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Page 23

P  

r2 20.0005 72.86  10 cos    cos(25)  264 N / m 2 r 0.0005

Kinetics of liquid metal flow in gating design of investment casting production Roy – 2U. K.264 Maity – A. K. Pramanick – P. K. Datta P 0S. 2 .5 0.5

v f  [v g 

 annular horizontal tube at room temperature r 0also .0005 vindicated the proposed mathematical model for fluid flow. (iii) In case of actual brass (60/40) castings in the hot clay molds P close to experimental  264 0.5 designed filling times were found 2tobe v f  [vg   ]0.5  [0.6262  ]  0.8m / s liq design in clay 1000molded ones. Hence, optimization of gating investment casting production could be guided from this vf kinetics of liquid flow modeling quality metal and C D  to achieve 1.27 v1 good surface quality in casting production.

Qi  Ag v g    0.0012  0.626  1.96  10 6 m3 / s

Nomenclature

Q f  C D Ag v g  1.27  1.96  10 6  2.48  10 6 m3 / s

Symbol Meaning

Unit V 1.143 10 5 tf  m   4.6 sec Q f 2.48 10 6 m/s

Velocity of fluid

P

Pressure at a point

N/m2

V

Volume

m3

m

Mass

kg

D

Diameter of gate or channel

m

Z α

Height

m

γ

Surface tension of liquid metal

N/m

µ

Viscosity of liquid

mPa · s

Re ρ

Reynold’s No Density of fluid

kg/m3

θ

Angle of liquid metal droplet

° (degree)

T

Temperature

K

RU

Universal gas constant

kJ · kg−1 · K−1

RC

Characteristics gas constant

kJ · kg−1 · K−1

A

Area of cross-section

m2

Energy correction factor

CD

Co-efficient of discharge Liquid metal pouring rate

m3/s

tf

Time of filling

s

vg

Velocity of fluid at gate

m/s

vf

Final velocity

m/s

Vm

Volume of casting

m3

Ag

Gate area

tf 

 4.6 sec0.0005 r 6 5 QVfm 21.48 1010 .143  P  264 0.5 tTherefore   2  06.5  4.6 sec is:2 final vf f Q[fvg 2.48  [0.626  ]  0.8m / s 10]velocity

∆Pliq − 2641000 2 v f = [vg − γ ]0.5 = [0.626 2 − ]0.5 = 0.8m / s  P  264 0.5 2 ρ  0.5 2 1000 liq v f  [v g  ]  0.8m / s ]  [0.626  1000 v f liq

CD 

 1.27 v1 v f v    0.0012  0.626  1.96  10 6 m 3 / s Qi  A g g 1.27 C D v1 Q  C A v  1.27  1.96  10 6  2.48  10 6 m 3 / s Qi f  Ag vDg g  g 0.0012  0.626  1.96  10 6 m3 / s Vm

1.143 10 5

t f  C   4.6 sec6  2.48  10 6 m 3 / s Q 110 .27 6 1.96  10 f Q D Ag 2v.g48 f

v

Q

PŘESNÉ LITÍ

]  [0.626  ]  0.8m / s 1000 264 0.5 2 P v f  [vg  liq  ]0.5  [0.6262  ]  0.8m / s 1000 Appendixliq A (i) Gating design calculations can be improved in Bernoulli’s vg  gzs  9.81 0.04 vf Calculation of experiment using water equation by considering (i) the kinetic energy correction factor C D   1.27 Velocity (α), (ii) pressure difference factor and (iii) surface tension factor vv f entered into the cavity (at gate): vg  gzs  9.81 0.04  0.62 C D  1  1.27 in the mold system. The pressure drop due to surface tension vgi  Agvgz 812 0.004  0.1626 / s63m3 / s Q v1g s 90.001 .626 .96 m 10 2  2  72 . 86  10 of liquid and gas pressure difference at high temperature 6 25 3 )  264 N / m 2 P Av  cos  2  0.626  1.96  10 cos( Q  0.001 3m / s Q if  C Dgof Aggrvresistance 1006.0005  2.surface 48 10 6 mtension: /s Effect due to provided better results in filling time for hot investment mold. g  1.27  1.96  6 3 6 3 (ii) Simulation experiments done 2by water2 and kerosene in  g v g  15.2721.72  10 Q f  C D2A 96.86  10  cos( 2.4825  10 72.86  10 3 cos   )  m 264/Ns / m 2 10 P   cos    cos(25)  264 N/Pm 2Vm  1.143

V 1.143 10 5 tf  m   4.6 sec Q f 2.48 10 6

Appendix B Calculation of gating design for wax pattern Details of casting as received (Fig. 6) Volume of the casting, × 10 6, m3 = 0.127; Heat dissipating area of the casting, × 103, m2 = 8.125 Weight of the casting, × 103, kg = 119 Gating design: Assumptions: Top gating Dimension of cup, sprue and gate are given in Fig. 6. For Brass: Viscosity, m · Pa · s = 4 and Liquid Density, kg/m3 = = 8400 Sprue area, × 10 6, m2 = 3.14 Effect of gas pressure: At ofair as ideal gas): P1 300 P atmKtemperature, RT  1.177  287pressure  300  1.01 105isN(considered / m2 P1  P atm RT  1.177  287  300  1.01105 N / m2 Let, pressure inside mold at 1250 is P2 0.2824 1250 T P  P ρ2 T2  P 0.2824 × 1250  0.9997 P1 P22 = P11 1T221T22 = 1P11.0177  300 .2824 1250 = 0.9997 P1 P2  P1 ρ1T1  P1 1.177 × 300  0.9997 P1 1.177  300 1T1  Pgas  P1  P2  30.3N / m2  Ppressure P2positive  30.3Nhere, / m2 i.e. accelerating the flow. gas  P1 is 2 Gas

m2 300  1.01105 N / m P1  P atm RT  1.177  287 Velocity Heat dissipating area of casting m2

