SFEE & Bernoulli's Equation Extended

Energy Concepts Revisited

• System Energy (E):

  • Total energy of a system is the sum of its internal (U), kinetic (KE), and potential (PE) energies, and sometimes strain energy (SE): $E = U + KE + PE + SE$

  • Change in system energy ($\Delta E$) equals work done (W) on it: $W = \Delta E = \Delta KE + \Delta PE + \Delta SE + \Delta U$

  • If no energy transfers occur, $E$ remains constant ($\Delta E = 0$).

Energy Equation

• Expresses conservation of energy.

• Energy change in a system equals energy change in surroundings.

• Energy transfers occur via heat transfer, work, or mass flow.

• General form: $Q - W = \Delta E$

  • $Q$ = energy supplied (heat).

  • $W$ = energy leaving (work).

  • $\Delta E$ = Change in energy, which can be further broken down into changes in kinetic, potential, strain, and internal energy. $Q - W = \Delta KE + \Delta PE + \Delta SE + \Delta U$

Open Systems and Control Volumes

• Open Systems: Allow mass transfer across boundaries, influencing energy change.

• Control Volume (CV): Region in space where energy and mass transfers analyzed.

• Energy equation for open systems (Steady Flow Energy Equation or SFEE):

  • $Q - W + \sum E<em>{mass</em>{in}} - \sum E<em>{mass</em>{out}} = E_{CV}$

Flow Work

• Work required to move mass into/out of a control volume.

• Also known as flow energy. $W_{Flow} = F \cdot L = PA \cdot L = PV$

• Specific flow work: $w_{Flow} = Pv$, where $v$ is specific volume ($v = \frac{1}{\rho}$).

Total Energy of Flowing Fluid

• Specific Energy (Non-Flowing): $e = u + KE + PE$

• Specific Energy (Flowing): $\theta = Pv + (u + KE + PE)$

• Including enthalpy term: $\theta = h + KE + PE$ where $h = Pv + u$

Steady Flow Energy Equation (SFEE)

• Conditions:

  • Properties within CV are constant over time.

  • Heat and work interactions with surroundings are time-invariant.

  • Volume, mass, and total energy within CV are constant.

• Equation:

  • General form: $\dot{Q} - \dot{W} + \sum \dot{E}<em>{mass</em>{in}} - \sum \dot{E}<em>{mass</em>{out}} = \dot{E}_{CV}$

  • Under steady flow ($\dot{E}<em>{CV} = 0$): $\dot{Q} - \dot{W} = \sum \dot{E}</em>{mass<em>{out}} - \sum \dot{E}</em>{mass_{in}}$

  • $\dot{Q} - \dot{W} = \dot{m}(h<em>2 + KE</em>2 + PE<em>2) - \dot{m}(h</em>1 + KE<em>1 + PE</em>1)$

  • $\dot{Q} - \dot{W} = \dot{m}\left[ (h<em>2 - h</em>1) + \frac{v<em>2^2 - v</em>1^2}{2} + g(z<em>2 - z</em>1) \right]$

• For perfect gas: $h<em>2 - h</em>1 = C<em>p(T</em>2 - T_1)$

Limitations of Bernoulli's Equation

• Specific Energy Form: $\frac{P}{\rho} + \frac{V^2}{2} + gz = constant$

• Limitations:

  • Steady flow.

  • Inviscid flow (negligible viscous effects).

  • No shaft work.

  • Incompressible flow.

  • Negligible heat transfer.

Energy Equation vs. Bernoulli Equation

• Energy equation accounts for energy transfers and changes in enthalpy.

• Bernoulli's equation is a special case with more limitations.

Extended Bernoulli Equation

• Includes terms for shaft work and energy losses.

• $\frac{P2}{\rho} + \frac{V2^2}{2} + gz2 = \frac{P1}{\rho} + \frac{V1^2}{2} + gz1 + w{shaft,net} - e{mech,loss}$

• In terms of "head" (dividing by $g$):

• $\frac{P2}{\rho g} + \frac{V2^2}{2g} + z2 = \frac{P1}{\rho g} + \frac{V1^2}{2g} + z1 + h{shaft} - h{losses}$

• Where:

  • $h{shaft} = \frac{w{shaft,net}}{g} = \frac{{\dot{W}}{shaft,net}}{{\dot{m}}g} = h{pump} - h_{turbine}$

  • $h{losses} = \frac{e{mech,loss}}{g} = \frac{{\dot{E}}_{mech,loss}}{{\dot{m}}g}$

Sign Convention

• Fluid mechanics: $Q<em>{in}$ = +ve, $W</em>{in}$ = +ve

• Pumps add energy (positive work), turbines remove energy (negative work).