# Thermophysical Fluid Models

## Overview

The ‘Thermophysical models’ define how the energy, heat and physical properties of the fluid are calculated. These variables are then determined based on the analysis type and the selected models. The models are relations of a Pressure-Temperature equation system using which the required fluid properties and variables are then calculated. [1]

The thermophysical model selection appears under ‘Model’ and then ‘Material’ sub-tree after the addition of a material.

As highlighted in the following figure, for any compressible or heat transfer analysis the set up requires 7 properties to be defined\(^1\). These settings are briefly described.

The thermophysical models are currently applicable to ‘Newtonian fluids’ only.

## 1. Type

The ‘Type’ of model is distinguished by the approach used to calculate the flow thermal variables. The available types on the SimScale platform are the ‘psiThermo’ and ‘rhoThermo’ types that are briefly detailed below.

### i- psiThermo

The “psiThermo” Thermophysical model is for a fluid with fixed chemical composition. It uses a variable ‘psi’, \(\psi\) that determines the compressibility as follows:

$$ \psi = \left ( RT \right )^{-1} \qquad [1]$$

where, \(R\) is the Gas Constant and \(T\) is Temperature. Based on this compressibility and the changes in pressure, the density is then determined by the selected Equation of State.

The “psiThermo” type is applicable for compressible flow analysis (i.e rhoSimpleFoam or rhoPimpleFoam OpenFoam solvers etc).

### ii- rhoThermo

The “rhoThermo” Thermophysical model is also for a fluid with fixed chemical composition. This type calculates the basic thermodynamic properties based on density, \(\rho\) variation of the fluid. This is used for Heat Transfer analysis where the density of the fluid is changing due to changes in temperature of the fluid.

The “rhoThermo” type is applicable for Heat Transfer ( natural Convection or coupled ) analysis of compressible fluids (i.e buoyantSimpleFoam or chtMultiRegionFoam OpenFoam solver etc)

## 2. Mixture

The ‘Mixture’ specifies the fluid mixture composition. Generally, thermophysical models without chemical reactions are categorised as pure mixtures, which represents a mixture with fixed chemical composition. Currently only ‘pure Mixture’ can be selected under ‘Material’ and sub-option ‘Mixture’.

## 3. Specie

Under specie the composition of the constituent is specified. As currently, a single constituent is available, so parameter values for one constituent are required. The following values are required for the specie:

### i- nMoles

This is the number of moles of the component. This parameter has a default value of 1 and generally is not required to be changed.

### ii- molWeight

This is the molecular weight of the component in units of kg/kmol and is dependent on the molecular structure of the fluid material.

## 4. Transport Model

The Transport model relates to the calculation of the transport variables dynamic viscosity \(\mu\), thermal conductivity \(\kappa\) and thermal diffusivity \(\alpha\) ( for energy and enthalpy equations). [1] Depending upon the problem, the following types of transport models are available:

### i- const (Constant parameters)

The ‘const’ type will assume a constant dynamic viscosity \(\mu\) and Prandtl number \(P_{r}= c_{p}\ \mu / \kappa\) . These parameters are then specified as Dynamic viscosity \(\mu\) in units of [Pa.s] and dimensionless Prandtl number.

### ii- sutherland

For the ‘sutherland’ type, the dynamic viscosity \(\mu\) is Not a constant and changes with temperature \(T\). The dynamic viscosity is then calculated as a function of temperature given by a the relation below:

$$ \mu = (A_{s}\sqrt{T})/(1+T_{s}/T) \qquad [1]$$

where, \(A_{s}\) is a sutherland coefficient with units \([kg/(m s K^{0.5})]\) and \(T_{s}\) is sutherland temperature with units [K].

### iii- polynomial

The user may also enter a custom relation for calculation of dynamic viscosity \(\mu\) and thermal conductivity \(\kappa\) as a function of temperature \(T\) by a polynomial of order \(N\) (maximum order N = 7). The relation is then given as follows:

$$ \mu = \sum_{i=0}^{N-1}a_{i}T^{i} \qquad [1]$$

This option is only available for ‘natural convection Heat Transfer’ cases with ‘rhoThermo’ models. For a ‘polynamial’ Transport model, the ‘Thermodynamic Model’ and ‘Equation of state’ then must also be specified by polynomials (see ‘hPolynomial and icoPolynomial below for details) as shown in figure below.

## 5. Thermodynamic Models

The thermodynamic models are used to calculate the specific heat \(c_{p}\) (at constant pressure) for the fluid, from which then the other properties are derived. The following methods are available for the evaluation of \(c_{p}\).

### i- hConst

This options assumes a constant value for specific heat \(c_{p}\) and the heat of fusion \(H_{f}\). These values are specified by the user in standard S.I units.

