Am struggling to solve this equation.
I want to find how much the temperature increases in a solar collector in time.
I have an equation for a vacuum type collector - its a polynomial...
Power (in watts) = AG(no - a1dT/G - a2(dT)^2/G)
A being aperture, G being solar incidence no being conversion factor, a1 and a2 being the loss coefficients - all these values are known. dT is difference between ambient and the collector temperature.
If the collector contains say 2 litres of water, you can use this equation, the energy to heat this is defined as
Power (in Joules) = mCdT
where m is mass, C is Specific heat of water and dT is temperature difference.
What I want is an equation that relates dT to t like
dT = somefunc(t)
So I can plot a graph and/or work out how the temperature of the collector increases in time given a known ambient temperature.
My maths just isn't up to the job...
Thanks in advance.
All what you have to do is an energy balance between the incoming energy for a time interval dt which you obtain as Power x dt and the incoming energy with the water flow the sum is the energy contained in the outgoing water flow. Knowing the values and assuming no leak (flow is at input and outlet the same) you get your temperature increase versus time. You said that your math is not going far enough, I would add that the notion of energy and power is not clear to you since you name power once in watt (correct) and in Joule (incorrect this is energy) power being energy/time.
If you have no flow but only a static volume to heat then you write the differential equation
dT/dt= (incoming power)/(mCp) It is asimple equation since yoy know the power function as you wrote above. So write it and make the integral.
Nick,
Thanks for that reply....
Yes I used power when I should have used energy - my mistake.
Assuming that E=pt (energy = power.time)
Then I get t=E/P and 1/t=P/E which is what you suggested.
Are you saying that the P/E needs to be integrated or differentiated - if you differentiate the a1 terms becomes zero which I cannot believe to be correct!!!!!
If you consider a "dt" time element the water mass will recieve the energy dE=P*dt and its temperature dependent energy content will increase with dE= c*m*dT where T is temperature. If you write that both quantities are equal then you obtain a differential equation you have to integrate. The solution will be the function you try to find T=f(t).
If you do not know how to integrate a differential equation then ask somebody to explain it to you. It is a simple one since the variables can be parted.
Sorry for the delay but it is not my major activity.
Just considering the water, you will have a first order system, wich temperature raise will follow the e-curve, meaning that by the heat capacity of water the temperature will reach its end temp after 5tau, assumed that the solar irradiation will be constant. Things will change if the water has a flow,. By the end of the collector the temp rise/sec (dT/dt) will slow down, in the meanwhile water is passing by, so th e flow rate will have a big influence on the systems final temperature. Another issue is the heat loss of the collector and the focus of the collector structure. Not every ray will be part in the process of heating, and the collector structure will have it's losses as well since the temperature of the water, has it's influence on the temperature of the collector. and the bigger de dT (collector vs ambient) the bigger the loss will be. This are three different linear differential equations you can combine into one. just analyze the system and its different processes. Desribe the processes in the linear way, and compose the new equation by the different linear parts of the process you already have analyzed.
A basic trick to describe a process is: The output is the input minus what left behind. Then you determine if what left behind, goed back into the system (as you see with buffering and temporary storage), or is it just lost into the environment.
Dear Roger I hope you will find an useful solution of your problem in the attachment of my message. Regards
Dan constantinescu
Dear Mr. Dan Constantinescu,
can you tell me how this document is code. Because I cannot get it readable.
Best regards,
Leo de Vos
Dear Mr. Leo de Vos
I am sorry but is a very simple doc. page ! If you can not read give me a message on [email protected] and I shall send you like an e mail attachment. Regards
Dan,
Thanks for taking the time with that doc.
From your equation (condensed form), should that not be a minus B3.v^2 ?
The original equation came from a vacuum tube specification where the first term was the conversion factor and the next two were minus's which are the losses.
By trade I am not a thermodynamic engineer nor a mathematician, I am an electronic engineer and I am trying to work out the most efficient way of transferring heat from a collector to a storage tank.
My reasoning goes like this, the efficiency of a vacuum tube collector goes down with an increase of temp(collector) w.r.t ambient.
So its not good to allow the temperature to rise too high. However if you pump when the temperature of the collector is not all that greater than ambient then you waste (electrical) energy on pumping (which will happen all too frequently) and bigger pipe loss as you will have water flowing through the pipes very often.
The electrical power and pipe loss are relatively easy to work out and these must be balanced with the collector efficiency There will be an optimum temperature that provides greatest transfer.
Going back to the equations in your doc, I plugged then into wolfram alpha (being the lazy engineer I am) and I got something different....
http://www.wolframalpha.com/input/?i=t%3D+integral+dv%2F%28B1-B2*v%2Bb3*v*v%29+
Now I am even more confused.....
Best Regards
Dan thank you for sending me that doc by e-mail. It is perfectly readable.
My appreciation for your time.
Best regards,
Dear Roger,
This evening I shell develop all the terms of equation sent you, and I'll try solve a numerical case in Excel, starting form a virtual flat collector and based on climatic data with one hour pace. If you are interested, it is possible to develop a more sofisticated problem based on double temperature variation, t(x, time) when flow is active inside of solar collector (depending on laminar / transient / turbulent developing and developed flow ).
Best regards,
Dan
Dan,
Thanks for going the 'extra mile' - great to see other with such keen interest as myself.
Look forward to what you come up with.
Regards,
Roger.
Dear Roger,
I hope you will can use EXCEL files sent to you. There are an application of mathematical model, no more. All inlet date (red color) you can change but after this you must solve the equations using solver from EXCEL (or Goal Seek). Good luck ! And was a pleasure to give you this small aid ! Best regards,
Dan
This is the second file - with water flow inside of collector (inside of collector equation !). Regards, Dan
Sorry Roger,
Now I seen that Solar collector 1.xls is disturbed. I hope the new one is ok !
Dan
Dan,
Thanks very much....
Something to 'chew' on over the weekend.
I will play with the inputs to get a feel for how it performs.
Roger.
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