Am

P

30.3

gas 0.m 2824 v1  gz1   9.81  0.P045   2T2  0.67 / s 1250  0.9997 P Data used 2  P1 8400  P1 1  liq 1.177  300 T

1 1

Properties

Value (unit)

Density (ρ) of air (at 300 K) [14]

 Pgas  P1  P2  30.3N / m

2

1.177 (kg/m3)

A1v1[14]  2.52  0.67 3 A1vof A2(at v2 1250 Density (ρ) 0.2824 kg/m 1 air  0.167 m / s) v2  K)  2  A  5 −1 Characteristic gas constant 2for air (R) 287 (kJ · kg · K−1)

Density (ρ) of water (at 300 K)

1000 (kg/m3)

Density (ρ) of kerosene (at 300 K) [15]

810 (kg/m3)

Surface energy (γ) of kerosene [17]

30 (mN/m)

3   4 2  0.167 A2(at ' v21250 Density (ρ) 8400  0.67 m / s(kg/m ) v g ofvbrass  K) [4] 2 3   A  2 Surface energy (γ) of 3water [16] 72.86 (mN/m)

Surface energy (γ) of liquid brass [4] Viscosity of liquid brass (µ) Contact angle (θ) of water [16]

2000 (mN/m)

v'2vgm · Pa · s  100 %  24 % v25° '

Contact angle (θ) of kerosene [16]

26°

Contact angle (θ) of liquid brass [4]

136°

2 2  0.2 cos    cos(136)  288N / m2 r 0.001

P  

2

v f  [v g  CD 

vf v1

P

liq

 0.95

]0.5  [0.67 2 

288 0.5 ]  0.64m / s 8400

at section -0: v0 = 0 (consider constant head) Velocity at -1: Pgassection 30.330.3 Pgas  P atm 300 0m.67 1/ .sm 01/ s105 N / m2 vP  9.181.9177 .81 0.045  30. gz1 PgasRT  0.287 045 11v1 gz 1  3  0.67  liq 9.81  0.045  84008400 v1  gz1  liqPgas 30.30.67m / s v1  gz1   liq  9.81  0.0458400   0.67m / s

 liq  1.177  287  300 8400 P1  P atm RT  1.01105 N / m2

0.2824 12502 Continuity Equation: Velocityat -2: Apply 2T2section 9997 P1 A11v1A1v1 2.522 .50.6700.. 67 1 vA v  P AP1v2 1A1vP A 22T m/m s /s v   0.167 1 . 177  1  2 2 v   0.167 2 .52 52 02.67 2 A1v1  A2 v21 v12  2AA1v2 1 A2   2300   5  0.167m / s 2 25.5  0.67 A1v1 1250 0A.2824  2Tv2 AP1v1  A  P1 2 2 v2P1 2  0.9997 P  0.167m / s 2Pgas  30.3N / m22 1 T1 P1  P 12.A 177 1 2  300   5 Velocity at section-3 is known as gate velocity: v3 = vg 2 4 2.4302N.167  Pgas A2P'1vA /0m v2230 .167 22 ' P 2 s /s v g v gv3v3A' v   4  02.167  0.670m .67/ m  2  2 2  0.67m / s v g  v3  2A3 2A AA23 ' v23  422 2 0.167

0.67 m / s sprue height, without v g  v3 of liquid  metal at2 gatefor Velocity same  2 A3 v' v' 9.819 .8 dam: v'  9.81  0.080  0.88m / s v'  9.81  v'vvg'v

v'

g % %24 % Velocity decrease using dam:  100 24 % v'vg  100

v' v' 100 %  24 % v' v'vg

Effect of pressure drop due to surface tension:  100 %  24 % v'

2γ 2 × 0.2 ∆Pγ = − 2 cos θ = − 2  0.2cos(136) = 288 N / m 2 2    0.001 2  0cos( .2 136)  288N / m2 P   r cos

 cos  2  0.2 cos(136)  288N 2/ m2  P 2 P   r cos cos(136)  288N / m r    0.001 0.001 r 0.001

P  

2 2  0.2 cos    cos(136)  288N / m2 r 0.001

 P .5 288288 2 2 0.5 2  0P v f v [f vg 2[vg   2288 P ]0.5 ]0.[50.67 [02.67  ]0.5 ]0.50.640m .64/ sm / s  8400 v f  [vg  liq ] liq  [0.67  8400 ]  0.64m / s S l évá re ns t v í . L X V . k v ě te n – č e r v e n 2017 . 5 – 6 8400 v f v liqP C  D[vvf 2 f 0.950.95]0.5  [0.67 2  288 ]0.5  0.64m / s C vCfD  vg D  1 v10.95 liq 8400 v1

153

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