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Modeling of air back pressure in die-casting dies
Journal of Materials Processing Technology 91 (1999) 264 – 269
Modeling of air back pressure in die-casting dies W.B. Lee *, H.Y. Lu Department of Manufacturing Engineering, The Hong Kong Polytechnic Uni6ersity, Hung Hom, Kowloon, Hong Kong Received 23 June 1998
Abstract In high-pressure die-casting, molten metal is injected into a die cavity at a high velocity. The metal entering into the cavity expels the air from the cavity through the vents. Air back pressure will be created when the volume of metal flow exceeds the volume of air escaping out of the die cavity. A new mathematical model is proposed for the calculation of the creation of the back pressure in a die cavity. Air discharge through vents is modeled by the Poiseuille flow. The differential equation of the time rate of change of the back pressure is derived. The creation of the back pressure is correlated with the configuration sizes of the vent, and computations are carried out for different configurations of the vent geometry. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Die-casting; Air back pressure; Poiseuille flow
1. Introduction During the stage of die cavity filling, molten metal can only advance as quickly as the air inside the cavity is expelled. As the die is impermeable to the air, vents are provided to allow the air to escape out of the cavity. Metal filling in a die cavity is a rapid transient process. The filling velocity can be as high as several tens of meters per second. The air inside the empty region is compressed and air back pressure can be created if there is not a sufficient venting area or if the die vents are sealed off before the die cavity is completely filled up. Air back pressure thus caused may be several times the atmospheric pressure or more [1,2]. The creation of the back pressure can lead to a change in the filling sequence [3]. If the back pressure being exerted on the forefront of the advancing molten metal is large enough to delay filling or to resist the pressure applied by the diecasting machine, incomplete die filling will occur in thin sections or corners. In addition, the back pressure would increase the amount of air dissolving into the molten metal and cause porosity in the pressure diecastings [4,5]. Therefore the problem of eliminating cavity
* Corresponding author. Tel.: 27666594.
+852-23659248; fax:
+ 852-
air and hence back pressure to facilitate metal flow is critical, especially for large thin-wall die-castings. An understanding on the process of the creation of back pressure is important for the production of quality castings. The effect of the vent configuration sizes and the air viscosity on the creation of the back pressure has not often been analyzed in the literature [6,7]. The air back pressure is a transient variable depending on the flow rate of the air discharged through the vents. The sizes of the vent and the air viscosity have an influence on the flow rate of the air. In this paper, an analysis of the air flow through die venting is made and the equation of the flow rate is obtained in the form of Poiseuille flow. A differential equation of the time rate of change of the back pressure is derived based on the assumption of uniform-state and adiabatic process. The transient value of the back pressure at different time intervals is calculated for different vent configurations with conventional model and the model proposed in this paper.
2. Air flow rate through vent As the metal die is impermeable to air, vents are provided for the escape of air, the vents are generally being along the parting line or ejector pin in the form
0924-0136/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 8 ) 0 0 4 1 8 - X
W.B. Lee, H.Y. Lu / Journal of Materials Processing Technology 91 (1999) 264–269
of shallow grooves. When the molten metal is injected into the cavity, air inside the cavity is expelled through the vents. A conventional model [3] often used in solving for mass flow rate of the air through vent and creation of the back pressure was derived from the idea of air flow through a nozzle. The variables used are shown in Fig. 1. The size of the container is usually much larger than that of the nozzle, and the velocity of the air inside the container is considered to be zero. Air inside the container discharges through the nozzle of area A. The velocity of the air escaping through the nozzle is considered to be much larger than that of heat transfer, such that the air discharge is assumed to be an adiabatic process. The energy balance equation about crosssections (1) and (2) in Fig. 1 can be written as [8]: k P1 k P0 m 20 = + k − 1 r1 k−1 r0 2
(1)
For an adiabatic process, r0 =r1(P0/P1)1/k, the exit velocity can be written as: m0 =
'
n
2k P1 P 1− 0 k −1 r1 P1
k−1 k
(2)
The mass flow can be calculated using the equation: m= ruA
(3)
Substituting the exit velocity and density into Eq. (3) yields: m= A or:
'
n
2k P1r1 k −1
'
P m= Ar1 0 P1
1 k
P0 P1
2 k
P − 0 P1
k+1 k
n
2k P1 P 1− 0 k −1 r1 P1
k−1 k
(4)
(5)
where m is the mass flow rate of air, P1 and P0 are the pressure inside and outside the die cavity, respectively. r1 is the air density inside the cavity, k is the adiabatic exponential and A is the cross-sectional area of the vent.
