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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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Convective heat transfer

Heat Transfer

Fluid Dynamics

Forced Convection

Free Convection

Mixed Convection

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5[4]



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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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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


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


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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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]


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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.


(((( )))) (((( ))))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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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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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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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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Conservation Equations

Mass Conservation:

For a control volume

CV

inlet outletports ports

Mm m

t

∂ = −∂

∑ ∑ɺ ɺ

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( ) ( ) ( ) ( )( ) ( )

( )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




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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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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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




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




Euler equation:

vg

Dp

Dt

ρ ρ= − +∇∇∇∇

Navier-Stokes equation: vg v

Dp

Dt

ρ ρ µ= − + + 2222∇ ∇∇ ∇∇ ∇∇ ∇

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Energy Conservation:

For a volume element we write a momentum balance in this form:


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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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



+∆+∆+∆+∆

+∆+∆+∆+∆

+∆+∆+∆+∆

∆ ∆ + − + +∆ ∆ + − + +∆ ∆ + − + +∆ ∆ + − + +

∆ ∆ + − + +∆ ∆ + − + +∆ ∆ + − + +∆ ∆ + − + +

∆ ∆ + − +∆ ∆ + − +∆ ∆ + − +∆ ∆ + − + 29




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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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



+∆+∆+∆+∆

+∆+∆+∆+∆

+∆+∆+∆+∆

∆ ∆ + + − + + +∆ ∆ + + − + + +∆ ∆ + + − + + +∆ ∆ + + − + + +

∆ ∆ + + − + + +∆ ∆ + + − + + +∆ ∆ + + − + + +∆ ∆ + + − + + +

∆ ∆ + + − + +∆ ∆ + + − + +∆ ∆ + + − + +∆ ∆ + + − + +

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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(((( ))))

(((( )))) (((( ))))

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




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34


Continuity Equation:



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After simplification:

For Newtonian fluids with constant density and thermal conductivity:

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37




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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

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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 + ∆+ ∆+ ∆+ ∆

∆∆∆∆ ∫∫∫∫

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(((( ))))(((( ))))

(((( ))))(((( ))))

(((( ))))(((( ))))

(((( ))))(((( ))))

(((( ))))(((( )))) (((( ))))

(((( )))) (((( ))))(((( ))))

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


+∆+∆+∆+∆

∆ →∆ →∆ →∆ → +∆ −+∆ −+∆ −+∆ −

∆ →∆ →∆ →∆ →

+ ∆ − + ∆ ++ ∆ − + ∆ ++ ∆ − + ∆ ++ ∆ − + ∆ + ∆∆∆∆ = + ∆ += + ∆ += + ∆ += + ∆ + ∆∆∆∆ + ∆ −+ ∆ −+ ∆ −+ ∆ − ∆∆∆∆

+ ∆ −+ ∆ −+ ∆ −+ ∆ − ∆∆∆∆

∫ ∫∫ ∫∫ ∫∫ ∫

∫∫∫∫

∫ ∫∫ ∫∫ ∫∫ ∫

∫∫∫∫ (((( ))))(((( )))) (((( ))))

(((( ))))

(((( ))))(((( )))) (((( ))))

(((( ))))(((( ))))

(((( ))))

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


+∆ −+∆ −+∆ −+∆ −

+∆ −+∆ −+∆ −+∆ −

∂∂∂∂= + ∆ += + ∆ += + ∆ += + ∆ +∆ ∂∆ ∂∆ ∂∆ ∂

= ∆= ∆= ∆= ∆

+ ∆ = + ∆ ∆+ ∆ = + ∆ ∆+ ∆ = + ∆ ∆+ ∆ = + ∆ ∆

∫ ∫∫ ∫∫ ∫∫ ∫

∫ ∫∫ ∫∫ ∫∫ ∫

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( ) ( )

( )

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.


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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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