Species Table - Density

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Introduction

The Density field is optional, but it should be filled in to ensure that the density calculations for any stream containing this compound are correct.

This is the density of the species in the defined phase. The unit for density is kg/m³.

Notes:

1. SysCAD works with mass flows and uses the density to convert mass flows to volume flows. Thus in order to get accurate volume flows, the user must specify accurate density values for their species.
2. The user may enter a constant value for species in all phases, but in the cases of aqueous and gas species it is preferable to use functions, as defined further on.
3. If this field is left blank SysCAD will assume the following:
   • A constant value of 2000 kg/m³ for solids,
   • A constant value of 1000 kg/m³ for liquids, and
   • Ideal gas density for gases. (Note: if a constant is entered, Linear Gas Density will be assumed)

The density provided here will also be used in volume calculations. The equation used is Volume = mass / density.

Refer to Stream Density for an example of how these individual densities are used to determine the density of a stream.

Summary of Density Options

The list of density functions that are available will change if the species is a Solid, a Liquid or a Gas. The table below shows the different functions that are available for the different phases:

Phase	Function Name	Description
SOLID	Constant Value	A constant density value that does not change with temperature. This is common for solid species.
	Density as a Function of T: Polynomial	The user enters a Polynomial function for Density as a function of Temperature.
	Density as a Function of MF: Polynomial	The user enters a Polynomial function for Density as a function of mass fraction of the solid in the solid phase.
	Spline under Tension	The user enters data points and these will be used in a Tspline function.
	Undefined	The user does not enter a density value for the species (in this case, SysCAD will assume a default density of 2000 kg/m3).
LIQUID	Constant Value	A constant density value that does not change with temperature. This is not very common for liquid species.
	Density as a Function of T: Polynomial	The user enters a Polynomial function for Density as a function of Temperature.
	Density as a Function of MF: Polynomial	The user enters a Polynomial function for Density as a function of mass fraction of the liquid in the liquid phase.
	Spline under Tension	The user enters data points and these will be used in a Spline function.
	Water Density Function	This will vary the density of the liquid species using the Water density function, i.e. it will always have a density = density of water.
	Density Correction as a Function of MF: Polynomial	The user enters a Polynomial function that will correct the Density of the solution as a function of the mass fraction of the liquid species in the solution. (this is often used for aqueous species).
	Density Correction as a Function of MF: Spline under Tension	The user enters 2 columns of data, Mass Fraction and Density Correction values. SysCAD will then use these values to calculate the required density correction values for the liquid species in the solution. (mainly used for aqueous species).
	Density Value as a Function of MF & T: Laliberté Values	The user enters the Laliberté constants for the liquid species. These will be used to calculate the solution density (this is only used for aqueous species).
	Undefined	The user does not enter a density value for the species (in this case, SysCAD will assume a default density of 1000 kg/m3).
Gas	Constant Value	A constant density value that does not change with temperature. This is not very common for gas species.
	Ideal Gas Density	SysCAD will use the Ideal Gas equations to calculate the density of the gas as a function of Temperature. This method is common for gases, and will give good correlations if the project does not have gases at high pressure.
	Linear Gas Density	SysCAD will calculate the density of the gas as a function of Temperature using a Linear equation.
	Undefined	The user does not enter a density value for the species (in this case, SysCAD will by default use the Ideal Gas equation to calculate density).

Constant Density

The user may enter a constant value for the density for a solid, liquid or gas species, by selecting 'Constant Value'.

In this case the density of the species will not change as a function of temperature or as the mass fraction of the species changes in a stream.

The constant density option is normally used for Solid species and pure Liquid species.

Density as a Function of Temperature

This option is only available for Solid or Liquid species.

If the user has data for changing density as a function of temperature, then they may enter that using either:

• Density as a function of T : Polynomial
• Spline under Tension

Polynomial Function

If the user has a polynomial function that describes density as a function of temperature, then they can select the option:

Density as a function of T : Polynomial

and enter the function in the following format: Poly_T(C0, C1, C2, C3, C4, C5), where

[math]\displaystyle{ Density = C0 + C1T + C2T^2 + C3T^3 + C4T^4 + C5T^5\, }[/math]

NOTES:

1. T is Temperature in K.
2. The user does not need to have all the constants, any that are not required may be set to zero. So, for example if you have a 3rd order equation, you would only need to enter C0, C1, C2 and C3. C4 and C5 would be zero.

