(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`
`0301
`
`EX1008 (Part 2 of 3)
`Yita v. MacNeil
`IPR2020-01139
`
`

`

`5.1
`
`Introduction
`
`5.2 Overall Cooling Heat Balance
`
`285
`
`Once the part has been heated and formed to shape by contact with the cool mold
`surface, it must be cooled andrigidified. Then the part and its web must be separated
`by trimming. When thermoforming a reactive polymer such as thermosetting
`polyurethane or a crystallizing one such as nucleated CPET', the formed shapeis
`rigidified by holding it against a heated mold to continue the crosslinking reaction or
`crystallization.
`In most cases, rigidifying implies cooling while in contact with a
`colder mold, For automatic thin-gage formers, the molds are usually actively cooled
`with water flowing through channels. Free surfaces of medium- and heavy-gage sheet
`are frequently cooled with forced air, water mist or water spray, For amorphous
`polymers, cooling of the formed part against a near-isothermal mold rarely controls
`the overall thermoforming cycle. For crystalline and crystallizing polymers such as
`PET, PP and HDPE,the cooling cycle can be long and can govern overall cycle time.
`When the sheet has cooled sufficiently to retain its shape and, to a large extent,
`its dimension,
`it
`is stripped from the mold and transferred to a trimming station,
`where the web is separated from the product. Requisite holes, slots and cut-outs are
`drilled, milled or burned into the part at this time. Thin-gage roll-fed sheet can be
`either trimmed on the mold surface immediately after forming or on an in-line
`mechanical trimming press. Heavier-gage formed sheet is usually removed from the
`mold and manually or mechanically trimmed on a remote station. Trimmingis really
`a solid phase mechanical process of crack propagation by brittle or ductile fracture.
`Care is taken when trimming brittle polymers such as PS or PMMA to minimize
`microcracks, With brittle polymers,
`the very fine sander, saw or microcrack dust
`generated by mechanical fracture can be a serious problem, Other polymers such as
`PP and PET are quite tough and require special cutting dies. Dull steel-rule dies
`cause fibers or hairs at the cutting edge of thin-gage fiber-forming polymers such as
`PET and PP. Slowly crystallizing polymers pose registry problems when in-line
`trimming presses are used. The speed of trim cutting and the nature of the cutting
`surface control
`the rate of crack propagation through the plastic. There are no
`exhaustive studies of the unique trimming and cutting characteristics of thermo-
`formed polymers and so much information must be inferred from other sources
`including extensive studies on the machining ofplastics.
`
`5.2 Overall Cooling Heat Balance
`
`it is assumed to be at an
`Asa first approximation, as the sheet touches the mold,
`average, equilibrated temperature, T,,,,,, a8 described in Section 3.13. When the
`
`The unique processing conditions for crystallizing polyethylene terephthalate (CPET) are de-
`scribed in detail in Chapter 9, Advanced Thermoforming Processes.
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`0302
`
`0302
`
`

