Investment casting converts a machined metal die into a wax pattern, then into a ceramic shell, and finally into a metal part. Every one of these conversions moves the dimension, and the movements do not cancel. For thin-walled aero-engine casings held to CT4–CT6 on key linear dimensions, with envelopes approaching 1.8 m, minimum wall sections below 2 mm and casting masses up to 1000 kg, the tolerance budget is set by five successive transformation steps rather than by any single one. Holding pressure chosen to limit wax shrinkage says nothing about the sintering contraction that follows; shell thickness established during slurry dipping governs both the dewaxing distortion and the thermal resistance seen by the solidifying alloy. Predicting the final dimension therefore requires a quantitative account of how a parameter error introduced at one stage becomes a dimensional error several stages later.
Why the tolerance problem is a process-chain problem
Classical stream-of-variation (SoV) theory was developed for machining and sheet-metal assembly, where the error inputs are geometric: fixture locator deviations, tool path errors, part-to-part mismatch. These inputs map onto part dimensions through rigid-body kinematics, so a state-space model with a system matrix and a control matrix describes the propagation exactly. Investment casting breaks both assumptions. The inputs are heterogeneous process variables — pressure, temperature, time, shell thickness — and they become dimensions only after solidification, sintering, relaxation and thermal expansion have acted. The part itself changes identity at each step: die, wax pattern, ceramic shell, casting. There is no repetition of the same operation on the same object, and the internal cavity of the shell cannot be touched by a probe.
What survives from the SoV framework is the structure of the problem: a chain of stages, each with its own parameter vector, each stage’s output dimension serving as the next stage’s input dimension, and errors accumulating multiplicatively along the chain. Retaining that structure, while replacing the geometric transfer with physically measured parameter-to-dimension sensitivities, gives a model that can be calibrated from a modest number of trials and used both for prediction and for process optimization.
Where the dimensional variation originates
Wax injection. Wax has low thermal conductivity, so the inner surface of a injected pattern solidifies well after the surface in contact with the die. Measured thermal data for a filled pattern wax (KC4017B, ring-and-ball softening point 64.4 °C, 32% filler) show conductivity rising and then falling with temperature, with the liquid phase distinctly less conductive. Rheologically the melt is shear-thinning and is described by a Cross-WLF viscosity model with a power-law index of 0.152 and a zero-shear viscosity referenced to 303.15 K; its pressure–volume–temperature behaviour follows a modified double-domain Tait equation fitted from 30 to 90 °C and 0–100 MPa. These material functions explain why holding pressure, holding time, melt temperature and die temperature dominate the stage: they set the density and the amount of material available to compensate shrinkage while the gate is still open. Clamping force matters only insofar as it keeps the halves closed.
Wax storage. A demoulded pattern continues to relax. The relaxation behaviour is captured by a generalized Maxwell spectrum; five to eight elements reproduce the measured decay, and eight elements fit better, although beyond roughly twenty elements linear regression no longer returns a physically admissible spectrum. Because of the time–temperature equivalence, storage temperature and storage time act as interchangeable knobs on the same relaxation process, which is why both appear as error sources rather than as a single “ageing” variable.
Shell build and dewaxing. Slurry viscosity, drain time and stucco grading determine layer thickness and, more importantly, its uniformity; sharp corners receive less stucco and therefore a thinner coat. For the shells used here (33 wt% mullite, 18 wt% Al2O3, 16 wt% ZrO2, 33 wt% SiO2), autoclave dewaxing at 170 °C and 755 kPa melts and partly vaporizes the pattern inside a confined cavity. Thin sections melt first but drain slowly, so the local pressure is highest where the wall is thinnest and the cavity expands most. Dewaxing time, dewaxing temperature and shell thickness are the controlling variables.
Shell firing. Sintering shrinkage is negative (the shell contracts on cooling) and grows sharply above 1100 °C: about −0.04% after 1 h at 1000 °C, −1.08% after 1 h at 1250 °C and −1.32% after 1 h at 1400 °C for a water-based shell, with firing temperature mattering more than holding time. The mechanism is the appearance of a liquid glass phase, whose viscosity falls with temperature, combined with cristobalite precipitation and its high-to-low inversion on cooling. Firing temperature and holding time therefore enter the error budget directly.
