NONWOVENS: MELT- AND SOLUTION BLOWING

NONWOVENS: MELT- AND SOLUTION BLOWING

A.L. Yarin

Department of Mechanical Eng. UIC, Chicago

Acknowledgement

• The Nonwovens Cooperative Research Center (NCRC), USA

Collaboration with:

PhD student S. Sinha-Ray

NC State: Prof. B. Pourdeyhimi

Outline

1. Meltblowing

2. Experimental: Solid flexible threadline in parallel high speed gas flow 3. Turbulence, bending perturbations, their propagation and flapping 4. Theoretical: Solid flexible threadline in parallel high speed gas flow 5. Comparison between theory and experiment

6. Theoretical: Polymer viscoelastic liquid jets in parallel high speed gas flow

7. Fiber-size distribution

8. Fiber orientation distribution in laydown 9. Spatial mass distribution in laydown

10. Experiments with solution blowing and co-blowing of core-shell and hollow fibers

My book published in 1993 by Longman (U.K.) and Wiley&Sons (New York) encompasses all relevant

equations and a number of relevant solutions for free liquid

jets moving in air

General quasi-one-dimensional equations of dynamics of free liquid jets moving in air

f fW t s 0

 

 

 

 

total

f fW 1 P

t s s f

  

  

   

    

V V Q

g + q

1. Continuity-Mass Balance- Equation

2. Momentum Balance Equation

All terms in Eq. (1) are of the order of a

All terms in Eq. (2) except the shearing force Q are of the order of a

;

Q is of the order of a

General quasi-one-dimensional equations of dynamics of free liquid jets moving in air

   

 

1 1 2 1 3 3

1 4

3

j W k

t

W 1 1

s s k



               

    

     

    

K U j U j j U j j

K j V M

Q- j g + m

3. The Moment-of-Momentum Balance- Equation

All terms in Eq. (3) are of the order of a

Equations (1)-(3) are supplemented with the geometric,

kinematic, and material relations. The material relations for P and M follow from the rheological constitutive equation

projected on the quasi-1D kinematics

Main results for highly viscous and

viscoelastic jets rapidly moving in air: flame throwers

4 2

2 a 0 2

2 3 2

0 0 0

0

3 U

4 a a a 0

where the dimensionless wavenumber 2 a /

  

 

             

   

The linear stability analysis: small 3D perturbations grow with the rate:

The bending instability sets in when:

 

*

0 0 a 0

U  U    / a

Main results for highly viscous and

viscoelastic jets rapidly moving in air: flame throwers

2

2 2

0 a 0

a U 1

 

 

The bending perturbations grow much faster than the capillary perturbations for highly viscous liquids when:

Linear spectrum:

Nonlinear-numerical-results for 2D bending

Newtonian jet

Nonlinear-numerical-results for 2D bending

Newtonian jet

Viscoelastic jet

Experimental: threadline blowing setup to probe turbulence

Meltblowing

S. Sinha-Ray, A.L. Yarin, B. Pourdeyhimi.

J. Appl. Phys

.

Air velocity at the nozzle exit

Chocking at pressure ratio of 47.91

Experimental: threadline configurations at different time moments

ARROWS DENOTE THE FLAPPING LENGTH

Threadline envelope: 2

method to

define flapping length

Measured threadline oscillations

FFT: Spectrum of the threadline

oscillations

Fourier reconstruction of the threadline oscillations with high frequency truncation above 167 Hz

Autocorrelation function: chaotic nature

of threadline oscillations

Turbulence, bending perturbations, their propagation and flapping

3 0

1/2 d

5 6

0

2

'2 1/2

' '

Large eddy frequency : U / L 10 Hz Taylor microscale : 1.23Re x

0.014 0.14 cm

Microscale frequency : U / 10 10 Hz Threadline oscillation frequency :10 10 Hz Multiple impacts of large eddies :

A [2 v t]

v u





 

 

 



   

 









'2 ' '

1

'2

t

v u v

( u / y)

v turbulent eddy viscosity!



  

   

  



 

Turbulence, bending perturbations, their propagation and flapping

1/2 t

t 0 0

threadline

Therefore, A (2 t)

In axisymmetric turbulent gas jets : 0.015U d const

Time t is restricted by bending perturbation propagation over the threadline :

t L / P / (S )

Threadline tension : P q L, which is imposed by a

 

 



 _







0.81

2 0 0

0 g 0

g

ir drag : q 0.65 a U 2U a

 

  

  



Turbulence, bending perturbations, their propagation and flapping

0 0

2 t

1

1/2 t

For U 230 m / s, d 0.05 cm in air : 17.25 cm / s, t 0.0256 s

Therefore, t 39Hz a remarkable agreement with the data!

