Transcript 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.

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 :
 u 
 
 x  y
u is a vector along the line of greatest slope and has a
magnitude equal to that slope.
u  i
u
u
 j  grad u
x
y
 u 
u will change by  xdue to the change in x, and by
 x 
u
i  u 
x
 u 
 y due to the change in y:
 y 
 u 
 u 
u   x   y
 x 
 y 
x  dx
y  dy
 u 
 u 
du   dx   dy
 x 
 y 
u  du
the “total differential” of u
 u 
 u 
 u 
dxn




dx1  
dx2  ...  
In general form : du  


 x1 
 x2 
 xn 
Important fact concerning
“partial derivative”
• The symbol “ g“ 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
 g 
 g 
dz   dx   dy
 x 
 y 
z  g ( x, y)
x   (u, v)
y   (u, v)
  
  
dx   du   dv
 u 
 v 
  
  
dy  
du  
dv
 u 
 v 
   g  
 
 g  
dz    du 
dv   
du 
dv
v   y  u
v 
 x  u
w.r.t. u
z  z  x   z  y 
       
u  x  u   y  u 
z  g     g   
      

u  x  u   y  u 
 z  xn 
z  z  x1   z  x2 












 ...  
In general form :






u1  x1  u1   x2  u1 
 xn  u1 
Independent variables not truly
independent
Vapour composition is a function of temperature, pressure and liquid composition:
y  f (T , P, x)
However, boiling temperature is a function of pressure and liquid composision:
T  g ( P, x)
Therefore
 f 
 f 
 f 
dy   dx  
dT   dP
 x 
 T 
 P 
 g 
 g 
dT   dx   dP
 x 
 P 
 f   f  g 
 f   f  g 
dy        dx        dP
 x   T  x 
 P   T  P 
Temperature increment of a fluid:
 T 
 T 
 T 
 T 
dT   dx   dy   dz   dt
 x 
 z 
 t 
 y 
dT  T  dx  T  dy  T  dz  T 
           
dt  x  dt  y  dt  z  dt  t 
Total time derivative
A special case when the path of a
fluid element is traversed
T
 T   T   T 
 T 
 u  T   u    v   w   
t
 x   y   z 
 t 
DT
Dt Substantive derivate of element of fluid
T
compare
partial derivate of element of space
t
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
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.
x
T
L
Rate of heat input at distance x and time t:
k
T
x
Rate of heat input at distance x and time t + t:
k
T  
T 
  k
t
x t 
x 
Rate of heat output at distance x + x and time t:  k
T  
T 
  k
x
x x 
x 
Rate of heat output at distance x + x and time t + t:
k
T  
T 
  T  
T  
  k
  k
t   k
t 
x t 
x 
x 
x t 
x  
 T 1  
T  
Average heat input during the time interval  t is  k


k

t t
x 2 t 
x  

Average heat output during the time interval  t is
 T  
T 
1   T  
T   

k


k

x


k


k



xt t


x x 
x 
2 t 
x x 
x   

Heat content of the element at time t is
C pTx
Heat content of the element at time t +  t is

 T  
C p T  
t x
 t  

Accumulation of heat in time  t is
C p 
 T 
tx
 t 
Conservation law
  

T 
1 2 
T 
 T 


k

x


k

x

t

t


C




tx
p


x 
2 tx 
x 
 t 
 x 

assuming k is constant
 2T 1  3T
 T 
k 2  k 2 t  C p  
x
2 x t
 t 
 2T 1  3T
 T 
k 2  k 2 t  C p  
x
2 x t
 t 
t  0
k  2T T

2
C p x
t
k
is the thermal diffusivity
C p
three dimensions
k
T
 2T 
C p
t
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


r
r
Area of face AB is
Area of face AD is
Volume of element is
2r sin  r 
2r sin  r 
2r sin  r r 
Material is transferred across each surface of the element by two mechanisms:
Bulk flow and molecular diffusion
Input rate across AB
Input rate across AD
c 

 uc  D 2r sin  r 
r 

D c 

vc


2r sin  r 
r  


c 
 
c 

 uc  D 2r sin  r    uc  D 2r sin  r r
r 
r 
r 



D c 
 
D c 

Output rate across BC  vc 
2r sin  r  
 vc 
2r sin  r 

r  
 
r  


c
Accumulation rate



2r sin  r r
t
Output rate across CD
Conservation Law:
input - output = accumulation

