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for Solutions Fundamentals of Modern Manufacturing, 3/e (published by Wiley) © MPGroover 2007
10 FUNDAMENTALS OF METAL CASTING
Review Questions
10.1 Identify some of the important advantages of shape casting processes. Answer. Advantages include (1) complex part geometries are possible; (2) some casting operations are net shape processes, meaning that no further manufacturing operations are needed to accomplish the final part shape; (3) very large parts are possible; (4) they are applicable to any metal that can be melted; and (5) some casting processes are suited to mass production. 10.2 What are some of the limitations and disadvantages of casting? Answer. Disadvantages include (1) limitations on mechanical strength properties; (2) porosity; (3) poor dimensional accuracy; (4) safety hazards due to handling of hot metals; and (5) environmental problems. 10.3 10.4 What is a factory that performs casting operations usually called?
foundry. What is the difference between an open mold and a closed mold? Answer. An open mold is open to the atmosphere at the top; it is an open container in the desired shape which must be flat at the top. A closed mold has a cavity that is entirely enclosed by the mold, with a passageway (called the gating system) leading from the outside to the cavity. Molten metal is poured into this gating system to fill the mold. 10.5 10.6 10.7 Name the two basic mold types that distinguish casting processes. Answer. The two mold types are (1) expendable molds and (2) permanent molds. Which casting process is the most important commercially? Answer. Sand casting is the most important casting process. What is the difference between a pattern and a core in sand molding? Answer. The pattern determines the external shape of the cast part, while a core determines its internal geometry if the casting includes a cavity. 10.8 What is meant by the term superheat? Answer. Superheat is the temperature difference above the melting point at which the molten metal is poured. The term also refers to the amount of heat that is removed from the molten metal between pouring and solidification. 10.9 Why should turbulent flow of molten metal into the mold be avoided? Answer. Turbulence causes the following problems: (1) it accelerates formation of oxides in the solidified metal, and (2) it causes mold erosion or gradual wearing away of the mold due to impact of molten metal. 10.10 What is the continuity law as it applies to the flow of molten metal in casting? Answer. The continuity law, or continuity equation, indicates that the volumetric flow rate is constant throughout the liquid flow. 10.11 What are some of the factors that affect the fluidity of a molten metal during pouring into a mold cavity?
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Solutions for Fundamentals of Modern Manufacturing, 3/e (published by Wiley) © MPGroover 2007
Answer. The factors include (1) pouring temperature above the melting point, (2) metal alloy composition, (3) viscosity of the liquid metal, and (4) heat transfer to the surroundings. 10.12 What does heat of fusion mean in casting? Answer. Heat of fusion is the amount of heat energy required to transform the metal from solid state to liquid state. 10.13 How does solidification of alloys differ from solidification of pure metals? Answer. Pure metals solidify at a single temperature equal to the melting point. Most alloys (exceptions are eutectic alloys) start to solidify at the liquidus and complete solidification occurs at the solidus, where the liquidus is a higher temperature than the solidus. 10.14 What is a eutectic alloy? eutectic alloy is a particular composition in an alloy system for which the solidus and liquidus temperatures are equal. The temperature is called the eutectic temperature. Hence, solidification occurs at a single temperature, rather than over a temperature range. 10.15 What is the relationship known as Chvorinov's Rule in casting? Answer. Chvorinov's Rule is summarized: TTS = Cm(V/A)2, where TTS = total solidification time, Cm = mold constant, V = volume of casting, and A = surface area of casting. 10.16 Identify the three sources of contraction in a metal casting after pouring. Answer. The three contractions occur due to (1) contraction of the molten metal after pouring, (2) solidification shrinkage during transformation of state from liquid to solid, and (3) thermal contraction in the solid state. 10.17 What is a chill in casting? chill is a heat sink placed to encourage rapid freezing in certain regions of the casting.
