In Engineering --- Designing a Pneumatic Pump

1
In Engineering --- Designing a Pneumatic Pump

• Introduction
• System characterization
• Model development
• Models 1, 2, 3, 4, 5 6
• Model analysis
• Time domain analysis
• Frequency domain analysis
• Discussions and Conclusions

2
Introduction

• In an aquarium, it is essential that there be a
  good supply of dissolved oxygen to maintain the
  health of the fish and other life inside the
  tank. This if often achieved by pumping air into
  it.
• The performance of the pump depends on its
  design, which in turn involves two phases.
• I Conceptual design in which a suitable scheme s
  developed ---- this involves identifying the
  different components and the coupling between
  components necessary for a solution
• Phase II Specification of components in terms of
  their size, characteristics and other functional
  parameters so that the pump performance meets
  some stated criterion.
• We concentrate the second phase!! Predicting the
  performance of a small pneumatic pump, suitable
  for domestic aquariums, as a function of the
  characteristics of its components.

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System Characterization
4
System Characterization

• The pump can be viewed as a system
• The boundary (interface) of system environment
  It receives electric power and air from the
  environment and supplies air to the aquarium
• Components

5
System Characterization

• Pump operating mechanism
• An alternating voltage from the main supply is
  applied across the coil, an alternating magnetic
  field is set up.
• This field interacts with the magnet to produce
  an oscillating motion (in a vertical plane) of
  the magnet.
• This motion is transmitted through the lever to
  produce a vertical motion of the bellows
• When the bellows move up, air is drawn through
  the inlet valve (with outlet valve closed) and
  when it moves down, air is pumped through the
  outlet valve (with inlet valve closed).
• Energy transformation Electrical energy (power)?
  mechanical energy (power) ? pneumatic energy
  (power)

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System Characterization

• The devices can be divided into three types
• (i) Energy (or power) transforming devices
  ---electromagnetic actuator
• (ii) Energy storage devices lever
• (iii) Energy dissipation devices bellows
• Variables
• effort variable
  flow variable
• Electrical voltage (e)
  current (i)
• Mechanical force (f)
  velocity (v)
• Pneumatic pressure (p)
  flow rate (q)

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System Charactization

• Electromagnetic actuator transform electrical
  power to mechanical power
• e1 voltage across the coil
• i1 current in the coil
• f1 force acting on the magnet
• v1 velocity with which the magnet is moving
• power conservation with H a constant

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System Characterization

• Lever is a coupler input and output power are
  mechanical
• L is the lever ratio
• Bellows
• A is the cross-sectional area of the bellows

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Model Development

• Model 1 simplest model
• Assumptions
• Ignoring all energy storage and dissipation
  elements
• The system is viewed as being made of three
  separate energy (or power) transforming elements
• The inputs to the model is ei and ii and the
  outputs are p0 and q0.

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Model 1

• Model
• In design, ei (the input voltage) and p0 (the
  pressure at which air is to be delivered) are
  specified
• The selection of system parameters (given by A, L
  and H) determine q0 and ii uniquely.
• Independent variables ei and p0
• Depend variables q0 and ii
• No energy is stored and dissipated

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Model 1

• Main drawbacks of model 1
• Since the model ignores all energy dissipative
  elements, it is highly unrealistic
• The variables q0 (ii) changes instantaneously as
  ei (p0) changes, this also is unrealistic
• As p0 increases, ii also increases. Thus, should
  a blockage occur on the output side, the current
  drawn (ii) approaches infinity. This follows
  because a blockage is equivalent to
  This is not true, and hence the model is
  unacceptable.

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Model 2

• This model includes two new features energy
  storage elements
• The mass of the magnet
• The compliance of the bellows
• The effect of including the mass of magnet is
  This follows as the total force
  generated by the electromagnetic actuator must
  now equal the force to move the magnet (given by
  mass times rate of change of velocity) plus the
  force to operate lever mechanism

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Model 2

• Model changes
• ?
• This can be interpreted as follows power
  generated to drive thelever and bellows (f2 v2)
  is equal to the power generated by the
  electromagnetic actuator (f1 v1) minus the power
  used in altering the kinetic energy stored in the
  mass of magnet.
• ?

