Differential equation

1
??????

• ???? ???

Partial Differentiation and Partial Differential
Equations
Lecture 7
2
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.

3
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

4
?u is a vector along the line of greatest slope
and has a magnitude equal to that slope.
u will change by due to the change in
x, and by due to the change in y
the total differential of u
In general form
5
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

6
Changing the independent variables
w.r.t. u
In general form
7
Independent variables not truly independent
Vapour composition is a function of temperature,
pressure and liquid composition
However, boiling temperature is a function of
pressure and liquid composision
Therefore
8
Temperature increment of a fluid
A special case when the path of a fluid element
is traversed
Total time derivative
Substantive derivate of element of fluid
compare
partial derivate of element of space
9
Formulating P.D.Es

• Identify independent variables
• Define the control volume
• Allowing one independent variable to vary at a
  time
• Apply relevant conservation law

10
Unsteady-state heat conduction in one dimension
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.
T
?x
x
Rate of heat input at distance x and time t
Rate of heat input at distance x and time t ?t
Rate of heat output at distance x ?x and time t
L
Rate of heat output at distance x ?x and time t
?t
11
Average heat input during the time interval ? t
is
Average heat output during the time interval ? t
is
Heat content of the element at time t is
Heat content of the element at time t ? t is
Accumulation of heat in time ? t is
Conservation law
assuming k is constant
12
is the thermal diffusivity
three dimensions
13
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 droplets
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
14
?
?
r
r
Area of face AB is
Area of face AD is
Volume of element is
15
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
Output rate across CD
Output rate across BC
Accumulation rate
Conservation Law input - output accumulation
16
Dividing throughout by the volume
17
The continuity equation
Input rate through ABCD
C
G
Input rate through ADHE
D
Input rate through ABFE
H
?z
B
Output rate through EFGH
F
?y
Output rate through BCGF
A
E
?x
Output rate through CDHG
Conservation Law input - output accumulation
18
Continuity equation for a compressible fluid
19
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.

20
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

21
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, ?)

22
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
G
z
D
The surface at x 0 is thermally insulated.
H
B
F
A
E
x
23
C
G
z
D
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
Accumulation of heat in time ? t is
24
Heat balance gives
size ? 0 ?x ? 0
at x 0
This is the required boundary condition.
25
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?
layer 1
Due to axial symmetry, no heat will flow across
the faces of the element given by ? constant
but will flow in the z direction.
B
C
?r
?
D
A
One boundary condition
a
layer 2
The rate of flow of heat just inside the boundary
of the first layer is
The rate of flow of heat into the element across
the face CD is
r a
Input across CD
26
layer 1
Input across CD
B
C
?r
?
D
A
Output across AB
a
layer 2
The heat flow rates in the z direction
Input at face z
Output at face z ?z
Accumulation within the element
27
The complete heat balance on the element
dividing by
This is the second boundary condition.
And...
28
If the heat balance is taken in either layer (say
layer 1)
Heat conduction in cylindrical polar coordinates
with axial symmetry.
29
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.

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
30
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.

31
The rate at which material diffuse to the surface
of a porous sphere of radius a is
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
N is the number of spheres.
For continuity of concentration, c C at r a
at r a,
Boundary condition
32
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.

Two boundary conditions are needed at fixed
values of x and one at a fixed value of t.
33
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).