Video Lecture

8/11/2019 Video Lecture

1/44

Video Lectures for MBA

By:

Video.edhole.com
http://school.edhole.com/http://school.edhole.com/


2/44

Appendices 10.A & 10.B:

An Educational Presentation

Presented By:

Joseph Ash

Jordan BaldwinJustin Hirt

Andrea Lance

Video.edhole.com


3/44

History of Heat Conduction

Jean Baptiste Biot

(1774-1862)

French Physicist

Worked on analysis ofheat conduction

Unsuccessful at dealing

with the problem of

incorporating external

convection effects in heatconduction analysis

Video.edhole.com


4/44


Jean Baptiste Joseph Fourier

(17681830)

Read Biots work

1807 determined how to solve theproblem

Fouriers Law

Time rate of heat flow (Q) through a

slab is proportional to the gradient of

temperature difference

Video.edhole.com
http://school.edhole.com/http://school.edhole.com/http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Fourier.html


5/44


Ernst Schmidt

German scientist

Pioneer in EngineeringThermodynamics

Published paper Graphical DifferenceMethod for Unsteady Heat Conduction

First to measure velocity andtemperature field in free convectionboundary layer and large heat transfercoefficients

Schmidt Number Analogy between heat and mass

transfer that causes a dimensionless

quantity

Video.edhole.com


6/44

Derivation of the Heat

Conduction Equation

A first approximation of the equations

that govern the conduction of heat in asolid rod.

Video.edhole.com


7/44

Consider the following:

A uniform rod is insulated on both lateral

ends.

Heat can now only flow in the axial direction.

It is proven that heat per unit time will pass

from the warmer section to the cooler one.

The amount of heat is proportional to the

area, A, and to the temperature differenceT2-T1, and is inversely proportional to the

separation distance, d.

Video.edhole.com


8/44

The final consideration can be expressed as the

following:

is a proportionality factor called the thermal

conductivity and is determined by material properties

Video.edhole.com


9/44

Assumptions

The bar has a length L so x=0 and x=L

Perfectly insulated

Temperature, u, depends onlyon position, x,

and time, t

Usually valid when the lateral dimensions are

small compared to the total length.

Video.edhole.com


10/44

The differential equation governing

the temperature of the bar is a

physical balance between two rates:

Flux/Flow term

Absorption term

Video.edhole.com


11/44

Flux

The instantaneous rate of heat transfer from left to

right across the cross sections x=x0where x0 is

arbitrary can be defined as:

The negative is needed in order to show a positive

rate from left to right (hot to cold)

Video.edhole.com


12/44

Flux

Similarly, the instantaneous rate of heat transfer

from right to left across the cross section x=x0+x

wherex is small can be defined as:

Video.edhole.com


13/44

Flux

The amount of heat entering the bar in a time span

oft is found by subtracting the previous two

equations and then multiplying the result byt:

Video.edhole.com


14/44

Heat Absorption

The average change in temperature,u, can be

written in terms of the heat introduced, Qt and

the massm of the element as:

where s = specific heat of the material= density

Video.edhole.com


15/44

Heat Absorption

The actual temperature change of the bar is simply

the actual change in temperature at some

intermediate point, so the above equation can also

be written as:

This is the heat absorption equation.

Video.edhole.com


16/44

Heat Equation

Equating the Qt in the flux and absorption

terms, we find the heat absorption equation to

be:

Video.edhole.com


17/44

If we divide the above equation byxt and allow

bothx andt to both go to 0, we will obtain the

heat conductionor diffusion equation:

where

and has the dimensions of length^2/time and calledthe thermal diffusivity

Video.edhole.com


18/44

Boundary Conditions

Certain boundary conditions may apply to the

specific heat conduction problem, for

example:

If one end is maintained at some constanttemperature value, then the boundary condition

for that end is u = T.

If one end is perfectly insulated, then the

boundary condition stipulates ux= 0.

Video.edhole.com


19/44

Generalized Boundary Conditions

Consider the end where x=0 and the rate of flow ofheat is proportional to the temperature at the end ofthe bar. Recall that the rate of flow will be given, from left to right, as

With this said, the rate of heat flow out of the bar from right toleft will be

Therefore, the boundary condition at x=0 iswhere h1is a proportionality constant

if h1=0, then it corresponds to an insulated end

if h1goes to infinity, then the end is held at 0 temp.

Video.edhole.com


20/44

Generalized Boundary Conditions

Similarly, if heat flow occurs at the end x = L, then the

boundary condition is as follows:

where, again, h2is a nonzero proportionality

factor

Video.edhole.com


21/44

Initial Boundary Condition

Finally, the temperature distribution at one

fixed instantusually taken at t = 0, takes the

form:

occurring throughout the bar

Video.edhole.com


22/44

Generalizations

Sometimes, the thermal conductivity, density,

specific heat, or area may change as the axial

position changes. The rate of heat transfer under

such conditions at x=x0is now:

The heat equation then becomes a partial

differential equation in the form:

or

Video.edhole.com


23/44

Generalizations

Other ways for heat to enter or leave a bar must

also be taken into consideration.

Assume G(x,t,u) is a rate per unit per time.