`

`286
`
`Cooling and Trimming the Part
`
`{Refs. on p. 379]
`
`average sheet temperature reaches the set temperature, T,.,, as given in Table 2.5,
`the sheet
`is assumed to be sufficiently rigid ta be removed from the mold
`surface. Typically,
`the amorphous polymer set
`temperature is about 20°C or
`40°F below its glass transition temperature, T,. The crystalline polymer set temper-
`ature is about 20°C or 40°F below its melting temperature, T,,.;, During the
`cooling time,
`the mold temperature is assumed to be essentially constant at
`Tinota’ Lhe amount of heat to be removed by the coolant flowing through the mold
`is given as:
`
`WYtense =V*- PC,(Teguit — Tet)
`
`=V* ‘pc, AT =V* + pAH
`
`(5.1)
`
`where V* is the volumeofplastic, p is its density and c, is its heat capacity’. Heat
`is removed from the free sheet surface by convection to the environmental air. Heat
`is removed from the sheet surface against the mold surface by conduction through
`the mold to the coolant. The coolant fluid removes the heat by convection. The heat
`load at any point on the mold surface depends on the sheet thickness, t),.,), at that
`point, Sheet thickness, as noted in Chapter4, is not uniform across the mold surface.
`The heat load at any point is given as:
`
`local = Pp 3 Cy i‘ Local ; AT
`
`(5.2)
`
`This is the heat to be removed from a given region during the cooling portion of the
`total cycle, 8,,.;. The local heat flux thenis:
`(5.3)
`Moet =Go = PG Lica —
`The units on qj.,..; are kW/m? or Btu/ft?- h- °F. The total heat load during this time
`1S given as:
`
`AT
`Oca
`
`Be
`oc!
`
`Quota= ata dA
`
`The total heat load per unit time on a steady-state process is given as:
`
`Qhreaay state = Qioiat ” N
`
`(5.4)
`
`(5.5)
`
`where N is the number of parts produced per unit time. The units on Qyready stare ATE
`kW or Btu/h.
`At steady-state conditions, this energy is removed by the coolant system and by
`convection to the environmental air. Mold, coolant and air temperatures increase
`until the steady state is reached,
`
`'
`
`The last equality relates the amount of heal removed to the differential enthalpy, AH, of the
`polymer. This expression should be used if the polymer is crystalline and moJten during the
`forming step andis solidifying during the cooling step,
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`0303
`
`0303
`
`

`

`5.3 Cooling the Formed Shape
`
`287
`
`5.3 Cooling the Formed Shape
`
`Consider a typical cooling step. The sheet of variable thickness but known tempera-
`ture is pressed against a slightly irregular surface of a mold, The properties of the
`mold material are known and uniform throughoutits volume, Chapter 6. Coolant of
`known properties fows through uniformly spaced channels in the mold, The free
`surface of the part is also cooled, At any instant, the temperature profile through the
`variouslayers of material is as shown in schematic in Fig. 5.1. The rate at which the
`energy is removed from the plastic to ambient air and coolant depends on the sum
`of resistances to heat transfer through each of these layers. Heat removal by the
`coolantis the primary way of cooling the formed part. Transfer to the environmental
`air is a secondary method but it can be quite important when trying to optimize cycle
`time. There are two aspects to heat removal from the formed polymer sheet to the
`coolant. The first deals with the overall heat transfer at steady state conditions. The
`second focuses on certain aspects ofcyclical transient heat transfer.
`
`Mold
`
`Surface Irreqularities
`
`Plastic Sheet
`
`Air Gap
`
`Coolant Film Resistance
`
`
`
`
`
`Free Surface Film Resistance
`
`To
`
`
`Ambient Environment
`
`
`
`Series Thermal Resistances
`
`Figure 5.1 Schematic of various thermalresistances for sheet cooling against thermoforming mold
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`0304
`
`0304
`
`

`

`288
`
`Cooling and Trimming the Part
`
`[Refs. on p. 379]
`
`Battle
`
`Mold Insert
`
`Flow Channel
`
`Coolant
`
`SIAN _N 1]
`
`NARS
`
`|_t
`
`a
`
`teAoP
`
`WINES
`
`Figure 5.2 Typical serpentine coolant flow channel through thermoforming mold
`
`5.4 Steady State Heat Balance
`
`Consider the limiting case where all the energy is transferred directly to the mold and
`thence to the coolant. Typically, coolant lines are drilled or cast on a discrete, regular
`basis parallel to the mold surface (Fig. 5.2), Heat removal by coolant depends on
`convection heat transfer, or fluid motion'. The total amount of energy removed by
`the coolant embedded in the mold is given as:
`
`Q=UA AT
`
`(5.6)
`
`where U is the overall heat transfer coefficient, A is the coolant surface area and AT
`is the increase in coolant temperature between inlet and outlet portions of the flow
`channel. The overall heat transfer coefficient includes all flow resistances between the
`sheet and the coolant. It is usually written as:
`
`lU
`
`(5.7)
`7 XR,
`where R,represents the ith resistance to heat transfer. As seen in Fig.5.1, for flowing
`fluids, there is a convective film resistance at the conduit surface,
`|/xDh,, where nD
`is the circumference of the conduit. If the coolantfluid is not kept clean, the coolant
`channel can become coated with residue, thus increasing thermal resistance. This
`
`'
`
`This section deals only with the convection heat transfer coefficient of coolant flowing through
`lines in the mold. Section 6.4 considers coolant pressure drop-fiow rate relationships for the
`specification of coolant line size,
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`0305
`
`0305
`
`

