Species Table 9.3 - Specific Heat (Cp)

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Related Links: Editing User Species Database 9.3, Species Properties (\$SDB)

Introduction

By selecting a Cp equation, together with H25 and S25, we are defining the equations to be used for thermodynamic calculations Cp, Enthalpy and Entropy. The main calculation used in SysCAD is Enthalpy.

The Cp field is optional, but if it is left blank it will have the following consequences:

• SysCAD will assume a constant value:
For solids and gases, SysCAD will assume a value of 1.0 kJ/kg.K,
For liquids, SysCAD will assume a value of 2.0 kJ/kg.K.
This assumption will be shown as a warning in the message window when loading a project.
• If the species is used in a Reaction then the user will either:
Have to define the Heat of Reaction (HOR) manually in the reaction file; or
Receive a warning that Cp is not defined in the database.
See Reaction Block (RB) for further information on reactions. (This is because Cp is used to calculate heats of reaction for chemical reactions specified in SysCAD

Notes:

1. The unit for Heat Capacity is J/mol.K.
2. All Cp equations must use temperature in degrees Kelvin (K). If you have a Cp relation expressed in terms of degrees Centigrade or Fahrenheit it will need to be converted to a function of degrees Kelvin.
3. Valid Cp equations must be supplied for all species used in the project to obtain a correct energy balance in SysCAD.
4. For Cp, when the temperature is outside of the specified temperature range, then a constant value for Cp at the temperature limit is used.
5. The integral of Cp gives H (Enthalpy) equation. The Cp equations are integrated over a temperature range to obtain the change of enthalpy around a unit operation.
$\Delta H = \int\limits_{T_1}^{T_2}Cp dT\,$
Where T1 is Initial temperature and T2 Final temperature
6. The enthalpy calculation when the species is outside of the defined temperature ranges is calculated using the Cp at the limit of the defined temperature range:
• If T1 < TLower Limit then:
$\Delta H _{T_1 \; \to \; T_{lowerlimit}} \quad = Cp_{T_{lowerlimit}} \quad *\left ( T_{lowerlimit} - T_1 \right)\,$
• Similarly, if T2 > TUpper Limit then:
$\Delta H _{T_{upperlimit} \quad \; \to \; T_2} \; = Cp_{T_{upperlimit}} \quad *\left ( T_2 - T_{upperlimit} \right)\,$
1. The integral of H (Enthalpy) gives S (Entropy) equation.

The user may choose from a number of different functions for the Cp value, as shown in the image below:

Each of these will be described in the sections below:

Constant Cp value

The first 2 functions represent constant Cp values:

• Constant Value; and
• Constant Value with Temperature Range.

With these 2 options Cp does not vary as a function of temperature. However, with the option that includes a Range, the user may define a specific temperature range for the value.

Cp Equation Format (Single or Multi segment)

The user may define a single temperature range for the Cp value, SingleSegment, OR multiple temperature ranges, MultiSegment. In both cases, the Cp equations that are used have the same form, which are described further on in this section.

Single Segment

If the user selects 'SingleSegment' then they need only enter a single Cp equation and the range for that equation. The form of the equation can be any of those shown below.

The forms of the equations are explained in the sections below.

Multi Segment

If the user selects 'MultiSegment':

1. They may enter a number of Cp equations covering different temperature ranges;
2. Each Cp equation must be specified with the temperature range for which it is valid;
3. Temperature ranges MUST NOT overlap;
4. Temperature ranges MUST be contiguous, i.e. there must not be any gaps in the temperature ranges.
5. Different forms of the Cp equation can be used for different temperature ranges.

See the example below:

To insert a new equation before the first equation, select the first equation and press the Insert button.

To insert a new equation after the last equation, press the Append button.

To insert a new equation inbetween current equations, select the nearest equation and press the Insert button. An additional equation with the same parameters will be added after the selected equation. The previous temperature range for the selected equation will be split equally over the selected and new equation.

Cp Equation Format

The format for the Cp equation and corresponding Temperature Range is the same if the user has chosen single or multi segment. The format is as follows:

Cp Equation Name(a,b,...):Range(C or K, TL, TH)
Where:
1. The each Cp Equation is described below (Note: The equation always uses K); and
2. The Range is the temperature range for which the Cp equation is valid -
• The range can be given in degrees Celsius (C) or Kelvin (K) (Note, as above, while the range can be given in C or K, the actual Cp equation always uses K),
• TL is the Lowest temperature at which the Cp equation is valid, and
• TH is the Highest temperature at which the Cp equation is valid

Please see Examples of Cp Equations below for some examples of actual data.

By selecting a Cp equation, the corresponding Enthalpy and Entropy equations are selected. The available Cp equation formats are given below.

