CHE315

2.7 Overall Energy Balance

Overall Energy Balance

Objectives1. To be able to derive overall energy balance from the 1st law of

thermodynamics2. To apply the OEB on engineering problems.3. To be able to derive mechanical energy balance (MEB) from the

1st law of thermodynamics4. To apply the MEB on engineering problems

1st Law of Thermodynamics:

zgv

UE 2

2WQE The Total Energy

CHE315

2.7 Overall Energy Balance

savav WQzzgvvHH 1221

2212 2

1

Overall Energy Balance

zgv

UE 2

2

WQE PVUH

Rate of Output – Rate of Input + Rate of Accumulation = 0

CHE315 Overall Energy Balance

2.7 Overall Energy Balance

savav WQzzgvvHH 1221

2212 2

1

a is usually 0.5 for laminar flow and close to 1.0 for

turbulent flow

(for more details, see section 2.7D)

CHE315

2.7 Overall Energy Balance

Overall Energy Balance

02

1 2

1

1221

22 FW

dpzzgvv s

p

p

avav

Group Activity:Starting with the MEB equation, proof that, for a static incompressible fluid, the pressure difference can be expressed as:

(P2-P1) = rg (Z1-Z2) =rgh

CHE315

2.7 Overall Energy Balance

Overall Energy Balance

0

2

1 1212

21

22

FW

ppzzgvv savav

02

1 2

1

1221

22 FW

dpzzgvv s

p

p

avav For incompressible Fluids:

012

12

pp

zzg

For incompressible Static Fluids:

CHE315

2.7 Overall Energy Balance

Overall Energy Balance

02

1 2

1

1221

22 FW

dpzzgvv s

p

p

avav

????????????????

For compressible Static Fluids:

CHE315

2.7 Overall Energy Balance

Overall Energy Balance

02

1 2

1

1221

22 FW

dpzzgvv s

p

p

avav

02

1

12 p

p

dpzzg

For compressible Static Fluids:

Ideal Gas:pV = nRT p/r = RT/M

CHE315

2.7G Bernoulli Equation

Overall Energy Balance

02

1 2

1

1221

22 FW

dpzzgvv s

p

p

avav

No mechanical energy is added (Ws = 0)

No friction (SF = 0)

Turbulent flow

Incompressible fluid

02

1 2

P

vzg

2

22

21

21

1 22

Pvgz

Pvgz

ConstantPv

zg 2

2

CHE315 Mechanical Energy Balance

02

1 2

1

1221

22 FW

dpzzgvv s

p

p

avav

MEB Equation:http://www.youtube.com/watch?v=e4uEFCtuNic

Example 2.7-4:Example 2.7-5:Example. 2.7-6:Example. 2.7-7:

------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

CHE315 Mechanical Energy Balance

2.6D Average Velocity1. Define the average velocity mathematically.2. Apply it to the following:

A) v (x) = 5x + 2B) v (r) = vmax(1-(r/R)2) (eq. 2.6-18) Where Vmax is the maximum velocity and R is the full radius of the pipe (take v and R as constants).

2.7D Kinetic Energy Velocity Correction Factor, a:3. Integrate the kinetic energy, KE , to express KE in average velocity.4. What is a?5. Prove that a equals 0.5 for laminar flow.6. What is the value of a for turbulent flow?2.7G Bernoulli Equation for MEB7. State the Bernoulli Equation (BE).8. In which case (conditions), the BE is applied?