`

`Table 5.1 Fouling Factors for Coolant Lines'
`
`5.4 Steady State Heat Balance
`
`289
`
`Coolant
`
`Condition
`
`
`
`Fouling factor
`(Biu/ft-h-°F)~'
`Velocity <3 ft/s
`
`Velocity >43 ft/s
`
`0
`
`Treated make-up
`cooling tower water
`
`Treated make-up
`cooling tower water
`City water
`City water
`River water
`River water
`Treated boiler
`feedwater
`Treated boiler
`feedwater
`Industrial heat
`transfer oil
`Ethylene glycol
`Glycerine-water
`Brine
`Brine
`Steam
`
`Water < 125°F
`
`Water > 125°F
`
`Water < 125°F
`Water > 125°F
`Water < 125°F
`Water > 125°F
`Water < 125°F
`
`Water > 125°F
`
`Temperature < |25°F
`Temperature > 125°F
`
`0.001
`
`0.002
`
`0.001
`0.002
`0.001
`0.002
`0.0005
`
`0.001
`
`0.001
`
`0.001
`0.001
`0.001
`0.002
`
`' Information extracted from [|] by permission of copyright holder
`
`resistance is called a fouling factor, ff Fouling factors are given in Table 5.1 [1].
`The resistance through the mold depends on the relative shapes of the mold surface
`and the coolant channels, and is usually described as 1/Sk,,, where S is a shape factor
`and k,,
`is the thermal conductivity of the mold material. Since polymer sheet does
`not press tightly against the mold surface, there is a conductive resistance owing to
`trapped air,
`|/h,. Although there may be other thermal resistances, these are the
`primary ones. So the overall heat transfer coefficient, U,
`is written as:
`
`I
`|
`1
`7 sph sth F
`
`es
`
`1U
`
`Interfacial Resistance
`
`In most heat transfer processes, intimate or perfect contact between the hot and cold
`solids is assumed. Imperfect contact causes resistance to heat flow. Mold surface
`waviness and microscopic roughness or asperities reduce physical contact (Fig. 5.3).
`Increasing pressure against
`the sheet
`increases physical contact and reduces the
`resistance to heat transfer. In general, energy is transmitted across the interstices by
`a combination of:
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`0306
`
`0306
`
`