Multiple Choice Quiz
There is a total of 15 correct answers in the following multiple-choice questions (some questions have multiple answers that are correct). To attain a perfect score on the quiz, all correct answers must be given. Each correct answer is worth 1 point. Each omitted answer or wrong answer reduces the score by 1 point, and each additional answer beyond the correct number of answers reduces the score by 1 point. Percentage score on the quiz is based on the total number of correct answers. 10.1 10.2 10.3 Sand casting is which of the following types: (a) expendable mold or (b) permanent mold? Answer. (a). The upper half of a sand casting mold is called which of the following: (a) cope or (b) drag? Answer. (a). In casting, a flask is which one of the following: (a) beverage bottle for foundrymen, (b) box which holds the cope and drag, (c) container for holding liquid metal, or (d) metal which extrudes between the mold halves? Answer. (b). 10.4 In foundry work, a runner is which one of the following: (a) channel in the mold leading from the downsprue to the main mold cavity, (b) foundryman who moves the molten metal to the mold, or (c) vertical channel into which molten metal is poured into the mold? Answer. (a).
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Solutions for Fundamentals of Modern Manufacturing, 3/e (published by Wiley) © MPGroover 2007
10.5
Turbulence during pouring of the molten metal is undesirable for which of the following reasons (two best answers): (a) it causes discoloration of the mold surfaces, (b) it dissolves the binder used to hold together the sand mold, (c) it increases erosion of the mold surfaces, (d) it increases the formation of metallic oxides that can become entrapped during solidification, (e) it increases the mold filling time, and (f) it increases total solidification time? Answer: (c) and (d). Total solidification time is defined as which one of the following: (a) time between pouring and complete solidification, (b) time between pouring and cooling to room temperature, (c) time between solidification and cooling to room temperature, or (d) time to give up the heat of fusion? Answer. (a). During solidification of an alloy when a mixture of solid and liquid metals is present, the solid-liquid mixture is referred to as which one of the following: (a) eutectic composition, (b) ingot segregation, (c) liquidus, (d) mushy zone, or (e) solidus? Answer. (d). Chvorinov's Rule states that total solidification time is proportional to which one of the following quantities: (a) (A/V)n, (b) Hf, (c) Tm, (d) V, (e) V/A, or (f) (V/A)2; where A = surface area of casting, Hf = heat of fusion, Tm = melting temperature, and V = volume of casting? Answer. (f). A riser in casting is described by which of the following (three correct answers): (a) an insert in the casting that inhibits buoyancy of the core, (b) gating system in which the sprue feeds directly into the cavity, (c) metal that is not part of the casting, (d) source of molten metal to feed the casting and compensate for shrinkage during solidification, and (e) waste metal that is usually recycled? Answer. (c), (d), and (e). In a sand casting mold, the V/A ratio of the riser should be (a) equal to, (b) greater than, or (c) smaller than the V/A ratio of the casting itself? Answer. (b). Which of the following riser types are completely enclosed within the sand mold and connected to the main cavity by a channel to feed the molten metal (two correct answers): (a) blind riser, (b) open riser, (c) side riser, and (d) top riser? Answer. (a) and (c).
