A machine element as seen in the figure is made of pure copper. It has an innerdiameter of D;=2.9 cm, outer diameter of Do=5.9 cm, and height of H=19 cm. The initialtemperature of the element is 749 K, and then suddenly placed in an environmentwhich has a temperature of Too=300O K. The temperature of the element is measuredas 489 K, 4.8 minutes after the cooling process. If the lumped system method isapplicable, determine the heat transfer coefficient of the environment, in W/(m²K).Note: All the surfaces should be considered to calculate the surface area.Not: Yüzey alanı hesaplamak için tüm yüzeyler dikkate alınmalıdır.Properties of pure copper:k=370 W/(mK), C=920 J/(kg°C), p=8933 kg/m³HDị

Answered: A machine element as seen in the figure…

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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A machine element as seen in the figure is made of pure copper. It has an inner
diameter of D;=2.9 cm, outer diameter of Do=5.9 cm, and height of H=19 cm. The initial
temperature of the element is 749 K, and then suddenly placed in an environment
which has a temperature of Too=300O K. The temperature of the element is measured
as 489 K, 4.8 minutes after the cooling process. If the lumped system method is
applicable, determine the heat transfer coefficient of the environment, in W/(m²K).
Note: All the surfaces should be considered to calculate the surface area.
Not: Yüzey alanı hesaplamak için tüm yüzeyler dikkate alınmalıdır.
Properties of pure copper:
k=370 W/(mK), C=920 J/(kg°C), p=8933 kg/m³
H
Dị

Expert Solution

Step 1

Given that,

$I n n e r d i a m t e r o f e l e m e n t, D_{i} = 2.9 c m, O u t e r d i a m e t e r o f e l e m e n t, D_{o} = 5.9 c m, H e i g h t o f e l e m e n t, H = 19 c m, I n i t i a l t m p e r a t u r e o f e l e m e n t, T_{o} = 749 K, T e m p e r a t u r e o f e n v i r o n e m e n t, T_{\infty} = 300 K, T e m p e r a t u r e o f e l e m e n t a f t e r c o o l i n g, T = 489 K, T i m e t o c o o l, t = 4.8 m i n u t e s,$

Properties of copper,

$T h e r m a l c o n d u c t i v i t y, k = 370 W / m \cdot K, S p e c i f i c h e a t, C = 920 J / k g \cdot ° C, D e n s i t y, ρ = 8933 k g / m^{3}$

Since lumped heat analysis is applicable then the temperature distribution of element can be given as,

$\frac{T - T_{\infty}}{T_{o} - T_{\infty}} = e^{\frac{- h A}{ρ V C} t}$

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