A (2 t) 0.94 cm a reasonable agreement with the data!



 

 

 

   





Turbulence, bending perturbations, their propagation and flapping

1/4 0.2025 1/4

0 0 0 0

1/4

g g

The shape of the threadline envelope in the case of distributed impacts of turbulent pulsations without distributed lift force is predicted as :

U d d d L A(x) 0.16

L (1 x) i.e.

A(x)

     

          

 



 

1/4

1 (1 x)

 

Turbulence vs. distributed aerodynamic

lift force

Distributed drag, lift and random forces

Straight unperturbed sewing threadline in high speed air flow

2

xx 0

0.81

2 0 0

0 g 0

g

The unperturbed momentum balance

dP q 0, P a

dx

The longitudinal aerodynamic drag force q 0.65 a U 2U a

Integrating the momentum balance, we obtain



  

 

  





 

  

xx 2

0

q (L x) a

 ^ 

 

Perturbed threadline in high speed air flow

2

2 2

n g 0 0 2

2 n

0 n

2

n 2

The lateral bending force

q U a

x

The linearized lateral momentum balance

a V kP q

t

The lateral velocity and thread curvature read

V , k

t x

  



  



 

 

 

  



 

Perturbed threadline in high speed air flow

2 2 2

g 0 xx

2 2

2 2

xx 0 0 g 0

* *

2

g 0

Then, the thread configuration is governed by

U (x)

0 (1)

t x

If q L /( a ) U , then Eq.(1) is

hyperbolic at 0 x x , and elliptic at x x L.

The transition cross section is found from q (L x)

U

  

     

 

 

   



 





 

 



  

  ²₀ ^*

x 0 0 x 0

0 @ x x a

The threadline is clamped and perturbed at x 0 H exp(i t), / x 0

 

 



 _   

  

Perturbed threadline in high speed air flow

0

1/ 2 2

g 0

2 2 0 0

1/ 2 2

g 0

2 0

0

Solution in the hyperbolic part is (x, t) exp(i t)cos[ I(x)]

where

q L U /

2 a a

I(x) q q (L x)

a U / Solution in the elliptic part is

(x, t) exp(i t) c



   

    

   

  

 

     

 

 

 

 







 



   

 

 

 



  

   

   

*

*

2 2 1/ 2

2

0 g 0

0

2 0

osh J(x) cos I(x ) isinh J(x) sin I(x ) where

q L /( a ) U 2 a q x

J(x) q a

 

 

 

 

 



 

   

 

 



 

 

 



 

Flapping of solid flexible Flapping of solid flexible

threadline

threadline

Theory vs. experiments for

threadline

Theory vs. experiments for

threadline

Polymeric liquid jet in high speed air flow

2 2 2

0 0

2

xx

0.81 2 g

g g

g

xx xx yy

The quasi one dimensional continuity and momentum balance equations for a straight jet

dfV 0, f a a V a V

dx

dfV d f

dx dx q where

2(U V )a q 0.65 a (U V )

and the stress is foun



 

   

 

  

   

 

 

 



 







 

  

 

  

   d from

Polymeric liquid jet in high speed air flow

xx xx

xx

yy yy

yy

The upper convected Maxwell model of viscoelasticity

d dV 2 dV

V 2

dx dx dx

d dV dV

V dx dx dx

which are solved numerically simultaneously with the transformed momentum balance.

The following dimen



  

   

 



 



   

 

  

  

0

g 0 0

0 g

g

sonless groups are used

L 2V a V

R , , Re , De

a L

E 2R , M

De Re M

   

 





 

 

 





Polymeric liquid jet in high speed air flow

2

xx yy

2

xx yy

xx xx

xx

yy yy

yy

The dimensionless equations for a straight polymeric jet solved numerically

E( ) /(DeV ) q dV

dx 1 E( 2 3) / V

d 1 dV dV

2 2

dx V dx dx De

d 1 dV dV

dx V dx dx De

The b

   

 

     

 

    

 

    

 

 





 



 



 

 

 



 



xx xx 0 yy

oundary conditions are

x 0 : V _ 1,    ,   0

Unperturbed polymer jet in melt blowing: velocity and Unperturbed polymer jet in melt blowing: velocity and

radius distributions

radius distributions

Unperturbed polymer jet in melt blowing: longitudinal Unperturbed polymer jet in melt blowing: longitudinal

deviatoric stress distribution

deviatoric stress distribution

Unperturbed polymer jet in melt blowing: lateral Unperturbed polymer jet in melt blowing: lateral

deviatoric stress distribution

deviatoric stress distribution

Unperturbed polymer jet in melt blowing: K(x) Unperturbed polymer jet in melt blowing: K(x)

distribution

distribution

Bending perturbations of polymeric liquid jet in high speed air flow

2 2 2 2

g g xx

2

2 2

2 2 2

2 2

g xx

2 2

The linearized lateral momentum balance reads (dimensional) :

(U V )

2V V 0

t x t x

and normalized :

2V V R(U V ) E 0

t x t x

All coefficients depend only on the unp

 

 

  

     

     

           



 

  

 

  



   

erturbed solution!!!

