Explanation Given: The temperatures of fluids in containers A and B are 0 ° C and 100 ° C respectively and the initial temperature of the bar is 100 ° C . Calculation: Consider the equation for the Newton’s Law of heating. d T d t = k ( T − T m ) (1) Here, T m is the ambient temperature, T is the temperature of the rod and k is the constant. Separate the variables and integrate the Equation (1). ∫ 1 ( T − T m ) d T = ∫ k d t ln | T − T m | = k t + c T − T m = c e k t T = c e k t + T m (2) When the bar is immersed in container A , substitute 0 for t , 0 for T m and 100 ° C for T in Equation (2). 100 = c e k ( 0 ) + 0 100 = c Substitute 100 for c and 0 for T m in Equation (2). T = 100 e k t + 0 (3) Substitute 1 for t and 90 for T in Equation (3). 90 = 100 e k ( 1 ) 9 10 = e k ln ( 9 10 ) = ln e k ln ( 9 10 ) = k Substitute ln ( 9 10 ) for k in Equation (3). T = 100 e ln ( 9 10 ) t = 100 ( 9 10 ) t (4) Substitute 99.9 for T in Equation (4). 99.9 = 100 ( 9 10 ) t 0.999 = ( 9 10 ) t ln ( 0.999 ) = t ln ( 9 10 ) ln ( 0.999 ) ln ( 9 10 ) = t Simplify the above equation

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 2.7, Problem 16E

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Find the time that will be taken by the bar to reach 99.9° C.

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Chapter 2.7, Problem 15E

Chapter 2.7, Problem 17E

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Find the time that will be taken by the bar to reach 99.9 ° C .

Solution Summary: The author analyzes the equation for the Newton's Law of Heating by comparing the temperatures of fluids in containers A and B.

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Find the time that will be taken by the bar to reach 99.9° C.

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Chapter 2 Solutions