nanofluids, rheology, mechanical engineering, chemical engineering, chemistry
The best way to obtain the values of n and K is to change the model equation in to log -log or ln-ln function, so the equation becomes a straight line and then draw it in excel, add trend line, click on "show the equation" box and you will get the equation of the straight line. And then make the anti-log of the y-intercept to get the K value and the value of n is equal to the slope. As suggested above, it is possible to do in Excel solver but if you try in different solver solution which is available in excel, you will get many more solutions for a single data. I have done so many experiments using rotational rheometer to measure the flow behaviours of the slurry and determine these indices. The accurate mechanism is the above mentioned one. Please see the attached files for more clarifications.
If you have some data and you want to fit the Herschel-Bulkley equation to it, you can use Excel's Solver add-in to do a fit. The fit will give you the parameters that you mention. Is that what you want?
Alan is right, once you are decided about the model, the fit procedure is simple. But from the question is not clear what you actually know and have: Do you know the model well?, Do you already have the experimental data?, What kind of fitting procedure are you looking for? A simple Excel Solver fit? Or are you required to rigorously justify the fitting method, confidence interval, etc?
Best regards.
Hello,
the EXCEL tool Solver is often a helpful one, but sometimes the problem is "badly conditioned" and the solver fails.
In those cases some procedure by hand is quite helpful to get good starting values for the solver. Change the dependent variable tau in (tau minus tauf) and try to fit the data with the Excel tool fitting trendlines with the power-law option. Observe the correlation coefficinet r^2 during changing tauf (value in an fixed cell of the excel sheet) manually. The value r^2 will change until reaching a climax and will getting worse again, if the tauf-value has passed the best value.
The procedure is valuable, but you have to choose the rheological model in advance and the problem has to be able to be rewritten in the offered fitting functions of the trendline feature of EXCEL.
Kind regards
Franz Werner
I think I have the same query. I have experimental data (Rheology) plotted log shear stress vs log shear rate. I have added a best-fit trend line from which I can get flow behaviour index (n) = slope, as in the file. How do I calculate K?
Kind regards,
Eimear
Dear Eimear:
For the Hersel-Bulkley model: t = t_y+KY^n,
t: shear stress, Y: Shear rate, you can define an objective function as the sum of the squares of the differences between the experimental values of log(t) experimental versus calculated values, and let the Excel Solver to fin the values of K, t_y and n that minimize that objective function. To have comparable magnitudes of all parameters, you can search for n, log(K) (instead of K), and log(t_y) (instead of t_y) and adjust your definition of objective function accordingly.
As for the initial values of the parameters you can use the ones obtained from a power law regression, which can be obtained from a line fit in an Excel graph of log(t) versus log(Y), n being the slope and log(K) being the intercept, use t_y=0 as a first guess.
The best way to obtain the values of n and K is to change the model equation in to log -log or ln-ln function, so the equation becomes a straight line and then draw it in excel, add trend line, click on "show the equation" box and you will get the equation of the straight line. And then make the anti-log of the y-intercept to get the K value and the value of n is equal to the slope. As suggested above, it is possible to do in Excel solver but if you try in different solver solution which is available in excel, you will get many more solutions for a single data. I have done so many experiments using rotational rheometer to measure the flow behaviours of the slurry and determine these indices. The accurate mechanism is the above mentioned one. Please see the attached files for more clarifications.
How to calculate the yield stress in HB model. can anyone guide with some example data of shear stress and shear rate. By using @Assefa Kebede sheet we can calculate the K and n, but what about Yield stress. Any book or paper mentioning complete procedure? Thanks
Good question, Sardar. Assefa's method is not well explained. He does a linear fit to the log-log data AFTER subtracting the yield stress from the measured stress. But we don't know the yield stress... So I don't see how it can work.
Assefa says that other solvers can give different answers, but this is due to the data, not the solver! When the problem is badly conditioned, then the result depends on the starting values.
I suggested Solver as an easy way to go, but the right way to go is to use a proper non-linear solver, for instance the function "nls" in the free R programming language (Google it!). Using powerful tools like this, you can also get error estimates on each parameter and do an automated grid search so that starting values are no longer a problem. However, you need to invest time in understanding what you are doing. There is no such thing as a free lunch :-).
As it is mentioned by Francisco Angel that for Yield Power Law (YPL), as an initial guess values of the parameters (n, K and Ty) we can use the ones obtained from a simple power law regression, and use t_y=0 as a first guess for yield T. Can anyone comment on this further with example.
thanks all for your kind attentions to this valuable question..
Regards,
Vishwas
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