Density Correction Calculations

From SysCAD Documentation

This page describes how density correction calculations are performed in SysCAD 9.2 and later. For SysCAD 9.1 and earlier see Density Correction Function and Example for Stream Density Calculations.

Density Correction Methods

The mass fraction of a species i, (MF_i), can be calculated in two separate ways. The user may define which method to use in the project - please see Plant Model-Species tab page.

The user may view the density of a solution containing ONLY the solvent and a single solute (at a user defined mass fraction) on the Species Data access window in a project.

Fraction in Solvent Only

Most data is obtained from single solute solutions. This method considers only the individual solute i and the solvent when calculating the MF value used in the density correction function:

$MF_i = \frac{Solute_i}{Solvent + Solute_i} \,$

Notes:

The solvent can be water or any other liquid, as selected in the Solvent section of the species database. (Please see Solution Data and Enter or Change Species data)
Since this method uses only the masses of the solvent and the individual solute, the calculated density correction factor will be Larger than the value calculated using the method below.

Fraction in Aqueous Solution

In this case the solvent MUST be water.

The calculation will consider water and ALL the aqueous species when calculating the MF value for i:

$MF_i = \frac{Solute_i}{Water + Sum(All Aqueous species)} \,$

Notes:

SysCAD will use species with an Individual Phase of (aq) or (a) when calculating the specie mass fraction.
ALL aqueous species will be included in the mass fraction calculations, including species that do not have density correction functions.
Since this method uses the mass of water and the masses of all of the aqueous species, the calculated density correction factor will be Smaller than the value calculated using the first method, and hence the solution density will be lower.

Implementation of the Density correction Function

1. The density correction factor for each solute is calculated as follows:

$DensityCorrection_i = DensCorrFn(MF)_i - DensCorrFn(0)_i \,$

Where DensCorrFn(MF)_i finds the density correction value at the Mass Fraction of i in the stream. Please see Density Correction for details of this function; and

DensCorrFn(0)_i is the density correction value at the Mass Fraction of 0

Note: The result of this is that the first term a in the Poly(a, b, c, d...) function has no effect.

2. The individual solute factors are then summed and multiplied by the Pure Solvent Density to determine the adjusted density of the solutes in solution:

$Density_{solution} = Density_{PureSolvent} * \big( 1 + \sum{DensityCorrection_i} \big) \,$

The pure solvent density is at stream temperature and pressure.

The density of the pure solvent may also be a function of temperature. In which case the density of the solvent at the relevant temperature is first calculated and then this value is multiplied by the solute correction factor.

Normally the solvent is water, so the density for water that is used will be at stream temperature and pressure.

The density correction function itself is independent of temperature and pressure.

Example of Solution Density Calculations

Consider the following example at a temperature of 25°C and 1 atmosphere.

Species	Mass kg
H₂O(l)	88.3
CuSO₄(aq)	6.1
NaCl(aq)	2.4
CaSO₄(aq)	3.2

The density correction functions have the following form:

$\mathbf{\mathit{DensCorrFn(MF)_i=a+bMF+cMF^2+dMF^3+eMF^4}}$

Please see Density Correction in the Species Table for further help on the density correction function.

Density of Water:

The Density of Water is a function of temperature and pressure. At the defined conditions Water Density = 997.048 kg/m³.

Density of CuSO₄(aq)

$\mathbf{\mathit{CuSO_4\ DensCorrFn = 1 + 0.9173x + 2.5987x^2 - 13.708x^3 + 35.212x^4}}$

Where x is the mass fraction of CuSO₄(aq) in solution.

(Ref: Lide D.R. Handbook of Chemistry and Physics 74^th Edition CRC Press)

Density of pure (100%) CuSO₄ = 3606 kg/m³.

Density of NaCl(aq)

$\mathbf{\mathit{NaCl\ DensityCorrFn = 1 + 0.7085x + 0.1214x^2 + 0.3702x^3}}$

Where x is the mass fraction of NaCl(aq) in solution.

(Ref: Lide D.R. Handbook of Chemistry and Physics 74^th Edition CRC Press)

Density of pure (100%) NaCl = 2165 kg/m³.

Density of CaSO₄(aq)

We do not have a density correction function for CaSO₄(aq) in this example.

Density of CaSO₄(aq) = 2960 kg/m³.