NOTES

1. All temperatures in the above formulae are in Kelvin. It does not matter if you give the range in units of C, K or F, the correlation, must use K.
2. If you want to use a correlation for Cp that is given as a function of C or F, set up a spreadsheet to generate the data for the range you want and then curve fit the data in degrees K and enter that into SysCAD (NB you should be able to reproduce an exact fit of your original correlation).

Refer to Stream Specific Heat values (Cp) for an example of how these individual heat capacities are used to determine the heat capacity of a stream.

CRC Equation

CRC_Cp(a,b,c,d)
$C_p = a + b.10^{-3} T + \dfrac{c.10^5}{T^2} + d.10^{-6} T^2\,$
where T - Temperature in K

CRC Equation (alternative format 1)

CRC1_Cp(a,b,c,d)
$C_p = a + b.10^{-3} T + \dfrac{c.10^5}{T^2} + d.10^{-6} T^2\,$
where T - Temperature in K

CRC Equation (alternative format 2)

CRC2_Cp(a,b,c)
$C_p = MolecularWeight * \Bigg( a + b.T - \dfrac{c}{T^2} \Bigg)\,$
where T - Temperature in K

HTE Equation format

HTE_Cp(a,b,c,d)
$C_p = 4.186 * \Bigg(b + 2*c.10^{-3} T - \dfrac{d.10^5}{T^2}\Bigg)\,$
where T - Temperature in K
Note: the first parameter a is required but is not used in the HTE equation and is ignored by SysCAD.

HSC Equation format

HSC_Cp(a,b,c,d)
$C_p = a + b.10^{-3} T + \dfrac{c.10^5}{T^2} + d.10^{-6} T^2\,$
where T - Temperature in K
Note: This equation is the same as the CRC format 1 equation.

Polynomial Equation format

Poly_Cp(a,b,c,d,e)
$C_p = a + bT + cT^2 + dT^3 + eT^4\,$
where T - Temperature in K

NOTE: The Poly_Cp will support upto 5 terms, user can ignore the later constants if the polynomial has lesser terms.

For example:

For Cp = 28.5 + 0.04T + 0.01T^2,
use Poly_Cp(28.5,0.04,0.01)

General Polynomial Equation format

GenPoly_Cp(c1,p1,c2,p2,c3,p3,c4,p4,c5,p5,...)
$C_p = c_1 T^{p_1} + c_2 T^{p_2} + c_3 T^{p_3} + c_4 T^{p_4} + c_5 T^{p_5} ...\,$
where T - Temperature in K
Notes:
1. This equation can have any number of terms, so the parameters are optional. You only need to enter as many parameters as are needed.
2. For Polynomials with 5 terms or less, we recommend using the Polynomial Equation format (Poly_Cp) format.

Shomate Equation

Shomate_Cp(a,b,c,d,e)
$C_p = a + b.10^{-3} T + c.10^{-6} T^2 + d.10^{-9} T^3 + \frac{e.10^6}{T^2}\,$
where T - Temperature in K
The Shomate Equation is used for many applications. Data can be obtained from NIST Chemistry WebBook. An example is to fit data from the NIST web site for gas components with higher temperature ranges. eg Oxygen

NASA Glenn Coefficients Equation

(This equation is available in Build 138)

NASAGlenn_Cp(a1, a2, a3, a4, a5, a6, a7, b1, b2)
$\frac{C_p(T)}{R} = a_1T^{-2} + a_2.T^{-1} + a_3 + a_4T + a_5T^2 + a_6T^3 + a_7T^4\,$
where
• $C_p(T)\,$ - molar heat capacity at constant pressure at temperature T for standard state
• T - Temperature in K
• R - universal gas constant 8.314510 J/(mol.K)
• b1 - integration constant for enthalpy
• b2 - integration constant for entropy

NOTE: H25 and S25 can be calculated directly using this equation. It is therefore recommended to specify "FromCp()" in fields for H25 and S25. If user specifies a value for H25 (or S25) and this does not match calculated value an error is given.

The NASA Glenn Coefficients Equation is used by the NASA Glenn computer program CEA (chemical Equilibrium with Applications). The NASA Data can be obtained from http://www.grc.nasa.gov/WWW/CEAWeb/.

General Gibbs Equation format

(This equation is available in Build 138 and later)

Gibbs_Cp(a, b, c, d, e, f)
$C_p(T) = -c - 2dT - 6eT^2 - \dfrac{2f}{T^2}$
where T - Temperature in K

The basic Gibbs function for constant pressure is G = H - TS, where H is Enthalpy and S Entropy. We can then derive the Cp equation above from the following general equation for gibbs function:

$G(T) = a + bT + cT.ln(T) + dT^2 + eT^3 + \dfrac{f}{T}$

NOTE: H25 and S25 can be calculated directly using this equation. It is therefore recommended to specify "FromCp()" in fields for H25 and S25. If user specifies a value for H25 (or S25) and this does not match calculated value an error is given.

