CaloricallyImperfectGas

Fluid properties for an ideal gas with imperfect caloric behavior.

This class implements fluid properties for a gas that behaves like an ideal gas except that the specific heat capacities are a function of temperature (as opposed to constants as for the ideal gas).

The relationship between pressure, density, and temperature is identical to an ideal gas:

$var element = document.getElementById("moose-equation-18646bc4-b0bc-4a93-b83f-6add25d3e157");katex.render(" pv = R_s T,", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-d2b5ca35-5f6c-408f-8bc7-d2519799eee9");katex.render("p", element, {displayMode:false,throwOnError:false});$ is pressure, $var element = document.getElementById("moose-equation-2eadb2a3-1b7e-4d84-9406-42e5c54315cf");katex.render("v", element, {displayMode:false,throwOnError:false});$ is specific volume, $var element = document.getElementById("moose-equation-456a6d56-972d-454c-baf6-c07b9a7b0f0f");katex.render("R_s", element, {displayMode:false,throwOnError:false});$ is the specific gas constant, and $var element = document.getElementById("moose-equation-ba4a8bbc-4ea8-420e-bf79-cc3243d02d34");katex.render("T", element, {displayMode:false,throwOnError:false});$ is temperature.

The internal energy $var element = document.getElementById("moose-equation-7ffb5e24-75e3-412e-a3ca-14f4c1a29767");katex.render("e", element, {displayMode:false,throwOnError:false});$ is a user-provided function of temperature:

$var element = document.getElementById("moose-equation-b4ab3f3e-86ac-424c-bd6c-ea33ae60cc1a");katex.render(" e = e(T).", element, {displayMode:true,throwOnError:false});$

The function $var element = document.getElementById("moose-equation-e897b341-5364-4d67-8d39-14c2606e0bed");katex.render("e(T)", element, {displayMode:false,throwOnError:false});$ is provided via parameter "e". The time argument is interpreted as temperature.

The enthalpy is computed by:

$var element = document.getElementById("moose-equation-d5a5fd39-e42d-449a-a595-9fce0df46ec6");katex.render(" h = e(T) + R_s T.", element, {displayMode:true,throwOnError:false});$

The specific heat capacities at constant volume and pressure $var element = document.getElementById("moose-equation-8f050167-d91e-439a-b3b6-587c35bd3193");katex.render("c_v", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-b3e7f532-5c34-4d7f-b04a-b3ff3e6c44a7");katex.render("c_p", element, {displayMode:false,throwOnError:false});$ are computed by:

var element = document.getElementById("moose-equation-5c862c00-f137-4ec5-9ba5-7699ccf16187");katex.render("\\begin{aligned} c_v &= \\frac{de}{dT} \\\\ c_p &= \\frac{de}{dT} + R_s. \\end{aligned}", element, {displayMode:true,throwOnError:false});

warning

The specific heat capacities are computed from $var element = document.getElementById("moose-equation-ee1596ba-37a2-429e-a5ea-2933b5359334");katex.render("e(T)", element, {displayMode:false,throwOnError:false});$ using the timeDerivative method of the Function class. The type of function that is used for $var element = document.getElementById("moose-equation-573d7497-8f7c-44f9-b78a-712c5df1005e");katex.render("e(T)", element, {displayMode:false,throwOnError:false});$ must implement the timeDerivative method.

The inverse functions $var element = document.getElementById("moose-equation-4199059b-79b4-40f4-ac91-fe62d3c159a4");katex.render("T(e)", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-fec9d2c2-02e6-43ba-910a-63f42986f901");katex.render("T(h)", element, {displayMode:false,throwOnError:false});$ are obtained as follows:

Ensure that within the acceptable temperature range (parameters min_temperature and max_temperature) $var element = document.getElementById("moose-equation-fd712c72-71f5-4f30-a072-0f6a2075a18c");katex.render("c_v > 0", element, {displayMode:false,throwOnError:false});$ .
Compute the minimum and maximum values of $var element = document.getElementById("moose-equation-e54dd02f-5930-4453-9d3e-6a8459694785");katex.render("e", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-e69ab43f-6e0e-4e19-a697-1fff50e8a855");katex.render("h", element, {displayMode:false,throwOnError:false});$ .

var element = document.getElementById("moose-equation-ad7e9db3-5bbc-4a86-9c9a-2ff43712296d");katex.render("\\begin{aligned} e_{min} &= e(T_{min}) \\\\ e_{max} &= e(T_{max}) \\\\ h_{min} &= h(T_{min}) \\\\ h_{max} &= h(T_{max}) \\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false});

