have been hydrated for 7 days and for 28 days. The differencebetween the heat of solution values between the dry and the partiallyhydrated cement specimens is taken as the heat of hydration for thattime period. Heat values range from 261 kJ/kg−1 to 468 kJ/kg−1. [Non-SI units (cal/g) may be converted to SI units (kJ/kg−1), multiplying bythe conversion factor 4.1868.] This test is time consuming, involvesa hazardous mixture of nitric and hydrofluoric acids, and has low pre-cision with the 95% limits on the difference between two test result(d2s) values being 48 kJ/kg−1 for measurements between different lab-oratories. As an alternative, the heat of hydration of cement can bemeasured via conduction calorimetry, with ASTM C1679, StandardPractice for Measuring Hydration Kinetics of Hydraulic Cementi-tious Mixtures Using Isothermal Calorimetry. This method hasproved useful in estimating total heat, in assessing early-age reac-tions and setting problems, and in measuring the influences of sulfateadditions and mineral admixtures on heat evolution (2).

The rate of hydration of a cement depends on its mineralogy,mass fraction of each phase, particle size distribution, water–cementmass ratio, and temperature and relative humidity during curing (3).Copeland et al. ascribe the total heat of hydration as emanating fromtwo processes: the chemical hydration reactions are thought to beresponsible for 80% of the heat, and the heat of wetting of the sub-sequent colloidal hydration product accounts for the remaining 20%(4). The C186 test uses a fixed water–cement ratio, but the curingtemperature depends on the rate of heat evolution.

Limits on composition and fineness in ASTM C150 and AASHTOM85 reflect their influence on heat of hydration. Type II cement withthe moderate heat option has two additional restrictions on eitherthe sum of mass % C3S (alite) + 4.75 � C3A (aluminate) ≤ 100 (heatindex equation); or a 7-day heat release amount of 290 kJ/kg−1 whenmeasured by ASTM C186. Type IV (low heat cement) has limits oneither phase mass fraction for C3S, C2S (belite), and C3A of 35%, 40%,and 7%, respectively, or a C186 heat value limit of 250 kJ/kg−1 at7 days. Type III cements generally have higher heats of hydration,and Type IV have the lowest (1).

Phase estimates are Bogue-calculated values as described in ASTMC150 and AASHTO M85. Errors in these estimates arise from thevariability of clinker phase chemistry relative to the assumed com-positions, from failure to account for minor constituents, and frominaccuracy in measured analytical values (5, 3). A standard testmethod for clinker and cement X-ray powder diffraction analysis maybe found in ASTM C1365. X-ray powder diffraction is ideally suitedfor direct phase analysis of fine-grained materials (such as clinker andcements), for each phase produces a unique diffraction pattern inde-pendent of the other phases, and the intensity of each phase is propor-tional to its concentration (6–8). Phase abundance directly determinedby quantitative X-ray powder diffraction and some nonphase vari-ables (particle size distribution, fineness) are considered in modeling7-day heat of hydration (HOH7).

Heat of Hydration for CementStatistical Modeling

Paul Stutzman, Stefan Leigh, and Kendall Dolly

1

The heat of hydration of hydraulic cements depends on a complex set ofphase dissolution and precipitation reactions following the addition ofwater. Heat of hydration is currently measured in one of two ways: aciddissolution of the raw cement and a hydrated cement after 7 days orisothermal calorimetry. In principle, the heat of hydration should be pre-dictable from knowledge of the cement composition and perhaps somemeasure of the cement fineness or total surface area. The improved min-eralogical estimates provided by quantitative X-ray powder diffraction,together with improved statistical data exploration techniques that exam-ine nonlinear combinations of candidate model constituents, were used toexplore alternative predictive models for the 7-day heat of hydration. Anexploratory tool, called “all possible alternating conditional expectations,”was created by combining all possible subsets regression with alternatingconditional expectation to judiciously select variables within an explana-tory variable class and subsets of variables across explanatory variableclasses exhibiting the highest potential predictive power for additive non-linear models for 7-day heat of hydration. Although a single, strongmodel for 7-day heat of hydration did not emerge from analyses, generalconclusions were drawn. Good-fitting models included a key structuralmineralogical phase (belite preferred); calcium sulfate phase (bassan-ite preferred); total fineness or surface area component (Blaine finenesspreferred); and ferrite in conjunction with iron oxide, or aluminate, orcubic aluminate.

