Thermal Engineering Cheatsheet

Cheatsheet Content

### Heat Transfer - Conduction - **Modes of Heat Transfer:** Conduction, Convection, Radiation. - **Conduction Assumptions:** Steady state, 1D, constant properties, no internal heat generation. - **Generalized Conduction Equation:** - Rectangular: $\frac{\partial}{\partial x}(k_x\frac{\partial T}{\partial x}) + \frac{\partial}{\partial y}(k_y\frac{\partial T}{\partial y}) + \frac{\partial}{\partial z}(k_z\frac{\partial T}{\partial z}) + \dot{q} = \rho c_p \frac{\partial T}{\partial t}$ - Cylindrical: $\frac{1}{r}\frac{\partial}{\partial r}(k r\frac{\partial T}{\partial r}) + \frac{1}{r^2}\frac{\partial}{\partial \phi}(k \frac{\partial T}{\partial \phi}) + \frac{\partial}{\partial z}(k\frac{\partial T}{\partial z}) + \dot{q} = \rho c_p \frac{\partial T}{\partial t}$ - **Steady State 1D Conduction:** - **Plane Wall:** $Q = \frac{k A (T_1 - T_2)}{L}$, $R = \frac{L}{kA}$ - **Composite Wall:** $Q = \frac{T_{total}}{\sum R_i}$ - **Cylinder:** $Q = \frac{2 \pi k L (T_1 - T_2)}{\ln(r_2/r_1)}$, $R = \frac{\ln(r_2/r_1)}{2 \pi k L}$ - **Electrical Analogy:** Heat flow ~ Current, Temperature difference ~ Voltage, Thermal Resistance ~ Electrical Resistance. - **Extended Surfaces (Fins):** - **Purpose:** Increase heat transfer area. - **Governing Equation (simplified):** $\frac{d^2\theta}{dx^2} - m^2\theta = 0$, where $\theta = T - T_\infty$, $m = \sqrt{\frac{hP}{kA_c}}$. - **Fin Effectiveness ($\epsilon_f$):** $\frac{Q_{fin}}{Q_{no-fin}} = \frac{\sqrt{hPkA_c}\theta_b \tanh(mL)}{hA_{base}\theta_b}$ - **Fin Efficiency ($\eta_f$):** $\frac{Q_{actual}}{Q_{ideal}} = \frac{\tanh(mL)}{mL}$ ### Heat Transfer - Convection and Radiation #### Convection - **Heat Transfer Coefficient ($h$):** $Q = h A (T_s - T_\infty)$. Determined experimentally or via correlations. - **Dimensional Analysis:** Used to group variables into dimensionless numbers. - **Dimensionless Numbers:** - **Nusselt (Nu):** $\frac{hL}{k}$ (ratio of convective to conductive heat transfer) - **Reynolds (Re):** $\frac{\rho VL}{\mu}$ (ratio of inertial to viscous forces, indicates flow regime) - **Prandtl (Pr):** $\frac{\nu}{\alpha} = \frac{\mu c_p}{k}$ (ratio of momentum diffusivity to thermal diffusivity) - **Grashof (Gr):** $\frac{g \beta (T_s - T_\infty) L^3}{\nu^2}$ (ratio of buoyancy to viscous forces, significant in natural convection) - **Rayleigh (Ra):** $Gr \cdot Pr$ (important for natural convection) #### Radiation - **Basic Laws:** - **Planck's Law:** Describes spectral radiance of electromagnetic radiation emitted by a black body. - **Stefan-Boltzmann Law:** $E_b = \sigma T^4$ (total emissive power of a black body). - **Kirchhoff's Law:** $\alpha = \epsilon$ (absorptivity equals emissivity for a given wavelength and temperature). - **Wien's Displacement Law:** $\lambda_{max} T = 2.898 \times 10^{-3} \text{ m} \cdot \text{K}$ (relates peak emission wavelength to temperature). - **Lambert's Cosine Law:** Radiation intensity from a diffuse surface is proportional to the cosine of the angle from the normal. - **Black Body:** Ideal emitter and absorber of radiation ($\epsilon = 1, \alpha = 1$). - **Radiation Intensity ($I$):** Power emitted per unit solid angle per unit projected area. - **Radiation Heat Exchange between Black Bodies:** $Q_{12} = A_1 F_{12} \sigma (T_1^4 - T_2^4)$, where $F_{12}$ is the view factor. ### Heat Exchanger - **Types:** Shell-and-tube, plate, double pipe, compact, etc. Classified by flow arrangement (parallel, counter, cross-flow). - **Overall Heat Transfer Coefficient ($U$):** $Q = U A \Delta T_{mean}$. Accounts for all thermal resistances. - **Fouling Factor ($R_f$):** Additional thermal resistance due to deposit formation on heat exchanger surfaces. $U_{dirty} = \frac{1}{\frac{1}{U_{clean}} + R_{f,i} + R_{f,o}}$. - **Analysis Methods:** - **Log Mean Temperature Difference (LMTD) Method:** - $\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}$ - $Q = U A \Delta T_{lm}$ - Applicable for constant overall heat transfer coefficient and specific flow arrangements. - **Effectiveness-NTU Method:** - **Effectiveness ($\epsilon$):** $\frac{Q_{actual}}{Q_{max}}$ - **Number of Transfer Units (NTU):** $\frac{UA}{C_{min}}$ - Useful when outlet temperatures are unknown. - **Correction Factor ($F$):** Used for multi-pass and cross-flow heat exchangers with LMTD method: $Q = U A F \Delta T_{lm, counter}$. ### Internal Combustion Engine - **Classification:** SI (Spark Ignition), CI (Compression Ignition); 2-stroke, 4-stroke; reciprocating, rotary. - **Parts & Materials:** Cylinder block (cast iron/aluminum), cylinder head, piston (aluminum alloy), connecting rod (steel), crankshaft (forged steel), valves, camshaft. - **Cycles of Operation:** - **4-stroke:** Intake, Compression, Power, Exhaust. One power stroke per two crankshaft revolutions. - **2-stroke:** Intake/Compression (simultaneous), Power/Exhaust (simultaneous). One power stroke per crankshaft revolution. Simpler design, higher power-to-weight ratio, less fuel efficient, higher emissions. - **Air-Standard Cycles:** - **Otto Cycle:** Constant volume heat addition/rejection (ideal for SI engines). - **Diesel Cycle:** Constant pressure heat addition (ideal for CI engines). - **Dual Cycle:** Combination of constant volume and constant pressure heat addition (more realistic for CI engines). - **Performance Parameters:** - **Brake Power (BP):** Power delivered at the crankshaft. - **Indicated Power (IP):** Power developed inside the cylinder. $IP = BP + FP$. - **Frictional Power (FP):** Power lost to friction. - **Fuel Consumption:** Mass of fuel consumed per unit time. - **Air Flow:** Mass of air consumed per unit time. - **Brake Mean Effective Pressure (BMEP):** Average pressure that, if acting on the piston during the power stroke, would produce the brake work. $BMEP = \frac{BP \times n \times 2}{V_d \times N}$ (for 4-stroke). - **Efficiencies:** - **Mechanical Efficiency ($\eta_m$):** $\frac{BP}{IP}$ - **Indicated Thermal Efficiency ($\eta_{it}$):** $\frac{IP}{\dot{Q}_{in}}$ - **Brake Thermal Efficiency ($\eta_{bt}$):** $\frac{BP}{\dot{Q}_{in}}$ - **Volumetric Efficiency ($\eta_v$):** $\frac{\text{Actual Air Volume}}{\text{Swept Volume}}$ - **Heat Balance Sheet:** Accounts for the distribution of fuel energy (BP, cooling water, exhaust gases, unaccounted losses). ### Steam Generator and Turbine #### Steam Generators (Boilers) - **Classification:** - **Fire Tube:** Hot gases pass through tubes surrounded by water (e.g., Lancashire, Locomotive). Lower pressure, simpler. - **Water Tube:** Water passes through tubes surrounded by hot gases (e.g., Babcock & Wilcox, Stirling). Higher pressure and capacity, safer. - **Low Pressure/High Pressure Boilers:** Based on operating pressure. - **Once-Through Boiler:** Water enters one end and exits as superheated steam at the other, no drum (e.g., Benson boiler). - **Mountings:** Safety valves, water level indicator, pressure gauge, fusible plug, stop valve. - **Accessories:** Economizer, superheater, air preheater, feed pump. - **Equivalent Evaporation:** Evaporation of water from and at 