265
Fig. 2. Schematic diagram of a die-casting die with vents.
A real rectangular vent is defined by its thickness (h), breadth (b) and its length (l) as shown in Fig. 2. It should be noted that the mass flow rate of the air relates only to the cross-sectional area of the vent and does not relate to the three individual sizes of the vent in Eq. (5). The sizes of the vent commonly used in die-casting practice are given in Table 1. The thickness of the vent is usually much smaller than its width (more than 100 times smaller), and the thickness of the vent is of the order of one tenth or one hundredth of a millimetre. The fluid flow in such a shallow groove is thus different from that found in a large channel of a piping system. For flow in a shallow groove, it is assumed that the velocity component is zero and the pressure is equi-distributed in the normal direction to the main flow (i.e. the y direction in Fig. 2). For a small elementary volume of the air flowing through the vent as shown in Fig. 3, the compressibility of the air in the vent is assumed to be negligible. The force balance on it may be written as: %Fx = 0
(6)
then: pdy− (p+ dp)dy = tdx − (t +dt)dx
(7)
or: dt dp = dy dx
(8)
Table 1 The vent size (rectangular shape) as recommended by different researchers [9]
Fig. 1. Example of air flow through a nozzle.
Recommended by
Thickness (mm)
Doehler Hoeb Street Stern Frommer
0.1270.152 0.0760.127 0.076 0.1270.254 0.05–1
Width (mm)
1325.4 13
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266
From Eq. (12), the rate of change of the air mass within cavity can be written as: dM rbh 3 Dp =− % dt vents 12 ml
(13)
The rate of change of the air volume within the cavity is thus given by: dV = − % Fu dt gates Fig. 3. The forces on a tiny elementary volume of flow.
Substituting t =m du/dy into Eq. (8):then: dp d2u =m 2 dy dx
(9)
m=
1 dp (y − h)y 2m dx
(10)
where p and m are the pressure and dynamic viscosity of the air, respectively. Eq. (10) shows that the velocity distribution in the vent is parabolic and dp/dx is constant. If the pressure inside and outside the cavity is P1 and P0, respectively, then: Dp dp −(p1 −p0) =− = l l dx
(11)
&
h
0
ubdy =
bh 3 Dp 12 ml
M VdM −MdV = V V2
(12)
where b, h and l are the width, thickness and length of the vent, respectively. Eq. (12) shows that the flow rate is directly proportional to the cube of the thickness of the vent, so that the thickness of the vent will has a strong influence on the air flow in the vent, and the flow is driven by differential pressure. This is a typical Poiseuille flow [10].
3. Creation of the air back pressure Cavity filling is an unsteady process. If the filling is supposed to be completed in a number of small time intervals, the state of the air within cavity can be assumed to be uniform in temperature, pressure and density in each interval, even though this uniform state may change with time. The state of the air passing through vents can also be assumed to be invariable although the mass-flow rate may change with time. The above assumptions of the uniform-state are widely used in engineering thermodynamics [11] to simplify the calculations.
(15)
where M is the mass of air inside the cavity and V is the volume occupied by the air. Since the die filling process is transient, the creation of the air back pressure can be considered to be an adiabatic process. For an adiabatic flow, p/r k = C, its differential form can be written as: dr dp= kp r
(16)
where k is an adiabatic exponential, and k= 1.4 for air. Substituting Eqs. (13) and (14) into Eq. (15), and then Eq. (15) into Eq. (16), yields:
dp kp rbh 3 Dp =− M % Fu−V % 2 dt rV gates vents 12 ml
The flow rate of the air can be written as: Q=
where F and u denote the area and velocity of the ingate, respectively. Since the mass and volume of the air inside the cavity change, its density will also change. The differential of density change is: dr= d
Integrating Eq. (9) and using boundary conditions y= 0, u = 0; y= h, u=0 yields:
(14)
(17)
Eq. (17) shows the rate of the change of the air back pressure during the filling process that can be used to calculate the back pressure. Compared with the conventional model for calculating back pressure, the present model correlates the creation of the back pressure with different configurations of the vent geometry.