Hydraulic cements react with water through a process called hydra-tion via a series of chemical reactions, ultimately resulting in the pre-cipitation of interlocking hydration products that provide strength tothe material. The hydration process produces heat that in some con-crete placements may cause expansion and, potentially, crackingat cooling. The temperature rise can also be beneficial in the caseof cold-weather concrete placements, where the heat facilitateshydration and keeps the concrete from freezing (1).

DETERMINING HEAT OF HYDRATION

ASTM C186, adopted in 1944 as a standard test method for determin-ing the heat of hydration of hydraulic cements, involves measurementof the heat of solution of cement specimens and cement pastes that

P. Stutzman, Materials and Construction Laboratory, and S. Leigh and K. Dolly,Statistical Engineering Division, National Institute of Standards and Technology,100 Bureau Drive, Gaithersburg, MD 20899-8615. Corresponding author: P. Stutzman, Stutz@nist.gov.

Transportation Research Record: Journal of the Transportation Research Board,No. 2240, Transportation Research Board of the National Academies, Washington,D.C., 2011, pp. 1–8.DOI: 10.3141/2240-01

Effects of Cement Phase Characteristics

Early work on developing predictive models for heat of hydrationfocused on the contributions of individual clinker phases, synergisticeffects of multiphase cement hydration, and heat of precipitation ofthe resulting hydration products (4, 9–11). Lerch concluded that gyp-sum (CaSO4�2H2O) retards early hydration of cements containinghigh C3A content; it accelerates the hydration of cements with lowC3A content (12); and more CaSO4�2H2O is required for alkali C3Ao

(orthorhombic form) cement than is needed for low-alkali C3Ac

(cubic form) cement. More recently, the accelerating effects ofpotassium oxide (K2O presumably from alkali sulfates and alkali-substituted C3A) on C3S, ferrite, and C3A have been demonstrated(3, 13, 14). Calcium sulfate additions have an accelerating effect onhydration of the silicates and ferrite while retarding the initial setand reactions of the C3A phases. CaSO4�2H2O has also been seen toretard heat development in mixtures of clinker phases. Bassanite[2CaSO4�(H2O); hemihydrate] may form through CaSO4�2H2Odehydration at grinding for some cements, an endothermic process.Rehydration at mixing to CaSO4�2H2O liberates heat at 192 kJ/kg−1

(9, 10). Overall, this reflects the complex synergy of mineralconstituents and portland cement during the hydration process.

Taylor summarized multilinear predictive models having the formof Equation 1 for heat (Ht) derived by least-squares regression analy-ses using Bogue potential phase mass fraction (μ), and a coefficientaccounting for degree of hydration of each phase for a specific age(Table 1) (3). The C3A reaction can result in ettringite (AFt) at earlyages and monosulfate (AFm) at later ages, with differing enthalpyof hydration, underscoring the complexity of the cement hydrationprocess. Taylor noted that the enthalpies of formation of clinkerphase hydration products would make the estimates of potential heatevolution more precise, but uncertainties in their values and variabil-ity in the reaction stoichiometry of hydration introduce errors into theestimates (3).

Poole summarized work relating Bogue phase composition toheat of hydration and developed a multilinear regression model(Equation 2) that used C3A and C3S with an R2 of 89% on the dataused to develop the model, exhibiting apparently little bias, and a±21 kJ/kg−1 95% confidence interval on the regression (15). Thecement fineness, as measured by Blaine permeability, was not foundto be a significant variable.

