Rate of change in temperature at everypoint on a heated circle

M_Usman · September 15, 2023, 1:53am

I solved a natural convection problem in a circle of radius r=0.1 with center (x0,y0)=(0.5,0.0.2*sqrt(3)). Now I am trying to find the rate of heat transfer i.e.
∂T/∂n
where n is normal at a point on a circle where we have to compute the rate of heat transfer as mentioned in the following figure.

Could you please guide me to find the rate of heat transfer on every point of a circle using a loop?
Thanks

cmd · September 16, 2023, 4:58am

Something like this?

mesh Th; // mesh
int BoundaryLabel = 1; // boundary label
real k = 1.0; // thermal conductivity 
int1d(Th,BoundaryLabel)(k*(N.x*dx(T) + N.y*dy(T)));

M_Usman · September 16, 2023, 7:04am

Thanks, Chris for your response.
Why you take int1d??? I think it is like grad(T)'[N.x, N.y] at some point on the circle. But the problem is how I can compute grad(T)'[N.x, N.y] at some specific point of the circle such as (x1,y1), where N.x and N.y should be the x and y compute of the normal “n” on the circle at point (x1,y1). Could you please guide me in this direction or any other which is appropriate?

M_Usman · September 16, 2023, 7:13am

@cmd May be this will help you to elaborate on my point.

cmd · September 16, 2023, 8:33am

Just create a vector like:

Vh Tx = dx(T);
Vh Ty = dy(T);

And then you can evaluate this at any point and compute the heat flux.

M_Usman · September 16, 2023, 8:56am

Thanks @cmd. What about the normal vector “n” at any point? It will be the same and just the change in dx(T) and dy(T)? if not then how I can evaluate N.x and N.y at any point on the circle?

cmd · September 16, 2023, 11:46am

You can compute it yourself using the known geometry of your mesh. Or you can do something like:

varf vHF(p,q) = on(BoundaryLabel, p = N.x*dx(T) + N.y*dy(T));
Vh HF;
HF[] = vHF(0,Vh,tgv=1);

marchywka · September 16, 2023, 2:03pm

I think the earlier point about trying to do a point wise result is well taken but
extending on your idea I guess you also flag the dofs that make up the boundary,
The IND result appears to select the boundary dofs, fwiw.

load "msh3"
load "medit"
load "isoline"
 load "mjm_dscope_freefem"
real szx=100;
real szy=100;
int nx=50;
int ny=50;
int nc=30;
real r=20;
real[int,int] cp(2,nc);
for(int i=0; i<nc; ++i) 
	{ real t=1.0*i/nc;  cp(0,i)=r*cos(t*2.0*pi); cp(1,i)=r*sin(t*2.0*pi); }
border left(t=0, 1){x=-szx/2; y=szy*(t-.5);label=1; }
border top(t=0, 1){x=szx*(t-.5);y=szy/2; label=2; }
border right(t=0, 1){x=szx/2; y=-szy*(t-.5);label=3; }
border bottom(t=0, 1){x=-szx*(t-.5);y=-szy/2; label=4; }
border mycircle(t=0, 1){
P=Curve(cp, 0, nc-1, t);
label=5;
}// EOM

mesh Th = buildmesh(left(-ny)+top(-nx)+right(-nx)+bottom(-ny)+mycircle(nc));
//plot(Th);
//medit("mesh",Th);
fespace Vh(Th,P1);
Vh p,q,ind;
Vh T=x*x;
// https://community.freefem.org/t/rate-of-change-in-temperature-at-everypoint-on-a-heated-circle/2708/6

varf vIND(ind,q) = on(5, ind = 5 );
varf vHF(p,q) = on(5, p = N.x*dx(T) + N.y*dy(T));
Vh HF,IND;
IND[] = vIND(0,Vh,tgv=1);
HF[] = vHF(0,Vh,tgv=1);
cout << " n = "<<HF[].n<<endl;
cout << " ind = "<<IND[]<<endl;
//medit("ind",Th,IND);

/*
meshL Th1=Sline(1024,[x]);
fespace Mh(Th1,P1);
Mh asol=uh(x,x);
Mh anex=uex(x,x);
Mh diff=anex-asol; // =uex(x,x);
//mjmdscopeL(Th1,"names asol anex diff "+nn,asol,anex,diff,"asdf"); // warning  do not forget () 
mjmdscopeL(Th1,"names asol anex diff "+nn,asol,anex,diff,"asdf"); // warning  do not forget () 

*/
//mjmdscopeL(Th1,"names asol anex diff "+nn,asol,anex,diff); // warning  do not forget ()

M_Usman · September 17, 2023, 5:32am

Thank you @cmd and @marchywka for your response. I have seen the normal on a curved body can be computed as:

where n_x and n_y are the x and y components respectively. Since the x and y component of a circle at any point is -cos(theta) and sin(theta) respectively. May, I compute the rate of heat transfer at any point (x_i,y_i) on the circle such x_i=0.5+rcos(theta), y_i=0.5+rsin(theta), where (0.5,0.5) is center of circle and r is radius, using the following expression:
-cos(theta)dx(T)(x_i,y_i)+sin(theta)dy(T)(x_i,y_i), for 0<theta<2pir

