(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 & MASS TRANSFER (LL) W/CONNECT CARD

6th Edition

ISBN: 9781266204166

Author: CENGEL

Publisher: MCG

See similar textbooks

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2, Problem 101P

Chapter 2, Problem 103P

Students have asked these similar questions

I am trying to code the solution to the problem in the image in MATLAB. I wanted to know what is the milankovich constraint equation that is talked about in part b.

mylabmastering.pearson.com Chapter 12 - Lecture Notes.pptx: (MAE 272-01) (SP25) DY... P Pearson MyLab and Mastering Scores

Air modeled as an ideal gas enters an insulated compressor at a temperature of 300 K and 100 kPa, and leaves at 600 kPa. The mass flowrate of air entering the compressor is 50 kg/hr, and the power consumed by the compressor is 3 kW. (Rair = 0.287 kJ/kg-K, k = 1.4, cp = 1.0045 kJ/kg-K, cv = 0.718 kJ/kg-K) Determine the isentropic exit temperature (Te,s) of the air in [K]. Determine the actual exit temperature (Te) of the air in [K]. Determine the isentropic efficiency of the compressor. (Answer: ηc,s = 93.3%) Determine the rate of entropy generated through the compressor in [kW/K]. (Answer: Ṡgen = 0.000397 kW/K)