More Case Studies in Materials Selection Material for a pressure vessel Short term thermal insulation Energy efficient kilns More info: Materials Selection in Mechanical Design, Chapters 5 and 6 ME 474-674 Winter 2008 Slides 6-1
Safe pressure vessels Cylindrical pressure vessels are containers for a fluid under pressure A safe design will be based on one of two factors Detectable plastic deformation (small pressure vessels) Leak before break (larger pressure vessels) The maximum principal stress is the hoop stress σ = pr t p R t 2a ME 474-674 Winter 2008 Slides 6-2
Safe pressure vessels Function Objective Constraints Free variables: Pressure vessel contain pressure p safely Maximize safety Yield before break or Leak before break Radius R is specified Material ME 474-674 Winter 2008 Slides 6-3
Safe pressure vessels Pressure vessels are usually examined for any flaws that may be present Ultrasonic or X-ray techniques have a detection limit of 2a * c There are no flaws larger than 2a * c Have to assume flaws up to size 2a * c are present The stress required to catastrophically propagate a crack in the presence of a flaw of size 2a * c is σ = CK where K ic is the fracture toughness of the material and C ( 1) is a constant that depends upon the shape and location of the crack 1C πa * c ME 474-674 Winter 2008 Slides 6-4
Safe pressure vessels Therefore, for safety p t = σ R t R K 1C πa * c The corresponding material index to be maximized is M = 1 K1 C ME 474-674 Winter 2008 Slides 6-5
Safe pressure vessels However, if one wanted to ensure that the material yielded before fracture, then it should be possible to reach the failure stress or yield stress even when the flaw size is greater than the detection limit of the NDE technique πa c C 2 K σ f 1C 2 In order to maximize the flaw size for with yield before break occurs, the material index to be maximized is M 2 = K1 C σ f ME 474-674 Winter 2008 Slides 6-6
ME 474-674 Winter 2008 Slides 6-7 Safe pressure vessels It may not be possible to subject a large pressure vessels to complete X- ray or ultrasonic examination to locate pre-existing flaws Therefore, if the vessel is designed such that critical flaw size (2a c ) is at least equal to the thickness of the wall the even when the stress reaches the yield stress, then the vessel will leak before break Under this situation, the material index to be maximized is f C f C K C R p or t pr K C t σ π σ σ π 2 1 2 2 1 2 2 2 = = = f K C M σ 2 1 3 =
Safe pressure vessels If one wanted to make a thin walled pressure vessel, the thinnest wall is obtained by having a high value of the yield strength. Therefore, there is a fourth index that needs to be maximized. Namely M 4 = σ f The following slides show the successive application of each of the indices to select a material ME 474-674 Winter 2008 Slides 6-8
Safe pressure vessels Summary of Material Performance parameters Parameter Equation Objective M 1 K 1C Maximize M 2 M 3 M 4 K1 C σ 2 K1 C σ f f σ f Maximize (minimize σ f?) Maximize (minimize σ f?) Maximize ME 474-674 Winter 2008 Slides 6-9
Safe pressure vessels 100 M 1 K 1C > 10 MPa.m 0.5 30 of 95 Materials All metals Ferrous and non- Ferrous Fracture toughness (MPa.m^1/2) 10 1 0.1 0.01 0.01 0.1 1 10 100 1000 Yield strength (elastic limit) (MPa) ME 474-674 Winter 2008 Slides 6-10
Safe pressure vessels 100 Non age-hardening wrought Al-alloys Copper Nickel Stainless steel M 2 = 0.4m 0.5 15 of 95 Materials Including Lead Polymer Foam Metal Foam Leather Fracture toughness (MPa.m^1/2) 10 1 0.1 Flexible Polymer Foam (MD) Metal foam Commercially pure lead Leather 0.01 0.01 0.1 1 10 100 1000 Yield strength (elastic limit) (MPa) ME 474-674 Winter 2008 Slides 6-11
