The problem was solved tp verify the performance by doing the calculations in the image. Redo the same problem, but by changing the design either to a Cylinder or Cube from Spherical.
The problem was solved tp verify the performance by doing the calculations in the image. Redo the same problem, but by changing the design either to a Cylinder or Cube from Spherical.
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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The problem was solved tp verify the performance by doing the calculations in the image. Redo the same problem, but by changing the design either to a Cylinder or Cube from Spherical.
Transcribed Image Text:### Designing the Next Generation Thermocouple
You have been asked to design the next generation thermocouple for your company's new product line. The existing thermocouple has the following design data:
- **Material**: AISI 304 Stainless Steel
- **Shape**: Spherical (2 mm diameter)
- **Heat Transfer Coefficient**: \( h = 2.2 \left(\frac{V}{D}\right)^{0.5} \)
- **Time to 0.99 ΔT**: 18 seconds
- **Minimum Flow Velocity**: 43 m/s
#### Verifying the Performance
To verify the performance of the existing thermocouple:
- **Check** that for a minimum velocity of 43 m/s, you can reach 99% of the \( T_{\infty} \) value at the center of the thermocouple in 18 seconds.
This information is crucial for ensuring that the thermocouple meets the necessary specifications for accurate temperature measurement and reliability in various applications.
![**Step 1**
**Solution:**
For AISI 304 stainless steel:
- **Density, \(\rho\)** = 8000 kg/m\(^3\)
- **Specific heat capacity, \(C\)** = 500 J/kg-K
- **Thermal Conductivity, \(K\)** = 16.2 W/m-K
**Given:**
\[
h = 2.2 \times \left( \frac{V}{D} \right)^{0.5}
\]
\[
= 2.2 \times \left(\frac{4}{3} \times 2 \times 10^{-3}\right)^{0.5}
\]
\[
= 322.583 \, \text{W/m}^2\cdot\text{K}
\]
**Biot Number:**
\[
\text{Biot number} = \frac{hL_c}{K} \quad ; \quad L_c = \text{Characteristic Length}
\]
\[
L_c = \frac{V}{A} = \frac{\frac{4}{3} \pi r^3}{4 \pi r^2} = \frac{r}{3}
\]
\[
= \frac{322.583 \times 1 \times 10^{-3}}{16.2 \times 3} = 0.0066 < 0.1
\]
So, Lumped System Approximation is applicable.
---
**Step 2**
\[
\frac{T-T_{\infty}}{T_i-T_{\infty}} = e^{-\frac{hA}{\rho CV} \cdot \tau}
\]
99% of \(T_{\infty}\):
\[
\ln\left(\frac{1}{100}\right) = -\frac{hA}{\rho CV} \cdot \tau
\]
\[
\text{or, } \frac{4.60517 \times \rho \times C \times V}{h \times A} = \tau
\]
\[
\text{or, } \frac{4.60517 \times 8000 \times 500 \times 1 \times 10^{-3}}{322.583 \times 3} = \tau
\]
\[
\text{or, } \tau](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ef60d13-9329-4571-8846-4a95d46f934f%2Fa807b58d-b84c-4f82-b46a-bdcf2dddaa40%2Fj3b4gp8_processed.png&w=3840&q=75)
Transcribed Image Text:**Step 1**
**Solution:**
For AISI 304 stainless steel:
- **Density, \(\rho\)** = 8000 kg/m\(^3\)
- **Specific heat capacity, \(C\)** = 500 J/kg-K
- **Thermal Conductivity, \(K\)** = 16.2 W/m-K
**Given:**
\[
h = 2.2 \times \left( \frac{V}{D} \right)^{0.5}
\]
\[
= 2.2 \times \left(\frac{4}{3} \times 2 \times 10^{-3}\right)^{0.5}
\]
\[
= 322.583 \, \text{W/m}^2\cdot\text{K}
\]
**Biot Number:**
\[
\text{Biot number} = \frac{hL_c}{K} \quad ; \quad L_c = \text{Characteristic Length}
\]
\[
L_c = \frac{V}{A} = \frac{\frac{4}{3} \pi r^3}{4 \pi r^2} = \frac{r}{3}
\]
\[
= \frac{322.583 \times 1 \times 10^{-3}}{16.2 \times 3} = 0.0066 < 0.1
\]
So, Lumped System Approximation is applicable.
---
**Step 2**
\[
\frac{T-T_{\infty}}{T_i-T_{\infty}} = e^{-\frac{hA}{\rho CV} \cdot \tau}
\]
99% of \(T_{\infty}\):
\[
\ln\left(\frac{1}{100}\right) = -\frac{hA}{\rho CV} \cdot \tau
\]
\[
\text{or, } \frac{4.60517 \times \rho \times C \times V}{h \times A} = \tau
\]
\[
\text{or, } \frac{4.60517 \times 8000 \times 500 \times 1 \times 10^{-3}}{322.583 \times 3} = \tau
\]
\[
\text{or, } \tau
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