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Wednesday, 15 January 2014

Numerical Code: Prediction of Temperature distribution within Aluminium Alloy chip during orthogonal turning process

clear all
clc

% Eng. Ammar Aziz   Zohaib Hassan     Ahmad Bilal      Ali Hassan
%-------------------------------------------------------------------------
% Modeling of heat generation and Temperature distribution in machining
%-------------------------------------------------------------------------


%--------------------------------------------------------------------------
    %--------------------------------------------------------------
    % Heat generated in the Primary and Secondary deformation Zones
    %--------------------------------------------------------------

% The tool is standard C2 WC tool
% The chip is Al2024-T351.
hehe = input('Press any numeric key and then enter to start working=');

% Rake face angle of tool is

    alpha=input('rake face angle of tool(0 Radians is recommended for turning)=');
    rake_face_angle = num2str(alpha);
    display(rake_face_angle);

% Coefficient of friction between tool and chp
u = 0.57271;
coefficient_of_friction = num2str(u);
display(coefficient_of_friction);

% Normal friction angle in Radians
% Depends upon the coefficient of friction b/w tool and chip
    beta = atan(u);
    normal_friction_angle = num2str(beta);
    display(normal_friction_angle);

% Thickness of the cutted chip in meters is (found experimentally....)
t2 = 0.000333;
thickness_of_cutted_chip = num2str(t2);
display(thickness_of_cutted_chip);
% Contact length
cntct = 1.5*t2;

% Feed per revolution(in case of turning), in meters is given by
h = input('feed per revolution(0.000165 m/rev is standard)=');
feed_per_revolution = num2str(h);
display(feed_per_revolution);

% Chip thickness ratio is
    rat = h/t2;
    chip_thickness_ratio = num2str(rat);
    display(chip_thickness_ratio);

% Shear angle in the shear plane
% Depends upon the rake face angle and chip thickness ratio
    phi = atan((rat*cos(alpha))/(1-rat*sin(alpha)));
    shear_angle = num2str(phi);
    display(shear_angle);

% Width of the cutted chip in meters
w = 0.00254;
width_of_cutted_chip = num2str(w);
display(width_of_cutted_chip);

% Entering the shear strength of the chip283e6
S = 276e8;
shear_strength_of_chip = num2str(S);
display(shear_strength_of_chip);

% Cutting force in Newtons is given by (Merchant's circle approximation ..)
Fc = 573;
Cutting_Force = num2str(Fc);
display(Cutting_Force);

% Feed or Thrust force is Newtons is given by (Merchant's approximation..)
Ft = 329;
thrust_force = num2str(Ft);
display(thrust_force);

% Shear Force
Fs = Fc*cos(phi) - Ft*sin(phi);
shear_force = num2str(Fs);
display(shear_force);

% Shear area is given by
A_s = (h*w)/sin(phi);
shear_area = num2str(A_s);
display(shear_area);

% Shear stress within the primary zone is
tau = Fs/A_s;
shear_stress = num2str(tau);
display(shear_stress);

% The cutting velocity in meters per second is
Vw= 1.36;
cutting_velocity = num2str(Vw);
display(cutting_velocity);

% heat generation rate per unit depth of cut in primary zone :
Qs =(tau*h*Vw*cos(alpha))/(sin(phi)*cos(phi - alpha));
rate_of_heat_generation_in_primary_zone = num2str(Qs);
display(rate_of_heat_generation_in_primary_zone);
% heat generation rate for, length of cut as thickness of chip, in primary zone
Qso = Qs*t2;

% Frictional force acting between the rake face and chip is given by
Ff = Fc*sin(alpha) + Ft*cos(alpha);
frictional_force = num2str(Ff);
display(frictional_force);

% Rate of heat generation in the secondary zone is:
Qf =(tau*h*Vw*sin(beta))/(cos(phi + beta - alpha)*sin(phi - alpha));
rate_of_heat_generation_in_secondary_zone = num2str(Qf);
display(rate_of_heat_generation_in_secondary_zone);
% Rate of heat generation within contact length of secondary zone is:
Qfo = Qf*cntct;

%--------------------------------------------------------------------------

%--------------------------------------------------------------------------
                %------------------------------------
                %Average temperature rise calculation
                %------------------------------------
% Thermal conductivity of the workpiece in watts per meter per kelvin
Kc=177;

% Mass density of chip in kilograms per cubic meter
p=2785;

% specific heat capacity of chip in joules per kilogram per kelvin
Cc=875;

% Thermal Diffusivity
Sa = Kc/(p*Cc);

% Thermal number
Rt = (h*Vw)/Sa;

% fraction of heat flowing into the tool
X = 0.5 - (0.35* (log10(Rt*tan(phi))));

% Average temperature rise of the (((chip))) per unit depth of cut due to shearing
delta_T = ((1-X)*Qs)/(p*Cc*h*Vw);
average_temp_rise = num2str(delta_T);
display(average_temp_rise);
%This average temperature rise on the shear plane is used as a boundary
%condition at the upper left corner of the chip

%-------------------------------------------------------------------------


%--------------------------------------------------------------------------
                    %------------------
                    %Mesh generation
                    %------------------

% Chip velocity, for unit depth of cut, is given as
Vc = Qf/Ff;
chip_velocity = num2str(Vc);
display(chip_velocity);

% No of grids in x-direction
nbx = input('please enter no. of grids in x-direction (4 is default)=');
%No of grids in y-direction
nby = input('please enter no. of grids in y-direction (4 is default)=');

% No of nodes in x-direction
nbnx = nbx + 1;
% No of nodes in y-direction
nbny = nby + 1;

% proportion of frictional heat entering into the tool
Tio = 0.03;

% Grid spacing in x-direction
grsx = cntct/nbx;

