Post on 02-Jul-2018
Convection Heat Transfer Department of Chemical Eng., Isfahan University of Technology,
Isfahan, Iran
Seyed Gholamreza Etemad
Winter 2013
Heat convection:
Difference between the temperature of the media and the fluid
Energy transfer from a media to a fluid over it.
Examples: Convective heat transfer occurs extensively in practice.
IntroductionIntroduction
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S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Convection Heat TransferConvection Heat Transfer
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Convective heat transfer
Heat Transfer
Fluid Dynamics
Forced Convection
Free Convection
Mixed Convection
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IntroductionIntroduction
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Convection Heat TransferConvection Heat Transfer
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Question: From a conceptual viewpoint, is the convection heat transfer a basic mode of heat transfer?
Several factors play major roles in convection heat transfer:
(i)fluid motion(ii)fluid nature and properties(iii)surface geometry(iv)boundary conditions
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IntroductionIntroduction
Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Focal Point in convection heat transfer the determination of the temperature distribution in a moving fluid
T =T(x, y, z, t)Convection heat transfer depends on material properties such as density, pressure, thermal conductivity, and specific heat.In the continuum model the characteristics of individual molecules are ignored and average or macroscopic properties become important. A continuum is a media composed of continuous matter. It is valid for sufficiently large number of molecules in a given volume.
Knudson number, Kn
- Molecular Mean Free Path - the characteristic length, such as the equivalent diameter or the spacing between parallel plates. 8
The Continuum and Thermodynamic Equilibrium ConceptsConvection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Continuum is valid for:
Is continuum valid for micro and/or nano-channels?
Thermodynamic equilibrium: fluid and the adjacent surface have the same velocity and temperature, no-velocity slip and no-temperature jump.
The condition for thermodynamic equilibrium:
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The Continuum and Thermodynamic Equilibrium ConceptsConvection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Eulerian and Lagrangian Approaches
There are two different points of view in analyzing problems in mechanics.
1- Eulerian method of description
2- Lagrangian method of description
The Eulerian view, appropriate to fluid mechanics, is to specify the fluid properties (e.g. density, velocity) at each point in space at each instant of time. The density, for example, is then specified by a function:
In the Euler picture, attention is focused on what is happening at a particular point in space, rather than on a particular fluid element.
(((( ))))x, y ,z ,tρ ρρ ρρ ρρ ρ====
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Examples:
--- When a pressure probe is introduced into a laboratory flow
--- Analysis of traffic flow along a freeway
--- Standing on a bridge and recording the variation of fish concentration below the bridge.
The Lagrangian approach, more appropriate for solid mechanics, follows an individual particle moving through the flow. Suppose we have a fluid element that is at position (xo, yo, zo) at time to. At later times, the position of this element is described by functions:
x=x(xo, yo, zo, t) , y=y(xo, yo, zo, t) , z=z(xo, yo, zo, t)
Therefore, any field variable is given as: V=f[x(t), y(t), z(t), t]
Eulerian and Lagrangian Approaches
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
These two descriptions are equivalent and there are relationships between the Lagrangian and Eulerian equations of fluid motion.
Eulerian description, the time derivative is the partial derivative with respect to t keeping x, y, and z fixed.
In the Lagrangian method, the time derivative is the total derivative:
Where V=(Vx, Vy, Vz) is the velocity of the fluid element.
Eulerian and Lagrangian Approaches
(((( )))) (((( ))))t 0
x, y ,z ,t t x , y ,z ,t
t tlim
ρ ρρ ρρ ρρ ρρρρρ∆ →∆ →∆ →∆ →
+ ∆ −+ ∆ −+ ∆ −+ ∆ −∂∂∂∂ ====∂ ∆∂ ∆∂ ∆∂ ∆
(((( )))) (((( ))))t 0
x y z
x x, y y,z z ,t t x , y ,z ,tddt td dx dy dz
V V Vdt t x dt y dt z dt t x y z
limρ ρρ ρρ ρρ ρρρρρ
ρ ρ ρ ρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρ ρ ρ ρ∆ →∆ →∆ →∆ →
+ ∆ + ∆ + ∆ + ∆ −+ ∆ + ∆ + ∆ + ∆ −+ ∆ + ∆ + ∆ + ∆ −+ ∆ + ∆ + ∆ + ∆ −= == == == =
∆∆∆∆∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + = + + += + + + = + + += + + + = + + += + + + = + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Eulerian and Lagrangian Approaches
The relationship between the derivatives for any field variables (A) is:
The operator d/dt is sometimes given a special name such as substantial derivative or material derivative and often assigned a special symbol such as D/Dt.