Bending perturbations of polymeric liquid jet in high speed air flow

0 1

2

x

1 0 2

xx g

x

2 0 2

xx g

2

1 x 0

Solution in the hyperbolic part :

exp[ i I (x)]

(x, t) exp(i t)

exp[ i I (x)]

1 where I (x) dx

V (x) E (x) R[U (x) V (x)]

I (x) dx

V (x) E (x) R[U (x) V (x)]

dI / dx dI / dx _

 

 

    

    

    





 

 

 

  











Bending perturbations of polymeric liquid jet in high speed air flow

0

1

1 * 2

2 * 2

x

1 2 2

x* g xx

2

g xx

2 2

Solution in the elliptic part : (x, t) H exp{i [t J (x)]}

1

exp[ i I (x )]exp[ J (x)]

exp[ i I (x )]exp[ J (x)]

where

V (x)

J (x) dx

V (x) R[U (x) V (x)] E (x) R[U (x) V (x)] E (x) J (x)

V

 



  

 

   

    

 







 





 



  

 





x

x* g 2 xx

0

g g

0

(x) R[U (x) V (x)] E (x)dx The velocity distribution in turbulent gas jet is U (x) U (0) 2.4d

x 2.4d

   

 

 

Bending perturbations of polymeric liquid jet Bending perturbations of polymeric liquid jet

in melt blowing

in melt blowing

Meltblowing: Nonlinear theory

Nonlinear model for predicting large perturbations on polymeric viscoelastic jets.

A.L. Yarin, S. Sinha-Ray, B. Pourdeyhimi. J. Appl. Phys. 108, 034913 (2010).





Basic vectorial equations

Scalar projections of the momentum balance equation in 2D

Numerical results

Basic vectorial equations

Scalar projections of the momentum balance equation in 2D

Numerical results



Nonlinear Model for

Isothermal Polymer and Gas

Jets

Basic equations: Momentless theory

f fW t s 0

 

 

 

total

f fW 1 P

t s s f

    

   

    

V V

g + q

Taking s to be a Lagrangian parameter of liquid elements in the jet, W=0, which gives    a

a

Momentum Equation Continuity Equation

- The integral of the continuity equation

Relation of the coordinate system associated with the jet axis and the laboratory coordinate system in 2D cases

(s, t) (s, t)

  

R = i j





    





k     

   





n 1 /

 

 

         



 



n

     

/    1    /   

Stretching ratio

Curvature of the

jet axis 

Polymer jet

Gas jet

Scalar projections of the momentum balance equation in 2D

total, n

n

V 1 V 1 P q

V kV g

t s f s f

 

 

                    

total,n

n n

n

V 1 V Pk q

V kV g

t

s

f f

                  

 

     

 

   

total total,n total,

2 2

,s ,ss ,s ,ss ,s ,s ,s ,s ,s

2

g g 2 2 5/2 2

,s ,s ,s ,s

0.81

2 g

g g

g

q q

/ sign /

U f a

1 /

2a U V

a U V 0.65





 

 

   

           

 

  

       

 

   

 

     

  

 

q n

n

The rheological constitutive viscoelastic model.

The mean flow field in the turbulent gas jet

1 1

2 2

t t t

 



    

   

      

Rheological Constitutive Equation :Upper Convected Maxwell Model



 



^{ }

( , ) , ( , )

4.8 / 1 / 8 0.05 4.8 /

          

     



 

L/a

, where a

is the nozzle radius and L is the distance between the nozzle and deposition screen

g g0

U ( , ) U       ( , )

The mean flow field in the turbulent gas jet

Dimensionless groups

Time, co- ordinates

and functions

Rendered dimensionless

by the scales

t L/U_g0

s,H, L

k L^-1

_ U_g0/L U_g0, U_g,V_,V_n U_g0

a a₀

q_total, _gU_g0²a₀

_t 

g0

g

2 1/2 g0

0 g0 a

g g0

Re LU

J Fr U

gL Re 2a U

De U

L

 



 



 

  

 

 

 

 

Dimensionless equations for numerical implementation

2 2

total,

2 2 2

2 1 q

t Re s Fr J f





   

    

  

   

 

2 2 2 2

,s ,s ,s ,s

2

2 2 2 2 2

,s ,s

/ sign /

1 n ( , )

J ( , ) J

t Re s Fr a 1 /

 

   



                           

where

    

^{1/ De}  

^,and  is found from :

t De



  

 

 

2

total, a

q _       a  _ ( , ) V_ 0.65 Re a     _ ( , ) V_ ^ and

(1)

(2)

(3)

Nature of the equations

Both the equations are basically wave equations. While (1) is –for the elastic sound (compression/stretching) wave propagation, (2) is nothing but bending wave propagation.

