From 654da49e2dbf0bcca7194cf9f686829d748ea412 Mon Sep 17 00:00:00 2001
From: Scott Mobley <Zott820@yahoo.com>
Date: Sat, 9 Nov 2013 01:39:18 -0800
Subject: [PATCH] Added a Bisection function that works for Biot values of 1,
 which unfortunately is not good enough. Better, but not much better than the
 previous transient model.

---
 js/CookingFormula.js | 81 ++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 79 insertions(+), 2 deletions(-)

diff --git a/js/CookingFormula.js b/js/CookingFormula.js
index 11895a0..45864fd 100644
--- a/js/CookingFormula.js
+++ b/js/CookingFormula.js
@@ -51,7 +51,7 @@ console.log("Simple Radius  " + simpleRadius + " Meters")
 console.log("Complex Radius  " + complexRadius + " Meters")
 }
 
-function LumpedCapacitanceMethod (radiusTotal,radiusInner tempInitial,tempInfini, t) {
+function LumpedCapacitanceMethod (radiusTotal,radiusInner,tempInitial,tempInfini, t) {
 name : "Skin"
 volume = (4/3)*Math.PI*Math.pow(radiusTotal,3) - (4/3)*Math.PI*Math.pow(radiusInner,3); //3D Sphere
 surfaceArea = 4*Math.PI*Math.pow(radiusTotal,2); //3D Sphere
@@ -77,7 +77,7 @@ Qdot = -1*heatConvection*surfaceArea*(tempAtTime-tempInfini) //Heat Transfer Rat
 console.log("The Heat Flux is " + Qdot )
 }
 
-function transientSphere (rPosition,rTotal,tempInitial,tempInfini,t) {
+function transientSphereOneTerm (rPosition,rTotal,tempInitial,tempInfini,t) {
 heatConvection = 12; // W/m2 K Some Reasonable estimate for natural Convection. Change as needed. 5-25
 thermalConduct = 0.412 // W/m K
 alpha = thermalConduct/(density*cp)
@@ -124,4 +124,81 @@ sinPortion= Math.sin(lambdaOne*rPosition/rTotal)/(lambdaOne*rPosition/rTotal);
 expotentialPortion = alphaOne*(1/Math.exp(Math.pow(lambdaOne,2)*Fourier))
 tempAtTimeAndRadius=(sinPortion*expotentialPortion*(tempInitial-tempInfini))+tempInfini
 console.log("The Temperature At radius " + rPosition +" m and time " + t + "  seconds is " + tempAtTimeAndRadius + " C or " + celsiusToFarenheit(tempAtTimeAndRadius) + " F" );
+}
+
+
+
+function findAllRoots (Biot) {
+limit = 11; //Terms to Compute too
+storage = [];
+	for (var k=0; k<=limit; k++) {
+		minK = (k+0.5)*Math.PI;
+		maxK = (k+1)*Math.PI;
+		answer = bisectionMethod(minK,maxK,Biot);
+		if (answer !=null) {
+		storage.push(answer);
+		}
+	}
+console.log(storage)
+return(storage)
+}
+
+
+function bisectionMethod(min,max,Biot) {
+errorTolerance = (1/Math.pow(10,8))
+result = 100 // some large value to ensure the calculation goes through.
+negativeTest =lambdaFormula(min, Biot)*lambdaFormula(max, Biot)
+if (negativeTest <=0 ) {
+	while (Math.abs(result) > errorTolerance) {	
+		lambdaN = (min+max)/2
+		result=lambdaFormula(lambdaN, Biot)
+	
+		if (Math.abs(result) <= errorTolerance) {
+			return (lambdaN); //At Root
+		}
+		else if ((lambdaFormula(min, Biot)*lambdaFormula(lambdaN, Biot))>=0) {
+		min=lambdaN;
+		}
+		else if ((lambdaFormula(max, Biot)*lambdaFormula(lambdaN, Biot))>=0) {
+		max=lambdaN;
+		}
+	}
+	return (lambdaN);
+}
+else
+{
+//console.log("No Bracketed Root " + negativeTest)
+}
+}
+
+function lambdaFormula(lambdaN, Biot) { 
+result = 1-lambdaN*(1/Math.tan(lambdaN))-Biot;
+return(result)
+}
+
+
+function transientSphereSeries (rPosition,rTotal,tempInitial,tempInfini,t) {
+heatConvection = 6000; // W/m2 K Some Reasonable estimate for natural Convection. Change as needed. 5-25
+thermalConduct = 20 // W/m K
+//alpha = thermalConduct/(density*cp)
+alpha = 6.66 * (1/Math.pow(10,6))
+console.log("Alpha is " + alpha)
+sum=0;
+
+Fourier = (alpha*t)/Math.pow(rTotal,2)
+console.log("Fourier is " +  Fourier)
+biotNum = heatConvection * rTotal/thermalConduct
+console.log("The Biot Value is " + biotNum)
+
+lambdaTerms = findAllRoots(biotNum)
+	for (var i = 0; i<lambdaTerms.length; i++) {
+		lambdaN = lambdaTerms[i]
+
+		sinPortion= Math.sin(lambdaN*rPosition/rTotal)/(lambdaN*rPosition/rTotal);
+		exponentialPortion = (1/Math.exp(Math.pow(lambdaN,2)*Fourier))
+		frontCoefficientPortion = 4*(Math.sin(lambdaN)-(lambdaN*Math.cos(lambdaN)))/ (2*lambdaN-Math.sin(2*lambdaN))
+		sum = frontCoefficientPortion*exponentialPortion*sinPortion + sum
+}
+tempAtTimeAndRadius=(sum*(tempInitial-tempInfini))+tempInfini
+console.log("The Temperature at radius " + rPosition + " m and time " + t + "  seconds is " + tempAtTimeAndRadius + " C or " + celsiusToFarenheit(tempAtTimeAndRadius) + " F");
 }
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