Source G(x,t,u) is added to the bar

G(x,t,u) is positive, non-zero, linear, and u does not depend on t

G(x,t,u) must be added to the left side of the heat equation

yielding the following differential equation

Video.edhole.com


24/44

Generalizations

Similarly,

Sink

G(x,t,u) is subtracted from the bar

G(x,t,u) is positive, non-zero, linear, and u does notdepend on t

G(x,t,u) then under this sink condition takes the

form:

Video.edhole.com


25/44

Generalizations

Putting the source and sink equations together

in the heat equation yields

which is commonly called the generalized

heat conduction equation

Video.edhole.com


26/44

Multi-dimensional space

Now consider a bar in which the temperature is

a function of more than just the axial x-

direction. Then the heat conduction equation

can then be written: 2-D:

3-D:

Video.edhole.com


27/44

Example 1: Section 10.6, Problem 9

Let an aluminum rod of length 20 cm be initially

at the uniform temperature 25C. Suppose that

at time t=0, the end x=0 is cooled to 0C while

the end x=20 is heated to 60C, and both arethereafter maintained at those temperatures.

Find the temperature distribution in

the rod at any time t

Video.edhole.com


28/44


Find the temperature distribution, u(x,t)

2uxx=ut, 0


29/44


Using Equations 16 and 17 found on page 614, we

find that

where

1112 sin,

2

222

n

L

tn

nL

xn

ecTL

x

TTtxu

L

n dxL

xn

TL

x

TTxfLc 0 112 sin

2

Video.edhole.com
http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/


30/44


Evaluating cn, we find that

n

nc

n

nnnnc

dxxnx

c

n

n

L

n

50cos70

5sin12cos71020

sin0

20

06025

20

2

2

0

Video.edhole.com
http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/


31/44


Now we can solve for u(x,t)

1

400

86.0

1

20

86.0

20sin

50cos703,

20

sin50cos70

0

20

060,

2

2

222

n

tn

n

tn

xne

n

nxtxu

xne

n

nxtxu

Video.edhole.com
http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/


32/44


Video.edhole.com


33/44

Derivation of the Wave Equation

Applicable for:

One space dimension, transverse vibrations on elastic string

Endpoints atx= 0 andx= L along thex-axis

Set in motion at t= 0 and then left undisturbed

Video.edhole.com


34/44

Schematic of String in Tension

Video.edhole.com


35/44

Equation Derivation

Since there is no acceleration in the horizontal direction

However the vertical components must satisfy

where is the coordinate to the center of mass and the

weight is neglected

Replacing Twith Vthe and rearranging the equation becomes

0cos),()cos(),( txTtxxT

),(sin),()sin(),( txxutxTtxxT tt x

),(),(),(

txux

txVtxxVtt

Video.edhole.com


36/44

Derivation continued

Letting , the equation becomes

To express this in terms of only terms of uwe note that

The resulting equation in terms of uis

and since H(t)is not dependant onxthe resulting equation is

0x

),(),( txutxV ttx

),()(tan)(),( txutHtHtxV x

ttxx uHu )(

ttxx uHu

Video.edhole.com


37/44

DerivationContinued

For small motions of the string, it is approximated that

using the substitution that

the wave equation takes its customary form of

TTH cos

/2 Ta

ttxx

uua 2

Video.edhole.com


38/44

Wave Equation Generalizations

The telegraph equation

where cand kare nonnegative constants

cut arises from a viscous damping force

kuarises from an elastic restoring forceF(x,t)arises from an external force

The differences between this telegraph equation and the customary

wave equation are due to the consideration of internal elastic

forces. This equation also governs flow of voltage or current in a

transmission line, where the coefficients are related to the electrical

parameters in the line.

),(2 txFuakucuuxxttt

Video.edhole.com


39/44

Wave Equations in Additional Dimensions

For a vibrating system with more than on significant space

coordinate it may be necessary to consider the wave equation in

more than one dimension.

For two dimensions the wave equation becomes

For three dimensions the wave equation becomes

ttyyxx uuua )(2

ttzzyyxx uuuua )(2

Video.edhole.com


40/44


Consider an elastic string of length L whose ends

are held fixed. The string is set in motion from

its equilibrium position with an initial velocity

g(x). Let L=10 and a=1. Find the stringdisplacement for any time t.

,4

,1

,4

L

xL

L

x

xg

LxL

Lx

L

Lx

4

34

3

4

40

Video.edhole.com
http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/


41/44


From equations 35 and 36 on page 631, we find

that

where

1

sinsin,n

n L

atn

L

xnktxu

L

n dxL

xnxgLkL

an

0 sin2

Video.edhole.com
http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/


42/44


Solving for kn, we find:

4sin

4

3sin

8

sin4

sin4

3sin

42

sin4

sinsin42

3

2

4

0

4

3

4 4

3

nn

na

Lk

nnn

n

L

ank

dx

L

xn

L

xLdx

L

xndx

L

xn

L

x

L

k

L

an

n

n

L L

L

L

Ln

Video.edhole.com
http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/


43/44


Now we can solve for u(x,t)

1 33

133

1

3

10sin

10sin

4sin

4

3sin

180,

sinsin4

sin4

3sin

18,

sinsin4

sin4

3sin

8,

n

n

n

tnxnnn

ntxu

L

atn

L

xnnn

n

Ltxu

L

atn

L

xnnn

na

Ltxu

Video.edhole.com
http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/http://school.edhole.com/


44/44

THE END
http://school.edhole.com/

Video Lecture

Documents

Transcript of Video Lecture