Dimensions and Other Considerations Before We Solve the Matrix

Yesterday’s post showed how some of the constants and parameters are initialized or calculated. Today we’ll describe a few more.

This exercise defines a cross-section of a steel billet that is 8 inches by 8 inches and divided into five nodes high and five nodes wide. I found it convenient to assume the surface nodes were only half the height or width of the middle nodes, and to assume the center of surface nodes is on the outer edge or corner of the node, or in the middle of the node for central nodes. This means that the centers are always equidistant both across and up. The surface nodes also receive conducted heat and radiated heat at the same location. Nodes on the corners have one quarter the cross-sectional area of the fully central nodes, and nodes on the surface but not on the corner have one half the area of the fully central nodes.

    var timeStep = 40.0;  //seconds
    
    var nodeWidth = 2.0;   //inches
    var nodeHeight = 2.0;  //inches
    
    var pieceCount = 50;
    
    var arrayTpieces = new Array;
    var arrayT = new Array;
    for (var i=0; i< pieceCount; i++) {
      arrayTpieces[i] = new Array;
      for (var j=0; j<nodesWide; j++) {
        arrayTpieces[i][j] = new Array;
        arrayT[j] = new Array;
        for (var k=0; k<nodesHigh; k++) {
          arrayTpieces[i][j][k] = 70.0;
          arrayT[j][k] = 0.0;
        }
      }
    }
    var arrayTfce = new Array;
    for (var i=0; i< pieceCount; i++) {
      arrayTfce[i] = 1800.0;
    }      
    
    var arrayRho = new Array;
    var arrayCp = new Array;
    var arrayKup = new Array;
    var arrayKright = new Array;
    var arrayDup = new Array;
    var arrayDright = new Array;
    var arrayArea = new Array;
    var arraySurfLength = new Array;
    var arrayVF = new Array;
    var arrayRad = new Array;
    var tempLength = 0.0;
    for (var i=0; i<nodesWide; i++) {
      arrayRho[i] = new Array;
      arrayCp[i] = new Array;
      arrayKup[i] = new Array;
      arrayKright[i] = new Array;
      arrayDup[i] = new Array;
      arrayDright[i] = new Array;
      arrayArea[i] = new Array;
      arraySurfLength[i] = new Array;
      arrayVF[i] = new Array;
      arrayRad[i] = new Array;
      for (j=0; j<nodesHigh; j++) {
        arrayRho[i][j] = 0.0;
        arrayCp[i][j] = 0.0;
        arrayKup[i][j] = 0.0;
        arrayKright[i][j] = 0.0;
        if (j < nodesHigh-1) {  
          arrayDup[i][j] = nodeHeight;
        } else {  
          arrayDup[i][j] = 0.0;
        }
        if (i < nodesWide-1) {  
          arrayDright[i][j] = nodeWidth;
        } else {  
          arrayDright[i][j] = 0.0;
        }
        arrayArea[i][j] = nodeHeight * nodeWidth;
        if ((i == 0) || (i == nodesWide-1)) {
          arrayArea[i][j] *= 0.5;
        }
        if ((j == 0) || (j == nodesHigh-1)) {
          arrayArea[i][j] *= 0.5;
        }
        arraySurfLength[i][j] = 0.0;
        arrayVF[i][j] = 0.0;
        if (j == nodesHigh-1) {
          if ((i == 0) || (i == nodesWide-1)) {
            arraySurfLength[i][j] += nodeWidth * 0.5;
          } else {
            arraySurfLength[i][j] += nodeWidth;
          }
          arrayVF[i][j] = 1.0 * arraySurfLength[i][j];
        }
        tempLength = 0.0;
        if ((i == 0) || (i == nodesWide-1)) {
          if ((j == 0) || (j == nodesHigh-1)) {
            arraySurfLength[i][j] += nodeHeight * 0.5;
            tempLength = nodeHeight * 0.5;
          } else {
            arraySurfLength[i][j] += nodeHeight;
            tempLength = nodeHeight;
          }
          if (j == 0) {arrayVF[i][j] += 0.26 * tempLength;}
          else if (j == 1) {arrayVF[i][j] += 0.41 * tempLength;}
          else if (j == 2) {arrayVF[i][j] += 0.55 * tempLength;}
          else if (j == 3) {arrayVF[i][j] += 0.67 * tempLength;}
          else if (j == 4) {arrayVF[i][j] += 0.78 * tempLength;}
          else if (j == 5) {arrayVF[i][j] += 0.87 * tempLength;}
          else if (j == 6) {arrayVF[i][j] += 0.95 * tempLength;}
        }
        if (arraySurfLength[i][j] > 0.0) {
          arrayVF[i][j] /= arraySurfLength[i][j];
        }
        arrayRad[i][j] = 0.0;        
      } 
    }
    