We will calculate the density of solution using three methods: The 2 density correction methods and then assuming no density correction functions, i.e. using the mass weighted method to find the solution density:

Description	Fraction in Solvent Only	Fraction in Aqueous Solution	Mass Weighted Mean (MF in aqueous solution * density)
Mass Fraction of H₂O(l)	NA	0.883	880.39 = 0.883 * 997.048
Mass Fraction of CuSO₄(aq)	0.0646	0.061	219.97 = 0.061 * 3606
Mass Fraction of NaCl(aq)	0.0264	0.024	51.96 = 0.024 * 2165
Mass Fraction of CaSO₄(aq)	NA	0.032	94.72 = 0.032 * 2960
Density Correction Factor for CuSO₄(aq)	0.067	0.063	NA
Density Correction Factor for NaCl(aq)	0.0188	0.0171	NA
Sum of Density Correction Factors	0.0859	0.0801	NA
Solvent Adjusted Density kg/m³	1082.67	1076.89	NA
Liquid Density kg/m³	1142.74	1137.15	1247.04

Density and Volume display for mixtures:

Consider the following example at 40 °C & atmospheric pressure; determine the volumetric flow of the stream.

Basis: 1000 kg/h

Component	Mass Fraction	Density (kg/m³)	Density Correction	Individual Volume Display (m³/h)
NaCl(aq)	0.0900	*	As per example 3	*
H2SO4(aq)	0.0900	*	As per example 3	*
H20	0.6300	992.216	As per example 3	0.6349
CuO(s)	0.0550	6450	-	0.0085
Cu(s)	0.0450	8920	-	0.0050
MgCO3(aq)	0.0450	1000	-	0.045
CuSO4(aq)	0.0450	1000	-	0.045

To determine the stream volumetric flow of the stream, we need to first compute the corrected solution density the mixture of Water-NaCl-H₂SO₄. See Example 3 above for equations.

Component	Mass flow (kg/h)	MF_i (Equation 2)	Correction Factor (Equation 3)
NaCl(aq)	90	0.125	0.0800
H2SO4(aq)	90	0.125	0.0832
H20	630

So using Equation 4 (see Example 3)

$\mathbf{\mathit{Density \ solution}}$	$\mathbf{\mathit{= Density_{ Water \ at \ 40^{\circ}\ C} * (1 + \sum density \ correction)}}$
	$\mathbf{\mathit{= 992.216 * (1 + 0.08 + 0.0832) = 1154.15 kg/m^3}}$

$\mathbf{\mathit{Liquid \ Density}}$	$\mathbf{\mathit{= (90+90+630+45+45) / [ (90+90+630)/1154.15 + 45/1000 + 45/1000 ]}}$
$\mathbf{\mathit{Liquid \ Density}}$	$\mathbf{\mathit{= 1136.63 kg/m^3}}$
$\mathbf{\mathit{Liquid \ Vol \ Flow}}$	$\mathbf{\mathit{= 900/1136.63 = 0.7918 m^3/h}}$

$\mathbf{\mathit{Solids \ Density}}$	$\mathbf{\mathit{= 100 / (55/6450 + 45/8920) = 7368.12 kg/m^3}}$
$\mathbf{\mathit{Solids \ Vol \ Flow}}$	$\mathbf{\mathit{= 100/7368.12 = 0.01357 m^3/h}}$

$\mathbf{\mathit{Stream \ Volume}}$	$\mathbf{\mathit{= 0.7918 + 0.01357 = 0.80537 m^3}}$
$\mathbf{\mathit{Stream \ Density}}$	$\mathbf{\mathit{= 1000 / 0.80537 = 1241.67 kg/m^3}}$

Density Correction for Solutions & Data Fitting

Given the following data for (NH₄)₂SO₄ @ 20°C (Source: CRC Handbook of Chemistry and Physics, 60th Edition), data fitting gives the following:

MF	Density (g/cm³)
0.005	1.003
0.01	1.0059
0.02	1.0119
0.03	1.0178
0.04	1.0238
0.05	1.0297
0.06	1.0356
0.07	1.0416
0.08	1.0475
0.09	1.0534
0.1	1.0593
0.12	1.071
0.14	1.0827
0.16	1.0943
0.18	1.1059
0.2	1.1174
0.22	1.1289
0.24	1.1403
0.26	1.1516
0.28	1.1629
0.3	1.1742
0.32	1.1854
0.34	1.1966
0.36	1.2077
0.38	1.2188
0.4	1.2298

Thus, density correction equation used in SysCAD is Poly(1, 0.5973, -0.0305, -0.1316, 0.1661). See Density Correction Function.

Density Correction Calculations

From SysCAD Documentation

Contents

Density Correction Methods

Fraction in Solvent Only

Fraction in Aqueous Solution

Implementation of the Density correction Function

Example of Solution Density Calculations

Density and Volume display for mixtures:

Density Correction for Solutions & Data Fitting

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