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Problems
Heating and Pouring 10.1 A disk 40 cm in diameter and 5 cm thick is to be cast of pure aluminum in an open mold casting operation. The melting temperature of aluminum = 660°C and the pouring temperature will be 800°C. Assume that the amount of aluminum heated will be 5% more than what is needed to fill the mold cavity. Compute the amount of heat that must be added to the metal to heat it to the pouring temperature, starting from a room temperature of 25°C. The heat of fusion of aluminum = 389.3 J/g. Other properties can be obtained from Tables 4.1 and 4.2 in the text. Assume the specific heat has the same value for solid and molten aluminum. Solution: Volume V = πD2h/4= π(40)2(5)/4 = 6283.2 cm3 Volume of aluminum to be heated = 6283.2(1.05) = 6597.3 cm3 From Table 4.1 and 4.2, density ρ = 2.70 g/cm3 and specific heat C = 0.21 Cal/g-°C = 0.88 J/g-°C
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Solutions for Fundamentals of Modern Manufacturing, 3/e (published by Wiley) © MPGroover 2007
Heat required = 2.70(6597.3){0.88(660-25) + 389.3 + 0.88(800-660)} = 17,812.71{558.8 + 389.3 + 123.2} = 19,082,756 J 10.2 A sufficient amount of pure copper is to be heated for casting a large plate in an open mold. The plate has dimensions: length = 20 in, width = 10 in, and thickness = 3 in. Compute the amount of heat that must be added to the metal to heat it to a temperature of 2150°F for pouring. Assume that the amount of metal heated will be 10% more than what is needed to fill the mold cavity. Properties of the metal are: density = 0.324 lbm/in3, melting point = 1981°F, specific heat of the metal = 0.093 Btu/lbm-F in the solid state and 0.090 Btu/lbm-F in the liquid state; and heat of fusion = 80 Btu/lbm. Solution: Volume V = (20 x 10 x 3)(1 + 10%) = 600(1.1) = 660.0 in3 Assuming To = 75 °F and using Eq. (12.1), H = 0.324 x 660{0.093(1981 - 75) + 80 + 0.090(2150 - 1981)} = 213.84{177.26 + 80 + 15.21} H = 58,265 Btu 10.3 The downsprue leading into the runner of a certain mold has a length = 175 mm. The cross-sectional area at the base of the sprue is 400 mm2. The mold cavity has a volume = 0.001 m3. Determine (a) the velocity of the molten metal flowing through the base of the downsprue, (b) the volume rate of flow, and (c) the time required to fill the mold cavity. Solution: (a) Velocity v = (2 x 9815 x 175)0.5 = (3,435,096)0.5 = 1853 mm/s (b) Volume flow rate Q = vA = 1853 x 400 = 741,200 mm3/s (c) Time to fill cavity TMF = V/Q = 1,000,000/741,200 = 1.35 s 10.4 A mold has a downsprue of length = 6.0 in. The cross-sectional area at the bottom of the sprue is 0.5 in2. The sprue leads into a horizontal runner which feeds the mold cavity, whose volume = 75 in3. Determine (a) the velocity of the molten metal flowing through the base of the downsprue, (b) the volume rate of flow, and (c) the time required to fill the mold cavity. Solution: (a) Velocity v = (2 x 32.2 x 12 x 6.0)0.5 = (4636.8)0.5 = 68.1 in/sec (b) Volume flow rate Q = vA = 68.1 x 0.5 = 34.05 in3/sec (c) Time to fill cavity TMF = V/Q = 75.0/34.05 = 2.2 sec. 10.5 The flow rate of liquid metal into the downsprue of a mold = 1 liter/sec. The cross-sectional area at the top of the sprue = 800 mm2 and its length = 175 mm. What area should be used at the base of the sprue to avoid aspiration of the molten metal? Solution: Flow rate Q = 1.0 l/s = 1,000,000 mm3/s Velocity v = (2 x 9815 x 175)0.5 = 1854 