Equations for the Conservation of Energy
Mechanical Energy Equation
Non-conservative form:
()
1
2
ij
ii i i i
j
Duu ug u
Dt x
τ
ρρ
∂
=+
∂
Conservative form:
(
)
()
1
21
2
ijii
jii iii
j
j
τ
ρuu uρuu ρug u
tx x
∂
∂∂
+=+
∂∂ ∂
or
()
ij
jiii
j
j
τ
EuE ρug u
tx x
∂
∂∂
+=+
∂
∂∂
where 2
1
2mV is the kinetic energy (the conserved quantity), 1
2ii
E
uu
ρ
≡ is the kinetic energy per unit volume,
and 1
2ii
uu is the kinetic energy per unit mass. The terms on the right are sources (or sinks) of kinetic energy per
unit volume.
Alternate conservative form:
() ()
j
jii iji
jj
u
EuE ρug τup
tx x x
j
φ
∂
∂∂ ∂
+
=+ +−
∂∂ ∂ ∂
where i
ij
j
u
σ
x
φ
∂
≡∂ = rate of viscous dissipation of kinetic energy per unit volume (increases internal energy at
expense of kinetic energy).
φ
is always positive since friction is an irreversible process. For a Newtonian fluid,
with Stokes’ assumption that 2
30λμ+=
,
φ
becomes 2
3
2
j
i
ij ij
ij
u
u
μee μ
x
x
φ
∂
∂
=−
∂
∂.
Thermal Energy Equation (Heat Equation – the “real” energy equation)
First Law of Thermodynamics for a Material Volume:
()
1
2ii ii iji j i i
AA
Dρeuud ρgud τudA qdA
Dt +=+−
∫∫∫∫
v
v
VV
VV
where e = internal energy per unit mass (what most Thermodynamics books call u). Or, using the RTT,
First Law of Thermodynamics for a Control Volume:
() ()
11
22
ii ii j j ii ij i j i i
CV CS CV CS CS
euud ρeuuudA ρgud τudA qdA
t
ρ
∂⎡⎤
+++ =+−
⎣⎦
∂
∫∫∫∫∫
v
vv
VV
.
First Law of Thermodynamics in Differential form:
()
()
1
2i
ii i i iji
j
i
q
D
ρeuu ρug τu
D
txx
∂
∂
+=+ −
∂
∂.
Or, combining this with the mechanical energy equation, after some algebra we get an alternate form:
Differential Thermal Energy Equation (Heat Equation):
ii
ii
qu
De
ρp
Dt x x
φ
∂
∂
=
−− +
∂∂,
I II III IV
where the terms are explained below:
I Rate of increase of internal energy of a fluid element per unit volume (following the fluid element). This
term can be positive or negative.
II Rate of heat flux into the fluid element per unit volume (negative because q
G
is defined as positive outward).
This term can be positive or negative.
III Rate of increase of internal energy per unit volume due to volumetric compression (negative because ui,i =
volumetric expansion, which is defined as positive, but compression increases the internal energy). This
term can be positive or negative.
IV Rate of increase of internal energy per unit volume due to viscous dissipation. This term is always positive.
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