Pouring and solidification. Shell thickness controls compliance, permeability and thermal resistance simultaneously; pouring temperature controls superheat and hence the thermal gradient. In trials with a K4169 nickel-based superalloy, shells were preheated to 1000 °C for 240 min and poured in 15 s under vacuum, with pouring temperature varied between 1450 and 1620 °C.
| Stage | Controlling parameters (range examined) | Change rate sj(k) |
|---|---|---|
| Die to wax pattern | Holding pressure 25–35 bar; holding time 5–25 s; die temperature 20–30 °C; melt temperature 70–74 °C | 0.9939 |
| Wax storage | Storage time 2–72 h; storage temperature 18–28 °C | 0.9890 |
| Shell dewaxing | Dewaxing time 20–40 min; shell thickness 8–16 mm | 1.0026 |
| Shell firing | Firing temperature 800–1200 °C; holding time 60–480 min | 1.0011 |
| Pouring | Shell thickness 8–16 mm; pouring temperature 1450–1620 °C | 0.9828 |
The dimensional change rate as the transfer variable
Between two consecutive stages the same nominal feature is measured before and after the transformation, and the ratio defines the stage gain:
$$s_j(k)=\frac{l_j(k)}{l_j(k-1)}$$
where lj(k) is the j-th feature size after stage k. A value below one indicates contraction, above one expansion, and the definition absorbs the different physical mechanisms — shrinkage, relaxation, thermal expansion, sintering — into a single dimensionless quantity that can be regressed against the stage’s parameter vector uj(k) by response-surface methods. Because a production parameter never sits exactly at its set point, the dimensional error follows from a first-order expansion about the target vector:
$$\Delta l_j(k)\approx l_j(k-1)\left.\frac{\partial f_j}{\partial \mathbf{u}_j}\right|_{\mathbf{u}_j^{t}}\Delta\mathbf{u}_j(k)$$
This is the mapping from heterogeneous parameter errors to a dimensional error. Its practical value depends on how well the sensitivity vector is identified; a Box–Behnken design on a 400 × 100 × 2.5 mm plate, with the four injection parameters varied and thickness measured at five positions along the flow path, produced a regression for the thickness change rate with R² = 0.978 and an adjusted R² = 0.958, adequate for the subsequent chain calculation.
State-space representation of the five-stage chain
The continuous process is discretized into five stages (injection, storage, dewaxing, firing, pouring); slurry dipping, stucco application and drying are absorbed into the shell-thickness variable, and air cooling after pouring is treated as part of the pouring stage. For each stage the feature sizes form a state vector, and one further geometric ingredient is required: a centre of dimensional change. A planar face contracts toward its centre line, a circular arc toward its centre, and a free-form surface of variable curvature toward a locally determined inscribed-circle centre found iteratively from the intersection of surface normals. This matters because a single global contraction factor reproduces only the average linear change and carries no information about form error, which is precisely what forces repeated die correction on casings and turbine blades.
With the state vector expressed in feature sizes, the propagation and observation equations are:
$$\Delta\mathbf{X}(k)=\mathbf{A}(k)\Delta\mathbf{X}(k-1)+\mathbf{B}(k)\Delta\mathbf{U}(k)+\mathbf{W}(k)$$
$$\Delta\mathbf{Y}(k)=\mathbf{R}(k)\Delta\mathbf{X}(k)+\mathbf{V}(k)$$
Here A(k) is diagonal and carries the previous stage’s error forward through the change rate, B(k) contains the sensitivities ∂fj/∂uj scaled by the incoming dimension, U(k) collects the parameter deviations, and W(k) absorbs unmodelled random effects and higher-order terms; R(k) rotates and translates each feature into the global measurement frame, and V(k) is measurement noise. Composing the stages gives the cumulative form:
$$\Delta\mathbf{X}(q)=\boldsymbol{\Phi}(q,0)\Delta\mathbf{X}(0)+\sum_{k=1}^{q}\boldsymbol{\Phi}(q,k)\mathbf{B}(k)\Delta\mathbf{U}(k)+\sum_{k=1}^{q}\boldsymbol{\Phi}(q,k)\mathbf{W}(k)$$
with the transition matrix Φ(q,k) = A(q−1)···A(k). The initial term ΔX(0) is the die manufacturing error, which is systematic for a given tool and therefore produces an asymmetric error distribution, while the parameter terms are random and produce the batch-to-batch scatter.
Observation of the internal cavity required computed tomography rather than a probe: an industrial CT system with 25 µm voxel resolution measured the shell cavity non-destructively, while the die and castings were measured on a bridge-type coordinate measuring machine at 4.5 µm and wax patterns by laser scanning at 80 µm, the last chosen to avoid the contact deformation that a thin wax wall cannot survive.
