che/me 109 heat transfer in electronics lecture 11 – one dimensional numerical models
Post on 21-Dec-2015
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NUMERICAL METHOD FUNDAMENTALS
• NUMERICAL METHODS FUNDAMENTALS• NUMERICAL METHODS PROVIDE AN ALTERNATIVE
TO ANALYTICAL MODELS• ANALYTICAL MODELS PROVIDE THE EXACT
SOLUTION AND REPRESENT A LIMIT• ANALYTICAL MODELS ARE LIMITED TO SIMPLE
SYSTEMS. • CYLINDERS, SPHERES, PLANE WALLS• CONSTANT PROPERTIES THROUGH THE SYSTEM• NUMERICAL MODELS PROVIDE APPROXIMATIONS• APPROXIMATIONS MAY BE ALL THAT IS AVAILABLE
FOR COMPLEX SYSTEMS• COMPUTERS FACILITATE THE USE OF NUMERICAL
MODELS; SOMETIMES TO THE POINT OF REPLACING ANALYTICAL SOLUTIONS
EXAMPLE USING NUMERICAL METHODS
• NEWTON-RAPHSON PROVIDES AN EXAMPLE TO MODEL A COMPLEX SYSTEM
• NEWTON-RAPHSON EXAMPLE• GIVEN: EQUATION OF THE FORM
NEWTON-RAPHSON EXAMPLE
• WANTED: FIND THE ROOTS OF THIS EQUATION• BASIS: USE ANALYTICAL OR NUMERICAL
METHODS• SOLUTION: A PLOT OF THIS EQUATION HAS THE
FORM
NEWTON-RAPHSON EXAMPLE
• SOLUTION: USING NEWTON-RAPHSON TO OBTAIN THE ROOTS, START BY EVALUATING THE FUNCTION AT x1= 1. THE VALUE OBTAINED IS y1 = 22.348.
• A SECOND CALCULATION IS COMPLETED AT x2 = 1.05 AND FROM
• THIS THE RESULT IS y2 = 17.099. USING THESE VALUES TO CALCULATE THE DERIVATIVE NUMERICALLY THE NEXT VALUE OF x CAN BE ESTIMATED:
• AND x3 = 1.213. USING THIS VALUE, THE RESULT IS y3 = 4.716. THIS VALUE IS STILL NOT ZERO, SO THE PROCESS IS REPEATED. RESULTS ARE SHOWN IN THE FOLLOWING TABLE.
NEWTON-RAPHSON EXAMPLE
FUNCTION HAS A STEEP SLOPE AND IS SENSITIVE TO SMALL CHANGES IN x, BUT THE METHOD STILL WORKS. TAKING ADDITIONAL VALUES COULD REDUCE THE VALUE OF y TO A TARGET LEVEL.
FORMULATION OF NUMERICAL MODELS
• DIRECT AND ITERATIVE OPTIONS EXIST FOR NUMERICAL MODELS
• DIRECT MODELS SET UP A MATRIX OF n LINEAR EQUATIONS AND n UNKNOWS
• FOR HEAT TRANSFER, THE EQUATIONS ARE TYPICALLY HEAT BALANCES
• ROOTS OF THESE ARE OBTAINED BY SOME REGRESSION TECHNIQUE
ITERATIVE MODELS • SET UP A SERIES OF RELATED
EQUATIONS • INITIAL VALUES ARE
ESTABLISHED AND THEN THE EQUATIONS ARE ITERATED UNTIL THEY REACH A STABLE “RELAXED” SOLUTION
• THIS METHOD CAN BE APPLIED TO EITHER STEADY-STATE OR TRANSIENT SYSTEMS.
• BASIC APPROACH IS TO DIVIDE THE SYSTEM INTO A SERIES OF SUBSYSTEMS.
• SYSTEMS ARE SMALL ENOUGH TO ALLOW USE OF LINEAR RELATIONSHIPS
• SUBSYSTEMS ARE REFERRED TO AS NODES
ONE DIMENSIONAL STEADY STATE MODELS
• THE GENERAL FORM FOR THE HEAT TRANSFER MODEL FOR A SYSTEM IS:
• FOR STEADY STATE, THE LAST TERM GOES TO ZERO
• SIMPLIFYING FURTHER TO ONE-DIMENSION, WITH CONSTANT k, AND A PLANE SYSTEM, THE EQUATION FOR
THE TEMPERATURE GRADIENT BECOMES (g’ = ė in text):
ONE DIMENSIONAL STEADY STATE• SYSTEM IS THEN DIVIDED INTO
NODES. WHICH SEPARATE THE SYSTEM INTO A MESH IN THE DIRECTION OF HEAT TRANSFER.
• THE NUMBER OF NODES IS ARBITRARY
• THE MORE NODES USED, THE CLOSER THE RESULT TO THE ANALYTICAL “EXACT SOLUTION”
• THE NUMERICAL METHOD WILL CALCULATE THE TEMPERATURE IN THE CENTER OF EACH SECTION
• THE SECTIONS AT BOUNDARIES ARE ONE-HALF OF THE THICKNESS OF THOSE IN THE INTERIOR OF THE SYSTEM
ONE DIMENSIONAL STEADY STATE• NUMERICAL METHOD REPRESENTS THE FIRST
TEMPERATURE DERIVATIVE AS: WHERE THE TEMPERATURES ARE IN THE CENTER OF THE ADJACENT NODAL SECTIONS
• SIMILARLY, THE SECOND DERIVATIVE IS REPRESENTED AS SHOWN IN EQUATION (5-9)
• SUBSTITUTING THESE EXPRESSIONS INTO THE HEAT BALANCE FOR AN INTERNAL NODE AT STEADY STATE AS PER EQUATION (5-11):
ONE DIMENSIONAL STEADY STATE
• FOR THE BOUNDARY NODES AT SURFACES, WHICH ARE ½ THE THICKNESS OF THE INTERNAL NODES AND INCLUDE THE BOUNDARY CONDITIONS, THE TYPES OF BALANCES INCLUDE:
• SPECIFIED TEMPERATURE - DOES NOT REQUIRE A HEAT BALANCE SINCE THE VALUE IS GIVEN
• SPECIFIED HEAT FLUX
• AN INSULATED SURFACE, q` = 0, SO
ONE DIMENSIONAL STEADY STATE
• OTHER HEAT BALANCES ARE USED FOR:• CONVECTION BOUNDARY CONDITION
WHERE:• RADIATION BOUNDARY WHERE
• COMBINATIONS (SEE EQUATIONS 5-26 THROUGH 5-28)
• INTERFACES WITH OTHER SOLIDS (5-29)
ONE DIMENSIONAL STEADY STATE
• WHEN ALL THE NODAL HEAT BALANCES ARE DEVELOPED, THEN THE SYSTEM CAN BE REGRESSED (DIRECTLY SOLVED) TO OBTAIN THE STEADY-STATE TEMPERATURES AT EACH NODE.
• SYMMETRY CAN BE USED TO SIMPLIFY THE SYSTEM
• THE RESULTING ADIABATIC SYSTEMS ARE TREATED AS INSULATED SURFACES
ITERATION TECHNIQUE
• THE ALTERNATE METHOD OF SOLUTION IS TO ESTIMATE THE VALUES AT EACH POINT AND THEN ITERATE UNTIL THE VALUES REACH STABLE VALUES.
• WHEN THERE IS NO HEAT GENERATION, THE EQUATIONS FOR THE INTERNAL NODES SIMPLIFY TO:
• ITERATIVE CALCULATIONS CAN BE COMPLETED ON SPREADSHEETS OR BY WRITING CUSTOM PROGRAMS.