class ii - conservation equations

8/8/2019 Class II - Conservation Equations

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Chapter 3

Conservation equations for Mass,

Momentum & Energy


2/16

CONVECTION: Heat transfer process that occurs between a

solid surface and a fluid medium when they are at different

temperatures and have a relative motion between them.

W/m2 (Newtons law of cooling)

h influenced by thermo physical properties of fluid, flow velocity

and surface geometry.

Value varies from point to point as the properties vary with

temperature & location.

Local heat transfer coefficient & Average heat transfer coefficient

Determination of the value ofh is difficult (but critical)

Recap.

)( g!! TThA

Qq sconv


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Mechanism of Convection:

By pure conduction at the surface or boundary

Macroscopic fluid motion in the rest of the region

Macroscopic fluid movement enhances heat transfer, since it brings

cooler chunks of fluid into surface contact continuously, initiatinghigher rates of heat transfer

Recap.

0!x

x!

yy

Tkq

g

!

x

x

!TT

y

Tk

hs

y 0

Value of are known

Value of needs to be estimated

gTTk s

0!x

x

yy

T


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Boundary layer -proposed by Prandtl in 1904

Types:

Hydrodynamic or Velocity boundary layer

defined as that distance from the boundary in which the velocityreaches 0 to 99 % of the free stream velocity

Thermal boundary layer

defined as the distance from the boundary in which the temperaturedifference varies from 0 to 99 % of the initial temperature difference

Flow types Laminar & Turbulent flowcharacterized by Reynolds's number

Recap.

Q

V

K

lxxU

forceviscous

forceInertial!!!Re

For a flow over flat plate:

Re < 5 x 105 - Laminar flow

Re > 5 x 105 - Turbulent flow


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Boundary Layer ConceptPrandtl Number (Pr): defined as the ratio of the momentum diffusivity to

the thermal diffusivity.

E

K!!!

k

c

ydiffusivitThermal

ydiffusivitMomentum pPr

Prandtl Number (physically) is the ratio of kinematic viscosity () to the thermal

diffusivity ()

Kinematic viscosity indicates the impulse transport through molecular friction

whereas thermal diffusivity indicates the heat energy transport by conduction

process

Significance:provides a measure of relative effectiveness of the momentum and energy

transport by diffusion

connecting link between the velocity field and temperature field and its value

strongly influences relative growth of velocity and thermal boundary layers.


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Main purpose of convective heat transfer analysis is to

determine:

- heat transfer coefficient, h

How to solve a convection problem ?

Solve governing equations along with boundary conditions Governing equations include

1. conservation of mass

2. conservation of momentum

3. conservation of energy

Solving all these equations is a tiresome task.

Steady, two dimensional incompressible flow of constant property

Convection Analysis


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Convection Equations

Consider the parallel flow of a fluid over a surface

Assumptions:

laminar flow,

steady two-dimensional flow

Newtonian fluid

constant properties

The fluid flows over the surface with a uniform free-stream velocity V, but the

velocity within boundary layer is two-dimensional (u=u(x,y), v=v(x,y)).

Three fundamental laws

1. conservation of mass - continuity equation

2. conservation of momentum - momentum equation

3. conservation of energy - energy equation


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Conservation of mass principle the mass can not be created or destroyed during aprocess.

In a steady flow

Rate of mass flow into control volume = Rate of mass flow out of control volume

The mass flow rate is equal to: uA

Considering unit thickness,

x directionFluid enters the control volume from the left surface at a rate of u(dy.1)

Fluid leaves the control volume from the right surface at a rate of (dy.1)

Continuity Equation

x

x dx

x

uu


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Substituting the results in the conservation equation, we get

Simplifying and dividing by (dx dy), we get

1.1.1.1. dxdyy

vvdydx

x

uudxvdyu

x

x

x

x! VVVV

Continuity Equation

0!x

x

x

x

y

v

x

u (Continuity Equation in cartesian system)

Repeating the same procedure in y-direction, we get

0!x

x

x

x

z

v

r

v

r

vzrr (Continuity Equation in cylindrical system)


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Momentum Equation

The differential forms of the equations of motion in the velocity boundarylayer are obtained by applyingNewtons second law of motion to a differential

control volume element in the boundary layer.

