Fluid Dynamics, Thermodynamics, and Electrical Formulas

Fluid Dynamics: Venturi and Bernoulli Principles

Venturi-s1 and s2 sections represent the areas before and after a narrowing. Constant points are maintained throughout the section by applying Bernoulli's theorem:
p1/gamma + z1 + v12 / 2g = p2/gamma + z2 + v22 / 2g.

By the principle of continuity, Q = v1s1 = v2s2. Assuming z1 = z2 and expressing the flow as a function of Q (where velocity is squared and cleared from the Bernoulli flow equation), we get:
Q = √[2g ((s1s2)2 / (s12 - s22))] · √(p1 - p2 / gamma).
Defining the first term as a constant K, the flow is Q = K · √(p1 - p2 / gamma).

Torricelli's Law and Discharge Velocity

At the Torricelli point, between the Bernoulli surface and the outlet:
Since p1 = p2 and v1 ≈ 0, the height difference z1 - z2 = h = v22 / 2g.
Thus, v2 = √(2gh), which is the expression for Torricelli's theorem.

Head Loss and Reynolds Number

Secondary Charge Loss (Head Loss): ΔHFs = ∑ (head · vi2 / 2g).
Primary Charge Loss: ΔHFp = f · (L / D) · (v2 / 2g).

• Reynolds Number (Re): Re = (f · 103 / water) · v · diameter / μ (10-3).
• Turbulent Flow: f = 0.25 / [log((ε / D) / 3.7 + 5.74 / Re0.9)]2 (Ensure same units are used!).
• Laminar Flow: f = 64 / Re.
• Kinematic Viscosity: Re = v · diameter / kinematic viscosity or v · diameter · ρ (specific gravity) / μ (dynamic viscosity).

For a static gauge: p1' = p2' = p1 + gamma1 · ∑ h1 (at point 2).

Heat Transfer and Thermal Resistance

Temperature Scales:
T(K) = T(°C) + 273.15; °C/100 = (T°F - 32) / 180.
For Temperature Differences (ΔT): Δ°C = ΔK = ΔF / 1.8.

Thermal Resistance (R):
R_wall = R_int (air barrier) + R1 + R2 + ... + R_ext (exterior air).

• Conduction: R = e / λ (in W/mK).
• Convection: R = 1 / h (in W/m²K).
• Total Transmission Coefficient (U): U = 1 / R_total.
• Fourier's Law: Q = ΔT / R_total.

Heat Transfer in Cylinders and Spheres

Through a Cylinder:
Convection: Q = S · ΔT · h.
Conduction: Q = 2πL · λ · ΔT / ln(r_ext / r_int). For multiple materials: ln(R2/R1)/λ1 + ln(R3/R2)/λ2.

Through a Sphere:
Q = h_int · ΔT · S_int = 4πR12 · h_int · (T_int - T1).
Conduction Material 1: 4π λ1 · (T1 - T2) / (1/R1 - 1/R2).
Conduction Material 2: 4π λ2 · (T2 - T3) / (1/R2 - 1/R3).
Convection Exterior: S_ext · h_ext · (T3 - T_ext).

Thermodynamics: Gas Laws and Processes

Ideal Gas Law: PV = nRT (1 atm·L = 101.3 J).

• Isobaric: V1/T1 = V2/T2; Work W12 = P(V2 - V1); Heat Q12 = nCp(T2 - T1) = W12 + ΔU12.
• Isochoric: P1/T1 = P2/T2; ΔU12 = nCv(T2 - T1); Q12 = ΔU12.
• Isothermal: P1V1 = P2V2; W12 = nRT · ln(V2/V1) or ln(P1/P2); Q12 = W12.
• Adiabatic: Q = 0; P1V1γ = P2V2γ; T1V1γ-1 = T2V2γ-1; W12 = (P1V1 - P2V2) / (γ - 1); ΔU12 = nCv(T2 - T1).

Specific Heats:
Monatomic: Cp = 5/2R, Cv = 3/2R.
Diatomic: Cp = 7/2R, Cv = 5/2R.
Adiabatic Coefficient: γ = Cp / Cv; Cp - Cv = R.

Electrical Circuits and Ohm's Law

Resistor Networks:
Parallel: Req = 1 / (1/R1 + 1/R2 + ...).
Series: Req = ∑ R_i.

Kirchhoff's Laws: ∑ E = ∑ IR.
Polarity: Enters by negative (-), exits by positive (+) = Generator. Enters by positive (+), exits by negative (-) = Motor.

Delta-Star Transformation: R12 = R1 + R2 + (R1 · R2 / R3).

Ohm's Law:
For a resistor: Vab = IR.
For a generator: Vab = Eg - I · rg.
General: V = IR.