### ii- eConst

This option does Not assume a constant specific heat (at constant pressure) \(c_{p}\). Rather, it assumes a constant specific heat (at constant volume) \(c_{v}\) and the heat of fusion \(H_{f}\). These values are specified by the user in standard S.I units.

### iii- janaf

The ‘janaf’ options provides a custom relation for the specific heat \(c_{p}\) (at constant pressure) as a function of temperature \(T\). This relation is defined by a set of coefficients from the JANAF tables of thermodynamics. The function is given as follows:

$$ c_p = R ((((a_4 T + a_3) T + a_2 )T + a_1)T + a_0). \qquad [1]$$

The function is evaluated between a lower and upper temperature limit \(T_{l}\) and \(T_{h}\). Therefore, two sets of coefficients are required. The first set is to define temperature range above a common temperature \(T_{c}\) and below \(T_{h}\). While the second set to define range for temperatures below \(T_{c}\) and above \(T_{l}\). [1]

The coefficients and variables are summarised below:

- Lower temperature limit \(T_{l}\) Tlow
- Upper temperature limit \(T_{h}\) Thigh
- Common temperature \(T_{c}\) Tcommon
- High/Low temperature coefficients \(c_{p}\) Coeffs (\(a_0, a_1, a_2, a_3, a_4\))
- High/Low temperature enthalpy offset \(a_{5}\)
- High/Low temperature entropy offset \(a_{6}\)

### iv- hPolynaomial

This option is available if a ‘polynomial’ Transport model is selected. Then the specific heat (\(c_{p}\)) is calculated as a function of temperature by a polynomial of order N as below.

$$ c_p = \sum_{i=0}^{N-1}a_{i}T^{i} \qquad [1]$$

## 6. Equation of State

An equation of state is a thermodynamic relation describing the interconnection between various macroscopic properties of a fluid. In OpenFoam solver, it describes the relation between density \(\rho\) of a fluid and the fluid pressure \(P\) and temperature \(T\). [1]

Based on the thermophysical model type, the following equations of state can be used :

### i- rhoConst

In this case, the fluid density \(\rho\) is kept constant and does not change by pressure \(P\) or temperature \(T\)

$$ \rho = constant$$

### ii- perfectGas

For the case of perfect gas the fluid is assumed to be an ‘Ideal Gas’ and obey the ‘Ideal Gas Law’, that is the equation of state as given by the following relation:

$$ \rho = P/(RT) \qquad [1]$$

where, \(P\) is the pressure, \(R\) is the specific gas constant and \(T\) is the temperature.

### iii- incompressiblePerfectGas

In this case, the fluid is assumed a perfect gas that is only incompressible with respect to changes in pressure \(P\) . The equation of state is then given as:

$$ \rho = P_{ref}/(RT) \qquad [1]$$

where, \(P_{ref}\) is the reference pressure, \(R\) is the specific gas constant and \(T\) is the temperature.

For ‘incompressiblePerfectGas’ the gas density ρ can vary due to changes in temperature T .

### iv- perfectFluid

In the perfect fluid case, the equation of state takes the form as given below:

$$ \rho=P/(RT) \ +\ \rho_{0} \qquad [1]$$

where, \(\rho_{0}\) is the density at \(T = 0\). Then the density of the fluid can change both due to pressure and temperature.

### v- icoPolynomial

This option is available if a ‘polynomial’ Transport model is selected. The user can define an ‘Incompressible polynomial equation of state’ based on the relation given below. Then the fluid density \(\rho\) is a function of temperature \(T\) only.

$$ \rho = \sum_{i=0}^{N-1}a_{i}T^{i} \qquad [1]$$

where, \(a_{i}\) are polynomial coefficients of any order \(N\). Thus, the user then must enter the coefficients for the incompressible fluid material that is to be modeled.

This option is recommended to model natural convection in liquids, e.g water, due to changes in temperature.

## 7. Energy

Under ‘Energy’ two options are available for the form of energy to be used in the solution. One is the ‘Internal Energy’ \(\rho\) and other is the ‘Enthalpy’ \(e\), respectively the user can select ‘sensible Internal Energy’ or ‘sensible Enthalpy’. The equations can have form that includes the heat of formation \(h\) or without it. So, absolute energy is where heat of formation is included, and sensible energy is where it is not. [1] Similarly, absolute enthalpy \(\Delta h_{f}\) is related to sensible enthalpy \(h\) for a single specie as follows:

$$ h=h_s+c\ \Delta h_{f} \qquad [1]$$

where, \(h_s\) is the molar fraction and \(c\) is the heat of formation.

In most cases it is generally recommended to use ‘sensible Enthalpy’, unless energy change due to reactions is expected.

Last updated: September 23rd, 2020

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