FLUID FLOW (P&T Ch. 14)
FLUID FLOW
Mechanical Energy Balance
[pic]
potential expansion kinetic work added/ sum of
energy change work energy change subtracted by friction losses
pumps or
compressors
Note that the balance is per unit mass. In differential form
Rewrite as follows
[pic]
divide by dL (L is the length of pipe)
[pic]
or :
[pic]
([pic] is usually ignored, as the equation applies to a section of pipe )
The above equation is an alternative way of writing the mechanical energy balance. It is not a different equation.
The differential form of the potential energy change is
[pic] [pic]
What about the friction losses?
1) Fanning or Darcy-Weisbach equation (Often called Darcy equation)
[pic]
This equation applies for single phase fluids !!!
The friction factor is obtained from the Moody Diagram (see P&T page 482).
Friction factor equations. (Much needed in the era of computers and excel)
[pic] Laminar Flow
[pic] smooth pipes: a=0.2
Iron or steel pipes a=0.16
[pic] Colebrook equation for
turbulent flow.
Equivalent length of valves and fittings.
Pressure drop for valves and fittings is accounted for as equivalent length of pipe. Please refer to P&T for a table containing these values (page 484).
SCENARIO I
Piping is known. Need pressure drop. (Pump or compressor is not present.)
Incompressible Flow
a) Isothermal (( is constant)
[pic]
for a fixed ( ( V constant ( dV = 0
[pic]
[pic]
b) Nonisothermal
It will not have a big error if you use ((Taverage), v(Taverage)
Compressible Flow (Gasses)
a) Relatively small change in T (known)
For small pressure drop (something you can check after you are done) can use Bernoulli and fanning equation as flows
[pic]
[pic]
but [pic]
V = Velocity
v = Specific volume (m3/Kg)
G = Molar flow (Kg/hr)
A = Cross sectional area
[pic]
Now put in integral form
[pic]
Assume
[pic]
[pic]
[pic]
[pic]
The integral form will be
[pic]
Now use [pic] M; Molecular weight
Then [pic]
[pic]
Therefore ;
[pic]
but,
[pic]
[pic]
This is an equation of the form [pic]
Algorithm
a) Assume [pic]
b) Use formula to get a new value [pic]
c) Continue using [pic]
until [pic]
OR BETTER: USE Solver in EXCEL, or even better use PRO II, or any other fluid flow simulator.
CAN THIS BE APPLIED TO LONG PIPES. What is the error ?
[pic]
===> If [pic] you will be OK. What to do if not. Use shorter sections of pipe.
What if temperature change is not known
Use total energy balance as your second equation
[pic]
[pic]
[pic]
Then, (ignore (wo ,will not use when pumps or compressors are not present)
[pic]
Integrate and solve for hout (use Tav in the heat transfer equation)
[pic]
But
[pic]
[pic]
Procedure :
a) Assume Tout, pout
b) Use mechanical energy balance to obtain [pic]
c) Use total energy balance to obtain [pic]
d) get temperature [pic]
e) continue until convergence
Heat Balance
Subtract mechanical energy balance from total energy balance to get
[pic]
Integrate to get the result (use averages as before)
[pic]
How is it done in simulators?
Pipe is divided in several "short" segments and either averaging is done, or the inlet temperature is used.
SCENARIO II
Have turbine or Compressor/pump need Wo
Easy : use total energy with (q = 0 and (z = 0
[pic]
[pic]
[pic]
(h is known for turbines but not for compressors.
Therefore we need to go back to the Mechanical Energy equation for pumps/compressors. Indeed, the Bernoulli equation gives
[pic]
Pumps (( is constant)
[pic]
[pic]
For compressors
pvn = constant (The evolution is nearly isentropic)
n = Cp/Cv (Ideal gas)
n ( Cp/Cv (Real gas)
Substitute [pic] integrate to get
[pic]
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