Global Variables for Heat and Energy Balance
This section describes variables defined by integrals. A concise notation denotes the different domains of integration: Ω is the geometry domain, ∂Ωext stands for the exterior boundaries, and ∂Ωint for the interior boundaries.
Total Accumulated Heat Rate
The total accumulated heat rate variable, dEiInt, is the variation of internal energy per unit time in the domain:
Total Net Heat Rate
The total net heat rate, ntfluxInt, is the integral of Total Heat Flux (Heat Transfer Interface) over all external boundaries. In the case of a fluid domain, it reads:
This indicates the sum of incoming and outgoing total heat flux through the system.
Total Heat Source
The total heat source, QInt, accounts for all domain sources; interior boundary, edge, and point sources; and radiative sources at interior boundaries:
Total Stress Power
The total stress power, WstrInt, corresponds to the work lost by a fluid by degradation of energy. These works are transmitted to the system through pressure work and viscous dissipation:
Total Accumulated Energy Rate
The total accumulated energy rate, dEi0Int, is the variation of total internal energy per unit time in the domain:
where the total internal energy, Ei0, is defined as
Total Net Energy Rate
The total net energy rate, ntefluxInt, is the integral of Total Energy Flux (Heat Transfer Interface) over all external boundaries. In the case of a fluid domain, it reads:
This indicates the sum of incoming and outgoing total energy flux through the system.
Total Work Source
The total work source, WInt, accounts for volume forces and boundary stresses as follows:
where K is the viscous stress vector defined as
and is the jump of viscous stress at interior boundaries.
Heat Balance
According to Equation 4-208, the following equality between COMSOL Multiphysics variables holds:
dEiInt + ntfluxInt = QInt - WstrInt
and the heat balance variable, heatBalance, is defined as
heatBalance = dEiInt + ntfluxInt + WstrInt - QInt
This is the most general form that can be used for time-dependent models. At steady-state the formula is simplified. The accumulated heat rate equals zero, so the total net heat rate (the sum of incoming and outgoing heat rates) should correspond to the heat and work sources:
ntfluxInt = QInt - WstrInt
The sign convention used in COMSOL Multiphysics for QInt is positive when energy is produced (as for a heater) and negative when energy is consumed (as for a cooler). For WstrInt, the losses that heat up the system are negative and the gains that cool down the system are positive.
For stationary models with convection by an incompressible flow, the heat balance becomes:
ntfluxInt = QInt
which corresponds to the conservation of convective and conductive flux as in:
Depending on the radiation discretization method, the contribution to the heat balance is handled differently. For the definition of ntfluxInt, the Optically Thick Participating Medium subnode uses the Rosseland approximation and defines qr, the radiative flux, as an extra contribution to the conductive heat flux. The P1 approximation and Discrete ordinates method, however, include the radiative source ∇ ⋅ qr to Q on the domain, in the variable QInt.
The heat balance can be evaluated assuming constant material properties using the following variables: heatBalance_cst, dEiInt_cst, ntfluxInt_cst, WstrInt_cst, and QInt_cst. The evaluation of these variables is faster and accurate for constant material properties or small temperature gradients.
Energy Balance
According to Equation 4-209, the following equality between COMSOL Multiphysics predefined variables holds:
dEi0Int + ntefluxInt = QInt + WInt
and the energy balance variable, energyBalance, is defined as
energyBalance = dEi0Int + ntefluxInt - WInt - QInt
In stationary models, dEi0Int is zero so the energy balance simplifies into:
ntefluxInt = QInt + WInt
At steady state, and without any additional heat source (QInt equal to zero), the integral of the net energy flux on all boundaries of the flow domain, ntefluxInt, vanishes. On the other hand, the corresponding integral of the net heat flux does not, in general, vanish. It corresponds instead to the losses from mass and momentum equations, such as WstrInt for pressure work and viscous dissipation in fluids. Hence, energy is the conserved quantity, not heat.
The heat and energy balance variables such as ht.ntfluxInt or ht.ntefluxInt do not take into account the flux contributions on thin structures. For example, if a Thin Layer node is applied on a boundary, the heat flux on this boundary is not included in ht.ntfluxInt. It is instead added to the variable ht.sls1.ntfluxInt, defined locally on the thin structure. Here, sls1 is the tag of the Thin Layer node.
The energy balance can be evaluated assuming constant material properties using the following variables: energyBalance_cst, dEi0Int_cst, ntefluxInt_cst, WInt_cst, and QInt_cst. The evaluation of these variables is faster and accurate for constant material properties or small temperature gradients.