Transformation matrix between species mass fraction and species density

learner00 · November 30, 2020, 02:34

The species mass fraction is defined as following $Y_s = \frac{\rho_s}{\rho}$ .

I want to compute the following derivative: $\frac{\partial Y_k}{\partial \rho_s}$
From my calculation, I thought that it should be $\frac{\partial Y_k}{\partial \rho_s} = \frac{\partial (\rho_k/\rho)}{\partial \rho_k} \frac{\partial \rho_k}{\partial \rho_s} = \dfrac{1}{\rho} \delta_{ks}$

However, I was told that the value should be $\frac{\partial Y_k}{\partial \rho_s} = \left(\delta_{ks}-Y_k\right) \frac{1}{\rho}$ .
Can you help me by pointing out where I made the mistake in the computation?

Eifoehn4 · December 3, 2020, 20:13

Your calculation is not correct. You are not allowed to put $\rho$ outside the bracket because $\rho$ is a function of $\rho_k$ or $\rho_s$ . Note that in general one suppose that the entities are $\rho_k$ and not $\rho$ .

The correct calculation is:

$\frac{\partial Y_1}{\partial \rho_1} = -\frac{\partial Y_2}{\partial \rho_1} =-\frac{\partial (\rho_2 / \rho) }{\partial \rho_1}= - \rho_2 \frac{\partial( 1 / \rho) }{\partial \rho_1}= - \rho_2 \frac{\partial ( 1 / (\rho_1 +\rho_2)) } {\partial \rho_1}=\dots$
$\dots= -\rho_2 \left(-\frac{1 }{( \rho_1 +\rho_2 )^2}\right) = -\rho_2 \left(-\frac{1 }{\rho^2}\right) = \frac{Y_2 }{\rho} = \frac{1-Y_1 }{\rho}$

The same can be done with:

$\frac{\partial Y_1}{\partial \rho_2} = -\frac{\partial Y_2}{\partial \rho_2}=\dots$

Regards

November 30, 2020, 02:34	Transformation matrix between species mass fraction and species density	#1
learner00 New Member Join Date: Nov 2020 Posts: 1 Rep Power: 0	The species mass fraction is defined as following $Y_s = \frac{\rho_s}{\rho}$ . I want to compute the following derivative: $\frac{\partial Y_k}{\partial \rho_s}$ From my calculation, I thought that it should be $\frac{\partial Y_k}{\partial \rho_s} = \frac{\partial (\rho_k/\rho)}{\partial \rho_k} \frac{\partial \rho_k}{\partial \rho_s} = \dfrac{1}{\rho} \delta_{ks}$ However, I was told that the value should be $\frac{\partial Y_k}{\partial \rho_s} = \left(\delta_{ks}-Y_k\right) \frac{1}{\rho}$ . Can you help me by pointing out where I made the mistake in the computation?

December 3, 2020, 20:13		#2
Eifoehn4 Senior Member - Join Date: Jul 2012 Location: Germany Posts: 184 Rep Power: 14	Your calculation is not correct. You are not allowed to put $\rho$ outside the bracket because $\rho$ is a function of $\rho_k$ or $\rho_s$ . Note that in general one suppose that the entities are $\rho_k$ and not $\rho$ . The correct calculation is: $\frac{\partial Y_1}{\partial \rho_1} = -\frac{\partial Y_2}{\partial \rho_1} =-\frac{\partial (\rho_2 / \rho) }{\partial \rho_1}= - \rho_2 \frac{\partial( 1 / \rho) }{\partial \rho_1}= - \rho_2 \frac{\partial ( 1 / (\rho_1 +\rho_2)) } {\partial \rho_1}=\dots$ $\dots= -\rho_2 \left(-\frac{1 }{( \rho_1 +\rho_2 )^2}\right) = -\rho_2 \left(-\frac{1 }{\rho^2}\right) = \frac{Y_2 }{\rho} = \frac{1-Y_1 }{\rho}$ The same can be done with: $\frac{\partial Y_1}{\partial \rho_2} = -\frac{\partial Y_2}{\partial \rho_2}=\dots$ Regards __________________ Check out my side project: A multiphysics discontinuous Galerkin framework: Youtube, Gitlab. Last edited by Eifoehn4; December 4, 2020 at 07:34.

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