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4 . F eeder Des ign and Analys is The solidification of metals continues to be a phenomenon of great interest to physicists, metallurgists, casting engineers and software developers. It directly affects the production cycle time, internal quality of castings and material utilization (yield). We will briefly review the solidification phenomenon and focus on three major factors affecting the solidification characteristics of a casting: freezing range, cooling rate and thermal gradient. Finally, we will list the different types of solidification shrinkage related defects and see why it is important to achieve controlled progressive directional solidification. 4.1 S olidification Phenomenon When molten metal enters a mold cavity, its heat is transferred through the mold wall. In the case of pure metals and eutectics, the solidification proceeds layer-by-layer (like onion shells) starting from the mold wall and proceeding inwards. The moving isothermal interface between the liquid and solid region is called the solidification front. As the front solidifies, it contracts in volume, and draws molten metal from the adjacent liquid layer. When the solidification front reaches the central hot spot, there is no more liquid metal left and a void – shrinkage cavity, is formed (Fig.5.1). This is avoided by attaching a feeder designed to solidify later than the hot spot. The shrinkage cavity shifts to the feeder, which is cut off after casting solidification and recycled. Understanding the solidification phenomenon will help us in predicting the type and location of shrinkage defects, and in overcoming them successfully by appropriate design of feeders.

Fig.4.1: Casting solidification in a mold The temperature history of a location inside the casting with respect to the neighboring locations governs the formation of shrinkage cavity as well as the macrostructure. This is

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difficult to determine even for a simple shape, since all modes of heat transfer are involved during casting solidification: by convection within the molten metal; by conduction in the solidified portion of the casting; by convection and radiation at the metal-mold interface; and by conduction in the mold material. Also, the release of latent heat has to be addressed; it increases the casting temperature at that instant and location, and has the effect of delaying the solidification. The most important factor affecting the rate of heat transfer from the casting to the mold is the interface heat transfer coefficient. It depends on the thickness of the oxide layer and the air gap. The air gap itself depends on the amount of gas generated (and retained) after metal-mold reaction, the roughness of the mold surface and the expansion of the mold and cores. The air gap is usually the largest near external surfaces at the top of the mold towards the end of solidification. Let us study three important factors that govern the solidification characteristics of castings: freezing range, cooling rate and thermal gradients. As we will see, these factors are primarily influenced by the casting metal, process and geometry, respectively. Freezing range: Most casting alloys do not have a distinct melting point; they solidify over a range of temperature. The difference between the liquidus (temperature above which the alloy is completely liquid) and solidus (temperature below which alloy is completely solid) is referred to as the freezing range, given by F = Tliq – Tsol. In such castings, there are three distinct zones: completely solid, completely liquid and intermediate mushy zone. This is because of the growth of dendrites, and the liquid being trapped in the branches. The freezing range is one of major factors affecting casting macrostructure, mainly the grain shape. Alloys with short freezing range behave like pure metals and eutectics, and the solidification proceeds layer-by-layer. The macrostructure comprises columnar grains growing along the direction of heat transfer (perpendicular to the mold wall) since they are hindered sideways by adjacent grains. In long freezing range alloys, the solidification is initiated at a large number of points, and the grains grow in size until the neighboring grains hinder them. Thus the macrostructure comprises equi-axed grains. The effective freezing range is greatly influenced by the cooling rate and thermal gradients inside the casting. A long freezing range alloy will behave like a short freezing range alloy (columnar structure) in a metal mold. Cooling rate: The cooling rate (R) at a given location inside the casting at a given instant of time is given by

R = • T / • ô Where, • T is the difference in temperature over a time period • ô. The cooling rate mainly depends on the mold material and the air gap formed at the metal-mold interface, which affect the rate at which heat is extracted from the metal. A metal mold will produce higher cooling rates than a sand mold. The cooling rates are higher near the metal-mold

