---
title: "General energy equation for control volume"
subject: "Fundamental of Thermodynamics & Heat Transfer"
chapter: 4
key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:general-energy-equation-control-volume"
source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/general-energy-equation-control-volume
site: BCT Engineering
---

# General energy equation for control volume

Fundamental of Thermodynamics & Heat Transfer · Chapter 4: First Law of Thermodynamics

- Asked in: 2070 Bhadra, 6 marks
- Video: [FLOW WORK & Energy Conservation in Open Systems in 11 Minutes!](https://www.youtube.com/watch?v=HleudqDDd9w) — Less Boring Lectures
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/general-energy-equation-control-volume
- Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-4

## Question

Derive the general energy equation for control volume.

## Short answer

The general energy equation for a control volume is dE_cv/dt = Q̇ - Ẇ_s + Σṁ_i(h_i + V_i²/2 + gz_i) - Σṁ_e(h_e + V_e²/2 + gz_e). For steady flow with one inlet and one exit, q - w = (h₂ - h₁) + (V₂² - V₁²)/2 + g(z₂ - z₁).

## Answer

**Derivation**

Consider a control volume (CV) with inlet i and exit e, over a time dt. Mass δm_i enters and δm_e leaves; heat δQ is received and shaft work δW_s is done by the CV.

```text
   m_i, P_i, v_i  ->[   CV   ]-> m_e, P_e, v_e
                     ^Q    |W_s
```

**Step 1: Apply the first law to a fixed mass (closed system).** At time t this mass is the CV contents plus the mass δm_i about to enter; at time t + dt the same mass is the CV contents plus the mass δm_e that has just left.

Energy at time t: E_cv(t) + δm_i e_i  
Energy at time t + dt: E_cv(t+dt) + δm_e e_e

where e = u + V²/2 + gz. The first law for this closed system:

δQ − δW_total = [E_cv(t+dt) + δm_e e_e] − [E_cv(t) + δm_i e_i]

**Step 2: Work terms.** The total work includes the shaft work and the flow work needed to push mass in and out:

δW_total = δW_s + δm_e P_e v_e − δm_i P_i v_i

**Step 3: Substitute** and collect terms:

δQ − δW_s = dE_cv + δm_e (e_e + P_e v_e) − δm_i (e_i + P_i v_i)

Since u + Pv = h:

δQ − δW_s = dE_cv + δm_e (h_e + V_e²/2 + g z_e) − δm_i (h_i + V_i²/2 + g z_i)

**Step 4: Divide by dt** (and sum over all inlets and exits):

**dE_cv/dt = Q̇ − Ẇ_s + Σṁ_i (h_i + V_i²/2 + g z_i) − Σṁ_e (h_e + V_e²/2 + g z_e)**

This is the general energy equation. For **steady flow** dE_cv/dt = 0, and with one inlet and one exit (ṁ constant):

q − w = (h₂ − h₁) + (V₂² − V₁²)/2 + g (z₂ − z₁)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The derivation applies the first law to:
   A. A closed system: CV contents plus the mass δm_i about to enter
   B. The CV only
   C. The exit mass only
   D. An isolated system
   Answer: A. This turns the CV into a control mass. (Key point: System choice)

2. Work terms include:
   A. Shaft work only
   B. Shaft work plus flow work δm_e P_e v_e - δm_i P_i v_i
   C. Boundary work only
   D. Zero flow work
   Answer: B. Flow work pushes mass in and out. (Key point: Flow work)

3. Combining u + Pv gives:
   A. u
   B. Ph
   C. h, so mass carries h + V²/2 + gz
   D. s
   Answer: C. Enthalpy replaces u + Pv. (Key point: Enthalpy)

4. For steady flow:
   A. dm/dt ≠ 0
   B. Q = 0
   C. W = 0
   D. dE_cv/dt = 0
   Answer: D. Gives q - w = Δh + ΔKE + ΔPE. (Key point: Steady reduction)

## Related in Fundamental of Thermodynamics & Heat Transfer

- [General mass and energy conservation for CV; unsteady](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Amass-energy-conservation-cv-unsteady) — Fundamental of Thermodynamics & Heat Transfer, chapter 4
- [Mass and energy equations for CV; gas filling process](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Amass-energy-cv-gas-filling-station) — Fundamental of Thermodynamics & Heat Transfer, chapter 4
- [Mass and energy equations for CV; gas cylinder in kitchen](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Amass-energy-cv-gas-cylinder-kitchen) — Fundamental of Thermodynamics & Heat Transfer, chapter 4
- [Mass and energy equations for CV; nozzle, condenser](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Amass-energy-cv-nozzle-condenser) — Fundamental of Thermodynamics & Heat Transfer, chapter 4
- [General displacement work; constant temperature](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Adisplacement-work-isothermal-piston) — Fundamental of Thermodynamics & Heat Transfer, chapter 2
- [First law for cyclic process; CV mass and energy equations](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Afirst-law-cyclic-mass-energy-cv) — Fundamental of Thermodynamics & Heat Transfer, chapter 4

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