2 2

total,

2 2 2

2 1 q

t Re s Fr J f





   

    

  

   

 

2 2 2 2

,s ,s ,s ,s

2

2 2 2 2 2

,s ,s

/ sign /

1 n ( , )

J ( , ) J

t Re s Fr a 1 /

 

   



                           

(1)

(2)

Boundary and initial conditions

origin origin 0

s s_

0,

exp(i t)

     

where

free end

,s s

1,

 

free end

,s s

0

 

 

0_ 0.06 / Re / _0

   

g0

L U

  

1)

2)

 

birth

2

0 0

t t_{ }

1/ De /

    

Boundary Conditions:

The initial condition for the longitudinal stress in the polymer

jet:

Numerical results for 2D: the isothermal case

Velocity flow field in gas jet

Snapshots of configurations

of polymer jet axis

Numerical results in 2D: the isothermal

case (continued)

Self-entanglement: Can lead to “roping”

and “fly”

The evolution points at possible self-intersection in meltblowing, even in the case of a single jet considered here

SEM image of solution blown PAN fiber mat obtained from single jet showing existence of

“roping”

Nonlinear Model for

Non-isothermal Polymer and

Gas Jets

Thermal variation of the rheological parameters

0

0 0

0 0

U 1 1 T U 1 1

exp , exp

R T T T R T T

       

               

   

   

where T

is the melt and gas jet temperature at the origin, µ

and θ

are the corresponding values of the viscosity and relaxation time, U

is the activation energy of viscous flow and R is the absolute gas

constant.

The additional and changed dimensionless equations; 2D, non-isothermal case





a g 0

T JC

2Nu T T

t Re Pr

 

  

  





A 2

a g 0 0 0

2Nu JC T T 1

T exp U 1

t Re Pr De T De

 

                     



   

   

t g

g g 2 2Pr

Pr 1/ 2 1 T 1 1

T ( , ) T

4.8 /

0.05 6 1 / 8





 

   

   

 

Thermal Balance Equation

Rheological Constitutive

Equation

Mean Temperature

Field in the Gas Jet

where





    

,

0

U U

De , U

L RT

  

, Nu 0.495Re Pr 

Numerical results for the 2D non- isothermal case

Snapshots of the axis configurations of polymer jet for the nonisothermal case The mean temperature field in the gas

jet

Numerical results for the 2D non- isothermal case

Snapshots of the axis configurations of the polymer jet for the nonisothermal

case Snapshots of the axis configurations

of the polymer jet in the isothermal case

Numerical results (continued)

Following material elements in the polymer jet in the

isothermal case

Following material elements in the polymer jet in the non-

isothermal case

The initial section of the jet-no bending

3D isothermal results: single jet

Three snapshots of the polymer jet axis in the isothermal three-dimensional blowing at the dimensional time moments t=15, 30 and 45

9 jets meltblown onto a moving screen-the beginning

of deposition

9 jets meltblown onto a moving screen-a later

moment

62 jets meltblown onto a moving screen

62 jets meltblown onto a moving screen

62 jets meltblown onto a moving screen: a higher

screen velocity

62 jets meltblown onto a moving screen: a higher

screen velocity

62 jets meltblown onto a moving screen: comparison

with experiment

62 jets meltblown onto a moving screen: comparison

with experiment

62 jets meltblown onto a moving screen

62 jets meltblown onto a moving screen

Experimental: solution blowing and co-

blowing setups

Solution

Solution - - blown polymer jet: Vigorous bending and flapping blown polymer jet: Vigorous bending and flapping

Solution

Solution - - blown and co blown and co - - blown nanofibers blown nanofibers and nanotubes

and nanotubes

Monolithic PAN nanofibers

Solution

Solution - - blown and co blown and co - - blown nanofibers blown nanofibers and nanotubes

and nanotubes

PMMA-PAN carbonized:

Hollow carbon nanotubes Optical image of PMMA-PAN

core-shell co-blown fibers

Solution

Solution - - blown monolithic nanofibers blown monolithic nanofibers

Solution

Solution - - blown core blown core - - shell nanofibers shell nanofibers

Close Relatives

Sea snakes, electrospun and meltblown jets, and flame thrower napalm jets extract energy from the surrounding medium via bending