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interface than the casting interior. The cooling rates are higher in the beginning and decrease as the solidification progresses. Also, the cooling rates are higher at mold bottom where the metal is in contact with the mold (zero air gap) than at the top. The cooling rate at the time of solidification affects the grain size. A higher cooling rate promotes solidification and produces fine grains. This is observed near the mold wall, where undercooling leads to almost instantaneous nucleation of crystals. It is also seen in metal molds, to a greater depth. On the other hand, the interior regions of a casting, where cooling rates are low, exhibit larger grains. The grain size affects the strength and hardness of the casting. Thermal gradient: The thermal gradient (G) between two points inside the casting at a given instant of time is given by

G = • T / • s Where, • T is the difference in temperature between the two points and • s is the distance between them. The gradients are greatly influenced by the casting geometry. In general, the gradients are highest in a direction normal to the solidification front, but gradually decrease as we move from the mold wall to the casting center. Thus thin castings are characterized by high gradients, whereas the middle regions of thick castings have low gradients. A change in section thickness increases the thermal gradient. The feed metal primarily moves along the direction of thermal gradients. Poor gradients, especially at an isolated hot spot, cause shrinkage porosity. Progressive solidification refers to solidification in a given cross-section of the casting: ideally starting from the mold wall and gradually progressing towards the center of the cross section. Directional solidification refers to sequence of solidification of different regions of the casting: ideally starting from thin regions at one end, followed by adjacent thicker regions, and finally ending at the thickest region (usually the feeder). A casting (along with feeders) should be designed to achieve controlled progressive directional solidification, so that the casting is free of solidification shrinkage defects. The solidification-related defects (Fig.5.2) in a casting mainly depend on the three factors described above. There are mainly four types of defects: shrinkage pipe (largest), shrinkage cavity, shrinkage porosity and micro-porosity. Shrinkage pipe: It occurs in the top half of a feeder in short freezing range alloys. It is usually in the form of an inverted cone. Its top diameter may almost approach the feeder diameter, gradually tapering to a point towards the feeder bottom. Shrinkage cavity: It occurs around isolated hot spots in short freezing range alloys. It appears as an irregular hole with rough surface.

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Shrinkage porosity: It occurs in a region of high temperature and low gradients in long freezing range alloys. It appears like small holes, usually during machining. In long thick sections, it appears as a dotted line and called as centerline porosity. Micro-porosity: It invariably occurs only in long freezing range alloys and may appear in any section with low temperature gradients. It may be barely visible to the naked eye, but affects the strength (and therefore the failure) of critical sections. Other defects: These include corner shrinkage, sink marks and cracks. Fig.4.2: Solidification shrinkage related defects: top row– macro porosity (left and right); middle– porosity (left) and sink (right); bottom– corner shrinkage (left) and crack (right). [Source: Atlas of Casting Defects, Institute of British Foundrymen, UK].

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4.2 S olidification T ime and Rate The solidification time of a casting depends on casting geometry, material and process. In this section, we will review the basic equations for estimating the casting solidification time and rate. We will also look at relationships between temperature, gradient and cooling rate that indicate the occurrence of solidification shrinkage defects. The following major assumptions are made for deriving an equation for the solidification time of a simple shaped casting: 1. The flow of heat is unidirectional, and the mold is semi-infinite (that is, neglect the

effect of finite thickness of mold). 2. The properties of the metal and mold material are uniform (throughout the bulk) and

remain constant over the range of temperature considered. 3. The metal is in complete contact with the mold surface (no air gap is formed) 4. The metal-mold interface temperature remains constant from the start to end of

solidification. The solidification time ôs can be determined by equating the heat given up by the casting Qcast to the heat transferred through the mold Qmold.

Qcast = ñcast V [ L + Ccast ( Tpour – Tsol ) ]

Qmold = • 0• ô (• Q/• ô) dô = 1.128 • (K mold ñmold Cmold) A ( Tint – Tamb ) • ôs

Equating both, we obtain the famous Chvorinov’s equation, as follows. • ôs = [ ñcast ( L + Ccast ( Tpour – Tsol ) ) / 1.128 • (K mold ñmold Cmold) ( Tint – Tamb ) ] ( V / A )

