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
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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?