A. Check whether the lumped system analysis is suitable. B. Determine the temperature of the copper block at t= 5 min after the immersion.
A. Check whether the lumped system analysis is suitable. B. Determine the temperature of the copper block at t= 5 min after the immersion.
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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![**Problem:**
A solid copper cube of 10-cm edge length (\(\rho\) = 8954 kg/m³, \(c\) = 383 J/kg.K, \(k\) = 386 W/m.K) initially at a uniform temperature \(T_i\) = 250°C is suddenly immersed in a well-stirred fluid which is maintained at a uniform temperature \(T_{\infty}\) = 50°C. The heat transfer coefficient between the cube and the fluid is \(h = 200 \, \text{W/m}^2.\text{K}\).
A. Check whether the lumped system analysis is suitable.
B. Determine the temperature of the copper block at \(t = 5\) min after the immersion.
---
**Graphical Explanation:**
1. **Diagrams:**
- The first diagram depicts the solid copper cube with each edge being 10 cm and the initial temperature of the cube \(T_i = 250°C\).
- The second diagram illustrates the cube being immersed in a well-stirred fluid maintained at a uniform temperature \(T_{\infty} = 50°C\), with the heat transfer coefficient \(h = 200 \, \text{W/m}^2.\text{K}\).
---
**Step-by-Step Solution:**
### Part A: Lumped System Analysis
**Lumped System Criterion:**
The lumped system analysis is applicable if the Biot number \( \text{Bi} \) is much less than 1.
\[ \text{Bi} = \frac{hL_c}{k} \]
Where:
- \( h \) = heat transfer coefficient
- \( L_c \) = characteristic length of the object
- \( k \) = thermal conductivity of the material
For a cube, the characteristic length \( L_c \) is given by:
\[ L_c = \frac{V}{A} \]
- \( V \) is the volume of the cube
\[ V = a^3 = (0.1\,m)^3 = 0.001\,m^3 \]
- \( A \) is the surface area of the cube
\[ A = 6a^2 = 6 \times (0.1\,m)^2 = 0.06\,m^2 \]
Therefore, the characteristic length \( L_c \) is:
\[ L_c =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e9db50a-4d56-435a-a058-414853800bca%2Fd00cdf2d-490e-49ff-b2fd-1df663fe1c3c%2Fiq2r0w8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
A solid copper cube of 10-cm edge length (\(\rho\) = 8954 kg/m³, \(c\) = 383 J/kg.K, \(k\) = 386 W/m.K) initially at a uniform temperature \(T_i\) = 250°C is suddenly immersed in a well-stirred fluid which is maintained at a uniform temperature \(T_{\infty}\) = 50°C. The heat transfer coefficient between the cube and the fluid is \(h = 200 \, \text{W/m}^2.\text{K}\).
A. Check whether the lumped system analysis is suitable.
B. Determine the temperature of the copper block at \(t = 5\) min after the immersion.
---
**Graphical Explanation:**
1. **Diagrams:**
- The first diagram depicts the solid copper cube with each edge being 10 cm and the initial temperature of the cube \(T_i = 250°C\).
- The second diagram illustrates the cube being immersed in a well-stirred fluid maintained at a uniform temperature \(T_{\infty} = 50°C\), with the heat transfer coefficient \(h = 200 \, \text{W/m}^2.\text{K}\).
---
**Step-by-Step Solution:**
### Part A: Lumped System Analysis
**Lumped System Criterion:**
The lumped system analysis is applicable if the Biot number \( \text{Bi} \) is much less than 1.
\[ \text{Bi} = \frac{hL_c}{k} \]
Where:
- \( h \) = heat transfer coefficient
- \( L_c \) = characteristic length of the object
- \( k \) = thermal conductivity of the material
For a cube, the characteristic length \( L_c \) is given by:
\[ L_c = \frac{V}{A} \]
- \( V \) is the volume of the cube
\[ V = a^3 = (0.1\,m)^3 = 0.001\,m^3 \]
- \( A \) is the surface area of the cube
\[ A = 6a^2 = 6 \times (0.1\,m)^2 = 0.06\,m^2 \]
Therefore, the characteristic length \( L_c \) is:
\[ L_c =
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