(a) Explanation Given information: The thermal conductivity of the wire 15.2 W / m ⋅ K , outer radius of the resistance wire is r 0 = 0.6 cm , the rate of generation of heat is 16.4 × 10 6 W / m 3 , the ambient temperature of the air is T ∞ = 100 ° C and the heat transfer coefficient of the wire is 3200 W / m 2 ⋅ K . Consider a small element of the diameter D of the wire. Write the expression for the area of the wire in the heat transfer. A = 2 π r L Here, the radius of the wire is r and length of the wire is L . Write the expression for the energy balance on the small element for time interval Δ t . Q ˙ r − Q ˙ r + Δ r + E ˙ gen,element = Δ E element Δ t ...... (I) Here, the rate of heat conduction at surface r is Q ˙ r , the rate of the heat conduction at the surface r + Δ r is Q ˙ r + Δ r , the rate of heat generation inside the element is E ˙ gen,element , the rate of the energy change of the element is Δ E element and the time period is Δ t . Write the expression for the rate of the energy change of the element. Δ E element = ρ C A Δ r ( T t + Δ t − T t ) Here, the specific heat of the element is C , the density of the element is ρ , the temperature of the element at time t is T t , the temperature of the element at time t + Δ t is T t + Δ t and the small change in the radius of the element is Δ r . Write the expression for the rate of heat generation inside the element. E ˙ gen,element = e ˙ gen A Δ r Here, the rate of small amount of heat generation inside the small element is g ˙ . Substitute ρ C A Δ r ( T t + Δ t − T t ) for Δ E element and e ˙ gen A Δ r for E ˙ gen,element in Equation (I). Q ˙ r − Q ˙ r + Δ r + e ˙ gen A Δ r = ρ C A Δ r ( T t + Δ t − T t ) Δ t Q ˙ r − Q ˙ r + Δ r + e ˙ gen A Δ r = ρ C A Δ r ( T t + Δ t − T t ) Δ t ...... (II) Divide the Equation (II) by A Δ r . Q ˙ r − Q ˙ r + Δ r A Δ r + e ˙ g e n A Δ r A Δ r = ρ C A Δ r ( T t + Δ t − T t ) Δ t A Δ r − 1 A Q ˙ r + Δ r − Q ˙ r Δ r + e ˙ g e n = ρ C ( T t + Δ t − T t ) Δ t ...... (III) Consider the limit conditions as Δ r → 0 , and Δ t → 0 . Using the definition of the derivative the heat conduction law of Fourier’s law the term Q ˙ r + Δ r − Q ˙ r Δ r on application of limits expressed as- lim Δ r → 0 Q ˙ r + Δ r − Q ˙ r Δ r = ∂ Q ∂ r = ∂ ∂ r ( − k A ∂ T ∂ r ) Take the limit Δ t → 0 and Δ r → 0 on the both side of the Equation (III). − 1 A lim Δ r → 0 Q ˙ r + Δ r − Q ˙ r Δ r + e ˙ gen = ρ C lim Δ t → 0 ( T t + Δ t − T t ) Δ t (b) To determine The relation for the variation of the temperature in the wire. (c) To determine The temperature at the center line of the wire.

Heat and Mass Transfer: Fundamentals and Applications

6th Edition

ISBN: 9781260440058

Author: CENGEL, Yunus

Publisher: MCGRAW-HILL HIGHER EDUCATION

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Question

Chapter 2, Problem 102P

(a)

To determine

The differential equation for the heat conduction through the wire.

The boundary conditions for the heat conduction through the wire.

(b)

To determine

The relation for the variation of the temperature in the wire.

(c)

To determine

The temperature at the center line of the wire.

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Chapter 2, Problem 101P

Chapter 2, Problem 103P

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