Mathematical Description of the
Physical Phenomena
MEL 807
Computational Heat Transfer (2-0-4)
Dr. Prabal Talukdar
Assistant Professor
Department of Mechanical Engineering
IIT Delhi
Conservation equations
• Nearly all physical processes of interests to us governed
by conservation equations
– Mass, momentum and energy conservation
• Written in terms of specific quantities (per unit mass
basis)
– Momentum per unit mass (velocity)
– Energy per unit mass e
• Consider a specific quantity φ and write conservation
statement for φ for control volume of size ∆x x ∆y x ∆z
Conservation Equations (cont’d)
Accumulation of φ in control volume over time ∆T =
Net influx of φ into control volume
- Net efflux of φ out of control volume
+ Net generation of φ inside control volume
Conservation Equations (cont’d)
Accumulation:
(ρφ∆i ) t + ∆t − (ρφ∆i ) t
Generation:
S ∆i ∆t
Influx and Efflux:
(J x − J x + ∆x )∆y∆z∆t + (J y − J y + ∆y )∆x∆z∆t + (J z − J z + ∆z )∆x∆y∆t
Combining all the terms and dividing by ∆V and ∆t
(ρφ) t + ∆t − (ρφ) t J x − J x + ∆x J y − J y + ∆y J z − J z + ∆z
=
+
+
+S
∆t
∆x
∆y
∆z
Taking limits as ∆x, ∆y, ∆z tends to 0
∂J x ∂J y ∂J z
∂ (ρφ)
=−
−
−
+S
∂t
∂x
∂y
∂z
Diffusive and Convective Fluxes
∂φ
= −Γ
∂x
• Diffusion Flux
J diffusion , x
• Convection Flux
J convection , x = ρuφ
• Net Flux
∂φ ⎞
⎛
J x = ⎜ ρuφ − Γ ⎟
∂x ⎠ x
⎝
J x + ∆x
∂φ ⎞
⎛
= ⎜ ρuφ − Γ ⎟
∂x ⎠ x + ∆x
⎝
General Scalar Transport Equation
∂
∂
∂
∂
(ρφ) +
(ρuφ) + (ρvφ) + (ρwφ)
∂t
∂x
∂y
∂z
∂ ⎛ ∂φ ⎞ ∂ ⎛ ∂φ ⎞ ∂ ⎛ ∂φ ⎞
=
⎜ Γ ⎟ + ⎜⎜ Γ ⎟⎟ + ⎜ Γ ⎟ + S
∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠
Or, in vector form:
∂ (ρφ)
+ ∇ ⋅ ρVφ = ∇ ⋅ (Γ∇φ) + S
∂t
General Scalar Transport Equation
∂ (ρφ)
+ ∇ ⋅ ρVφ = ∇ ⋅ (Γ∇φ) + S
∂t
Storage
φ
Γ
ρ
S
Convection
Diffusion
Generation
specific quantity (say energy per unit mass)
diffusion coefficient
density
source term (Generation per unit volume W/m3)
Continuity Equation
∂ (ρφ)
+ ∇ ⋅ ρVφ = ∇ ⋅ (Γ∇φ) + S
∂t
With φ = 1, Γ = 0 and S = 0
∂ρ
+ ∇.(ρV) = 0
∂t
Energy Equation
(∂ρh )
+ ∇ ⋅ (ρVh ) = ∇ ⋅ (k∇T ) + S h
∂t
h sensible enthalpy per unit mass, J/kg
K thermal conductivity
Sh energy generation W/m3
How to cast in the form of general transport equation?
φ
Γ
Equation of state
dh = C p dT
Substitute to find
⎛ k
⎞
∂ρh
+ ∇ ⋅ (ρVh ) = ∇ ⋅ ⎜
∇h ⎟ + S h
⎟
⎜ Cp
∂t
⎝
⎠
S
Momentum Equation
• X-momentum equation
φ
Γ
S
∂ρu
∂p
+ ∇ ⋅ (ρVu ) = ∇ ⋅ (µ∇u ) −
+ Su
∂t
∂x
Species Transport Equation
∂ρYi
+ ∇ ⋅ (ρVYi ) = ∇ ⋅ (Γi ∇Yi ) + R i
∂t
Yi
kg of species/kg of mixture
Γi
diffusion coefficient of i in mixture
Ri
Reaction source
Different Forms of Continuity Eq.
Different Forms of Continuity Eq.
Classifications of PDE
• Consider the second order partial differential equation
for φ(x,y):
aφ xx + bφ xy + cφ yy + dφ x + eφ y + fφ + g = 0
Coefficients a,b,c,d,e,f and g are linear- not functions of φ,
but can be functions of x,y
Discriminant:
W<0
W =0
W >0
W = b2-4ac
Elliptic PDE (No real characteristics)
Parabolic PDE (one real characteristic)
Hyperbolic PDE (Two real characteristics)
Elliptic PDEs
•
•
•
Protoype is Laplace equation
Irrotational flow of an incompressible fluid
Steady state heat conduction problem
Consider 1-D heat conduction in a plane wall
∂ ⎛ ∂T ⎞
⎜k
⎟=0
∂x ⎝ ∂x ⎠
To
Boundary conditions:
T(0) = T0
Solution:
T(L) = TL
(TL − T0 )
T( x ) = T0 +
x
L
TL
x
L
Elliptic PDEs (cont’d)
(TL − T0 )
T ( x ) = T0 +
x
L
• T(x) is influenced by both
the boundaries
• In the absence of source term,
T(x) is bounded by the values on
both the boundaries
Parabolic PDEs
• Describes time dependent problems
which involves significant amount of
dissipation
• Unsteady viscous flows, unsteady
heat conduction
To
Ti
• Consider 1-D unsteady conduction
∂T
∂ T
=α 2
∂t
∂x
2
Initial condition
Boundary conditions
x
L
T(x,0) = Ti
T(0,t) = To T(L,t) = To
To
Parabolic PDEs (cont’d)
• The
solution at T(x,t) is influenced by the
boundaries, just as Elliptic PDEs.
• There is only one characteristic
direction through point P(x,t)
• We need only initial condition T(x,0).
We don’t need any future condition.
• Initial condition affect only future
condition, not the past conditions.
•Marching solutions are possible.
Region influenced
by P
Parabolic PDEs (cont’d)
• The variation in t admits only one way
influences, whereas the variable x admits twoway influences.
• Spatial variables also may behave in this way,
for example, the axial direction in a pipe flow
Hyperbolic PDEs
• Hyperbolic PDEs have a special behavior which is
associated with finite speed at which information travels thro’
the medium
• They appear in time-dependent processes with negligible
amount of dissipation
• Dominates the analysis of vibration problem
• The prototype hyperbolic equations is the wave equation
2
∂ 2φ
∂
φ
2
=c
2
∂t
∂x 2
With initial conditions
φ( x ,0) = f ( x )
and
∂φ
( x,0) = g ( x )
∂t
Hyperbolic PDEs (cont’d)
Solution:
1
1 x + ct
φ( x , t ) = [f ( x − ct ) + f ( x + ct )] +
∫ g (s)ds
2
2c x − ct
Relationship to scalar transport
equation
∂ (ρφ)
+ ∇ ⋅ ρVφ = ∇ ⋅ (Γ∇φ) + S
∂t
• Contains all three canonical PDE terms
• If Re is low and situation is steady we get an
elliptic equation
• If diffusion coefficient is zero, we get a hyperbolic
equation
• If Re is low and situation is unsteady, we get a
parabolic equation
• For mixed regime, we get mixed behavior