H a b c dt C S C S C A C AF= + + +i i i iμ μ μ μ3 2 3 4

1( )

2 Transportation Research Record 2240

Phases not included in these predictive models, such as thealkali and calcium sulfates and the different forms of C3A, canexert significant influences on hydration characteristics of a cement.It is reasonable to ask whether a complete accounting for cementphases and fineness measures would provide a better set of pre-dictive variables. More generally, would one expect relativelysimple multilinear models to work well given the complexity ofhydration processes?

Data

Variables analyzed in this work are organized into logical groupings,as shown in Table 2. Data were collected for 22 cements from theCement and Concrete Reference Library proficiency test program,18 cements from NCHRP Program 18-05 on cement performance(16), and 4 cements provided by the U.S. Army Corps of Engineers.The X-ray powder diffraction data were the average of three repli-cates each of a bulk cement and an extraction residue after a salicylicacid–methanol extraction to facilitate identification and quantitativeestimates. Bulk oxide and Bogue-calculated values were used forcomparative purposes and setting times, and 3-day strengths wereselected, for they had been mentioned in studies as being relevant (3).

Fineness Measures

All other variables held equal, a more finely ground cement might beexpected to react more rapidly than a coarser cement. However, thisdoes not always seem to be the case in practice. The Blaine finenessis an indirect measure of total particle surface area, denominated byvolume of material, based on time for unit volume air to flow througha cement powder packed cylinder (ASTM C204-7). An alternativemeans of characterizing cement fineness is particle size distribution.Laser diffraction of cements provides estimated size distributionsincluding D10, D50, and D90, which are indirect measures of the10th, 50th, and 90th percentiles of particle size distribution. Spanis another distributional statistic characterizing the size range ofparticles in the distribution.

HOH kJ kg C A C S7

1133 9 9 36 2 13 2

3 3

−( ) = + +. . . ( )i iμ μ

TABLE 1 Heat of Hydration Values for Clinker Phases and at 7 Days and 28 Days (3)

Value of Coefficient (kJ/kg) for Age (days)

Enthalpy of CompleteCompound Coefficient 7 days 28 days Hydration (kJ/kg)

C3S a 222 126 −517 ± 13

C2S b 42 105 −262

C3A c 1,556 1,377 −1,144; −1,672 (AFm and AFt reactions)

C4AF d 494 494 −418

NOTE: C4AF = tetracalcium aluminoferrite.

TABLE 2 Predictor Variables and Classes Used in Exploratory Data Analysis

Group Variable or Class

Mineral phase C3S, C2S, C3A, C3Ac, C3Ao, ferrite, MgO, by XRD alkali sulfates, CaSO4�2H2O, 2CaSO4�(H2O),(% mass fraction) CaSO4

Bulk oxide content(% mass fraction)

Fineness

Extras (other physicalmeasurements)

NOTE: XRD = X-ray powder diffraction analysis; CaSO4 = anhydrite; CaO = calcium oxide; SiO2 = silicon dioxide; Al2O3 = aluminum oxide; SO3 = sulfur trioxide; MgO = magnesium oxide; Na2O = sodium oxide; TiO2 = titanium dioxide; P2O5 = phosphorus dioxide; ZnO = zinc oxide; Mn2O3 = manganese oxide.

CaO, SiO2, Fe2O3, Al2O3, SO3, MgO, Na2O, K2O,TiO2, P2O5, ZnO, Mn2O3

Blaine; particle size by laser diffraction: D10,D50, D90, span calcite

Set time (Vicat)3-day strength

Time of Setting

The Vicat test measures the setting time of a cement by measuring thepenetration depth of a standardized needle into hydrating cement atdifferent times. A penetration depth of less than 25 mm provides ameasure of initial set time whereas a depth penetration of 0 mm pro-vides an estimate of the final set time (ASTM C191). Because the dif-ference between the final and initial times (VicatF − VicatI) is themeaningful test for the speed of setting, HOH7 might be expected tocorrelate with [VicatF − VicatI] or [1/(VicatF − VicatI)]. This simpleexample illustrates the occasional need to transform raw variables toachieve meaningful variable response. A potential confounding fac-tor with this test procedure is the use of normal consistency paste, inwhich the prescribed water content for the Vicat test varies by cement,which may affect setting times and may be different from the heat ofhydration test conditions.