Systems and Control Volumes:System is defined as an arbitrary quantity of mass of fixed identity.
The Lagrangian method of fluid mechanics is used in the mathematical description of a system.
(((( ))))x y z
dA A A A A AV V V V. A
dt t x y z t∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂= + + + = + ∇= + + + = + ∇= + + + = + ∇= + + + = + ∇∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Eulerian and Lagrangian Approaches
At a system, neglecting nuclear reactions, the quantity of mass is fixed. Thus the mass of the system is conserved and does not change.
---If the surroundings exert a net force F on the system, Newton’s second law states that the mass will began to accelerate.
In fluid mechanics Newton’s law is called the conservation of linear momentum or alternately, the momentum principle.
---If heat Q is added to the system or work dW is done by the system, the system energy dE must change according to the energy relation, or the first law of thermodynamics.
(((( ))))dV dF ma m mV
dt dt= = == = == = == = =
dQ dW dEdQ dW dE ,
dt dt dt− = − =− = − =− = − =− = − =
syst
dmm const
dt= == == == = , 0
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Eulerian and Lagrangian Approaches
Control Volume is the same as a system, except that the rest of the continuum may cross the fixed or deformable boundaries of the control volume at one or more places. This is the only difference between a control volume and a system. The Eulerian method of fluid mechanics is used in the mathematical description of a control volume.
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
Mass Conservation:
For a control volume
CV
inlet outletports ports
Mm m
t
∂ = −∂
∑ ∑ɺ ɺ
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
( ) ( ) ( ) ( )( ) ( )
( )By dividing both sides of the equation by and taking the limit
as these dimensions approach zero, we get:
x x y yx x x y y y
z zz z z
x
x y z y z v v x z v vt
x y v v
x y z
v
t x
ρ ρ ρ ρ ρ
ρ ρ
ρρρ
+∆ +∆
+∆
∂ ∆ ∆ ∆ = ∆ ∆ − + ∆ ∆ − + ∂ ∆ ∆ −
∆ ∆ ∆
∂∂∂ = − +∂ ∂
( ) ( )
( )
This equation is called the continuity equation.
We may write the continuity equation in vector form:
v , v
For a fluid of constant density:
v
y zv v
y z
D. .
t Dt
. 0
ρ
ρ ρρ ρ
∂+ ∂ ∂
∂ = − = −∂
=
∇ ∇∇ ∇∇ ∇∇ ∇
∇∇∇∇17
Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
Momentum Conservation:
For a volume element we write a momentum balance in this form:x y z∆ ∆ ∆
rate of rate of rate of sum of forces
momentum momentum momentum acting on
accumulation in out system
= − +
( )n CVn n n
inlet outletports ports
MvF mv mv
t
∂= + −
∂∑ ∑ ∑ɺ ɺ
Where n is the direction chosen for analysis and (vn, Fn) are the projections of fluid velocity and forces on the n direction. This equation is the control volume formulation of Newton’s second law of motion and is recognized in the literature as the momentum principle.
