Viscosity and Polymer Melt Flow - University of Thessalycan be used for direct determination of...
Transcript of Viscosity and Polymer Melt Flow - University of Thessalycan be used for direct determination of...
Rheology-Processing / Chapter 2 1
Viscosity and
Polymer Melt Flow
Rheology-Processing / Chapter 2 2
Viscosity: a fluid property → resistance to flow(a more technical definition resistance to shearing)
Remember that:
dy
duμτ
shear stress F/A
viscosity
shear rate: ≈ U/H
Rheology-Processing / Chapter 2 3
POLYMER MELTS
THE VISCOSITY DECREASES AS THE SHEAR RATE (du/dy) INCREASES
(due to molecular alignments and disentanglements)
A characteristic value is the limiting viscosity for zero shear rate(frequently referred to as zero shear viscosity) usually denoted withηo.
Rheology-Processing / Chapter 2 4
ZERO SHEAR VISCOSITY is a function of POLYMER MOLECULAR WEIGHT
Critical molecular weight Mc ....... forentanglements to become effective.
Mc critical molecular weight for entanglements.Mc ≈ 4,000 for PE, Mc ≈ 36,000 for PS
Most (NOT all) commercial polymers have Mw=50,000 to 500,000
Rheology-Processing / Chapter 2 5
In characterization of polymers, the solution viscosity is often used.
For a polymer dissolved in a solvent:viscosity ↑ as concentration ↑
(solution viscosity definitions in pages 2.4 and 2.5)
For instance, the specific viscosity:
or the intrinsic viscosity:
The intrinsic viscosity is found by plotting ηsp/C against C andextrapolating to zero concentration
solvent
solventsolutionsp
η
ηηη
0
][
C
sp
C
ηη
Rheology-Processing / Chapter 2 6
The so called viscosity average molecular weight, introduced in chapter1, can be found from the Mark-Houwink equation:
[η]: intrinsic viscosityM: (viscosity) average molecular weightK, a: experimentally determined
constants (depend on polymer solventsystem)
αΚΜη ][
Rheology-Processing / Chapter 2 7
SHEAR-THINNING BEHAVIOR OF POLYMERS
From Newton’s law of viscosity
sPasrateshear
Pastressshear
dy
du
τμ
1
Viscosity is constant for Newtonian fluidsBUT
it DECREASES as SHEAR RATE INCREASES for polymer solutions and melts(non-Newtonian fluids)
This behavior is called: SHEAR-THINNING(due to molecular alignments and disentanglements )
Rheology-Processing / Chapter 2 8
We cannot talk about a constant viscosity μ in polymers, but rather about
srateshear
Pastressshearityvis
1cos
We usuallysymbolize itwith theGreek letterη
Rheology-Processing / Chapter 2 9
Most popular model to express the shear-thinning behavior: POWER-LAW
• m is called consistency index (the larger the m the more viscous the melt)
• n indicates the degree of Non-Newtonian behaviorn=1 means Newtoniann<1 for SHEAR THINNING POLYMERS
nγmτ (shear stress) = m (shear stress)n
γητ or or1 nγmη viscosity is a
function of shear rate
Rheology-Processing / Chapter 2 10
The power-law relation gives
γnmη log1loglog Note that consistency index m is equal to viscosity η atOn a log-log paper η vs is a straight line and the slope is equal to n-1
11 sγ
γ
• The power-law fit is OK for high shear rates, butfor low shear rates polymers exhibit aNewtonian plateau (e.g. at values say )
• The power-law model is good for most polymerprocessing operations because the shear rate isbetween 100 s-1 and 5000 s-1
13 sγ
Rheology-Processing / Chapter 2 11
THE MOST POPULAR MODELS:
Carreau–Yasuda: a
na
1
1
where ηo is the viscosity at zero shear and λ, α and n are fitted parameters
Cross model: n
11
UNFORTUNATELY NO ANALYTICAL SOLUTIONS ARE POSSIBLEWITH THESE VISCOSITY MODELS
USED IN COMPUTER SIMULATIONS FOR DETAILED LOCALFLOW ANALYSIS
WHAT ABOUT FITTING THE WHOLE VISCOSITY CURVE?
Rheology-Processing / Chapter 2 12
The consistency index m is sensitive to temperature. A commonrepresentation is:
oTTb
oemm
where mo the consistency index at a reference temperature To
Rheology-Processing / Chapter 2 13
Typical values:
For polymer melts at usual processing conditions m=1000 – 100000 Pa·sn
n=0.2 – 0.8b=0.01 – 0.1 oC-1
For example, for a commercially available polystyrene (PS) the followingparameters were obtained by curve fitting of viscosity data
mo=10800 Pa·sn, n=0.36, To=200oC, b=0.022 oC-1
The above value b corresponds to a viscosity reduction ~20% for 10oCtemperature rise. For isothermal flows we can use the power-law viscositymodel to solve problems of practical importance using analytical methodsof solution.