STATISTICAL MODELING

Historically, modeling responses such as HOH7 typically haveinvolved multilinear modeling in which raw (test data) or transforma-tions of raw data were used as input into a statistical model (e.g.,multilinear Bogue models relating oxide compositions to mineralog-ical phase compositions). Typically, models were derived by straight-forward multilinear fitting, or by forward or backward selectiontechniques. Over the past 40 years, however, sophistication in modelselection techniques has increased tremendously for both multilinearcandidate models and nonlinear extensions of multilinear models.

In the initial approach to modeling HOH7 based on the mineralog-ical, fineness, and particle size distribution data, all possible subsetsregression (APSR) was adopted (17) and a nonlinear-addend multi-linear fit obtained by the use of alternating conditional expectation(ACE) (18). Although these techniques are not new, their separateor combined use offers significant improvements over backward andforward selection techniques.

Here, all possible alternating conditional expectations (APACE)was chosen, and the tool was created by combining APSR with ACE.With the use of APACE, one can easily explore which variables withinan explanatory variable class and which subsets of variables acrossexplanatory variable classes exhibit the highest potential predictivepower for additive nonlinear models for HOH7. An additive nonlinearmodel is a weighted sum of nonlinear function summands. APACElinked with the use of automated parametric fitting tools is found to bean easy-to-use, easy-to-interpret, and insightful approach to meaning-ful model search that offers enormous extension of the classical auto-mated model search employing only multilinear functions. (Certaincommercial materials and equipment are identified to adequately spec-ify experimental procedures. In no case does such identification implyrecommendation or endorsement by the National Institute of Standardsand Technology, nor does it imply that the items identified are neces-sarily the best available for the purpose.) (TableCurve 2D, http://www.sigmaplot.com/products/tablecurve2d/tablecurve2d.php).

Because APSR exhaustively assays every possible combinationof variables, limits had to be set on the number of variables assayed.Here, the maximum number of combinations assayed was limited toa maximum of 10 variables. This limitation does not seem undulyrestrictive. Nonetheless, because a number of the basic techniquesused in developing and selecting multilinear models are still appli-cable, in practice or at least in motivating more modern approaches,APSR and associated statistics are discussed.

Stutzman, Leigh, and Dolly 3

Prescreening Variables: Scatterplots

A fundamental principle of modern exploratory data analysis, includ-ing model selection, is to prescreen the data. Graphical prescreeningenables the modeler to scan for outlying data or obviously anomalouspatterns in data and to gauge the potential statistical explanatory powerand potential model meaningfulness of each explanatory variableassessed against the response (variable) of interest (HOH7).

A cross-correlation table of the variables serves an additionalpurpose. Variables that cross-correlate highly often contain muchthe same explanatory information for the proposed model. Incorpo-rating both in a model may lead to either redundancy, or overfitting,in the model, or to numerical instabilities (e.g., multicollinearities)in the numerical fitting procedure. Variables C3S and C2S are highlyanticorrelated. This is a natural result of their physical concur-rence as calcium silicates at the expense of one another. The nat-ural modeling solution is to employ one or the other, selecting thevariable for the model situation that gives the best goodness-of-fitstatistics.

All Possible Subsets Regression

APSR regresses a response variable (HOH7) against all possible mul-tilinear combinations of a preselected set of explanatory variables,typically screening for the best model fits from among the many fit-ted using criteria such as R2 and exhibiting only the best models fit asjudged by the goodness-of-fit criteria. In standard software, the manyregressions performed in APSR are multilinear regressions with up to32 input variables. Although it is a linear tool, it is still an excellentscreening device, and to be preferred to forward or backward modelsubset selection.