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
The convective flow of x-momentum must be considered across all six faces and that the net convective x-momentum flow into the volume element is:
( ) ( )( )
x x x x y x y xx x x y y y
z x z xz z z
y z v v v v x z v v v v
x y v v v v
ρ ρ ρ ρ
ρ ρ+∆ +∆
+∆
∆ ∆ − + ∆ ∆ − +
∆ ∆ −
The x-momentum by molecular transport:
( ) ( ) ( )xx xx yx yx zx zxx x z z zx y y yy z x z x yτ τ τ τ τ τ+∆ +∆+∆
∆ ∆ − + ∆ ∆ − + ∆ ∆ −
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
The forces related to pressure and gravity in x-direction will be:
( ) xx x xy z p p g x y zρ
+∆∆ ∆ − + ∆ ∆ ∆
By dividing the entire resulting equation by and taking the limit as
approach zero, we obtain the x-component of the equation of motion:
xvx y z
t
ρ∂ ∆ ∆ ∆ ∂
The rate of accumulation of x-momentum within the element is:
x y z∆ ∆ ∆
x, y, and z∆ ∆ ∆
y x yxx x x z x xx zxx
v vv v v v v pg
t x y z x y z x
ρ τρ ρ ρ τ τ ρ∂ ∂∂ ∂ ∂ ∂ ∂ ∂= − + + + + + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
The y- and z-components are as following:
y x y y y z y xy yy zyy
v v v v v v v pg
t x y z x y z y
ρ ρ ρ ρ τ τ τρ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= − + + + + + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 21
Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
y z yzx z xzz z z zzz
v vv vv v v pg
t x y z x y z z
ρ τρ τρ ρ τ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= − + + + + + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
Where:
( ) ( )vvv g. . p
t
ρ ρ ρ∂ = − + − +∂
∇ ∇ τ ∇∇ ∇ τ ∇∇ ∇ τ ∇∇ ∇ τ ∇
( )vg
D. p
Dt
ρ ρ= − +∇ τ ∇∇ τ ∇∇ τ ∇∇ τ ∇
It is convenient to combine them to give the single vector equation:
, ,
,
yx zxx yy zz
yx xzxy yx xz zx
y zyz zy
vv v2 2 2
x y z
vv vv
y x x z
v v
z y
τ µ τ µ τ µ
τ τ µ τ τ µ
τ τ µ
∂∂ ∂= = =∂ ∂ ∂
∂ ∂ ∂∂ = = + = = + ∂ ∂ ∂ ∂
∂ ∂= = + ∂ ∂ 22
Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
Euler equation:
vg
Dp
Dt
ρ ρ= − +∇∇∇∇
Navier-Stokes equation: vg v
Dp
Dt
ρ ρ µ= − + + 2222∇ ∇∇ ∇∇ ∇∇ ∇
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Energy Conservation:
For a volume element we write a momentum balance in this form:
Conservation Equations
x y z∆ ∆ ∆
rate of rate of rate of
accumulation internal and internal and
of internal and kinetic energy kinetic energy
kinetic energy in by convection out by convection
net rate of
heat addition
= − +
net rate of
work done by
system on by conduction
surroundings
+
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
The rate of accumulation of internal and kinetic energy within is:x y z∆ ∆ ∆
21e u
2x y zt
ρ ρρ ρρ ρρ ρ ∂ +∂ +∂ +∂ + ∆ ∆ ∆∆ ∆ ∆∆ ∆ ∆∆ ∆ ∆
∂∂∂∂Where e is the internal energy per unit mass of the fluid and u is the magnitude of the local velocity.
The net rate of convection of internal and kinetic energy into the element is:
2 2x x
x x x
2 2y y
y y y
2 2z z
z z z
1 1y z u e u u e u
2 2
1 1x z u e u u e u
2 2
1 1y x u e u u e u
2 2
ρ ρ ρ ρρ ρ ρ ρρ ρ ρ ρρ ρ ρ ρ
ρ ρ ρ ρρ ρ ρ ρρ ρ ρ ρρ ρ ρ ρ
ρ ρ ρ ρρ ρ ρ ρρ ρ ρ ρρ ρ ρ ρ
+∆+∆+∆+∆
+∆+∆+∆+∆
+∆+∆+∆+∆
∆ ∆ + − + +∆ ∆ + − + +∆ ∆ + − + +∆ ∆ + − + +
∆ ∆ + − + +∆ ∆ + − + +∆ ∆ + − + +∆ ∆ + − + +
∆ ∆ + − +∆ ∆ + − +∆ ∆ + − +∆ ∆ + − + 29
Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
The net rate of energy input by conduction is:
{{{{ }}}} {{{{ }}}} {{{{ }}}}x x y y z zx x x z z zy y yy z q q x z q q y x q q
+∆+∆+∆+∆ +∆+∆+∆+∆+∆+∆+∆+∆∆ ∆ − + ∆ ∆ − + ∆ ∆ −∆ ∆ − + ∆ ∆ − + ∆ ∆ −∆ ∆ − + ∆ ∆ − + ∆ ∆ −∆ ∆ − + ∆ ∆ − + ∆ ∆ −
Where qx, qy, qz are the x, y, and z components of the heat flux vector q.