mm/s Area at base A = 1,000,000/1854 = 540 mm2 10.6 The volume rate of flow of molten metal into the downsprue from the pouring cup is 50 in3/sec. At the top where the pouring cup leads into the downsprue, the cross-sectional area = 1.0 in2. Determine what the area should be at the bottom of the sprue if its length = 8.0 in. It is desired to maintain a constant flow rate, top and bottom, in order to avoid aspiration of the liquid metal. Solution: Velocity at base v = (2gh)0.5 = (2 x 32.2 x 12 x 8)0.5 = 78.6 in/sec Assuming volumetric continuity, area at base A = (50 in/sec)/(78.6 in/sec) = 0.636 in2 10.7 Molten metal can be poured into the pouring cup of a sand mold at a steady rate of 1000 cm3/s. The molten metal overflows the pouring cup and flows into the downsprue. The cross-section of the sprue is round, with a diameter at the top = 3.4 cm. If the sprue is 25 cm long, determine the proper diameter at its base so as to maintain the same volume flow rate. Solution: Velocity at base v = (2gh)0.5 = (2 x 981 x 25)0.5 = 221.5 cm/s Assuming volumetric continuity, area at base A = (1000 cm/s)/(221.5 cm/s) = 4.51 cm2 Area of sprue A = πD2/4; rearranging, D2 = 4A/π = 4(4.51)/π = 5.74 cm2
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Solutions for Fundamentals of Modern Manufacturing, 3/e (published by Wiley) © MPGroover 2007
D = 2.39 cm 10.8 During pouring into a sand mold, the molten metal can be poured into the downsprue at a constant flow rate during the time it takes to fill the mold. At the end of pouring the sprue is filled and there is negligible metal in the pouring cup. The downsprue is 6.0 in long. Its cross-sectional area at the top = 0.8 in2 and at the base = 0.6 in2. The cross-sectional area of the runner leading from the sprue also = 0.6 in2, and it is 8.0 in long before leading into the mold cavity, whose volume = 65 in3. The volume of the riser located along the runner near the mold cavity = 25 in3. It takes a total of 3.0 sec to fill the entire mold (including cavity, riser, runner, and sprue. This is more than the theoretical time required, indicating a loss of velocity due to friction in the sprue and runner. Find (a) the theoretical velocity and flow rate at the base of the downsprue; (b) the total volume of the mold; (c) the actual velocity and flow rate at the base of the sprue; and (d) the loss of head in the gating system due to friction. Solution: (a) Velocity v = (2 x 32.2 x 12 x 6.0)0.5 = 68.1 in/sec Flow rate Q = 68.1 x 0.60 = 40.8 in3/sec (b) Total V = 65.0 + 25.0 + 0.5(0.8 + 0.6)(6.0) + 0.6(8.0) = 99.0 in3 (c) Actual flow rate Q = 99.0/3 = 33.0 in3/sec Actual velocity v = 33.0/0.6 = 55.0 in/sec (d) v = (2 x 32.2 x 12 x h)0.5 = 27.8 h0.5 = 55.0 in/sec. h0.5 = 55.0/27.8 = 1.978 h = 1.9782 = 3.914 in Head loss = 6.0 - 3.914 = 2.086 in Shrinkage 10.9 A mold cavity has the shape of a cube, 100 mm on a side. Determine the dimensions and volume of the final cube after cooling to room temperature if the cast metal is copper. Assume that the mold is full at the start of solidification and that shrinkage occurs uniformly in all directions. Use the shrinkage values given in Table 10.1. Solution: For copper, solidification shrinkage is 4.9%, solid contraction during cooling is 7.5%. Volume of cavity V = (100)3 = 1,000,000 mm3 Volume of casting V = 1,000,000(1-0.049)(1-0.075) = 1,000,000(.951)(.025) = 879,675 mm3 Dimension on each side of = cube (879,675)0.333 = 95.82 mm 10.10 The cavity of a casting mold has dimensions: L = 250 mm, W = 125 mm and H = 