b) Three plates having thermal conductivities k2
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
Related questions
Question
Three plates having thermal conductivities k2 < k1 < k3 have excellent contact between
each other and arranged as indicated on the figure below.
The left side of the first solid and the right side of the third solid are kept at constant
temperatures T1 and T2 respectively. Sketch;
i) the qualitative temperature distribution curves at steady-state
ii) the heat fluxes at steady-state.
The left side of the first solid and the right side of the third solid are kept at constant temperatures \( T_1 \) and \( T_2 \) respectively.
**Problem Statement:**
Sketch:
i) The qualitative temperature distribution curves at steady-state.
ii) The heat fluxes at steady-state.
**Answer Explanation:**
1. **Qualitative Temperature Distribution at Steady-State:**
As heat flows from the hot side (\( T_1 \)) to the cold side (\( T_2 \)) through the three plates, the temperature distribution will be influenced by their respective thermal conductivities. Since \( k_2 < k_1 < k_3 \), the plate with \( k_2 \) (the least conductivity) will have the steepest temperature gradient, while the plate with \( k_3 \) (the highest conductivity) will have the smallest temperature gradient.
- For Plate 1 (conductivity \( k_1 \)), the temperature drop will be moderate.
- For Plate 2 (conductivity \( k_2 \)), the temperature drop will be higher.
- For Plate 3 (conductivity \( k_3 \)), the temperature drop will be the smallest.
**Sketch:**
The qualitative temperature distribution curve will show a linear decrease in temperature from \( T_1 \) to \( T_2 \), with sharper drops representing the lower conductivity material.
2. **Heat Fluxes at Steady-State:**
In a steady-state scenario, the heat flux (\( q \)) through each plate must be equal as there is no accumulation of energy within any of the plates. The heat flux can be determined using Fourier’s law:
\[
q = -k \frac{dT}{dx}
\]
Given \( k_2 < k_1 < k_3 \), the heat flux through the three plates will be the same, but since the thermal conductivities differ, the temperature gradients (\( \frac{dT}{dx} \)) will adjust accordingly to maintain a constant heat flux through all three plates.
By](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83fb07a4-f3da-4008-a352-8f832b9b60bd%2F4b3a7e36-a05a-4d14-9a7e-0205118ca184%2Fidiaa1v_processed.png&w=3840&q=75)
Transcribed Image Text:### Thermal Conductivity and Steady-State Heat Transfer
**Question:**
Three plates having thermal conductivities \( k_2 < k_1 < k_3 \) have excellent contact between each other and are arranged as indicated in the figure below.
The left side of the first solid and the right side of the third solid are kept at constant temperatures \( T_1 \) and \( T_2 \) respectively.
**Problem Statement:**
Sketch:
i) The qualitative temperature distribution curves at steady-state.
ii) The heat fluxes at steady-state.
**Answer Explanation:**
1. **Qualitative Temperature Distribution at Steady-State:**
As heat flows from the hot side (\( T_1 \)) to the cold side (\( T_2 \)) through the three plates, the temperature distribution will be influenced by their respective thermal conductivities. Since \( k_2 < k_1 < k_3 \), the plate with \( k_2 \) (the least conductivity) will have the steepest temperature gradient, while the plate with \( k_3 \) (the highest conductivity) will have the smallest temperature gradient.
- For Plate 1 (conductivity \( k_1 \)), the temperature drop will be moderate.
- For Plate 2 (conductivity \( k_2 \)), the temperature drop will be higher.
- For Plate 3 (conductivity \( k_3 \)), the temperature drop will be the smallest.
**Sketch:**
The qualitative temperature distribution curve will show a linear decrease in temperature from \( T_1 \) to \( T_2 \), with sharper drops representing the lower conductivity material.
2. **Heat Fluxes at Steady-State:**
In a steady-state scenario, the heat flux (\( q \)) through each plate must be equal as there is no accumulation of energy within any of the plates. The heat flux can be determined using Fourier’s law:
\[
q = -k \frac{dT}{dx}
\]
Given \( k_2 < k_1 < k_3 \), the heat flux through the three plates will be the same, but since the thermal conductivities differ, the temperature gradients (\( \frac{dT}{dx} \)) will adjust accordingly to maintain a constant heat flux through all three plates.
By
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