Mathematically determined solidification for timing the injection of die castings

A method of improving the die casting process by determining a thermal conductivity, ka, of die cast metals at elevated temperatures for use in an equation for heat conduction, Q(t), that establishes a time range for setting and controlling a fill time, wherein ka is determined in a computer software program by varying ka in a calculated plot of an equation for a boundary surface, ub(t), between the solidifying metal and the die, to fit an Isoclines boundary plot of data from a die surface-mounted thermocouple, where the equation for a boundary of the calculated plot is, 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:
• ub(t)=boundary temperature or alloy-die surface temperature as a function of time,
• t=time
• uai=metal temperature (initial) or alloy pouring temperature of the molten alloy,
• ul=liquidus temperature of the molten alloy,
• usi=temperature (initial) of die at t=0,
• cpls=liquidus-solidus molten alloy specific heat
• cpliq=liquidus molten alloy specific heat,
• ρa=molten alloy density,
• d=the half-thickness of an alloy wall of 2 times d,
• kst=die material thermal conductivity,
• cst=die material specific heat,
• ρst=die material density,
• l=a thickness of die material >20×d,
• π=3.1416,
• Exp(n)=base en or 2.71828182845904n,
• Σ=summation from 0 to n and,
• where:
• λalliq=αalliq*(2*n+1)*π/(2*d) and, αalliq=(kst/cpliq*ρa)1/2
• and,
• λalls=αalls*(2*n+1)*π/(2*d) and, αalls=(kst/cpls*ρa)1/2
• and,
• λst=αst*(2*n+1)*π/(2*l) and, αst=(kst/(cst*ρst))1/2
• with said calculated boundary being used to determine ka as a function of temperature from the molten phase through the solidus temperature, using additional thermal properties of specific heat and density for said metal stored in the software program, and using stored thermal properties of specific heat, thermal conductivity, and density for the die material, whereby a fitting of said calculated boundary, ub(t), to the plot of a stored boundary temperature ub(t) data obtained by the alloy-die surface thermocouple is from a casting operation, in which the properties of the alloy material cast and the properties of the die material used in constructing said die are those of the calculated plot, where a wall thickness of 2 times d in the calculated plot is equal to a wall thickness at the location where the thermocouple boundary data was taken, and where an initial metal temperature uai for a calculated plot is the same as an initial metal temperature uai for the thermocouple plot and an initial die temperature usi for a calculated plot equals an initial die temperature usi for the thermocouple plot.

A method of improving a pressure die casting process by controlling an injection phase of an operation of a die casting machine through application of equations derived by equating a heat flow from an injected solidifying alloy wall to heat entering a die material, said derived equations are for calculating temperature vs. time at varying depths in a solidifying alloy wall, for calculating temperature vs. time at varying depths in a die, for calculating temperature vs. time at a boundary surface between the alloy wall and the die, and a quantity of heat transferred from the alloy to the die vs. time, with said equations used to determine a maximum and minimum time for injection of metal into a cavity by using a computer software program to calculate said equations for determining a solidification progression or solidification pattern, Isoclines, determining a dynamic liquidus and a dynamic solidus in a selected wall of a casting, and calculating an amount of heat remaining Q(t) in said wall above the dynamic solidus, where said wall is formed by a stationary cover die and an ejector die assembly which define the cavity therebetween when a die is in its closed position, wherein said process a thermocouple in a path of molten metal poured into a metal injection sleeve is used to detect and store in said computer program an initial molten metal temperature as a value of uai in said equations, with one or more locations of said cavity's surface probed with one or more pyrometers an instant before the machine closes to detect said location's surfaces temperature, using a means for preparing and transmitting transmitting one or more pyrometer signals for digital storage as data in the software program and this data is to be used as usi, and having a set of alloy thermal properties consisting of specific heats, an alloy density, and an alloy thermal conductivity, and having a set of die material thermal properties consisting of specific heat, density and conductivity stored in the computer, and at least one wall thickness of 2 times d for a pyrometer sensing location/s is manually entered and stored in the software program to provide a value d, whereby values for three variables uai, usi, d, and thermal properties for the alloy and the die material stored in the software program are all used for calculating and plotting the equation for ub(t), an Alloy-Die Surface Boundary, at selected wall location/s, 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* n1)*π/(2*d) and, αalliq=(kst/cpliq*ρal)1/2
• and,
• λal=αalls*(2*n1)*π/(2*d) and, αalls=(kst/cpls*ρal)1/2
• and,
• λst=αst*(2*n+1)*π/(2*l) and, αst=(kst/(cst*ρst))1/2
• t=time
• ub(t)=calculated boundary temperature or alloy-die surface temperature as a function of time,
• uai=metal temperature (initial) or alloy pouring temperature of the molten alloy,
• ul=liquidus temperature of molten alloy,
• usi=temperature (initial) of die at t=0 from a pyrometer,
• d=half-thickness of alloy wall of 2 times d,
• cpls=specific heat of the liquidus-solidus molten alloy
• cpliq=specific heat of the liquidus molten alloy,
• ρa=density of the molten alloy,
• αst=kst/(cst*ρst)=diffusivity of the die material,
• kst=thermal conductivity of a die material,
• cst=specific heat of a die material,
• ρst=density of a die material,
• l=a thickness of die material >20×d,
• π=3.1416,
• Exp(n)=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 wall solidification pattern, Isoclines, using the equation, u a ⁡ ( x <= d , t ) = ⁢ u b + 4 / π * { ( u ai - u l ) * ∑ ⁢ ( ( - 1 ) n / ( 2 * n + 1 ) ) * ⁢ 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 ) ) * [AutoLeftMatch] 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 line of the alloy wall equal to or less than d,
• ua(x<=d, t)=calculated temperature of an alloy as a function of depth & time,
• Cos=trigonometric cosine,
• and for calculating and plotting a set of heating lines in a die material using the equation,
• us(d=<x<l,t)=ub+4*(usi−ub)/π*Σ(1/(2n+1))*EXP(−((αst*(2*n+1)*π/(2*l)){circumflex over ( )}2*t))*SIN(((2*n+1)*π*(xs−d)/(2*l))
• where:
• xs=a depth in a die material equal to or greater than d,
• us(d=<x<l, t)=temperature of die material as a function of depth & time,
• Sin=trigonometric sine,
• and calculating an amount of heat remaining above the dynamic solidus temperature, using the equation, Q ⁡ ( t ) = 2 * k a / d ⁢ { ( u ai - 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, Q(t), being used for establishing a maximum and minimum limit for the fill time and whereby, for comparison purposes with that fill time, a pressure transducer means is provided in a die casting die or in a die casting machine injection cylinder for detection of an injection pressure whereupon this pressure signal is converted to digital data for entry and storage in said software program and this data is used by the computer program to plot an injection pressure-time graph for fill time, with said fill time to be controlled and set for the selected wall location automatically for a next cycle by said computer program within the Isoclines determination of the dynamic solidus and the dynamic liquidus thereby setting limits for the maximum and minimum fill time and/or using the heat calculation, Q(t), for establishing the time range within which fill time is to be controlled and set, all based on a calculated boundary ub(t).