Equation 1.64 is the statement of the conservation of momentum principle.. Note that the conservation of momentum principle is stated in terms of the proper-ties of particles and not in
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F= m d ˆV
dt =
d(m ˆV)
wheremV is the momentum Equation (1.64) is the statement of the conservation of
momentum principle.
Note that the conservation of momentum principle is stated in terms of the proper-ties of particles and not in terms of the properproper-ties of a field To derive the momentum theorem, a region in a fluid confined by the control surfaceS1, shown in Fig 1.7, is employed The surfaceS1contains a definite and fixed number of particles at timet1
At timet2, these particles will have moved to a region bounded by the control surface
S2, which is shown as a dashed curve to distinguish it fromS1 The control surfacesS1andS2enclose three separate and distinct regions, desig-nated bya,b, and c Let the momentum in the three regions be P a , P b, and Pc, re-spectively At timet1the particles within surfaceS1will possess momentum Pa+Pb1
At timet2these particles will have momentum, Pb2+ Pcbecause they have moved into the region enclosed by surfaceS2 Hence the momentum change during the time intervalt2− t1may be described by
(P b2+ Pc ) − (P b1+ Pa ) = (P b2− Pb1 ) + (P c− Pa )
and the time rate of change of momentum will be
lim
t2→t1
Pb2− Pb1
t2− t1
+Pc− Pa
t2− t1
(1.65)
Ast2 approachest1 as a limit, the control surfaceS2 will coincide withS1 The first term in eq (1.65) is therefore the time rate of change of momentum of the fluid contained within region 1, R1, contained withinS1 This may be written as the integral overR1 Because the mass of fluid contained inR1is
R1
ρ dR1
the time rate of change of momentum of the fluid contained within region 1 will be
∂
∂t
R1
ρ ˆV dR1
The second term in eq (1.65) is the momentum efflux through the control surface
S1 If the flux in the outward direction is taken as positive, this efflux can be expressed
by the integral
S1
ρ ˆV ˆV n dS1 whereV nis the component of velocity normal toS1
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c b a
Figure 1.7 Regions bounded by control surfaces used for the development of the momentum theorem
The conservation of momentum principle then becomes
F = m d ˆV
dt =
∂
∂t
R1
ρ ˆV dR1+
S1
ρ ˆV ˆV n dS1 (1.66)
or, by rearrangement of terms,
∂
∂t
R1
ρ ˆV x dR1= F x−
S1
ρ ˆV x ˆV n dS1 (1.67a)
∂
∂t
R1
ρ ˆV y dR1= F y−
S1
ρ ˆV y ˆV n dS1 (1.67b)
∂
∂t
R1
ρ ˆV z dR1 = F z−
S1
ρ ˆV x ˆV n dS1 (1.67c)
in the three rectangular coordinate directions
The foregoing development leads to the statement of the momentum theorem: The
time rate of increase of momentum of a fluid within a fixed control volumeR will
be equal to the rate at which momentum flows intoR through its confining surface
S, plus the net force acting on the fluid within R When the flow is incompressible, the viscosity is constant, and the flow is laminar, the Navier–Stokes equations result.