For example, if there are 10 candidate explanatory variables, APSRsoftware performs

or 1,023 regressions, where 10C1 means all possible 1-variable-at-a-time models, 10C2 means all possible 2-variable-at-a-time-combination models, and so forth. Goodness-of-fit is typicallyassessed by residual sum of squares (RSS).

where the sum is taken over all the data points, N is the number of datapoints, and P is the number of parameters being fitted (typically eithernumber of explanatory variables or number of explanatory variables+ 1 if an additive constant is being fitted as well). Because an RSS ofzero denotes a perfect errorless fit, the RSS having a value as close aspossible to zero is better (19).

The coefficient of determination, or R2,

quantifies the improvement of a proposed model’s predictive perfor-mance over the most primitive model prediction where pred (HOH7) is

R2

2

=( ) − ( )( )

−∑ pred HOH mean HOH

HOH mean HOH

7 7

7 77( )( )∑ 25( )

RSSHOH model prediction7

alldata

=−[ ]

−( )∑2

4N P

( ))

2 1 31010 1 10 2 10 10−( ) = + + +C C C. . . ( )

the model-based prediction of HOH7. It is often referred to as quantify-ing the percentage of variation in the data explained by the model; thestatistic is multiplied by 100 to express it in percent. Expressed that way,it is clear that the closer the value of R2 is to 1 or 100, the better is thevalue of 1 or 100, signifying a perfect fit.

Misspecification and Model Bias

In comparing the performance of the candidate models, it is not enoughto restrict attention to goodness-of-fit statistics. In general, the morevariables (parameters) one adds to a model, the better the fit. Alsoexisting is the issue of protecting against model misspecification, ormodel bias, referring to the possible inclusion of too few or too manypredictor variables in the model. For example, on any given pass ofan all possible subsets routine, if variables are being included thatshould not be in the model, the variances (noise levels) of the extramodel coefficients and predictions increase and push predictions offtarget. However, if too few variables are being included in the model,the model will also be biased.

Because simply increasing the number of variables incorpo-rated in a model will automatically tend to improve such good-ness-of-fit statistics, reference must also be made to an adjustmentfor bias statistic, such as Mallow’s Cp, in which the bias refers tothe biasing of a model by incorporation of too few or too manyexplanatory variables.

Mallow’s Cp

Mallow’s Cp statistic is very useful for assessing misspecificationswhen comparing multilinear models. It assesses the balance betweenbias owing to too few variables and too many variables by evaluatingthe following:

where

p = number of parameters in the candidate model,s2 = residual mean square for the candidate model,n = number of multivariate data points, and

σ̂2 = estimate of the true model variance.

A Cp value close to p indicates minimal misspecification in acandidate model. The use of Cp in conjunction with a goodness-of-fit metric such as R2 is probably the single most reliableapproach to fitting blind, that is, searching by statistical trial anderror for a model where no scientifically derived candidate modelexists (20–22).

ENSURING VALIDITY OF MODEL

To ensure validity of the model, generally one seeks a number ofobjectives:

1. Physical meaningfulness—use of scientifically meaningfulvariables in the model,

Cp =( )( )⎢

⎣⎢

⎥

⎦⎥ +

(∑ var pred HOH bias pred HOH7

2

7

σ))( )⎢

⎣⎢

⎥

⎦⎥

= +−( ) −( )

∑σ

σσ

2

ps n p2 2

26

ˆ

ˆ( )

i

4 Transportation Research Record 2240

2. Goodness of fit—in RSS or R2 metrics,3. Parsimony—employing as few variables as possible in the

model without overly underfitting and without sacrificing too muchgoodness of fit,

4. Avoiding of misspecification, and5. Cross-validation—testing the goodness of model achieved on

a set of training data by cross-validating against a nontraining set(not performed in this study).

