BIEN 301 Individual Project Presentation

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BIEN 301Individual Project Presentation

• The Linear Momentum Equation
• Hsuan-Min Huang

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The Linear Momentum Equation

• Equation

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The Linear Momentum Equation

• The fluid velocity relative to an inertial
  coordinate system
• The sum of force is the vector sum off all forces
  acting on the system considered as a free body
• The entire equation is a vector relation

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Noninertial Reference Fram

• Equation
• Where

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Pressure Condition at a Jet Exit

• Only two effects could maintain a pressure
  difference between the atmosphere and a free
  exit.
• Surface tension
• Supersonic
• In this problem, both effects are negligible

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PROBLEM p3.98

• As an extension of example 3.10, let the plate
  and its cart be unrestrained horizontally, with
  frictionless wheels. Derive (a) the equation of
  motion for cart velocity Vc(t) and (b) a formula
  for the time required for the cart to accelerate
  from rest to 90 percent of the jet velocity
  (assume the jet continues to strike the plate
  horizontally). (c) Compute numerical value for
  part (b) using the condition of example 3.10 and
  a cart mass of 2 kg.

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APPROACH

• To find the equation of motion for cart velocity
  Vc, we apply equation
• To find the formula for the time required for the
  cart to accelerate from rest to 90 percent of the
  jet velocity, we use the mass analysis in the
  condition of uc 0.9 uj.
• Compute numerical value from example 3.10 into
  the formula we got in

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SKETCH

• Jet stricking a moving plate normally

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SKETCH

• Control volume fixed relative to the plate

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ASSUMPTION

• The system is steady static, so the force acts on
  the cart would not be lost.
• In fact, the sum of the force is zero.
• The fluid is incompressible, so the jet force
  will be applied on the cart completely. However,
  if the fluid is compressible, the pressure will
  not be constant. As a result, some force may be
  absorbed.
• The fluids density and viscosity is constant.
• The wheels of the cart are frictionless, so there
  is no lost in the jet force.

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GIVEN

• From the example 3.10, we got Vj 20 m/s Vc
  15 m/s jet density is 1000 kg/m3
• V1V2Vj-Vc20-155 m/s
• S Fx Rx m1u1m2u2-mjuj
• -?jAj (Vj-Vc)(Vj-Vc)
• Where m1 m2 1/2 mj 1/2?jAj (Vj-Vc)

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Solution

• Part a
• Fx Rx m1u1m2u2-mjuj
• m1 m2 1/2 mj 1/2?jAj (Vj-Vc)
• Vj-Vc uj, and u1 u2 0
• Fx Rx - mjuj
• - 1/2?jAj (Vj-Vc) (Vj-Vc)
• - 1/2?jAj (Vj-Vc)2

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Solution

• Part b
• uc 0.9 uj
• mass analysis, mj /t ?jAj uj
• uj mj /?jAj t
• uc 0.9 mj /?jAj t
• t 0.9 mj /?jA uc

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Solution

• Part c
• t 0.9 mj /?jAj uc
• 0.92(kg)/1000(kg/m3)(0.0003m2)(20m/s) 0.3s

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Computational Results

• Part a Fx Rx - mjuj
• - 1/2?jAj (Vj-Vc)2
• Part b t 0.9 mj /?jAj uc
• Part c t 0.3s

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Applications to Biofluids

• Blood cells move along with blood
• We can apply the concept of the linear momentum
  to measure the erythrocytes, leukocyte, and
  thrombocytes move along with the blood. When our
  heart beats are increased, the blood cells speed
  will be increased by blood. It would be the same
  situation as the problem3.98. We can find the
  speed for the blood cells from the same concept.