ΜΕΡΟΣ V ΡΕΟΛΟΓΙΑ ΤΗΓΜΑΤΩΝ ΠΟΛΥΜΕΡΩΝ. Σημειώσεις Μαθήματος «Ρεολογία & Μορφοποίηση Πολυμερών Υλικών» Α.. Παπαθανασίου, Ανοιξη 2012
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- Φόρκυς Ουζουνίδης
- 8 χρόνια πριν
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1 ΜΕΡΟΣ V ΡΕΟΛΟΓΙΑ ΤΗΓΜΑΤΩΝ ΠΟΛΥΜΕΡΩΝ Α.. Παπαθανασίου, Ανοιξη 2012
2 ΖΗΤΟΥΜΕΝΟ ΠΡΟΣ ΙΟΡΙΣΜΟΣ ΤΟΥ ΙΞΩ ΟΥΣ (μ) ΤΗΓΜΑΤΩΝ ΠΟΛΥΜΕΡΩΝ ΚΑΙ ΤΩΝ ΠΑΡΑΜΕΤΡΩΝ ΠΟΥ ΤΟ ΕΠΗΡΕΑΖΟΥΝ ΟΡΙΣΜΟΣ Μονάδες Pa/(1/s)=Pa.s ΓΕΝΙΚΑ, (,T,P)
3 ΙΞΩ ΕΣ ΜΕΡΙΚΩΝ ΙΑΛΥΜΑΤΩΝ ΠΟΛΥΜΕΡΩΝ Ψευδοπλαστικά ρευστά
4 ΤΟ ΓΕΝΙΚΕΥΜΕΝΟ ΝΕΥΤΟΝΙΚΟ ΡΕΥΣΤΟ 1: Zero-shear-rate viscosity μ=μ 0 ιξώδες σε μηδενικό ρυθμό διάτμισης 2: Power-law region μ~γ n-1 n<1 Περιοχή εκθετικής συμπεριφοράς 3: Transition zone Μεταβατική περιοχή
5 ΜΟΝΤΕΛΑ ΓΙΑ ΤΟ ΙΞΩ ΕΣ Carreau Cross 1 2 (1n ) / 2 0 [1 ( t1 ) ] 1 (1n ) 0 [1 ( t 1 )]
6 ΜΟΝΤΕΛΑ ΓΙΑ ΤΟ ΙΞΩ ΕΣ Carreau model is recovered at a=2 and b=(n-1)/2, where (n) is the power-law index
7 ΑΛΛΑ ΜΟΝΤΕΛΑ ΓΙΑ ΤΟ ΙΞΩ ΕΣ
8 ΠΑΡΑ ΕΙΓΜΑ: PEO solution in water
9 ΕΡΩΤΗΣΗ ΠΟΙΑ ΕΊΝΑΙ Η ΜΟΡΦΗ ΤΗΣ ΚΑΜΠΥΛΗΣ ΤΑΣΗΣ - ΡΥΘΜΟΥ ΙΑΤΜΙΣΗΣ (σ-γ) ΓΙΑ EΝΑ ΝΕΥΤΩΝΙΚΟ ΡΕΥΣΤΟ ΚΑΙ ΓΙΑ EΝΑ ΡΕΥΣΤΟ POWER-LAW???
