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Differential equations can also be developed and solved to describe Fanno flow property ratios with respect to the values at the choking location. The ratios for the pressure, density, temperature, velocity and stagnation pressure are shown below, respectively. They are represented graphically along with the Fanno parameter.
A relatively simple version [1] of the vertical fluid pressure variation is simply that the pressure difference between two elevations is the product of elevation change, gravity, and density. The equation is as follows: =, where P is pressure, ρ is density, g is acceleration of gravity, and; h is height.
If mass density is ρ, the mass of the parcel is density multiplied by its volume m = ρA dx. The change in pressure over distance dx is dp and flow velocity v = dx / dt . Apply Newton's second law of motion (force = mass × acceleration) and recognizing that the effective force on the parcel of fluid is −A dp.
The pressure value that is attempted to compute, is such that when plugged into momentum equations a divergence-free velocity field results. The mass imbalance is often also used for control of the outer loop. The name of this class of methods stems from the fact that the correction of the velocity field is computed through the pressure-field.
ρ p is the mass density of the sphere [kg/m 3] ρ f is the mass density of the fluid [kg/m 3] g is the gravitational acceleration [m/s 2] Requiring the force balance F d = F e and solving for the velocity v gives the terminal velocity v s.
The equation to calculate the pressure inside a fluid in equilibrium is: + = where f is the force density exerted by some outer field on the fluid, and σ is the Cauchy stress tensor. In this case the stress tensor is proportional to the identity tensor:
In the Boussinesq approximation, variations in fluid properties other than density ρ are ignored, and density only appears when it is multiplied by g, the gravitational acceleration. [2]: 127–128 If u is the local velocity of a parcel of fluid, the continuity equation for conservation of mass is [2]: 52
Assuming conservation of mass, with the known properties of divergence and gradient we can use the mass continuity equation, which represents the mass per unit volume of a homogenous fluid with respect to space and time (i.e., material derivative) of any finite volume (V) to represent the change of velocity in fluid media ...