化工應用數學
授課教師： 郭修伯
Lecture 7
Partial Differentiation and Partial DifferentialEquations
Chapter 8
•Partial differentiation and P.D.E.s
–Problems requiring the specification of morethan one independent variable.
–Example, the change of temperaturedistribution within a system:
•The differentiation process can be performedrelative to an incremental change in the spacevariable giving a temperature gradient, or rate oftemperature rise.
Partial derivatives
•Figure 8.1 (contour map for u)
–If x is allowed to vary whilst y remains constantthen in general u will also vary and the derivateof u w.r.t. x will be the rate of change of urelative to x, or the gradient in the chosendirection :
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:
In general form :
the “total differential” of u
Important fact concerning“partial derivative”
•The symbol “ “ cannot be cancelled out!
•The two parts of the ratio defining a partialderivative can never be separated andconsidered alone.
–Marked contrast to ordinary derivatives wheredx, dy can be treated separately
Changing the independent variables
w.r.t. u
In general form :
Independent variables not trulyindependent
Vapour composition is a function of temperature, pressure and liquid composition:
However, boiling temperature is a function of pressure and liquid composision:
Therefore
Temperature increment of a fluid:
Total time derivative
A special case when the path of afluid element is traversed
Substantive derivate of element of fluid
compare
partial derivate of element of space
Formulating P.D.Es
•Identify independent variables
•Define the control volume
•Allowing one independent variable to varyat a time
•Apply relevant conservation law
Unsteady-state heat conduction inone dimension
L
x
T
x
Considering the thermal equilibrium of a slice of the wall betweena plane at distance x from the heated surface and a parallel plane atx+x from the same surface gives the following balance.
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:
Rate of heat output at distance x + x and time t + t:
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
Average heat input during the time interval t is
Average heat output during the time interval t is
Conservation law
assuming k is constant
is the thermal diffusivity
three dimensions
Mass transfer example
A spray column is to be used for extracting one component from a binary mixturewhich forms the rising continuous phase. In order to estimate the transfer coefficientit is desired to study the detailed concentration distribution around an individualdroplet of the spray. (using the spherical polar coordinate)
During the droplet’s fall through the column, the droplet moves into contact withliquid of stronger composition so that allowance must be made for the time variationof the system. The concentration will be a function of the radial coordinate (r) andthe angular coordinate ()
r
r
A
B
D
C
Area of face AB is
Area of face AD is
Volume of element is
r
r
Output rate across CD
Output rate across BC
Accumulation rate
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
Input rate across AD
Dividing throughout by the volume
The continuity equation
x
z
y
x
z
y
A
C
B
D
E
F
G
H
Input rate through ABCD
Input rate through ADHE
Input rate through ABFE
Output rate through EFGH
Output rate through BCGF
Output rate through CDHG
Conservation Law:
input - output = accumulation
Continuity equation for a compressible fluid
Boundary conditions
•O.D.E.
–boundary is defined by one particular value of theindependent variable
–the condition is stated in terms of the behaviour of thedependent variable at the boundary point.
• P.D.E.
–each boundary is still defined by giving a particularvalue to just one of the independent variables.
–the condition is stated in terms of the behaviour of thedependent variable as a function of all of the otherindependent variables.
Boundary conditions for P.D.E.
•Function specified
–values of the dependent variable itself are given at allpoints on a particular boundary
•Derivative specified
–values of the derivative of the dependent variable aregiven at all points on a particular boundary
•Mixed conditions
•Integro-differential condition
Function specified
•Example 8.3.1 (time-dependent heat transfer in onedimension): The temperature is a function of both x and t.The boundaries will be defined as either fixed values of xor fixed values of t:
–at t = 0, T = f (x)
–at x = 0, T = g (t)
•Steady heat conduction in a cylindrical conductor of finitesize: The boundaries will be defined as by keeping one ofthe 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 eletricalheating of a surface), the heat flow rate is known but notthe surface temperature.
•The heat flow rate is related to the temperature gradient.
•Example:
A
F
E
D
C
B
H
G
The surface at x = 0 is thermally insulated.
x
z
A
F
E
D
C
B
H
G
x
z
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
Heat balance gives
size 0
x 0
at x = 0
This is the required boundary condition.
Example
A cylindrical furnace is lined with two uniform layers of insulting brick of differentphysical properties. What boundary conditions should be imposed at the junctionbetween the layers?
r
a
A
D
C
B
layer 1
layer 2
Due to axial symmetry, no heat will flow across the faces of theelement given by = constant but will flow in the z direction.
One boundary condition:
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
Input across CD =
r = a
r
a
A
D
C
B
layer 1
layer 2
Input across CD =
Output across AB =
The heat flow rates in the z direction
Input at face z =
Output at face z + z =
Accumulation within the element
The complete heat balance on the element
dividing by
This is the second boundary condition.
And...
Heat conduction in cylindrical polar coordinates with axial symmetry.
If the heat balance is taken in either layer (say layer 1)
Mixed conditions
•The derivative of the dependent variable is relatedto the boundary value of the dependent variable bya linear equation.
•Example: surface rate of heat loss is governed by aheat 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
Integro-differential boundarycondition
•Frequently used in mass transfer
–materials crossing the boundary either enters or leaves arestricted volume and contributes to a modified drivingforce.
•Example: a solute is to be leached from a collectionof porous spheres by stirred them as a suspension ina solvent. Determine the correct boundary conditionat the surface of one of the spheres.
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
Two boundary conditions are needed at fixed values of x and one at a fixed value of t.
Initial value and boundary valueproblems
•Numer of conditions:
–O.D.E.
•the number of B.C. is equal to the order of thedifferential 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 isspecified at one fixed value of that variable.
–The behaviour of the dependent variable is restricted at the beginning of a rangebut no end is specified. The range is “open”.
•When two or more conditions are needed, they can all bespecified at one value of the variable, or some can be specified atone value and the rest at another value.
–When conditions are given at both ends of a range of values of an independentvariable, the range is “closed” by conditions at the beginning and the end of therange.
–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 boundaryvalue problem (or, a jury problem).
•The range of any independent variable is open: an initial valueproblem (or, a marching problem).