ôs = k ( V / A )2

where, V is the casting volume (representing the heat content) and A is the cooling surface area (through which heat is extracted). The ratio V/A is referred to as the casting modulus. Thus, if two different shapes (say, a cube and a plate) have the same volume, the one with the larger cooling surface area (the plate) will solidify first. The Chvorinov’s equation is very useful for comparing the relative solidification time of two or more simple shaped castings (same metal and mold material), but with different volume and cooling surface area. This principle can even be applied to determine the order of solidification of different regions of a casting, by dividing it into simple shapes and determining the volume and cooling surface area of each region. The region with the highest modulus is considered to solidify last and identified as a hot spot. Feeders are designed so that their modulus is more than the modulus of the hot spot region. This is a simple yet effective criterion to ensure that the feeder remains liquid

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long enough to supply the feed metal to compensate the volumetric shrinkage of the casting. Rate of solidification: This can be estimated for skin freezing alloys in the following manner. Let d be the thickness of casting solidified near a mold wall of area A after time ô from the start of solidification. Thus we have,

ô = k ( V / A )2 = k d2

d = k1 • ô The above relation has been experimentally verified by ‘pouring-out’ a set of castings each after a different length of time, by researchers such as Briggs as early as 1935. The relation between solidification time and casting modulus has been verified by a large number of researchers including Chvorinov, Wlodawer, Ruddle and Pellini between 1940-60. The most widely used method involved placing thermocouples in a mold and obtaining the cooling curves from each. The equations for solidification time and rate have limited application in practice, due to the geometric complexity of the casting, significant variation in metal and mold properties from pouring to solidus temperature and the effect of varying resistance at the metal-mold interface (due to air gap and oxide layer). Various researchers have attempted to derive improved equations with limited success. They are further hindered by the unavailability of accurate thermo-physical data for different casting and mold materials, which need to be determined from experiments. The most probable locations for shrinkage porosity inside a casting are characterized by high temperature, coupled with low gradient and high cooling rate. High Temperature (could be a peak, a ridge or even a plateau) signifies fewer directions from where liquid metal can flow in to compensate for solidification shrinkage. Low gradient implies that even if liquid metal is available at a neighboring section, there is insufficient thermal ‘pressure’ for the flow to actually take place. High cooling rate implies that even if liquid metal and sufficient gradients are available, the time available is too short and the liquid metal freezes before reaching the hot spot. In the centerlines of thick sections in short freezing range alloys (for example, steel), the shrinkage porosity can be predicted using the Niyama criterion given by

G / • R < 1 Where G is the thermal gradient in K/s and R is the rate of solidification in mm/s.

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While Chvorinov’s equation is useful to identify the most probable regions of shrinkage porosity, we require the temperature history T=T (x,y,z,ô) of those regions, especially towards the end of solidification. Based on this, we can determine the temperature peaks, gradients and cooling rates, and thereby predict the location and occurrence of shrinkage cavity. 4.3 Feeder Location and S hape Feeders are designed to compensate the solidification shrinkage of a casting, so that it is free of shrinkage porosity. We will first review the concept of feed path and feeding distance, which influence the location and number of feeders. Different options for feeder position, type and shape are described, followed by the design criteria for determining the dimensions of feeder and its neck, and finally the design of feedaids. The direction of solidification starts from end regions that solidify first, to intermediate regions and finally ends at the last freezing regions inside a casting. The feed metal flows in the reverse direction: from regions at a higher temperature (containing liquid metal) to adjacent solidifying regions. The entire path, starting from a local hot spot to an end region is referred to as the feed path. It follows that any intermediate point on a feed path has only one adjacent point with a higher temperature. The exception is the hot spot, which is a local temperature maxima. The hot spot effectively feeds all regions along the feed path. Ideally, the hot spot must be inside a feeder. The distance from a feeder to the farthest point along the feed path is referred to as the feeding distance. Several researchers such as Pellini and Bishop have experimentally established the relationship between feeding distance and section thickness for simple shaped steel castings in sand molds. The feeding distance is represented by two terms: feeder effect and end effect. For steel plate castings in sand molds, the total feeding distance is given by 4.5 t (from the feeder edge), where t is the section thickness. Of this, the feeder effect is 2 t and end effect is 2.5 t. Other researchers have expressed feeding distance in terms of modulus instead of thickness. The feeding distance is not very well established for other metals, particularly long freezing range alloys, and does not appear to directly relate to section thickness (as in the case of steel plate castings). In complex shaped castings, it is difficult to estimate the feeding distance by the above relationships. One way to overcome this is by dividing the casting into a number of simple shaped regions and calculating the modulus of each (the ratio of volume to cooling surface area). If two adjacent regions have different modulus, then the one with the higher modulus may be assumed to feed the other region. The thermal gradient along the path must be greater than a minimum critical value for feeding to take place. A value of about 0.5 K/mm for steel castings and 2 K/mm for aluminum castings (both in sand molds) has been suggested by some researchers. The critical value is affected by the casting shape: for example, circular sections require