Sample
$var element = document.getElementById("moose-equation-863bfda0-8080-4f20-bc48-ff24a48a8305");katex.render("e_j = e_{min} + j \\frac{e_{max}-e_{min}}{N},\\qquad j=0,..,N.", element, {displayMode:true,throwOnError:false});$
and for each $var element = document.getElementById("moose-equation-4d633d61-90d4-4f4d-be1a-399594bcc461");katex.render("e_j", element, {displayMode:false,throwOnError:false});$ solve
$var element = document.getElementById("moose-equation-563e92a8-219d-4890-ab41-b83a3a05a291");katex.render("e_j - e(T_j)= 0,", element, {displayMode:true,throwOnError:false});$
for $var element = document.getElementById("moose-equation-f4db2728-d10c-4f34-9a62-b6ad24720bd5");katex.render("T_j", element, {displayMode:false,throwOnError:false});$ . Create tabulation $var element = document.getElementById("moose-equation-bfc9bbb3-f903-4404-a76c-16346ad9113a");katex.render("\\{e_j, T_j\\}", element, {displayMode:false,throwOnError:false});$ . Create a similar tabulation for enthalpy.
Evaluating $var element = document.getElementById("moose-equation-328513af-d7f0-4330-80f8-77a2c6fb4b00");katex.render("T(e)", element, {displayMode:false,throwOnError:false});$ uses linear interpolation in $var element = document.getElementById("moose-equation-31ab4865-a04f-490e-b05b-461ef7d1444f");katex.render("\\{e_j, T_j\\}", element, {displayMode:false,throwOnError:false});$ .

Evaluating Entropy

From the first and second law of thermodynamics is follows that:

var element = document.getElementById("moose-equation-afff3b71-dcc3-44b5-a41d-86e42f548dd8");katex.render("\\begin{aligned} dq &= de + pdv \\\\ ds &= dq/T, \\end{aligned}", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-a7b0392b-c5df-40a3-8858-974b0839a714");katex.render("dq", element, {displayMode:false,throwOnError:false});$ is an infinitesimal amount of heat. Solving for $var element = document.getElementById("moose-equation-7d38ce4b-adbb-4678-8087-feea6e6e2e16");katex.render("ds", element, {displayMode:false,throwOnError:false});$ and using the ideal gas law leads to:

$var element = document.getElementById("moose-equation-c583aedd-0faf-43bc-8e64-7660e724e6d9");katex.render(" ds = \\frac{de}{T} + R_s \\frac{dv}{v}.", element, {displayMode:true,throwOnError:false});$

Integrating this equation leads to an expression for the entropy:

$var element = document.getElementById("moose-equation-5f858a5f-e616-46ff-b620-ed1af32afdca");katex.render(" s(e, v) - s(e_0, v_0) = \\int\\limits_{e_0}^e \\frac{de'}{T'} + R_s \\log \\frac{v}{v_0},", element, {displayMode:true,throwOnError:false});$

where the prime indicates that these variables are dummy variables of integration. We are free to select a zero-point for entropy and we select $var element = document.getElementById("moose-equation-8bcb345f-7d54-4b12-a197-f6bb1c6b0de6");katex.render("s(e_0, v_0) = 0", element, {displayMode:false,throwOnError:false});$ . We also select $var element = document.getElementById("moose-equation-78ad6ace-7237-4fe1-9a58-5bd9b14b71dc");katex.render("v_0 = 1", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-d5699e5b-2806-41de-b9a7-f81ded5ed8b7");katex.render("e_0 = e(T_{min})", element, {displayMode:false,throwOnError:false});$ . Then we define:

$var element = document.getElementById("moose-equation-8d856fdf-a580-4cb4-b6ce-363031486c1a");katex.render(" Z(T) = \\int\\limits_{e_0}^{e(T)} \\frac{de'}{T'} = \\int\\limits_{T_{min}}^{T} \\frac{c_v(T')}{T'} dT'.", element, {displayMode:true,throwOnError:false});$

Entropy is computed from the expression:

$var element = document.getElementById("moose-equation-90d54612-7087-4420-a463-ffea081eda90");katex.render(" s(e, v) = Z(T(e)) + R_s \\log v,", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-aff28ad0-5494-44bc-93c8-d1601c6bdcf1");katex.render("Z(T)", element, {displayMode:false,throwOnError:false});$ is computed using a trapezoidal rule and tabulated between $var element = document.getElementById("moose-equation-dfea034b-dacc-407e-b8b2-8bb264083dbb");katex.render("T_{min}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-d91fed61-31bb-45c8-899a-961d21540d2c");katex.render("T_{max}", element, {displayMode:false,throwOnError:false});$ . It is linearly interpolated.

