ENGI 2102 Thermo-Fluid Engineering I
Chapter 1: Introduction to the Mechanics and
Thermodynamics of Fluids
G. Mazzanti
Process Engineering and Applied Science
Dalhousie University
Fall 2019
Slides by Michele Hastie, 2016
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ENGI 2102 Thermo-Fluid Engineering I
Chapter 1: Introduction to the Mechanics and
Thermodynamics of Fluids
G. Mazzanti Process Engineering and Applied Science Dalhousie University Fall 2019 Slides by Michele Hastie, 2016
1.4 Classification of Fluid Flows
A fluid is a form of matter that is unable to withstand an applied shear stress ( τ ). Given sufficient time, the smallest shear stress is capable of producing any change of shape. This is in contrast to the behaviour of a solid , in which a definite value of the shearing stress must be applied to produce a given deformation.
Solid
Fluid
τ τ τ
τ τ τ
Time τ
τ
When the final state of steady motion has been achieved, a constant force is needed to keep the plate in motion. Provided the flow is laminar, the force is given by Newton's law of viscosity :
Fluids that obey Eq. (1.3) are termed Newtonian fluids All gases and most simple liquids are Newtonian fluids Fluids such as pastes, slurries and high polymers, which do not obey Eq. (1.3), are termed non-Newtonian fluids
(d) Largestate t , steady
u (^) plate
t >> 0 u (^) x = u (^) x (y)
y
u
F A x
= −μ
F
y
u
A
F x
yx ∂
Shear stress
y
x z
Shear rate ( γ̇ )
yx
x
y
u
A
F
μ = τ
The fluid molecules at the plate are moving in the x -direction at a velocity of u (^) plate By random molecular movement , some of these molecules will move in the y -direction
As a result, the x -directed momentum will be transferred in the y - direction from the faster- to the slower-moving layer (momentum flux) The momentum transferred to the next fluid layer causing that layer to move in the x -direction (at a lower velocity) The resulting velocity profile will depend on the viscosity of the fluid
(d) Largestate t , steady
u (^) plate
t >> 0 u (^) x = u (^) x (y)
y
x z
Direction of the velocity ( x )
Direction of momentum transfer ( y )
1.5 Viscosity (Appendix B – pages 292-296, 299)
Viscosity ( μ ) is a function of temperature ( T )
Use the value for μ at the system T
As the temperature of a liquid increases: The kinetic energy of the molecules increases. The degree of molecular motion increases. This decreases the short- range attractive forces between the molecules. This results in a decrease in the viscosity of the liquid.
http://chemwiki.ucdavis.edu/Physical_Chemistry/Physical_Properties_of_Matter/Bulk_Properties/Viscosity
As the temperature of a gas increases: The kinetic energy of the molecules increases. The degree of molecular motion increases. This leads to more interaction between molecules causing the viscosity to increase. However, the viscosity of gases is not as sensitive to temperature as that of liquids.
Shear-rate thinning fluids (pseudoplastics) – viscosity decreases with increased shear stress Ketchup, molasses, syrups, honey, paint, blood, polymer solutions, etc.
http://alcheme.tamu.edu/?page_id=5933 http://www.dailymail.co.uk/sciencetech/article-2034273/Why-ketchup-squirts-Scientists-uncover-mystery-non-Newtonian-liquids.html
Shear-rate thickening fluids (dilatants) – viscosity increases with increased shear stress Cornstarch in water Used in some four-wheel-drive vehicles Currently being researched for use in body armour
http://www.dandelionsonthewall.com/2012/01/liquid-or-solid-make-and-oobleck.html http://en.wikipedia.org/wiki/Viscous_coupling_unit
Newtonian
- Shear-rate thinning (pseudoplastic)
- Structural
- Shear-rate thickening (dilatant)
𝛾𝛾̇
𝜇𝜇(𝛾𝛾̇ )^ =
Viscosity
- Bingham
Fig. 1.7: Viscosity vs. shear rate for various time-independent fluids.
Example 1.1: As a junior design engineer you are explaining possible designs for a parallel plate viscometer. Your initial design consists of a vertical, rectangular box with a centrally located plate inside. The fluid to be tested is placed in the box and the force necessary to remove the plate at constant velocity is measured.
Calculate the viscosity in cP for the following conditions: weight of the plate is negligible the plate is located equidistant between the walls the clearance between the plate and each wall is 0.50 cm the area of the plate immersed at the instant of reading is 70 cm^2 on each side when the plate is moved at 1 cm/s, the force required is 5.6×10-4^ N end effects may be neglected
u = 1 cm/s (0.01 m/s)^ F = 5.6 x 10-4^ N
(5 x 100.5 cm -3^ m)
A = 2 x 70 cm= 140 cm 22 (1.4 x 10 -2^ m^2 )
Viscometer – A device that measures a fluid’s viscosity
The Couette or cup-and-bob viscometer consists of two concentric cylinders an outer rotating “cup” an inner stationary “bob”
The torque on the inner cylinder is:
Torque = force × arm = shear stress × surface area × radius
T = ( )(τ 2 π Ri L )( Ri )
R L
T 2 π i^2
τ =
Ro Ri
Ω^ L
T
Angular velocity (rad/s)
If the annular gap is small, the curvature can be neglected, and the average shear rate is given by:
Recall that:
Ro Ri
Ω^ L
T
o i
o o i
o i R R
R
R R
u u r
u dr
du −
γ = ≅
0
o i
o o i
o R R
R n R R
R
γ 2 π
= ^ ⇒ = ∂
∂
y
u
Rotation frequency (s -1)
1.6 Surface Tension and Capillary Effects (page 18)
A surface is an interface between two phases (solid, liquid, or gas)
Surface tension is observed at this interface
Molecules within the bulk liquid are attracted equally in all directions
Molecules at the surface are drawn toward the centre because there are no liquid molecules to pull them outward
Free surface
Molecule at surface
Molecule within bulk
1.6 Surface Tension and Capillary Effects (page 18)
The result is a pulling force that acts parallel to the surface and causes the liquid to minimize its surface area The force imbalance at the surface has the effect of an elastic film
http://chemwiki.ucdavis.edu/Wikitexts/Simon_Fraser_Chem1%3A_Lower/States_of_Matter/Liquids_and_their_Interfaces