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higher gradients than flat rectangular sections. It also depends on the quality requirement: critical castings (or sections) require higher gradients to be free of even micro-porosity. The temperature and gradients at any point along the feed path influence the type of feeding at that location. If both temperature and gradient are high (near the feeder), mass feeding takes place by movement of liquid. If temperature is high, but gradient is low (near the center of long thick sections), inter-dendritic feeding takes place. Finally, if temperature is low, but gradient is high (thin end sections), solid feeding takes place. Improper feeding in the above three zones usually leads to macro porosity, micro-porosity and surface sink, respectively. If there is only one major hot spot inside a casting, the feeder must be connected to the casting face closest to the hot spot. Two or more isolated hot spots located far apart will require multiple feeders, one for each hot spot. If there are several hot spots, with different solidification times, the feeder can be first designed for the hottest one, followed by analysis to verify if the same feeder can also feed any other hot spot. Then a feeder is designed for the next largest hot spot, and so on. A minor hot spot may be eliminated by using chills (described later). Depending on the position, feeders may be classified as top and side. The top feeders are placed above the hot spot, whereas the side feeders are placed at the side of the hot spot, usually at the parting line. A top feeder is more effective because of the additional effect of gravity. It may however, require a core for producing the undercut at its neck. On the other hand, side feeders do not require a core; they can be directly fed by hot metal from the ingates and can remain liquid longer. Feeders are also classified as open or blind, depending on whether the top of the feeder is open to atmosphere or not. Open feeders lose more heat than blind feeders and therefore are less efficient. Open feeders are also referred to as risers, since the liquid metal can be seen rising in them, servicing as useful indicators that the mold has filled completely. The blind feeders also require an opening to the atmosphere, to enable feed metal flowing down to the hot spot. This is ensured by placing a special core above a blind feeder. The feeder location must facilitate fettling. This implies connecting a feeder to a flat surface rather than a curved face of the casting. Also, there must be sufficient gap around the feeder for ease of fettling as well as for minimizing its influence on other sections of the casting. The ideal shape of a feeder is spherical. This has the lowest surface area for a given volume and therefore the longest solidification time compared to other shapes. In practice, other shapes are used because of the formation of shrinkage pipe (which may extend into the casting) and molding constraints (mainly undercuts). Taller feeders are used for steel castings (H/D = 2), which exhibit shrinkage pipe, whereas in iron and aluminum castings, H/D can be about 1.5. For small castings, cylindrical feeders are widely used. For larger castings, cylindrical feeders with spherical bottom (side location)

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or spherical top (top position, blind type) are widely used. Another shape reported in literature but not widely used, is the cruciform feeder. The shape of the feeder neck depends on the feeder shape, feeder position and the connected portion of the casting. The most widely used neck shapes are cylindrical (for top cylindrical feeders) and rectangular (mainly for side feeders). The neck may be tapered down towards the casting. A single or double V-notch may be included in the neck to facilitate fettling. This does not affect the neck modulus (or its solidification time) because of poor heat transfer from the sharp reentrant corner. Another major feeder design parameter is the use of insulating or exothermic sleeves and covers. They essentially increase the effective modulus of the feeder, so that a smaller feeder can be used and the yield is increased. The shape of the feedaid depends on the feeder shape. Often the reverse is true, since feedaids are available in standard shape/size. 4.4 Feeder and Feedaid Des ign A feeder designed for a given hot spot has to satisfy three major requirements as follows. Solidification time: The feeder must solidify later than the nearest hot spot, expressed by the following criterion:

