Maths question RE summation symbol

[HarleyE30]

Hi everyone,

I'm integrating a PV diagram. I have successfully done the integration, but now just want to show that the final value is the sum of all the values collected for each crank angle, from let's say for instance 1 degree to 180 degrees.

I could just not bother using the symbol and just explain in words, but I'd like to know how to properly use the summation symbol. The image below is my guess at how I could use it in this situation. Please tell me if this is incorrect ! Thanks

 

[ivymike]

whew, it has been a long time... but based on what you've written I gather that:
1) you've not done the integral, you've done a step-by-step summation of values by crank angle
2) you want to show THAT in equation form

My proposal is that you want Sum_{(theta from 1 to 180)}[P(theta)*(V(theta)-V(theta-1)]

 

[ivymike]

(otherwise you're talking about the finite integral from vmin to vmax of P, and no summation sign is needed?)

 

[HarleyE30]

I've done a trapezoidal numerical integration of the pressure and volume data I have, for crank angles in 0.5 degree increments.

The image below is a screenshot of the spreadsheet:

 

[HarleyE30]

So for instance I wanted to find the work done per unit volume from cells A5 to A8. I would sum cells D58 and divide by the cylinder's total displacement volume. I wish to show this as an equation using the summation symbol

 

[GregLocock]

It is wrong as you've used a summation and an integral sign, which indicates a double summation.

So, lose the algebraic integral sign, and add a subscript theta to the P. Change the d to a small delta.

So it should be

sigma super 180 sub theta=0 P sub theta dot small_delta V





Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

http://eng-tips.com/market.cfm?

 

[HarleyE30]

Many thanks for your help. Is this now correct?

 

[IRstuff]

I you are attempting to show an integration, then it should be a definite integral, so, the integral symbol with limits, rather than summation. The summation is the mathematical approach to solving for the integral in question.

TTFN
faq731-376

 

[ivymike]

the more recent summation was closer, but deltaV is dependent on theta and should show it. That's why I suggested (1 to 180) and (V_theta-V_theta-1)

 

[SomptingGuy]

You may also find that since P and V both vary during the time-step that a more appropriate value for dV_theta might be required so that the approximated dV is closer to the measured value of P. Something like (V_theta+1 -Vt_theta-1).






- Steve