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Formula Sheet - Basics of Hydraulics

The following are the formulas used for exams and assignments.

Course

Fluid Power Control Systems (ME 561)

University

University of Waterloo

Academic year: 2019/2020

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Warning: TT: undefined function: 32 Warning: Popup annotation has a missing or invalid parent annotation. Warning: Popup annotation has a missing or invalid parent annotation. Warning: Popup annotation has a missing or invalid parent annotation. Chapter 1 - Basics of Hydraulics:

Force ܹൌ݃݉ൌܨ ሾ݂ܾ݈ሿሾܰሿ

Density ൌߩ

௠

௏

ݐ݂/ܾ݈ሾ

ଷ

݉/݃݇ሿሾ

ଷ

ሿ

Specific weight ൌߛ

ௐ

௏

ݐ݂/݂ܾ݈ሾ

ଷ

݉/ܰሿሾ

ଷ

ሿ

Specific gravity ൌ ܩܵ

ఊ ೚೔೗

ఊ ೢೌ೟೐ೝ

ሾ ሿ

Pressure ൌ݌

ி

஺

ቂ

௟௕௙

௙௧

మ

ቃ ሾ

௟௕௙

௜௡

మ

ሾ ሿ݅ݏ݌ൌ

ே

௠

మ

ሿܽܲൌ

Pressure differential ݖ݀ߛെ ൌ ݌݀

Work ܨൌܷ

റ

݀∙

റ

ሿ݉ܰ ൌ ܬሾ

Power ൌܲ

௎

௧

ܨൌ

௟

௧

ݒܨൌ ቂ

ௐ

଻ସ଺

ܲܪൌቃሾ

௙௧

೗್೑

ೞ

ହହ଴

ሿܲܪൌ

Torque ܶ

ሬറ

ܨൌ

റ

റݎൈ

ሾ

݉ܰ ൌ ݐ݂ ݂ܾ݈

ሿ

Torque horsepower ൌ ܲܪܶ

ே்

ହଶହଶ

ሾ

௥௘௩

௠௜௡

ሿݐ݂ ݂ܾ݈

ൌ ܲܪܶ

ே்

଻ଵଶଵ

ሾ

௥௘௩

௠௜௡

ሿ݉ܰ

Bulk modulus ൌെߚ

ଵ

ట

ൌെ

ఘ

೏ഐ

೏೛

ൌെ

ௗ௣

೏ഐ

ഐ

ൌെ

ௗ௣

೏ೇ

ೇ

ሾ

݅ݏ݌

ሿ

Absolute viscosity ൌߤ

ఛ

೏ೠ

೏೤

ቂ݂ܾ݈

௦

௙௧

మ

ቃቂ ݊ݕ݀

௦

௖௠

మ

ቃܰሾ

௦

௠

మ

ሿ

Kinematic viscosity ൌ߭

ఓ

ఘ

ሾ

ݐ݂

ଶ

ݏ/

ሿ

ቂ

௠

మ

௦

ቃൌ ݁݇݋ݐݏሾ

௖௠

మ

௦

ሿ

Reynolds number ܴ

௘

ൌ

ఘ௩஽

ఓ

ൌ

௩஽

జ

ሾ ሿ

Hydraulic diameter ܦ

௛

ൌ

ସ஺

೎ೝ೚ೞೞ೐೎೟೔೚೙

௪௘௧ ௣௘௥௜௠௘௧௘௥

ሾ

ݐ݂

ሿሾ

݉

ሿ

Pascal’s law ܲ

௜

ܲൌ

௞

applied pressure is transmitted

Conservation of energy ܧ

௧௢௧௔௟

݃݉൅ݖ݃݉ൌ

௉

ఊ

൅

ଵ

ଶ

ݒ݉

ଶ

Volume flowrate ܣൌܳ

ଵ

ݒ

ଵ

ܣൌ

ଶ

ݒ

ଶ

ሾ

݉݌݃

ሿ

ቂ

௙௧

య

௠௜௡

ቃሾ

௠

య

௦

ሿ

Hydraulic work ܸ݌ൌ݈ܨൌܷ ሿ݉ܰ ൌ ܬሾ

Hydraulic power ܳ݌ൌݒܨൌܲ ቂ

ௐ

଻ସ଺

ܲܪൌቃሾ

௙௧

೗್೑

ೞ

ହହ଴

ሿܲܪൌ