4. Determination of the air back pressure In order to solve for the air back pressure in different time intervals, Eq. (17) should be rewritten into an increment form. The filling is supposed to be completed in a number of small time intervals, and the back pressure is calculated step by step. When the new values, p, r, M and V are solved, the calculation of the next time step will be done and these new values will become the old values to be used for the next calculation cycle. It should be note that Dp is the differential pressure between the cavity and the ambient and is time-dependent. Dp is just equal to the back pressure inside the cavity. The initial value of the back pressure is normalized at 1 atmospheric pressure, i.e. the initial pressure inside and outside the cavity is equal. For a real die-casting situation, the vents can be partly or
W.B. Lee, H.Y. Lu / Journal of Materials Processing Technology 91 (1999) 264–269
267
entirely sealed off during the filling stage. The change of the venting area should be considered in the calculation of the back pressure. For the partly sealed case, the venting area is reduced. When the vent is entirely sealed, dM/dt = 0 and dr = − MdV/V 2, the last term in the parenthesis of the Eq. (17) should be deleted. In addition, the local cross-sectional area of the cavity and the velocity of the molten metal can be used as the values of the F and u, respectively for the calculation of the local back pressure. Based on the model of the creation of back pressure and the above algorithm, a computer program has been developed. The program flowchart is shown in Fig. 4. The input parameters required for the calculation are as follows: (i) sizes of the die cavity; (ii) sizes of the vents; (iii) sizes of the ingate; and (iv) the filling velocity. 5. Example of calculation and discussion With the program developed, examples of calculation in a rectangular die cavity are demonstrated. The sizes of the cavity and the gate velocity used in the calculations are as follows: thickness 3.0 mm; width 400 mm; length 400 mm; gate area 100 mm2; gate velocity 15 m s − 1. The results are shown in Figs. 5 – 8. In the figures, the abscissa represents the ratio of the volume filled by molten metal and the initial cavity volume (V/V0), and the ordinate represents the back pressure in relative values, i.e. the origin of the ordinate denotes unit atmospheric pressure. The results are calculated on the assumption that the vents remain unsealed.
Fig. 4. The program flowchart for the calculation of air back pressure.
Fig. 5. The creation of back pressure (p) with four vent thickness (h) in different stages of die filling from start V/V0 =0 to finish V/V0 = 0.9 (vent length l= 50 mm; vent breadth b= 100 mm).
Fig. 5 illustrates the results of the creation of air back pressure with different vent-thickness at different stages of the cavity filling process. Rectangular vents of 50 mm length (l) and 100 mm width (b) with thickness (h) of 0, 0.03, 0.05 and 0.10 mm are used in the calculation. The rates of change of the back pressure are shown for different vent thickness. The maximum rate of change of the back pressure and the maximum final back pressure occur for curve 1 in the case of h= 0 mm. In this case, no air can escape through the vent as the air inside the cavity is adiabatically compressed, so that the maximum rate of change and high back pressure are created. The fourth curve, corresponding to h= 0.1 mm is smooth, which shows that the air inside the cavity
Fig. 6. The creation of back pressure (p) with four vent breadths (b) in different stages of die filling from start V/V0 =0 to finish V/V0 = 0.9 (vent thickness h = 0.05 mm; vent length l= 40 mm).
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W.B. Lee, H.Y. Lu / Journal of Materials Processing Technology 91 (1999) 264–269
Fig. 7. The creation of back pressure (p) with four vent lengths (l) in different stages of die filling from start V/V0 = 0 to finish V/V0 =0.9 (vent breadth b = 100 mm; vent thickness h= 100 mm).
can discharge easily through a vent of this thickness, and only a low back pressure is created. The other curves lie between the two extreme cases. A considerable difference of the creation of the back pressure can be seen for different vent thickness. A small increment in the vent thickness can cause a notable reduction in the back pressure and the results demonstrate the significant influence of the thickness of the vent on the escape of air. Figs. 6 and 7 give the results for different breadths and lengths of the vents. Reverse effect of the two factors on the creation of back pressure can be found. The back pressure decreases with an increase in the vent breadth or with a decrease in the vent length, as a
Fig. 8. A comparison of results calculated using the traditional model and present model, with the same area of vent but with different breaths, thickness or lengths (area of vent = 5.0 mm2).
