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refining_setting_weight_percents_directly [2011/08/25 08:46]
alancoelho created
refining_setting_weight_percents_directly [2020/07/16 11:29] (current)
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 +====== Refining/​Setting Weight Percents directly ======
 +==== Version 5 ====
 +
 +The macro Known_Weight_Percent can be used to either set a weight percent or to refine a weight percent. For example:
 +
 +   ​str...
 +      ' scale parameter must not be defined for this phase
 +      Known_Weight_Percent(10) ' set weight percent to 10
 +
 +To refine the weight percent then the following could be used:
 +
 +   ​str...
 +      prm wt1 10 min 0 max 100
 +      Known_Weight_Percent(wt1)
 +
 +Only one phase can have its weight percentage set by this macro. All remaining phases must have a scale factor.
 +
 +==== Version 4.1 ====
 +
 Symbols used here are defined in the Technical Reference entry for [cell_mass !E] [cell_volume !E] [weight_percent !E]  ​ Symbols used here are defined in the Technical Reference entry for [cell_mass !E] [cell_volume !E] [weight_percent !E]  ​
  
 Ignoring the Brindley correction we can write  Ignoring the Brindley correction we can write 
- 
    Sp = Q_sum  wp / (Mp Vp (1 – wp))    Sp = Q_sum  wp / (Mp Vp (1 – wp))
- 
 where Qsum is Qp as written in the manual except that Sp Mp Vp is excluded. where Qsum is Qp as written in the manual except that Sp Mp Vp is excluded.
  
 Thus if you had four phases and you knew the weight percents w1, w2, w3, w4 then you could write: Thus if you had four phases and you knew the weight percents w1, w2, w3, w4 then you could write:
- 
    s1 = (s2 m2 + s3 m3 + s4 m4) w1 / (m1 (1 - w1))    s1 = (s2 m2 + s3 m3 + s4 m4) w1 / (m1 (1 - w1))
    s2 = (s1 m1 + s3 m3 + s4 m4) w2 / (m2 (1 - w2))    s2 = (s1 m1 + s3 m3 + s4 m4) w2 / (m2 (1 - w2))
    s3 = (s1 m1 + s2 m2 + s4 m4) w3 / (m3 (1 - w3))    s3 = (s1 m1 + s2 m2 + s4 m4) w3 / (m3 (1 - w3))
    s4 = (s1 m1 + s2 m2 + s3 m3) w4 / (m4 (1 - w4))    s4 = (s1 m1 + s2 m2 + s3 m3) w4 / (m4 (1 - w4))
- 
 where    ​ where    ​
- 
    m1 = M1 V1    m1 = M1 V1
    m2 = M2 V2    m2 = M2 V2
    m3 = M3 V3    m3 = M3 V3
    m4 = M4 V4    m4 = M4 V4
- 
 simplifying:​ simplifying:​
- 
    s1 = s2 m2 x1 + s3 m3 x1 + s4 m4 x1    s1 = s2 m2 x1 + s3 m3 x1 + s4 m4 x1
    s2 = s1 m1 x2 + s3 m3 x2 + s4 m4 x2    s2 = s1 m1 x2 + s3 m3 x2 + s4 m4 x2
    s3 = s1 m1 x3 + s2 m2 x3 + s4 m4 x3    s3 = s1 m1 x3 + s2 m2 x3 + s4 m4 x3
    s4 = s1 m1 x4 + s2 m2 x4 + s3 m3 x4    s4 = s1 m1 x4 + s2 m2 x4 + s3 m3 x4
- 
 where where
- 
    x1 = w1 / (m1 (1 - w1))    x1 = w1 / (m1 (1 - w1))
    x2 = w2 / (m2 (1 - w2))    x2 = w2 / (m2 (1 - w2))
    x3 = w3 / (m3 (1 - w3))    x3 = w3 / (m3 (1 - w3))
-   x4 = w4 / (m4 (1 - w4))                                   +   x4 = w4 / (m4 (1 - w4))     
 +    
 +    
 +    
 +If you have need for up to 9 phases, please see [[refine_weight_percentage|this macro]] for help.    
 +                                 
 We need to solve the four equations for three of the scale parameters. We will choose to solve for s1, s2 and s3.  We need to solve the four equations for three of the scale parameters. We will choose to solve for s1, s2 and s3. 
  
