A method developed by A. Heuvers that is used to determine and correct casting cross sections which require sufficient feeding via a riser. The theoretical basis for this method is the requirement that the solidificationmodulus (see Casting modulus) of a cross-sectional area has to progressively increase in the direction of the riser in order to achieve a uniform heat gradient.

According to Chvorinov’s formula (**eq. 1**, see also Chvorinov’s rule), the solidification time t_{E} of a casting in the mold is proportional to its square solidificationmodulus M:**Eq. 1: **

t_{E} | Solidification time in s |

k | Metal-, mold-material- und temperature-dependent coefficient (s/cm^{2}) |

M | Modulus M = V/A in cm |

V | Casting volume in cm^{3} |

A | Surface area of the casting in cm^{2} |

n | Exponent with n = 1.5 to 2.0 |

The solidificationmodulus M in cm denotes the ratio of the casting volume in cm3 to the heat-dissipating surface area of the casting in cm2. The seal feeding of a casting requires a riser volume having a greater solidification time tE than the casting. Therefore, Heuvers’ simple circle method is a rough approximation to Chvorinov’s rule for level solidification issues.

Heuvers’ circle method can be easily understood when looking at the directed solidification of a wedge by means of a riser (Fig. 1). Here, the riser’s task is to enable post-feeding of the casting. In order to do so, it must be correctly dimensioned and positioned. It may only solidify once the casting is already solid, i.e. it should be located at the largest cross section, if possible in the area of the mold joints. Simple dimensioning of the wall thicknesses and risers can then be done by Heuvers’ circle method.

Starting at the lowest wall thickness at A1 in Fig. 1, circles touching each other are drawn into the wall cross section, with the areas of these circles increasing up to riser A6 by a constant factor, kH, the so-called Heuvers’ factor. Table 1 contains proven Heuvers’ factors kH for the seal feeding of certain ferrous casting materials during green sand and sand casting. As the factor increases, the seal feeding requirements become higher, meaning that the effective riser length decreases. With a favorably engineered casting, the ratio (A1/A2) of adjacent inscribed circular cross sections should therefore be close to one or slightly greater than one in feed direction.

The connection between the riser and the casting, i.e. the gate area, must be designed such that the riser can be removed by spruing, sawing, separating or gas cutting with appropriate ease. An effective riser will exhibit a deep surface cavity and shrink marks upon casting. In addition, the riser height must be sufficient to provide the appropriate pressure head h on the casting as the driving force for feeding. The gate area should therefore also correspond to the nearest circular cross section according to Heuvers, while the sand edge effect should be taken into account as a positive influence (i.e. reducing the cross section).

Thus, according to the method described by A. Heuvers, circles are drawn in such a way that they correspond to the casting cross sections resulting from the pattern elevation. The cross sections must be completely filled by the circles and their diameters must progressively increase in the direction of the riser. Drawing in Heuvers’ circles can also be called “circling”. If required, reinforcement allowances are to be added to the casting if the circles no longer fit into the cross sections due to their increasing diameter as can be seen in Fig. 2. These allowances will have to be removed during processing of the casting. Cross sections which prove to be too large based on the method should be reduced or forced to solidify more quickly by applying external chills (denseners). Sand edges at transitions (junctions) in the cross section lead to an accumulation of heat which is the larger, the smaller the rounding radius. This sand edge effect should be taken into account by drawing the fillets at the cross-sectional transitions with a larger assumed radius in estimation of the anticipated heat accumulation, which needs to be tangent to the Heuvers’ circles (Fig. 2).

3-dimensionally, the circles should be regarded as spheres which exhibit a solidificationmodulus that increases as the diameter becomes larger. The drawing should also take into account machining allowances; therefore, it is most convenient to use a pattern elevation as a basis instead of a cross-sectional drawing of the casting.

Warm cracking (see Warm crack, Hot crack) on a reinforced cover or plate is shown in Fig. 3a. If the rib thickness is oversized and thesolidification front is growing in the direction of the thermal center against the heat flux in a columnar fashion, warm cracks may occur due to the shrinkage impairment of the mold material in the radii. When redesigning the ribs as shown in Fig. 3b by applying Heuvers’ circle method, the propensity to warm cracking can be reduced.

The modulus theory permits a refined mathematical calculation of the solidification behavior and dimensioning of risers for castings. It was primarily developed by Wlodawer for cast steel.