Non Linear Multivariable Function

[Ernits]

Hello,

I´m having problems while solving the system of equations bellow,
I tried the symbolic solution and the numerical, by given to the constants values, but I didn´t suceed... It says that I have to use the submatrix function because the solution is too big to be displayed, but I don´t know how to use it, I really would thank a lot if somebody could help me,

the system of equations:

[(5.6*10^-9)*Tpa^4] - (11.27 * Tpc^4) - 90*Tpc + 30*Tpa +17849.28 = 0

660.72 - [5.6*10^-9*(Tpa^4 - Tpc^4)]-4200*Tms + 1255800=0

(4200*Tms-1255800)*[ln[(Tpa-Tms)/(Tpa-299)]= 27.75*Tms - 7400.25

thanks!

Ernits

 

[IRstuff]

I count 4 variables and only 3 equations?

TTFN

 

[Ernits]

Hi,

no there are only 3 variables, Tpa, Tpc and Tms,

maybe i didn´t, made the LN clear in the third equation, sorry

Do you have any idea how could I solve it?

thanks for the attencion
Ernits

 

[IRstuff]

Sorry, my fault for mis-reading.

Is your 2nd equation correct? Why are there two separate constants?

Also, is not the 5.6*10^-9 the Stefan-Boltzman constant, which should be 5.67*10^-8 W/m^2K^4? Not that it matters, since the equation is highly unstable as constructed. If you are calculating thermal radiance, then why do the emissivities appear to be the same in one equation yet different in another?


Why are all three temperature allowed to float? Isn't one of them supposed to the background temperature? If not, then what object is at 299?

I dinked around with CTOL and TOL to 0.05, each, and got answers of 1006.365,9.8319,285.3103, for Tpa,Tpc,Tms, but the solution is unstable.

TTFN

 

[Ernits]

Hi,
yes, the second equation is correct, I didn´t add them, was just that.

It represents the thermal balance for a solar collector, wich is going to have a varible flow (but for now I´m just considering 1, but it´s just for testing the equation).

The temperature considered as 299K is the entrance of the fluid inside the collector, the other variables are Tms(Temperature that leaves the collector), Tpc(Temperature of the cover plate - glass) and Tpa (Temperature of the collecting plate - black surface),

You are right that was used the Stefan´s Boltzmann constant, but inside an equation, so that´s why it changed its value.

So the idea is by changing the flow inside the collector the temperatures are going to change as well (sure there are many other considerations, like the global coeficient of heat transfer that will change too, but I tried to simplificate the most). I wanted to feel the problem just giving some numbers, and see what I could achieve. But in the end nothing appeared.

I don´t know if there is some argument that could be used to simplificate more the equation.

So you used the CTOL, how can I use that? Did the message "This aray is too large to be displayed...try to use the submatrix function" appeared to you?

thanks a lot for the attencion!

Ernits

 

[IRstuff]

CTOL and TOL are changed through the menu settings for Mathcad options.

The error messages might have been due to the fact that you probably used T[pa, T[pc, etc. to create subscripts. This actually creates indexed variables. If you use T.pa, T.pc, T.ms, you'll create variables with literal subscripts.

Based on your description, the solution I got is completely wrong. That suggests that there might be a sign error lurking in the equations.

It might be easier to treat the problem as a thermal resistance network problem, just to see if you can get it to work. I'd blow off the radiated effect for the time being, except for the actual black collector panel and just use a derated power input to account for the radiation back to ambient. That will at least get rid of the T^4 problems for the time being.

TTFN

 

[jghrist]

Using a Solve block, I get:

T_pa=2644
T_pc=13.457
T_ms=234.01

You probably used indexed variables instead of subscripts as IRstuff said.

 

[IRstuff]

The problem is that based on his description, none of the temperatures should be lower than 299K.

TTFN