CFD%20Solution%20of%20Ice%20Melting%20Problem%20on%20Transmission%20Lines%20in%20Cold%20Climate%20

1
CFD Solution of Ice Melting Problem on
Transmission Lines in Cold Climate A Parallel
Computing Approach

• Prof. P.N. Shivakumar
• Department of Mathematics
• (http//home.cc.umanitoba.ca/shivaku/)
• Dr. Ruppa K. Thulasriam
• Department of Computer Science
• (www.cs.umanitoba.ca/tulsi)
• University of Manitoba
• Winnipeg, Manitoba, Canada

2
Outline

• Problem statement
• Related work
• Definitions
• Implementation details
• Experimental platform
• Results and discussions
• Ice Storm 98

3
Problem Statement

• During winter ice forms on electrical
  transmission lines, and due to the wind and
  increase in its weight, the lines with the
  electrical poles are likely to be pulled down
  causing heavy casualties.
• Every winter, power lines and communication
  towers collapse in freezing rain storms.
• The glaze ice load on trees,wires,and structural
  members,combined with wind acting on the
  increased projected area, causes these failures.
• Repair costs after a severe storm can be hundreds
  of millions f dollars, and electrical outages may
  deprive residents and businesses of
  heat,water,and power for extended periods.
• This was one of the major problems in the Ice
  Storm 98 of Montreal.
• A part of the ice melts because of the heat from
  transmission itself.

4
Background

• Mathematical models for this study have been
  developed before One model is developed by
  S.Yu.Sadov and P.N.ShivaKumar.
• Their model was separated into four material
  regions and heat transfer equations were
  developed with appropriate Boundary Conditions in
  Cartesian coordinates.
• Finally, a 2 dimensional cross-sectional model
  for melting ice on an electrical transmission
  line due to applied fault current was developed.

5
Our Model and the General Factors

• We do not consider the effects of gravity,
• convection and also the exterior climatic
  conditions.
• Hence, the regions are shown in concentric
  circles
• rather than in ellipses.

6
Governing Equations
The governing are equations are
Here -- metal (electric wire)
-- water melted off ice -- ice
-- atmosphere

• The Boundary Conditions are
• Wire-Water Boundary
• T and heat flux are continuous
• b)

Water-Ice a) b) Stefan condition
(melting temperature)
7

• where is the velocity of ice melting in the
  normal direction given by
• where is a position of a point on
  water-ice boundary

8
Finite-Difference Equations
The above equations are used to calculate the
values of temperature in different regions
9
Variables Used in the Equations
1-wire 2-water 3-ice

• C1, C2 and C3 are the specific heat values.
• T is the Temperature
• n Number of time steps
• ?t 0.0001
• ? 0.00729727    
• Grid size(?x) (?y) 0.002
• i, j are the axes, with i pointing upwards
• Q1 internal heat source in the wire a constant
• Q2 Q3 0

10
Implementation Details

• Based on the mathematical model, Parabolic PDEs
  were subject to central differencing and
  transformed into Finite-Difference Equations
  (FDEs).
• Cartesian coordinates were used and grid size of
  1000 x 1000 was super-imposed onto the problem
  domain, such that the boundary of wire does not
  change with respect to time and each node of the
  grid is considered to be a computational node.
• The values of temperature for different regions
  are calculated by solving these FDEs at each
  processor, until steady state is reached as t ?
  infinity
• for example in the current project the time steps
  n ?106.

11
Implementation Details

• Since the current model is a set of concentric
  circles, only one quadrant is considered for
  computation and analysis due to symmetric nature
  of the problem.
• This quadrant is distributed among four
  processors, each having 250 computational grid
  points in i and j direction individually.
• Each processor computes the values of temperature
  in a SIMD (Single Instruction Multiple Data)
  manner.
• Since there is a constant heat source in the wire
  (Q1), wire equations are not computed explicitly.

12
Our Model
P1
P2
P3
P4
4 x 4 grid
Processors (P)
13
Focus

• Focus of the current aspect of the overall
  project is addressing computational issues such
  as partitioning, load balancing and
  communication/synchronization as recaptured here
• Decomposition or Domain Decomposition is the
  process of integrating the equations on
  rectangular domains, such that data can be
  decomposed uniformly by assigning rectangular sub
  domains to each processor.
• Load Balancing is the process of ensuring equal
  load among all processors in the domain, so that
  each processor has enough work and they finish
  execution of a task at the same time.
• Communication and Synchronization between the
  processors are necessary as each processor
  performs only a segment of the main task,
  individually.

14
Experimental Platform
Beowulf is a class of parallel workstations,
developed to evaluate and characterize the design
space of single user dedicated systems in price
and performance.
Cluster of Workstations (COW)
Beowulf Workstation
15
Several steps in the current study

• Fluid Dynamics/Partial Differential Equations
• Basic ideas of fluid mechanics and differential
  equations
• Physics of heat transfer issues and boundary
  conditions
• Implications of changing a boundary condition is
  quite large. Importance of BC as well as how
  these BC affect the physics of the problem is
  addressed in this step
• Finite-Difference method
• Though simple this technique has its own
  intricacies for implementation to capture the
  physics.
• Implementation issues
• The machine characteristics and assigning the
  evenly distributed load on individual processors
  were addressed.

16
Discussions

• The Finite Difference code is producing
  acceptable results in the coarser grid, which is
  computationally less challenging. Physics is not
  fully captured in the coarser grid.
• A finer computational grid takes more than one
  million time steps, for steady state solutions.
• The code is yet to be tried on the Beowulf
  cluster.
• The overall execution time is expected to reduce
  significantly, when compared to the sequential
  code.
• More results are being generated
• Fine tuning the code to capture the physics is
  going to take much longer time.

17
Parallel Implementation of the Ice Melting
problem
18
Ice Storm 98
Ice Rolling Techniques
19
Ice Storm 98
20

• Thank You