Laplace transform isotherm .pdf - University of Hertfordshire ...

More documents

Recommendations

Info

I study mathematics with the Open University when my knowledge was found to be lacking during an interview for an astronomy course which I was hoping to join in 1978. I think neither he nor I would have imagined then that I would have come so far and been awarded a PhD for my work in mathematics. ii
Abstract There are many different methods available for the solution of the heat equation and the choice of which to use is dependent upon the nature of the problem and the specific regions of the domain where the temperature is required. In the case of melting or freezing problems it is usual for the position of the boundary, at which change of physical state (phase change) occurs, to be of greater interest than the temperature at particular points. Again there are several solution methods enabling the tracking of the moving interface between the physical states of the material. For this work we begin with the isotherm migration method, which first appeared in the 1970s but is less frequently cited now. We first solve problems in one dimension with no phase change using the isotherm migration method, which is in itself new work, since all references we have found al- lude to it as a tool for the solution of phase change problems. We test the method using a variety of examples to explore the difficulties and challenges it produces, and we find it to be robust and tolerant of errors. We then combine it with the Laplace transform method, a well-established technique for solving ordinary and partial differential equations, in which the number of independent variables is reduced by one. The solution is then transformed back into the time domain using a suitable numerical process. The Laplace transform isotherm migration method is a new process, not mentioned previously to our knowledge, and it produces results which are comparable with the isotherm migration method. The new process is applied to one-dimensional phase change problems,where we find that due to the mathematics at the phase change boundary, we are required to make a modification to the usual manner of operating the Laplace transform. This is novel as far as we are aware. Our method is applied to a variety of problems and produces satisfactory iii
Page 1 and 2: Aspects of the Laplace transform is
Page 3: Acknowledgements It is my greatest
Page 7 and 8: Contents 1 Introduction 1 2 The con
Page 9 and 10: 4.7.1 Contribution . . . . . . . .
Page 11 and 12: 9.2.3 To extend the use of Laplace
Page 13 and 14: 3.11 The positions of isotherm 8 fo
Page 15 and 16: 7.10 Diagram showing the position o
Page 17 and 18: List of Tables 3.1 The position of
Page 19 and 20: 6.1 The positions of the freezing f
Page 21 and 22: u temperature ũ dimensionless u co
Page 23 and 24: Chapter 1 Introduction We begin our
Page 25 and 26: system. In 1998 the Department was
Page 27 and 28: at a point per unit area. If we hav
Page 29 and 30: where G(x,y) is some function indep
Page 31 and 32: condition may vary from one point o
Page 33 and 34: We can find the solution in differe
Page 35 and 36: this is the initial condition. The
Page 37 and 38: Figure 2.2: Typical grid for the fi
Page 39 and 40: fundamental solution which satisifi
Page 41 and 42: tending the technique to non-homoge
Page 43 and 44: 2.2.7 The method of separation of v
Page 45 and 46: to a set of linear equations of the
Page 47 and 48: labelled Qk, where k = 1,2,...,n
Page 49 and 50: in the case of melting ice (Crank a
Page 51 and 52: and the moving boundary differs. In
Page 53 and 54: 2.3.5 Fixed-domain methods In certa
Page 55 and 56:
2.4 Summary of Chapter 2 In this ch
Page 57 and 58:
e easily evaluated, topological cha
Page 59 and 60:
We wish to write the heat flow equa
Page 61 and 62:
Equations (3.8), (3.9), (3.10) and
Page 63 and 64:
In this example we consider the cas
Page 65 and 66:
and ˜x 0.8 0.7 0.6 0.5 0.4 0.3 0.2
Page 67 and 68:
