First project – Step 2-C: Final model

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Determination of the final mathematical model for the test.

In the previous article we saw that our flask calibration test could be represented by the following diagram;

Breaking down the input magnitudes in their contributions we would get:

  • Ml = Ml_cal + SMl_res + SMl_rep, where the suffixes “cal”, “res” and “rep” will be by calibration, instrument resolution and repeatability respectively. The prefixes “S” indicate that this component will have zero value since it is added only for the purpose of evaluation of uncertainties.
  • Mv = Mv_cal + SMv_res
  • The temperature, as we said before will be obtained from an average, so we can break down the equation in this average of readings. On the other hand, this thermometer has corrections. We could add this correction value (corr_t) to the average as a constant value without uncertainty, since this will be associated to the original values ​​of temperature.
    t = (ti + tf) / 2 + corr_t
  • but each one of these values ​​will be affected by the calibration and resolution of the thermometer. So:
    ti = ti_cal + Sti_res
    tf = tf_cal + Stf_res

Therefore to conclude this step 2:

  1. We write the complete model in the text area for the Set of equations:
    V20 = ((Ml-Mv) / (Dens_w-Dens_a)) * (1- (Dens_a / Dens_b)) * (1-CDT * (t-20))
    Ml = Ml_cal + SMl_res + SMl_rep
    Mv = Mv_cal + SMv_res
    t = (ti + tf) / 2 + corr_t
    ti = ti_cal + Sti_res
    tf = tf_cal + Stf_res
  2. We fill the parameters grid with the following data:
V20
ml
Volume contained at 20 ºC
Ml
ml
Mass of the flask full with water up to calibration mark
Mv
ml
Mass of empty flask
Dens_w
g / ml
Filling water density
Dens_a
g / ml
Density of the air
Dens_b
g / ml
Density of weighing scale adjustment masses
CDT
1 / ºC
Coefficient of thermal deformation
t
ºC
Average temperature along the test
Ml_cal
ml
Flask mass including calibration uncertainty
SM_res
ml
Contribution of uncertainty of Ml due resolution of the weighing scale
SM_rep
ml
Contribution of uncertainty of Ml due repeatability
Mv_cal
ml
Empty flask mass including calibration uncertainty
SMv_res
ml
Contribution of uncertainty of Mv due resolution of the weighing scale
ti
ºC
Initial temperature of the test
tf
ºC
Final temperature of the test
corr_t
ºC
Temperature correction
ti_cal
ºC
Initial temperature with calibration uncertainty of the termometer
Sti_res
ºC
Contribution of uncertainty of ti due thermometer resolution
tf_cal
ºC
Final temperature with calibration uncertainty of the termometer
Stf_res
ºC
Contribution of uncertainty of tf due thermometer resolution

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