20 mm. Determine the dimensions of the final casting after cooling to room temperature if the cast metal is aluminum. Assume that the mold is full at the start of solidification and that shrinkage occurs uniformly in all directions. Use the shrinkage values given in Table 10.1. Solution: For aluminum, solidification shrinkage = 6.6%, solid contraction during cooling = 5.6%. Total volumetric contraction = (1-0.066)(1-0.056) = 0.8817 Linear contraction = (0.8817)0.333 = 0.9589 Final casting dimensions: L = 250(0.9589) = 239.725 cm W = 125(0.9589) = 119.863 cm H = 20(0.9589) = 19.178 cm 10.11 Determine the scale of a "shrink rule" that is to be used by pattern-makers for low carbon steel. Express your answer in terms of decimal fraction inches of elongation per foot of length compared to a standard rule. Use the shrinkage values given in Table 10.1. Solution: Low carbon steel: solidification shrinkage = 2.75%, solid contraction = 7.2%. Total volumetric contraction = (1-0.0275)(1-0.072) = 0.9025 Linear contraction = (0.9025)0.333 = 0.9664 Shrink rule elongation = (0.9664)-1 = 1.0348
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Solutions for Fundamentals of Modern Manufacturing, 3/e (published by Wiley) © MPGroover 2007
Elongation of a 12 inch rule = 12(1.0348 - 1.0) = 0.418 in/ft 10.12 Determine the scale of a "shrink rule" that is to be used by pattern-makers for brass which is 70% copper and 30% zinc. Express your answer in terms of millimeters of elongation per meter of length compared to a standard rule. Use the shrinkage values given in Table 10.1. Solution: For brass, solidification shrinkage is 4.5%, solid contraction during cooling is 8.0%. Total volumetric contraction = (1-0.045)(1-0.080) = 0.8786 Linear contraction = (0.8786)0.333 = 0.9578 Shrink rule elongation = (0.9578)-1 = 1.0441 Elongation of a 1 meter rule = 1000(1.0441 - 1.0) = 44.1 mm/m 10.13 Determine the scale of a "shrink rule" that is to be used by pattern-makers for gray cast iron. The gray cast iron has a volumetric contraction of -2.5%, which means it expands during solidification. Express your answer in terms of millimeters of elongation per meter of length compared to a standard rule. Use the shrinkage values given in Table 10.1. Solution: For gray CI, solidification shrinkage = -2.5%, solid contraction during cooling = 3.0%. Total volumetric contraction = (1+0.025)(1-0.030) = 0.99425 Linear contraction = (0.99425)0.333 = 0.9981 Shrink rule elongation = (0.9981)-1 = 1.00192 Elongation of a 1 meter rule = 1000(1.00192 - 1.0) = 1.92 mm/m 10.14 The final dimensions of a disk-shaped casting of low carbon steel are: diameter = 12.0 in and thickness = 0.75 in. Determine the dimensions of the mold cavity to take shrinkage into account. Assume that shrinkage occurs uniformly in all directions. Use the shrinkage values given in Table 10.1. Solution: For low carbon steel, solidification shrinkage is 3.0%, solid contraction during cooling is 7.2%. Total volumetric contraction = (1-0.03)(1-0.072) = 0.90016 Linear contraction = (0.90016)0.333 = 0.9656 Oversize factor for mold = (0.9656)-1 = 1.0356 Mold cavity dimensions: D = 12.00(1.0356) = 12.428 in and t = 0.750(1.03927) = 0.777 in Solidification Time and Riser Design 10.15 In the casting of steel under certain mold conditions, the mold constant in Chvorinov's Rule is known to be 4.0 min/cm2, based on previous experience. The casting is a flat plate whose length = 30 cm, width = 10 cm, and thickness = 20 mm. Determine how long it will take for the casting