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13.2.1 Heat Transfer Theory



The Energy Equation


FLUENT solves the energy equation in the following form:


 \frac{\partial}{\partial t} (\rho E) + \nabla \cdot ({\vec v... ...rline{\overline{\tau}}}_{\rm eff} \cdot {\vec v})\right) + S_h (13.2-1)

where $k_{\rm eff}$ is the effective conductivity ( $k + k_t$, where $k_t$ is the turbulent thermal conductivity, defined according to the turbulence model being used), and ${\vec J}_j$ is the diffusion flux of species $j$. The first three terms on the right-hand side of Equation  13.2-1 represent energy transfer due to conduction, species diffusion, and viscous dissipation, respectively. $S_h$ includes the heat of chemical reaction, and any other volumetric heat sources you have defined.

In Equation  13.2-1,


 E = h - \frac{p}{\rho} + \frac{v^2}{2} (13.2-2)

where sensible enthalpy $h$ is defined for ideal gases as


 h = \sum_j Y_j h_j (13.2-3)

and for incompressible flows as


 h = \sum_j Y_j h_j + \frac{p}{\rho} (13.2-4)

In Equations  13.2-3 and 13.2-4, $Y_j$ is the mass fraction of species $j$ and


 h_j = \int^{T}_{T_{\rm ref}} c_{p,j}\; dT (13.2-5)

where $T_{\rm ref}$ is 298.15 K.



The Energy Equation for the Non-Premixed Combustion Model


When the non-adiabatic non-premixed combustion model is enabled, FLUENT solves the total enthalpy form of the energy equation:


 \frac{\partial}{\partial t} (\rho H) + \nabla \cdot (\rho {\... ... H) = \nabla \cdot \left(\frac{k_t}{c_p} \nabla H\right) + S_h (13.2-6)

Under the assumption that the Lewis number (Le) = 1, the conduction and species diffusion terms combine to give the first term on the right-hand side of the above equation while the contribution from viscous dissipation appears in the non-conservative form as the second term. The total enthalpy $H$ is defined as


 H = \sum_j Y_j H_j (13.2-7)

where $Y_j$ is the mass fraction of species $j$ and


 H_j = \int^{T}_{T_{{\rm ref},j}} c_{p,j} dT + h_j^{0} (T_{{\rm ref},j}) (13.2-8)

$h_j^{0} (T_{{\rm ref},j})$ is the formation enthalpy of species $j$ at the reference temperature $T_{{\rm ref},j}$.



Inclusion of Pressure Work and Kinetic Energy Terms


Equation  13.2-1 includes pressure work and kinetic energy terms which are often negligible in incompressible flows. For this reason, the pressure-based solver by default does not include the pressure work or kinetic energy when you are solving incompressible flow. If you wish to include these terms, use the define/models/energy? text command to turn them on.

Pressure work and kinetic energy are always accounted for when you are modeling compressible flow or using the density-based solver.



Inclusion of the Viscous Dissipation Terms


Equations  13.2-1 and 13.2-6 include viscous dissipation terms, which describe the thermal energy created by viscous shear in the flow.

When the pressure-based solver is used, FLUENT's default form of the energy equation does not include them (because viscous heating is often negligible). Viscous heating will be important when the Brinkman number, Br, approaches or exceeds unity, where


 {\rm Br} = \frac{\mu U^{2}_{e}}{k \Delta T} (13.2-9)

and $\Delta T$ represents the temperature difference in the system.

When your problem requires inclusion of the viscous dissipation terms and you are using the pressure-based solver, you should activate the terms using the Viscous Heating option in the Viscous Model panel. Compressible flows typically have ${\rm Br} \geq 1$. Note, however, that when the pressure-based solver is used, FLUENT does not automatically activate the viscous dissipation if you have defined a compressible flow model.

When the density-based solver is used, the viscous dissipation terms are always included when the energy equation is solved.



Inclusion of the Species Diffusion Term


Equations  13.2-1 and 13.2-6 both include the effect of enthalpy transport due to species diffusion.

When the pressure-based solver is used, the term


\nabla \cdot \left ( \sum_j h_j {\vec J}_j \right )

is included in Equation  13.2-1 by default. If you do not want to include it, you can turn off the Diffusion Energy Source option in the Species Model panel.

When the non-adiabatic non-premixed combustion model is being used, this term does not explicitly appear in the energy equation, because it is included in the first term on the right-hand side of Equation  13.2-6.

When the density-based solver is used, this term is always included in the energy equation.



Energy Sources Due to Reaction


Sources of energy , $S_h$, in Equation  13.2-1 include the source of energy due to chemical reaction:


 S_{h,{\rm rxn}} = - \sum_j \frac{h_j^0}{M_j} {\cal R}_j (13.2-10)

where $h^0_j$ is the enthalpy of formation of species $j$ and ${\cal R}_j$ is the volumetric rate of creation of species $j$.

In the energy equation used for non-adiabatic non-premixed combustion (Equation  13.2-6), the heat of formation is included in the definition of enthalpy (see Equation  13.2-7), so reaction sources of energy are not included in $S_h$.



Energy Sources Due To Radiation


When one of the radiation models is being used, $S_h$ in Equation  13.2-1 or 13.2-6 also includes radiation source terms. See Section  13.3 for details.



Interphase Energy Sources


It should be noted that the energy sources, $S_h$, also include heat transfer between the continuous and the discrete phase. This is discussed further in Section  22.9.1.



Energy Equation in Solid Regions


In solid regions, the energy transport equation used by FLUENT has the following form:


 \frac{\partial}{\partial t} (\rho h) + \nabla \cdot ({\vec v} \rho h) = \nabla \cdot ( k \nabla T) + S_h (13.2-11)


where $\rho$ = density
  $h$ = sensible enthalpy, $\int_{T_{\rm ref}}^T {c_p dT}$
  $k$ = conductivity
  $T$ = temperature
  $S_h$ = volumetric heat source

The second term on the left-hand side of Equation  13.2-11 represents convective energy transfer due to rotational or translational motion of the solids. The velocity field ${\vec v}$ is computed from the motion specified for the solid zone (see Section  7.18). The terms on the right-hand side of Equation  13.2-11 are the heat flux due to conduction and volumetric heat sources within the solid, respectively.



Anisotropic Conductivity in Solids


When you use the pressure-based solver, FLUENT allows you to specify anisotropic conductivity for solid materials. The conduction term for an anisotropic solid has the form


 \nabla \cdot \left( k_{ij} \nabla T \right) (13.2-12)

where $k_{ij}$ is the conductivity matrix. See Section  8.5.5 for details on specifying anisotropic conductivity for solid materials.



Diffusion at Inlets


The net transport of energy at inlets consists of both the convection and diffusion components. The convection component is fixed by the inlet temperature specified by you. The diffusion component, however, depends on the gradient of the computed temperature field. Thus the diffusion component (and therefore the net inlet transport) is not specified a priori.

In some cases, you may wish to specify the net inlet transport of energy rather than the inlet temperature. If you are using the pressure-based solver, you can do this by disabling inlet energy diffusion. By default, FLUENT includes the diffusion flux of energy at inlets. To turn off inlet diffusion, use the define/models/energy? text command.

Inlet diffusion cannot be turned off if you are using the density-based solver.


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