Analysis of All Possible Subsets Regression

Various combinations of the chief untransformed variable classeswere subjected to APSR analysis. Results of the best for each com-bination are summarized in Table 3. From these combinations, keyvariables were selected, generating Group 7 in Table 3, noted as thebest of best and reflecting the comparatively high R2 and the Cp statis-tic having a value that is close to the number of variables. The combi-nation of C3S and C3A that form the basis for the ASTM and AASHTOheat index equation does not appear in this table, although C3S doesappear in combination with other phases, fineness measures, andoxides. Results suggest that Blaine fineness may be the best predic-tor from among the fineness variables considered. Although the R2

values are not strong for these models, they indicate potentiallyinteresting candidate variables for the subsequent step of nonlineartransformation of the data.

In each instance of class combination, the combination with the beststatistics is reported. In categories where the mineral variables wereincluded (five of seven), some form of calcium sulfate is selected as akey variable, specifically 2CaSO4�(H2O) or CaSO4, or both. TheFe2O3 and TiO2 significance may lie in their occurrence primarily inthe ferrite phase, and to a lesser extent concurrence with the alumi-nates and C2S (3). When variables on particle size distribution areincluded among the candidate predictor variables, most often theBlaine fineness is selected. Interestingly, TiO2 recurs frequently inconjunction with Fe2O3 and ferrite in good models. That will be seenagain when nonlinear transformations with APACE-selected modelswith R2 > .90 are explored.

Alternating Conditional Expectation

ACE is a technique that greatly extends the scope of classical multi-linear model selection techniques (23). Given HOH7 data and k asso-ciated candidate predictor variables Xk, the ACE algorithm findstransformations of the predictor variables and of the HOH7 response

TABLE 3 Results of APSR by Untransformed VariableClass Combinations

Group No. and Variable Class Combination Cp R2

1. C3S, ferrite, 2CaSO4�(H2O), Fe2O3, TiO2 5.55 .50

2. C3S, 2CaSO4�(H2O), span –0.92 .32

3. SiO2, Fe2O3, MgO, SO3, TiO2 2.95 .37

4. C3Ao, CaSO4, 2CaSO4�(H2O), 3-day strength 2.52 .42

5. SiO2, Fe2O3, MgO, SO3, TiO2, Blaine 3.40 .39

6. Bassanite, 3-day strength, Blaine 3.19 .35

7. C3S, ferrite, CaSO4, 2CaSO4�(H2O), Blaine, 7.44 .55Fe2O3, TiO2 (best of best)

NOTE: No. = number.

variable that maximize the correlation between f (HOH7), the trans-formed HOH7, and ∑gk(Xk), the sum of the transformed predictor vari-ables. The transformations are produced nonparametrically in the formof pictures (24), relating transformed to original variable for each ofthe variables, including HOH7 response. Such pictures can be paramet-rically modeled either from first principles or by the use of auto-mated software. Because the transformation graphs can assume manyforms, they need to be critically evaluated before they are incorporatedinto the predictive model. Analyzing each transformation picture forsmoothness performs this evaluation. Transforms that would appear tobe approximately straight lines, low order polynomials, exponen-tial or logarithmic functions, or circular functions are candidates forincorporation. Transforms with severe inflection points or that have theappearance of multiple distinct behaviors adjoined, for example, mightnot be candidates for incorporation in the model, or they might bemodeled distinctly for each simple submodel regime.

In using the ACE algorithm, one notices very quickly that ACEdrives R2 up dramatically. If one feeds seven or eight candidateexplanatory variables into a model of HOH7, one can easily obtainR2 values ranging from .80 to .95, possibly regardless of how mean-ingful the incorporated variables are for the true prediction of HOH7.Some of the transformations will look smooth and easy to parame-terize, but some will not. Because reapplying the ACE algorithm tosubsets of variables can give different transformations from apply-ing it to the original full set, it is clear that the appropriate way to pro-ceed for optimal variable selection purposes is to perform an APACEof the original full set of variables—to match HOH7 response to onevariable at a time, then two variables at a time, then three vari-ables at a time, and so on, using ACE for each distinct combina-tion. That is what has been done, using a nested loop of S-pluscode (24–26). Doing this guarantees that the ACE transform out-puts for each distinct combination of variables are mathematicallymeaningful and complete. For future work, one might consideraugmenting the APACE statistics with a Cp analog and then out-putting only the highest R2 with Cp closest to p combinations. Indoing so, however, one might easily ignore interesting subsets ofvariables that are consistent contributors to good models and thatconsistently transform cleanly.

