3. Heat Capacity

class gasthermo.cp.CpIdealGas(dippr_no: str = None, compound_name: str = None, cas_number: str = None, T_min_fit: float = None, T_max_fit: float = None, n_points_fit: int = 1000, poly_order: int = 2, T_units='K', Cp_units='J/mol/K')

Heat Capacity \(C_{\mathrm{p}}^{\mathrm{IG}}\) [J/mol/K] at Constant Pressure of Inorganic and Organic Compounds in the Ideal Gas State Fit to Hyperbolic Functions [RWO+07]

(1)\[C_{\mathrm{p}}^{\mathrm{IG}} = C_1 + C_2\left[\frac{C_3/T}{\sinh{(C_3/T)}}\right] + C_4 \left[\frac{C_5/T}{\cosh{(C_5/T)}}\right]^2\]

where \(C_{\mathrm{p}}^{\mathrm{IG}}\) is in J/mol/K and \(T\) is in K.

Computing integrals of Equation (1) is challenging. Instead, the function is fit to a polynomial within a range of interest so that it can be integrated by using an antiderivative that is a polynomial.

Parameters

• dippr_no (str, optional) – dippr_no of compound by DIPPR table, defaults to None

• compound_name (str, optional) – name of chemical compound, defaults to None

• cas_number (str, optional) – CAS registry number for chemical compound, defaults to None

• MW (float, derived from input) – molecular weight in g/mol

• T_min (float, derived from input) – minimum temperature of validity for relationship [K]

• T_max (float, derived from input) – maximum temperature of validity [K]

• T_min_fit – minimum temperature for fitting, defaults to Tmin

• T_max_fit – maximum temperature for fitting, defaults to Tmax

• C1 (float, derived from input) – parameter in Equation (1)

• C2 (float, derived from input) – parameter in Equation (1)

• C3 (float, derived from input) – parameter in Equation (1)

• C4 (float, derived from input) – parameter in Equation (1)

• C5 (float, derived from input) – parameter in Equation (1)

• Cp_units (str, optional) – units for \(C_{\mathrm{p}}^{\mathrm{IG}}\), defaults to J/mol/K

• T_units (str, optional) – units for \(T\), defaults to K

• n_points_fit (int, optional) – number of points for fitting polynomial and plotting, defaults to 1000

• poly_order (int, optional) – order of polynomial for fitting, defaults to 2

cp_integral(T_a, T_b)

Evaluate integral

(2)\[\int_{T_a}^{T_b}C_{\mathrm{p}}^{\mathrm{IG}}(T^\prime) \mathrm{d}T^\prime\]

Parameters

• T_a – start temperature in K

• T_b – finish temperature in K

Returns

integral

eval(T, f_sinh=<ufunc 'sinh'>, f_cosh=<ufunc 'cosh'>)

Evaluate heat capacity

Parameters

• T – temperature in K

• f_sinh (callable) – function for hyperbolic sine, defaults to np.sinh

• f_cosh (callable) – function for hyperbolic cosine, defaults to np.cosh

Returns

\(C_{\mathrm{p}}^{\mathrm{IG}}\) J/mol/K (see equation (1))

get_numerical_percent_difference()

Calculate the percent difference with numerical integration

numerical_integration(T_a, T_b) → tuple

Numerical integration using scipy

class gasthermo.cp.CpStar(T_ref: float, **kwargs)

Dimensionless Heat Capacity at Constant Pressure of Inorganic and Organic Compounds in the Ideal Gas State Fit to Hyperbolic Functions [RWO+07]

The dimensionless form is obtained by introducing the following variables

\[\begin{split}\begin{align} C_{\mathrm{p}}^\star &= \frac{C_\mathrm{p}^\mathrm{IG}}{\text{R}} \\ T^\star &= \frac{T}{T_\text{ref}} \end{align}\end{split}\]

where R is the gas constant in units of J/mol/K, and \(T_\text{ref}\) is a reference temperature [K] input by the user (see T_ref)

The heat capacity in dimensionless form becomes

(3)\[C_{\mathrm{p}}^\star = C_1^\star + C_2^\star \left[\frac{C_3^\star/T^\star}{\sinh{(C_3^\star/T^\star)}}\right] + C_4^\star \left[\frac{C_5^\star/T^\star}{\cosh{(C_5^\star/T^\star)}}\right]^2\]

where

\[\begin{split}\begin{align} C_1^\star &= \frac{C_1}{\text{R}} \\ C_2^\star &= \frac{C_2}{\text{R}} \\ C_3^\star &= \frac{C_3}{T_\text{ref}} \\ C_4^\star &= \frac{C_4}{\text{R}} \\ C_5^\star &= \frac{C_5}{T_\text{ref}} \end{align}\end{split}\]

Parameters

T_ref (float) – reference temperature [K] for dimensionless computations

eval(T, f_sinh=<ufunc 'sinh'>, f_cosh=<ufunc 'cosh'>)

Parameters

• T – temperature in K

• f_sinh (callable) – function for hyperbolic sine, defaults to np.sinh

• f_cosh (callable) – function for hyperbolic cosine, defaults to np.cosh

Returns

\(C_{\mathrm{p}}^{\star}\) [dimensionless] (see equation (3))

class gasthermo.cp.CpRawData(T_raw: list, Cp_raw: list, T_min_fit: float = None, T_max_fit: float = None, poly_order: int = 2, T_units='K', Cp_units='J/mol/K')

Obtain heat capacity relationships from raw data

Using input raw data # fit to polynomial of temperature # fit polynomial to antiderivative

Parameters

• T_min_fit (float, optional) – minimum temperature for fitting function [K]

• T_max_fit (float, optional) – maximum temperature for fitting function [K]

• poly_order (int, optional) – order of polynomial for fitting, defaults to 2

• T_raw (list) – raw temperatures in K

• Cp_raw (list) – raw heat capacities in J/K/mol

• Cp_units (str, optional) – units for \(C_{\mathrm{p}}\), defaults to J/mol/K

• T_units (str, optional) – units for \(T\), defaults to K

get_max_percent_difference()

Get largest percent difference