Bernoulli’s equation

௉

భ

ఘ௚

൅

௩

భ

మ

ଶ௚

ݖ൅

ଵ

݄൅

௣

݄െ

௙

݄െ

௠௜௡௢௥

݄െ

௠௢௧௢௥

ൌ

௉

మ

ఘ௚

൅

௩

మ

ଶ௚

ݖ൅

ଶ

Chapter 2 – Frictional Losses in Hydraulic Systems

Darcy’s equation ݄

௅

݂ൌ

௅

஽

௩

మ

ଶ௚

ሾ

ݐ݂

ሿሾ

݉

ሿ

Laminar flow ൌ݂

଺ସ

ோ

೐

݄ ;

௅

ൌ

ଵଶ଼ఓ௅ொ

గఘ௚஽

ర

ൌ

଺ସఓ௅௩

ଶఘ௚஽

మ

ሾ

ݐ݂

ሿሾ

݉

ሿ

Turbulent flow

ఌ

஽

; ܴ

௘

and determine ݂ from Moody Table

Losses in valves and fittings ݄

௅

ܭൌ

௩

మ

ଶ௚

ሾ

ݐ݂

ሿሾ

݉

ሿ

Equivalent length ܮ

௘

ൌ

௄஽

௙

; k from handbooks

ሾ

ݐ݂

ሿሾ

݉

ሿ

Chapter 3 – Pumps

Theoretical flowrate ܳ

்

ܸ ൌ

஽

ܰ ቂ

௠

య

௥௘௩

௦

ቃሾ

௜௡

య

௥௘௩

௠௜௡

ሿ

Rotational rate ܰ ቂ

௥௘௩

௦௘௖

ቃቂ

௥௘௩

௠௜௡

ቃ

Rotational velocity ܰߨൌ2߱ ቂ

௥௔ௗ

௦௘௖

ቃ

Theoretical torque ܶ

்

ൌ

௣௏ ವ

ଶగ

ሿܰ ݉ሾሾ݂ܾ݈ ݊݅ሿ

Volumetric efficiency ߟ

௏

ൌ

ொ

ಲ

ொ ೅

ܳ

஺

actual flowrate pushed through outlet of the pump

ܳ

்

theoretical flowrate (what pump could push, no leakage)

Mechanical efficiency ߟ

௠

ൌ

௉ ೅

௉

ಲ

ߟ

௠

ൌ

ܶ

்

ܶ

஺

ߟ

௠

ൌ

ܳ݌

்

ܶ

஺

߱

ܲ

்

theoretical power (what pump could deliver, no leakage)

ܲ

஺

actual power delivered to the pump

ܶ

்

theoretical torque (what pump could deliver, no leakage)

ܶ

஺

actual torque delivered to the pump by electrical motor

Overall efficiency ߟ

௢

ߟൌ

௏

ߟ

௠

ൌ

௣ொ

ಲ

்

ಲ

ఠ

Pumps: Gear Pumps, Vane Pumps, Piston Pumps

Motor

Pump

INLET

OUTLET

஺

߱,ܰ,

஺

݌,

ࡽ

ࢀ

ࢀ,

ࢀ

Chapter 6 – Hydraulic Systems - Ancillary Devices

Additional Example:

Figure 1 shows the duty cycle of a hydraulic circuit. Design an accumulator if the size of the pump is

required to be minimum. The maximum system pressure is 200bar, the working pressure is 150 bar and ߛ

is 1. If the overall efficiency of the pump is 85%, what is the required horsepower of the electric motor?