The work done by the fluid element against its surroundings consists of two parts:
--- the work against the volume forces (body forces) e.g. gravity
--- the work against the surface forces i.e. pressure and viscous forces
Work = (Force) (Distance in the direction of the force)
Rate of doing work = (Force) (Velocity in the direction of the force)
The rate of doing work against the gravitational force per unit mass is:
(((( ))))x x y y z zx y z u g u g u gρρρρ− ∆ ∆ ∆ + +− ∆ ∆ ∆ + +− ∆ ∆ ∆ + +− ∆ ∆ ∆ + +
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
The rate of doing work against the pressure at different faces is:
(((( )))) (((( )))){{{{ }}}} (((( )))) (((( )))){{{{ }}}}(((( )))) (((( )))){{{{ }}}}
x x y yx x x y y y
z zz z z
y z pu pu x z pu pu
y x pu pu
+∆+∆+∆+∆ +∆+∆+∆+∆
+∆+∆+∆+∆
∆ ∆ − + ∆ ∆ − +∆ ∆ − + ∆ ∆ − +∆ ∆ − + ∆ ∆ − +∆ ∆ − + ∆ ∆ − +
∆ ∆ −∆ ∆ −∆ ∆ −∆ ∆ −
The rate of doing work against the viscous forces is:
(((( )))) (((( )))){{{{ }}}}(((( )))) (((( )))){{{{ }}}}(((( )))) (((( )))){{{{ }}}}
xx x xy y xz z xx x xy y xz zx x x
yx x yy y yz z yx x yy y yz zy y y
zx x zy y zz z zx x zy y zz zz z z
y z u u u u u u
x z u u u u u u
x y u u u u u u
τ τ τ τ τ ττ τ τ τ τ ττ τ τ τ τ ττ τ τ τ τ τ
τ τ τ τ τ ττ τ τ τ τ ττ τ τ τ τ ττ τ τ τ τ τ
τ τ τ τ τ ττ τ τ τ τ ττ τ τ τ τ ττ τ τ τ τ τ
+∆+∆+∆+∆
+∆+∆+∆+∆
+∆+∆+∆+∆
∆ ∆ + + − + + +∆ ∆ + + − + + +∆ ∆ + + − + + +∆ ∆ + + − + + +
∆ ∆ + + − + + +∆ ∆ + + − + + +∆ ∆ + + − + + +∆ ∆ + + − + + +
∆ ∆ + + − + +∆ ∆ + + − + +∆ ∆ + + − + +∆ ∆ + + − + +
By substituting the foregoing expressions into main energy equation and Dividing the entire equation by while the dimensions approach zeroThe energy equation is obtained:
x y z∆ ∆ ∆
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
(((( ))))
(((( )))) (((( ))))
2
2 2 2x y z
y yz zx xx x y y z z
xx x xy y xz z yx x yy y yz z zx
1e u
t 2
1 1 1u e u u e u u e u
x 2 y 2 z 2
q puq puq puu g u g u g
x y z x y z
u u u u u ux y z
ρ ρρ ρρ ρρ ρ
ρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρ
ρρρρ
τ τ τ τ τ τ ττ τ τ τ τ τ ττ τ τ τ τ τ ττ τ τ τ τ τ τ
∂∂∂∂ + =+ =+ =+ = ∂∂∂∂
∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ − + + + + + −− + + + + + −− + + + + + −− + + + + + − ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂
∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂ ∂ ∂∂ ∂∂ ∂∂ ∂+ + + + + − + + ++ + + + + − + + ++ + + + + − + + ++ + + + + − + + + ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂+ + + + + ++ + + + + ++ + + + + ++ + + + + +∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ (((( ))))x zy y zz zu u uτ ττ ττ ττ τ + ++ ++ ++ +
In vector-tensor notation:
(((( )))) (((( ))))
(((( )))) (((( ))))(((( ))))
2 21 1e u . u e u .q u.g
t 2 2
.pu . .u
ρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρ∂∂∂∂ + = − ∇ + − ∇ − −+ = − ∇ + − ∇ − −+ = − ∇ + − ∇ − −+ = − ∇ + − ∇ − − ∂∂∂∂
∇ − ∇ τ∇ − ∇ τ∇ − ∇ τ∇ − ∇ τ32
Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
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Convection Heat TransferConvection Heat Transfer
Continuity Equation:
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Conservation Equations
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
After simplification:
For Newtonian fluids with constant density and thermal conductivity:
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
For Newtonian fluids with constant density and thermal conductivity:
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
For Newtonian fluids with constant density and thermal conductivity:
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Reynolds Transport Theorem (R.T.T.)
In order to convert a system analysis into a control volume analysis we must convert our mathematics to apply a specific region rather than to individual masses. This conversion is called Reynolds Transport Theorem and can be applied to all the basic laws.
The next figure presents the system and control volume. At time t the volume is occupied by system is identical to the control volume. At time t+ t system moves to another location and the system and control volume possess different volumes.