Table of Data

If the user has a table with Density versus Temperature, then they may select the option:

Spline under Tension.

And enter the data in 2 columns. SysCAD will interpolate the data using the TSpline method. The easiest way to enter the data is to copy the data from a spreadsheet.

Note: A fitted equation will be processed more quickly within SysCAD and hence is preferred to entering a table of data.

Density as a Function of Mass Fraction

This option is only available for Solid or Liquid species.

If the user has a polynomial function that describes density as a function of mass fraction, then they can select the option:

Density as a function of MF : Polynomial

and enter the function in the following format: Poly_MF(C0, C1, C2, C3, C4, C5), where

[math]\displaystyle{ Density = C0 + C1*MF + C2*MF^2 + C3*MF^3 + C4*MF^4 + C5*MF^5\, }[/math]

NOTES:

1. MF is Mass Fraction in %.
2. The user does not need to have all the constants, any that are not required may be set to zero. So, for example if you have a 2nd order equation, you would only need to enter C0, C1 and C2. C3, C4 and C5 would be zero.

Liquid Density Correction Functions

The following methods are only available for liquid species, and are normally used for dissolved aqueous species, for example, NaCl(aq).

The density of a solution containing aqueous or ionic species changes according to the mass fraction of dissolved species in solution. The Density Correction functions describe the solution density as a function of solute mass fraction.

The user may enter a mix of density correction functions for aqueous species that will occur in a solution. SysCAD will calculate apparent densities for each species and then use these to calculate the overall solution density. However, where possible it is recommended to use functions from the same set for all species (e.g. Laliberté values for all aqueous species).

Emulate Water Density

If the user has defined a species that will normally exist in the aqueous form, but the user does not have a density correction function, then it is recommended that the density of the species be set to the water density using the special density function LiqH2ORho(), by choosing the 'Water Density Function' option. This will ensure that the species has the same density as water and hence the species will not change the density of the solution. For further info on this equation see Water Density.

Polynomial Function

If the user has a polynomial function that describes density correction factors as a function of mass fraction, then they can select the option:

Density Correction as a function of MF : Polynomial

and enter the function in the following format: Poly_MF(a, b, c, d, e, f)

Once the data has been entered in the user data dialog box, the data will be summarised in the Species Database as:

Poly(a, b, c, d, e, f), Limit(Limit Fraction, Pure Value, Generate Warnings)

This consists of 2 parts:

1. The Polynomial, which represents the density correction function:

   [math]\displaystyle{ DensCorrFn(MF)_i = a + b.MF + c.MF^2 + d.MF^3 + e.MF^4 + f.MF^5 \, }[/math]

   Where MF is the mass fraction of the species in solution.

   Notes:

   • The a value in the above polynomial is usually approximately 1.0 as it represents the pure solution density based on the data for the solvent - usually water. Please see Implementation of the Density correction Function for further information.
   • The user only needs to enter the required number of parameters. For example, if the polynomial is 2nd order you need only enter a,b and c.
   • As a minimum, the user must enter values for a and b, as this will then be first order with respect to MF.
   • If the user only enters a value for a, then the species will have the same density as the solvent, usually water.

2. The Limiting Values, which are described below:
   • Limit Fraction is the maximum mass fraction of the solute for which the equation is valid.

   If the mass fraction of the solute in the solvent is above the Limit Fraction value, then SysCAD calculates the density of the solution by linearly interpolating between the density at the Limit Fraction and the density of the pure solute (Pure Value).

   • Pure Value is only used if MF is greater than Limit Fraction. The purpose of Pure Value is to give a linear extension of the curve, so that reasonable densities are calculated when MF is greater than Limit Fraction. The Pure Value used is typically the density of the pure solute in kg/m^3, but can be any suitable value for linear interpolation between the Limit Fraction and a MF of 100%.
   • Generate Warnings is either On or Off. If it is On (recommended), SysCAD will warn the user if the mass fraction of the solute is above the specified Limit Fraction.

