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23/09/2021

Chemical Engineering Processes 1 ENG_5_474 3. Modified Bernoulli Equation

Dr Donglin Zhao Office: FW301 [email protected]

Bernoulli's Theorem For an incompressible and inviscid fluid, the total mechanical energy of the fluid is constant at all cross-section in the streamline flow

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Mechanical energy includes: pressure energy (PV) potential energy (mgh = ρVgh)   kinetic energy ( 𝑚𝑢   𝜌𝑉𝑢  󰇜

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

𝜌𝑢 𝑃   𝜌𝑔𝑧  𝜌𝑔𝑧  𝑃  2 2 𝜌𝑢

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23/09/2021

Limitation of Bernoulli's Theorem 1. Real fluid has viscosity. 2. The friction between the fluid and pipe wall consumes energy. 3. In a turbulent flow, some kinetic energy is converted into heat. 4. Fluid mechanical energy can also be added by pumps and extracted by turbines. 5. As a liquid is flowing in a fitting such as elbows, bends, valves…there is an energy loss due to centrifugal forces.

Flow from tank

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23/09/2021

Flow from tank

Modified Bernoulli Equation 𝑃 𝑢 𝑃 𝑢  𝑧  𝑧  ℎ    𝜌𝑔 2𝑔 𝜌𝑔 2𝑔

Friction head

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23/09/2021

Head loss in pipes – Major loss No. of velocity head

𝐿 𝑢 ℎ  4𝑓 𝑑 2𝑔

Velocity head, m

f: fanning factor L: pipe length d: pipe diameter u: fluid average velocity

Friction factor or Fanning factor Define

𝑓

𝑅

1  𝜌𝑢 2

Laminar flow,

𝑅  ∆𝑃

𝑑 4𝐿

∆𝑃 32𝜇   𝑢 𝐿 𝑑

∴𝑅

8𝜇𝑢 𝑑

𝑓

16 𝑅𝑒

Turbulent flow in smooth pipe: Blasius equation for 2100...