The goal of this page is to derive 1D mass, momentum and energy equations of a homogeneus two phase liquid-vapor flow in a pipe. The homogeneus approximation permit us to describe the two fluids as one with average properties mediated with the mass quality \chi = \Gamma_g / \Gamma.
Nomenclature:
\Gamma Mass flow rate
G Mass flux
u Velocity
\rho Density
\nu Specific volume
p Pressure
h Enthalpy
\alpha Void fraction
S Pipe surface
\Omega Pipe Cross section
\theta Pipe inclination
Subscripts
g vapor
l liquid
f friction
Steady state equations:
For simplicity derivation is performed in steady state condition, then will be generalized for a time dependent behavior.
Mass equation
Mass conservation is achived by imposing a costant mass flow rate along the pipe
\frac{d\Gamma}{dz} = 0
This can be rewritten in terms of mass flux G = \Gamma/\Omega , considering a constant section we obtain:
\frac{dG}{dz} = 0
momentum equation
Momentum equation can be derived according to the momentum inflow and outflow as in the picture:
The terms in brakets can be rearranged expliciting mixture properties as total mass flux G, for example:
\Gamma_g u_g = \frac{\Gamma_g^2\nu_g}{\Omega_g} = \nu_g\frac{\chi^2\Gamma^2}{\alpha \Omega} = \nu_g\frac{\chi^2}{\alpha}G^2\Omega
By doing the same with the other term we get what is called momentum specific volume:
\nu_m = \nu_v \frac{\chi^2}{\alpha} + \nu_l \frac{(1-\chi)^2}{(1-\alpha)}
It can be demonstrated that in Homogeneus Flow Model (HFM) \nu_m coincide with bulk specific volume \nu_b = \nu_g\chi + \nu_l(1-\chi)
Forces acting on the fluids are of three kind:
- Gravitational forces
- Pressure forces
- Wall shear
By considering an inclination \theta with respect the horizontal plane we can compute gravitational forces along the direction of the two phase flow adopting the photoraphic density \rho* . Also in this case is possible to demonstrate that in HFM \rho* = \rho_b
Finally in picture below is schematized the action of pressure and wall shear stress:
Finally making the necessary simplifications we obtain the final momentum equation:
G^2 \frac{d}{dz}\left ( \nu_g \frac{\chi^2}{\alpha} + \nu_l \frac{(1-\chi)^2}{(1-\alpha)}\right) = \frac{dp}{dz} - \tau_w \frac{S}{\Omega} - g\rho sin\theta
Changing a bit the notation becomes:
G^2 \frac{d\nu}{dz} = \frac{dp}{dz} - \frac{dp_f}{dz}(\nu,G) - \frac{dp_g}{dz}(\nu)
energy equation
Using the same resoning of the momentum balance we obtain the energy balance equation:
Where “e” is the bulk internal energy and “k” is the bulk kinetic energy.
\frac{de}{dz} + g\frac{dH}{dz} + \frac{dk}{dz} = - \frac{d}{dz}(p\nu) + \frac{1}{\Gamma}\frac{d\dot Q}{dz}
Finally we can exploit enthalpy h = e + p\nu and manipulating momentum and energy equations together:
\frac{dh}{dz} = \frac{1}{ \rho}\left [ \frac{dp}{dz} - \frac{dp_f}{dz}\right] +q_s''\frac{S}{\Omega G}
In this last step one additional simplification is adopted: the heat term is substituted with an external heat flux q”
final set of equations
finally mass, momentum and energy equations are:
\begin{cases}\frac{dG}{dz}=0 \\ \frac{dp}{dz} = \frac{dp_f}{dz} - G^2\frac{d\nu}{dz} - g\bar \rho_b sin\theta \\ \frac{dh}{dz} = \frac{1}{ \rho}\left [ \frac{dp}{dz} - \frac{dp_f}{dz}\right] +q_s''\frac{S}{\Omega G}\end{cases}