to solidify. Solution: Volume V = 30 x 10 x 2 = 600 cm3 Area A = 2(30 x 10 + 30 x 2 + 10 x 2) = 760 cm2 Chvorinov’s Rule: TTS = Cm (V/A)2 = 4(600/760)2 = 2.49 min 10.16 Solve for total solidification time in the previous problem only using an exponent value of 1.9 in Chvorinov's Rule instead of 2.0. What adjustment must be made in the units of the mold constant? Solution: Chvorinov’s Rule: TTS = Cm (V/A)1.9 = 4(600/760)1.9 = 2.55 min The units for Cm become min/in1.9 - strange units but consistent with Chvorinov’s empirical rule. 10.17 A disk-shaped part is to be cast out of aluminum. The diameter of the disk = 500 mm and its thickness = 20 mm. If the mold constant = 2.0 sec/mm2 in Chvorinov's Rule, how long will it take the casting to solidify? Solution: Volume V = πD2t/4 = π(500)2(20)/4 = 3,926,991 mm3 Area A = 2πD2/4 + πDt = π(500)2/2 + π(500)(20) = 424,115 mm2
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Solutions for Fundamentals of Modern Manufacturing, 3/e (published by Wiley) © MPGroover 2007
Chvorinov’s Rule: TTS = Cm (V/A)2 = 2.0(3,926,991/424,115)2 = 171.5 s = 2.86 min 10.18 In casting experiments performed using a certain alloy and type of sand mold, it took 155 sec for a cube-shaped casting to solidify. The cube was 50 mm on a side. (a) Determine the value of the mold constant the mold constant in Chvorinov's Rule. (b) If the same alloy and mold type were used, find the total solidification time for a cylindrical casting in which the diameter = 30 mm and length = 50 mm. Solution: (a) Volume V = (50)3 = 125,000 mm3 Area A = 6 x (50)2 = 15,000 mm2 (V/A) = 125,000/15,000 = 8.333 mm Cm = TTS /(V/A)2 = 155/(8.333)2 = 2.232 s/mm2 (b) Cylindrical casting with D = 30 mm and L = 50 mm. Volume V = πD2L/4 = π(30)2(50)/4 = 35,343 mm3 Area A = 2πD2/4 + πDL = π(30)2/2 + π(30)(50) = 6126 mm2 V/A = 35,343/6126 = 5.77 TTS = 2.232 (5.77)2 = 74.3 s = 1.24 min. 10.19 A steel casting has a cylindrical geometry with 4.0 in diameter and weighs 20 lb. This casting takes 6.0 min to completely solidify. Another cylindrical-shaped casting with the same diameter-to-length ratio weighs 12 lb. This casting is made of the same steel and the same conditions of mold and pouring were used. Determine: (a) the mold constant in Chvorinov's Rule, (b) the dimensions, and (c) the total solidification time of the lighter casting. The density of steel = 490 lb/ft3. Solution: (a) For steel, ρ = 490 lb/ft3 = 0.2836 lb/in3 Weight W = ρV, V = W/ρ = 20/0.2836 = 70.53 in3 Volume V = πD2L/4 = π(4)2L/4 = 4πL = 70.53 in3 Length L = 70.53/4π = 5.61 in Area A = 2πD2/4 + πDL = 2π(4)2/4 + π(4)(5.61) = 95.63 in2 (V/A) = 70.53/95.63 = 0.7375 Cm = 6.0/(0.7353)2 = 11.03 min/in2 (b) Find dimensions of smaller cylindrical casting with same D/L ratio and w = 12 lb. Weight is proportional to volume: V = (12/20)(70.53) = 42.32 in3 D/L ratio = 4.0/5.61 = 0.713; thus L = 1.4025D Volume V = πD2L/4 = π(4)2(1.4025D)/4 = 1.1015D3 D3 = (42.32 in3)/1.1015 = 38.42 in3 D = (38.42)0.333 = 3.374 in L = 1.4025(3.374) = 4.732 in (c) V = πD2L/4 = π(3.374)2(4.732)/4 = 42.32 in3 A = 2πD2/4 + πDL = 0.5π(3.374)2 + π(3.374)(4.732) = 68.04 in2 V/A = 42.32/68.04 = 0.622 in. TTS = 11.03(.622)2 = 4.27 min. 10.20 The total solidification times of three casting shapes are to be compared: (1) a sphere with diameter = 10 cm, (2) a cylinder with diameter and length both = 10 cm, and (3) a cube with each side = 10 cm. The same casting alloy is used in the three cases. (a) Determine the relative solidification times for each geometry. (b) Based on the results of part (a), which geometric element would make the best riser? (c) If the mold constant = 3.5 min/cm2 in Chvorinov's Rule, compute the total solidification time for each casting. Solution: For ease of computation, make the substitution 10 cm = 1 decimeter (1 dm) (a) Chvorinov’s Rule: TTS = Cm(V/A)2
