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It is simply a matter of symbolically reversing a equation with respect to a variable.

Part of the answer in red [attached]
Will be back later on. Inversing a function
is sometimes difficult, but always possible.

Cheers, Jean4.2.5 Steel temperature development.sm (25 KiB) downloaded 66 time(s).

Wrote

Attach the file with the comments you mentioned. I hope we will figure out what was wrong

Regards,
Radovan

P.S. By the way, Martin Kraska prepared the Portable version with quite a lot of plugins included.
Try it if you find this way more appropriate for you/

Thanks!

Attached is the workbook in a compressed format (deleted/removed a lot of irrelevant stuff).

I have written some comments here and there, and added pictures of examples i am using to verify the DE.

425specificheat.sm (763 KiB) downloaded 59 time(s).

Wrote

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It is simply a matter of symbolically reversing a equation with respect to a variable.

Part of the answer in red [attached]
Will be back later on. Inversing a function
is sometimes difficult, but always possible.

Cheers, Jean4.2.5 Steel temperature development.sm (25 KiB) downloaded 66 time(s).

Thank you for helping me.

I just added the Maple plugin. It now tells me the input format is not valid...

... solving for muo is almost constant, still puzzling ?
Will be back tonight.

4.2.5 Steel temperature development.sm (34 KiB) downloaded 48 time(s).

Wrote

... solving for muo is almost constant, still puzzling ?
Will be back tonight.

4.2.5 Steel temperature development.sm (34 KiB) downloaded 48 time(s).

This is the result I get.. I don't understand..

Wrote

This is the result I get.. I don't understand..

Ah ! Your attempt is incorrect, why ?
Power function X^Y is devil to maths. It involves ln(x) & exp(x),
Thus too reflexive for any symbolic solver, you must generalize.
See attached all, correct now.

Cheers, Jean

4.2.5 Steel temperature development.sm (41 KiB) downloaded 49 time(s).

Problem 1: How can I solve the differential equation 4.25 with respect to
θa(t)? I want to be able to calculate the development of temperature
in a steel member as a function of time.

Not sure I understand, θa(t) does not appear in 4.25 ?

Wrote

Wrote

This is the result I get.. I don't understand..

Ah ! Your attempt is incorrect, why ?
Power function X^Y is devil to maths. It involves ln(x) & exp(x),
Thus too reflexive for any symbolic solver, you must generalize.
See attached all, correct now.

Cheers, Jean

4.2.5 Steel temperature development.sm (41 KiB) downloaded 49 time(s).

I'm sorry, but I really don't understand your approach to solving the equation 4.22 for my.0. But the results in the Check-matrix are indeed correct. Can you please explain what you did there?

Thanks

Wrote

Problem 1: How can I solve the differential equation 4.25 with respect to
θa(t)? I want to be able to calculate the development of temperature
in a steel member as a function of time.

Not sure I understand, θa(t) does not appear in 4.25 ?

You are correct. θa is not included in 4.25.

Please have a look at the latest workbook from post #22. Radovan has helped me arrange all the expressions into a ODE solver, however, there is still one problem.

I appreciate all your help!

Wrote

I'm sorry, but I really don't understand your approach to solving the equation 4.22 for my.0. But the results in the Check-matrix are indeed correct. Can you please explain what you did there?

By inspection, if μ is < 1, we know in advance the result will be complex.
The "check" was to confirm this assumption. When you say solve for μ,
you are correct in your own words. In fact the matter is to inverse the roles.
From the explicit f(x):= ... where 'x' is given on range, now you want the
inverse of the procedure, i.e: given f(x) solve for 'x'. The procedure is
called "inversing". The Maple solution is the InverseFunction.
On that, Smath/Maple companions are far superior to Mathcad 11/Maple.

Please, read more on the revisited work sheet.
I will look at Radovan.

Cheers, Jean

4.2.5 Steel temperature development.sm (38 KiB) downloaded 48 time(s).

Wrote

Attached is the workbook in a compressed format (deleted/removed a lot of irrelevant stuff).

I have written some comments here and there, and added pictures of examples i am using to verify the DE.

You are solving for θ.a wrt c.a which in turn depends upon θ.a itself
That's too "circular". You can solve iteratively within a loop for
some predefined "c.a" around the peak 735

425specificheat.sm (772 KiB) downloaded 48 time(s).

Wrote

Wrote

Attached is the workbook in a compressed format (deleted/removed a lot of irrelevant stuff).

I have written some comments here and there, and added pictures of examples i am using to verify the DE.

You are solving for θ.a wrt c.a which in turn depends upon θ.a itself
That's too "circular". You can solve iteratively within a loop for
some predefined "c.a" around the peak 735

425specificheat.sm (772 KiB) downloaded 48 time(s).

But how is that done in the Excel Spreadsheets the examples refers to?

Wouldn't it be possible to have 4 ODE's, one for θ.a≤600, one for 600≤θ.a≤735, one for 735≤θ.a≤900 and for 900≤θ.a?

I have tried to illustrate my approach in the attachment, but alas - it does not work....


425specificheat-reply.sm (21 KiB) downloaded 37 time(s).

Try this one with another solver - dn_GearsBDF (it will escape from the critical regions - possible division by zero). Actually, it will work for all the four c.a() but not if you put them in the condition statement.

425specificheat-2-corr.sm (738 KiB) downloaded 47 time(s).

Regards,
Radovan

Wrote

Try this one with another solver - dn_GearsBDF (it will escape from the critical regions - possible division by zero). Actually, it will work for all the four c.a() but not if you put them in the condition statement.

425specificheat-2-corr.sm (738 KiB) downloaded 47 time(s).