The derivatives of entropy with respect to $var element = document.getElementById("moose-equation-19120116-c5ba-44ad-8f16-be46d4f30d6c");katex.render("T", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-4b314135-368b-4463-a707-f759fce8d966");katex.render("e", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-1c3b7918-de98-491a-979c-8e896e295fc2");katex.render("h", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-fbed7e31-5b18-42e6-b639-e952b418aafe");katex.render("v", element, {displayMode:false,throwOnError:false});$ are given by:

var element = document.getElementById("moose-equation-5eb49e72-6472-4ebe-9bce-42d70ed0b238");katex.render("\\begin{aligned} \\left(\\frac{\\partial s}{\\partial e}\\right)_v &= \\frac{1}{T(e)} \\\\ \\left(\\frac{\\partial s}{\\partial v}\\right)_e &= \\frac{R_s}{v} \\\\ \\left(\\frac{\\partial s}{\\partial p}\\right)_T &= -\\frac{R_s}{p} \\\\ \\left(\\frac{\\partial s}{\\partial T}\\right)_p &= \\frac{c_p(T)}{T} \\\\ \\left(\\frac{\\partial s}{\\partial p}\\right)_h &= -\\frac{R_s}{p} \\\\ \\left(\\frac{\\partial s}{\\partial h}\\right)_p &= \\frac{1}{T}. \\end{aligned}", element, {displayMode:true,throwOnError:false});

These derivatives are implemented in the 5 argument versions of the s_from_x_y functions.

Input Parameters

eSpecific internal energy as a function of temperature [J/kg]
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Specific internal energy as a function of temperature [J/kg]
kThermal conductivity, [W/(m-K)]
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Thermal conductivity, [W/(m-K)]
max_temperatureLargest temperature for lookup tables [K]
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Largest temperature for lookup tables [K]
min_temperatureSmallest temperature for lookup tables [K]
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Smallest temperature for lookup tables [K]
molar_massConstant molar mass of the fluid [kg/mol]
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Constant molar mass of the fluid [kg/mol]
muDynamic viscosity, [Pa-s]
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Dynamic viscosity, [Pa-s]

Required Parameters

T_c0Critical temperature, [K]
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Critical temperature, [K]
e_c0Specific internal energy at the critical point, [J/kg]
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Specific internal energy at the critical point, [J/kg]
emit_on_nannoneWhether to raise a warning, an exception (usually triggering a retry with a smaller time step) or an error (ending the simulation)
Default:none
C++ Type:MooseEnum
Options:none, warning, exception, error
Controllable:No
Description:Whether to raise a warning, an exception (usually triggering a retry with a smaller time step) or an error (ending the simulation)
out_of_bound_errorTrueError if out of bounds
Default:True
C++ Type:bool
Controllable:No
Description:Error if out of bounds
rho_c0Critical density, [kg/m3]
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Critical density, [kg/m3]
temperature_resolution1Step size in temperature for creating the inverse lookup T(e)
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Step size in temperature for creating the inverse lookup T(e)

Optional Parameters

T_initial_guess400Temperature initial guess for Newton Method variable set conversion
Default:400
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Temperature initial guess for Newton Method variable set conversion
max_newton_its100Maximum number of Newton iterations for variable set conversions
Default:100
C++ Type:unsigned int
Controllable:No
Description:Maximum number of Newton iterations for variable set conversions
p_initial_guess200000Pressure initial guess for Newton Method variable set conversion
Default:200000
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Pressure initial guess for Newton Method variable set conversion
tolerance1e-08Tolerance for 2D Newton variable set conversion
Default:1e-08
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Tolerance for 2D Newton variable set conversion

Variable Set Conversions Newton Solve Parameters

allow_imperfect_jacobiansFalsetrue to allow unimplemented property derivative terms to be set to zero for the AD API
Default:False
C++ Type:bool
Controllable:No
Description:true to allow unimplemented property derivative terms to be set to zero for the AD API
control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
fp_typesingle-phase-fpType of the fluid property object
Default:single-phase-fp
C++ Type:FPType
Controllable:No
Description:Type of the fluid property object

Advanced Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

References

No citations exist within this document.

Evaluating Entropy
Input Parameters
References