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Solutions for Fundamentals of Modern Manufacturing, 3/e (published by Wiley) © MPGroover 2007
(1) Sphere volume V = πD3/6 = π(1)3/6 = π/6 dm3 Sphere surface area A = πD2 = π(1)2 = π dm2 V/A = (π/6)/π = 1/6 = 0.1667 dm Chvorinov’s Rule TTS = (0.1667)2Cm = 0.02778Cm (2) Cylinder volume V = πD2H/4 = π(1)2(1)/4 = π/4 = 0.25π dm3 Cylinder area A = 2πD2/4 + πDL = 2π(1)2/4 + π(1)(1) = π/2 + π = 1.5π dm2 V/A = 0.25π/1.5π = 0.1667 dm Chvorinov’s Rule TTS = (0.1667)2Cm = 0.02778Cm (3) Cube: V = L3 = (1)3 = 1.0 dm3 Cube area = 6L2 = 6(1)2 = 6.0 dm2 V/A = 1.0/6.0 = 0.1667 dm Chvorinov’s Rule TTS = (0.1667)2Cm = 0.02778Cm (b) All three shapes are equivalent as risers. (c) If Cm = 3.5 min/cm2 = 350 min/dm2, then TTS = 0.02778(350) = 9.723 min. Note, however, that the volumes of the three geometries are different: (1) sphere V = 0.524 dm3 = 524 cm3, cylinder V = 0.25π = 0.7854 dm3 = 785.4 cm3, and (3) cube V = 1.0 dm3 = 1000cm3. Accordingly, we might revise our answer to part (b) and choose the sphere on the basis that it wastes less metal than the other shapes. 10.21 The total solidification times of three casting shapes are to be compared: (1) a sphere, (2) a cylinder, in which the length-to-diameter ratio = 1.0, and (3) a cube. For all three geometries, the volume = 1000 cm3. The same casting alloy is used in the three cases. (a) Determine the relative solidification times for each geometry. (b) Based on the results of part (a), which geometric element would make the best riser? (c) If the mold constant = 3.5 min/cm2 in Chvorinov's Rule, compute the total solidification time for each casting. Solution: For ease of computation, make the substitution 10 cm = 1 decimeter (1 dm). Thus 1000 cm3 = 1.0 dm3. (1) Sphere volume V = πD3/6 = 1.0 dm3. D3 = 6/π = 1.910 dm3. D = (1.910)0.333 = 1.241 dm Sphere area A = πD2 = π(1.241)2 = 4.836 dm2 V/A = 1.0/4.836 = 0.2067 dm Chvorinov’s Rule TTS = (0.2067)2Cm = 0.0428Cm (2) Cylinder volume V = πD2H/4 = πD3/4 = 1.0 dm3. D3 = 4/π = 1.273 dm3 Therefore, D = H = (1.273)0.333 = 1.084 dm Cylinder area A = 2πD2/4 + πDL = 2π(1.084)2/4 + π(1.084)(1.084) = 5.536 dm2 V/A = 1.0/5.536 = 0.1806 dm Chvorinov’s Rule TTS = (0.1806)2Cm = 0.0326Cm (3) Cube: V = L3 =1.0 dm3. L = 1.0 dm Cube area = 6L2 = 6(1)2 = 6.0 dm2 V/A = 1.0/6.0 = 0.1667 dm Chvorinov’s Rule TTS = (0.1667)2Cm = 0.02778Cm (b) Sphere would be the best riser, since V/A ratio is greatest. (c) Given that Cm = 3.5 min/cm2 = 350 min/dm3 Sphere: TTS = 0.0428(350) = 14.98 min Cylinder: TTS = 0.0326(350) = 11.41 min Cube: TTS = 0.02778(350) = 9.72 min
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.
48
Solutions for Fundamentals of Modern Manufacturing, 3/e (published by Wiley) © MPGroover 2007
