Mathematically determined solidification for timing the injection of die castings

A pressure die casting process wherein the thermal property of alloy conductivity ka is determined using the preferred computer software program to fit the calculated plot of the inventors equation for ub(t) i.e.

• 7 u b = ( 2 * ( u ai - u l ) * k a / d * ∑ ⁢ Exp ⁡ ( - λ al liq ⋀ 2 * t ) + 2 * u l * k a / d * ∑ ⁢ Exp ⁡ ( - λ al ls ⋀ 2 * t ) + 2 * u si * k st / l * ∑ ⁢ Exp ⁡ ( - λ st ⋀ 2 * t ) ) / ( 2 * k st / l * ∑ ⁢ Exp ⁡ ( - λ st ⋀ 2 * t ) + 2 * k a / d * ∑ ⁢ Exp ⁡ ( - λ al ls ⋀ 2 * t ) )
• where:
• λal=αalliq*(2*n+1)*π/(2*d) and, αalliq=(kst/cpliq*ρal)½
• and,
• λal=αalls*(2*n+1)*π/(2*d) and, αalls=(kst/cpls*ρal)½
• and,
• λst=αst*(2*n+1)*π/(2*l) and, αst=(kst/(cst*ρst))½
• using other stored thermal properties for the alloy and stored thermal properties for the die materials, fitting said calculated plot to a plot of the boundary temperature ub(t) as a function of time from data obtained by the alloy-die surface thermocouple stored in the computer using the same alloy pouring temperature uai and the same initial die temperature usi for both plots.

A method of improving the die casting process by applying the inventors mathematics and physics in the operation of a pressure die casting machine wherein a thermocouple in the path of the molten metal poured into the metal injection sleeve is used to detect and store the metal temperature in the preferred software program as the value of uai for calculating the inventor's equations where said machine contains a die comprised of a stationary cover die and an ejector die assembly movable along a longitudinal or vertical axis of said machine between a die open and a die closed position cooperatively defining a cavity there between in said closed position, with one or more locations of said cavity's surface probed with a pyrometer, either optical or surface type, the instant before the machine closes to detect said surface's temperature, means for preparing and transmitting the pyrometer signal for digital storage directly in the preferred software program of this/these value/s as usi for calculating the inventor's equations using the physical thermal properties of the alloy and die material stored in the computer in conjunction with a wall thickness for the location/s of 2d previously stored in the preferred computer software program but more specifically calculating the inventors equation for ub(t), the Alloy-Die Surface Boundary,

• 8 u b ⁡ ( t ) = ( 2 * ( u ai - u l ) * k a / d * ∑ ⁢ Exp ⁡ ( - λ al liq ⋀ 2 * t ) + 2 * u l * k a / d * ∑ ⁢ Exp ⁡ ( - λ al ls ⋀ 2 * t ) + 2 * u si * k st / l * ∑ ⁢ Exp ⁡ ( - λ st ⋀ 2 * t ) ) / ( 2 * k st / l * ∑ ⁢ Exp ⁡ ( - λ st ⋀ 2 * t ) + 2 * k a / d * ∑ ⁢ Exp ⁡ ( - λ al ls ⋀ 2 * t ) )
• where:
• λal=αalliq*(2*n+1)*π/(2*d) and, αalliq=(kst/cpliq*ρal)½
• and,
• λal=αalls*(2*n+1)*π/(2*d) and, αalls=(kst/cpls*ρal)½
• and,
• λst=αst*(2*n+1)*π/(2*l) and, αst=(kst/(cst*ρst))½
• uai=metal temperature (initial) or alloy pouring temperature of die cast molten alloy as determined by a thermocouple,
• ul=liquidus temperature of die cast molten alloy,
• usi=temperature (initial) of die at t=0 from a pyrometer,
• d=the half-thickness of alloy wall,
• cpls=the specific heat of the liquidus-solidus molten alloy
• cpliq=the specific heat of the liquidus molten alloy,
• ρa=the density of the molten alloy,
• αst=kst/(cst*ρst)=the diffusivity of the die material,
• kst=the thermal conductivity of the die material as a function of temperature,
• cst=the specific heat of the die material,
• ρst=the density of the die material.
• l=thickness of die material >20×d,
• π=3.1416,
• Exp(n)=the base en or 2.71828182845904n,
• Σ=summation from 0 to n and
• with said Alloy-Die Surface Boundary used for calculating and plotting the injected alloy solidification paftem, Isoclines, using the inventor's equation,
• 9 u a ⁡ ( x <= d , t ) = u b + 4 / π * { ( u ai - u l ) * ∑ ⁢ ( ( - 1 ) n / ( 2 * n + 1 ) ) * [AutoLeftMatch] EXP ⁡ ( - ( ( ( k st / ( c pliq * ρ ⁢ ⁢ a ) ) .5 * ( 2 * n + 1 ) * π / ( 2 * d ) ) ⋀ 2 * t ) ) * COS ⁡ ( ( ( 2 * n + 1 ) * π * x ) / ( 2 * d ) ) + ( u l - u b ) * ∑ ⁢ ( ( - 1 ) n / ( 2 * n + 1 ) ) * EXP ⁡ ( - ( ( ( k st / c pls * ρ ⁢ ⁢ a ) ) .5 * ( 2 * n + 1 ) * π / ( 2 * d ) ) ⋀ 2 * t ) ) * COS ⁡ ( ( ( 2 * n + 1 ) * π * x ) / ( 2 * d ) ) } .
• where:
• x=distance from center of the alloy wall equal to or less than d,
• ua(x<=d, t)=calculated temperature of die cast alloy as a function of depth & time,
• ub(t)=calculated boundary temperature for the alloy-die surface temperature as a function of time,
• Cos=the trigonometric cosine,
• and for calculating and plotting, but not necessarily, the cooling lines in the die material using the inventor's equation,
• us(d=<x<l,t)=ub+4*(usi−ub)/π*Σ(1/(2n+1))*EXP(−((αst*(2*n+1)* π/(2*l)ˆ 2*t))*SIN(((2*n+1)*π*(xs−d))/(2*l)),
• where:
• xs=depth in the die material equal to or greater than d,
• us(d=<x<l,t)=temperature of die material as a function of depth & time,
• Sin=the trigonometric sine.
• and calculating the amount of heat remaining above the dynamic solidus using the inventor's equation,
• 10 Q = 2 * k a / d ⁢ { ( u a - u l ) * [ ∑ ⁢ e - λ l 2 * t - λ l 2 - ∑ ⁢ 1 - λ l 2 ] + ( u l - u b ) * [ ∑ ⁢ e - λ ls 2 * t - λ ls 2 - ∑ ⁢ 1 - λ ls 2 ] }
• where:
• λl=(kst/(cpliq*ρa))0.5*(2*n+1)*π/(2*d)
• and
• λls=(kst/(cpls*ρa))0.5*(2*n+1)*π/(2*d)
• with said solidification pattern and heat calculations being used for establishing a maximum and minimum limit for the optimum cavity fill time and a pressure transducer means is provided in the die casting die or in the die casting machine injection cylinder for the detection of the injection pressure whereupon the pressure signal is converted to digital data for direct entry and storage in the preferred software program and said data is used by the computer to plot the classical well-known and often used injection pressure-time graph which reveals the fill time, to be controlled and set for the selected cavity location somewhat automatically for the next cycle by the computer program within the Isoclines determined limits for the maximum and minimum fill time and/or within the die surface thermocouple boundary determinations of the dynamic liquidus and the dynamic solidus.