**2. Feeder System Design for Solidification Contraction**

Attention is focused on the second stage, i.e. solidification contraction in the mushy zone, which is the root cause of shrinkage porosity. Initially, as stated above, a feeder system is required to compensate for solidification contraction, but also to ensure that shrinkage porosity formation is minimised. This requires that the feeder/s (a) must contain sufficient melt volume to compensate for the volume contraction of the part, and, more importantly, (b) solidify later than the casting (see **Figure 4**). The latter effectively implies that the thermal centre of the total casting should eventually migrate to the feeder/s, where high porosity formation due to hotspots or in the form of pipe shrinkage will be experienced.

Applying the two requirements for feeder design, firstly, the volume of the feeder/s is given by:

$$V\_f \approx aV\_c$$

where *α* = shrinkage fraction, *V <sup>f</sup>* = feeder/s volume and *Vc* = cavity volume. Taking account of shrinkage within the feeder itself:

$$V\_f - aV\_f = aV\_c$$

However, feeders in sandcasting moulds have low levels of feeding efficiency due to gravitational feeding and low design heads, giving:

#### **Figure 4.**

*Solid fraction predictions for a valve body casting at two different times, showing the feeders (with exothermic sleeves) solidifying later than the casting.*

*Shrinkage Porosity in Steel Sand Castings: Formation, Classification and Inspection DOI: http://dx.doi.org/10.5772/intechopen.94392*

$$eV\_f = a(V\_\varepsilon + V\_f) \tag{2}$$

where the feeder efficiency *e* is generally taken to be in the range of 15–30%. It is necessary to balance the feeder efficiency with feeder size to avoid high waste and energy consumption. Secondly, the solidification time of the feeder/s must be greater than that of the casting, typically by a "safety factor" of 1.2, giving:

$$\left.t\_s\right|\_{cating} < t\_s|\_{feeder} \implies \left.t\_s\right|\_{feeder} = \mathbf{1}.\mathbf{2}t\_s|\_{cating}\tag{3}$$

The well-known Chvorinov's equation allows the above relation to be expressed in terms of the mould geometry, and can be expressed as:

$$t\_s = B\left(\frac{V}{A}\right)^n\tag{4}$$

where *V* is the casting volume, *A* is the surface area of the casting through which heat is conducted, *n* is an empirical exponent (equal to 2 for simply shaped castings in silica sand moulds), and B is the moulding constant which is dependent on (a) process parameters (the equilibrium solidification temperature *Tm*, the initial temperature *T*<sup>0</sup> of the mould, and the superheat *ΔTs* ¼ *Tpour* � *Tm*, all in °K), (b) mould material properties (thermal conductivity *<sup>k</sup>* in W�m�<sup>1</sup> �K�<sup>1</sup> , density *<sup>ρ</sup>* in kg�m�<sup>3</sup> , and specific heat capacity *<sup>c</sup>* in J�kg�<sup>1</sup> �K�<sup>1</sup> ), and (c) casting metal properties (latent heat of fusion *<sup>L</sup>* in J�kg�<sup>1</sup> , density *<sup>ρ</sup><sup>m</sup>* in kg�m�<sup>3</sup> , and specific heat capacity *cm* in <sup>J</sup>�kg�<sup>1</sup> �K�<sup>1</sup> ), given by:

$$B = \left[\frac{\rho\_m L}{T\_m - T\_0}\right]^2 \left[\frac{\pi}{4k\rho c}\right] \left[1 + \left(\frac{c\_m \Delta T\_s}{L}\right)^2\right] \tag{5}$$

Although Eq. (4) may appear complex, it is relatively straightforward to evaluate given the casting process parameters deployed and the materials used for the sandmould and the casting, the latter being readily available in metal casting databases. It is therefore relatively easy to estimate the solidification time for a specific cast geometry. However, in the design of the feeder system, this is not necessary, since substituting Eq. (3) into Eq. (2) simply requires that:

$$\frac{V\_f}{A\_f} = 1.2 \frac{V\_c}{A\_c} \Longrightarrow m\_f = 1.2 m\_c \tag{6}$$

where the modulus *m* has been introduced. Solving Eqs. (1) and (5) simultaneously for a given mould cavity allows the dimensioning of the system of feeders.

**Figure 5.**

melt, which results in either pore nucleation (due to volumetric shrinkage) or solid collapse (due to negative pressure), as shown in **Figure 3**. As will be shown later, the freezing range (difference between the liquidus and solidus temperatures) is an important factor in determining the morphology of

• Thirdly, the solid state contraction of the metal. Even though solid-state solidification values may be used in foundries (especially for pattern design), metals are rarely free to contract due to various microstructural constraints – this leads to internal stresses, and where these stresses overcome resistance, the material can undergo plastic or viscoplastic strain to adapt to the thermal and

transformational density changes. Hence, the part may be larger than

hot tears during solidification contraction).

*Casting Processes and Modelling of Metallic Materials*

Taking account of shrinkage within the feeder itself:

due to gravitational feeding and low design heads, giving:

**2. Feeder System Design for Solidification Contraction**

due to hotspots or in the form of pipe shrinkage will be experienced.

predicted. These constraints can lead to localised cold cracking (initiated from

Attention is focused on the second stage, i.e. solidification contraction in the mushy zone, which is the root cause of shrinkage porosity. Initially, as stated above, a feeder system is required to compensate for solidification contraction, but also to ensure that shrinkage porosity formation is minimised. This requires that the feeder/s (a) must contain sufficient melt volume to compensate for the volume contraction of the part, and, more importantly, (b) solidify later than the casting (see **Figure 4**). The latter effectively implies that the thermal centre of the total casting should eventually migrate to the feeder/s, where high porosity formation

Applying the two requirements for feeder design, firstly, the volume of the

*V <sup>f</sup>* ≈*αVc*

where *α* = shrinkage fraction, *V <sup>f</sup>* = feeder/s volume and *Vc* = cavity volume.

*V <sup>f</sup>* � *αV <sup>f</sup>* ¼ *αVc*

However, feeders in sandcasting moulds have low levels of feeding efficiency

*Solid fraction predictions for a valve body casting at two different times, showing the feeders (with exothermic*

shrinkage pores.

feeder/s is given by:

**Figure 4.**

**136**

*sleeves) solidifying later than the casting.*

*Hotspot formation in the feeder junction regions and the gate area of a steel disc casting for a modulus ratiom <sup>f</sup> =mc* ¼ 1*:*2 *.*

The effect of the modulus ratio *m <sup>f</sup> =mc* on shrinkage porosity is quite significant – with no feeding (i.e. modulus ratio = 0), shrinkage porosity as high as 7% can result. However, whilst modulus ratios higher than 1.2 can significantly reduce shrinkage porosity, it will not be entirely eliminated due to interdendritic phenomena. That is, the inclusion of oversized feeders does not guarantee sufficient pressure differentials to feed the solidifying dendritic network. This will be the subject of further investigation in the next section.

Furthermore, the modulus ratio of 1.2 does not guarantee the formation of hotspots in the junction region between the part and the feeder, as shown in **Figure 5**. This can be resolved through increasing the modulus ratio, albeit at the cost of production. However, a better approach would be to use exothermic sleeves around the feeders, as can be seen in **Figure 4**.