10.22 A cylindrical riser is to be used for a sand casting mold. For a given cylinder volume, determine the diameter-to-length ratio that will maximize the time to solidify. Solution: To maximize TTS, the V/A ratio must be maximized. Cylinder volume V = πD2L/4. L = 4V/πD2 Cylinder area A = 2πD2/4 + πDL Substitute the expression for L from the volume equation in the area equation: A = πD2/2 + πDL = πD2/2 + πD(4V/πD2) = πD2/2 + 4 V/D Differentiate the area equation with respect to D: dA/dD = πD – 4 V/D2 = 0 Rearranging, πD = 4V/D2 3 D = 4 V /π D = (4 V/π)0.333 From the previous expression for L, substituting in the equation for D that we have developed, L = 4V/πD2 = 4V/π(4V/π)0.667 = (4V/π)0.333 Thus, optimal values are D = L = (4V/π)0.333, and therefore the optimal D/L ratio = 1.0 10.23 A riser in the shape of a sphere is to be designed for a sand casting mold. The casting is a rectangular plate, with length = 200 mm, width = 100 mm, and thickness = 18 mm. If the total solidification time of the casting itself is known to be 3.5 min, determine the diameter of the riser so that it will take 25% longer for the riser to solidify. Solution: Casting volume V = LWt = 200(100)(18) = 360,000 mm3 Casting area A = 2(200 x 100 + 200 x 18 + 100 x 18) = 50,800 mm2 V/A = 360,000/50,800 = 7.0866 Casting TTS = Cm(7.0866)2 = 3.50 min Cm = 3.5/(7.0866)2 = 0.0697 min/mm2 Riser volume V = πD3/6 = 0.5236D3 Riser area A = πD2 = 3.1416D2 V/A = 0.5236D3/3.1416D2 = 0.1667D TTS = 1.25(3.5) = 4.375 min = 0.0697(0.1667D)2 = 0.001936D2 D2 = 4.375/0.001936 = 2259.7 mm2 D = 47.5 mm 10.24 A cylindrical riser is to be designed for a sand casting mold. The length of the cylinder is to be 1.25 times its diameter. The casting is a square plate, each side = 10 in and thickness = 0.75 inch. If the metal is cast iron, and the mold constant = 16.0 min/in2 in Chvorinov's Rule, determine the dimensions of the riser so that it will take 30% longer for the riser to solidify. Solution: Casting volume V = tL2 = 0.75(10.0)2 = 75 in3 Casting area A = 2L2 + 4Lt = 2(10.0)2 + 4(10.0)(0.75) = 230.0 in2 V/A = 75/230 = 0.3261 Casting TTS = 16(0.3261)2 = 1.70 min Riser TTS = 1.30(1.70) = 2.21 min Riser volume V = πD2H/4 = 0.25πD2(1.25D) = 0.3125πD3 Riser area A = 2πD2/4 + πDH = 0.5πD2 + 1.25πD2 = 1.75πD2 V/A = 0.3125πD3/1.75πD2 = 0.1786D Riser TTS = 16.0(0.1786D)2 = 16.0(0.03189)D2 = 0.5102D2 = 2.21 min D2 = 2.21/0.5102 = 4.3316 D = (4.3316)0.5 = 2.081 in H = 1.25(2.081) = 2.602 in. 10.25 A cylindrical riser with diameter-to-length ratio = 1.0 is to be designed for a sand casting mold. The casting geometry is illustrated in Figure P10.25, in which the units are inches. If the mold constant = 19.5 min/in2 in Chvorinov's Rule, determine the dimensions of the riser so that the riser will take 0.5 minute longer to freeze than the casting itself.
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.
49
Solutions for Fundamentals of Modern Manufacturing, 3/e (published by Wiley) © MPGroover 2007
Solution: Casting volume V = V(5 in x 10 in rectangular plate) + V(5 in. half disk) + V(upright tube) - V(3 in x 6 in rectangular cutout). V(5 in x 10 in rectangular plate) = 5 x 12.5 x 1.0 = 62.5 in3 V(5 in. half disk) = 0.5π(5)2(1)/4 = 9.817 in3 V(upright tube) = 3.0π(2.5)2/4 - 4π(1.5)2/4) = 7.657 in3 V(3 in x 6 in rectangular cutout) = 3 x 6 x 1 = 18.0 in3 Total V = 62.5 + 9.817 + 7.657 - 18.0 = 61.974 in3 Total A = 1 x 5 + 1(12.5 + 2.5π + 12.5) + 2(6+3) + 2(5 x 12.5 - 3 x 6) + 2(.5π(5)2/4) - 2(1.5)2π/4 + 2.5π(3) + 1.5π(3+1) = 203.36 in2 V/A = 61.974/203.36 = 0.305 in Casting TTS = 19.5(0.305)2 = 1.81 min Riser design: specified TTS = 1.81 + 0.5 = 2.31 min Riser volume V = πD2L/4 = πD3/4 = 0.25πD3 Riser area A = πDL + 2πD2/4 = πD2 + 0.5πD2 = 1.5πD2 V/A = 0.25πD3/1.5πD2 = D/6 TTS = Cm(V/A)2 2.31 = 19.5(D/6)2 = 0.5417D2 D2 = 2.31/0.5417 = 4.266 in2 D = 2.065 in and L = 2.065 in
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.
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