 
c 

  uc  D 2r sin  r r 
r 
r 


 2r sin  r r 
c
t


D c 
 vc  r  2r sin  r 




 
c 

  uc  D 2r sin  r r 
r 
r 


 2r sin  r r 


D c 
 vc  r  2r sin  r 



c
t
Dividing throughout by the volume

c
1   2
c 
1
 
D c 
2
 2
r
uc

r
D

vc

sin








t
r r 
r  r sin   
r  

The continuity equation
z
C
y
x
D
z
G
H
B
F
y
A
x
E
Input rate through ABCD
uyz
Input rate through ADHE
vxz
Input rate through ABFE
wxy
Output rate through EFGH
uyz 
Output rate through BCGF
Output rate through CDHG

uyz x
x

vxz  vxz y
y

wxy  wxy z
z
Conservation Law:
input - output = accumulation




 uyz x  vxz y  wxy z 
xyz
x
y
z
t
 


 u   v   w  0
t x
y
z
 


 u   v   w  0
t x
y
z

u
v
w




 
u
v
w
0
t
x
y
z
x
y
z
D
     
     
 u   v   w    
Dt
 x   y 
 z   t 
1 D u v w

 
0
 Dt x y z
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:
C
z
D
G
The surface at x = 0 is thermally insulated.
H
B
F
A
E
x
C
z
D
G
H
B
F
A
E
x
Input rate through ADHE
Input rate through ABFE
Output rate through BCGF
Output rate through DCGH
Output rate through EFGH
T
k
xz
y
T
k
xy
z

T
 
T
k
xz    k xz y
y
y 
y

T

T

k
xy    k
xy z
z
z 
z

T
 
T

k
yz    k
yz x
x
x 
x

Accumulation of heat in time  t is
 T
 t
C p 

xyz

Heat balance gives
 2T
 2T
 2T
 2T
 T 
k 2 xzy  k 2 xzy  k 2 zy  k 2 xzy  C p 
xzy
y
z
x
x
 t 
  2T  2T  2T
k  2  2  2
z
x
 y

T
 T 
x  k
 C p 
x
x
 t 

size  0
x  0
T
0
x
This is the required boundary condition.
at x = 0
Example
A 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?
B
r
layer 1
C

D
A
Due to axial symmetry, no heat will flow across the faces of the
element given by  = constant but will flow in the z direction.
a
One boundary condition:
T1  T2
at r  a
layer 2
T1
a z 
The rate of flow of heat just inside the boundary of the first layer is  k1
r
The rate of flow of heat into the element across the face CD is
r=a
 k1
T1
rz      k1 T1 rz  1 r
r
r 
r
2
T1 1
  T1 
 k1 rz   r
 r a
Input across CD =  k1 az 
r 2
r  r 
B
r
layer 1
C
T1 1
  T1 
 k1 az 
 k1 rz   r
 r a
r 2
r  r 

D
A
Input across CD =
a
Output across AB =
T2 1
  T2 
 k2 az 
 k2 rz   r
 r a
r 2
r  r 
layer 2
The heat flow rates in the z direction
1

 a r  k1T1  k 2T2 
Input at face z =
2
z
1

  1


Output at face z + z =  ar  k1T1  k2T2    ar  k1T1  k2T2 z
2
z
z  2
z

Accumulation within the element
1
T 1
T
1C p1 a rz  1   2C p 2 a rz  2
2
t 2
t
The complete heat balance on the element
1
arz   1C p1T1   2C p 2T2  
2
t

T1 1
  T1 





k
a

z

k

r

z
r

 1
1
r 2
r  r 


T 1
  T 
  k 2 az  2  k 2 rz   r 2 
r 2
r  r 

  1

 
   ar  k1T1  k 2T2 z 
z
 
 z  2
dividing by
a z
r  0
T1
T2
k1
 k2
r
r
This is the second boundary condition.
And...
1

arz  1C p1T1   2C p 2T2  
2
t

T1 1
  T1 
 k1 rz   r

 k1 az 
r 2
r  r 


T2 1
  T2 
  k 2 az 
 k 2 rz   r

r 2
r  r 

  1

 
   ar  k1T1  k 2T2 z 
z
 
 z  2
If the heat balance is taken in either layer (say layer 1)
subscript 2  1
ar
1   T1   2T1 C p T
r
 2 
r r  r  z
k t
Heat conduction in cylindrical polar coordinates with axial symmetry.
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.
T
k
 h(T  T0 )
x
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:
c
 4a D
r
2
a
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:
C
c
2
V
 4 Na D
t
r
a
N is the number of spheres.
For continuity of concentration, c = C at r = a :
at r = a,
4 Na 2 D c
C  
a dt
0
V
r
t
Boundary condition
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.
 2T T
 2 
x
t
Two boundary conditions are needed at fixed values of x and one at a fixed value of t.
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).