the Standard species model method - Density Calculations using the Standard Species Model

### Enthalpy Calculations

With the Lalilberte model the user has a choice of Enthalpy models:

• The Laliberte model will calculate the Enthalpy as a full integral of Cp with respect to Temperature, i.e. $\Delta H = \int\limits_{T_1}^{T_2}Cp dT\,$
• The Laliberte Low model with calculate Enthalpy as $\Delta H = Cp * (T_2 - T_1)$.
This will be less accurate, but faster computationally.

Note: Both methods will calculate Cp using the equations show below.

### Heat Capacity Calculations

#### Liquid Heat Capacity

With the Lalilberte model the liquid specific heat is calculated using the water specific heat, Cpw and the solutes heat capacity using following equation:

$\mathbf{\mathit{Cp_m = m_wCp_w + \sum{m_iCp_i}}}$

The heat capacity of each solute in solution is calculated from:

$\mathbf{\mathit{Cp_i= a_1e^{\alpha}+a_5(1-m_w)^{a_6}}}$

Where

$\mathbf{\mathit{\alpha= a_2*T +a_3e^{0.01T}+a_4(1-m_w)}}$

 mw = mass fraction of water mi = mass fraction of solute species i Cpw = Heat capacity of water (at stream temperature and pressure), kJ/kg.K Cpi = Heat capacity of solute i, kJ/kg.K Cpm = solution Heat capacity, kJ/kg.K T = Temperature in °C a1 to a6 = dimensionless empirical constants for each solute species.

Notes:

1. The constants for most of the aqueous species are valid for temperatures between approximately 5 and 1200C.
2. If the unit temperature is outside of the species temperature range, then SysCAD will use the values at the temperature limit.
3. The constants for CaSO4(aq) are only valid for a single temperature, 250C, and hence these values are suspect.
4. Water heat capacity is calculated in SysCAD as described here: Water and Steam Properties.

#### Solids and Vapours Heat Capacity

Solids Cp (Cps) and Vapours Cp (Cpv) are calculated from Cp values as given in the species database, please see Specific Heat values (Cp) Calculations using the Standard Species Model.

#### Stream Heat Capacity

$\mathbf{\mathit{Cp=\frac{SolidsMass*Cp_s+LiquidsMass*Cp_L+VapoursMass*Cp_V}{SolidsMass+LiquidsMass+VapoursMass}}}$

### Viscosity

With the Laliberte method, the liquid viscosity is calculated using the water viscosity, vw and the solutes viscosity using following equation:

$\mathbf{\mathit{\ln{n_m} = m_w * \ln{n_w} + \sum{m_i * \ln{n_i}}}}$

The viscosity for each solute is defined by:

$\mathbf{\mathit{\ln{n_i} = \frac{v_1(1-m_w)^{v_2}+v_3}{(v_4*T+1)(v_5(1-m_w)^{v_6}+1)}}}$

 nm = Solution Viscosity, mPa.s nw = Viscosity of water, mPa.s ni = Viscosity of solute i, mPa.s mw = mass fraction of water mi = mass fraction of solute i T = Temperature in °C v1 to v6 = dimensionless empirical constants.

Notes:

1. The constants for most of the aqueous species are valid for temperatures between approximately 5 and 1200C.
2. If the unit temperature is outside of the species temperature range, then SysCAD will use the values at the temperature limit.
3. CaSO4(aq) does not have any data for viscosity.

### Boiling Point Elevation (BPE)

Using the Fabuss Korosi method, the Boiling Point Elevation (BPE) of a brine is calculated by first determining the vapour pressure lowering of a solution containing KCl and NaCl relative to pure water. The activity coefficient is determined by a semi-empirical correlation.4,5

$\mathbf{\mathit{k = \frac{p^0 - p}{mp^0}}}$

and

$\mathbf{\mathit{k = a + bu^{0.5}}}$

where: k - Relative molar vapour pressure depression for a species;

m - molality of the solution;

p - vapour pressure of the solution;

p0 - vapour pressure of water at the given conditions;

u - Species ionic strength;

a and b - Temperature dependant constants, calculated as follows:

$\mathbf{\mathit{a = a_1 + a_2t + a_3t^2}}$

$\mathbf{\mathit{b = b_1 + b_2t + b_3t^2}}$

t - Temperature

The constants in the above equations, a1, a2, a3, b1, b2 and b3 are empirical values determined by experimentation.