Example of Explicit Parameterizationof ACE Outputs

Running APACE on a candidate predictor set that includes C3A,ferrite, 2CaSO4�(H2O), Blaine, and 1/Vicat yields four high-R2

subclusters (Table 4). It is interesting that the ACE transform for1/Vicat is close to linear for all subclusters until bassanite is added.This is an example of the kind of persistence of inclusion in high-R2

models coupled with persistence of transformation shape that mightsignal the importance of a variable. Why might a model incorporat-ing these variables be good? Blaine fineness is an indirect measureof the total particle surface area, which should affect the rate of heatrelease. Ferrite occurs as medium- to fine-grained crystals that shouldexhibit greater surface area than some of the other structural mineralphases, and its enthalpy of complete hydration ranks just below that ofC3S. In addition, at 7 days, the reaction coefficient is almost twice thatof C3S (Table 1). Also, as explained, high HOH7 may also promote arapid setting time, associated with a low Vicat.

For illustration purposes only, it is shown how using automatedsoftware can parameterize one of the APACE-suggested models.The second example from Table 4 combines relatively smooth orlinear transforms with a relatively high R2 (.88); the plots of each

Stutzman, Leigh, and Dolly 5

transformed variable are shown in Figure 1. From a large menu ofpotential parametric fits to any given ACE transform, the modelthat seems to give the best combination of high R2, visual goodnessof fit, and is parsimonious in the number of selected parameters isgenerally selected.

If ACE (HOH7) can be modeled, either as a simple linear function(which in practice it is often) or more generally as an explicitlyinvertible function (e.g., log-to-exp or sin-to-arcsin), a completelyexplicitly parameterized model is obtained for HOH7 in regard toferrite, 2CaSO4�(H2O), and (1/Vicat). The three-variable ACE modeldeveloped here, without the explicit parameterization, gives an R2

of .88 on the modeled data. The two-variable Poole model in Equa-tion 2 gives an R2 of .76 (15). However, the model considered herefor illustrative purposes only is just one among dozens of high R2

(R2 > .90) candidate models that APACE has identified.As illustrated in Figures 2 through 4, a simplified cubic fit coarsely

captures the main structure in the ferrite transform, an x-(1/x) quadraticcaptures Blaine structure, and a line captures some of the descent ofthe (1/Vicat) transform. It should be clear that approximate parametricfitting of each ACE transformation will contribute to a diminution ofthe overall model fit’s R2 (.88), as each approximate parametric fitreduces that variable’s contribution to R2.

Substituting the best-fit parameterizations into the APACEcandidate model gives the following:

where

F = ferrite,V = 1/Vicat, andB = Blaine.

ACE HOH ACE ACE ACE7

20 06 0 009

( ) = ( ) + ( ) + ( )= +

F B V

F. . −−( ) +

+ − + ( ) −⎛ −

0 0007

535 0 12 9 10773336

3

6 2

.

.

F

B BB

i⎝⎝⎜

⎞⎠⎟

+ − +( )120 1 3 7V . ( )

TABLE 4 Small Clusters of Variables Providing High R2

and Smooth Transformations

Subcluster and Variable Transform Shape R2

1 .78C3A Smooth, with edgesBlaine Smooth, with cusp1/Vicat Almost linear

2 .88Ferrite Smooth, with drop-offBlaine Smooth, with asymptote1/Vicat Almost linear

3 .90C3A RoughFerrite Line with tentBlaine Smooth1/Vicat Almost linear

4 .86C3A MultimodalFerrite RoughBassanite High–low disconnectBlaine Rough1/Vicat Tent