Example:

For FeSO4(aq): Poly(0.998,0.951,0.62), Limit(0.2, 2200, On)

Notes:

1. If the user does not have a function relating the change in solvent density with solute concentration, then SysCAD will use the density in the species database in a pure mass weighted mean calculation. However, this does not usually produce the correct liquid density or volumetric flow rates.
2. Please see Density Correction Calculations for the implementation method for density correction, a description of the 2 methods of calculating the species Mass Fraction, MFi and Density Calculation Examples.
3. The user may view the corrected density of a solution consisting of ONLY the solvent and the solute on the Species Properties ($SDB) access window.

Table of Data

If the user has a table with density correction factors versus mass fraction in solution, then they may select the option:

Density Correction as a function of MF : Spline under Tension

And enter the data in 2 columns. SysCAD will interpolate the data using the TSpline method. The easiest way to enter the data is to copy the data from a spreadsheet.

Note: A fitted equation will be processed more quickly within SysCAD and hence is preferred to entering a table of data.

Laliberté Function

NOTE: This method is only valid for aqueous species.

If the user has appropriate constants for the Laliberté equation, then they can select the option:

Density Value as a function of MF & T: Laliberté values

and enter the function in the following format: Laliberte_Rho(c₀, c₁, c₂, c₃, c₄)

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

[math]\displaystyle{ \mathbf{\mathit{p_m=\cfrac{1}{\cfrac{m_w}{p_w}+\sum{\cfrac{m_i}{p_{app,i}}}}}} }[/math]

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

[math]\displaystyle{ p_{app,i}=\cfrac{ c_0\left(1-m_w \right)+c_1 }{\left(1-m_w\right)+c_2+c_3 T}\times e^ {10^{-6} \; \left( \; T \;+ \; c_4 \;\right) \;^2} }[/math]

Where:

mw	=	mass fraction of water
mi	=	mass fraction of solute species i
pw	=	density of water (at stream temperature and pressure), 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. If the user wishes to use the Laliberté method, then it is recommended that this method is used for ALL aqueous species in a project, as a mixture of Laliberté and other aqueous density methods may result in inaccurate solution density values.
2. It is important to note that the constants for many of the aqueous species are valid for temperatures between 0 and approximately 100°C.
3. If the unit temperature is outside of the species temperature range, then SysCAD will use the values at the temperature limit.
4. If the species Mass Fraction in a unit exceeds the maximum mass fraction, then SysCAD will continue to use the Laliberté equation, but will log a warning that the values are questionable.
5. Water density is calculated in SysCAD as described here: Water and Steam Properties.

Reference

Laliberté M. and Cooper W.E. Model for Calculating the Density of Aqueous Electrolyte Solutions J. Chem. Eng. Data 2004, 49.

Gas Density

For gases the user may use one of the following three input formats (the formulation for Ideal Gas is also shown here):

• Constant Value
• Ideal Gas Density - The density value will be calculated based on the Ideal gas law. Equations used are:

  (1) [math]\displaystyle{ \mathbf {\mathrm{Density_{T,P} = \frac{m}{V}}} }[/math] and (2) [math]\displaystyle{ \mathbf {\mathrm{PV = nRT}} }[/math] and (3) [math]\displaystyle{ \mathbf {\mathrm{m = nM}} }[/math]

  Rearranging the above equations will give:

  [math]\displaystyle{ \mathbf {\mathrm{Density_{T,P} = \frac{PM}{RT}}} }[/math]

  Where:

  m = mass of compound

  V = Volume of compound

  P = Partial Pressure of species

  R = Universal Gas Constant = 8.314 472 J/mol.K (Reference: National Institute of Standards and Technology)

  T = Temperature in Kelvin

  n = number of moles of compound

  M = molecular weight of compound

• Linear Gas Density - the density value provided (in brackets) is expected to be at 0°C and Std. Pressure. Density @ T, P will be corrected based on:

  [math]\displaystyle{ \mathbf {\mathrm{Density_{T,P} = Density_{0,StdP}*\frac{P}{StdP}*\frac{273(K)}{T(K)}}} }[/math]