In Cartesian coordinates, withF x,F y, andF ztaken as the components of the body force per unit volume, the Navier–Stokes equations are
ρ
∂ ˆV x
∂t + ˆV x
∂ ˆV x
∂x + ˆV y
∂ ˆV x
∂y + ˆV z
∂ ˆV z
∂z
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= −∂P ∂x + µ
∂2ˆV x
∂x2 +∂2ˆV x
∂y2 +∂2ˆV x
∂z2
+ F x (1.68a)
ρ
∂ ˆV y
∂t + ˆV x
∂ ˆV y
∂x + ˆV y
∂ ˆV y
∂y + ˆV z
∂ ˆV y
∂z
= −∂P
∂y + µ
∂2ˆV y
∂x2 +∂2ˆV y
∂y2 +∂2ˆV y
∂z2
+ F y (1.68b)
ρ
∂ ˆV z
∂t + ˆV z
∂ ˆV x
∂x + ˆV y
∂ ˆV z
∂y + ˆV z
∂ ˆV z
∂z
= −∂P
∂z + µ
∂2ˆV z
∂x2 +∂2ˆV z
∂y2 +∂2 ˆV z
∂z2
+ F z (1.68c)
In cylindrical coordinates withF r Fθ, andF ztaken as the components of the body
force per unit volume, the Navier–Stokes equations are
ρ
∂ ˆV r
∂t + ˆV r
∂ ˆV r
∂r +
ˆVθ
r
∂ ˆV r
∂θ −
ˆV2
θ
r + ˆV z
∂ ˆV r
∂z
= −∂P
∂r
+ µ
∂2ˆV r
∂r2 +1r ∂ ˆV ∂r r − ˆV r
r2 +r12∂2ˆV r
∂θ2 −r22 ˆVθ
∂θ+
∂2ˆV r
∂z2
+ F r (1.69a)
ρ
∂ ˆVθ
∂t + ˆV r
∂ ˆVθ
∂r +
ˆVθ
r
∂ ˆVθ
∂θ −
ˆV r ˆVθ
r + ˆV z
∂ ˆVθ
∂z
= −1
r
∂P
∂θ
+ µ
∂2ˆVθ
∂r2 +1r ∂ ˆV ∂rθ− ˆVθ
r2 +r12∂2 ˆVθ
∂θ2 +r22∂ ˆV ∂θ r +∂2 ˆVθ
∂z2
+ Fθ (1.69b)
ρ
∂ ˆV z
∂t + ˆV r
∂ ˆV z
∂r +
ˆVθ
r
∂ ˆV z
∂θ + ˆV z
∂ ˆV z
∂z
= −∂P
dz
+ µ
∂2ˆV z
∂r2 +1
r
∂ ˆV z
∂r +
1
r2
∂2ˆV z
∂θ2 +∂2ˆV z
∂z2
Finally, in spherical coordinates withF r Fθ, andFφ taken as the components of the body force per unit volume and with
D
DT =
∂
dt + ˆV r
∂
∂r +
ˆVφ
r
∂
∂φ +
ˆVθ
r sin φ
∂
∂θ
∇2= 1
r2
∂
∂r
r2 ∂
∂r
+ 1
r2sinφ
∂
∂φ
sinφ ∂
∂φ
+ 1
r2sin2φ
∂2
∂θ2
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the Navier–Stokes equations are
ρ
D ˆV r
Dt −
ˆV2
φ + ˆV2
θ
r
= −∂P ∂r + µ
∇2ˆV r−2 ˆV r
r2 − 2
r2
∂ ˆVφ
∂φ −
2 ˆVθcotφ
r2 − 2
r2sinφ
∂ ˆVθ
∂θ
+ F r (1.70a)
ρ
D ˆVφ
Dt −
ˆV r ˆVφ
r −
ˆV2
θ cotφ
r
= −1
r
∂P
∂φ
+ µ
∇2ˆVφ+r22∂ ˆV ∂φ r − ˆVφ
r2sinφ−
2 cosφ
r2sin2φ
∂ ˆVθ
∂θ
+ Fφ (1.70b)
ρ
D ˆVθ
Dt +
ˆVθˆV r
r +
ˆVφ ˆVθcotφ
r
= − ˆVθ
r sin φ
∂P
∂θ
+ µ
∇2ˆVθ− ˆVθ
r2sin2φ+
2
r2sinφ
∂ ˆV r
∂θ +
2 cosφ
r2sin2φ
∂ ˆVθ
∂θ
+ Fθ (1.70c)
In Fig 1.8, an imaginary two-dimensional control volume of finite size∆x ∆y with
flow velocity ˆV = ex ˆV x+ ey ˆV y, heat flux q= ex q
x+ ey q
y, specific internal energy
u, and rate of internal heat generation q, the first law of thermodynamics requires
that
rate of energy
accumulation within the control volume
=
net transfer
of energy by fluid flow
+
net heat
transfer by conduction
×
rate of
internal heat generation
−
net work transfer from the control volume to the environment
(1.71)
Four of the five terms indicated do not involve work transfer from the control volume
to the environment
• The rate of energy accumulated in the control volume is
∆x ∆y ∂t ∂ (ρu) (1.72a)
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∂
∂
( )ρe
t ⌬ ⌬x y
y
q⬙x⌬y q x⌬ ⌬y
q⬙y ⌬x
q⬙ ⫹x ∂q ∂x x x⬙ ⌬ ⌬y
(
(
)
)
ρV uˆy ⫹∂y V u y ∂ (ρˆy )⌬ ⌬ x
ρV uˆx ⫹∂y V u x ∂ (ρˆx )⌬ ⌬ y
ρ ˆV u y x
ρ ˆV u x y
Figure 1.8 First law of thermodynamics applied to an imaginary control volume in two-dimensional flow
• The net transfer of energy by fluid flow is
