When can I use... ENGRD 2210

First Law (Closed System)

ΔU = Q − W. Applies to closed systems with no mass flow. U depends only on temperature and state. W is work BY the system (+). Use for piston, rigid tank, unsteady systems.

First Law (Steady-Flow Device)

ṁ(h₁ + ke₁ + pe₁ + q̇ − ẇs) = ṁ(h₂ + ke₂ + pe₂). Applies to steady devices with constant mass flow: turbines, compressors, nozzles, heat exchangers.

Negligible Kinetic & Potential Energy Assumption

Drop ke and pe terms when flow speeds and elevation changes are small. Use for pumps, compressors, throttling valves, heat exchangers.

Turbine Work Equation

ẇt ≈ h₁ − h₂. Assumes steady flow, adiabatic, negligible KE/PE. Use for actual and isentropic turbines.

Compressor Work Equation

ẇin ≈ h₂ − h₁. Assumes steady flow, adiabatic, negligible KE/PE. Apply to compressors, pumps (use vΔP for liquids).

Isentropic Efficiency - Turbine

ηt = (h₁ − h₂_actual)/(h₁ − h₂_s). Use when given turbine inlet state and exit pressure.

Isentropic Efficiency - Compressor

ηc = (h₂_s − h₁)/(h₂_actual − h₁). Use when inlet state and final pressure are known.

Throttling Process (Valve)

h₁ = h₂. Process is isenthalpic. Always irreversible, entropy increases. KE/PE negligible. Use for expansion valves, capillary tubes.

Nozzle/Diffuser Energy Equation

h₁ + ke₁ ≈ h₂ + ke₂. Adiabatic, negligible PE. Use for jets, rockets, flow speed changes.

Heat Exchanger (Steady Flow)

∑ ṁ h_in = ∑ ṁ h_out (adiabatic overall). No work, no kinetic changes. Use for condensers, boilers, radiators.

Ideal Gas Law

PV = mRT. Applies when gas is far from saturation (high T, low P).

Ideal Gas Internal Energy

ΔU = mCvΔT. Apply to ideal gases only. Cv constant only over small T ranges.

Ideal Gas Enthalpy

ΔH = mCpΔT. Cp constant only over limited T ranges.

Constant Specific Heats Assumption

Cp, Cv are treated as constant when temperature range is small (< ~300 K). Use for air-standard cycles.

k = Cp/Cv Relation

k = Cp/Cv. Applies for ideal gases.

Isentropic Relations (Ideal Gas)

T₂/T₁ = (P₂/P₁)^((k−1)/k), T₂/T₁ = (v₁/v₂)^(k−1), P₂/P₁ = (v₁/v₂)^k. Use for reversible adiabatic compression/expansion of ideal gases.

Entropy of Ideal Gas

Δs = Cp ln(T₂/T₁) − R ln(P₂/P₁) OR Δs = Cv ln(T₂/T₁) + R ln(v₂/v₁). Use for entropy changes when no tables are available.

Entropy Balance (Closed System)

ΔS = ∫ δQrev/T + Sgen. Use to determine reversibility or minimum heat transfer.

Entropy Balance (Control Volume)

ṁ(s₂ − s₁) = ∑ Q̇/T + Ṡgen. Use for turbines, compressors, HX, nozzles, valves.

Reversible Process Condition

Sgen = 0. Indicates ideal isentropic behavior.

Irreversible Process Condition

Sgen > 0. Applies to real compressors, turbines, throttling valves, frictional flows.

Phase-Change Enthalpy Equation

ΔH = m·hfg. Applies during boiling/condensation at constant P/T.

Vapor Quality Definition

x = (v − vf)/(vg − vf). Use for saturated mixtures. Applies to v, u, h, s.

Saturated Mixture Property Equation

y = yf + x(yfg). For any property y ∈ {v, h, u, s}. Applies in two-phase region.

Liquid Pump Equation

w_pump ≈ vΔP. Use when liquid properties do not change significantly with pressure.

Heat Engine Efficiency

η = Wout/Qin. Applies to cycles.

Carnot Efficiency

ηrev = 1 − (Tc/Th). Only valid for reversible cycles.

Refrigeration COP

COP_R = QL/Wnet,in. Use for refrigerators & heat pumps.

Heat Pump COP

COP_HP = QH/Wnet,in. Use for heating systems.

Perfect Gas + Steady Flow Combination

Drop KE/PE, use h = CpT. Applies to air compressors, air turbines, gas turbines.