# Homework Help: Checking partial derivative

1. Oct 14, 2012

### germana2006

1. The problem statement, all variables and given/known data

Hi!

I have to derivate the two phase multiplier R which is a function of the following parameters: $\dot{x}$ the steam quality, $\rho_b$ and $\zeta_b$ the density and the friction coefficient at the bubble point respectively, $\rho_d$ and $\zeta_d$ the density and the friction coefficient at the dew point respectively

2. Relevant equations

The two-phase multiplier equation is:
$$R = 1 + \dot{x} \left( \frac{\rho_b \zeta_d}{\rho_d \zeta_b} -1 \right) + 3 \dot{x}^{2,637}(1-\dot{x})^{0,565} \left( \frac{\rho_b-\rho_d}{\rho_d} \right)$$

3. The attempt at a solution

My attempt:

$$\frac{{\partial R}}{{\partial \dot{x}}} = \left( \frac{\rho_b \zeta_d}{\rho_d \zeta_b} -1 \right) - 3 \left( \frac{\rho_b-\rho_d}{\rho_d} \right) \dot{x}^{1,637} (1-\dot{x})^{0,565-1} (2,637 (\dot{x}-1)+0,565 \dot{x})$$

$$\frac{{\partial R}}{{\partial \rho_b}} = \dot{x} \left( \frac{\zeta_d}{\rho_d \zeta_b} \right) + 3 \dot{x}^{2,637} (1 - \dot{x}) ^{0,565} (\frac{1}{\rho_d})$$

$$\frac{{\partial R}}{{\partial \rho_d}} = -\dot{x} \frac{\rho_b \zeta_d}{\rho_d^2 \zeta_b} - \frac{3 \rho_b \dot{x}^{2,637} (1 - \dot{x}) ^{0,565}}{\rho_d^2}$$

$$\frac{{\partial R}}{{\partial \zeta_d}} = \frac{\dot{x} \rho_b}{\rho_d \zeta_b}$$

$$\frac{{\partial R}}{{\partial \zeta_d}} = - \frac{\dot{x} \rho_b \zeta_d}{\rho_d \zeta_b^2}$$