Validation on a stepped thin-walled ring
The test part abstracts the ring-within-ring, variable-section, thin-wall topology of a rear casing: an inner ring of 134 mm diameter, an outer ring stepped in five 25 mm increments with outer diameters D1–D5 of 304, 308, 316, 332 and 348 mm and a 300 mm inner diameter D0, wall thicknesses T1–T5 of 2, 4, 8, 16 and 24 mm, and four 30° struts. The geometric similarity ratio to the full casing is 1:3.56, and the corresponding stress similarity ratio is 1:12.67.
Each stage was calibrated with a Latin hypercube design — 15 injection trials (30 patterns), 15 storage trials, 9 dewaxing trials, 6 firing trials and 9 pouring trials — and the model was then tested in 9 complete repetitions of the chain under fixed set points with realistic parameter scatter (holding pressure ±0.5 bar, holding time ±0.5 s, storage ±0.5 h and ±0.5 °C, dewaxing time ±0.25 min, shell thickness ±0.1 mm, firing ±3 °C and ±1 min, pouring ±3 °C). The die itself measured 348.0847 mm against a 348 mm nominal.
| Stage | Calculated mean | Measured mean | Calculated range | Measured range |
|---|---|---|---|---|
| Die | 348.0847 | 348.0847 | 0.0847 | 0 |
| Wax injection | 345.9691 | 345.6767 | 0.4723 | 0.5876 |
| Wax storage | 342.1736 | 342.5188 | 0.5584 | 0.6968 |
| Shell dewaxing | 343.0463 | 343.1020 | 0.6322 | 0.7918 |
| Shell firing | 343.4349 | 343.0370 | 0.6716 | 0.8515 |
| Pouring | 337.5154 | 337.5450 | 0.6781 | 0.9018 |
The predicted final mean was 0.0296 mm below the measured mean, and the predicted 6σ range 0.2237 mm narrower than the observed range — about 80% agreement. The bias is systematic and expected: the model retains the four to six principal parameters per stage and assigns the remainder to the noise term, so it under-reports scatter while tracking the mean closely.
Decomposing the final variance by stage identifies where control effort pays back. Acting alone, the die error contributes 0.1939 mm (22.24%) and injection 0.4606 mm (52.83%); storage, dewaxing, firing and pouring contribute 10.33%, 8.17%, 4.36% and 2.07% respectively. At parameter level, holding time alone accounts for 57.52% of the final scatter, holding pressure for 10.43%, storage time 8.58%, shell thickness at dewaxing 6.58%, firing temperature 5.14% and storage temperature 4.70%, with the remaining parameters below 4%. This ranking is what makes a large optimization tractable: only two stages and four parameters need to be tuned to address roughly 80% of the problem.
Robust control of the parameter set
Because a casting carries several toleranced features at once, the stages interact: a parameter level that centres the outer diameter may push the wall thickness out of tolerance, since the two features see different constraint conditions and therefore different change rates. Optimizing the mean alone is not sufficient, and neither is a mean-only desirability function, which ignores how flat the response surface is around the chosen level. The robust formulation therefore replaces the mean response with the range of the response over the actual parameter scatter:
$$R_j(k)=\max s_j(k)-\min s_j(k),\qquad \left|\alpha_p(k)-\alpha_p^{o}(k)\right|\le \delta_p,\ \ \mathbf{u}(k)\in\Omega(k)$$
where δp is the observed fluctuation of parameter p and Ω(k) is the feasible experimental region that keeps the stage physically valid — no shell cracking during dewaxing, no shrinkage porosity during pouring. Range desirabilities drj(k) are aggregated geometrically, and the mean and robustness objectives are combined with weights:
$$D_r(k)=\left[\prod_{j=1}^{m}d_{rj}(k)\right]^{1/m},\qquad D_{\text{overall}}(k)=D(k)^{W_1}D_r(k)^{W_2},\qquad W_1+W_2=1$$
A genetic algorithm searches the constrained space for the maximum, avoiding the local optima that gradient methods readily fall into on these multi-modal surfaces. Applied to injection and storage only, with D5 and T5 as simultaneous responses, this produced holding pressure 30.47 bar, holding time 17.77 s, storage time about 21.8 h and storage temperature 21.66 °C, and reduced the final D5 range by 0.6 mm and the T5 range by 0.14 mm. Extending the same procedure to dewaxing, firing and pouring gave a D5 range reduction of 0.58 mm and a T5 reduction of 0.18 mm, moving the achieved tolerance grades from CT6 to CT3 and from CT5 to CT3 respectively. The mean shifted slightly in each case, which is acceptable because the mean can be corrected at the die-design stage while the scatter cannot.