− (∆x ∆y)
∂
∂x
ρ ˆV x u+∂y ∂ ρ ˆV y u (1.72b)
• The net heat transfer by conduction is
− (∆x ∆y)
∂q
x
∂x +
∂q
y
∂y
(1.72c)
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• The rate of internal heat generation is
The origin of the term involving the net work transferred from the control volume
to the environment is shown in Fig 1.9, where the normal and tangential stresses are sketched For example, the work done per unit time by the normal stressσxon the left
ρV Vˆ ˆx y⫹∂y V V ∂ (ρˆ ˆx y) ⌬ ⌬y x
∂
∂t (ρˆV x) ⌬ ⌬y x
ρˆ2⫹ ∂( )ρ∂xˆ2 ⌬x⌬y
⫹xy ∂ ∂yxy ⌬ ⌬y x
xy ⌬x
x ⌬y
ρˆV x2⌬y
ˆ ˆ
V V x y ⌬x
⫹x ∂ ∂xx ⌬ ⌬x y
X x y⌬ ⌬
(
(
)
)
y
x
Figure 1.9 Force balance in thex direction of an imaginary control volume in two-dimensional
flow
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side of the∆x ∆y element is negative and equal to the force acting on the boundary
σy, multiplied by the boundary displacement per unit time ˆV x This yields− ˆV xσx∆y.
Similarly, the work transfer associated with normal stresses acting on the right side
of the element is positive and equal to
σx+∂σ ∂x x ∆x
ˆV x+∂ ˆV ∂x x ∆x
∆y
The net work transfer rate due to these two contributions is
σx ∂ ˆV x
∂x + ˆV x
∂σ x
∂x
(∆x ∆y)
Three more work transfer rates can be calculated in the same way by examining the effect of the remaining three stresses,τxyin thex direction and σ yandτyxin the
y direction This gives
(∆x ∆y)
σx ∂ ˆV x
∂x − τxy
∂ ˆV x
∂y + σy
∂ ˆV y
∂y − τyx
∂ ˆV y
∂x
+ (∆x ∆y)
ˆV x ∂σ x
∂x − ˆV x
∂τ xy
∂y + ˆV y
∂σ y
∂y − ˆV x
∂τ yx
∂x
where the eight terms have been separated into two groups It can be shown that the second group reduces to
− ρDt D ˆV2
x + ˆV2
y
2 which represents the change of kinetic energy of the fluid in the control volume This change may be considered negligible relative to the internal energy change,∂(ρu)/∂t,
so that the work transfer becomes
(∆x ∆y)
σx ∂ ˆV x
∂x − τxy
∂ ˆV x
∂y + σy
∂ ˆV y
∂y − τyx
∂ ˆV y
∂x
(1.72e)
The stressesσxandτxycan be related to the flow field via the constitutive relations given by Rohsenow and Choi (1961):
σx = P − 2µ ∂ ˆV x
∂x +
2
3µ
∂ ˆV x
∂x +
∂ ˆV y
∂y
(1.73a) and
τxy = µ
∂ ˆV x
∂y +
∂ ˆV y
∂x
(1.73b)
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The bookkeeping required by eq (1.71) dictates the assembly of eqs (1.72), using eqs (1.73), into
ρDu
Dt + u
Dp
Dt + ρ∇ · ˆV
= −∇ · q+ q− P∇ · ˆV + µΦ (1.74)
whereµ is the dynamic viscosity of the fluid and Φ is the viscous dissipation function,
which is detailed subsequently in rectangular, cylindrical, and spherical coordinates
However, eq (1.53) shows that the term in parentheses on the left-hand side of eq
(1.74) is equal to zero, so that eq (1.74) reduces to
ρDu
Dt = − ∇ · q+ q− P∇ · ˆV + µΦ (1.75)
In the special case where the flow can be modeled as incompressible and two-dimensional, the viscous dissipation function reduces to
Φ = 2
∂ ˆV x
∂x
2
+
∂ ˆV y
∂y
2
+
∂ ˆV x
∂y +
∂ ˆV y