Mf = kf Mh Where, is the Mf modulus of the feeder, Mh is the modulus of the casting region around the hot spot and kf is the feeder design factor, always more than 1 (more than 1.1 for iron casting, and more than 1.2 for aluminum and steel castings). If there is an intermediate section of casting between the feeder and the hot spot, a larger factor may be needed. Note that the modulus of the hot spot region will increase after connecting the feeder, because of reduced heat transfer area corresponding to the feeder neck. Feed path: There must be a clear feed path between the feeder and the hot spot. Essentially, sufficient thermal gradients must exist for the liquid metal to flow from the feeder to the hot spot. If the feeder is connected to the casting through a neck, it must be designed such that the following criteria are satisfied:

Mf = kf Mn and Mn = kf Mh Where, Mn is the modulus of the feeder neck. If the feeder cannot be connected to a casting face near the hot spot, but farther away to another intermediate section i with modulus Mi, then the above criterion is modified as follows:

Mf = kf Mn , Mn = kf Mi , Mi = kf Mh Feed metal volume: The feeder must compensate solidification shrinkage of the hot spot region. This requirement is satisfied by the criterion:

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ηf Vf = α ( Vc + Vf )

where, Vf and Vc are the volume of the feeder and the casting, respectively; ηf is the feeder efficiency (ratio of volume of available feed metal to feeder volume); and α is the volumetric shrinkage of the cast metal. The feeding efficiency comes into picture because the feeder itself is solidifying and all of its volume is not available for feeding the casting. The efficiency depends on the feeder shape, type (open or blind) and application of feedaids (insulation or exothermic). For an open cylindrical feeder with height = 1.5 times diameter, the efficiency is 14%. The volumetric shrinkage ranges from zero for irons, to 3-4% for steels and 6-7% for aluminum alloys. The feeder design follows these steps: 1. Estimate the modulus of region around the hot spot in casting. 2. Determine the feeder modulus based on the solidification time criterion. 3. Select the feeder shape, aspect ratio, and then its dimensions based on its modulus. 4. Design the feeder neck based on feed path criterion 5. Recalculate modulus of hot spot region (because of neck) and redesign the feeder. 6. Check the feed metal volume criterion and increase feeder dimensions, if necessary. Feedaids – including chills, insulation and exothermic – are used when progressive directional solidification cannot be achieved by feeders alone. The feedaids are kept in contact with a particular face of the casting or feeder, altering the local solidification characteristics. The chills increase the local rate of heat transfer (compared to other surfaces of the casting in contact with mold), reducing the local solidification time. Insulating materials (which reduce the rate of heat transfer) and exothermic materials (which add heat) both increase the solidification time of the local section. Chills are usually made of copper, iron/steel or graphite. They are in the form of rectangular blocks or cylinders or contoured to match the casting surface (form chills). The insulation and exothermic materials are usually applied to feeders and are in the shape of sleeves or covers. There are three major considerations in feedaid design: the distance to which the feedaid must be effective, the initial rate of heat transfer required, and the actual amount of heat to be transferred. We explain these by taking the example of a chill. Effective distance: The distance to which a chill is effective mainly depends on the thermal conductivity of the casting material, assuming that the chill is not undersized (a small chill that gets saturated with heat is less effective). Experimental investigations have shown that in iron castings (K=73 J/mKs), the chill effect is visible for a distance equal to 1-1.5 times the section thickness, whereas in aluminum castings (K=238 J/mKs),

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it is visible for a distance up to 4 times the section thickness. Beyond this distance, there is no significant change in local cooling rate or solidification time. Heat transfer rate: It primarily depends on the thermal conductivity K of the chill material and the area of contact A. An iron chill (K=73 J/mKs) can conduct heat several orders of magnitude faster than a sand mold, and a copper chill (K=397 J/mKs) is 5 times more conductive than an iron chill. The rate reduces as the chill becomes hotter. Heat absorption: It depends on the specific heat C and the mass of the chill. Given the specific heats of sand, iron and copper (1130, 456 and 386 J/kgK, respectively) and their densities (1500, 7800 and 8900 kg/m3, respectively), it is clear that the actual heat transferred for either iron or copper chill (of same size) is nearly equal, and only about twice as much as a sand mold. In other words, a chill reduces the effective modulus of the casting section to half of the original modulus. This has been experimentally proven. A simplified approach to estimate the effect of a feedaid on solidification characteristics of a casting is based on modulus extension factor (MEF). Typical values of MEF for chill, insulation and exothermic materials are 0.5, 1.4 and 1.8. In other words, a smaller feeder (with insulation) will be required for the same solidification time as a larger feeder (without insulation), thereby improving the yield.