narrow vent reduces the venting area and a long venting distance increases the flow resistance of the air, which both lead to a reduction in the venting capability. A comparison of the creation of the back pressure between the conventional model and the present model is shown in Fig. 8. The sizes and area of the vents used in this calculation are given in Table 2. Curves (1), (2) and (3) are calculated from the present model and curve (4) from the conventional model. All of the results are obtained with the same venting area but with different thickness-to-breadth ratios. Although the venting areas are the same, the creation of the back pressure is different for all cases. A high back pressure is created in thin vents, i.e. curve (1). There is a greater resistance to the escape of air in this configuration. Curve (4) calculated from the conventional model shows a lower back pressure, as the conventional model does not involve the effect of the sizes of the vent. The conventional model always gives the same curve provided the vent area is the same for a given problem. Curve (3) shows a similar result with that of curve (4), since the calculation of curve (3) uses a thicker vent thickness and a smaller vent length. The difference between the present model and the traditional model in the calculation of the back pressure is clearly demonstrated. The venting capability of a die is related to the configuration of its vents. The sizes of a vent, i.e. the thickness, the breadth and the length, all affect the venting capability. The change of the back pressure is directly proportional to the venting length, inversely proportional to the venting breadth and in proportion to the third power of the venting thickness. Since the thickness of a vent is a key factor in controlling the venting capability of dies, it should be as large as possible in die design. However, it should be noted that too thick a vent can cause flashes of the metal. Vents with variable cross-sections or a stepped vent can be used to improve the venting capability, since the resistance to air can be reduced in such a design. In general, vents are located in the far end from the ingate and in a position to be filled last to prevent the vent from being sealed off early. If the vents can be kept unblocked during the entire filling stage, the volume of molten metal entering the cavity is equal to that of the air leaving the cavity. The air back pressure will increase only slightly, and air entrapment in the diecastings can be minimized. In recent years, many studies have focused on the computer simulation of cavity filling. Free surfaces of molten metal occur during the filling stage, and back pressure is exerted on these free surfaces. In the flow simulation, the back pressure boundary condition is generally set to zero or taken as a simple function. The validity of the computer simulation will be improved if the values of the calculated back pressure is used as the pressure boundary condi-
W.B. Lee, H.Y. Lu / Journal of Materials Processing Technology 91 (1999) 264–269
269
Table 2 The sizes and areas of the vents for contrast calculation Curve no.
Thickness (h) (mm)
Breadth (b) (mm)
Length (l) (mm)
Area (h×b) (mm2)
1 2 3 4
0.05 0.1 0.1
100 50 50
50 50 20
5 5 5 5
tion in the flow simulation. In addition, a knowledge of the back pressure is useful for the prediction of the entrapment of air and the formation of porosity. When the back pressure exceeds the pressure of adjacent molten metal, the area occupied by the air cannot be filled and porosity will result.
6. Conclusion A model for calculating back pressure is proposed which involves the influence of the vent geometry. Computations are carried out for a rectangular vent with different sizes. The results of the computation were analyzed and some useful findings were obtained. Venting capability is directly proportional to the width and inversely proportional to the length of the vent. The vent thickness exerts the greatest influence on the venting capability. There is a considerable difference between the present model and the traditional model in the calculation of the back pressure. In order to have an optimum venting condition, the vent thickness should be close to the possible maximum in the vent design of diecasting dies.
Acknowledgements The authors wish to thank the Research Committee
.
of the Hong Kong Polytechnic University for granting a scholarship to carry out this research project.
