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The conservation laws

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Title: The conservation laws


1
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  • ???? ??? ????

Lecture 3
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2
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  • The conservation laws
  • material balance
  • heat balance
  • enery balance
  • Rate equations
  • the relationship between flow rate and driving
    force in the field of fluid flow
  • heat transfer
  • diffusion of matter

3
??????
  • The conservation laws
  • material balance
  • heat balance
  • enery balance
  • (rate of) input - (rate of) output (rate of)
    accumulation

4
????
A single-stage mixer settler is to be used for
the continuous extraction of benzoic acid from
toluene, using water as the extracting
solvent. The two streams are fed into a tank A
where they are stirred vigorously, and the
mixture is then pumped into tank B where it is
allowed to settle into two layers. The upper
toluene layer and the lower water layer
are removed separately, and the problem is to
find what proportion of the benzoic acid has
passed into the solvent phase.
toluene benzoic acid
water
toluene benzoic acid
water benzoic acid
5
??(???)
Rate equation for the extraction efficiency y
mx
Material Balance Input of benzoic acid
output of benzoic acid
Rc Rx Sy
Same method can be applied to multi-stages.
6
?????
  • Funtion of time

7
????? (unsteady state)
In unsteady state problems, time enters as a
variable and some properties of the system become
functions of time. Similar to the previous
example, but now assuming that the mixer is so
efficient that the compositions of the two liquid
streams are in equilibrium at all times. A stream
leaving the stage is of the same composition as
that phase in the stage. The state of the system
at a general time t, wher x and y are now
functions of time.
8
Material balance on benzoic acid
Input - output accumulation
???????
t 0, x 0
9
Mathematical Models
  • Salt accumulation in a stirred tank

t 0 Tank contains 2 m3 of water
Q Determine the salt concentration in the
tank when the tank contains 4 m3 of brine
10
??????
  • V and x are function of time t
  • During ?t
  • balance of brine
  • balance of salt

11
??????
  • Solve
  • x 20 - 20 (1 0.005 t)-2
  • V 2 0.01 t

12
Mathematical Models
  • Mixing

t 0 Tank 1 contains 150 g of chlorine dissolved
in 20 l water Tank 2 contains 50 g of chlorine
dissolved in 10 l water
Q Determine the amount of chlorine in each tank
at any time t gt 0
13
??????
  • Let xi(t) represents the number of grams of
    chlorine in tank i at time t.
  • Tank 1 x1(t) (rate in) - (rate out)
  • Tank 2 x2(t) (rate in) - (rate out)
  • Mathematical model

x1(t) 3 0 3 x2/10 - 2 x1/20 - 4
x1/20
x2(t) 4 x1/20 - 3 x2/10 - 1 x2/10
14
??????
  • How to solve?
  • Using Matrices
  • X AX X(0) X0 where
  • x1(t)120e-t/1030e-3t/5
  • x2(t)80e-t/10-30e-3t/5

15
Mathematical Models
  • Mass-Spring System
  • Suppose that the upper weight is pulled down one
    unit and the lower weight is raised one unit,
    then both weights are released from rest
    simultaneously at time t 0.

Q Determine the positions of the weights
relative to their equilibruim positions at any
time t gt 0
16
??????
  • Equation of motion
  • weight 1
  • weight 2
  • Mathematical model

m1 y1(t) - k1 y1 k2 (y2 - y1)
m2 y2(t) - k2 (y2 - y1) - k3 y2
17
??????
  • How to solve?
  • y1(t)-1/5 cos (2t) 6/5 cos (3t)
  • y2(t)-2/5 cos (2t) - 3/5 cos (3t)

18
?????
  • Funciotn of position

19
Mathematical Models
  • Radial heat transfer through a cylindrical
    conductor

Temperature at a is To Temperature at b is T1
r dr
r
a
b
Q Determine the temperature distribution as a
function of r at steady state
20
??????
  • Considering the element with thickness ? r
  • Assuming the heat flow rate per unit area Q
  • Radial heat flux
  • A homogeneous second order O.D.E.

where k is the thermal conductivity
21
??????
  • Solve

22
?? (Flow systems) - Eulerian
  • The analysis of a flow system may proceed from
    either of two different points of view
  • Eulerian method
  • the analyst takes a position fixed in space and a
    small volume element likewise fixed in space
  • the laws of conservation of mass, energy, etc.,
    are applied to this stationary system
  • In a steady-state condition
  • the object of the analysis is to determine the
    properties of the fluid as a function of position.