Safe pressure vessels 100 Copper Nickel Lead alloys M 3 = 4 MPa.m 22 of 95 Materials Lead is still an option Fracture toughness (MPa.m^1/2) 10 1 0.1 Flexible Polymer Foam (MD) Metal foam Commercially pure lead Leather 0.01 0.01 0.1 1 10 100 1000 Yield strength (elastic limit) (MPa) ME 474-674 Winter 2008 Slides 6-12
Safe pressure vessels M 4 = 100 MPa 100 CFRP, epoxy matrix (isotropic) Copper Nickel Medium carbon steel 36 of 95 Materials Lead and foams are gone but we have picked up a bunch of ceramic materials Fracture toughness (MPa.m^1/2) 10 1 0.1 Aluminum nitride Low alloy steel Silicon nitride Tungsten carbides Silicon 0.01 0.01 0.1 1 10 100 1000 Yield strength (elastic limit) (MPa) ME 474-674 Winter 2008 Slides 6-13
Safe pressure vessels All stages 100 Copper Zinc die-casting alloys Nickel Stainless steel Non age-hardening wrought Al-alloys Bronze 8 materials Fracture toughness (MPa.m^1/2) 10 1 0.1 Cast Al-alloys Zinc die-casting alloys 0.01 0.01 0.1 1 10 100 1000 Yield strength (elastic limit) (MPa) ME 474-674 Winter 2008 Slides 6-14
Safe pressure vessels Select Materials - All Stages Brass Cast Al-alloys Commercially pure zinc Copper Nickel Non age-hardening wrought Al-alloys Stainless steel Zinc die-casting alloys ME 474-674 Winter 2008 Slides 6-15
Short term thermal insulation An application for short term thermal insulation is the rescue beacons for military aircraft pilots These electronic devices do not function if the temperature drops below a critical value Therefore, to give the rescue operation the greatest chance of being effective, the temperature of the electronics in the radio beacon must not fall below a critical value for the longest period of time even when exposed to cold temperatures The temperature of most of the earth s oceans is around 4ºC The electronics have to be wrapped in an insulating blanket ME 474-674 Winter 2008 Slides 6-16
Short term thermal insulation Function Objective Constraints Free variables: Short term thermal insulation Maximize time before which internal temperature drops below critical value Wall thickness must not exceed w Material Insulating material of wall thickness w Electronic circuits packaged in this space ME 474-674 Winter 2008 Slides 6-17
Short term thermal insulation Model 1 Minimize heat flux out of the containment area First law of heat conduction dt q = λ λ dx ( T T ) Where q is heat flux, λ is thermal conductivity Therefore, minimize λ to minimize heat flow Best materials are polymer foams i w o ME 474-674 Winter 2008 Slides 6-18
Short term thermal insulation Ceramics 100 Foams and Thermal conductivity (W/m.K) 10 1 Hybrids Metals Polymers Rigid Polymer Foam (HD) 0.1 Flexible Polymer Foam (MD) Rigid Polymer Foam (LD) Rigid Polymer Foam (MD) ME 474-674 Winter 2008 Slides 6-19
Short term thermal insulation But is this the answer we are looking for? The answer is no! The problem requires that the time that it takes for the electronic package to cool down be maximized. This is not a steady state problem. Therefore use 2 nd law of heat conduction If the temperature at the surface is decreased suddenly, as in dropping the pilot and his radio beacon into a cold ocean, the distance x from the surface at which a certain temperature is reached changes with time t as Where a is the thermal diffusivity x 2at λ a = ρ C p ME 474-674 Winter 2008 Slides 6-20
Short term thermal insulation ρ is the density and C p is the specific heat of the material. We can replace x in the above equation by the wall thickness to get 2 w t 2a Therefore, we seek the material with the smallest a to maximize the time t, if the thickness of the insulation w is fixed The best materials are therefore elastomers ME 474-674 Winter 2008 Slides 6-21