References [1] Yoshiaki Yamamoto, Yasushi Iwata, Metal flow and solidification behavior of aluminum die-castings, Foundry 60 (1988) 770 – 776. [2] Yoshiaki Yamamoto and Yasushi Iwata, Molten metal velocities and gas pressure in die cavity and defects in commercial aluminum pressure die-castings, 15th Int. Die-casting Congress & Exposition, Paper no. G-T 89-081, St. Louis, October 1989. [3] Koichi Tanaka, Kazuhiko Terashima, Melt filling behavior during die-casting under the reduced pressure, Foundry 65 (1993) 277 – 283. [4] Lee Wing-bun, The high pressure die-casting of quality thinwall components, Dissertation, Brunel University, 1976. [5] Li Xianchen, On the filling theory and porosity forming in die-castings, Trends of Die Casting, Shanghai Diecasting Society, (3/4) (1990). [6] B. Sachs, An analytical study of die-casting process, ASTM Bulletin, September 1953. [7] A.B. Draper, An analytical approach to die-casting, Proc. 2nd National Die Casting Congress, Detroit, 1962. [8] Li Shijiu, Engineering fluid mechanics, Engineering Industry Press, 1980. [9] Guanye Youxin, Metal Diecasting Technology, Fuhan Press, 1980. [10] Sheng Jingchao, Hydraulic Fluid Mechanics, Mechanical Industrial Press, 1980. [11] Jui Sheng Hsieh, Engineering thermodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1993.
Modeling of air back pressure in die-casting dies W.B. Lee *, H.Y. Lu Department of Manufacturing Engineering, The Hong Kong Polytechnic Uni6ersity, Hung Hom, Kowloon, Hong Kong Received 23 June 1998
Abstract In high-pressure die-casting, molten metal is injected into a die cavity at a high velocity. The metal entering into the cavity expels the air from the cavity through the vents. Air back pressure will be created when the volume of metal flow exceeds the volume of air escaping out of the die cavity. A new mathematical model is proposed for the calculation of the creation of the back pressure in a die cavity. Air discharge through vents is modeled by the Poiseuille flow. The differential equation of the time rate of change of the back pressure is derived. The creation of the back pressure is correlated with the configuration sizes of the vent, and computations are carried out for different configurations of the vent geometry. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Die-casting; Air back pressure; Poiseuille flow
1. Introduction During the stage of die cavity filling, molten metal can only advance as quickly as the air inside the cavity is expelled. As the die is impermeable to the air, vents are provided to allow the air to escape out of the cavity. Metal filling in a die cavity is a rapid transient process. The filling velocity can be as high as several tens of meters per second. The air inside the empty region is compressed and air back pressure can be created if there is not a sufficient venting area or if the die vents are sealed off before the die cavity is completely filled up. Air back pressure thus caused may be several times the atmospheric pressure or more [1,2]. The creation of the back pressure can lead to a change in the filling sequence [3]. If the back pressure being exerted on the forefront of the advancing molten metal is large enough to delay filling or to resist the pressure applied by the diecasting machine, incomplete die filling will occur in thin sections or corners. In addition, the back pressure would increase the amount of air dissolving into the molten metal and cause porosity in the pressure diecastings [4,5]. Therefore the problem of eliminating cavity
* Corresponding author. Tel.: 27666594.
+852-23659248; fax:
+ 852-
air and hence back pressure to facilitate metal flow is critical, especially for large thin-wall die-castings. An understanding on the process of the creation of back pressure is important for the production of quality castings. The effect of the vent configuration sizes and the air viscosity on the creation of the back pressure has not often been analyzed in the literature [6,7]. The air back pressure is a transient variable depending on the flow rate of the air discharged through the vents. The sizes of the vent and the air viscosity have an influence on the flow rate of the air. In this paper, an analysis of the air flow through die venting is made and the equation of the flow rate is obtained in the form of Poiseuille flow. A differential equation of the time rate of change of the back pressure is derived based on the assumption of uniform-state and adiabatic process. The transient value of the back pressure at different time intervals is calculated for different vent configurations with conventional model and the model proposed in this paper.