∂x
2
(1.76)
To express eq (1.74) in terms of enthalpy, the definition
h = u + P v = u + P
ρ
is invoked Hence,
Dh
Dt =
Du
Dt +
1
ρ
DP
Dt −
P
ρ2
Dρ
Moreover, the heat fluxesq
x andq
y can be expressed in terms of local temperature
gradients through use of Fourier’s law:
Thus, the combination of eqs (1.74), (1.77), and (1.78) results in
ρDh
Dt = ∇ · (k ∇T ) + q+
DP
Dt + µΦ −
P
ρ
Dρ
Dt + ρ ∇ · ˆV
(1.79)
Here, too, eq (1.53) points out that the terms in parentheses in eq (1.79) are equal to zero, so that eq (1.75) reduces to
ρDh
Dt = ∇ · (k ∇T ) + q+
DP
Bejan (1995) points out that the change in specific enthalpy for a single-phase fluid
is given by
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dh = T ds + v dP = T ds + dP
whereT is the absolute temperature and ds is the specific entrophy change:
ds =
ds dT
P dT +
ds dP
The last of the Maxwell relations given by Bejan (1997)
∂s
∂P
T = −
∂(1/ρ)
∂T
P = ρ12
∂ρ
∂T
P = −βρ (1.83) whereβ is the volumetric coefficient of thermal expansion,
β = −1 ρ
∂ρ
∂T
P
(1.84)
Withc p taken as the specific heat at constant pressures, it can be shown that
ds
dT
P = c p
and eqs (1.81) through (1.85) can be combined,
dh = c p dT + 1
ρ(1 − βT ) dP (1.86)
so that the left-hand side of eq (1.80) can be written as
ρDh
DT = ρc p
DT
Dt + (1 − βT )
DP
Thus, the temperature formulation of the first law of thermodynamics is
ρc p DT
Dt = ∇ · (k ∇T ) + q+ βT
DT
with the special forms for the ideal gas whereβ = 1/T ,
ρc p DT
Dt = ∇ · (k ∇T ) + q+
DP
and for an incompressible fluid whereβ = 0,
ρc p DT
Dt = ∇ · (k ∇T ) + q+ µΦ (1.89b)
Most convection problems concern an even simpler model where the fluid has constant thermal conductivityk, neglible viscous dissipation Φ, zero internal heat
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generation (q = 0), and a negligible compressibility effect, βT (DP /Dt) ≈ 0 The
energy equation for this model is simply
ρc p DT
Dt = k ∇2T (1.90)
or, in the rectangular coordinate system,
ρc p
∂T
∂t + ˆV x
∂T
∂x + ˆV y
∂T
∂y + ˆV z
∂T
∂z
= k
∂2T
∂x2 +∂2T
∂y2 +∂2T
∂z2
(1.91)
in the cylindrical coordinate system,
ρc p
∂T
∂t + ˆV r
∂T
∂r +
ˆVθ
r
∂T
∂θ + ˆV z
∂T
∂z
= k
1
r
∂
∂r
r ∂T ∂r
+r12∂2T
∂θ2 +∂ ∂z2T2
(1.92) and in the spherical coordinate system,
ρc p
∂T
∂t + ˆV r
∂T
∂r +
ˆVφ
r
∂T
∂φ +
ˆVθ
r sin φ
∂T
∂θ
= k
1
r2
∂
∂r
r2∂T
∂r
+r2 1
sinφ
∂
∂φ
sinφ∂T ∂φ
+ 1
sin2φ
∂2T
∂θ2
(1.93)
If the fluid can be modeled as incompressible then, as in eq (1.89b), the specific heat at constant pressure c p is replaced by c And when dealing with extremely
viscous flows, the model is improved by taking into account the internal heating due
to viscous dissipation,
ρc p DT
Dt = k ∇2T + µΦ (1.94)
In the rectangular coordinate system, the viscous dissipation can be expressed as
Φ = 2
∂ ˆV x
∂x
2
+
∂ ˆV y
∂y
2
+
∂ ˆV z
∂z
2
+
∂ ˆV x
∂y +
∂ ˆV y
∂x
2
+
∂ ˆV y
∂z +
∂ ˆV z
∂y
2
+
∂ ˆV z
∂x +
∂ ˆV x
∂z
2
−2
3
∂ ˆV x
∂x +
∂ ˆV y
∂y +
∂ ˆV z
∂z
2
(1.95)