Modeling Fluid Systems

Dr. Nhut Ho

ME584

Chp5 1

Agenda

• Introduction

• Properties of Fluid and Reynolds Number Effects

• Passive Components

• Case Study: Spring-Loaded Diaphragm Actuator

• Active Learning: Pair-share Exercises, Case Study

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Introduction

• Fluid Systems– Operate through effects of either liquids or gases

– Have wide range of applications, e.g., vehicle suspension systems, hydraulic servomotors, and chemical processing systems

• Hydraulics (fluid is incompressible) and pneumatic (fluid is compressible) systems– Common modeling principle is conservation of mass

– Key advantages relative to electro-mechanical systems

• Power density of pump/actuators (1 order of mag. higher 200 psi electromagnetic actuator vs. 3000-8000 psi hydraulic actuators)

• Circulating fluid removes heat generated by actuator (instead of free or forced convection)

– Have more nonlinearities -> challenging for modeling and simulation

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Properties of Fluid

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Fluid Density

• Incompressible: density of fluid (e.g., liquid) remains constant despite changes in fluid pressure (an approximation, but simpler modeling)

• Compressible: density of fluid (e.g., gas) changes with pressure

• Liquids have higher density, absolute viscosity, bulk modulus, and exhibits surface tension effects

• Density of a fluid: mass m per unit volum V under pressure P0

and temperature T0

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Equation of State: Liquids

• Equation of state: relationship between ρ,P, and T :

• Substitute with Bulk modulus β and Coefficient of thermal expansion α

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Reference conditions: ρ0,P0, T0,

Equation of State: Liquids

• Isothermal bulk modulus: pressure changes occur at a slow enough rate during heat transfer to maintain constant temperature

• Adiabatic bulk modulus: pressure change is significant, preventing heat transfer. Specific heats ratio Cp/Cv ~ 1

• Thermal expansion coefficient: incremental change in volume with temperature changes ~ 0.5x10-3/0F for most liquid

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(Cp and Cv specific heat at constant pressure and temperature)

Equation of State: Gases

• Ideal gas:

• Gas undergoing polytropic process:

• Bulk modulus:

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(βliquid ~ 5 to 15 Kbar >> β gas ~ 1 to 10 Bar)

Viscosity

• Absolute viscosity μ

• Kinematic viscosity: ν=μ/ρ

• Liquids: λL constant depends on liquid

• Gases: λG constant depends on gas

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Speed of Sound, Specific Heat Ratio, and Reynolds Number

• Speed of sound or propagation

• Specific ratio

• Reynolds Number: inertial forces / viscous forces

Laminar: Nr < 1400, transition: 1400<Nr<3000, turbulent: Nr > 3000

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~ 1370 m/s in oil at 250 C, ~ 347 m/s in air at 250C

~ 1.4 for gases and 1.04 for liquids

Passive Components: Capacitance, Inductance, Resistance

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Chap5 12

Fluid CapacitanceChp5 13

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Example 1: Spring-Loaded Piston Capacitance

• Find total capacitance for this system

• Accumulators: liquid capacitors– Spring-loaded pistons, bellows, gas-filled bladders

– Use mechanical capacitors when β is large for incompressible fluids

– Use large V to get large compressibility capacitance

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Mechanical capacitance

Compressibility capacitance

(βliquid ~ 10 Kbar >> β gas ~ 10 Bar)

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Capacitance for GasesChp5 15

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Example 2: Capacitance of Thin-Walled Tube

• A circular tube of length l is used to hold fluid pressure. If the tube has an internal diameter di, a wall thickness t, and a Young’s modulus E,

– Derive the capacitance of the tube, using an incompressible fluid,

– Derive the total capacitance CT, which includes the volume capacitance of the fluid, CF (with a fluid of bulk modulus β), and the mechanical capacitance, CM.

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Example 2: Capacitance of Thin-Walled Tube Chp5 17

Example 2: Capacitance of Thin-Walled Tube Chp5 18

Example 3: Pair-Share: Capacitance of a Balloon

The radius expansion, R-R0, of a balloon filled with a gas is directly proportional to the internal pressure of the gas. Let us write this proportionality as δP=K(R-R0). Derive an expression for the total capacitance of the balloon that considers the change in volume of the balloon and the effect of compressibility of the gas. (Volume = 4πR3/3)

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Example 3: Pair-Share: Capacitance of a Balloon

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CV

Fluid Inductance

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Fluid Resistance

• Laminar flow: viscous-dominated flow– Low enough flow rates or pressure drop in long capillary

tubes -> viscous flow

– Viscous terms dominate -> Reynolds is low (Nr < 1000)

• Orifice-type or head loss resistance: inertia-dominated flow– Orifice with short length in direction of flow

– Head loss with turbulent flow

• Compressible flow resistance– Similar to orifice type, but includes density variation of gas

– Flow equations have high degree of nonlinearity

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Laminar Flow Resistance

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Example 4: A Liquid-Level System

The tank shown has a mass inflow rate of . The liquid height above the orifice is h. Compute the time constant of the system, assuming that the flow is laminar. The tank contains fuel oil at 700F with a mass density ρ of 1.82 slug/ft3 and a viscosity μ = 0.02 lb-sec/ft2. The outlet pipe diameter D is 1 in., and its length L is 2 ft. The tank is 2 ft in diameter.

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Example 4: A Liquid-Level System

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Orifice Flow ResistanceChp5 26

Accelerating Flow

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Example 5: Tank with an Orifice

• The tank shown has an orifice in its side wall. The orifice area is A0 and the bottom area of the tank is A. The liquid height above the orifice is h. The volume inflow rate is Qv. Develop a model of the height h with Qv as the input.

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Qv

AA0

h

Example 5: Tank with an Orifice

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This is a nonlinear equation, how can it be analyzed?