M_Usman · September 17, 2023, 5:46am

This gives the following error:
Compile error :
line number :19, Curve

M_Usman · September 17, 2023, 7:00am

Hi @cmd @marchywka the following is the Poisson equation solved in a circle of radius 0.1 and center (0.5, ):

real Mesh=60;
real ra=0.1;
border C(t=0,1){x=0.5+racos(2pit);y=0.2sqrt(3)+rasin(2pit);}
mesh Th=buildmesh(C(Mesh));
plot(Th,wait=1);
fespace Vh(Th,P2);
func f= xy;
Vh U,V;
problem Poisson(U,V,solver=LU)=
int2d(Th)(dx(U)*dx(V) + dy(U)*dy(V))

int2d(Th)( f*V )

on(C,U=1) ;
Poisson;
real n=11;
real[int] xx(n),flux(n);
for(int i=0;i<n;i++)
{
xx[i] = (2pii)/(n-1);
flux[i]=-cos(xx[i])dx(U)(0.5+racos(xx[i]),0.2sqrt(3)+rasin(xx[i]))
-sin(xx[i])dy(U)(0.5+racos(xx[i]),0.2sqrt(3)+rasin(xx[i]));
cout<<raxx[i]<<" "<<flux[i]<<endl;
}
macro grad(u) [dx(u),dy(u)]//
cout<<"flux1 = "<<((grad(U))'[N.x,N.y])<<endl;
plot(U);

Now I have computed ∂U/∂n on 500 points on the circle. My way is correct? If yes then how can get the same values on every 500 points using (grad(U))'*[N.x,N.y] command?

M_Usman · September 18, 2023, 12:45pm

@cmd @marchywka could you please give your valuable suggestions on the above posts please. Thanks!

marchywka · September 18, 2023, 1:07pm

Can you upload the code? Did it run? I guess you can do it pointwise
uising u(x,y) and FF should give you the value at that arbitrary point
with interpolation If you want to calculate the normal yourself
its easy for a circle but difficult in general.

M_Usman · September 18, 2023, 1:12pm

Yes, @marchywka you are right this is easy for the circle that is why I mentioned this code and computed the required expression using some mathematics, now I want to know how I can get the same results using ((grad(U))'* [N.x, N.y]) command or some other way which is appropriate for every shape not only circle.

cmd · September 18, 2023, 1:33pm

The code I gave here should do what you’re asking. If not, please explain why it doesn’t do what you need.

M_Usman · September 18, 2023, 1:39pm

How I can extract flux[1], flux[2],…,flux[500] (as I obtained in above), where flux[i] are the rate of change on the circle for 500 different points.

cmd · September 18, 2023, 1:43pm

The code I provided should give you the flux at each point on the boundary in the array HF[]

M_Usman · September 18, 2023, 2:02pm

Yes sir @cmd , I got an array for this circle problem (posted by me) as bellow.
0.009976449806 0.01022786813 0 0.01017127374 0
0 0.01036609768 0.01010309073 0 0.009996923058
0 0 0.01026848656 0 0
0.01005449145 0.01028976586 0 0 0
0 0 0 0.00988521446 0.00983777796
0 0.009760014285 0 0 0.009681407371
0.009626978686 0.009370266281 0 0.009495604429 0
0 0.00950908328 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0.01026828858 0 0 0
0.01023723867 0 0.0100778128 0 0.01049083483
0.01028432381 0.01026328347 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0.009178976079 0 0.008948200174
0.009317974872 0 0 0 0
0.009620942578 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0.01017972185
0.009804091975 0 0.01008043608 0.0103381028 0
0.01020221113 0 0 0 0.01025413732
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
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0 0 0 0 0.01003440996
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0.01015735372 0.009878638772 0 0.009768710773 0
0 0.009638180034 0 0 0
0 0 0.009850216655 0 0
0.009638531588 0.009303229429 0 0.009486680878 0
0 0.008808276496 0.00892237405 0 0
0.008736047786 0.008641869733 0 0.008553388845 0.008482261106
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0.009640006896 0 0.009363356954 0 0
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0 0.009034063396 0 0 0
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0 0.008781110423 0 0.008897509133 0
0 0 0 0.008030449285 0
0.007771554442 0 0 0 0
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0.008762036982 0 0.008709343918 0.007504656477 0.00766564131
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0 0 0.008616690592 0 0.008551253631
0.00772690838 0 0.007563498948 0 0
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0 0 0.008481464341 0.008401306223 0
0.007601874843 0 0.007479486991 0 0
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0.008348593026 0 0.008253466641 0.007498606579 0
0.007430781148 0 0 0 0
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0 0 0 0.008178010639 0
0.008169095043 0.007451143256 0 0.007342576625 0
0 0 0 0 0
0 0 0 0 0.007081512038
0.00731168822 0 0 0 0
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0 0.008000198325 0 0 0
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0.007384942941 0 0 0 0
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0 0.008004415351 0 0.00789793697 0
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0.00709932628 0.007246035103 0 0 0
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0.007883428924 0 0.007768981524 0 0
0 0 0 0.007245020798 0.007392743923
0 0 0 0 0
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0 0 0.007796203849 0 0.007662102185
0 0 0 0 0.007232847062
0.007291578599 0 0 0 0
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0 0 0 0 0
0 0 0 0 0
0 0 0.00772438426 0 0.00756676888
0 0 0 0.007308923286 0
0.007145375743 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0.007654159697 0.007484197262 0 0.007355853744
0.00716194191 0 0 0 0
0 0 0 0 0
0 0 0.007611156238 0 0
0.007549765583 0 0.007423663465

M_Usman · September 18, 2023, 2:04pm

@cmd I want to plot in 2D on the x-axis surface length means 0 to 2pira and on the y-axis value of flux. How can I get appropriate data in a .dat file so that I can plot in tecplot.?

cmd · September 18, 2023, 3:16pm

It looks like the data is there. You will have to post-process it to plot it.
https://doc.freefem.org/documentation/visualization.html

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Rate of change in temperature at everypoint on a heated circle

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