Energy Equations applied to pipe flow PDF

Preview

Summary

Download Energy Equations applied to pipe flow PDF

Description

Energy equations for Pipe flow problems Reynolds Transport applied to Energy: 1st Law of Thermodynamics Energy E of a system is a scalar extensive propertry the relative intensive property i.e. the value of Energy per unit mass is e =

E M

Reynolds Transport theorem states that DE Dt

(

=

∂ ∂t

 ∫CV e ρ ⅆ + ∫CSρ e v . n ⅆA

DE )system= Dt

v

D Dt

∫Systeme ρ ⅆ

v

By 1 st law of Thermodynamics

(

DE )system= Dt

D Dt

) ) ) ) ∫Systeme ρ ⅆ = ∑ Qin - ∑ Qout)system + ∑ Win - ∑ W out)system

v

) Note ∑ Win is total work done ON the system per unit time ) Note ∑ W out is total work done BY the system per unit time ) Note ∑ Qin is total heat added to the system per unit time ) Note ∑ Qout is total heat taken from the system per unit time Energy of a system is the sum of : Internal Energy U v2 Kinetic Energy M 2

Potential Energy M g z Specific energy (energy per unit mass) is e = u + ) ) ) ) ∑∑ Q in - ∑ Qoutsystem + ∑ W in - ∑ Woutsystem = v2  ∫CS u + 2 + g z ρ v . n ⅆA

Work of Pressure Forces

∂ ∂t

v2 + 2

gz

∫CV u +

v2 2

+ g z ρ ⅆ +

v

2

Energy Equations applied to pipe flow..nb

 If the boundary of the system coinciding with the CV is moving with velocity V under the action of pressure p, then the work of pressure at any point of the CS of are dA is:

dW = - p  n. v dA Hence above equation may be written ) ) ) ) ∑Qin - ∑Q out system + ∑Wshaft in - ∑W shaft out system +∫CS -p  n ⅆA = v.  ∂ v2 u + 2 ∂t ∫CV

+ g z ρ ⅆ

v

+ ∫CS u +

v2 2

+ g z ρ  v . n ⅆA

n ⅆA to the right hand side: bringing the term ∫CS -p  v.

) ) ) ) Q in - Qout system + Wshaft in - Wshaft out system = ∂ v2  u+ +g z ρ ⅆ 2 ∂t CV

p v2 u + + + g z ρ v .  n ⅆA ρ 2

v+

CS ) Where Wshaft means work EXCLUDING pressure work on the system usually this will be shaft work

Special Cases: Steady Flow ) ) ) ) Q in - Qout system + Wshaft in - Wshaft out system = 

CS

u+

p v2 n ⅆA + + g z ρ v .  ρ 2

Adiabatic Steady Flow

) ) W in - Wout system = 

CS

u+

p v2 n ⅆA + +gz ρ v. ρ 2

Application to pipe flows Starting with the steady flow equation derived above and written for a pipe with inlet and outlet:

) ) ) ) (Q in - Qout )system + (W shaft in - Wshaft out )system = -

pipe in



p v2 u + + + g z ρ v .  n ⅆA + ρ 2

pipe out

p v2 n ⅆA u + + + g z ρ v .  ρ 2

) ) ) ) where for simplicity we replaced Q and W by Q and W respectively.

Energy Equations applied to pipe flow..nb

If the flow takes place in a pipe system, we can replace velocities by their average values in the pipe, and hence the equation becomes:

) ) ) ) (Q in - Qout )system + (W shaft in - Wshaft out )system = V2 ) u+ p +gz -m +α 2 ρ

p V2 + m) u + + α + g z 2 ρ in

out

Which can be simplified further for adiabatic flow, i.e no heat exchange:

) ) (W shaft in - Wshaft out )system = p V2 ) - m u+ +α +gz 2 ρ

p V2 ) + m u+ +α +g z 2 ρ in

out

where V is the average value of v across the pipe section:

VA = ) m = α=

pipe section

pipe section

∫pipe section V2 2

v ⅆA

i.e. V =

1 v ⅆA  A pipe section

ρ v ⅆA = ρ V A

v2 2

ⅆA

A

=

We can write the energy equation for a unit mass going through the pipe as:

(wshaft in - wshaft out )system = - u+

p V2 +α +gz ρ 2

in

+

u+

V2 p +α +gz ρ 2

out

where

) W w= ) m We can write the equation as:

p V2 +α + g z + (wshaft in - wshaft out )system = ρ 2 in Which we write more simply as:

V2 p +α +gz ρ 2

out

+ uout - uin

3

4

Energy Equations applied to pipe flow..nb

p

ρ

+α

V2 + g z + wshaft net in = 2 in

the internal energy increase uout losses per unit mass flow,

this can be written as

1 ρ

V2 p +gz +α 2 ρ

+ uout - uin out - uin is due to the losses due to friction uout - uin =

ploss = f

L D

+ Σ KL

p V2 +α + g z + wshaft net in = ρ 2 in

V2 2

V2 p +α +gz ρ 2

V2 L + f + Σ KL  D 2 out

In this form, each term of the energy equation has the dimension

Joule Kg

" =

M L T -2 L M

" = L2 T -2 , let us

check: p # = M L T-2 L-2 M-1 L3 =# L2 T-2 ρ α

V2 # 2 -2 =L T 2

# L2 T-2 g z =# L T-2 L = -2 # N m s =# Joule =# M L T L =# L2 T-2 wshaft net in = M s Kg Kg

f

L D

+ Σ KL

V2 # 2 -2 V2 # = same dimensions than =L T 2 2

The equation can also be written as:

p+αρ

V2 +γz 2

in

+ ρ wshaft net in = p + α ρ

Where each term has the dimension

N m2

" =

V2 +γz 2

out

+ f

V2 L + Σ KL  ρ D 2

M L T -2 L2

" = M L -1 T -2

Or also in terms of head:

p V2 +α +z γ 2g

in

+

wshaft net in = g

V2 p +α +z γ 2

out

+ f

V2 L + Σ KL  D 2g

Where each term has the dimension of length L or head. wshaft net in > 0 in the case of a pump, the term is called pump head, energy is added to the flow g wshaft net in < 0 in the case of a turbine, g the term is called turbine head, energy is subtracted from the flow The above equation is also written

Energy Equations applied to pipe flow..nb

p

+α

V2 +z 2g

V2 p +z +α 2 γ

+ hp = in where hp is the pump head in meters, or

γ

p V2 +α +z γ 2g

in

=

V2 p +α + z γ 2

V2 L + Σ K  + f L 2g D out

V2 L + ht +  f + Σ KL  D 2g out

where ht is the turbine head in meters Note that the pump head and turbine head are related to the pump power and turbine power by the relations

) = h γ volume flowrate Pp = hp g m p ) = h γ volume flowrate Pt = ht g m t

5...