100°C per kg of fuel. - **Boiler Performance:** Measured by steam generation rate, fuel consumption, and efficiency. - **Boiler Efficiency ($\eta_b$):** $\frac{\text{Heat in steam}}{\text{Heat in fuel}}$ #### Steam Turbine - **Basics:** Converts thermal energy of steam into mechanical work via rotating blades. - **Classification:** Impulse, Reaction, Impulse-Reaction. - **Compounding:** Methods to reduce rotor speed (e.g., velocity compounding, pressure compounding, pressure-velocity compounding). - **Impulse Turbine:** - Steam expands completely in fixed nozzles, then strikes moving blades. - Velocity Diagram: Inlet and outlet triangles for steam velocity, blade velocity. - Condition for Max Efficiency (single stage): $u/V_1 = \cos\alpha_1 / 2$ (for equiangular blades). - **Reaction Turbine:** - Steam expands partly in fixed blades (nozzles) and partly in moving blades. - Velocity Diagram: Similar to impulse, but includes relative velocity changes due to expansion in moving blades. - **Degree of Reaction (R):** Ratio of enthalpy drop in moving blades to total enthalpy drop per stage. - **Parsons Turbine:** A common type of reaction turbine with 50% degree of reaction. - Condition for Max Efficiency (Parsons type): $u/V_1 = \cos\alpha_1$. ### Refrigeration - **Methods:** Vapor Compression, Vapor Absorption, Air Refrigeration, Thermoelectric, Vortex tube. #### Vapour Compression Refrigeration (VCR) System - **Components:** Compressor, Condenser, Expansion Valve, Evaporator. - **Simple VCR Cycle:** 1. **Compression (1-2):** Isentropic compression of low-pressure vapor to high-pressure vapor. 2. **Condensation (2-3):** Constant pressure heat rejection in condenser, high-pressure vapor to high-pressure liquid. 3. **Expansion (3-4):** Isenthalpic expansion of high-pressure liquid through expansion valve to low-pressure liquid-vapor mixture. 4. **Evaporation (4-1):** Constant pressure heat absorption in evaporator, low-pressure liquid-vapor mixture to low-pressure vapor. - **Effect of Subcooling:** Cooling the liquid refrigerant below saturation temperature before expansion. Increases COP and refrigerating effect. - **Effect of Superheating:** Heating the vapor refrigerant above saturation temperature before compression. - **Useful Superheating:** Superheating in evaporator increases refrigerating effect. - **Harmful Superheating:** Superheating in suction line increases compressor work. - **P-h Charts:** Pressure-enthalpy diagrams used for analyzing VCR cycles, determining COP, and state points. #### Vapour Absorption Refrigeration (VAR) System - **Importance:** Uses heat energy (waste heat, solar) instead of mechanical work for compression. - **Components:** Generator, Absorber, Pump, Condenser, Evaporator, Expansion Valve, Heat Exchanger (optional). - **COP of Ideal VAR System:** $\frac{T_E (T_G - T_C)}{T_G (T_C - T_E)}$ (for heat sources at $T_G$, heat sinks at $T_C$, evaporator at $T_E$). - **Ammonia-Water VAR System:** Ammonia is refrigerant, water is absorbent. Used for industrial applications, lower temperatures. - **Lithium Bromide – Water VAR System:** Water is refrigerant, lithium bromide is absorbent. Used for air conditioning, higher evaporator temperatures. - **Single Effect vs. Double Effect:** Double effect systems use the heat rejected from the first generator to drive a second generator, increasing COP but requiring higher heat source temperature. - **Psychrometric Chart:** Not directly related to VAR system operation, but used for air conditioning applications which VAR systems often serve.