solids handouts 10

8/2/2019 Solids Handouts 10

http://slidepdf.com/reader/full/solids-handouts-10 1/8

HANDOUT 10-1

xxσ

θ

xyσ

yxσ

yyσ

yyσ

xyσ

yxσ

xxσ

(a)

Normal

Shear

Min

Max(b)

MAX NORMAL

NORMAL

AXIS

S H E A R

S T R E S

S A X I S

MIN NORMAL

MOHR

CIRCLE

(c)



HANDOUT 10-2

Normal Stress

S h e a r S t r e s s

Critical Point

(End Point)

Normal Stress


c f

Normal Stress

Critical Point

(End Point)

1σ

JYL


JYL

JYL



HANDOUT 10-3

Normal Stress


Critical Points

δ

MFF

Non-cohesive

Less cohesive

More cohesive

Multiple JYL

curves

c f

1σ



HANDOUT 10-4

0

10

20

30

40

50

0 10 20 30 40 50 60 70

Semi-included angle, degrees

W a l l F r i c t i o n , d

e g r e e s

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

δ=30

δ=40

δ=60

δ=50

δ=60

δ=40

δ=30

δ=50

δ=60

δ=50

δ=45

δ=40

δ=35

δ=30

F l o w F a

c t o r , f f

Flow Factor Curves

Wall Friction Curves

Figure 10-17. Design chart for symmetrical slot outlet hoppers. For example (dashed arrows),

and gives and

o22=wδ o50=δ o5.30=θ 19.1= ff .



HANDOUT 10-5

0

10

20

30

40

0 5 10 15 20 25 30 35 40 45

Semi-included angle, degrees

W a l l F r i c t i o n ,

d e g

r e e s

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

δ=30

δ=40

δ=60

δ=50

δ=60

δ=40

δ=30

δ=50

δ=60

δ=50

δ=40

δ=35

δ=30

F l o w F a c t o

r , f f

Flow Factor Curves

Wall Friction Curves

Figure 10-18. Design chart for conical outlet hoppers. For example, and gives

and .

o22=wδ o50=δ

o5.20=θ 29.1= ff



HANDOUT 10-6

Steps on using the shear stress data to design a hopper.

1. Rotating Shear Test

Plot SHEAR STRESS

vsNORMAL STRESS

Internal Friction

Get f c vs σ 1Get δ

δ

SHEARSTRESS

f c σ 1

NORMAL STRESS

JYL

SHEAR

STRESS

wδ

2. Rotating Shear Test

Wall material

Get wδ

NORMAL STRESS

3. Fit φ w and δ to hopper correlation

Get θ and ff .

(θ is the theoretical

angle of the hopper;in final design subtract

3 degrees for margin of

safety)

4. Plot 1/ ff on mff curve ( f c vs σ 1) to get CAS

Get CAS

δ

δ

wφ

ff

θ

MFFCAS

f c

Slope 1/ ff

σ 15. Use CAS and θ in correlations to select opening size.



HANDOUT 10-7

D

θ Semi included angle

For conical hoppers, Figure 10-20, the opening

diameter, , is given by D

c

ogg

CAS H D

/ )( ρ

θ = (10-12)

602)(

θ θ += H (10-13)

Where θ is in degrees, from the charts in

Figures 10-17 or 10-18. Typical values for H are

about 2.4.

Figure 10-20. Conical Hopper with outlet size D

and semi included angle θ .

L

W

For symmetrical slot outlet hoppers the opening

size is determined from

c

ogg

CAS H W

/ )( ρ

θ = (10-14)

180

1)(θ

θ += H (10-15)

W L 3> (10-16)

Figure 10-21. Symmetrical slot outlet h

of opening size W x L.

opper



HANDOUT 10-8

10.4.1 COARSE PARTICLES (particles > 500 microns in diameter)

MASS FLOW – JOHANSON EQUATION

)()1(2 θ ρ

Tanm

Bg Am

o

+=& (10-17)

where θ = semi included angle of the hopper

m& = discharge rate (kg/sec)o ρ = bulk density (kg/m3)

g = gravity acceleration (9.807 m/s2)

Table 10-3. Parameters in the Johanson Equation, Eq.(10-18)

Parameter Conical hopper Symmetric slot hopper

B D, diameter of outlet W

A 2

4 D

π

WL

m 1 0

FUNNEL FLOW – BEVERLOO EQUATION

5.25.0 )(58.0 p

okd Dgm −= ρ & (10-18)

where d p = particle diameter (m)

k = constant, typically 1.3 < k < 2.9 with k = 1.4 if not

discharge rate data are available.

)perimeteroutlet()areasectionalcross(4= D

10.4.2 FINE PARTICLES (d p < 500 microns)

CARLETON EQUATION

g

d

V

B

V

p p

oo=+

35

34

32

31

2

15sin4

ρ

µ ρ θ (10-19)

o

o AV m ρ =&

where = average velocity of solids discharging (m/s)oV A, B = given in Table 10-3

µ ρ , = air density and viscosity

p ρ = particle density

o ρ = bulk density of the powder bed

solids handouts 10

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