Die compensation and its verification
Inverting the chain gives the die dimensions needed for a specified casting. The inverse computation supplies the initial cavity size and tolerance for each key feature; stage simulation then refines the free-form regions, with wax shrinkage predicted in a mould-filling package, dewaxing and firing in a structural solver, and solidification in a casting package, each result exported as a mesh and re-imported so that the deformation of one stage becomes the starting geometry of the next. If a die already exists, measured intermediate parts can be substituted for the simulated ones at any stage, which raises the accuracy of the prediction exactly where the free-form geometry is hardest to model.
A new die was designed by this route for a target casting and tested with the same parameter set as the original trial. The die was machined to ±0.025 mm; results are given in Table 3.
| Feature | Die design size | Casting target | Measured casting mean | Achieved grade |
|---|---|---|---|---|
| D0 | 306.32 | 300.00 | 300.52 | CT6 |
| D1 | 309.84 | 304.00 | 303.86 | CT5 |
| D2 | 313.94 | 308.00 | 308.55 | CT5 |
| D3 | 322.70 | 316.00 | 316.45 | CT5 |
| D4 | 340.01 | 332.00 | 332.48 | CT5 |
| D5 | 357.32 | 348.00 | 348.41 | CT5 |
All key means fell within 0.55 mm of target, and every feature except D0 lay inside the assigned tolerance band under the original, unoptimized process settings. D0 sat at the band edge, which is consistent with its restrained shrinkage condition.
How thin walls and stepped sections deform
The stage gains are not constants; they depend on the local section. For wall thickness the dependence is well described by exponential or saturating forms (T in mm):
| Transformation | Fitted relation | Parameters | R² |
|---|---|---|---|
| Die to wax | S = a − b·cT | a = 0.94853, b = 0.20160, c = 0.74326 | 0.995 |
| Wax to shell cavity | S = S0 + A·exp(−T/ξ) | S0 = 1.04162, A = 1.68895, ξ = 1.1684 | 0.971 |
| Shell cavity to casting | S = a − b·cT | a = 1.00957, b = 0.08483, c = 0.87807 | 0.923 |
| Die to casting | S = S0 + A·exp(−T/ξ) | S0 = 0.98333, A = 0.92536, ξ = 1.75499 | 0.976 |
The physics behind the trend is a surface-to-volume argument. A thin wax wall cools and solidifies first, after which molten wax from the thicker sections cannot feed it; short shots and incomplete filling become likely at low holding pressure, short holding time, low die temperature and low melt temperature. The same argument applies during solidification of the alloy, except that the gating system continues to feed liquid metal, so the casting’s thickness loss is smaller than the wax pattern’s — hence the die-to-casting gain of 0.983 for thick sections against 0.948 for thin wax. In absolute terms the thickest step (T5 = 24 mm) lost about 1.5 mm during injection.
Diameters behave differently again. Wax and casting both contract, with the change rate approaching unity as the diameter grows because a larger ring solidifies later and receives more feed; the outer diameters follow exponential fits in D with R² of 0.990 and 0.997. During dewaxing the cavity expands slightly (D3 by 0.35%), and no monotonic fit against diameter survives, because the expansion is driven by local vapour pressure rather than by section size. Free versus restrained shrinkage explains the split between the outer and inner rings: the outer diameter contracts freely against the die or shell, whereas the inner diameter D0 is held by the four struts and by the inner shell wall, so it contracts less during injection (0.996 against 0.994 for D5, both well above the free-shrinkage value) and more during pouring, where the compliant ceramic shell allows a partial release. Across the whole chain D0 contracts at every stage and contracts most during pouring, at 0.9805.
The practical consequence is that a single global shrinkage factor is inadequate for stepped, thin-walled geometry. A factor tuned on the thick sections over-compensates the thin ones, and a factor tuned on free outer diameters under-compensates the restrained inner diameter — which is the origin of the repeated cut-and-try die correction that dominates lead time for casing tooling.
Scope and open issues
The predictions reported here rest on four to six parameters per stage and on response surfaces fitted from 6 to 15 trials per stage; additional parameters would narrow the gap between predicted and observed scatter, which is currently biased low by construction. Form and position error are represented in the feature-point formulation through multiple centres of dimensional change, but the production calculations were carried out on feature sizes, because the local parameter values at individual surface points are not measurable in a real furnace or autoclave. The five-stage discretization also treats a continuous, time-varying process as a sequence of lumped transformations; resolving the dimension at intermediate times within a stage would require a time-dependent formulation. Within these limits, the chain model gives a defensible answer to the question that matters on the shop floor: given a die and a set of process set points with realistic scatter, what dimension and what spread will the casting have, and which two parameters should be moved first to reduce the spread without losing the mean.