10 Materials with yield (ΠΛΑΣΤΙΚΑ ΡΕΥΣΤΑ, ΡΕΥΣΤΑ BINGHAM) Herschel-Buckley fluid 0 m n1 As goes to zero, the viscosity becomes infinite (solid-like behavior)
11 ΤΙ ΑΚΡΙΒΩΣ ΕΊΝΑΙ ΤΟ () ΣΕ ΑΠΛΗ ΙAΤΜΗΤΙΚΗ ΡΟΗ, ΕΙΝΑΙ u y Ο ΓΕΝΙΚΟΣ ΟΡΙΣΜΟΣ ΤΟΥ () ΕΙΝΑΙ II 2 ΟΠΟΥ II tr( ) ( ) είναι ο τανυστής του ρυθμού παραμόρφωσης ( u u T ) / 2
12 O ΤΑΝΥΣΤΗΣ ΤΟΥ ΡΥΘΜΟΥ ΠΑΡΑΜΟΡΦΩΣΗΣ (Τhe rate of deformation tensor ()) (Cartesian coordinates)
13 O ΤΑΝΥΣΤΗΣ ΤΟΥ ΡΥΘΜΟΥ ΠΑΡΑΜΟΡΦΩΣΗΣ
14 Ο ΤΑΝΥΣΤΗΣ () ΣΕ ΥΣ ΙΑΣΤΑΤΗ ΡΟΗ A11 A21 A31 A12 A22 A32 A13 A23 A33 A11 A21 A31 A12 A22 A32 A13 A23 A33 = A11 2 A21A11 A31A11 A12A21 A22A21 A32A21 A13A31 A23A31 A33A31 A11A12 A12A22 A12A21 A22 2 A31A12 A32A22 A13A32 A23A32 A33A32 A11A13 A21A13 A13A31 A12A23 A22A23 A13A33 A23A33 A23A32 A33 2 If A 3j =0 (and A=) tr( ) Note: ij = ji
15 ΒΡΕΙΤΕ ΤΟ () ΣΕ ΜΙΑ ΑΠΛΗ ΙΑΤΜΗΤΙΚΗ ΡΟΗ
16 ΒΡΕΙΤΕ ΤΟ () ΣΕ ΡΟΗ ΜΕΣΑ ΣΕ ΣΩΛΗΝΑ
17 ΒΡΕΙΤΕ ΤΟ () ΣΕ ΜΙΑ ΡΟΗ ΑΝΑΜΕΣΑ ΣΕ ΥΟ ΠΑΡΑΛΛΗΛΟΥΣ ΙΣΚΟΥΣ (injection molding) 3Q 1 z 2 u( r, z) [1 8 H r H 2 ] This problem yields a () tensor with a non-zero diagonal component ( 11 )
18 ΕΠΙ ΡΑΣΗ ΘΕΡΜΟΚΡΑΣΙΑΣ ΣΤΟ ΙΞΩ ΕΣ B=-E/R, E~10 5 (J/mol) A*exp( B / T ) E RT 2 ( T ) E RT T T Μικρά λάθη στην μέτρηση της Τ (T/T) μεγεθύνονται μέσω του όρου (E/RT) 1% T-error at ~400K -- 30% -error
19 ΕΠΙ ΡΑΣΗ ΘΕΡΜΟΚΡΑΣΙΑΣ ΣΤΟ ΙΞΩ ΕΣ ΙΣΟ ΥΝΑΜΙΑ ΧΡΟΝΟΥ-ΘΕΡΜΟΚΡΑΣΙΑΣ g T T T g log( ) T 17.4( T 51.6 ( T T g T g ) ) By such data shift, measuring viscosity over ~2 decades of () and at several temperatures, results in a master curve spanning several decades of shear rate ()
20 ΓΕΝΙΚΕΥΜΕΝΗ ΚΑΜΠΥΛΗ ΙΞΩ ΟΥΣ ΓΙΑ ΤΗΓΜΑΤΑ LDPE Reduced viscosity curve for an LDPE at 150 o C
21 ΕΠΙ ΡΑΣΗ ΠΙΕΣΗΣ ΣΤΟ ΙΞΩ ΕΣ ~ exp( P () μεταξύ 2x10-8 και 6x10-8 (Pa -1 ) ) Example: if =3.3x10-8, find the change in viscosity when 30<P<3000 (atm) 1atm~100 kpa Shape of the viscosity curve largely unaffected
22 ΣΥΝΗΘΕΙΣ ΠΙΕΣΕΙΣ ΣΕ injection molding Easily within the range at which the Pressure effect on () becomes visible
23 ΡΕΟΜΕΤΡΙΑ ΣΤΑ ΠΟΛΥΜΕΡΗ Ασχολείται με την μέτρηση του ιξώδους τηγμάτων (και διαλυμάτων) πολυμερών Πάντοτε ισχύει
24 ΠΑΝΤΟΤΕ Το πρόβλημα είναι πώς να συσχετίσουμε τα () and () με κάτι μετρήσιμο Στην πράξη, μετρούμε ύναμη, Ροπή, Παροχή και Πίεση Το () και το () συνδέονται μέσω ενός μοντέλου
