Formulas for Hot Isostatic Pressing

• Labor time standard: specified as the machine property Labor Time Standard. This multiplier is used to account for otherwise unaccounted for factors that affect labor time, such as operator fatigue or time spent by the operator for cleaning or maintenance.

• Labor handling time (see formula)

Labor Handling Time =

(Load Time + Unload Time) * Number of Handling Operators * Labor Time Standard

This is the labor time required to rack or wire up one part to be placed in the heat treatment furnace:

• Load time: time to rack or wire up one part to be placed in the heat treatment furnace. This time is interpolated from values looked up by mass in the lookup table tblHandlingTimes. The interpolated value is based on the table rows whose Max Weight values straddle the mass that must be handled. If the mass falls below the minimum mass listed in the table, the handling time is considered negligible, and is set to 0. If the mass exceeds the maximum mass listed in the table, aPriori uses the values in the row with the maximum mass.

• Unload time: time to remove a part from the heat treatment furnace. This is assumed to be the same as load time (see above).

• Number of handling operators: number of operators loading and unloading parts. By default in starting point VPEs, this is looked up by mass in the lookup table tblHandlingTimes. Users can override the default on a part-by-part basis with the setup option Labor Handling Number of Operators.

Amortized Batch Setup =

(Setup Time * (Labor Rate + Direct Overhead Rate)) / Batch Size

Batch setup cost per part depends on the following:

• Setup time (specified by the cost model variable hipProgrammingTime—0.25 hours in starting point VPEs)

• Labor rate (specified by the machine property Labor Rate)

• Direct overhead rate (see Direct and Indirect Overhead)

• Batch size (specified in the Production Scenario screen of the Cost Guide)

Cycle Time Formulas for Hot Isostatic Pressing

The calculation of cycle time relies on the formulas below.

Cycle Time = Process Time * Cycle Time Adjustment Factor

Cycle time is the product of the following:

• Process time (see formula)

• Cycle Time Adjustment Factor: value of the cost model variable cycleTimeAdjustmentFactor. This is set to 1 in aPriori starting point VPEs. VPE administrators can change the cost model variable to globally adjust cycle times.

Process Time =

(Vacuum Purge and Equalize Time +

Pressurize System Time +

Heat Time +

Holding Time +

Cool Time) / Number of Parts

Process time per part depends on the following:

• Vacuum purge and equalize time: time to fill the vessel with the inert, process gas, and equalize vessel pressure with the pressure in the gas inlet lines. By default, this is specified by the cost model variable vacuumPurgeEqualize (9 minutes in starting point VPEs). You can override the default on a part-by-part basis with the setup option Vacuum Purge and Equalize Pressure.

• Pressurize system time: time to bring the vessel to the required hold pressure. By default, this is specified by the cost model variable pressurizingVesselTime (40 minutes in starting point VPEs). You can override the default on a part-by-part basis with the setup option Pressurize Vessel Time.

• Heat time: by default, heat time is given by the formula for Default Heating Time, below. Users can override the default on a part-by-part basis with the setup option Heating Time.

• Holding time: by default, this is looked up by material type in the lookup table tblMaterialTypeProperties. If the material type is not found, default holding time is given by the cost model variable averageHoldTime (2.6 hours in starting point VPEs). Users can override the default on a part-by-part basis with the setup option Hold Time.

• Cool time: by default, cool time is given by the formula for Default Cooling Time, below. Users can override the default on a part-by-part basis with the setup option Cooling Time.

• Number of parts: number of parts per batch--see Number of Parts for Hot Isostatic Pressing.

Default Heating Time =

(Material Hold Temperature – Room Temperature) / Heating Rate

Default heating time depends on the following:

• Material hold temperature: looked up by material type in the lookup table tblMaterialTypeProperties. Specified in degrees Celsius.

• Room temperature: specified by the cost model variable defaultRoomTemperature (15°C in starting point VPEs).

• Heating rate: by default, this is looked up by material type in the lookup table tblMaterialTypeProperties. If the material type is not found, default heating rate is given by the cost model variable averageHeatingRate (7.2°C per minute, in starting point VPEs). Users can override the default on a part-by-part basis with the setup option Heating Rate.

Default Cooling Time =

(Material Hold Temperature – Room Temperature) / Cooling Rate

Default heating time depends on the following:

• Material hold temperature: looked up by material type in the lookup table tblMaterialTypeProperties.

• Room temperature: specified by the cost model variable defaultRoomTemperature (15°C in starting point VPEs).

• Cooling rate: by default, this is looked up by material type in the lookup table tblMaterialTypeProperties. If the material type is not found, default cooling rate is given by the cost model variable averageCoolingRate (13.3°C per minute, in starting point VPEs). Users can override the default on a part-by-part basis with the setup option Cool Rate.

Process Gas Cost for Hot Isostatic Pressing

The cost for the process gas is accounted for in Additional Direct Costs:

Additional Direct Costs = Actual Gas Cost Per Part / Final Yield

This is the per-part cost for process gas, taking into account process yield. It depends on the following:

• Actual gas cost per part (see formula)

• Final yield (see Yield Formulas for Hot Isostatic Pressing).

Actual Gas Cost Per Part = Actual Gas Cost / Number of Parts

This is the per-part cost of process gas. It depends on the following:

• Actual gas cost (see formula)

• Number of parts: number of parts processed per machine cycle. See Number of Parts for Hot Isostatic Pressing.