Figure 1

4 7 10

1 cycle

Chapter 7 – Hydraulic Systems - Circuit Design

Example 1: (Case Study) A single acting up stroking moldings press has a cylinder diameter of 400 mm,

a stroke of 250 mm of which 225 mm is to close the dies onto the work-piece at a cylinder pressure of 20

bar. The final 25 mm stroke is to be at a cylinder pressure of 350 bar. The press is to produce 60

components per hour, The breakdown of the required operating time is:

 Rapid approach 225 mm = 5 seconds

 Pressing over final 25 mm = 5 seconds

 Cure time (i. hold under full thrust)=25 seconds

 Return time= 15 seconds

 Unload and reload press= 10 seconds

 Total cycle time=60 seconds

Solution (1)-Analyze the shown proposed circuit and find the efficiency for this circuit.

Solution (2)-Other circuit has been designed using an accumulator? How can it increase the efficiency?

Hint: In this study, ignore the required electrical control circuit!

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Formula Sheet - Basics of Hydraulics

Course: Fluid Power Control Systems (ME 561)

University: University of Waterloo

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Chapter 1 - Basics of Hydraulics:

Force  󰇟󰇠󰇟󰇠

Density 

 󰇟/󰇠󰇟/󰇠

Specific weight 

 󰇟/󰇠󰇟/󰇠

Specific gravity   

 󰇟󰇠

Pressure 

 󰇣

󰇤󰇟

󰇠󰇟

󰇠

Pressure differential   

Work 

∙

 󰇟  󰇠

Power 



 󰇣

 

󰇤󰇟



 󰇠

Torque 

󰇍





 󰇟  󰇠

Torque horsepower   

 󰇟

 󰇠

  

 󰇟

 󰇠

Bulk modulus 





 









󰇟󰇠

Absolute viscosity  





󰇣 

󰇤󰇣 

󰇤󰇟 

󰇠

Kinematic viscosity 

 󰇟/󰇠󰇣

󰇤󰇟  

󰇠

Reynolds number 



 󰇟󰇠

Hydraulic diameter 

 󰇟󰇠󰇟󰇠

Pascal’s law 

 applied pressure is transmitted

Conservation of energy  





Volume flowrate 



 󰇟󰇠󰇣

󰇤󰇟

󰇠

Hydraulic work  󰇟  󰇠

Hydraulic power  󰇣

 

󰇤󰇟



 󰇠

Bernoulli’s equation 

 

 







 

 

 

 



Formula Sheet - Basics of Hydraulics

Fluid Power Control Systems (ME 561)

University of Waterloo

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ݐ݂/ܾ݈ሾ

݉/݃݇ሿሾ

ሿ

ݐ݂/݂ܾ݈ሾ

݉/ܰሿሾ

ሿ

ሾ ሿ

ቂ

ቃ ሾ

ሾ ሿ݅ݏ݌ൌ

ሿܽܲൌ

റ

݀∙

റ

ሿ݉ܰ ൌ ܬሾ

ܨൌ

ݒܨൌ ቂ

ܲܪൌቃሾ

ሿܲܪൌ

ሬറ

ܨൌ

റ

റݎൈ

ሾ

݉ܰ ൌ ݐ݂ ݂ܾ݈

ሿ

ሾ

ሿݐ݂ ݂ܾ݈

ൌ ܲܪܶ

ሾ

ሿ݉ܰ

ൌെ

ൌെ

ൌെ

ሾ

݅ݏ݌

ሿ

ቂ݂ܾ݈

ቃቂ ݊ݕ݀

ቃܰሾ

ሿ

ሾ

ݐ݂

ݏ/

ሿ

ቂ

ቃൌ ݁݇݋ݐݏሾ

ሿ

ൌ

ൌ

ሾ ሿ

ൌ

ሾ

ݐ݂

ሿሾ

݉

ሿ

ܲൌ

݃݉൅ݖ݃݉ൌ

൅

ݒ݉

ݒ

ܣൌ

ݒ

ሾ

݉݌݃

ሿ

ቂ

ቃሾ

ሿ

ܲܪൌቃሾ

ሿܲܪൌ

൅