˜x 0.5 0.4 0.3 0.2 0.1 Isotherm 0
Page 69 and 70:
w 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0
Page 71 and 72:
ũ 10 9 8 7 6 5 0 0.2 0.4 0.6 0.8 1
Page 73 and 74:
Table 3.5: Effect of errors in star
Page 75 and 76:
Table 3.7: Effect of errors in star
Page 77 and 78:
˜x 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Page 79 and 80:
it is much more straightforward to
Page 81 and 82:
Chapter 4 The Laplace transform iso
Page 83 and 84:
available in tables. We do have an
Page 85 and 86:
N� A = − 3. Gauss quadrature, w
Page 87 and 88:
The value of M is chosen by the use
Page 89 and 90:
are found where wj are the Stehfest
Page 91 and 92:
where h is the difference in the va
Page 93 and 94:
Table 4.3: Comparison of the result
Page 95 and 96:
α = 1 + u. For the steady-state so
Page 97 and 98:
Table 4.5: The positions of the iso
Page 99 and 100:
To start this problem, we use an ac
Page 101 and 102:
4.7 Summary of Chapter 4 In this ch
Page 103 and 104:
differences, and we shall show how
Page 105 and 106:
˜x = 0, ũ = ũL, ˜t � 0 As in
Page 107 and 108:
However, we find that doing this ca
Page 109 and 110:
front and isotherm 8 respectively u
Page 111 and 112:
straightforward and might require o
Page 113 and 114:
freezing front is given by x0 = 2φ
Page 115 and 116:
x 1 0.8 0.6 0.4 0.2 Freezing front
Page 117 and 118:
ũ(˜x, ˜t) = 0, ˜x = ˜x0, ˜t
Page 119 and 120:
so that when X = 0 Eliminating ũw
Page 121 and 122:
˜x 1 0.8 0.6 0.4 0.2 Steady Convec
Page 123 and 124:
˜x 1 0.8 0.6 0.4 0.2 Steady Convec
Page 125 and 126:
Page 127 and 128:
∂u ∂y = 0 y = 0 0 � x � 1 t
Page 129 and 130:
6.3 The finite difference form We e
Page 131 and 132:
Results We see from table 6.1 that
Page 133 and 134:
where t0 is the initial time at whi
Page 135 and 136:
x 1 0.8 0.6 0.4 0.2 Freezing front
Page 137 and 138:
Page 139 and 140:
grid line x = xM = 1 for all t > 0.
Page 141 and 142:
7.2 Solidification of a square pris
Page 143 and 144:
which is our previous equation (6.8
Page 145 and 146:
Figure 7.2: Diagram showing possibl
Page 147 and 148:
Poots (1962) defines a function �
Page 149 and 150:
y 1 0.8 0.6 0.4 0.2 t=0.05 t=0.1 t=
Page 151 and 152:
so that we have ¯y (n) = y (t0) λ
Page 153 and 154:
y 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0
Page 155 and 156:
y 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0
Page 157 and 158:
Table 7.3: Values of the y co-ordin
Page 159 and 160:
y 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6
Page 161 and 162:
y 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6
Page 163 and 164:
y 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.
Page 165 and 166:
espaced using a given algorithm. We
Page 167 and 168:
Chapter 8 The use of multiple proce
Page 169 and 170:
In considering parallel computing m
Page 171 and 172:
The issue with shared memory system
Page 173 and 174:
Figure 8.3: Diagram showing a distr
Page 175 and 176:
are several MPI implementations ava
Page 177 and 178:
communication in the program adds e
Page 179 and 180:
Sp 4 3.5 3 2.5 2 1.5 1 1 1.5 2 2.5
Page 181 and 182:
work. Because the calculation invol
Page 183 and 184:
Table 8.3: Speed-up when allocating
Page 185 and 186:
not occur when only one process is
Page 187 and 188:
Therefore, as in examples 8.1 and 8
Page 189 and 190:
upon which we focus. We mention the
Page 191 and 192:
heating of a rod at zero degrees, n
Page 193 and 194:
described by an equation involving
Page 195 and 196:
grid points. An example, described
Page 197 and 198:
simple. 9.2 Research objectives Our
Page 199 and 200:
Laplace transform, new complexities
Page 201 and 202:
orthogonal flow lines. To implement
Page 203 and 204:
Crank J (1957) Two methods for the
Page 205 and 206:
Davies A J and Mushtaq J (1998) Par
Page 207 and 208:
Kreider D L, Kuller R G, Ostberg D
Page 209 and 210:
Price R H and Slack M R (1954) The
show all

Laplace transform isotherm .pdf - University of Hertfordshire ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?