Governing Equations of Energy Conservation
For fluid flow, the energy equation can be written in different forms based on the total/static enthalpy or internal energy. In Simerics-MP, the governing equation solved is the conservative form of the total energy conservation, which has the following general form:
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where 
is the total enthalpy; 
is the effective heat conductivity; 
and 
are the static enthalpy and diffusion flux of the 
chemical species (also called component). The second diffusion term represents the transport of enthalpy due to the chemical species diffusion in an n-component flow, described in detail in the module of Multicomponent Mixing. The term (
) indicates the work due to viscous stresses and is called the viscous dissipation/heating. It represents the internal heating by viscosity in the fluid and is negligible in most flows; and the last term 
includes all the remaining source terms such as phase change heat sources, effects of buoyancy and body forces, radiation, and external/user sources.
The species diffusion induced heat transfer only appears when species transport is considered. The viscous heating is often negligible, in particular, for incompressible flows. For the standard heat conduction term, the effective heat conductivity, 
, is computed as:
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where 
is the heat conductivity. For a single fluid, it is a material property. For a mixture, it is usually volumetric-averaged heat conductivity of all the species in the thermal system. As for the turbulent heat conductivity 
, it is estimated by the turbulent viscosity 
and thermal Prandtl number 
. By default, 
.
To close equation 5.357, the relationships among the energy-related variables: 
and 
need to be defined based on the fluid or the mixture in the thermal flows. Specifically, they have the following definitions and relations:
Static Enthalpy
It is the measurement of energy per unit mass for a material property or in a thermodynamic system. It is defined as the internal energy of a fluid and the fluid state. For a single material or fluid flow, the static enthalpy is defined as:
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where 
is the internal energy, depending on the material property and the thermal state.
equation 5.359 indicates that the enthalpy generally varies with changes in both temperature and pressure. The differential relationship can be given in the following:
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where 
is the specific heat at constant pressure.
Standard State Reference Enthalpy
To calculate and tabulate the thermodynamic properties of a material or substance, a reference point, called the Standard State, is commonly defined as the condition: the pressure is 
Pa, and the temperature has the value of 
K. The formation enthalpy at this reference point is called Standard State Reference Enthalpy. In general, it is written as:
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For a material property, its value can be easily found from references such as the NIST tables/curves.
Sensible Enthalpy
Sensible Enthalpy is the energy required for heating a unit mass from the reference point (
,
) to a given condition (
,
). Integrating equation 5.360, we have the sensible enthalpy in the following general formulation:
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with equation 5.362, the following two limiting conditions are obtained:
Ideal Gas Flow
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The sensible enthalpy for ideal gas is thus:
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Incompressible Flow
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The sensible enthalpy for incompressible flow has the form:
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With the standard state reference enthalpy, 
, and the sensible enthalpy, 
, the static enthalpy h is usually computed as:
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Total Enthalpy
With the static enthalpy computed from equation 5.367, the total enthalpy 
in the energy equation can be easily obtained:
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Mixture Enthalpies
For a mixture with multiple materials (species), the mixture static and total enthalpies are computed as:
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|
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where 
is the mass fraction of species "
" and 
is total number of the species.
Energy Equation in a Non-Inertial Reference Frame
In a non-inertial reference frame, the motion of the reference frame can be translation or/and rotation. Considering a coordinate system with a translational linear velocity 
and rotational angular velocity 
relative to the stationary (inertial) reference frame, we have the non-inertial reference frame moving with the velocity:
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where 
is the vector connecting the point of the interest (cell center, face or vertices) to the center of the rotating axis.
In Simerics-MP, when the governing equations are solved in the moving reference frame, the equations are formulated based on the absolute velocity formulation. As a result, the energy equation solved has the following form:
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And the additional term in the energy convection, 
, is considered as a source term.
Conjugate Heat Transfer
When modelling heat transfer in a practical system, it is often necessary or desirable for users to include solid regions in a computational domain, consisting of either only solids or both solid and fluid zones. This is commonly known as conjugate heat transfer. In solid regions, the energy transport equations solved has the following form:
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where 
is solid density; 
is heat conductivity. For solid materials, it can be isotropic (
) or anisotropic (
); and 
is the volumetric heat source. In equation 5.373, the solved variable "
" is the enthalpy of the solid material, which is defined as:
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Also, the second term on the left-hand side of equation represents convective energy transfer due to rotational or translational motion of the solids. The velocity 
is computed from the motion specified for the solid zone with respect to the reference frame. Without solid motion, the heat transfer in solids is purely heat conduction, represented by the first term on the right-hand side of equation 5.373.
Natural Convection
Natural convection is a heat transfer mechanism, in which the flow is induced purely by the fluid density differences (buoyancy) occurring due to temperature gradients. In natural convection, the fluid surrounding a heat source receives energy and undergoes thermal expansion. Under the effect of gravity force, the less dense (lighter) fluid rises, while the surrounding cooler (heavier) fluid moves down to fill the host spot and then becomes heated again. The process continues and forms the buoyancy-driven flow to transfer the heat energy from bottom to top.
To account for the natural convection, a source term is added in the momentum equations to capture the buoyancy-driven flow:
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where 
is the reference density; and 
is gravity.
Following the Boussinesq model, we can approximate the density difference in equation 5.375 with the temperature variations:
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And 
is the thermal expansion coefficient:
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Substituting equation 5.376 and equation 5.377 in equation 5.375, we have the Boussinesq buoyancy model:
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The strength of the natural convection can be determined by the following non-dimensional parameters:
Pure Natural Convection
It is characterized by the Rayleigh number (
), defined as:
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where
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Magnitude of the gravity (m/s2) |
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Characteristic length (m) |
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Thermal diffusivity (m2/s) |
Typically, if 
is the buoyancy-induced flow is laminar, while the flow transition to turbulence is considered to occur over the range of 
.
Mixed Convection
In cases where both natural and forced convection mechanisms act together to transfer heat, the importance of buoyancy forces relative to pressure forces is measured by the ratio of Grashof number to the square of the Reynolds number:
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When this ratio approaches or exceeds unity, strong buoyancy contributions to the flow would be expected. Otherwise, if it is very small, buoyancy forces may be ignored in the simulation.