2. Air flow rate through vent As the metal die is impermeable to air, vents are provided for the escape of air, the vents are generally being along the parting line or ejector pin in the form
0924-0136/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 8 ) 0 0 4 1 8 - X
W.B. Lee, H.Y. Lu / Journal of Materials Processing Technology 91 (1999) 264–269
of shallow grooves. When the molten metal is injected into the cavity, air inside the cavity is expelled through the vents. A conventional model [3] often used in solving for mass flow rate of the air through vent and creation of the back pressure was derived from the idea of air flow through a nozzle. The variables used are shown in Fig. 1. The size of the container is usually much larger than that of the nozzle, and the velocity of the air inside the container is considered to be zero. Air inside the container discharges through the nozzle of area A. The velocity of the air escaping through the nozzle is considered to be much larger than that of heat transfer, such that the air discharge is assumed to be an adiabatic process. The energy balance equation about crosssections (1) and (2) in Fig. 1 can be written as [8]: k P1 k P0 m 20 = + k − 1 r1 k−1 r0 2
(1)
For an adiabatic process, r0 =r1(P0/P1)1/k, the exit velocity can be written as: m0 =
'
n
2k P1 P 1− 0 k −1 r1 P1
k−1 k
(2)
The mass flow can be calculated using the equation: m= ruA
(3)
Substituting the exit velocity and density into Eq. (3) yields: m= A or:
'
n
2k P1r1 k −1
'
P m= Ar1 0 P1
1 k
P0 P1
2 k
P − 0 P1
k+1 k
n
2k P1 P 1− 0 k −1 r1 P1
k−1 k
(4)
(5)
where m is the mass flow rate of air, P1 and P0 are the pressure inside and outside the die cavity, respectively. r1 is the air density inside the cavity, k is the adiabatic exponential and A is the cross-sectional area of the vent.
265
Fig. 2. Schematic diagram of a die-casting die with vents.
A real rectangular vent is defined by its thickness (h), breadth (b) and its length (l) as shown in Fig. 2. It should be noted that the mass flow rate of the air relates only to the cross-sectional area of the vent and does not relate to the three individual sizes of the vent in Eq. (5). The sizes of the vent commonly used in die-casting practice are given in Table 1. The thickness of the vent is usually much smaller than its width (more than 100 times smaller), and the thickness of the vent is of the order of one tenth or one hundredth of a millimetre. The fluid flow in such a shallow groove is thus different from that found in a large channel of a piping system. For flow in a shallow groove, it is assumed that the velocity component is zero and the pressure is equi-distributed in the normal direction to the main flow (i.e. the y direction in Fig. 2). For a small elementary volume of the air flowing through the vent as shown in Fig. 3, the compressibility of the air in the vent is assumed to be negligible. The force balance on it may be written as: %Fx = 0
(6)
then: pdy− (p+ dp)dy = tdx − (t +dt)dx
(7)
or: dt dp = dy dx
(8)
Table 1 The vent size (rectangular shape) as recommended by different researchers [9]
Fig. 1. Example of air flow through a nozzle.
Recommended by
Thickness (mm)
Doehler Hoeb Street Stern Frommer
0.1270.152 0.0760.127 0.076 0.1270.254 0.05–1
Width (mm)
1325.4 13
W.B. Lee, H.Y. Lu / Journal of Materials Processing Technology 91 (1999) 264–269
266
From Eq. (12), the rate of change of the air mass within cavity can be written as: dM rbh 3 Dp =− % dt vents 12 ml
(13)
The rate of change of the air volume within the cavity is thus given by: dV = − % Fu dt gates Fig. 3. The forces on a tiny elementary volume of flow.
Substituting t =m du/dy into Eq. (8):then: dp d2u =m 2 dy dx
(9)
m=
1 dp (y − h)y 2m dx
(10)
where p and m are the pressure and dynamic viscosity of the air, respectively. Eq. (10) shows that the velocity distribution in the vent is parabolic and dp/dx is constant. If the pressure inside and outside the cavity is P1 and P0, respectively, then: Dp dp −(p1 −p0) =− = l l dx
(11)
&
h
0
ubdy =
bh 3 Dp 12 ml
M VdM −MdV = V V2
(12)
where b, h and l are the width, thickness and length of the vent, respectively. Eq. (12) shows that the flow rate is directly proportional to the cube of the thickness of the vent, so that the thickness of the vent will has a strong influence on the air flow in the vent, and the flow is driven by differential pressure. This is a typical Poiseuille flow [10].
3. Creation of the air back pressure Cavity filling is an unsteady process. If the filling is supposed to be completed in a number of small time intervals, the state of the air within cavity can be assumed to be uniform in temperature, pressure and density in each interval, even though this uniform state may change with time. The state of the air passing through vents can also be assumed to be invariable although the mass-flow rate may change with time. The above assumptions of the uniform-state are widely used in engineering thermodynamics [11] to simplify the calculations.
