THERMODYNAMICS
GATE/ESE READY
32 FORMULAS
Basic Laws & Properties
First Law
Energy Balance (Closed System)
Q - W = ΔU = mCᵥΔT
Q = Heat transfer (kJ)
W = Work done (kJ)
ΔU = Change in internal energy
Cᵥ = Specific heat at const. vol.
First Law
Steady Flow Energy Equation (SFEE)
Q - W = Δh + ΔKE + ΔPE
h = Enthalpy (kJ/kg)
KE = V²/2 (m²/s²)
PE = gz (J/kg)
Ideal Gas
Ideal Gas Law
PV = mRT = nR̄T
P = Pressure (Pa)
V = Volume (m³)
R = Specific gas constant
R̄ = 8.314 kJ/kmol·K
Specific Heats
Mayer's Relation
Cₚ - Cᵥ = R | γ = Cₚ/Cᵥ
Cₚ = Specific heat at const. pressure
γ = 1.4 (diatomic), 1.67 (monatomic)
Processes & Cycles
Polytropic
Polytropic Process Work
W = (P₁V₁ - P₂V₂)/(n-1)
n = 0: Isobaric | n = 1: Isothermal
n = γ: Isentropic | n = ∞: Isochoric
PVⁿ = const
Carnot
Carnot Efficiency & COP
η = 1 - T_L/T_H | COP = T_L/(T_H-T_L)
T_H = Source temperature (K)
T_L = Sink temperature (K)
COP_HP = T_H/(T_H-T_L)
Otto Cycle
Otto Cycle Efficiency
η_Otto = 1 - 1/rᵞ⁻¹
r = Compression ratio = V₁/V₂
γ = 1.4 for air
Diesel Cycle
Diesel Cycle Efficiency
η_Diesel = 1 - (rᶜʸ - 1)/[γ·r^(γ-1)·(rc-1)]
r = Compression ratio
rc = Cut-off ratio = V₃/V₂
Entropy
Entropy Change (Ideal Gas)
Δs = Cᵥ·ln(T₂/T₁) + R·ln(v₂/v₁)
Also: Δs = Cₚ·ln(T₂/T₁) - R·ln(P₂/P₁)
Isentropic: Δs = 0
Isentropic
Isentropic Relations
T₂/T₁ = (P₂/P₁)^((γ-1)/γ) = (V₁/V₂)^(γ-1)
Valid for reversible adiabatic process
PVᵞ = constant
Heat Transfer
Conduction
Fourier's Law
Q = -kA(dT/dx) = kA(T₁-T₂)/L
k = Thermal conductivity (W/m·K)
A = Cross-section area (m²)
Convection
Newton's Law of Cooling
Q = hA(T_s - T_∞)
h = Convective heat transfer coefficient
T_s = Surface temp | T_∞ = Fluid temp
Radiation
Stefan-Boltzmann Law
Q = εσA(T₁⁴ - T₂⁴)
σ = 5.67×10⁻⁸ W/m²K⁴
ε = Emissivity (0 to 1)
LMTD
Log Mean Temperature Difference
LMTD = (ΔT₁ - ΔT₂)/ln(ΔT₁/ΔT₂)
Q = U·A·LMTD
U = Overall heat transfer coeff.
STRENGTH OF MATERIALS
SOM / MOS
28 FORMULAS
Stress & Strain
Hooke's Law
Stress-Strain Relation
σ = E·ε | τ = G·γ
σ = Normal stress (Pa)
E = Young's modulus
G = E/[2(1+ν)]
ν = Poisson's ratio
Principal Stresses
Mohr's Circle, Principal Stress
σ₁,₂ = (σₓ+σᵧ)/2 ± √[(σₓ-σᵧ)²/4 + τ²]
τ_max = √[(σₓ-σᵧ)²/4 + τ²]
tan 2θ = 2τ/(σₓ-σᵧ)
Volumetric
Volumetric Strain
εᵥ = εₓ + εᵧ + ε_z = ΔV/V
K = E/[3(1-2ν)] (Bulk modulus)
εᵥ = p/K (Hydrostatic pressure)
Thermal
Thermal Stress
σ_thermal = -E·α·ΔT (if constrained)
α = Coefficient of thermal expansion
Free expansion: δ = αLΔT
Beams & Bending
Bending
Flexure Formula (Bending Stress)
M/I = σ/y = E/R
M = Bending moment (N·m)
I = Second moment of area (m⁴)
y = Distance from neutral axis
R = Radius of curvature
Shear
Shear Stress in Beams
τ = VQ/(Ib)
V = Shear force
Q = First moment of area
b = Width at the point
Deflection
Beam Deflection (Simply Supported, UDL)
δ_max = 5wL⁴/(384EI)
w = UDL (N/m)
Point load at center: δ = PL³/(48EI)
Cantilever: δ = PL³/(3EI)
Column
Euler's Column Buckling Load
Pₑ = π²EI/(Lₑ)²
Lₑ = Effective length
Both ends pinned: Lₑ = L
Fixed-free: Lₑ = 2L
Fixed-fixed: Lₑ = L/2
Torsion & Cylinders
