Fundamentals of Engineering Exam (FE) Thermal and Fluid Systems Syllabus

Define the Mach number and the Froude number.

Mach number (compressibility): $$Ma = \frac{V}{c}$$ where $c$ is the speed of sound. Froude number (gravity/free-surface flows): $$Fr = \frac{V}{\sqrt{gL}}$$

What conditions define the critical point and the triple point of a pure substance?

The critical point is the state at which the saturated liquid and saturated vapor states become identical (above it, no distinct phase change). The triple point is the unique state where solid, liquid, and vapor phases coexist in equilibrium.

Define quality $x$ of a liquid-vapor mixture and write a property (e.g., specific volume) in terms of it.

Quality is the mass fraction of vapor: $$x = \frac{m_{vapor}}{m_{total}}$$ Any specific property: $$v = v_{f} + x \, v_{fg} = v_{f} + x (v_{g} - v_{f})$$

How is specific enthalpy $h$ defined in terms of internal energy?

$$h = u + p v$$ where $u$ is specific internal energy, $p$ is pressure, and $v$ is specific volume. It is the convenient property for flow (open) systems.

State the ideal gas equation of state in both molar and specific (mass) forms.

$$pV = n R_{u} T \qquad \text{and} \qquad p v = R T$$ where $R_{u} = 8.314\ \text{J/(mol·K)}$ is the universal gas constant and $R = R_{u}/M$ is the specific gas constant.

Relate $c_{p}$, $c_{v}$, $R$, and the specific heat ratio $k$ for an ideal gas.

$$c_{p} - c_{v} = R, \qquad k = \frac{c_{p}}{c_{v}}, \qquad c_{p} = \frac{kR}{k-1}, \qquad c_{v} = \frac{R}{k-1}$$

State the first law of thermodynamics for a closed system.

The change in internal energy equals heat added minus work done by the system: $$\Delta U = Q - W$$ or in differential form $dU = \delta Q - \delta W$.

Write the steady-flow energy equation (first law for a control volume) per unit mass.

$$q - w_{s} = \left(h_{2} - h_{1}\right) + \frac{V_{2}^{2}-V_{1}^{2}}{2} + g(z_{2}-z_{1})$$ where $q$ is heat per unit mass and $w_{s}$ is shaft work per unit mass.

For an ideal gas, how are changes in internal energy and enthalpy computed with constant specific heats?

$$\Delta u = c_{v} \, \Delta T \qquad \Delta h = c_{p} \, \Delta T$$ Both depend only on temperature for an ideal gas.

Give the boundary (moving) work for a closed system and its value for a constant-pressure process.

General: $$W = \int_{1}^{2} p \, dV$$ For constant pressure: $$W = p(V_{2} - V_{1})$$

What is the boundary work for an isothermal process of an ideal gas?

$$W = m R T \ln\!\frac{V_{2}}{V_{1}} = p_{1}V_{1}\ln\!\frac{V_{2}}{V_{1}}$$

State the Kelvin-Planck and Clausius statements of the second law of thermodynamics.

Kelvin-Planck: No cycle can produce net work while exchanging heat with only a single thermal reservoir (no 100% efficient heat engine). Clausius: Heat cannot spontaneously flow from a colder to a hotter body without external work input.

What is the thermal efficiency of a heat engine, and the maximum (Carnot) efficiency?

$$\eta_{th} = \frac{W_{net}}{Q_{H}} = 1 - \frac{Q_{L}}{Q_{H}}$$ Carnot (max) efficiency between reservoirs: $$\eta_{Carnot} = 1 - \frac{T_{L}}{T_{H}}$$ with temperatures in absolute units.

Define the coefficient of performance (COP) for a refrigerator and a heat pump.

Refrigerator: $$COP_{R} = \frac{Q_{L}}{W_{net}} = \frac{Q_L}{Q_H - Q_L}$$ Heat pump: $$COP_{HP} = \frac{Q_{H}}{W_{net}} = COP_{R} + 1$$

State the Clausius inequality and the definition of entropy change.

Clausius inequality: $$\oint \frac{\delta Q}{T} \leq 0$$ Entropy change is defined via the reversible path: $$dS = \left(\frac{\delta Q}{T}\right)_{rev}, \qquad \Delta S = \int_{1}^{2}\frac{\delta Q}{T}$$

Write the two Tds (Gibbs) relations.

$$T\,ds = du + p\,dv$$ $$T\,ds = dh - v\,dp$$

Give the entropy change of an ideal gas (constant specific heats) using temperature and pressure.

$$\Delta s = c_{p}\ln\!\frac{T_{2}}{T_{1}} - R\ln\!\frac{p_{2}}{p_{1}}$$

What is an isentropic process, and what does it require?

An isentropic process has constant entropy ($\Delta s = 0$). It is both reversible and adiabatic. It is the ideal model for pumps, turbines, compressors, and nozzles.

State the isentropic relations for an ideal gas with constant specific heat ratio $k$.

$$\frac{T_{2}}{T_{1}} = \left(\frac{p_{2}}{p_{1}}\right)^{\frac{k-1}{k}} = \left(\frac{v_{1}}{v_{2}}\right)^{k-1}, \qquad p v^{k} = \text{const}$$

Define isentropic (adiabatic) turbine efficiency.

$$\eta_{T} = \frac{w_{actual}}{w_{isentropic}} = \frac{h_{1} - h_{2a}}{h_{1} - h_{2s}}$$ the ratio of actual work output to the ideal (isentropic) work output.

List the four processes of the ideal Carnot cycle.

