Basics of Fluid Mechanics and how it is helpful in simulation(Part1)

Topic1:
Basic Equations:
Lets start with a continuity equation which is A1V1=A2V2 , if area(A) is small the velocity(V) will be more, application would be gardening pipe..

There is an extension of this equation is Bernoulis Equation most famous energy balance equation.

P1+Rho1*g*h1+1/2Rho1*V1^2 = P2 + Rho2*g*h2+1/2Rho2*V2^2

1st term is pressure energy term , 2nd term is potential energy term and third is kinetic energy term.
Pressure in Pascals,
Density in Kg/m3,
Velocity in m/s,
h1 in meters,
g is 9.8m/s^2.
This equation can be modified in to simple cases i.e many terms will get cancel according to the given problem..

One example is if inlet and outlet of a pipe is in a same height then Potential energy term will cancel out.
P1+1/2Rho1*V1^2 = P2 +1/2Rho2*V2^2

Second Example is Leak from a tank: V1(Velocity of tank) =0 because the fluid wont move and only the V2 is in motion.
P1+Rho1*g*h1 = P2 + Rho2*g*h2+1/2Rho2*V2^2
If the tank is open to atmosphere then pressure terms will get cancel.
Lets do a simple example,
Inlet conditions of water flowing in a pipe with a conditions of 1.2 m/s and having 130 Kpa and its height is 2m and the outlet of the pipe is at 4m height and the outlet velocity is 7m/s. Calculate Outlet pressure.

It is at certain height and the velocity is more so we can say the outlet pressure will be less

130000+1000(9.8)(2)+1/2(1000)(1.2)^2 = P2 + 1000(9.8)(4)+1/2(1000)(7)^2

On substituting we will get P2 = 86.62 Kpa which is lower than the inlet pressure.

Continuity and Bernouli equations are basic equations that are used in calculating Flow meters velocities (Venturi / Orifice meter).

How to find pressure drop if U tube Manometer is installed and we know reading of Manometer.
Delp=delrho*g*h this is similar to rho*g*h

Here h is the Manometer reading.
In general, fluid in U tube Manometer will be Mercury, density of Mercury is 13600kg/m3.
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Topic2
Pumps: The objective of pump is to trasfer liquid from source to a destination.

This may be filling a tank at higher level or circulating a liquid. In both the cases pressure is required to make this work.. This is generally referred as HEAD.

HEAD = STATIC HEAD + FRICTION HEAD.

Head(feet)/2.31 = Pressure(Psi).

Please Note: Supplier/vendor terminology is HEAD and users terminology is pressure.

STATIC HEAD is the vertical distance that the liquid has to be lifted.
FRICTION HEAD is due to the fittings and size of the pipe.

The centrifugal pumps which generally creates energy due to rotation of impellers - the created energy is converted in to fluid pressure.

Pump power is P = delp*Q/eff

Q in m3/sec
P in Watts = kgm2s-3
Pressure in Pa = kg/m. s2
Eff is efficiency.

DelH = delp/rho

In steady state simulation inlet stream conditions will be given(pressure, temperature, flow  and composition) and we can provide outlet pressure or pressure rise or pressure ratio and efficiency in pump Unitop, the simulator will predict required power. This is the basic equation.

In dynamic simulation we generally provide curves, head vs flow and efficiency vs flow/power vs flow. And when pump is running at its design conditions we verify the power / efficiency.


Pump Curves:
There are 4 types of curves related to pump,
1) Head vs Flow,
2) Efficiency vs Flow
3) Power vs Flow
4) NPSH Required vs Flow

Pump Curves data extraction in OTS modelling is one of the most important work to get a proper discharge pressures of the pump.

In general there will be controller (Indirect action ) arrangement at the minimum circulation line, unless the flow develops the pump will be in minimum circulation. If flow starts developing the forward flow controller will start opening The forward controllers has to kept in auto which is a direct action controller.

When you need additional head use pumps in series and when you need additional flow use pumps in parallel arrangement.
Affinity Laws:
Volume Proportional to Speed
Head Proportional to Speed2
Power Proportional to Speed3

Small change in Speed can give significant change to the parameters.
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Topic3:
FRICTION HEAD -> Calculated based on Darcy Equation.

Darcy friction factor =4f

Hf= fd* L/D * V^2/2g

Hf-> Friction Head in Meters.
f ->fanning friction factor,
D->Pipe Diameter,
L->Length of the pipe.
V->Mean velocity of fluid.

Head loss due to fittings is calculated based on K Method.

H_fittings = K V2/2g

Example the K value for 90degree elbow having R/D=1.5 is 0.45. This is a long radius elbow.
Note: "K "value will be different for different fittings. In general these will be saved in softwares.
If fluid velocity is 4m/s
Then head loss due to fittings is 0.367m

Now the total head loss is Ht=H_fittings + Hf
For Laminar flow f = 16/Re
For Turbulent and e/d(Relative pipe Roughness)<0.001 we have f = 0.0079/Re^1/4
For greater values of e/d(Relative pipe Roughness) we need to use Moody Chart.

How to read Moody chart: 
https://www.youtube.com/watch?v=tISdp_394Bw


Steps involved in calculating Hf/Pressure Drop in a pipe is
1) Calculate Reynolds number-> Decide the region->if Re<2000 it is laminar or else Turbulent.
Ignore transition region.
2) Use the proper correlation/Moody chart for calculating f(friction factor)
and substitute all the value in Darcy Equation. to get the values like Hf and also Pressure Drop(Rho*g*Hf).Power required etc.

Simulators clearly mention the pressure drop is calculated based on Darcy friction factor.. Beggs and Brill is one of the Equation. As per these experiments they have divided in to 3 flow regimes, defined as segregated, intermittent and distributed flow - the term liquid hold up will vary based on three flow regimes. 

There Will be 3 terms in pressure drop calculations in a pipe..  friction, elevation and acceleration term. 



The drawbacks for this equation is.. it won't predict properly for high pressure gas condensate systems. As it is derived based on water - Air system. 

I think most important part is how Moody chart is converted to equation format.- Answer for this is there are many equations in literature to find the f in turbulent flow region one of them is
Colebrook - White Equation

https://en.m.wikipedia.org/wiki/Darcy_friction_factor_formulae


Reynolds Number = Intertial Force/Viscous Force = Rho*d*V/Viscosity

Rho in Kg/m3
d in Meters
V in m/s --> If flow rate is given divide it by cross section area to get in m/s
Viscosity in kg/m-s which is equivalent to N-s/m2 =Pa.s

1c.p = 1 Mpa.s = 0.001 Pa.s=0.001N-s/m2 =0.001kg/m-s.


Power units 1 Watt = 1 J/Sec = Kg m^2 s^-2
1 Hp= 736 Watts.
Valve coefficient Cv=q*(sg/delp) 0.5

Reviewed by Prof KV Rao