Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional...

24
Dimensional analysis The Buckingham π Theorem - If there are m physical quantities con- taining r fundamental units, then there are π = m - r dimensionless groups that are independent. Procedure: 1. Write down all relevant physical quantities and their fundamental units. 2. Determine π. 3. Write down the π dimensionless groups. Example: Flow through a circular pipe dP dx = f (Q, R, μ) m =4 dP dx dynes cm 3 = g cm 2 s 2 Q (cm 3 /s) R (cm) μ dynes cm 2 s= g cm s g, cm, s r =3 π = m - r =1 1 μ dP dx 1 cm s only way to get rid of grams 1 μQ dP dx 1 cm 4 only way to get rid of seconds R 4 μQ dP dx (dimensionless) Dimensional Analysis dP dx μQ R 4

Transcript of Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional...

Page 1: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Dimensional analysisThe Buckingham π Theorem - If there are m physical quantities con-

taining r fundamental units, then there are π = m− r dimensionless groupsthat are independent.

Procedure: 1. Write down all relevant physical quantities and theirfundamental units.2. Determine π.3. Write down the π dimensionless groups.

Example: Flow through a circular pipe

dP

dx= f(Q, R, µ) m = 4

dP

dx

(dynes

cm3=

g

cm2s2

)Q (cm3/s)

R (cm)

µ

(dynes

cm2s =

g

cm s

)g, cm, s r = 3

∴ π = m− r = 1

1

µ

dP

dx

(1

cm s

)only way to get rid of grams

1

µQ

dP

dx

(1

cm4

)only way to get rid of seconds

R4

µQ

dP

dx(dimensionless)

Dimensional Analysis ⇒ dP

dx∼ µQ

R4

1

Page 2: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Dimensional analysis

Dimensional Analysis ⇒ dP

dx∼ µQ

R4

Not as useful as the exact result:

∆P

L=

8

π

µQ

R4

because exact result has the prefactor.

How did we get so much information? We cheated!

Used “experience” to guess that ∆P and L would only enter as dP/dx =∆P/L, and that the velocity v would only enter though Q.

“Honest” Treatment:

∆P = f(Q,R,L, µ, v)

∆P (g/cm s2)

Q (cm3/s)

R (cm)

L (cm)

µ (g/cm s)

v (cm/s)

m = 6 r = 3 ∴ π = m− r = 3

2

Page 3: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Dimensional analysisEXAMPLE: FLOW IN AN EXTRUDER

dP

dx= f(Q, Ds, hs, Ns, φs, µ)

( g

cm2s2

)Q (cm3/s) volumetric flow rate

Ds (cm) barrel diameter

hs (cm) flight depth

Ns (1/s) screw rotation rate

φs (dimensionless) helical angle of screw

µ (g/cm s) viscosity

m = 7, r = 3 ⇒ π = m− r = 4

Four dimensionless groups

φs,Ds

hs

,Q

NsD3s

1

µ

dP

dx

(1

cm s

)(cancels grams)

Ds

µNs

dP

dx(dimensionless pressure gradient)

4

Page 4: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Dimensional analysisEXAMPLE: FLOW IN AN EXTRUDER

Ds

µNs

dP

dx= f

(φs, Ds/hs, Q/(NsD

3s)

)Optimum φs = 23◦ (universally utilized)Most extruders are geometrically similar and designed to provide a certain

pressure at the die:

∴Ds

hs

∼= constant

∴Ds

µNs

dP

dx∼= constant

If other three dimensionless groups are constant, then the last one mustalso be constant.

Expect Q ∼ NsD3s (not obvious)

5

Page 5: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Melt Fiber-Spinning

1

Page 6: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Melt vs. Solution Fiber-Spinning

2

Page 7: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

PIPE EXTRUSION: Die Swell is Crucial!

Mandrel Supports:

1

Page 8: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Profile Extrusion

3

Page 9: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Sheet ProcessingSHEET EXTRUSION

1

Page 10: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Sheet ProcessingCLAM SHELLING IN A SHEET DIE

Figure 1: Pressure is very high (100atm) at the center of the die.

Figure 2: High pressure causes die to creep. After extensive extrusion, sheetwill be thicker in the center.

2

Page 11: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Sheet ProcessingPOST-DIE SHEET FORMING

Calendering Thin Sheet:

3

Page 12: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

COEXTRUSION LAYER PACKS

9

Page 13: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

COEXTRUSIONBARRIER POLYMERS

EVOH (ethylene-vinyl alcohol) random copolymer with 30%EPVDC (polyvinylidene chloride) [CH2CCl2]

8

Page 14: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

COEXTRUSION

5

Page 15: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

COEXTRUSION

6

Page 16: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

COEXTRUSION

7

Page 17: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

FILM BLOWING

3

Page 18: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

FILM BLOWING

4

Page 19: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Profile

Extru

sion

2

Page 20: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Wire Coating

Combination of Poiseuille flow (pressure driven) and Couette (drag) flow dueto the moving wire.

First we do the Couette Flow problem assuming ∆P = 0.

vr = vθ = 0 vz = vz(r)dP

dr=

dP

dθ=

dP

dz= 0

Continuity is identically satisfied

N.-S.: µ1

r

d

dr

(rdvz

dr

)= 0

d

dr

(rdvz

dr

)= 0 r

dvz

dr= C1

dvz

dr=

C1

rvz = C1 ln r + C2

5

manias
Text Box
More on Wire-Coating in the NEXT LECTURE
Page 21: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

Sheet ProcessingTHERMOFORMING

R.G. Griskey, Polymer Process Engineering, chapter 10

J. Florian, Practical Thermoforming (Marcel Dekker, 1987)

Extrude a Sheet, Clamp it, Heat it up, and ...

4

Page 22: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

THERMOFORMINGVACUUM FORMING

∆P = 1atm.

5

Page 23: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

THERMOFORMINGVENTING

6

Page 24: Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional analysis EXAMPLE: FLOW IN AN EXTRUDER dP dx = f(Q,D s,h s,N s,φ s,µ) g cm 2s Q (cm3/s)

THERMOFORMINGPRESSURE-ASSISTED

To exceed ∆P = 1 atm. use pressurized air.

Avoid trapped air by using vacuum.

7