Conservation of Internal Energy

Conservation of internal energy is used to model temperature dependent material behavior. It also allows an energy balance evaluation.

However, internal energy is only calculated if it is turned on, to reduce computation time in problems not involving heat transfer.

The conservation of energy is given by:

∂ ρ e ∂ t − ( ( w i − v i ) ⋅ ∂ ρ e ∂ x i ) + ( ρ e + P ) ∂ v K ∂ x K = 0

Where,
e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaaaa@36E0@
Internal energy in Joules (Nm)
P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaaaa@36E0@
Fluid pressure

Applying a Galerkin variation form for the solution gives:

∫ V ψ ( ∂ ρ e ∂ t − ( ( w i − v i ) ⋅ ∂ ρ e ∂ x i ) + ( ρ e + P ) ∂ v K ∂ x K ) = 0

Making the following assumptions:

ψ =1

ρ e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaadwgaaeqaaaaa@38CC@ = constant over control volume V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaaaa@36E0@

Equation 2 reduces to:

∫ V ∂ ρ e ∂ t d V + ∫ V ( ρ e + P ) ∂ v K ∂ x K d V = 0

Applying the divergence theorem gives:

∫ V ∂ ρ e ∂ t d V + ∫ S ( ρ e ( v j ⋅ n j ) d S ) + ∫ V P ∂ v K ∂ x K d V = 0

Hence:

d d t ∫ V ρ e d V = ∫ S ρ e ( w j − v j ) n j d S − ∫ V P ∂ v K ∂ x K d V

This formula is still valid if e is not assumed uniform over volume V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaaaa@36E0@ .

New formulation based on total energy conservation are under investigation.