`

`290
`
`Cooling and Trimming the Part
`
`[Refs. on p. 379]
`
`Cold, Rigid or Heavy-Gage Sheet
`
`Trapped Air
`
`Hot, Flexible or Thin-Gage Sheet
`
`Wiking
`
`AQYMREK
`
`Cold Mold
`
`Mold Asperities
`
`Hot Mold
`
`Figure 5.3 Interfacial resistance between sheet and thermoforming mold surface. Left shows sub-
`stantial thermal resistance owing to large air gap. Right shows reduced thermal resistance
`
`® Conduction at the asperities,
`« Conduction through the interstitia] fluid, and
`® Radiation.
`
`The resistance is thus a function of:
`
`e® The contacting material properties such as
`Relative hardness,
`Thermal conductivity,
`Surface roughness and
`Flatness,
`* The conductivity and pressure of the interstitial fluid, and
`« The pressure applied against the free surface of the sheet.
`
`The interface coefficient, h,, is a measure of thermal resistance across the gap.It is
`similar in concept to the confection heat transfer in that resistance to heat flow
`decreases with increasing value of h,. For perfect contact, h, 00,
`In thermoforming, the interstitial fluid is air, perhaps at a substantially reduced
`pressure. If the interface is a uniform air gap of 6= 0.025 cm or 0.010 in and air
`thermal conductivity is k,;,=0.029 W/m-°C or 0.0167 Btu/ft: h- °F, the value for
`h,
`is about 114 W/m?-°C or 20 Btu/ft?-h-°F. Contact heat transfer coefficient
`values between flowing polymer melts and mold surfaces of about h, = 568
`W/m?) °C or 100 Btu/ft?-h-°F have been reported in injection molding [2-4].
`Similar values are expected here. For two surfaces in contact, h, =h,,p", where p is
`the applied pressure and h,, depends on the relative waviness and roughness of the
`two surfaces, Table 5.2 [5]. As seen, n has a value of about 2/3 for both rigid-rigid
`and rigid-flexible material contact in vacuum. Values for h,, are typically 50 times
`greater in air than in hard vacuum.
`There are no available data for interfacial resistance during thermoforming of
`softened plastic sheet against various types of mold surfaces. For hot sheet pressed
`against a relatively smooth, heated mold surface at a relatively high differential
`pressure,
`it
`is expected that an appropriate value for h, would be about 4568
`W/m?:°C or 100 Btu/ft?-h:+°F. For rapidly cooling and rigidizing plastic sheet
`pressing against a highly textured, cold mold surface with a modest differential
`pressure, an appropriate value range for h, of 114 to 284 W/m*-°C or 20 to
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`0307
`
`0307
`
`

`

`Table 5.2 Contact Resistance and Conductance [5] h,=h.,. p"
`(Units on h,=W/m?- °C or Btu/ft?- h- °F)
`(Units on p are MPa orIb,/in*)
`
`5.4 Steady State Heat Balance
`
`29]
`
`2/3
`
`Material
`contact
`
`Elastic deformation
`theory
`Hard-to-hard
`Hard-to-hard
`Hard-to-soft
`
`None
`
`Vacuum
`Air
`Vacuum
`
`Contact coefficient, h,.
`
`(W/m? - C)
`
`(Btu/ft? -h- °F)
`
`?
`
`;
`
`n
`
`2/3
`
`2/3
`1/6
`
`50 Btu/ft? - h - °F should be consideredfor first cooling time estimates. Section 5.6 on
`computer simulation of the cooling process explores the relative effect of interfacial
`resistance on time-dependent sheet cooling.
`
`Shape Factor
`
`For thin metal molds, heat is conducted very rapidly from the plastic to the coolant.
`Forrelatively thick molds of:
`
`Plaster,
`Wood,
`Epoxy,
`Glass fiber-reinforced unsaturated polyester resin (FRP),
`Pressed fiberboard, or
`Any other nonmetallic material,
`
`heat transfer is slowed by low mold material thermal conductivity. If the coolant
`system is considered to be coplanar with the mold surface (Fig. 5.4), the resistance to
`heat transfer per unit length (L= 1), across a mold D units thick is given as:
`D
`5.9
`Lee
`(3.9)
`k
`R,,
`1
`where k,, is the mold material thermal conductivity, Thermal conductivity values for
`many mold materials are given in Table 2.7, For round discrete conduits (Fig. 5.5)
`{6}, a shape factor, 8, is used:
`
`where § is given by:
`
`l
`Rea
`
`a.ee
`In or ak a
`D
`P
`
`(5.10)
`
`(5.11)
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`0308
`
`0308
`
`

`

`[Refs. on p. 379]
`
`292
`
`Cooling and Trimming the Part
`
`
`
`Figure 5.4 Schematic of coplanar coolant flow channel and mold surface, concept frequently called
`flooded cooling
`
`Yy
`
`Coolant Channel
`
`P
`}-——__——+
`
`S ©
`
`HotSheet
`
`D
`
`Mold
`
`Figure 5.5 Geometric factors for mold shape factor analysis
`
`the thermal
`that
`is apparent
`It
`Figure 5.6 gives this equation in graphic form,
`resistance of the mold decreases with increasing value of S, which is achieved with
`many large-diameter coolant lines placed relatively close to the mold surface. The
`typical value range for S is 2<S<3. Example 5.1 illustrates the relative effects of
`these parameters on mold thermal resistance,
`
`Example 5.1 Shape Factors and Mold Thermal Resistance
`
`Determine the relative thermal resistances for the following two molds:
`
`Mald 1: Thin-walled aluminum mold with k,,= 131 Btu/ft h °F, having d= 1/2-in
`water lines on P= 2 in centers, with the centerline being D=1 in from the mold
`surface.
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`0309
`
`0309
`
`