Mf-effective = (MEF) Mf = kf Mh where, Mf is the feeder modulus (without feedaid) and Mf-effective is the effective modulus. 4.5. S olidification Analys is The feeder design can be verified by casting trials to find the location and distribution of shrinkage porosity. Besides being expensive and time-consuming, shop floor trials may not provide a complete and correct picture, leading to unexpected defects during regular production. This can be overcome by virtual casting trials (using simulation software) for defect prediction and yield optimization. Solidification of castings is a non-linear transient phenomenon, posing a challenge in terms of modeling and analysis. It involves a change of phase with liberation of latent heat from a moving liquid-solid boundary. The heat is transferred from the molten metal to solidified portion of the casting, then through the air gap at casting-mold interface and finally through the mold. All the three modes of heat transfer: conduction, convection and radiation are involved. The influence of the location of the ingate and the pouring rate, as well as varying rates of heat transfer in different parts of the mold, owing to cores, feeding aids and variation in mold thickness have to be accounted for. The properties of casting and mold materials, which change non-linearly over the range of temperatures involved, are not easily available and have to be obtained through detailed experiments. The casting geometry and multiple-cavity molds make the analysis even more difficult.

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The most important result sought from the solidification analysis is the location and extent of shrinkage porosity defects. This requires an analysis of heat flow within the casting, as well as from the casting to the mold, and finally the temperature history of all points inside the casting. One way is to obtain the temperature history of all points inside the casting, plot the progress of solidification fronts (isothermal contours) at different instants of time, and identify the last freezing regions. This approach is implemented using either Finite Difference Method (FDM) or Finite Element Method (FEM), which essentially involves dividing the space and time domain into small elements or steps, and solving the governing equations. The numerical simulation of solidification process using either Finite Difference or Finite Element methods (FDM/FEM) involves the following steps: 1. Formulating an accurate mathematical model of the solidification process. 2. Use of accurate values for thermal properties of material involved. 3. Performing the analysis to obtain the temperature history of casting and mold points. 4. Post-processing the results to visualize the solidification pattern and identify defects. The unsteady state heat transfer involved in solidification of metal in a mold is expressed as follows.

2 2 2

2 2 2p

T T T TC K

x y zρ

τ ∂ ∂ ∂ ∂ = + + ∂ ∂ ∂ ∂

There is loss of heat even as the metal enters the gating system, and during its rise in the mold cavity. We will however, assume that the mold cavity is instantaneously filled with molten metal with an initial temperature. The outer surface of the mold is initially assumed to be at ambient temperature. The bottom surfaces of the casting are always in contact with the mold, and the vertical surfaces are in contact with the mold until the air gap forms. The heat flux across the metal-mold interface is given by the product of heat transfer coefficient hg and temperature difference • T across the interface. The boundary conditions in different regions of the casting and the mold are described next. Solid-liquid interface: The energy balance is obtained by equating the rate of heat removed from the solid phase to the sum of the rate of heat supplied to the interface from the liquid phase and the rate of heat liberated at the interface during solidification. Here Ksc and Klc are the thermal conductivity of the solid and liquid metal, respectively. The L denotes latent heat, and n denotes the normal to the surface (direction of heat transfer).

( )sc scsc lc sc

T T sK K L

n nτρτ

∂ ∂ ∂− = − +∂ ∂ ∂

Casting-mold interface: Before air gap formation, heat is transferred by conduction. Given Tc and Tm are the temperature of the casting and mold, the temperature at casting mold interface can be found from heat flux w

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c mc m

T Tw K K

n n∂ ∂= =∂ ∂

After air gap formation, heat transfer is by convection and radiation. Here Tcs and Tms are the temperature at the casting and mold side of the interface, ó is the Boltzmann’s constant, å is the emmissivity and F is the form factor. The heat flux is:

( ) ( ){ }4 4273 273 *cs ms g

Tw F T T h T K

nσε ∂ = + − + + ∆ = − ∂

Outer surface of mold: Heat transfer is by convection. Here Tmo is the temperature of the outer surface of mold and Ta is the ambient temperature.

