The conservation laws

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Lecture 3
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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

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

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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
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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.
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• Funtion of time

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????? (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.
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Material balance on benzoic acid
Input - output accumulation
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t 0, x 0
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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
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• V and x are function of time t
• During ?t
• balance of brine
• balance of salt

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

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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
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• 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
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• 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

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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
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• 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
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• How to solve?
• y1(t)-1/5 cos (2t) 6/5 cos (3t)
• y2(t)-2/5 cos (2t) - 3/5 cos (3t)

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• Funciotn of position

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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
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• 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
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• Solve

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?? (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.

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?? (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.

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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.
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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
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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.
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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????
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???? (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

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????? (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

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?? (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

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• 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.

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???? (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.

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• ?? (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.

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???????(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.

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