(a) Explanation Given information: The temperature of hydrocarbon fluid leaving the reservoir is 70 ° C , the surrounding temperature is 5 ° C , the inner diameter of the pipe is 0.5 m , the wall thickness is 8 mm , the average temperature is 70 ° C , the convective heat transfer at inner surface is 250 W / m 2 ⋅ K , the convective heat transfer at outer surface is 150 W / m 2 ⋅ K and the thermal conductivity is 60 W / m ⋅ K . The heat transfer in the pipe through the conduction is equal to the heat transfer through the convection. Write the expression of energy balance on the thin cylindrical shell. − k ( d T d r ) = h ( T ∞ − T r ) ......... (I) Here, the thermal conductiveis k , the temperature gradient is d T d r , the convective heat transfer coefficient is h , the temperature of water is T ∞ and the temperature of water at radius r is T r . Write the expression for one dimensional steady state heat conduction equation without internal heat generation for the cylindrical coordinates. d d r ( r d T d r ) = 0 ...... (II) Here, the radius is r . Calculation: Integrate Equation (II) on the both side. ∫ d d r ( r d T d r ) = ∫ 0 r d T d r = C 1 d T d r = C 1 r ...... (III) Here, the arbitrary constant is C 1 . Integrate Equation (III) on the both side. ∫ ( d T d r ) = ∫ C 1 r ∫ d T = ∫ C 1 ( d r r ) T ( r ) = C 1 ln r + C 2 ...... (IV) Here, the arbitrary constant is C 2 Apply the boundary condition that is at, r = r 1 , T ( r ) = T r 1 . Substitute r 1 for r and T r 1 for T ( r ) in Equation (IV). T r 1 = C 1 ln r 1 + C 2 Substitute r 2 for r and T r 2 for T ( r ) in Equation (IV). T r 2 = C 1 ln r 2 + C 2 Substitute C 1 r 1 for d T d r , h 1 for h , T ∞ , 1 for T ∞ and T r 1 for T r in Equation (I). − k ( C 1 r 1 ) = h 1 ( T ∞ , 1 − T r 1 ) ...... (V) Substitute C 1 r 2 for d T d r , h 2 for h , T r 2 for T ∞ and T ∞ , 2 for T r in Equation (I). − k ( C 1 r 2 ) = h 2 ( T r 2 − T ∞ , 2 ) ...... (VI) Substitute ( C 1 ln r 1 + C 2 ) for T r 1 in Equation (V). − k ( C 1 r 1 ) = h 1 ( T ∞ , 1 − ( C 1 ln r 1 + C 2 ) ) ∑ i = 1 n X i 2 − k ( C 1 r 1 h 1 ) = ( T ∞ , 1 − C 1 ln r 1 − C 2 ) ...... (VII) Substitute ( C 1 ln r 2 + C 2 ) for T r 2 in Equation (VI). − k ( C 1 r 2 ) = h 2 ( ( C 1 ln r 2 + C 2 ) − T ∞ , 2 ) − k ( C 1 r 2 h 2 ) = ( C 1 ln r 2 + C 2 − T ∞ , 2 ) ...... (VIII) Solve Equation (VII) and Equation (VIII) by adding both sides. − k ( C 1 r 1 h 1 ) − k ( C 1 r 2 h 2 ) = ( T ∞ , 1 − C 1 ln r 1 − C 2 ) + ( C 1 ln r 2 + C 2 − T ∞ , 2 ) − k ( C 1 r 1 h 1 ) − k ( C 1 r 2 h 2 ) = ( T ∞ , 1 − C 1 ln r 1 ) + ( C 1 ln r 2 − T ∞ , 2 ) − C 1 [ k r 1 h 1 + k r 2 h 2 + ln r 2 r 1 ] = ( T ∞ , 1 − T ∞ , 2 ) C 1 = − ( T ∞ , 1 − T ∞ , 2 ) [ k r 1 h 1 + k r 2 h 2 + ln r 2 r 1 ] Substitute − ( T ∞ , 1 − T ∞ , 2 ) [ k r 1 h 1 + k r 2 h 2 + ln r 2 r 1 ] for C 1 in equation (VII) (b) To determine The inner surface temperature of the pipeline. (c) To determine The expression for the rate for the rate of heat transfer through the pipeline. (d) To determine The expression for the rate of heat flux through the outer surface of the pipeline.

6th Edition

ISBN: 9780073398198

Author: CENGEL

Publisher: RENT MCG

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Question

Chapter 2, Problem 73P

(a)

To determine

The temperature variation in the pipe.

(b)

To determine

The inner surface temperature of the pipeline.

(c)

To determine

The expression for the rate for the rate of heat transfer through the pipeline.

(d)

To determine

The expression for the rate of heat flux through the outer surface of the pipeline.

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Chapter 2, Problem 72P

Chapter 2, Problem 74P

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