( )mm mo a

TK h T T

n∂− = −∂

The model equations can be solved numerically by using simple explicit finite difference method. In this method the casting and mold regions are subdivided into small intervals of constant space (• x, • y, • z in x, y and z direction, respectively) and time interval (• t). The Fig.5.3 below explains the discretization for 2D.

Fig.4.3 Space and time discretization in 2D.

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The equation can be written using FTCS (forward in time and central in space) explicit finite difference method as:

( ) ( ) ( ), , , , 1, , , , 1, , , 1, , , , 1, , , 1 , , , , 1 2

2 2 2

2 2 2( )i j k i j k i j k i j k i j k i j k i j k i j k i j k i j k i j k

n

T T T T T T T T T T TkO x

C x y z

τ τ τ τ τ τ τ τ τ τ τ τ

τ ρ

+∆+ − + − + −

− − + − + − += + + + ∆ ∆ ∆ ∆ ∆

The first term on the right hand side is a central finite difference form for second order derivative of temperature T with respect to space coordinate x, y and z at grid point (i,j,k). The other term constitutes the truncation error. We can get the solution from above equation in terms of temperature distribution with respect to space coordinates in casting and mold region, at the desired time. The solution can be obtained by imposing the boundary conditions listed earlier, in the basic equation, and marching along the time axis in a suitable step. The solution becomes unstable if the errors grow while marching. The appropriate time step (to avoid error accumulation) is determined by applying the stability criterion given by:

( ) ( ) ( )2 2 2

1 1 1 12

KC x y z

τρ

∆ + + ≤ ∆ ∆ ∆

The results are post-processed to display a color-coded map of temperatures inside the casting at any instant of time. The temperature map at the end of solidification points out the last-freezing regions, which are the most probable locations of shrinkage porosity. 4.6 Vector Element Method This method is based on determining the feed path passing through any point inside the casting and following the path back to the local hot spot. The feed path is assumed to lie along the maximum thermal gradient. The gradient can be determined from Fourier’s law of heat conduction as follows:

q = – K A ∆T / ∆s G = ( –1 / K ) w

Where, G = ∆T / ∆s is the thermal gradient and w = q / A is the heat flux at any given point inside the casting, in any given direction. The gradient (as well as the heat flux) is zero in a tangential direction to the isotherm passing through the point, and the maximum in perpendicular direction. The magnitude and direction of the maximum thermal gradient at any point inside the casting is proportional to the vector resultant of thermal flux vectors in all directions originating from that point.

wr = ∑i wi The casting volume is divided into a number of pyramidal sectors originating from the given point, each with a small solid angle. For each sector, the heat content (proportional to volume) and cooling surface area is determined to compute the flux vector. We take a

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step along the resultant flux vector, reach a new location and repeat the computation, until the resultant flux vector is zero (or close to zero, for computational purpose). The final location is the hot spot (Fig.5.4). The locus of points along which iterations are carried out is the feed path (Fig.5.5). Multiple hot spots inside a casting can be identified by starting the computation from a number of ‘seed points’, each in a different region of the casting. The method can be easily verified for a 2D shape. The length of a flux vector is given by a/2, where a is the distance of intersection of a ray from the given point with the casting boundary. The direction of the ray as well as the flux vector for any sector can be taken along the angle bisector of the sector.

Fig.4.4: The resultant flux vector points to the hot spot

The method is robust compared to FDM or FEM, since minor errors in computing the flux vector at any point (arising due to lack of accurate thermo-physical data) are automatically corrected in subsequent iterations. The VEM has also proved to be much more efficient (lower memory requirement and 10-100 times faster) than FDM or FEM, for identifying hot spots in even complex shaped castings.

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Fig4.5: Top: Simple casting with feeder; middle: directional solidification (feed paths);

bottom: progressive solidification in the central section.