Actual Gas Cost = Hot Zone Gas Cost * (1 – Gas Reuse Percentage)

Cost of the process gas used for one machine cycle (that is, for one batch of parts), taking into account gas recycling. It depends on the following:

• Hot zone gas cost (see formula)

• Gas reuse percentage: this is the fraction of gas from one machine cycle that is recycled and used in the next machine cycle. By default in starting point VPEs, this is 0.95 (that is, 95% of the gas is reused). Administrators can customize the default with the cost model variable hipDefaultGasReusePercentage. Users can override the default on a part-by-part bases with the setup option Gas Reuse Percentage.

Hot Zone Gas Cost = Required Gas Volume * Argon Gas Cost Per Cubic Meter

Cost of the process gas used for one machine cycle. This does not take into account gas recycling, but rather reflects what the cost would be in the absence of recycling. It is the product of the following:

• Required gas volume (see formula)

• Argon gas per cubic meter (specified by the machine property Argon Gas Cost)

Required Gas Volume =

Net Gas Volume In Hot Zone *

(Machine Hold Pressure / Gas Cylinder Pressure) *

(Gas Cylinder Temperature / Machine Hold Temperature)

This formula is an application of the combined gas law. During the hold phase of a machine cycle (when parts are held for a period of time at high pressure and temperature), the portion of the hot zone that is not occupied by parts is filled with the process gas (assumed to be Argon). A certain number, n, of molecules of process gas is required to fill this space at hold pressure and temperature. Required Gas Volume is the volume of this number, n, of molecules of process gas at standard pressure and ambient temperature (or whatever is the pressure and temperature inside the gas cylinder from which the process gas is fed into the vessel). Required Gas Volume depends on the following:

• Net gas volume in hot zone: volume of gas in the hot zone at hold pressure and temperature. See formula.

• Machine hold pressure: pressure in the hot zone during the hold phase of the machine cycle. By default in starting point VPEs, this is 100 MPa. Administrators can customize the default with the cost model variable hipDefaultHoldPressure. Users can override the default on a part-by-part basis with the setup option Hold Pressure.

• Gas cylinder pressure: pressure inside the gas cylinder from which the process gas is fed into the vessel. This is specified by the cost model variable hipGasCylinderPressure (in starting point VPEs, this is standard pressure, 0.101325 MPa).

• Gas cylinder temperature: temperature inside the gas cylinder from which the process gas is fed into the vessel (which presumably is the ambient temperature). This is specified in degrees Celsius by the cost model variable defaultRoomTemperature (15 degrees Celsius in starting point VPEs), and is converted to Kelvin for this calculation.

• Machine hold temperature: temperature of the hot zone during the hold phase of the machine cycle. This temperature is material-specific, and is looked up by material type in the lookup table tblMaterialTypeProperties (converted to Kelvin for this calculation).

Net Gas Volume In Hot Zone =

Machine Hot Zone Volume – Total Loaded Parts Volume

This is the volume of the portion of the hot zone that is not occupied by parts. It is the difference between the following volumes:

• Machine hot zone volume: the hot zone is assumed to be cylindrical with diameter specified by the machine property Hot Zone Diameter and height specified by the machine property Hot Zone Height.

• Total loaded parts volume: this is the volume of the portion of the hot zone that is occupied by parts. It is the product of the GCD property Volume and the number of parts per machine batch (see Number of Parts for Hot Isostatic Pressing).

Yield Formulas for Hot Isostatic Pressing

Yields are calculated with the formulas below.

Final Yield = Final Output Volume / Local Input Volume

Final yield affects material, labor, and overhead costs per part. It is the fraction of parts created by this process in the current production scenario that will survive as good parts once any secondary processes are completed. That is, final yield is the fraction of parts created by this process that are not discarded as scrap parts, either by this process or by a downstream process. Final yield depends on the following:

• Final output volume is the number of parts created by this process that are not discarded as scrap parts, either by this process or by a downstream process. It is the product of the annual volume and number of production years, specified in the Production Scenario screen of the Cost Guide.

• Local Input volume (see the formula below). This is the total number of parts produced by this process, including all parts discarded as scrap from this process or downstream processes.

Local Input Volume = Local Output Volume + Number of Scrap Parts for This Process

Local input volume is the total number of parts produced by this process in the current scenario, including all parts discarded as scrap from this process or downstream processes. It is the sum of the following:

• Local output volume (see the formula below). This is the number of parts produced by this process, excluding parts discarded as scrap by this process, but including parts that are discarded as scrap by downstream processes.

• Number of scrap parts for this process (see the formula below)

Local Output Volume = Final Output Volume + Number of Scrap Parts Downstream

Local output volume is the number of parts produced by this process in the current scenario, excluding parts discarded as scrap by this process, but including parts that are discarded as scrap by downstream processes. It is the sum of the following:

• Number of Scrap Parts Downstream. This the number of parts discarded as scrap by downstream processes.

Number of Scrap Parts for This Process = (Local Output Volume / Local Good Part Yield) – Local Output Volume

Number of scrap parts for this process is the number of parts produced by this process in the current scenario that are discarded as scrap by this process, before any downstream process is performed on them. It is a function of the following:

• Local output volume (see the formula above). This is the number of parts produced by this process, excluding parts discarded as scrap by this process, but including parts that are discarded as scrap by downstream processes.

• Local good part yield (specified by the machine property Good Part Yield). This is the fraction of parts produced by this process that are not discarded as scrap by this process but may be discarded as scrap by downstream processes.