Rheology-Processing / Chapter 2 14
To solve general flow problems we must set a momentum balance. It turnsout that the momentum balance can be written verbally
Rheology-Processing / Chapter 2 15
• Polymers are characterized by extremely high viscosities (about amillion times more viscous than water) in molten state.
Rheology-Processing / Chapter 2 16
Therefore:
• Pressure p is a scalar
• Velocity is vector V i.e. Vx, Vy, Vz having components in the
x, y and z directions
• Stress is defined as the ratio Force/Area and can be normal or
tangential
• Stress is a tensor having nine components:
(τxx, τyy, τzz) → normal stresses
The rest components → shear stresses
τp 0
Rheology-Processing / Chapter 2 17
For planar unidirectional flows the equation
is simplified to
y
τ
x
p yx
0
A good way to remember it, is symbolically:
flowtonormal
stressshear
flowofdirection
p
)(0
τp 0
Rheology-Processing / Chapter 2 18
PRESSURE DRIVEN FLOW OF A POWER-LAW FLUID BETWEEN TWO FLAT PLATES (book page 2.15)
The absolute value || is needed because sometimes is negative and n-1<0 for polymers.yVx
Rheology-Processing / Chapter 2 19
←boundary condition
Delicate point: the right-hand side is negative and thusmust be negative.yVx
Rheology-Processing / Chapter 2 20
Apply NO-SLIP boundary condition: Vx=0 at y=b
This is the velocity profile!
Rheology-Processing / Chapter 2 21
The maximum velocity Vmax is at y=0
The average velocity Vavg is calculated by
Rheology-Processing / Chapter 2 22
The volume flow rate per unit width W (plate width) is given by
The pressure drop may then be easily calculated
Rheology-Processing / Chapter 2 23
From a previous relation we saw that
which means that the stress varies linearly in the gap.
The maximum value is at the wall (i.e. at y=b)
the w subscript refers to…. “wall”
The negative sign simply indicates that when this quantity is multiplied by thearea, it gives a force (i.e. Fw=τw·A) that is exerted by the plate on the fluid which is,of course, in the negative x direction. The force exerted by the fluid on the wettedplate should therefore be positive.
Rheology-Processing / Chapter 2 24
Another important quantity that we need to calculate is the shear rate
We may then calculate the shear rate at the wall (absolute value)
The maximum shear stress at the wall can also be expressed as
Rheology-Processing / Chapter 2 25
PRESSURE DRIVEN FLOW OF A POWER-LAW FLUID IN A TUBE (book page 2.19)
governing equation
Rheology-Processing / Chapter 2 26
Boundary conditions
The solution gives the following
velocity profile
maximum velocity
average velocity
volume flow rate
pressure drop
Rheology-Processing / Chapter 2 27
• In the previous equations if we set n=1 we end up with the well-known Hagen-Poiseuille formula for Newtonian fluids.
• Pressure driven flows are also referred to as Poiseuille flows.
For SHEAR-THINNING fluids, the velocity profiles are more flat than the parabolic profiles of Newtonian fluids.
Rheology-Processing / Chapter 2 28
From the previous equations it is easy to see that the stress varies linearly
and the maximum value is at the wall (i.e. at r=R)
The linearity of the stress is often expressed as
τw is taken as positive
Rheology-Processing / Chapter 2 29
The shear rate is:
And the shear rate at the wall:
The maximum shear stress at the wall can be then calculated from
Rheology-Processing / Chapter 2 30
CAPILLARY VISCOMETER ANALYSIS (book page 2.23)
For Newtonian fluids, the Hagen-Poiseuille formula
can be used for direct determination of viscosity μ, from the measurement of pressuredrop ΔP at flow rate Q, through a tube of length L and radius R.