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 Substitute s2 into equation s1:      Substitute s2 into equation s1:     
- 
    s1 = (s1 m1 x2 + s3 m3 x2 + s4 m4 x2) m2 x1 + s3 m3 x1 + s4 m4 x1    s1 = (s1 m1 x2 + s3 m3 x2 + s4 m4 x2) m2 x1 + s3 m3 x1 + s4 m4 x1
 or,          ​ or,          ​
    s1 = (s3 (m3 x2 m2 x1 + m3 x1) + s4 (m4 x2 m2 x1 + m4 x1)) / (1 - m1 x2 m2 x1)    s1 = (s3 (m3 x2 m2 x1 + m3 x1) + s4 (m4 x2 m2 x1 + m4 x1)) / (1 - m1 x2 m2 x1)
- 
 Let Let
    t13 = (m3 x2 m2 x1 + m3 x1) / (1 - m1 x2 m2 x1)    t13 = (m3 x2 m2 x1 + m3 x1) / (1 - m1 x2 m2 x1)
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 or,          ​ or,          ​
    s1 = s3 t13 + s4 t14                                                        ​    s1 = s3 t13 + s4 t14                                                        ​
-                
 Substitute s2 into equation s3:      Substitute s2 into equation s3:     
- 
    s3 = s1 (m1 x3 + m1 x2 m2 x3) + s3 m3 x2 m2 x3 + s4 (m4 x2 m2 x3 + m4 x3)    s3 = s1 (m1 x3 + m1 x2 m2 x3) + s3 m3 x2 m2 x3 + s4 (m4 x2 m2 x3 + m4 x3)
 or,          ​ or,          ​
    s3 = (s1 (m1 x3 + m1 x2 m2 x3) + s4 (m4 x2 m2 x3 + m4 x3)) / (1 - m3 x2 m2 x3)    s3 = (s1 (m1 x3 + m1 x2 m2 x3) + s4 (m4 x2 m2 x3 + m4 x3)) / (1 - m3 x2 m2 x3)
- 
 Let          Let         
    t31 = (m1 x3 + m1 x2 m2 x3) / (1 - m3 x2 m2 x3)    t31 = (m1 x3 + m1 x2 m2 x3) / (1 - m3 x2 m2 x3)
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 or,          ​ or,          ​
    s3 = s1 t31 + s4 t34                                                        ​    s3 = s1 t31 + s4 t34                                                        ​
-                
 Substituting s1 into s3 we get:      ​ Substituting s1 into s3 we get:      ​
-                
    s3 = s3 t13 t31 + s4 t14 t31 + s4 t34    s3 = s3 t13 t31 + s4 t14 t31 + s4 t34
 or,          ​ or,          ​
    s3 = s4 (t14 t31 + t34) / (1 - t13 t31)    s3 = s4 (t14 t31 + t34) / (1 - t13 t31)
- 
 Then we set s4 to a starting refinement value and then calculate s1, s2 and s3 using the above equations, ie. Then we set s4 to a starting refinement value and then calculate s1, s2 and s3 using the above equations, ie.
- 
    s3 = s4 (t14 t31 + t34) / (1 - t13 t31)    s3 = s4 (t14 t31 + t34) / (1 - t13 t31)
    s1 = s3 t13 + s4 t14                                                        ​    s1 = s3 t13 + s4 t14                                                        ​
    s2 = s1 m1 x2 + s3 m3 x2 + s4 m4 x2    s2 = s1 m1 x2 + s3 m3 x2 + s4 m4 x2
- 
 Refining on s4 works well and the weight percents are kept to their known values. Similar equations can be written for three phases etc... Refining on s4 works well and the weight percents are kept to their known values. Similar equations can be written for three phases etc...
  
 Below is and INP file ZRO2-WT.INP (with data file ZRO2.DAT) that shows the above working nicely on Zro2. It uses the helper macro wt_dets ​ to hide some of the details. At the top of the file we see: Below is and INP file ZRO2-WT.INP (with data file ZRO2.DAT) that shows the above working nicely on Zro2. It uses the helper macro wt_dets ​ to hide some of the details. At the top of the file we see:
- 
 <code topas>​prm s4  0.01 min 1.0e-12 ​ ' REFINABLE <code topas>​prm s4  0.01 min 1.0e-12 ​ ' REFINABLE
 prm w1  5 min 1.0e-6 '​ Refine of fix depending on prior knowledge prm w1  5 min 1.0e-6 '​ Refine of fix depending on prior knowledge
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 If all of the weight percents were known then the three weight percents should be fixed to their known values. If all of the weight percents were known then the three weight percents should be fixed to their known values.
 +Here's the ZRO2-WT.INP
 <code topas>​macro wt_dets(n) <code topas>​macro wt_dets(n)
 { {
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       wt_dets(4)</​code>​       wt_dets(4)</​code>​
  
 + --- //​[[alan.coelho@bigpond.com|Alan Coelho]] 2011/08/25 19:09//
 +
 +Weights can be restrained using:
  
 +<code topas>'​---------------------------
 +'​Example of restraining weight percent to a known value
 +penalties_weighting_K1 .2
 +restraint = (Cubic_Zirconia_wt_percent - 36); : 0
 +'​---------------------------</​code>​

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