Thermodynamic Analysis of an Electrical Heater Heating Water: Entropy Change and Entropy Generation

Introduction

The transfer of heat from one body to another is a fundamental concept in thermodynamics. In this article, we will analyze the process of heating 20 kg of water with a 1500 W electrical heater, focusing on the entropy change of the heater. This analysis is crucial for understanding the energy balance and entropy generation in various thermal systems.

Problem Statement and Objective

The problem involves determining the change in entropy for the heater as it transfers heat to a 20 kg water body with a specific heat capacity of 4186 J/kg·K, heated from 30°C to 80°C.

Calculation of Heat Transfer to Water

Firstly, we need to calculate the heat ((Q)) transferred to the water. Using the formula:

[Q m cdot C_p cdot Delta T]

where:

(m 20 , text{kg}) - mass of water, (C_p 4186 , text{J/kg}cdottext{K}) - specific heat capacity of water, (Delta T 80 - 30 50 , text{K}) - change in temperature.

Substituting the values:

[Q 20 , text{kg} cdot 4186 , text{J/kg}cdottext{K} cdot 50 , text{K} 4,186,000 , text{J} 4.186 , text{MJ}]

Entropy Change for the Heater

Assuming the heater temperature ((T_h)) is infinitesimally larger than the water temperature, we denote the average temperature of the water during heating as (T_w). The entropy change ((Delta S)) for the heater is given by:

[Delta S_{text{heater}} frac{Q}{T_h}]

Since (T_h) is infinitesimally larger than (T_w), we approximate (T_h) as being close to the final temperature of the water. The average temperature of the water during heating can be calculated as:

[T_w frac{T_{text{initial}} T_{text{final}}}{2} frac{30 80}{2} 55^circtext{C} 328.15 , text{K}]

Substituting the values to find the change in entropy:

[Delta S_{text{heater}} frac{4,186,000 , text{J}}{328.15 , text{K}} approx 12,756.9 , text{J/K}]

Conclusion

The change in entropy for the heater is approximately 12,756.9 J/K. This positive entropy change indicates that the heater supplies heat to the water, increasing the disorder (entropy) of the water molecules as they gain thermal energy and transition to higher energy states.

The heaters temperature remains slightly above that of the water, allowing heat to flow into the water while maintaining a small temperature difference. This process is consistent with the second law of thermodynamics, which states that heat flows from a hotter body (the heater) to a cooler body (the water).

Implications

This analysis is crucial for understanding energy efficiency and entropy generation in various thermal systems, such as residential heating, industrial processes, and renewable energy technologies. It highlights the importance of maintaining temperature differences in heat transfer processes to maximize efficiency and minimize entropy generation.