Derive the energy balance equation that has K, the heating rate, on the left-hand side of the equation.
Derive the energy balance equation that has K, the heating rate, on the left-hand side of the equation.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![**Energy Balance Equation Derivation**
**Problem Statement:**
Derive the energy balance equation that has \( K \), the heating rate, on the left-hand side of the equation.
**Solution Steps:**
1. **Energy Balance in a System:**
- The fundamental principle of energy balance in a closed system is written as:
\[
\Delta E_{system} = Q - W
\]
where \( Q \) is the heat added to the system and \( W \) is the work done by the system.
2. **Internal Energy Change:**
- For a process at a constant volume, the change in internal energy (\( \Delta U \)) of the system is equivalent to the heat added:
\[
\Delta U = Q
\]
3. **Define Heating Rate (K):**
- Heating rate \( K \) can be defined as the rate at which heat is added to the system.
\[
K = \frac{dQ}{dt}
\]
4. **Equation in Terms of Heating Rate:**
- The energy balance equation can now be written as:
\[
K = \frac{dU}{dt}
\]
where \( \frac{dU}{dt} \) is the rate of change of internal energy with respect to time.
By adhering to these steps, we can derive an energy balance equation that incorporates the heating rate \( K \) on the left-hand side.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd4f4b624-9180-41b5-94fa-c093ec7455d7%2F03000991-ffac-4449-95eb-6acf6d4260e2%2Fjt77lx9_processed.png&w=3840&q=75)
Transcribed Image Text:**Energy Balance Equation Derivation**
**Problem Statement:**
Derive the energy balance equation that has \( K \), the heating rate, on the left-hand side of the equation.
**Solution Steps:**
1. **Energy Balance in a System:**
- The fundamental principle of energy balance in a closed system is written as:
\[
\Delta E_{system} = Q - W
\]
where \( Q \) is the heat added to the system and \( W \) is the work done by the system.
2. **Internal Energy Change:**
- For a process at a constant volume, the change in internal energy (\( \Delta U \)) of the system is equivalent to the heat added:
\[
\Delta U = Q
\]
3. **Define Heating Rate (K):**
- Heating rate \( K \) can be defined as the rate at which heat is added to the system.
\[
K = \frac{dQ}{dt}
\]
4. **Equation in Terms of Heating Rate:**
- The energy balance equation can now be written as:
\[
K = \frac{dU}{dt}
\]
where \( \frac{dU}{dt} \) is the rate of change of internal energy with respect to time.
By adhering to these steps, we can derive an energy balance equation that incorporates the heating rate \( K \) on the left-hand side.
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