% Grid spacing in y-direction
grsy = t2/nby;

% No. of nodes in the mesh
nod = nbnx*nbny;

% heat flow rate in chip's primary control zone due to primary zone heat source is
Qrs = ((1-Tio)*Qs*grsy)/t2;
heat_flow_in_chip_node_due_to_shear_source=num2str(Qrs);
display(heat_flow_in_chip_node_due_to_shear_source);

% heat generation rate for a secondary node in chip due to frictional heat source
Qrf = ((1 - Tio)*Qf*grsx)/cntct;
heat_flow_in_chip_due_to_frictional_source=num2str(Qrf);
display(heat_flow_in_chip_due_to_frictional_source);

% Heat generation array
hga = [];
for i= 1:((sqrt(nod)));
hga(i,sqrt(nod)) = Qrf;
end

for j= 1:((sqrt(nod))-1)
    for k = 1:(sqrt(nod)-1)
    hga(k,j) = 0;
    end
end

for m=1:(sqrt(nod)-1)
hga(sqrt(nod),m) = Qrs;
end

hga(sqrt(nod),sqrt(nod)) = Qrf + Qrs;
display(hga);

%temperature distribution within the chip

Am0 = (p*Cc*Vc)/(Kc);
Am1 = Am0/grsx;
Am2 = - (2/(grsx^2)) - (2/(grsy^2)) + Am1;
Am3 = (1/(grsx^2)) - (Am1);
Am4 = (grsx^2)*(grsy^2);
Am5 = Am4 * Am2;
Am6 = Am4 * Am3;
Am7 = grsy^2;
Am8 = grsx^2;
Am9 = Qrf/Kc;
A_1 = Am6/Am5;
A_2 = Am7/Am5;
A_3 = Am8/Am5;
A_4 = Am9/Am5;
A1 = -A_1;
A2 = -A_2;
A3 = -A_3;
A4 = -A_4;

for M = 1: (nbx - 1)
xM = M*grsx;
end

for N = 1: (nby - 1)
xN = N*grsy;
end

tempi = [];

% Boundary Condition
tempi(sqrt(nod),1) = delta_T;

% Data is imported from Solidworks 2012 by applying the boundary conditions
tempi = [344.1 345 346.1 347.3 350.3; 343 344.2 345.4 347.8 350.3; 339 336.9 343.9 348 351.2; 329.9 335.6 341.3 354.4 356.1; 234.5 343.1 352.7 362.4 370.6];
display(tempi);

% matrix of coefficients is given by
coeffik = (inv(tempi))*hga;
display(coeffik);

% Temperature distribution within the chip is given by multiplying inverse
% of coefficient matrix with heat generation array.

% plotting temperature
[x,y]=meshgrid(1:sqrt(nod),1:sqrt(nod));
surf(x,y,tempi)

%--------------------------------------------------------------------------
%THE END
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------

Prediction of Temperature distribution within Aluminium Alloy chip during orthogonal turning process

By: Engr. Ammar Aziz, Engr. Zohaib Hassan, Engr. Ahmad Bilal, Engr. Ali Hassan
COMSATS University of Science and Technology Sahiwal, Pakistan

Abstract
In this paper, a numerical model based on finite difference method is presented to predict chip temperature field in orthogonal turning process. Orthogonal turning can be studied by modeling the heat transfer between tool and chip at tool-rake face contact zone. The mathematical model and simulation results, presented here, are in satisfactory agreement with experimental temperature measurement reported in the literature.

1- Preparing Numerical Model
First of all consider the orthogonal turning process. Heat is generated within the chip due to its interaction with the tool. This heat generation is sub-divided into primary heat generation zone and secondary heat generation zone, as shown in the following figure:

Here ‘phi’ is the shear angle. It is the angle of the shear plane with the horizontal axis.

Rate of heat generation, per unit depth of cut within primary shear zone is given by:

Rate of heat generation within secondary shear zone is given by:

Average temperature rise of chip per unit depth of cut due to shearing:

This temperature rise is used as a boundary condition at the point shown in the figure:

Finite Difference Method and Derivative Concept:
Before proceeding to further modification, it is first necessary to clear that what is the meaning of a derivative and why we are relating it with finite difference concept. Also what does mean by partial derivative?

Meshing of our chip with 25 nodes

First law of thermodynamics implementation
Supposing a control zone within the mesh and applying the law of conservation of energy there we get:

According to Fourier’s law of heat conduction:

Substituting the values from Fourier’s law into heat balance equation we get:

Now by using the concept of Taylor series expansion about a point….

Approximating partial derivatives using forward and backward difference simultaneously, we get:

So our final model is ……

Heat flow in differential chip control zone from the frictional heat source is:

So heat generation array can be constructed by putting the value from above formula into all entries of last column and putting the value of heat generated in primary zone, in last row. All the entries except which described above are zero because here there is no heat generation as shown in the following figure:

At last we determine the coefficients matrix which is obtained by solving the final model equation for each node of the chip and determining the coefficients through algebraic manipulations.

The purpose of above picture is to show you something interesting. As we write
the equation for each node the coefficients jump to next variable by one step. So if we try to write the above equations in matrix form than the matrix of
coefficients contain the entries which move diagonally. Siedal algorithm is best
to write the matrix of this type. The inverse of coefficient matrix is when multiplied by the heat generation array then we get the temperature distribution array which is our goal.

Plot of temperature against the distances along the edges of the chip in MATLAB:

Contour Plot:

Simulation of chip:
Solid works 2012 is used to accomplish the simulation of chip.

Description
Prediction of temperature distribution within chip of Aluminium. The tool which is used for turning process is made up of ‘C2 WC’.

Meshing…………..

Comment: The results are very close to those which are practically observed

Tuesday, 14 January 2014

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