Torsion
Torsion Formula
T/J = τ/r = Gθ/L
T = Torque (N·m)
J = Polar moment of inertia
J_solid = πd⁴/32
θ = Angle of twist (rad)
Thin Cylinder
Thin-Walled Pressure Vessel
σ_h = pd/(2t) | σ_l = pd/(4t)
σ_h = Hoop (circumferential) stress
σ_l = Longitudinal stress
p = Internal pressure | d = dia | t = thickness
FLUID MECHANICS
FM / HYDRAULICS
26 FORMULAS
Fluid Properties & Statics
Viscosity
Newton's Law of Viscosity
τ = μ(du/dy)
μ = Dynamic viscosity (Pa·s)
ν = μ/ρ (Kinematic viscosity, m²/s)
du/dy = Velocity gradient
Hydrostatics
Hydrostatic Pressure & Force
P = ρgh | F = ρg·ȳ·A
ȳ = Depth of centroid
Center of pressure: yₚ = ȳ + I_G/(ȳ·A)
Buoyancy
Archimedes' Principle
F_B = ρ_fluid·g·V_submerged
Floatation: F_B = W_body
Metacentric height: GM = I/V - BG
Flow Equations
Continuity
Continuity Equation
A₁V₁ = A₂V₂ = Q (incompressible)
Q = Volume flow rate (m³/s)
ρ₁A₁V₁ = ρ₂A₂V₂ (compressible)
Bernoulli
Bernoulli's Equation
P/ρg + V²/2g + z = const = H
P/ρg = Pressure head
V²/2g = Velocity head
z = Datum head
Reynolds No.
Reynolds Number
Re = ρVD/μ = VD/ν
Re < 2000: Laminar flow
Re > 4000: Turbulent flow
2000-4000: Transition zone
Pipe Flow
Darcy-Weisbach Equation
h_f = f·L·V²/(D·2g)
f = Darcy friction factor
Laminar: f = 64/Re
Turbulent: Use Moody chart
Flow Measurement
Venturimeter / Orifice Meter
Q = Cd·A₂√[2g(h)/(1-(A₂/A₁)²)]
Cd = Discharge coefficient (≈0.98 venturi)
Orifice: Cd ≈ 0.62
Boundary Layer
Boundary Layer Thickness (Laminar)
δ = 5x/√(Reₓ) = 5x/√(Vx/ν)
C_D = 1.328/√(Re_L) (flat plate, laminar)
C_D = 0.074/Re_L^0.2 (turbulent)
MACHINE DESIGN
DESIGN OF MACHINE ELEMENTS
24 FORMULAS
Static & Fatigue Failure
Factor of Safety
FOS, Static Loading
FOS = S_ut / σ_max or S_yt / σ_max
S_ut = Ultimate tensile strength
S_yt = Yield tensile strength
Von Mises
Von Mises Criterion
σ' = √(σ² + 3τ²) ≤ S_yt/FOS
Combined loading criterion
Most conservative for ductile materials
Soderberg
Soderberg Line (Fatigue)
σ_a/S_e + σ_m/S_yt = 1/FOS
σ_a = Alternating stress
σ_m = Mean stress
S_e = Endurance limit
Goodman
Modified Goodman Line
σ_a/S_e + σ_m/S_ut = 1/FOS
More realistic than Soderberg
S_e = 0.5·S_ut (steel, unmodified)
Power Screws & Springs
Power Screw
Torque to Raise / Lower Load
T_raise = W·d_m/2 · (l+πμd_m)/(πd_m-μl)
μ = Coefficient of friction
l = Lead | d_m = Mean thread dia
η = Wl/(2πT_raise)
Helical Spring
Spring Stiffness & Stress
k = Gd⁴/(8D³n) | τ = K_w·8WD/(πd³)
C = D/d (Spring index)
K_w = (4C-1)/(4C-4) + 0.615/C (Wahl factor)
δ = 8WD³n/(Gd⁴)
Gears & Bearings
Spur Gear
Lewis Bending Equation
σ_b = W_t/(b·m·Y)
W_t = Tangential force
b = Face width | m = Module
Y = Lewis form factor
Bearing
Dynamic Load Capacity & Life
L₁₀ = (C/P)³ × 10⁶/60N (hours)
C = Dynamic load capacity (kN)
P = Equivalent dynamic load
p = 3 (ball), 10/3 (roller)
Belt Drive
Belt Tension Ratio
T₁/T₂ = e^(μθ)
T₁ = Tight side | T₂ = Slack side
Power = (T₁-T₂)·V
θ = Angle of wrap (rad)
Clutch / Brake
Disc Clutch Torque (Uniform Pressure)
T = (2/3)μW(R₁³-R₂³)/(R₁²-R₂²)
W = Axial force
Uniform wear: T = μW(R₁+R₂)/2