Respuesta :
Answer:
The value of K is 90461 Pa·s^(0.456) and the value of n is 0.456 (with no units).
Compleated data:
η ∂vx / ∂y
0,02 750000
0,05 450000
0,1 350000
0,2 200000
0,5 130000
1 100000
2 60000
5 35000
10 28000
20 17000
50 10000
100 8000
Step-by-step explanation:
To solve this problem we can use a Least Squares Approximation with a power function to approximate the data sample.
In this case, we have to do some mathematical work to linearize the function.
For the function selected:
[tex]\eta=K(\partial v_x/\partial y)^{n-1}\\ln(\eta)=ln(K(X)^{m})=ln(K)+ln((\partial v_x/\partial y)^{n-1})\\ln(\eta)=ln(K)+(n-1)ln((\partial v_x/\partial y))[/tex]
Now we can do the next change of variables:
[tex]ln(\eta)=Y\\ln(K)=C\\(n-1)=m\\ln((\partial v_x/\partial y))=X\\[/tex]
Therefore:
[tex]Y=C+mX[/tex]
the matrix resultant of Least Squares Approximation with the data above is:
[tex]Y=\left[\begin{array}{c}-3.91&-3&-2.3\\-1.61&-0.69&0\\0.69&1.61&2.3\\3&3.91&4.61\end{array}\right][/tex] and [tex]\left[\begin{array}{cc}1&13.53\\1&13.02\\1&12.77\\1&12.21\\1&11.78\\1&11.51\\1&11\\1&10.46\\1&10.24\\1&9.74\\1&9.21\\1&8.99\end{array}\right] \cdot \left[\begin{array}{c}C&M\end{array}\right]=A\cdot x[/tex]
We then solve the equation:
[tex]A\cdot x=Y\\A^tA\cdot x=A^tY=b[/tex]
Solving this system of 2x2, we obtain:
C=11.4126741 and m=-0.544
Therefore
[tex]C=11.4126741=ln(K)\rightarrow K=90461\\m=-0.544=(n-1)\rightarrow n=0.456[/tex]
Knowing that the viscosity has as units Pa·s and the shear rate s⁻¹, the units of the constant k is:
[tex]K=90461 Pa\cdot s^{0.456}\\[/tex]
The constant n has no units.
Answer and Step-by-step explanation
η = K ((∂Vx / ∂y)^(n-1))
-Take the natural logarithm of both sides
In η = In {K ((∂Vx / ∂y)^(n-1))}
In η = In K + In ((∂Vx / ∂y)^(n-1))
In η = In K + (n-1) In (∂Vx / ∂y)
In η = (n-1) In (∂Vx / ∂y) + In K
-Compare this relation to the equation of a straight line, y = mx + C
y = In η
m = (n-1)
x = In (∂Vx / ∂y)
C = In K
So, the data missing must be for the Viscosity, η and the shear rate, ∂Vx / ∂y
- First step in the data treatment is to take the natural logarithm of these data sets.
- This leads to a new table of data with In η and In (∂Vx / ∂y).
- Plot this new set of data on a graph with In η on the y-axis and In (∂Vx / ∂y) on the x-axis.
- The slope of this graph, m = (n-1) from the power law relation. Therefore, n = slope + 1
- And the intercept on the y-axis, c = In K, that is, K = (e^c)
So, there goes the answers to the questions, n and K.
n has no units and K has varying units depending on the value of n.
