化工應用數學

授課教師： 郭修伯

Lecture 7 Partial Differentiation and Partial Differential Equations

Chapter 8

• Partial differentiation and P.D.E.s– Problems requiring the specification of more

than one independent variable.– Example, the change of temperature

distribution within a system:• The differentiation process can be performed

relative to an incremental change in the space variable giving a temperature gradient, or rate of temperature rise.

Partial derivatives

• Figure 8.1 (contour map for u)– If x is allowed to vary whilst y remains constant

then in general u will also vary and the derivate of u w.r.t. x will be the rate of change of u relative to x, or the gradient in the chosen direction :

yx

u

u is a vector along the line of greatest slope and has amagnitude equal to that slope.

ugrady

u

x

uu

jix

uu

i

u will change by due to the change in x, and by due to the change in y: xx

u

yy

u

yy

ux

x

uu

dxx dyy

duu

dyy

udx

x

udu

nn

dxx

udx

x

udx

x

udu

...22

11

In general form :

the “total differential” of u

g

Important fact concerning “partial derivative”

• The symbol “ “ cannot be cancelled out!

• The two parts of the ratio defining a partial derivative can never be separated and considered alone.– Marked contrast to ordinary derivatives where

dx, dy can be treated separately

Changing the independent variables

),(

),(

vuy

vux

dvv

duu

dy

dvv

duu

dx

),( yxgz dyy

gdx

x

gdz

dvv

duuy

gdv

vdu

ux

gdz

uy

g

ux

g

u

z

w.r.t. u

u

y

y

z

u

x

x

z

u

z

11

2

21

1

11

...u

x

x

z

u

x

x

z

u

x

x

z

u

z n

n

In general form :

Independent variables not truly independent

Vapour composition is a function of temperature, pressure and liquid composition:

),,( xPTfy However, boiling temperature is a function of pressure and liquid composision:

),( xPgT

Therefore

dPP

fdT

T

fdx

x

fdy

dPP

gdx

x

gdT

dPP

g

T

f

P

fdx

x

g

T

f

x

fdy

Temperature increment of a fluid:

dtt

Tdz

z

Tdy

y

Tdx

x

TdT

t

T

dt

dz

z

T

dt

dy

y

T

dt

dx

x

T

dt

dT

Total time derivative

t

T

z

T

y

T

x

TT

t

Twvuu

A special case when the path of a fluid element is traversed

Dt

DTSubstantive derivate of element of fluid

comparet

T

partial derivate of element of space

Formulating P.D.Es

• Identify independent variables

• Define the control volume

• Allowing one independent variable to vary at a time

• Apply relevant conservation law

Unsteady-state heat conduction in one dimension

L

x

T

x

Considering the thermal equilibrium of a slice of the wall between a plane at distance x from the heated surface and a parallel plane at x+x from the same surface gives the following balance.

Rate of heat input at distance x and time t:x

Tk

Rate of heat input at distance x and time t + t: tx

Tk

tx

Tk

Rate of heat output at distance x + x and time t:

Rate of heat output at distance x + x and time t + t:

tx

Tk

tx

Tk

xt

x

Tk

tx

Tk

xx

Tk

xx

Tk

Heat content of the element at time t is xTC p

Heat content of the element at time t + t is xtt

TTC p

Accumulation of heat in time t is xtt

TC p

Average heat input during the time interval t is ttx

Tk

tx

Tk

2

1

Average heat output during the time interval t is

ttxx

Tk

xx

Tk

tx

x

Tk

xx

Tk

2

1

Conservation law

xtt

TCttx

x

Tk

xtx

x

Tk

x p

2

2

1

t

TCt

tx

Tk

x

Tk p

2

3

2

2

2

1assuming k is constant

t

TCt

tx

Tk

x

Tk p

2

3

2

2

2

1

0t

t

T

x

T

C

k

p

2

2

is the thermal diffusivity

three dimensions

t

TT

C

k

p

2

pC

k

Mass transfer example

A spray column is to be used for extracting one component from a binary mixture which forms the rising continuous phase. In order to estimate the transfer coefficient it is desired to study the detailed concentration distribution around an individual droplet of the spray. (using the spherical polar coordinate)During the droplet’s fall through the column, the droplet moves into contact with liquid of stronger composition so that allowance must be made for the time variation of the system. The concentration will be a function of the radial coordinate (r) and the angular coordinate ()