Now, consider an arbitrary flow field . We want to calculate the following integral:
∆∆∆∆
(((( ))))(((( ))))
(((( ))))(((( ))))
(((( ))))(((( ))))
(((( ))))
s
s s s
V t
t 0V t V t t V t
d x , y ,z ,t dV
dt
The above equation can be written in the following form:
d 1 dV lim t t dV t dV
dt t
αααα
α α αα α αα α αα α α∆ →∆ →∆ →∆ → +∆+∆+∆+∆
= + ∆ −= + ∆ −= + ∆ −= + ∆ − ∆∆∆∆
∫∫∫∫
∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫
( )x, y,z,tα
39
Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Reynolds Transport Theorem
(((( ))))(((( ))))sV t
In the right hand side of the equation we add and subtract the following term:
1 t t dV
tαααα + ∆+ ∆+ ∆+ ∆
∆∆∆∆ ∫∫∫∫
40
Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Reynolds Transport Theorem
(((( ))))(((( ))))
(((( ))))(((( ))))
(((( ))))(((( ))))
(((( ))))(((( ))))
(((( ))))(((( )))) (((( ))))
(((( )))) (((( ))))(((( ))))
s s
s s
s s
s
V t t V t
t 0 V t t V t
V t V t
t 0V t
The right hand side of the equation is as follows:
1t t dV t t dV
t 1 lim t t dV
t1t t dV t dV
t
1lim t t t dV
t
α αα αα αα ααααα
α αα αα αα α
α αα αα αα α
+∆+∆+∆+∆
∆ →∆ →∆ →∆ → +∆ −+∆ −+∆ −+∆ −
∆ →∆ →∆ →∆ →
+ ∆ − + ∆ ++ ∆ − + ∆ ++ ∆ − + ∆ ++ ∆ − + ∆ + ∆∆∆∆ = + ∆ += + ∆ += + ∆ += + ∆ + ∆∆∆∆ + ∆ −+ ∆ −+ ∆ −+ ∆ − ∆∆∆∆
+ ∆ −+ ∆ −+ ∆ −+ ∆ − ∆∆∆∆
∫ ∫∫ ∫∫ ∫∫ ∫
∫∫∫∫
∫ ∫∫ ∫∫ ∫∫ ∫
∫∫∫∫ (((( ))))(((( )))) (((( ))))
(((( ))))
(((( ))))(((( )))) (((( ))))
(((( ))))(((( ))))
(((( ))))
s s C
s s s
V t t V t V
V t t V t A t
1t t dV dV
t t
dV n.u t dA
t t dV t t n.u t dA
αααααααα
α αα αα αα α
+∆ −+∆ −+∆ −+∆ −
+∆ −+∆ −+∆ −+∆ −
∂∂∂∂= + ∆ += + ∆ += + ∆ += + ∆ +∆ ∂∆ ∂∆ ∂∆ ∂
= ∆= ∆= ∆= ∆
+ ∆ = + ∆ ∆+ ∆ = + ∆ ∆+ ∆ = + ∆ ∆+ ∆ = + ∆ ∆
∫ ∫∫ ∫∫ ∫∫ ∫
∫ ∫∫ ∫∫ ∫∫ ∫
41
Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
( ) ( )
( )
Gauss-Ostrogradskii Divergence Theorem :
If V is a closed region in space surrounded by a surface A, then:
u n u
Using this theory the final form of the equation is as following:
∇. .∇. .∇. .∇. .
s
V A
V
dV dA
dt dV
dt
α α
α
=∫ ∫
( )( )u
This equation is called Reynolds Transport Theorem .
This equation states that the rate of increase of a m aterial quantity
is equal to the rate of increase of that quantity in thos
∇.∇.∇.∇.Ct V
dVt
α α∂ = + ∂ ∫ ∫
e particles inside
fixed control volum e plus the net flux of the quantity through the
boundaries of the control volum e.
Reynolds Transport Theorem
42
Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013
Reynolds Transport Theorem
The conservation equations can be derived using the R.T.T.
by substitution of with appropriate param eters as following.
For m ass conservation equation =
For m om entum conservation
αα ρ
equation = u
For m om entum conservation equation = 21e u
2
α ρ
α ρ +
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Convection Heat TransferConvection Heat Transfer
S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013S. Gh. Etemad, Dept. of Chem. Eng., Isfahan University of Technology, Winter 2013