For non-Newtonian fluids, the viscosity η is a function of the shear rate (i.e. )and special treatment is necessary. We will determine the viscosity from its basicdefinition
The wall shear stress can be calculated from
γ γη
γ
τγη
Rheology-Processing / Chapter 2 31
The shear rate requires special manipulations. Again for Newtonian fluids (n=1) theshear rate can be obtained by differentiating the velocity profile
and using the Hagen-Poiseuille formula we get
For non-Newtonian fluids we will develop a general expression for the shear rate at thewall by starting from the definition of the volume rate of flow through a tube
(see subsequent steps followed in pages 2.24 and 2.25. Next slide goes directly to the result)
Rheology-Processing / Chapter 2 32
This equation is usually referred to as the Rabinowitsch equation. It gives the shearrate in terms of Q, R and τw. The term in the parentheses may be considered as a“correction” to the Newtonian expression which is simply 4Q/πR3. To obtain wemust plot Q versus τw on logarithmic coordinates to evaluate the derivative dlnQ/dlnτw
for each point of the curve.
wγ
Rheology-Processing / Chapter 2 33
The previous method can be simplified if POWER-LAW fluid is assumednγmτ
We may then write an empirical expressionn
Rπ
Qm΄τ
3
4
in which n is the slope of the logτw versus log(4Q/πR3) plot, that is
3
4log
log
Rπ
Qd
τdn w
It turns out that for the derivative dlnQ/dlnτw we have dlnQ/dlnτw=1/n, therefore
n
n
Rπ
Q
nRπ
Qγw
4
134
4
1
4
3433
n
n
nm΄m
13
4
This means that the relation between the apparent m΄ and the true m is
Rheology-Processing / Chapter 2 34
STEPS FOR DETERMINATION OF m and n
1. Determine τw from pressure drop Δp data
2. Determine
3. Plot logτw versus log(4Q/πR3) to get n (slope)and m΄ (intercept at )
4. Correct m΄ using
3
4
Rπ
Qγapp
11 sγapp
n
n
nm΄m
13
4
Rheology-Processing / Chapter 2 35
Tapered dies (slits or truncated cones) are used very often in polymer meltprocessing
(a) Flow through a tapered slit
Start from the pressure drop expression derived for flow between parallel plates
Rheology-Processing / Chapter 2 36
For an infinitesimal slit of length dz
Where for a tapered slit
Integration between z=0 and z=L gives
Po PL
note that ΔP=Po-PL
Rheology-Processing / Chapter 2 37
Further noting that
We get
The derivation of the above expression was based on the implicit assumption of nearlyparallel flow. It is a very good approximation for half angles θ of up to 15o.
Rheology-Processing / Chapter 2 38
(b) Flow through a truncated cone
We start from
It gives the pressure for flow through a tube of length dz. We integrate from z=0 to z=L,noting that
or
cotθ=1/tanθ
This expression, which is based on the nearly parallel flow assumption, is a very goodapproximation for half angle θ up to 15o.
Rheology-Processing / Chapter 2 39
DRAG FLOW BETWEEN FLAT PLATES
0
So the velocity profile will be
Velocity profile:
The average velocity is: Vavg=(1/2)Vo
The volumetric flow rate, for a channelof width W will be Q=(1/2)VoBWThis gives the amount of fluid that isdragged by the moving plate
Governing equation:
Rheology-Processing / Chapter 2 40
COMBINED PRESSURE AND DRAG FLOW BETWEEN FLAT PLATES
The pressure gradient may be aiding or opposing to the drag flow:
V
V
Rheology-Processing / Chapter 2 41
Governing equation
The power-law expression
must be introduced into the equation. Unfortunately, the resulting equation DOES NOThave a simple closed-form solution. Since this type of flow is of considerable practicalimportance, we present the Newtonian solution, i.e. n=1. We have
where η=μ is a constant Newtonian viscosity.
Rheology-Processing / Chapter 2 42
B.C.s
The pressure gradient is in general
Solving the governing equation of theprevious slide we obtain thevelocity profile
V
V
Rheology-Processing / Chapter 2 43
The volume flow rate for width W
integrationgives
or(in terms ofpressure gradient)
This may be integrated to give thepressure as a linear function of x:
Rheology-Processing / Chapter 2 44
INTRODUCTION OF AN EQUIVALENT NEWTONIAN VISCOSITY
• Combined pressure and drag flow→ no closed-form solution• For problem solution: apply numerical methods or literature for analytical (series
???) solution.• It is also possible to come up with a Newtonian approximation (for back-of-the-
envelope calculations)
Viscosity:
We can determine an equivalent Newtonian viscosity by choosing a characteristic shearrate
Then and further we may carry out calculations using the ηref in theNewtonian solution
Rheology-Processing / Chapter 2 45
For a reference shear rate it might be a good idea to use the apparent viscosity for flowin tubes with for some problems
For slit dies of gap H and width W, the reference (apparent) shear rate can easily beobtained from
Calculations involving equivalent Newtonian viscosity should be carried out withcaution for (rough) approximation purposes only.