r

r

A

B

D

C

Area of face AB is rr sin2Area of face AD is rr sin2Volume of element is rrr sin2

r

r

Output rate across CD rrrr

cDuc

rrr

r

cDuc

sin2sin2

Output rate across BC

rrc

r

Dvcrr

c

r

Dvc sin2sin2

Accumulation rate t

crrr

sin2

Conservation Law:input - output = accumulation

Material is transferred across each surface of the element by two mechanisms:Bulk flow and molecular diffusion

Input rate across AB rrr

cDuc sin2

Input rate across AD rrc

r

Dvc

sin2

t

crrr

rrc

r

Dvcrrr

r

cDuc

r

sin2

sin2sin2

t

crrr

rrc

r

Dvcrrr

r

cDuc

r

sin2

sin2sin2

sin

sin

11 222

c

r

Dvc

rr

cDrucr

rrt

c

Dividing throughout by the volume

The continuity equation

x

z

y

x

z

yA

C

B

D

E

F

G

H

Input rate through ABCD zyu

Input rate through ADHE zxv

Input rate through ABFE yxw

Output rate through EFGH xzyux

zyu

Output rate through BCGF yzxvy

zxv

Output rate through CDHG zyxwz

yxw

Conservation Law:input - output = accumulation

zyxt

zyxwz

yzxvy

xzyux

0

wz

vy

uxt

0

wz

vy

uxt

0

zw

yv

xu

z

w

y

v

x

u

t

tz

wy

vx

uDt

D

01

z

w

y

v

x

u

Dt

D

Continuity equation for a compressible fluid

Boundary conditions

• O.D.E.– boundary is defined by one particular value of the independent

variable

– the condition is stated in terms of the behaviour of the dependent variable at the boundary point.

• P.D.E.– each boundary is still defined by giving a particular value to

just one of the independent variables.

– the condition is stated in terms of the behaviour of the dependent variable as a function of all of the other independent variables.

Boundary conditions for P.D.E.

• Function specified– values of the dependent variable itself are given at all

points on a particular boundary

• Derivative specified– values of the derivative of the dependent variable are

given at all points on a particular boundary

• Mixed conditions

• Integro-differential condition

Function specified

• Example 8.3.1 (time-dependent heat transfer in one dimension): The temperature is a function of both x and t. The boundaries will be defined as either fixed values of x or fixed values of t:– at t = 0, T = f (x)

– at x = 0, T = g (t)

• Steady heat conduction in a cylindrical conductor of finite size: The boundaries will be defined as by keeping one of the independent variables constant:– at z = a, T = f (r, )

– at r = r0, T = g (z, )

Derivative specified

• In some cases, (e.g., cooling of a surface and eletrical heating of a surface), the heat flow rate is known but not the surface temperature.

• The heat flow rate is related to the temperature gradient.

• Example:

AF

E

D

C

B

H

G

The surface at x = 0 is thermally insulated.

x

z

AF

E

D

C

B

H

G

x

z

Input rate through ADHE zxy

Tk

Input rate through ABFE yxz

Tk

Output rate through BCGF yzxy

Tk

yzx

y

Tk

Output rate through DCGH zyxz

Tk

zyx

z

Tk

Output rate through EFGH xzyx

Tk

xzy

x

Tk

Accumulation of heat in time t is zyxt

TC p

Heat balance gives

yzxt

TCyzx

x

Tkyz

x

Tkyzx

z

Tkyzx

y

Tk p

2

2

2

2

2

2

2

2

xt

TC

x

Tkx

x

T

z

T

y

Tk p

2

2

2

2

2

2

size 0x 0

0

x

Tat x = 0

This is the required boundary condition.