General Gibbs Equation format 2

(This equation is available in Build 139 and later)

Gibbs2_Cp(a, b, c, d, e, f)
$C_p(T) = -c - 2dT - \dfrac{2e}{T^2} - \dfrac{6f}{T^3}$
where T - Temperature in K

The basic Gibbs function for constant pressure is G = H - TS, where H is Enthalpy and S Entropy. We can then derive the Cp equation above from the following general equation for gibbs function:

$G(T) = a + bT + cT.ln(T) + dT^2 + \dfrac{e}{T} + \dfrac{f}{T^2}$

NOTE: H25 and S25 can be calculated directly using this equation. It is therefore recommended to specify "FromCp()" in fields for H25 and S25. If user specifies a value for H25 (or S25) and this does not match calculated value an error is given.

Gibbs ChemApp Equation format

(This equation is available in Build 138 and later)

GibbsChemApp_Cp(a, b, c, d, e, f, P1, E1, P2, E2, P3, E3, P4, E4, P5, E5, P6, E6)
$C_p(T) = -c - 2dT - 6eT^2 - \dfrac{2f}{T^2} + \sum\nolimits P_{i}E_{i}T^{E_{i}-1}(1-E_{i}) + \dfrac{P_{lg}}{T}$
where T - Temperature in K

The basic Gibbs function for constant pressure is G = H - TS, where H is Enthalpy and S Entropy. We can then derive the Cp equation above from the following general equation for gibbs function:

$G(T) = a + bT + cT.ln(T) + dT^2 + eT^3 + \dfrac{f}{T} + \sum\nolimits P_{i}T^{E_{i}} + P_{lg}lnT$

NOTE: H25 and S25 can be calculated directly using this equation. It is therefore recommended to specify "FromCp()" in fields for H25 and S25. If user specifies a value for H25 (or S25) and this does not match calculated value an error is given.

NOTE: if EN = 0, the corresponding PN is interpreted as a logarithmic term, Plg.

Laliberte´ Cp Function

The user may choose to use the Laliberte´ method for calculating the specific heat of an aqueous species. This method uses the constants as calculated by Laliberte´ based on data obtained from many references.

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. Important: This method does NOT include a calculation for Entropy.
2. The constants for most aqueous species are valid for temperatures between approximately 5 and 1200C.
3. If the unit temperature is outside of the species temperature range, then SysCAD will use the values at the temperature limit.
4. Water heat capacity is calculated in SysCAD as described here: Water and Steam Properties.
5. The user MUST define all 6 constants and also the valid range for the function.
6. To remove the temperature dependence, set a2 = 0 AND a3 = 0 (or a1 = 0).
7. Due to the method of solving, a2 = 0 is not allowed unless a3 = 0 or a1 = 0.
8. This method calculates the Cp of a solution based on ALL of the aqueous species dissolved in the solution. Therefore, if the user wishes to use this method, then it is recommended that all aqueous species in a project use the Laliberte´ method for the results to be consistent.

Reference:

Laliberte´ M. A Model for Calculating the Heat Capacity of Aqueous Solutions, with Updated Density and Viscosity Data. J.Chem Eng. Data 2009.

Cp as a Function of Temperature - Spline under Tension

Select this option if Cp varies as a function of Temperature and the user has data points for this. SysCAD will then use a Spline method to interpolate from the data in the table.

Notes:

1. A spline is a special function defined piecewise by polynomials and is used in interpolating problems.
2. A fitted equation will be processed more quickly within SysCAD and hence is preferred to entering TSpline data.

See the example below:

Examples of Cp Equations

 Species Cp Total Temperature Range (C) CaCO3(s) HTE_Cp(-9122, 23.8351, 3.2146, 5.1569):Range(C, 25, 927) 25 to 927 CO2(g) Shomate_Cp(19.7961, 0.07344, -5.60221e-005, 1.71541e-008):Range(K, 298.15, 1000) 25 to 726.85 Fe2(SO4)3(aq) Poly_Cp(-22429, 57.4101, 29.6643, 7.9582):Range(K, 298.15, 900) 25 to 626.85 2CaO*SiO2(s) HSC_Cp(145.896, 40.752, -26.192, 0.000):Range(C, 25, 848), HSC_Cp(134.557, 46.108, 0.000, 0.000):Range(C, 848, 1439), Const(205.016):Range(C, 1439, 2130) 25 to 2130