    function calcParams(index) {  //index says which piece
      var i; var j; var wide; var high;
      var a; var d; var k1; var k2;

      wide = nodesWide;
      high = nodesHigh;

      for (i = 0; i < wide; i++) {
        for (j = 0; j < high; j++) {
          //nodal energy
          arrayRho[i][j] = coeffRho * arrayArea[i][j] / timeStep;
          var t = arrayTpieces[index][i][j];
          arrayCp[i][j] = findCpofT(t);

          //conduction up
          //if (arrayDup[i][j] > 0.0) {
          if (j < high-1) {
            a = nodeWidth;
            if ((i == 0) || (i == wide-1)) {
              a *= 0.5;
            }
            d = arrayDup[i][j];  //nodeHeight
            k1 = findKofT(arrayTarrayTpieces[index][i][j]);
            k2 = findKofT(arrayTarrayTpieces[index][i][j+1]);
            arrayKup[i][j] = (k1 + k2) * 0.5 * a / d;
          } else {
            arrayKup[i][j] = 0.0;
          }

          //conduction to right
          //if (arrayDright[i][j] > 0.0) {
          if (i < wide-1) {
            a = nodeHeight;
            if ((j == 0) || (j == high-1)) {
              a *= 0.5;
            }
            d = arrayDright[i][j];  //nodeWidth
            k1 = findKofT(arrayTarrayTpieces[index][i][j]);
            k2 = findKofT(arrayTarrayTpieces[index][i+1][j]);
            arrayKright[i][j] = (k1 + k2) * 0.5 * a / d;
          } else {
            arrayKright[i][j] = 0.0;
          }

          //radiation
          var z = coeffRad * arrayVF[i][j] * arraySurfLength[i][j];
          arrayRad[i][j] = z;
          var z = k1 + k2;
        }
      }
    }     //calcParams

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

var timeStep = 40.0; //seconds

var nodeWidth = 2.0; //inches

var nodeHeight = 2.0; //inches

var pieceCount = 50;

var arrayTpieces = new Array;

var arrayT = new Array;

for (var i=0; i< pieceCount; i++) {

arrayTpieces[i] = new Array;

for (var j=0; j<nodesWide; j++) {

arrayTpieces[i][j] = new Array;

arrayT[j] = new Array;

for (var k=0; k<nodesHigh; k++) {

arrayTpieces[i][j][k] = 70.0;

arrayT[j][k] = 0.0;

}

var arrayTfce = new Array;

for (var i=0; i< pieceCount; i++) {

arrayTfce[i] = 1800.0;

}

var arrayRho = new Array;

var arrayCp = new Array;

var arrayKup = new Array;

var arrayKright = new Array;

var arrayDup = new Array;

var arrayDright = new Array;

var arrayArea = new Array;

var arraySurfLength = new Array;

var arrayVF = new Array;

var arrayRad = new Array;

var tempLength = 0.0;

for (var i=0; i<nodesWide; i++) {

arrayRho[i] = new Array;

arrayCp[i] = new Array;

arrayKup[i] = new Array;

arrayKright[i] = new Array;

arrayDup[i] = new Array;

arrayDright[i] = new Array;

arrayArea[i] = new Array;

arraySurfLength[i] = new Array;

arrayVF[i] = new Array;

arrayRad[i] = new Array;

for (j=0; j<nodesHigh; j++) {

arrayRho[i][j] = 0.0;

arrayCp[i][j] = 0.0;

arrayKup[i][j] = 0.0;

arrayKright[i][j] = 0.0;

if (j < nodesHigh-1) {

arrayDup[i][j] = nodeHeight;