`

`5.4 Steady State Heat Balance
`
`293
`
`Mold 2: Thick-walled plaster mold with k,,= 1.0 Btu/ft h °F, having d= 1/2-in
`water lines on P= 4 in centers, with the centerline being D=2 in from the mold
`surface.
`
`Mold !
`
`Mold 2
`
`P/d=4, D/d =2. From Fig. 5.6, 8 = 2.
`I
`]
`
`P/d=8, D/d =4. From Fig. 5.6, 8 = 1.6.
`
`angi
`®
`™ Sk, 1.6-1.0
`The plaster mold has more than 160 times the thermal resistance to heat
`transfer than the aluminum mold.
`
`= 0.625
`
`R
`
`Factor,S
`
`Shape
`
`Coolant Channel Spacing to Diameter Ratio, P/d
`
`Figure $.6 Effect of coolant line location on mold shape factor
`
`Convection Heat Transfer Coefficient
`
`roll-fed thin-gage thermoformers and on many
`The metal molds on automatic,
`heavy-gage forming operations are actively cooled, with water being the primary
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`0310
`
`0310
`
`

`

`294
`
`Cooling and Trimming the Part
`
`(Refs. on p. 379]
`
`coolant. There is a thermal resistance between the cool bulk flowing fluid and the
`warmer tube wall (Fig. 5.1). The primary dimensionless group used in fluid mechan-
`ics is the Reynolds number, Re:
`
`Re=—
`
`(5.12)
`
`where D is the tube diameter, v is the fluid velocity, p is the density of the fluid and
`jt
`is its Newtonian viscosity [7]. The Reynolds number is the ratio of inertial to
`viscous forces for the fluid. Slowly moving fluids are laminar when Re < 2000.
`Convection heat transfer to slowly moving fluids is poor. Rapidly flowing fluids are
`fully turbulent when Re> 10,000 and heat
`transfer is very rapid. Example 5.2
`illustrates the interaction between flow rate and Reynolds number.
`
`Example 5.2 Water as Coolant—Flow Rates
`
`Consider 21°C or 70°F water flowing through 0.5-in or 1.27-cm diameter coolant
`channels. Determine the Reynolds number and the flow characteristic if the velocity
`is a) 0.52ft/s or 0.16 m/s and b) 2.6ft/s or 0.79 m/s. What are the volumetricflow
`rates at these velocities?
`
`The water density is 62.4 lb/ft*. The viscosity is 0.658 x 10~? Ib,,/ft+s. The
`Reynolds numberis:
`RS age e624 0.658 x 10- ip2 ts
`Dvp_0.5,
`| ft
`ft:
`
`For v = 0.52 ft/s: Re = 2050 and the water is laminar.
`For v= 2.6 ft/s: Re = 10,300 and the water is turbulent.
`The volumetric flow rate is given as:
`Zz
`3
`ya y=0.00136-y t= 0.612-y St
`4
`8
`min
`
`For v=0.52 ft/s, the flow rate is 0.32 GPM.
`For v=2.6 ft/s, the flow rate is 1.6 GPM.
`
`As discussed in Section 3.6, energy interchange between solid surfaces and
`flowing fluids is by convection. The proportionality between heat flux and thermal
`driving force is the convection heat transfer coefficient. The convection heat transfer
`coefficient is obtained from standard heat transfer theory and experiments. There are
`many methodsfor calculating values of h.. In general, however, the Chilton-Colburn
`analogy between resistance to fluid flow andresistance to thermal energy flow yields
`adequate results [8]. The analogy states:
`
`St - Pr2? =£/8
`
`(5.13)
`
`where St is the Stanton number, Pr is the Prandtl number andfis the coefficient of
`friction or friction factor. The Stanton and Prandtl numbers are:
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`0311
`
`0311
`
`