Regards,
Radovan

Hi Radovan,

Forgive me my ignorance, but it still seems to me your result does not consider c.a as θ.a develops.

Is this completely impossible using SMath?

Best,

Wrote

Wrote

Try this one with another solver - dn_GearsBDF (it will escape from the critical regions - possible division by zero). Actually, it will work for all the four c.a() but not if you put them in the condition statement.

425specificheat-2-corr.sm (738 KiB) downloaded 47 time(s).

Regards,
Radovan

Hi Radovan,

Forgive me my ignorance, but it still seems to me your result does not consider c.a as θ.a develops.

Is this completely impossible using SMath?

Best,

Your differential equation is dθ.a/dt = D(t,θ.a) = f(t,θ.a,c(θ.a)). Your goal is to find θ.a(t) by solving this initial problem in the range t[0,150] provided the initial condition θ.a(0)=20 finding θ.a(0+..150). I do not know how else to explain this to you, sorry.

Regards,
Radovan

P.S. The main problem here IMHO is why the solvers can not accept conditional statements for c(θ.a) taking into account the numerical nature of the solving this problem but, on the other hand, the individual c(θ.a) causes no problem.

Wrote

Wrote

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Try this one with another solver - dn_GearsBDF (it will escape from the critical regions - possible division by zero). Actually, it will work for all the four c.a() but not if you put them in the condition statement.

425specificheat-2-corr.sm (738 KiB) downloaded 47 time(s).

Regards,
Radovan

Hi Radovan,

Forgive me my ignorance, but it still seems to me your result does not consider c.a as θ.a develops.

Is this completely impossible using SMath?

Best,

Your differential equation is dθ.a/dt = D(t,θ.a) = f(t,θ.a,c(θ.a)). Your goal is to find θ.a(t) by solving this initial problem in the range t[0,150] provided the initial condition θ.a(0)=20 finding θ.a(0+..150). I do not know how else to explain this to you, sorry.

Regards,
Radovan

P.S. The main problem here IMHO is why the solvers can not accept conditional statements for c(θ.a) taking into account the numerical nature of the solving this problem but, on the other hand, the individual c(θ.a) causes no problem.

I understand this part I think. It makes sense


What I don't understand is whether or not SMath is able to solve this. If it is able to do so, how do you formulate/structure the equation/solution?

If SMath is not able to solve this, what other approaches is viable to solve this problem?

I am sure this is a rather common problem in the world of chemistry and chemical engineering?

Best regards,


Edit: found a pdf which have tabulated θ.a(t) for different values of k.sh*A.m/V (=0.9V/A.m,b)

Page 7-10

tables for theta.a


Edit2: found another pdf which discusses the differential equation. It is apparently a Fourier equation, if that helps matters any.

Page 12-16

Fourier equation

Wrote

What I don't understand is whether or not SMath is able to solve this. If it is able to do so, how do you formulate/structure the equation/solution?

Yes, Smath solves but there are few points to understand.
The conditional 'ca' shows simply the system has 4 different DE solutions that depend
upon θ ... in other words the DE is not consistent. In Mathcad 11 Tom G. did a setup
for ordinary RK to take conditional. Not sure if I can retrieve, not sure either if
possible to code Smath ... from recollection, the system was consistent, i.e: a single
solution. Say otherwise, in there is like solving for different "phase phenomenon'

Excel is just given the data table to get a plot. Ordinary people understand Excel
because it's not a math tool, they understand like a grocery list. Any and all standards
are tabulated, thus instead of using tabulation from Peter, Bill, Arthur ,,, Excel
is convenient to get the hard copy.

The sol900 does not survive very far !

Cheers, Jean

425specificheat-2-corr [Piecewise].sm (772 KiB) downloaded 48 time(s).

Wrote

Wrote

What I don't understand is whether or not SMath is able to solve this. If it is able to do so, how do you formulate/structure the equation/solution?

Yes, Smath solves but there are few points to understand.
The conditional 'ca' shows simply the system has 4 different DE solutions that depend
upon θ ... in other words the DE is not consistent. In Mathcad 11 Tom G. did a setup
for ordinary RK to take conditional. Not sure if I can retrieve, not sure either if
possible to code Smath ... from recollection, the system was consistent, i.e: a single
solution. Say otherwise, in there is like solving for different "phase phenomenon'

Excel is just given the data table to get a plot. Ordinary people understand Excel
because it's not a math tool, they understand like a grocery list. Any and all standards
are tabulated, thus instead of using tabulation from Peter, Bill, Arthur ,,, Excel
is convenient to get the hard copy.

The sol900 does not survive very far !

Cheers, Jean

425specificheat-2-corr [Piecewise].sm (772 KiB) downloaded 48 time(s).

I have to admit, I am not sure I understand what you mean. A lot of this goes way above my head.

I have fiddeled a bit more.. I have taken a iterative approach, however it is manual.. Not automated like I would have wanted it to be.
Do you think I am onto something here?


Edit: I have cleaned up the workbook a bit.

testing 4 ode v3.sm (30 KiB) downloaded 43 time(s).

Wrote

Do you think I am onto something here?

Well done ! just add some of the EuroCode equations number ...

Cheers, Jean

testing 4 ode [Gorgeous].sm (24 KiB) downloaded 53 time(s).

Wrote

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Do you think I am onto something here?

Well done ! just add some of the EuroCode equations number ...

Cheers, Jean

testing 4 ode [Gorgeous].sm (24 KiB) downloaded 53 time(s).

Thank you for your compliment!

This would not have been possible without the help from you and Radovan!

BTW, I think I uploaded a later version than the one you editen. My latest version is v3, which includes two expressions that returns temperature and time for any given point, across the 4 ODE's.