} else {

arrayDup[i][j] = 0.0;

}

if (i < nodesWide-1) {

arrayDright[i][j] = nodeWidth;

} else {

arrayDright[i][j] = 0.0;

}

arrayArea[i][j] = nodeHeight * nodeWidth;

if ((i == 0) || (i == nodesWide-1)) {

arrayArea[i][j] *= 0.5;

}

if ((j == 0) || (j == nodesHigh-1)) {

arrayArea[i][j] *= 0.5;

}

arraySurfLength[i][j] = 0.0;

arrayVF[i][j] = 0.0;

if (j == nodesHigh-1) {

if ((i == 0) || (i == nodesWide-1)) {

arraySurfLength[i][j] += nodeWidth * 0.5;

} else {

arraySurfLength[i][j] += nodeWidth;

}

arrayVF[i][j] = 1.0 * arraySurfLength[i][j];

}

tempLength = 0.0;

if ((i == 0) || (i == nodesWide-1)) {

if ((j == 0) || (j == nodesHigh-1)) {

arraySurfLength[i][j] += nodeHeight * 0.5;

tempLength = nodeHeight * 0.5;

} else {

arraySurfLength[i][j] += nodeHeight;

tempLength = nodeHeight;

}

if (j == 0) {arrayVF[i][j] += 0.26 * tempLength;}

else if (j == 1) {arrayVF[i][j] += 0.41 * tempLength;}

else if (j == 2) {arrayVF[i][j] += 0.55 * tempLength;}

else if (j == 3) {arrayVF[i][j] += 0.67 * tempLength;}

else if (j == 4) {arrayVF[i][j] += 0.78 * tempLength;}

else if (j == 5) {arrayVF[i][j] += 0.87 * tempLength;}

else if (j == 6) {arrayVF[i][j] += 0.95 * tempLength;}

}

if (arraySurfLength[i][j] > 0.0) {

arrayVF[i][j] /= arraySurfLength[i][j];

}

arrayRad[i][j] = 0.0;

}

function calcParams(index) { //index says which piece

var i; var j; var wide; var high;

var a; var d; var k1; var k2;

wide = nodesWide;

high = nodesHigh;

for (i = 0; i < wide; i++) {

for (j = 0; j < high; j++) {

//nodal energy

arrayRho[i][j] = coeffRho * arrayArea[i][j] / timeStep;

var t = arrayTpieces[index][i][j];

arrayCp[i][j] = findCpofT(t);

//conduction up

//if (arrayDup[i][j] > 0.0) {

if (j < high-1) {

a = nodeWidth;

if ((i == 0) || (i == wide-1)) {

a *= 0.5;

}

d = arrayDup[i][j]; //nodeHeight

k1 = findKofT(arrayTarrayTpieces[index][i][j]);

k2 = findKofT(arrayTarrayTpieces[index][i][j+1]);

arrayKup[i][j] = (k1 + k2) * 0.5 * a / d;

} else {

arrayKup[i][j] = 0.0;

}

//conduction to right

//if (arrayDright[i][j] > 0.0) {

if (i < wide-1) {

a = nodeHeight;

if ((j == 0) || (j == high-1)) {

a *= 0.5;

}

d = arrayDright[i][j]; //nodeWidth

k1 = findKofT(arrayTarrayTpieces[index][i][j]);

k2 = findKofT(arrayTarrayTpieces[index][i+1][j]);

arrayKright[i][j] = (k1 + k2) * 0.5 * a / d;

} else {

arrayKright[i][j] = 0.0;

}

//radiation

var z = coeffRad * arrayVF[i][j] * arraySurfLength[i][j];

arrayRad[i][j] = z;

var z = k1 + k2;

}

} //calcParams

Dimensions and Other Considerations Before We Solve the Matrix

Leave a Reply Cancel reply

Recent Posts

Recent Comments

Categories

Meta