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4.7 Optimization and Validation The feeding system must be designed to obtain the desired solidification characteristics in a casting, essentially to avoid solidification shrinkage related defects. At the same time, the yield must be maximized and fettling problems must be minimized. The feeding design can be assessed using the following simple criteria. All criteria have been normalized and have to be maximized. Internal Porosity: The size of internal porosity in a critical section of the casting must be less than the acceptable size. Porosity may refer to macro-porosity (usually more than 1 mm size), or micro-porosity (0.01-0.1 mm), which is barely visible to the naked eye. We can also introduce a middle term called mini-porosity for intermediate sizes (0.1-1 mm). The criterion is written as:

CF1 = 1 - maxi (di) / dmax Where, maxi (di) gives the maximum size of porosity in the casting and dmax is the maximum allowable size of porosity (quality specification, determined from functional requirements). Feeder efficiency: The feeder efficiency is the ratio of total feed metal required to the total volume of feeders. This is compared with the maximum possible efficiency of the feeder. The criterion is given by:

CF2 = α ( Vc + Σi Vfi ) / (ηf-max Σi Vfi ) where, Vc is the casting volume, Vfi is the volume of feeder i and α is the volumetric shrinkage of the cast metal. The maximum efficiency of a feeder depends on its shape and use of feedaids. Open cylindrical feeders have low efficiency (less than 15%); an exothermic cover and sleeve increases its efficiency to 70% or more. Feeder yield: The volume of the feeders must be minimized to increase the yield. The criterion is given by:

CF3 = Nc vc / ( Nc vc + Σi vfi ) Where, Nc is the number of casting cavities per mold, vc is the volume of each cavity and vfi is the volume of feeder i. Fettling: The size of the feeder connection (neck) must be small compared to the connected portion of the casting to avoid breakage or cracks in casting during fettling. When several feeders are present, the feeder that is most likely to cause damage to the casting determines the criteria assessment value.

CF4 = mini ( 1 – ( tfi / tci ) Where, tfi is the smallest dimension of the neck of feeder i and tci is the thickness of the connected potion of casting.

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A poor design of feeding system (feeders, necks and feedaids) can lead to solidification shrinkage related defects in the casting. These include macro, ‘mini’ or micro -porosity, shrinkage pipe (extending into the casting) and surface sink. Other defects, caused by subsequent cooling of the casting, include casting distortion and cracks. Based on their location, the defects can be classified as external, subsurface or internal. The most widely used experimental techniques for feeding design validation are briefly described below. Thermocouple method: In this method, thermocouples are embedded in the mold at strategic points: end sections, center of thick sections, along the feeder axis and along the centerline of long thick sections. Then the metal is poured into the mold and for each thermocouple, the temperature history is recorded. The results can be used for plotting the time-temperature curves for different locations inside the casting, indicating the progress of solidification. The thermocouples must be chosen to minimize heat absorption. The method is more suitable for theoretical studies in a lab. Non-destructive testing: The casting is inspected using radiography (for internal defects) and dye penetration (for sub-surface defects with some opening to the surface). Other methods include magnetic particle and ultrasound, but these are more indirect methods and require considerable expertise for interpreting the readings accurately. Sectioning and machining: This is the most widely used method in practice for industrial castings. All suspected regions of the casting are cut through, polished and visually inspected. The sections are usually made through the center planes of feeders and their necks, thick sections of the casting (example, bosses) and junctions of two or more walls. Machining and drilling of specified features is also carried out. The method is however, not as reliable as it seems. It is possible to cut a section and assume that the region is defect-free, when a major porosity may be lying in a parallel plane just a few mm away. Also, the machined or drilled surface may appear perfect, but further machining may bring out porosity.

Fig.4.6: Feeder design – over, borderline and robust

In general, successful experimental validation of sample castings does not guarantee defect-free production castings. This may happen owing to ‘borderline’ optimization of feeding design, when the feeders (especially their connection with casting) does not leave any safety margin for variation in process parameters (such as metal composition and pouring temperature). The feeding must be slightly over-designed and made sufficiently robust to avoid such surprises during regular production.