(15)
where M is the mass of air inside the cavity and V is the volume occupied by the air. Since the die filling process is transient, the creation of the air back pressure can be considered to be an adiabatic process. For an adiabatic flow, p/r k = C, its differential form can be written as: dr dp= kp r
(16)
where k is an adiabatic exponential, and k= 1.4 for air. Substituting Eqs. (13) and (14) into Eq. (15), and then Eq. (15) into Eq. (16), yields:
dp kp rbh 3 Dp =− M % Fu−V % 2 dt rV gates vents 12 ml
The flow rate of the air can be written as: Q=
where F and u denote the area and velocity of the ingate, respectively. Since the mass and volume of the air inside the cavity change, its density will also change. The differential of density change is: dr= d
Integrating Eq. (9) and using boundary conditions y= 0, u = 0; y= h, u=0 yields:
(14)
(17)
Eq. (17) shows the rate of the change of the air back pressure during the filling process that can be used to calculate the back pressure. Compared with the conventional model for calculating back pressure, the present model correlates the creation of the back pressure with different configurations of the vent geometry.
4. Determination of the air back pressure In order to solve for the air back pressure in different time intervals, Eq. (17) should be rewritten into an increment form. The filling is supposed to be completed in a number of small time intervals, and the back pressure is calculated step by step. When the new values, p, r, M and V are solved, the calculation of the next time step will be done and these new values will become the old values to be used for the next calculation cycle. It should be note that Dp is the differential pressure between the cavity and the ambient and is time-dependent. Dp is just equal to the back pressure inside the cavity. The initial value of the back pressure is normalized at 1 atmospheric pressure, i.e. the initial pressure inside and outside the cavity is equal. For a real die-casting situation, the vents can be partly or
W.B. Lee, H.Y. Lu / Journal of Materials Processing Technology 91 (1999) 264–269
267
entirely sealed off during the filling stage. The change of the venting area should be considered in the calculation of the back pressure. For the partly sealed case, the venting area is reduced. When the vent is entirely sealed, dM/dt = 0 and dr = − MdV/V 2, the last term in the parenthesis of the Eq. (17) should be deleted. In addition, the local cross-sectional area of the cavity and the velocity of the molten metal can be used as the values of the F and u, respectively for the calculation of the local back pressure. Based on the model of the creation of back pressure and the above algorithm, a computer program has been developed. The program flowchart is shown in Fig. 4. The input parameters required for the calculation are as follows: (i) sizes of the die cavity; (ii) sizes of the vents; (iii) sizes of the ingate; and (iv) the filling velocity. 5. Example of calculation and discussion With the program developed, examples of calculation in a rectangular die cavity are demonstrated. The sizes of the cavity and the gate velocity used in the calculations are as follows: thickness 3.0 mm; width 400 mm; length 400 mm; gate area 100 mm2; gate velocity 15 m s − 1. The results are shown in Figs. 5 – 8. In the figures, the abscissa represents the ratio of the volume filled by molten metal and the initial cavity volume (V/V0), and the ordinate represents the back pressure in relative values, i.e. the origin of the ordinate denotes unit atmospheric pressure. The results are calculated on the assumption that the vents remain unsealed.
Fig. 4. The program flowchart for the calculation of air back pressure.
Fig. 5. The creation of back pressure (p) with four vent thickness (h) in different stages of die filling from start V/V0 =0 to finish V/V0 = 0.9 (vent length l= 50 mm; vent breadth b= 100 mm).
Fig. 5 illustrates the results of the creation of air back pressure with different vent-thickness at different stages of the cavity filling process. Rectangular vents of 50 mm length (l) and 100 mm width (b) with thickness (h) of 0, 0.03, 0.05 and 0.10 mm are used in the calculation. The rates of change of the back pressure are shown for different vent thickness. The maximum rate of change of the back pressure and the maximum final back pressure occur for curve 1 in the case of h= 0 mm. In this case, no air can escape through the vent as the air inside the cavity is adiabatically compressed, so that the maximum rate of change and high back pressure are created. The fourth curve, corresponding to h= 0.1 mm is smooth, which shows that the air inside the cavity
Fig. 6. The creation of back pressure (p) with four vent breadths (b) in different stages of die filling from start V/V0 =0 to finish V/V0 = 0.9 (vent thickness h = 0.05 mm; vent length l= 40 mm).