25 Παράδειγμα ΙΞΩ ΕΣ ΑΠΌ ΜΕΤΡΗΣΕΙΣ ΠΑΡΟΧΗΣ ΚΑΙ ΠΤΩΣΗΣ ΠΙΕΣΗΣ ΣΕ ΣΩΛΗΝΑ 2000 DP Issues: - model MUST correspond to experiment - entry effects - non-newtonian viscosity Q
26 Η ΚΑΜΠΥΛΗ ΡΟΗΣ (Flow Curve) - vs. γ<0.01, slope=1, Newtonian 0.1<γ<1, slope ~0.56 Viscosity curve The difference between various rheometric methods is how () and () are measured and the achievable range of () NOTE: =m* (n-1) *m n, therefore the slope of ln() vs. ln() is (n)
27 ΜΕΡΙΚΕΣ ΑΠΛΕΣ ΡΟΕΣ ΠΟΥ ΣΥΝΑΤΩΝΤΑΙ ΣΤΗΝ ΡΕΟΜΕΤΡΙΑ ΠΟΛΥΜΕΡΩΝ simple shear flow flow in a pipe (basis for capillary viscometry) flow in a slit (basis for slit viscometry) flow between concentric cylinders flow in a cone-and-plate rheometer Schematic diagram of a SIMPLE SHEAR flow v z (y) v o y h Q v ohw 2
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30 u R r ( ) R r R 1 ( ) 1 Tr R r ( ) R o Torque 4LR Ω R i ΤΟ ΙΞΩΔΟΜΕΤΡΟ Couette
31 ΤΟ ΙΞΩ ΟΜΕΤΡΟ Cone-and-Plate Model (fluid mechanics) M 2R 3sin( ) 3 tan( ) 3M cos( ) 3 2R Schematic diagram of a cone-plate rheometer tan( )
32 ΤΟ ΤΡΙΧΩΕΙ ΕΣ ΙΞΩ ΟΜΕΤΡΟ (CAPILLARY VISCOMETER) Heater Insulation METΡOYME ΠΑΡΟΧΗ (Q) ΠΙΕΣΗ (P) in position of pressure transducer Pressure Transducer Polymer sample Extrudate Σχηματικό διάγραμμα ενός capillary viscometer
33 L R p 1 p 2 Δp=p 1 -p 2 Σχηματικό διάγραμμα ροής σε σωλήνα power-law fluid m n-1 v z (r) Q R 1 s R3 Rp s 32mL s Rp 1 r 2mL R s s1 Where s =1/n Για Νευτωνικό ρευστό (n=1), το ιξώδες είναι η κλίση της καμπύλης (RP/2L) vs. 4Q/R 3. Note that units of the second term in Q= are s -1 this is a shear rate, actually the Wall Shear Rate ( w ). The term RP/2L has units of stress and is the Wall Shear Stress ( w ). Η παραπάνω σχέση είναι γραμμική για ένα Νευτωνικό ρευστό
34 ΡΕΟΜΕΤΡΙΑ ΤΡΙΧΩΕΙ ΟΥΣ ΣΩΛΗΝΑ ΣΕ ΤΗΓΜΑΤΑ ΠΟΛΥΜΕΡΩΝ - ΜΟΝΤΕΛΟ ur () R R DP s 1 1 s 2m L r R s1 d dr ur () r DP 2 s ml s Obviously =(du/dr) is function of (r). Remember, Evaluate at r=r ----> get Wall Shear Rate ( w ) w RP 2m L s Where (m) is evaluated at the wall. This value is UNKNOWN (this is what we are after), so. v z (r) R s Rp 1 r 1 s 2mL R Q R3 Rp s s 32mL s1 w ( s 3) Q!!!!!!! R 3
35 ΡΕΟΜΕΤΡΙΑ ΤΡΙΧΩΕΙ ΟΥΣ ΣΩΛΗΝΑ ΣΕ ΤΗΓΜΑΤΑ ΠΟΛΥΜΕΡΩΝ The wall shear stress w RP 2L Why? Therefore, the viscosity is found by relating Wall Shear Stress (R.P/2L) to Wall Shear Rate [(s+3)*q/r 3 ]