23
?? (Flow systems) - Lagrangian
  • the analyst takes a position astride a small
    volume element which moves with the fluid.
  • In a steady state condition
  • the objective of the analysis is to determine the
    properties of the fluid comprising the moving
    volume element as a function of time which has
    elapsed since the volume element first entered
    the system.
  • The properties of the fluid are determined solely
    by the elapsed time (i.e. the difference between
    the absolute time at which the element is
    examined and the absolute time at which the
    element entered the system).
  • In a steady state condition
  • both the elapsed time and the absolute time
    affect the properties of the fluid comprising the
    element.

24
Eulerian ??
A fluid is flowing at a steady state. Let x
denote the distance from the entrance to an
arbitrary position measured along the centre line
in the direction of flow. Let Vx denote the
velocity of the fluid in the x direction, A
denote the area normal to the x direction, and ?
denote the fluid density at point x. Apply the
law of conservation of mass to an infinitesimal
element of volume fixed in space and of length dx.
25
If Vx and ? are essentially constant across the
area A, The rate of input of mass is
The rate of mass output is
0
Rate of input - rate of output rate of
accumulation
Equation of continuity
26
Lagrangian ??
Consider a similar system. An infinitesimal
volume element which moves with the fluid through
the flow system. Let ? denote the elapsed time
? t -t0 where t is the absolute time at which
the element is observed and t0 is the absolute
time at which the element entered the system. At
elapsed time ?, the volume of the element is A?a,
the density is ?, and the velocity of the element
relative to the stationary wall is Vx. Apply the
law of conservation of mass to the volume element.
27
x
Mass balance of the element at steady-state
t integral
The elapsed time ?
The difference between the relative velocity of
the forward face and the relative velocity of the
trailing face is the change rate of the length of
the element
?Eulerian????
28
???? (independent variable)
  • These are quantities describing the system which
    can be varied by choice during a paticular
    experiment independently of one another.
  • Examples
  • time
  • coordinates

29
????? (dependent variable)
  • These are properties of the system which change
    when the independent variables are altered in
    value. There is no direct control over a
    dependent variable during an experiment.
  • The relationship between independent and depend
    variables is one cause and effect the
    independent variable measures the cause and the
    depend variable measures the effect of a
    particular action.
  • Examples
  • temperature
  • concentration
  • efficiency

30
?? (Parameter)
  • It consists mainly of the charateristics
    properties of the apparatus and the physical
    properties of the materials.
  • It contains all properties which remain constant
    during an individual experiment. However, a
    different constant value can be taken by a
    property during different experiments.
  • Examples
  • overall dimensions of the apparatus
  • flow rate
  • heat transfer coefficient
  • thermal conductivity
  • density
  • initial or boundary values of the depent variables

31
????????
  • A dependent variable is usually differentiated
    with respect to an independent variable, and
    occasionally with respect to a parameter.
  • When a single independent variable is involved in
    the problem, it gives rise to ordinary
    differential equations.
  • When more than one independent variable is needed
    to describe a system, the usual result is a
    partial differential equation.

32
???? (Boundary conditions)
  • There is usually a restriction on the range of
    values which the independent variable can take
    and this range describes the scope of the
    problem.
  • Special conditions are placed on the dependent
    variable at these end points of the range of the
    independent varible. These are natually called
    boundary conditions.

33
???????
  • ?? (heat transfer)
  • Boundary at a fixed temperature, T T0.
  • Constant hear flow rate through the boundary,
    dT/dx A.
  • Boundary thermally insulated, dT/dx 0.
  • Boundary cools to the surroundings through a film
    resistance described by a heat transfer
    coefficient, k dT/dx h (T-T0).
  • k is the thermal conductivity h is the heat
    transfer coefficient and T0 is the temperature
    of the surrendings.

34
???????(Boundary value and initial value)
  • Specifying conditions on a solution and its
    derivative at the ends of an interval (boundary
    value problem) is quite different from specifying
    the value of a solution and its derivative at a
    given point (initial value problem).
  • Boundary value problems usually do not have
    unique solutions, and it is this lack of
    uniqueness which makes certain boundary value
    problems important in solving P.D.E. of physics
    and engineering.

35
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