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Model 2

• Define
• The model
• Drawbacks of this model
• q0 changes instantaneously with changes in ei
• If ei changes suddenly (e.g. a step change), then
  ii assumes unbounded. This does not happen in
  real life hence this model is still inadequate
  and needs to be further modified!!

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Model 3

• In model 2, ignore all energy losses in the
  system.
• In this model, we include a feature to account
  for one such loss. It treats the coil (or the
  electromagnetic actuator) as having a non-zero
  resistance R. As a consequence, energy is
  dissipated in the form of heat.
• ?
• The Model

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Model 3

• This model overcomes all the drawbacks of model
  1.
• However, it is still not adequate as it ignores
  various other features compressibility, air
  friction, etc.
• The underlying mathematical formulation for both
  model 2 and 3 is the same in that they each
  involve two coupled ODEs. However, in model 1,
  the underlying formulation is algebraic!!
• In general, as we make the system
  characterization more detailed, i.e. include more
  features, the complexity of the underlying
  formulation use in the model also increases.

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Model 4

• System characterizationinclude effect of air
  compressibility -- we have one more energy
  storage element
• ?
• The model
• Properties If q0 is held constant, then p4 must
  fluctuate with time as v1 fluctuates. Since
  p0p4, this implies that both the output pressure
  and flow rate cannot be held constant. If the
  output pressure is held constant, then the output
  flow rate fluctuates and is given by

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Model 5

• System characterization includes losses due to
  air friction resistance and coil resistance (R0
  is the resistance coefficient)
• ?
• The model
• With (three coupled ODEs)

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Model 6

• System characterization includes two new
  features damping in the bellows and inductance
  of the coil
• ?
• ?
• The Model (4 ODEs)
• 
  

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Model analysis

• The input voltage is the line voltage which is
  sinusoidal, the output pressure p0 is a constant,
  the current drawn (ii) and the flow rate (q0) are
  also sinusoidal.
• The variables of interest are the amplitudes of
  ii and q0 as functions of model parameters and
  input line frequency.
• Define

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Model analysis

• ODEs in matrix form
• with

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Model analysis

• The model can be solved either analytically or
  numerically to obtain q0(t) and ii(t) for
  specified parameter values.
• Steady states and their stability
• Time domain analysis
• Input voltage ei is sinusoidal
• with an amplitude of 220 volts
• and frequency of 50 cycles/s
• The output pressure is assumed
• to be 100 kilo Pascals

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Model analysis

• Solve it numerically via RK4

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Model analysis

• Frequency domain analysis
• Laplace transform (1785, Pierre-Simon Laplace)
• Very powerful method to analysis 1st order linear
  ODEs
• Take Laplace transform
• Since

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Model analysis

• From
• After some simplification, we get
• with

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Model analysis

• We can study the behavior
• Define
• Good design requires that Gain be maximum at
  f50, implying that the pump develops maximum
  output flow rate f50

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Discussions and conclusions

• Discussions
• From model 1 up to model 6, each model is an
  improvement over the previous one. Thus, models
  1-5 are special cases of model 6.
• Model 1 uses a static formulation with only 3
  parameters, model 6 uses a dynamic one with 4
  coupled 1st-order linear ODEs.
• All the relationships are linear. Thus they are
  justified only when changes in the variables are
  small. When the changes become large, the linear
  relationships are no longer valid and one needs
  to use nonlinear relationships.

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Conclusions

• A design problem where mathematical model plays
  an important role in the engineering task of
  designing.
• Further question is to transform the problem to
  an optimization problem by defining the
  optimization criterion as the maximization of
  Gain at f50.
• Here we illustrate the iterative nature of model
  building starting from the simplest model and
  increasing the complexity till an adequate model
  of least complexity is obtained!!