36 ΡΕΟΜΕΤΡΙΑ ΤΡΙΧ0ΕΙ ΟΥΣ ΣΩΛΗΝΑ ΣΕ ΤΗΓΜΑΤΑ ΠΟΛΥΜΕΡΩΝ Q ( s 3) w R 3 But we do not know (s=1/n)!!!!! 4Q 3 R Apparent or Newtonian shear rate (n=s=1) It turns out (n) can be obtained as the slope of the curve ln( w ) vs. ln( ) (prove it!) THEN the shear rate is corrected to be (Rabonowitz correction) FINALLY the w vs. w data are re-plotted The viscosity is obtained as the ratio ( w / w ) w Q ( s 3) R 3
37 Capillary viscometry for non-newtonian fluids (=m n-1, s=1/n) Αρχικά δεδομένα για 232 o C Data plotted as suggested by (RP/2L) s =m s (Q/R 3 )(s+3) (flow curve from capillary data) From the right figure, we get the curve vs., by observing
38 Παράδειγμα ΡΕΟΜΕΤΡΙΑ ΤΡΙΧOΕΙ ΟΥΣ ΣΩΛΗΝΑ ΣΕ ΤΗΓΜΑ ΠΟΛΥΜΕΡΟΥΣ P (kpa) (4Q/R 3 ) (s -1 ) L/D=
39 Παράδειγμα ΡΕΟΜΕΤΡΙΑ ΤΡΙΧOΕΙ ΟΥΣ ΣΩΛΗΝΑ ΣΕ ΤΗΓΜΑΤΑ ΠΟΛΥΜΕΡΩΝ Always P (kpa) (4Q/R 3 ) (s -1 ) w how? ln(w) ln( a ) how? slopes w (corre cted) ln(w) L/D=
40 S w =wall shear stress, a =apparent wall shear rate Sw lsw a la
41 S w =wall shear stress, a =apparent wall shear rate w =corrected wall shear rate lsw lsw Notice: Same slopes, but different curves!!!!! la lw
42 Viscosity vs. shear rate curve at L/D= w lw w lw
43 lw16 Αποτελέσματα μ=μ(γ) για τρία Ιξωδόμετρα διαφορετικού μήκους (L/D=4,12,16) lw12 lw lw16 lw12 lw At this stage, we have considered a power-law fluid Remaining issue: the impossibility of measuring pressure along the length of a tube
44 ΡΕΟΜΕΤΡΙΑ ΤΡΙΧOΕΙ ΟΥΣ ΣΩΛΗΝΑ ΣΕ ΤΗΓΜΑΤΑ ΠΟΛΥΜΕΡΩΝ Επίδραση περιοχής εισόδου Ps Με αυτόν τον τρόπο «διορθώνουμε» την διατμητική τάση στο τοίχωμα (Bagley correction)
45 ΠΩΣ ΒΡΙΣΚΟΥΜΕ ΤΟ ΜΗΚΟΣ ΕΙΣΟ ΟΥ (L) Experiments with capillaries of various lengths and various radii (L/D) w/out end (Bagley) correction, data obtained for different (L/D) are NOT the same (even though the polymer is evidently the same) The Bagley correction (add the entry length in the denominator of w ) corrects this Recall: various () are obtained by using capillaries of different (R) OR by changing (Q) Point where P=Po
46 Παράδειγμα ΡΕΟΜΕΤΡΙΑ ΤΡΙΧOΕΙ ΟΥΣ ΣΩΛΗΝΑ ΙΟΡΘΩΣΗ BAGLEY 5 lw16 lw12 4 lw4 3.5 lsw16 3 lsw12 lsw4 2.5 No end correction lw16 lw12 lw lw16 lw12 lw4 5 lw16 lw12 4 lw4 3 lsw16 lsw lsw4 End correction with N= lw16 lw12 lw lw16 lw12 lw4
47 Παράδειγμα ΡΕΟΜΕΤΡΙΑ ΤΡΙΧOΕΙ ΟΥΣ ΣΩΛΗΝΑ ΙΟΡΘΩΣΗ BAGLEY pg DP.356 DP.189 DP DP LoD interceptdp LoD e interceptdp LoD 16 Correction from the intercept (DP=0) of DP vs. L/D graphs lw16 4 lw12 lw lw16 lw12 lw4