Short term thermal insulation 100 Thermal conductivity (W/m.K) 10 1 Isoprene (IR) Polychloroprene (Neoprene, CR) 0.1 Butyl Rubber Isoprene (IR) 1e-7 1e-6 1e-5 1e-4 Thermal Diffusivity ME 474-674 Winter 2008 Slides 6-22
Energy efficient kiln Kilns used for firing pottery are heated up from room temperature to the firing temperature during each cycle Unbaked pottery is placed in the furnace The heating mechanism, electric or gas, is turned on and the kiln is heated up to the firing temperature After the requisite time at temperature, the kiln is allowed to cool down Once cooled, the pottery is removed and the cycle is repeated There are two major factors that consume energy The energy to heat up the kiln The energy lost through conduction through the walls The first can be minimized by reducing the thermal mass of the system, i.e. minimize the wall thickness The second can be minimized by reducing the heat loss through the wall by increasing its thickness ME 474-674 Winter 2008 Slides 6-23
Energy efficient kiln How can these apparently contradictory requirements be reconciled? Is there a material index that can capture both requirements? Wall thickness w Insulation T-con λ Density ρ T o T i Sp-heat C p ME 474-674 Winter 2008 Slides 6-24
Energy efficient kiln Function Objective Constraints Free variables: Thermal insulation for kiln (cyclic heating and cooling) Minimize energy consumed in each cycle Hard: Max operating temp = 1000 C Soft: Wall thickness due to space limitation Material Wall thickness ME 474-674 Winter 2008 Slides 6-25
Energy efficient kiln Analysis There are two sources of heat loss Heat lost by conduction through walls Q 1 dt T = λ t = λ dx T w i o t Heat required to increase temperature of insulating material ( T T ) Total heat loss is Ti To Q = Q1 + Q2 = λ t + C pρw w Q 2 = C ρw p i 2 o ( T T ) i 2 o ME 474-674 Winter 2008 Slides 6-26
Energy efficient kiln To minimize total heat loss, differentiate the above equation and set equal to zero and find w 2 t w λ = C pρ 1/ 2 ( 2at) 1/ 2 Substituting back into the equation for Q gives = Q = ( T T )( 2t) 1/ 2 ( λc ρ) 1/ 2 i o p The material index to be maximized is ( λc ρ ) 1/ 2 ( 1/ 2) a M = p = λ ME 474-674 Winter 2008 Slides 6-27
Energy efficient kiln Select Materials - Stage 1 limit stage Min operating temperature - 1000 C 14 materials Alumina Aluminum nitride Boron carbide Brick Ceramic foam Glass ceramic Nickel-based superalloys Nickel-chromium alloys Silica glass Silicon carbide Silicon nitride Tungsten alloys Tungsten carbides Zirconia ME 474-674 Winter 2008 Slides 6-28
Energy efficient kiln Stage 2 100 High values of M = 1/ a λ 2 Thermal conductivity (W/m.K) 10 1 0.1 1e-7 1e-6 1e-5 1e-4 Thermal diffusivity ME 474-674 Winter 2008 Slides 6-29
Energy efficient kiln Both Stages 100 Thermal conductivity (W/m.K) 10 1 0.1 1e-7 1e-6 1e-5 1e-4 Thermal diffusivity ME 474-674 Winter 2008 Slides 6-30
Energy efficient kiln Select Materials - All Stages 5 materials Brick Ceramic foam Glass ceramic Silica glass Zirconia Switching to the larger database gives over 60 materials. ME 474-674 Winter 2008 Slides 6-31
Energy efficient kiln Additional criteria can be imposed such as cost, oxidation 100 resistance, flammability, etc. to screen out certain materials like carbon Thermal conductivity (W/m.K) 10 1 Plaster of Paris Glass Ceramic - Slipcast Mullite (Al2O3-SiO2 alloys) Graphite (perpendicular to plane) Glass Ceramic (N11) Alumina Foam (99.8%)(0.4) Carbon (Vitreous) Carbon Fiber Reinforced Carbon Matrix Composite (Vf:50%) 0.1 Graphite Foam (0.12) Carbon Foam (Reticulated, Vitreous)(0.05) 1e-7 1e-6 1e-5 1e-4 Thermal Diffusivity = Thermal conductivity / Specific heat / Density ME 474-674 Winter 2008 Slides 6-32