Laminate T and %RH Curl Calculator

//One universal basic required here to get things going once loaded
window.onload = function () {
    //restoreDefaultValues(); //Un-comment this if you want to start with defaults
    Main();
};

//Main() is hard wired as THE place to start calculating when inputs change
//It does no calculations itself, it merely sets them up, sends off variables, gets results and, if necessary, plots them.
function Main() {
    //Save settings every time you calculate, so they're always ready on a reload
    saveSettings();

    //Send all the inputs as a structured object
    //If you need to convert to, say, SI units, do it here!
    const inputs = {
        Trise1:sliders.SlideTrise1.value,
        CTE1:sliders.SlideCTE1.value,
        Thick1: sliders.Slideh1.value /1e6, //μm to m
        Mod1: sliders.SlideM1.value *1e9, //GPa to Pa
        Hrise1:sliders.SlideHrise1.value,
        CHE1:sliders.SlideCHE1.value,
        Trise2:sliders.SlideTrise2.value,
        CTE2:sliders.SlideCTE2.value,
        Thick2: sliders.Slideh2.value /1e6, //μm to m
        Mod2: sliders.SlideM2.value *1e9, //GPa to Pa
        Hrise2:sliders.SlideHrise2.value,
        CHE2:sliders.SlideCHE2.value,
    };

    
    //Send inputs off to CalcIt where the names are instantly available
    //Get all the resonses as an object, result
    const result = CalcIt(inputs);

    document.getElementById('S1').value = result.S1;
    document.getElementById('S2').value = result.S2;
    document.getElementById('R').value = result.R;

    if (result.plots) {
        for (let i = 0; i < result.plots.length; i++) {
            plotIt(result.plots[i], result.canvas[i]);
        }
    }

    //You might have some other stuff to do here, but for most apps that's it for Main!
}

//Here's the app calculation
//The inputs are just the names provided - their order in the curly brackets is unimportant!
//By convention the input values are provided with the correct units within Main
function CalcIt({ Trise1, CTE1, Hrise1, CHE1, Thick1, Mod1,Trise2, CTE2, Hrise2, CHE2, Thick2, Mod2}) {
    const Stretch1=(Trise1*CTE1+Hrise1*CHE1)/1000000 //m
    const Stretch2=(Trise2*CTE2+Hrise2*CHE2)/1000000
    
    let TheCurl=0
    const D1=Mod1*Thick1*Thick1*Thick1/12
    const D2=Mod2*Thick2*Thick2*Thick2/12
    const Top=2*(D1+D2)
    const H12=Thick1+Thick2
    const tmp=(Stretch1-Stretch2)/(1/(Mod1*Thick1)+1/(Mod2*Thick2)+H12*H12/(2*Top))
    if (tmp !=0) TheCurl=1000*Top/(H12*tmp)
    if (TheCurl>9999 || TheCurl<-9999) TheCurl=0

        
    //Now we return everything - text boxes, plot and the name of the canvas, which is 'canvas' for a single plot
    return {
        S1:(100*Stretch1).toPrecision(3),
        S2:(100*Stretch2).toPrecision(3),
       R:TheCurl.toFixed(0),
    };
}

                        

What does this app do?

This app is a complement to the Lamination Curl Calculator. The LCC finds out the effects of the differential tensions going into a laminating nip - depending on the thickness and modulus of the two webs, labelled 1 and 2. The THC calculates the effects that differential temperature (e.g. one web being hot, the other cold) or differential humidity (one web "wet" the other "dry") might have on curl. Strain.

A web will stretch under increased temperature or increased humidity. The % Strain (the scientific name for degree of stretching) gets larger for:

• Higher temperature
• Higher coefficient of thermal expansion
• Higher humidity
• Higher coefficient of hygroscopic expansion

Expansion and Wrinkles

Although people are familiar with thermal expansion they are generally unfamiliar or, perhaps, unaware of the hydgroscopic expansion. For convenience the units are given in µm/m/°C or µ/m/%RH. Suppose we have a web 1000mm wide and a temperature rise of 10°C. If the expansion coefficient is 15 (typical for PET film) then the web will be 150µm wider, i.e. 1000.15mm wide. For PET the hygroscopic expansion coefficient happens to be the same, 15µm/m/%RH. So if the web happened to be at 0%RH (e.g. just after a hot oven) and then was exposed to 75%RH it would expand by 75*15µm which is 1.125mm, i.e. to 1001.13mm. For PMMA where both values are in the 100 range, the new widths are 1001mm and 1007.5mm.

The curl induced by these effects can be significant. But it's worth looking at another important, and little-known fact.

Suppose your PC web (thermal coefficient=30) is 2000mm wide and goes onto a heated roller which is 50°C hotter. The web will expand by 30*2*50=3000µm=3mm. Suppose the web tension stops the web from sliding sideways on the roller. The expansion then forces it vertically, creating a wrinkle. What is the height of that wrinkle? Intuitively, because it is only 3mm in 2000mm the wrinkle will be very small. The reality is shockingly different. Imagine (for simplicity) that the web distorts into a triangle. The half width of the triangle is 1000mm and the length of each of the two sides is 1001.5mm. By Pythagoras, the height of the triangle is given by the square root of (1001.5² - 1000²) which is 54.8mm. Yes! The wrinkle will be 5.4cm high! Everyone thinks that there is an error in the calculation, but if you check it out you'll find it's correct. So webs hitting a hot roller can create very severe wrinkles.

Curl

If the two %Strain are not the same then the laminate will tend to curl once it is out of the nip. The curl can be calculated in terms of its radius. The calculation is surprisingly complex and sometimes it takes a while to figure out why changing one of the inputs (such as modulus) produces the calculated result. This is because the curl depends not only on the differential stretch but also on the ability of the laminate as a whole to be bent - which depends on the combined modulus and thickness of the two webs.

A positive value of curl means that it curls with Laminate1 inside and a negative value means that it curls with Laminate2 on the inside. This applies if the webs were hot/wet before lamination. If the webs were neutral before lamination and then temperature/humidity changes, the curl will be in the opposite direction (a push from expansion rather than a pull from contraction).

A smaller radius means a tighter curl which may mean trouble further into production - when the roll is cut into sheets or even when the roll is in storage as that stretch likes to even itself out. Note that in these calculations, any curl with a radius >9999mm is shown as "0", meaning there is no curl.

Typical Moduli and Expansion Coefficients

The modulus of a web represents the strength of the web under tension and is reported in GPa (Metric) and Mpsi(US). The CTE (Coefficients of Thermal Expansion) are hard to track down and are often approximate. The CHE (Coefficients of Hygroscopic Expansion) are even less known, so where values are shown they are mostly informed guesses. Values for paper/board are all set to a nominal 100. The coefficients are all expressed in terms of °C.