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Fig. 7. The creation of back pressure (p) with four vent lengths (l) in different stages of die filling from start V/V0 = 0 to finish V/V0 =0.9 (vent breadth b = 100 mm; vent thickness h= 100 mm).
can discharge easily through a vent of this thickness, and only a low back pressure is created. The other curves lie between the two extreme cases. A considerable difference of the creation of the back pressure can be seen for different vent thickness. A small increment in the vent thickness can cause a notable reduction in the back pressure and the results demonstrate the significant influence of the thickness of the vent on the escape of air. Figs. 6 and 7 give the results for different breadths and lengths of the vents. Reverse effect of the two factors on the creation of back pressure can be found. The back pressure decreases with an increase in the vent breadth or with a decrease in the vent length, as a
Fig. 8. A comparison of results calculated using the traditional model and present model, with the same area of vent but with different breaths, thickness or lengths (area of vent = 5.0 mm2).
narrow vent reduces the venting area and a long venting distance increases the flow resistance of the air, which both lead to a reduction in the venting capability. A comparison of the creation of the back pressure between the conventional model and the present model is shown in Fig. 8. The sizes and area of the vents used in this calculation are given in Table 2. Curves (1), (2) and (3) are calculated from the present model and curve (4) from the conventional model. All of the results are obtained with the same venting area but with different thickness-to-breadth ratios. Although the venting areas are the same, the creation of the back pressure is different for all cases. A high back pressure is created in thin vents, i.e. curve (1). There is a greater resistance to the escape of air in this configuration. Curve (4) calculated from the conventional model shows a lower back pressure, as the conventional model does not involve the effect of the sizes of the vent. The conventional model always gives the same curve provided the vent area is the same for a given problem. Curve (3) shows a similar result with that of curve (4), since the calculation of curve (3) uses a thicker vent thickness and a smaller vent length. The difference between the present model and the traditional model in the calculation of the back pressure is clearly demonstrated. The venting capability of a die is related to the configuration of its vents. The sizes of a vent, i.e. the thickness, the breadth and the length, all affect the venting capability. The change of the back pressure is directly proportional to the venting length, inversely proportional to the venting breadth and in proportion to the third power of the venting thickness. Since the thickness of a vent is a key factor in controlling the venting capability of dies, it should be as large as possible in die design. However, it should be noted that too thick a vent can cause flashes of the metal. Vents with variable cross-sections or a stepped vent can be used to improve the venting capability, since the resistance to air can be reduced in such a design. In general, vents are located in the far end from the ingate and in a position to be filled last to prevent the vent from being sealed off early. If the vents can be kept unblocked during the entire filling stage, the volume of molten metal entering the cavity is equal to that of the air leaving the cavity. The air back pressure will increase only slightly, and air entrapment in the diecastings can be minimized. In recent years, many studies have focused on the computer simulation of cavity filling. Free surfaces of molten metal occur during the filling stage, and back pressure is exerted on these free surfaces. In the flow simulation, the back pressure boundary condition is generally set to zero or taken as a simple function. The validity of the computer simulation will be improved if the values of the calculated back pressure is used as the pressure boundary condi-
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Table 2 The sizes and areas of the vents for contrast calculation Curve no.
Thickness (h) (mm)
Breadth (b) (mm)
Length (l) (mm)
Area (h×b) (mm2)
1 2 3 4
0.05 0.1 0.1
100 50 50
50 50 20
5 5 5 5
tion in the flow simulation. In addition, a knowledge of the back pressure is useful for the prediction of the entrapment of air and the formation of porosity. When the back pressure exceeds the pressure of adjacent molten metal, the area occupied by the air cannot be filled and porosity will result.
6. Conclusion A model for calculating back pressure is proposed which involves the influence of the vent geometry. Computations are carried out for a rectangular vent with different sizes. The results of the computation were analyzed and some useful findings were obtained. Venting capability is directly proportional to the width and inversely proportional to the length of the vent. The vent thickness exerts the greatest influence on the venting capability. There is a considerable difference between the present model and the traditional model in the calculation of the back pressure. In order to have an optimum venting condition, the vent thickness should be close to the possible maximum in the vent design of diecasting dies.
Acknowledgements The authors wish to thank the Research Committee
.
of the Hong Kong Polytechnic University for granting a scholarship to carry out this research project.
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