6 Transportation Research Record 2240

6

0.0

-0.5

-1.0A

CE

dF

ER

RIT

E

-1.5

0-1

-2

AC

Ed

BLA

INE

-3-4

FINENESS OF BLAINE

8

MASS PERCENTAGE OF FERRITE

10 12 14 16 18 3500 4000 4500 5000

(a) (b) A

CE

dIN

VIC

AT

0.01 0.02

INVICAT

0.03 0.04

10

-1-2

-3

(c)

FIGURE 1 ACE transforms for (a) ferrite, (b) Blaine, and (c) 1/Vicat yielding combination of smooth transform and high R2 of.88 (combination chosen to illustrate explicit parameterization).

y = a + bx2 + cx3

r2 = 0.91a = 0.06b = 0.009c = -0.0007

FIGURE 2 Ferrite ACE transform approximately described by simplified cubic function.

Stutzman, Leigh, and Dolly 7

y = a + bx2 + cx2 + d/x

r2 = 0.96a = 535b = -0.12c = -9e-06d = -773336

FIGURE 3 Transformed Blaine fineness that can be described by mixed x-(1/x) quadratic.

y = a + bx

r2 = 0.93a = 1.3b = -120

FIGURE 4 Transformed 1/Vicat resulting in almost linear structure.

If ACE(HOH7) were easily parametrically invertible (approximatelya straight line), the application of the inverse of ACE(HOH7) to theright-hand side of the equation would yield a parametric equation forHOH7 in regard to ferrite, Blaine, and inverse Vicat. In this example,however, the ACE transform of HOH7 is sufficiently irregular so thata simply invertible parameterization is unavailable.

SUMMARY

Quantitative X-ray powder diffraction provides a more complete andmore accurate accounting for the phase compositions of portlandcements compared with the traditional Bogue calculation estimates.The Cement and Concrete Reference Library data set is being contin-ually updated with the mineralogical and fineness data, augmentingthe chemical and physical test data for today’s North Americancements. This data set will provide the basis for exploring perfor-mance properties relative to the mineralogical and fineness character-istics of cement, facilitating the refinement of specification limits fortoday’s cements using the more accurate means for characterization.

This work has not resulted in a simple parametric model for HOH7.Instead, simple conclusions were found concerning the variables andthe data considered that offer general guidance on the modeling ofHOH7. Strongly emphasized is that the one explicitly parameterizedmodel presented is meant to illustrate the power of the statisticalmethodology only.

From the analyses, however, good-fitting models for HOH7 oftenincorporate a number of attributes:

1. Structural mineralogical phase component (C2S preferred),2. Sulfate phase component [2CaSO4�(H2O) preferred],3. Total fineness or particle surface area component (Blaine

preferred), and4. Ferrite in conjunction with Fe2O3, and possibly TiO2 or

C3A or C3Ac, but not C3Ao.

The prevalence of noisy and multistructured ACE plots in thisstudy can possibly have multiple potential causes:

1. Inclusion of all cement types in the data set is inappropriate.In the future, attention should be focused for this kind of modelingon one type of cement, representing possibly different heat evolutionmechanisms.

2. The occasionally sharp transition, or inflection, points in theACE plots could correspond to transition points in the variable beingtransformed by the ACE from a range of values characteristic of onetype of cement to a range of values characteristic of another type.The break in the Blaine transform curve that consistently occurs isat a value corresponding approximately to the differences betweenType III (mean Blaine of 556 m2/kg) and Types I, II, and V (approx-imately 380 m2/kg). This factor suggests that these types should bemodeled separately.

3. Variables included in a model are not inappropriate. That is,variables are included that have no true heat of hydration modelingcontent; and variables that should be included, such as precipitationproducts known to influence heat evolution, have not been selectedby the statistical procedure.

ACKNOWLEDGMENTS

This work was sponsored by AASHTO, in cooperation with FHWA,and was conducted through NCHRP, which is administered by TRBof the National Academies.

8 Transportation Research Record 2240

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The Basic Research and Emerging Technologies Related to Concrete Committeepeer-reviewed this paper.