`

`5.4 Steady State Heat Balance
`
`295
`
`Table 5.3. Prandtl Number Values For Several Coolants
`[9,10]
`
`Coolant
`
`Temperature
`(°F)
`
`(Cc)
`
`Prandtl no.
`
`
`
`Air
`Air
`
`Steam
`Steam
`
`Water
`Water
`Water
`Water
`Water
`
`SAE 30 Oil
`SAE 30 Oil
`SAE 30 Oi
`SAE 30 Oil
`SAE 30 Oil
`SAE 30 Oil
`
`Glycerine
`Glycerine
`Glycerine
`Glycerine
`Glycerine
`Air
`Air
`Air
`
`Light Oi
`Light Oil
`Light Oil
`Light Oil
`
`0.72
`0.72
`
`0.96
`0.94
`
`13.7
`6.82
`4.52
`2.74
`1.88
`
`1170
`340
`122
`62
`35
`22
`
`31,000
`12,500
`5,400
`2,500
`1,600.
`0.72
`0.71
`0.685
`
`340
`62
`35
`22
`
`st=
`
`
`h
`
`pe,V
`
`fra
`a
`
`(5.14)
`
`(5.15)
`
`the
`transfer coefficient, p is the fluid density at
`where h is the convective heat
`appropriate temperature, c, is the fluid heat capacity, v is the average fluid velocity,
`v= /p, is the kinematic viscosity, and «=k/pc,, is the fluid thermal diffusivity. In
`essence, Pr is the ratio of inertial to thermal properties and St is the ratio of fluid to
`thermal resistances. Table 5.3 gives appropriate Prandt! number values for several
`coolants [9,10]. As examples, Pr 7 for room temperature water and Pr ~ 300 for
`
`(cid:926)(cid:3)(cid:1005)(cid:1013)(cid:1013)(cid:1010)(cid:3)(cid:18)(cid:258)(cid:396)(cid:367)(cid:3)(cid:44)(cid:258)(cid:374)(cid:400)(cid:286)(cid:396)(cid:3)(cid:115)(cid:286)(cid:396)(cid:367)(cid:258)(cid:336)(cid:856)(cid:3)(cid:4)(cid:367)(cid:367)(cid:3)(cid:396)(cid:349)(cid:336)(cid:346)(cid:410)(cid:400)(cid:3)(cid:396)(cid:286)(cid:400)(cid:286)(cid:396)(cid:448)(cid:286)(cid:282)(cid:856)(cid:3)
`© 1996 Carl Hanser Verlag. All rights reserved.
`(cid:69)(cid:381)(cid:3)(cid:437)(cid:374)(cid:258)(cid:437)(cid:410)(cid:346)(cid:381)(cid:396)(cid:349)(cid:460)(cid:286)(cid:282)(cid:3)(cid:282)(cid:349)(cid:400)(cid:272)(cid:367)(cid:381)(cid:400)(cid:437)(cid:396)(cid:286)(cid:3)(cid:381)(cid:396)(cid:3)(cid:396)(cid:286)(cid:393)(cid:396)(cid:381)(cid:282)(cid:437)(cid:272)(cid:410)(cid:349)(cid:381)(cid:374)(cid:854)(cid:3)(cid:367)(cid:349)(cid:272)(cid:286)(cid:374)(cid:400)(cid:286)(cid:282)(cid:3)(cid:410)(cid:381)(cid:3)(cid:393)(cid:437)(cid:396)(cid:272)(cid:346)(cid:258)(cid:400)(cid:286)(cid:396)(cid:3)(cid:381)(cid:374)(cid:367)(cid:455)(cid:856)
`No unauthorized disclosure or reproduction; licensed to purchaser only.
`
`031