ExampleA cylindrical furnace is lined with two uniform layers of insulting brick of different physical properties. What boundary conditions should be imposed at the junction between the layers?

r

a

AD

CB

layer 1

layer 2

Due to axial symmetry, no heat will flow across the faces of the element given by = constant but will flow in the z direction.

One boundary condition: aratTT 21

The rate of flow of heat just inside the boundary of the first layer is zar

Tk

11

The rate of flow of heat into the element across the face CD is

rzrr

Tk

rzr

r

Tk

2

111

11

Input across CD = arr

Tr

rzrk

r

Tzak

11

11 2

1

r = a

r

a

AD

CB

layer 1

layer 2

Input across CD =

arr

Tr

rzrk

r

Tzak

11

11 2

1

Output across AB =

arr

Tr

rzrk

r

Tzak

22

22 2

1

The heat flow rates in the z direction

Input at face z = 22112

1TkTk

zra

Output at face z + z = zTkTkz

raz

TkTkz

ra

22112211 2

1

2

1

Accumulation within the element

t

TzraC

t

TzraC pp

2

221

11 2

1

2

1

The complete heat balance on the element

zTkTkz

raz

r

Tr

rzrk

r

Tzak

r

Tr

rzrk

r

Tzak

TCTCt

zra pp

2211

22

22

11

11

222111

2

1

2

1

2

1

2

1

dividing by za0r

r

Tk

r

Tk

2

21

1 This is the second boundary condition.

And...

zTkTkz

raz

r

Tr

rzrk

r

Tzak

r

Tr

rzrk

r

Tzak

TCTCt

zra pp

2211

22

22

11

11

222111

2

1

2

1

2

1

2

1

t

T

k

C

z

T

r

Tr

rrp

21

211

Heat conduction in cylindrical polar coordinates with axial symmetry.

If the heat balance is taken in either layer (say layer 1)

ra

subscript

12

Mixed conditions

• The derivative of the dependent variable is related to the boundary value of the dependent variable by a linear equation.

• Example: surface rate of heat loss is governed by a heat transfer coefficient.

)( 0TThx

Tk

rate at which heat is removed from the surface per unit area

rate at which heat is conducted to the surface internally per unit area

Integro-differential boundary condition

• Frequently used in mass transfer– materials crossing the boundary either enters or leaves a

restricted volume and contributes to a modified driving force.

• Example: a solute is to be leached from a collection of porous spheres by stirred them as a suspension in a solvent. Determine the correct boundary condition at the surface of one of the spheres.

The rate at which material diffuse to the surface of a porous sphere of radius a is:

ar

cDa

24

D is an effective diffusivity and c is the concentration within the sphere.

If V is the volume of solvent and C is the concentration in the bulk of the solvent:

ar

cDaN

t

CV

24

N is the number of spheres.

For continuity of concentration, c = C at r = a :

dtr

c

V

DaNC

t

a

0

24 at r = a,

Boundary condition

t

T

x

T

2

2

Two boundary conditions are needed at fixed values of x and one at a fixed value of t.

Initial value and boundary value problems

• Numer of conditions:– O.D.E.

• the number of B.C. is equal to the order of the differential equation

– P.D.E.• no rules, but some guild lines exist.

Initial value or boundary value?• When only one condition is needed in a particular variable, it is specified at one

fixed value of that variable.– The behaviour of the dependent variable is restricted at the beginning of a range but no end is specified.

The range is “open”.

• When two or more conditions are needed, they can all be specified at one value of the variable, or some can be specified at one value and the rest at another value.

– When conditions are given at both ends of a range of values of an independent variable, the range is “closed” by conditions at the beginning and the end of the range.

– When all conditions are stated at one fixed value of the variable, the range is “open” as far as that independent variable is concerned.

• The range is closed for every independent variable: a boundary value problem (or, a jury problem).

• The range of any independent variable is open: an initial value problem (or, a marching problem).