(a) Explanation Given information: The length of the stack is 10 m , outer diameter of the stack is 1 m ,thickness of the stack wall is 10 cm , discharge rate of the exhaust gases is 1 ⋅ 2 kg / s , thermal conductivity of the wall is 40 W / m ⋅ K , temperature drop between inlet and exhaust of the stack is 30 ° C , specific heat of the exhaust gases at constant pressure is 1600 J / kg ⋅ K , average heat convection transfer coefficient is 8 W / m 2 ⋅ K , temperature of the surrounding is 27 ° C , temperature of the ambient air is 27 ° C , solar absorptivity of the outer surface of the stack is 0 ⋅ 9 , and the rate of solar radiation on the outer surface of the stack is 150 W / m 2 . The below figure shows the heat transfer through the stack, surrounding temperature, and dimensions of the stack. Figure-(1) Write the expression of the heat carried out by the exhaust gases. Q l o s t = m c p Δ T g a s ...... (I) Here, mass flow rate of the exhaust of the gases is m , specific heat at constant pressure is c p , and temperature drop between inlet and exhaust of the gases is Δ T g a s . Write the expression of heat transfer rate through the stack . q ˙ = h ( T 2 − T ∞ ) + ε σ ( ( T 2 ) 4 − ( T ∞ ) 4 ) − α q s o l a r ...... (II) Here, average heat convection transfer coefficient is h , outer surface temperature the stack is T 2 , temperature of the surrounding is T ∞ , solar absorptivity of the outer surface of the stack is α , rate of solar radiation on the outer surface of the stack is q s o l a r , emissivity of the outer surface is ε , and Stefan Boltzmann’s constant is σ . Write the expression of the one dimensional heat conduction in cylindrical coordinates. d d r ( r d T d r ) = 0 ...... (III) Write the expression of heat transfer rate through the outer surface of stack. q ˙ = Q l o s t A s , o ...... (IV) Here A s , o is outer surface area of the stack. Write the expression of heat transfer rate through the inner surface of stack. q ˙ = Q l o s t A s , i ...... (V) Here A s , i is inner surface area of the stack. Write the expression for the outer radius of the stack. r 2 = d 2 2 ...... (VI) Here outer diameter is d 2 . Write the expression for inner radius of the stack. r 1 = r 2 − t ...... (VII) Here, inner radius of the stack is r 1 , and thickness of the stack is t . Write the expression of inner surface area of the stack. A s , i = 2 π r 1 L Write the expression of outer surface area of the stack. A s , o = 2 π r 2 L Here, length of the stack is L . Write the expression for heat conduction through inner surface of the stack. q ˙ = − k d T 1 d r ...... (VIII) Here, thermal conductivity of the pipe is k , and inner surface temperature of the stack is T 1 . Calculation: Substitute Q l o s t A s , i for q ˙ in Equation (VIII). Q l o s t A s , i = − k d T 1 d r ...... (IX) Substitute 2 π r 1 L for A s , i in Equation (IX) (b) To determine The inner surface temperature of the exhaust stack

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 153P

(a)

To determine

The variation of temperature in the exhaust stack wall.

(b)

To determine

The inner surface temperature of the exhaust stack

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Chapter 2, Problem 152P

Chapter 2, Problem 154EP

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Heat and Mass Transfer: Fundamentals and Applications

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