48 The Melt Flow Indexer Schematic diagram of an extrusion plastometer used to measure melt flow index The Melt Flow Index measures the ease of flow of a polymer melt. It is defined as the weight of polymer in grams flowing in 10 minutes through a capillary of specific diameter and length, by a pressure applied via prescribed gravimetric weights for prescribed temperatures. The method is given in ASTM D1238 Example:
49 Foods and other materials that cannot be melted Measures (Q) and (P) P is measured directly Flush-mounted transducers Slit Die Viscometry
50 Slit-die variations: Adjustable gap
51 P-transducers can be mounted with an offset Can measure N!!
52 Cylindrical Die with offset mounted transducers for direct measurement of pressure drop
53 Combination of flush- and offset-mounted pressure transducers to evaluate melt elasticity
54 Slit dies: Summary of pressure readings
55 Velocity profile and flowrate (Q) in a pressure-driven SLIT FLOW s h h z s s z ml p h s Wh Wdy y v Q h y ml p h s h y u 2 2) 2( ) ( ) ( 2 2 / 2 / 1 y uy ( ) d d 1 2 h DP ml s n n Wh Q n n a w ) 6 (
56 εν είναι δυνατή η προβολή αυτής της εικόνας αυτή τη στιγμή. Slit-Die Viscometry: Basic Equations w h 2 P x a w 6Q Wh a 2 2n 1 3n Derive!!!! n [ln( )] w [ln( )] a
57 Use of slit-die viscometry to measure N 1 N 1 =T 11 -T 22 N 1 4P h N 1 2P h (ln( P )) h (ln( )) w
58 Pressure effects in slit-die viscometry When pressure is high, its effect on () is manifested in non-linear pressure profiles Q o Px ( ) 1 ln[ 1 12 x] 3 Wh Q 3 Wh 12 o L
59 Effect of a pressure-dependent viscosity (~0.001 bar-1) P 1 ( x) P 2 ( x) PL 1 ( x) x PL 2 ( x) x P 3 ( x) P 4 ( x) PL 3 ( x) PL 4 ( x) x x
60 Finite slit width (h/w>0.1): Correction for edge effects - Newtonian uxy ( ) DP y 2 ( y h) 4h n 0 1 ( 2n 1) 3 sin cosh ( 2n 1) y h cosh ( 2n 1) x h ( 2n 1) E 2h -E/2<x<+E/2 0<y<h ( c) w 6Q h a h ( ) w ( ) 2 Wh W W QSS( L) L
61 Effect of die body deflection
62 Schematic diagram of a sliding plate rheometer (measure directly shear stress) Shear stress transducer Capacitance proximeter Cantilever beam h- 1mm Sliding plate Active face (1 cm 2 ) Polymer
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Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
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