gen-define-the-energy-balance-equation-and-give-one-short-worked-79b5

Define the energy balance equation and give one short worked example.

# Energy Balance Equation ## Definition The **energy balance equation** states that the change in energy of a system equals the energy transferred into the system minus the energy transferred out: $$\Delta E_{system} = E_{in} - E_{out}$$ For an **open system** (with mass flow), the general form is: $$\frac{dE_{system}}{dt} = \dot{Q} - \dot{W} + \sum_{in} \dot{m}h_{in} - \sum_{out} \dot{m}h_{out}$$ Where: - $\dot{Q}$ = heat transfer rate into the system - $\dot{W}$ = work done by the system - $\dot{m}h$ = energy flow with mass (enthalpy) For a **closed system** (no mass flow), this simplifies to the **First Law of Thermodynamics**: $$\Delta U = Q - W$$ --- ## Worked Example **Problem:** A rigid tank contains 2 kg of water. A heater adds 500 kJ of heat to the water while a stirrer does 100 kJ of work on it. Find the change in internal energy. **Solution:** Since the tank is rigid (no volume change), no boundary work is done by the system. Given: - Q = +500 kJ (heat added to system) - W = -100 kJ (work done ON system, negative in our convention) - System is closed (no mass flow) Applying the energy balance: $$\Delta U = Q - W = 500 - (-100) = 600 \text{ kJ}$$ **Answer:** The internal energy of the water increases by **600 kJ**.