Finding a numerical value for a partial differential

[NEGATIVE_40]

I have a function $$Z = f(P,T)$$
and would like to calculate the partial differentials $$\left ( \frac{\partial Z}{\partial P} \right )_T$$ and $$\left ( \frac{\partial Z}{\partial T} \right )_P$$ at values of P and T.

The function Z is compressibility factor (Lee and Kessler equation of state), P and T refer to pressure and temperature. The equation for Z is quite large;
$$Z = \frac{P_rv_r}{T_r} = 1 + \frac{B}{v_r} + \frac{C}{v_r^2} + \frac{D}{v_r^5}+\frac{c_4}{T_r^3v_r^2}\left ( \beta +\frac{\gamma}{v_r^2} \right )\exp\left ( -\frac{\gamma}{v_r^2} \right )$$
T_r and P_r refer to reduced pressures and temperatures:
$$T_r = \frac{T}{T_c},~~P_r = \frac{P}{P_c}$$ Other variables are constants and other expressions with T_r and P_r in them.

The solution procedure is a bit weird, but the gist of it is you solve the right hand side of the Z equality for v_r (which is pseudo variable);
$$\frac{P_rv_r}{T_r} = 1 + \frac{B}{v_r} + \frac{C}{v_r^2} + \frac{D}{v_r^5}+\frac{c_4}{T_r^3v_r^2}\left ( \beta +\frac{\gamma}{v_r^2} \right )\exp\left ( -\frac{\gamma}{v_r^2} \right )$$
then use some other relations and definitions to find Z. I have this part working correctly.

Although the expression for Z is long, I can find the derivatives manually.

Now, my question is this: can I treat v_r as being a constant when I take these derivatives manually? The value of v_r depends on both P and T when I am solving for it, so I'm not sure that I can.


Finally, if I do figure this out, how would I calculate the numerical value of:
$$\left ( \frac{\partial Z}{\partial P} \right )_T$$
Since we are holding T constant, would it be equivalent to
$$\frac{\partial Z}{\partial P}$$ , treating T as constant when taking the derivatives?

Hopefully this makes sense, and my apologies if this isn't the appropriate place to ask.

 

[arildno]

Well, if I've understood you correctly, you have introduced a pseudo-variable v_r=v_r(p,T), so that your expression for Z may be rewritten as Z=Z(v_r(P,T), P,T).
Am I correct?

Written like this, a fixed P, and differentiation of Z with respect to T will need to account for TWO effects, the change in Z as the result of change in V_r, plus the "direct" contribution in the change of Z due to T.

To take a trivial example, set Z=v_r+P+T, v_r=P+2T, you'll see you need BOTH contributions to the total change in Z.

 

[Chestermiller]

What I would suggest would be to write:
$$v_r=\frac{ZT_r}{P_r}$$
$$dv_r=\frac{T_r}{P_r}dZ+\frac{Z}{P_r}dT_r-\frac{ZT_r}{P_r^2}dP_r$$

So, $dZ = (-\frac{B}{v_r^2}-2\frac{C}{v_r^3}+...)dv_r$+ dT_r term


Substitute the second equation into this one, and solve for dZ as a function of dP_r and dT_r.