Two type of forces:

body forces and surface forces.

Newtons second law of motion for a control volume is given by

Mass x (Acceleration in the specified direction)

= Net force (body & surface) acting in that direction

i.e

where the mass of the fluid element within the control volume is

dm = (dx.dy.1)

)()()(. xsurfacexbodyx FFa !x


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dyy

udx

x

udu

x

x

x

x!

Momentum EquationThe flow is steady and two-dimensional and thus u=u( x, y), the total

differential ofu is

Then the acceleration of the fluid element in the x direction becomes

The forces acting on a surface are due to pressure & viscous effects and areproportional to the area

Viscousstresscan be resolved into two perpendicular components Normalstress

Shearstress

Normal stress should not be confused with pressure

yuv

xuu

dt

dy

yu

dt

dx

xu

dt

duax

xx

xx!

xx

xx!!


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Momentum Equation

Neglecting the normal stresses, the netsurface force is given as

)1..(

)1..()1.()1.(

2

2

)(

dydxx

P

y

u

dydxx

P

ydydx

x

Pdxdy

yF xsurface

x

x

x

x!

x

x

x

x!

x

x

x

x!

Q

XX

The body forces is external force (like gravity) acting on the fluid particle andis proportional to the volume

If Bx is the body force per unit volume in the x direction, then body force in x-

direction is Bx.(dx.dy.1)

Substituting for mass, acceleration, surface force & body force (for x-direction) in Newton's law of motion, we get


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)1.()1..()1..( 2

2

dyxP

yudydxB

yuv

xuudydx x

xx

xx!

xx

xx QV

2

2

y

u

x

PB

y

uv

x

uu x

x

x

x

x!

x

x

x

xQV

Momentum Equation

Dividing by (dx dy)

x direction

2

2

x

u

y

PB

y

vv

x

vu y

x

x

x

x!

x

x

x

xQV y direction

The above sets of equation are known as Navier stokes equations for asteady, two dimensional flow of an incompressible, constant property

fluid

The above 2 equations along with continuity equation presents a set of 3equations to solve the three unknowns u, v and p.


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t

Tc

y

T

x

Tk p

x

x!

x

x

x

xV

2

2

2

2

x

x

x

x!

x

x

x

x

y

Tv

x

Tuc

y

T

x

Tk pV2

2

2

2

Energy EquationThe concept of conservation of energy has been already discussed in the

conduction chapter

We know that, for a two dimensional flow (without heat generation), theequation is given as

x

x

x

x!

x

x

y

Tv

x

Tu

t

T

But, the total time derivative of temperature is given as

Hence the energy balance equation is given as


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*

*

x

x

x

x!

x

x

x

xQV y

T

vx

T

ucy

T

x

T

k p2

2

2

2

Energy Equation

In certain cases (highly viscous fluid), certain work dW is done by fluid to

overcome the viscous effect which results in energy dissipation due to friction.

Accounting for the energy given out by viscous dissipation we have

222

2

-

x

x

x

x

-

x

x

x

x!*

x

v

y

u

y

v

x

u

Where is given as (after a lengthy analysis)


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Convective AnalysisTotal number of unknowns (Factors influencing the value ofh):

1.Three velocity components (u, v, w)2.Temperature (T)

3.Pressure (P)

4.Density ()

5.Viscosity ()

6.Thermal conductivity (k)

By assumption andsimplification, the number of unknowns are reduced as

1.Two velocity components (u, v)

2.Temperature (T)

3.Pressure (P)which requires a system of4 equations

The 4 equations are given by continuity equation, x momentum & y momentum equations and energy equation

class ii - conservation equations

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