Explanation Given information: Diameter of the fin is 10 mm , length of the fin is 50 mm , surface temperature of the wall is 350 ° C ,thermal conductivity of the fin material is 240 W / m ⋅ K , ambient air temperature is 25 ° C and heat transfer coefficient of air is 250 W / m 2 ⋅ K . Write the expression for the cross sectional area of the fin. A c s = π 4 d 2 ...... (I) Here cross sectional area of the fin is A c s diameter of the fin is d . Write the expression for the perimeter of the circular fin. p = π d ...... (II) Here perimeter of the fin is p diameter of the fin is d . Write the expression for value m . m = h p k A c s ...... (III) Here heat transfer coefficient of the air is h , perimeter of fin is p , thermal conductivityof the fin is k and cross sectional area of the fin is A c s . Write the expression for m L . m L = h p k A c s L ...... (IV) Here length of the fin is L . Write the expression for h p k A c s . γ = h p k A c s ...... (V) Write the expression for the heat transfer rate of infinitely long fin. Q · inf , f i n = γ ⋅ ( T b − T ∞ ) ...... (VI) Here heat transfer rate of infinitely long fin is Q inf , f i n , heat transfer coefficient of the air is h , perimeter of fin is p , thermal conductivity of the fin is k cross sectional area of the fin is A c s ,base temperature of fin is T b and ambient temperature is T ∞ . Write the expression for the temperature distribution or temperature variation for the infinitely long fin. T ( x ) − T ∞ T b − T ∞ = e − m x ...... (VII) Here temperature at a distance of x is T ( x ) , ambient temperature is T ∞ ,and base temperature of the fin is T b . Write the expression for the heat transfer rate of adiabatic tip fin. Q · a d i a b t i c , f i n = γ ⋅ ( T b − T ∞ ) tanh ( m L ) ...... (VIII) Here heat transfer rate of adiabatic finis Q a d i a b a t i c , f i n , length of the fin is L . Write the expression for the temperature distribution or temperature variation for the adiabatic tip fin. T ( x ) − T ∞ T b − T ∞ = cosh [ m ( L − x ) ] cosh ( m L ) ...... (IX) Here temperature at a distance of x is T ( x ) , ambient temperature is T ∞ , wall or base temperature is T b and length of the fin is L . Write the expression for the loss of heat when tip temperature of fin is 250 ° C . Q · s p e c i f i c t e m p = γ ⋅ ( T b − T ∞ ) cosh ( m L ) − [ ( T L − T ∞ ) ( T b − T ∞ ) ] sinh ( m L ) ...... (X) Here heat transfer rate of specified temperature fin is Q · s p e c i f i c t e m p . Write the expression for the temperature distribution or temperature variation for the specified fin temperature. T ( x ) − T ∞ T b − T ∞ = [ ( T L − T ∞ ) ( T b − T ∞ ) ] sinh ( m x ) + sinh [ m ( L − x ) ] sinh ( m L ) ...... (XI) Here specified temperature is T L . Write the expression for the rate of heat transfer by the fin when fin loosing heat by the convection. Q · c o n v . = γ ⋅ ( T b − T ∞ ) sinh ( m L ) + ( h m k ) cosh ( m L ) cosh ( m L ) + ( h m k ) sinh ( m L ) ...... (XII) Here rate of heat transfer of fin loosing heat by convection is Q · c o n v . , thermal conductivity of the fin is k and length of the fin is L . Write the expression for the temperature distribution or temperature variation along thefin loosing heat by convection. T ( x ) − T ∞ T b − T ∞ = cosh [ m ( L − x ) ] + ( h m k ) sinh [ m ( L − x ) ] cosh ( m L ) + ( h m k ) sinh ( m L ) To determine (b) The heat transfer rate of adiabatic tip fin. To determine (c) The heat transfer rate of fin when tip temperature is 250 ° C . To determine (d) The heat transfer rate of the fin loosing heat by convection.

Heat and Mass Transfer: Fundamentals and Applications

5th Edition

ISBN: 9780073398181

Author: Yunus A. Cengel Dr., Afshin J. Ghajar

Publisher: McGraw-Hill Education

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Question

Chapter 3, Problem 185P

To determine

(a)

The heat transfer rate of infinitely long fin.

To determine

(b)

The heat transfer rate of adiabatic tip fin.

To determine

(c)

The heat transfer rate of fin when tip temperature is 250 °C.

To determine

(d)

The heat transfer rate of the fin loosing heat by convection.

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Chapter 3, Problem 184P

Chapter 3, Problem 186P

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