1) Reversible isothermal heat addition at $T_{H}$. 2) Reversible adiabatic (isentropic) expansion. 3) Reversible isothermal heat rejection at $T_{L}$. 4) Reversible adiabatic (isentropic) compression.

Name the four ideal processes of the Rankine (steam power) cycle and the components.

1) Isentropic compression in the pump. 2) Constant-pressure heat addition in the boiler. 3) Isentropic expansion in the turbine. 4) Constant-pressure heat rejection in the condenser.

List the four processes of the air-standard Otto cycle and give its thermal efficiency.

Processes: isentropic compression, constant-volume heat addition, isentropic expansion, constant-volume heat rejection. Efficiency: $$\eta_{Otto} = 1 - \frac{1}{r^{k-1}}$$ where $r$ is the compression ratio.

What distinguishes the Diesel cycle from the Otto cycle, and what is the Diesel efficiency?

The Diesel cycle adds heat at constant pressure (instead of constant volume). Efficiency: $$\eta_{Diesel} = 1 - \frac{1}{r^{k-1}}\left[\frac{r_{c}^{k}-1}{k(r_{c}-1)}\right]$$ where $r_{c}$ is the cutoff ratio.

Name the four ideal processes of the Brayton (gas turbine) cycle and its efficiency.

Isentropic compression, constant-pressure heat addition, isentropic expansion (turbine), constant-pressure heat rejection. Efficiency: $$\eta_{Brayton} = 1 - \frac{1}{r_{p}^{(k-1)/k}}$$ where $r_{p}$ is the pressure ratio.

Describe the four components/processes of the ideal vapor-compression refrigeration cycle.

1) Isentropic compression of vapor (compressor). 2) Constant-pressure heat rejection / condensation (condenser). 3) Throttling through an expansion valve (constant enthalpy). 4) Constant-pressure heat absorption / evaporation (evaporator).

For a mixture of ideal gases, state Dalton's law of partial pressures and the mole fraction relation.

Total pressure equals the sum of component partial pressures: $$p = \sum_{i} p_{i}$$ and each partial pressure is $$p_{i} = y_{i}\, p$$ where $y_{i} = n_{i}/n$ is the mole fraction.

Write the stoichiometric (theoretical) combustion reaction of methane with air and define the air-fuel ratio.

$$\ce{CH4 + 2(O2 + 3.76 N2) -> CO2 + 2H2O + 7.52 N2}$$ Air-fuel ratio (mass basis): $$AF = \frac{m_{air}}{m_{fuel}}$$

Distinguish the higher heating value (HHV) from the lower heating value (LHV) of a fuel.

HHV is the heat released when the water product is condensed to liquid (latent heat recovered). LHV is when the water remains vapor. They differ by the enthalpy of vaporization of the product water: $HHV = LHV + m_{H_2O}\, h_{fg}$.

State Fourier's law of one-dimensional steady heat conduction.

$$\dot{Q} = -k A \frac{dT}{dx}$$ where $k$ is thermal conductivity, $A$ is area, and $\frac{dT}{dx}$ is the temperature gradient. The negative sign indicates heat flows down the gradient.

Write the conduction thermal resistance for a plane wall and for a cylindrical shell.

Plane wall: $$R = \frac{L}{kA}$$ Cylinder (radial): $$R = \frac{\ln(r_{2}/r_{1})}{2\pi k L}$$ Heat rate: $\dot{Q} = \Delta T / R$.

State Newton's law of cooling for convection and define the convection resistance.

$$\dot{Q} = h A (T_{s} - T_{\infty})$$ where $h$ is the convection coefficient, $T_{s}$ surface temperature, $T_{\infty}$ fluid temperature. Convection resistance: $$R_{conv} = \frac{1}{hA}$$

Define the Nusselt, Prandtl, and Biot numbers used in heat transfer.

Nusselt (convection vs conduction in fluid): $Nu = \frac{hL}{k_{fluid}}$. Prandtl (momentum vs thermal diffusivity): $Pr = \frac{\nu}{\alpha} = \frac{\mu c_{p}}{k}$. Biot (internal vs surface resistance): $Bi = \frac{hL_{c}}{k_{solid}}$.

State the Stefan-Boltzmann law for radiation from a real surface.

$$\dot{Q} = \varepsilon \sigma A T_{s}^{4}$$ where $\varepsilon$ is emissivity ($0 \leq \varepsilon \leq 1$), $\sigma = 5.67\times10^{-8}\ \text{W/(m}^{2}\text{·K}^{4})$ is the Stefan-Boltzmann constant, and $T_{s}$ is absolute surface temperature.

Write the net radiation heat exchange between a small surface and large surroundings.

$$\dot{Q}_{net} = \varepsilon \sigma A \left(T_{s}^{4} - T_{surr}^{4}\right)$$ Temperatures must be absolute (Kelvin or Rankine).

For a heat exchanger, write the heat transfer rate using overall coefficient $U$ and the log mean temperature difference (LMTD).

$$\dot{Q} = U A \, \Delta T_{lm}, \qquad \Delta T_{lm} = \frac{\Delta T_{1} - \Delta T_{2}}{\ln(\Delta T_{1}/\Delta T_{2})}$$ where $\Delta T_1,\Delta T_2$ are end temperature differences.

Compare parallel-flow and counter-flow heat exchangers in terms of effectiveness.

In parallel flow both fluids enter at the same end and outlet temperatures approach each other (limited). In counter flow they move in opposite directions, giving a larger and more uniform $\Delta T_{lm}$, higher effectiveness, and the ability for the cold outlet to exceed the hot outlet temperature.