Dairy Science and Food Technology Homepage
Home  Cheese starters Donate support Site  Probiotics Cheese yield  Cheese quality Cheese science  Italian cheese Anti-microbial  proteins  Phages Primary production  Ice cream Wine Calculators  Harvard referencing  Packaging  Jobs
Forum Home Forum Home > Site calculators and spreadsheet queries > Spreadsheet and site calculators
  New Posts New Posts RSS Feed - Area under lethality curve
  FAQ FAQ  Forum Search   Events   Register Register  Login Login


Area under lethality curve

 Post Reply Post Reply
Author
Message Reverse Sort Order
formula123 View Drop Down
Admin Group
Admin Group

Administrator

Joined: 01 Sep 2009
Location: N. Ireland
Status: Offline
Points: 390
Post Options Post Options   Thanks (0) Thanks(0)   Quote formula123 Quote  Post ReplyReply Direct Link To This Post Topic: Area under lethality curve
    Posted: 07 Apr 2011 at 11:11pm
I have taken 7 sets of time temperature data and calculated Fo using the Trapezoid rule, Simpson's rule and the Romberg method. The results are summarised below.
 
For 3 of the data sets the Trapezoid rule slightly overestimated Fo compared with the more accurate methods. However, it underestimated Fo for 4 of the datasets.
 

Fo  values obtained using published data and three methods of numerical integration

Integration method

Data set 1

Data set 2

Data set 3

Data set 4

Data set 5

Data set 6

Data set 7

Simpson's rule

21.322

9.865

6.783

15.828

9.833

3.908

11.020

Trapezoid rule

21.588

9.821

6.906

15.276

9.319

3.872

11.412

Romberg

21.242

9.851

6.783

15.828

ND

ND

11.083

 
I will provide the full analysis including all the data,  the trend line equations and the spreadsheets as a download at a future time if there is enough interest.
 
So Beachgirl it would be reasonable to predict that there will be instances when your company's  F values have been underestimated when  the Trapezoid rule has been used. BUT unless you have used large time intervals and few data points this underestimation should be too small to cause any concern.
Back to Top
Sponsored Links


Back to Top
formula123 View Drop Down
Admin Group
Admin Group

Administrator

Joined: 01 Sep 2009
Location: N. Ireland
Status: Offline
Points: 390
Post Options Post Options   Thanks (0) Thanks(0)   Quote formula123 Quote  Post ReplyReply Direct Link To This Post Posted: 30 Mar 2011 at 8:52pm
I have been wondering why you have kept asking questions about calculating the area under the lethality curve!
 
I am aware that others have solved a range of equations (with known solutions) using several numerical methods; based on their findings the order of accuracy in obtaining the area under the curve is 1, Romberg, 2 Simpson's and 3 the trapezoidal rule. Based on most of the work published the trapezoidal rule is slightly less accurate and sometimes overestimates area (F value) compared with more accurate methods (e.g. Simpson's rule) - in thermal processing.
 
Having calculated a few F values using the three methods (using actual data) I have found that as expected  the trapezoidal rule gave lower F values compared with Simpson's rule for many data sets. 
Thank you for being persistent.
 
 
Back to Top
formula123 View Drop Down
Admin Group
Admin Group

Administrator

Joined: 01 Sep 2009
Location: N. Ireland
Status: Offline
Points: 390
Post Options Post Options   Thanks (0) Thanks(0)   Quote formula123 Quote  Post ReplyReply Direct Link To This Post Posted: 27 Mar 2011 at 8:42pm
You certainly are interested in this topic!
 
I will prepare a spreadsheet over the next week or so and answer your question more fully.
 
The trapezoidal rule is expected to slightly underestimate area with concave down curves. However if the spacing (or interval between readings) is small and a large number of readings are taken the error becomes very small and to all practical purposes (thermal processing calculations) is not significant.
 
Now to theory. The theoretical error of the simple trapezoidal rule is proportional to sample interval raised to the third power. The error for the composite rule is proportional to the sample interval squared. If you increase the number of samples so as to decrease the interval by a factor of 2, then the error will decrease by 2 x 2 , a  factor of 4.  If you are using 4 min intervals then moving to 2 or 1 will markedly affect accuracy. With a small number of test results and larger time intervals Simpson's rules (there are 2) will give a more accurate estimate. But in many practical situations this does not really matter.
 
You have succeeded in me wanting to close this topic! So over the next few weeks I will prepare a spreadsheet showing F values obtained using trapezoidal rule, Simpson's two rules and the equation for a lethality curve. Hopefully this will give you the information you need?
Back to Top
beachgirl View Drop Down
Newbie
Newbie


Joined: 07 Nov 2010
Status: Offline
Points: 10
Post Options Post Options   Thanks (0) Thanks(0)   Quote beachgirl Quote  Post ReplyReply Direct Link To This Post Posted: 27 Mar 2011 at 11:28am
Hi u might said this bfor what are the rrors with the t rule & how u control?
Back to Top
formula123 View Drop Down
Admin Group
Admin Group

Administrator

Joined: 01 Sep 2009
Location: N. Ireland
Status: Offline
Points: 390
Post Options Post Options   Thanks (0) Thanks(0)   Quote formula123 Quote  Post ReplyReply Direct Link To This Post Posted: 22 Mar 2011 at 10:00pm
Yes welcome back.
 
I think I have responded to this query under thermal processing before? Perhaps the easiest way to get the equation you want is to use the trendline function in Excel. If the r value is > 0.9 (you can use lower r values) and your knowledge of calculus is good you can integrate the equation to get the area. The

Wolfram|Alpha: Computational Knowledge Engine

 can be used to check your calculation.
 
There are a few really good Excel textbooks written by scientists for scientists and engineers and provide a good introduction to the computing power of Excel. Look up the  LINEST function it can used instead of the trendline function to find the equation for a curve.
 
What's wrong with numerical integration using e.g. the trapezoidal rule; the error is usually less than

0.5% and can be reduced further?

 
Back to Top
beachgirl View Drop Down
Newbie
Newbie


Joined: 07 Nov 2010
Status: Offline
Points: 10
Post Options Post Options   Thanks (0) Thanks(0)   Quote beachgirl Quote  Post ReplyReply Direct Link To This Post Posted: 21 Mar 2011 at 10:20pm
Hi back again! How do you get equation for lethal curve and how to solve it?
 
Thnks!
Back to Top
 Post Reply Post Reply
  Share Topic   

Forum Jump Forum Permissions View Drop Down

Forum Software by Web Wiz Forums® version 12.03
Copyright ©2001-2019 Web Wiz Ltd.