ch11 - PROBLEM 11.1 KNOWN Initial overall heat transfer...

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PROBLEM 11.1 KNOWN: Initial overall heat transfer coefficient of a fire-tube boiler. Fouling factors following one year’s application. FIND: Whether cleaning should be scheduled. SCHEMATIC: ASSUMPTIONS: (1) Negligible tube wall conduction resistance, (2) Negligible changes in h c and h h . ANALYSIS: From Equation 11.1, the overall heat transfer coefficient after one year is f,i f,o i o 1 1 1 R R . U h h ′′ ′′ = + + + Since the first two terms on the right-hand side correspond to the reciprocal of the initial overall coefficient, ( 29 2 2 2 1 1 0.0015 0.0005 m K / W 0.0045 m K / W U 400 W / m K = + + = 2 U 222 W / m K. = COMMENTS: Periodic cleaning of the tube inner surfaces is essential to maintaining efficient fire- tube boiler operations.
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PROBLEM 11.2 KNOWN : Type-302 stainless tube with prescribed inner and outer diameters used in a cross-flow heat exchanger. Prescribed fouling factors and internal water flow conditions. FIND: (a) Overall coefficient based upon the outer surface, U o , with air at T o =15 ° C and velocity V o = 20 m/s in cross-flow; compare thermal resistances due to convection, tube wall conduction and fouling; (b) Overall coefficient, U o , with water (rather than air) at T o = 15 ° C and velocity V o = 1 m/s in cross- flow; compare thermal resistances due to convection, tube wall conduction and fouling; (c) For the water-air conditions of part (a), compute and plot U o as a function of the air cross-flow velocity for 5 V o 30 m/s for water mean velocities of u m,i = 0.2, 0.5 and 1.0 m/s; and (d) For the water-water conditions of part (b), compute and plot U o as a function of the water mean velocity for 0.5 u m,i 2.5 m/s for air cross-flow velocities of V o = 1, 3 and 8 m/s. SCHEMATIC : ASSUMPTIONS: (1) Steady-state conditions, (2) Fully developed internal flow, PROPERTIES: Table A.1 , Stainless steel, AISI 302 (300 K): k w = 15.1 W/m K; Table A.6 , Water ( m,i T = 348 K): ρ i = 974.8 kg/m 3 , µ i = 3.746 × 10 -4 N s/m 2 , k i = 0.668 W/m K, Pr i = 2.354; Table A.4, Air (assume f,o T = 315K, 1 atm): k o = 0.02737 W/m K, ν o = 17.35 × 10 -6 m 2 /s, Pr o = 0.705. ANALYSIS: (a) For the water-air condition, the overall coefficient, Eq. 11.1, based upon the outer area can be expressed as the sum of the thermal resistances due to convection (cv), tube wall conduction (w) and fouling (f): o o tot cv,i f ,i w f ,o cv,o 1 U A R R R R R R = = + + + + cv,i i i cv,o o o R 1 h A R 1 h A = = f ,i f ,i i f ,o f,o o R R A R R A ′′ ′′ = = and from Eq. 3.28, ( ) ( ) w o i w R ln D D 2 Lk π = The convection coefficients can be estimated from appropriate correlations. Continued...
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PROBLEM 11.2 (Cont.) Estimating i h : For internal flow, characterize the flow evaluating thermophysical properties at T m,i with m,i i D,i 4 2 3 i u D 0.5m s 0.022m Re 28,625 3.746 10 N s m 974.8kg m ν × = = = × For the turbulent flow, use the Dittus-Boelter correlation, Eq. 8.60, 0.8 0.4 D,i D,i i Nu 0.023Re Pr = ( ) ( ) 0.8 0.4 D,i Nu 0.023 28,625 2.354 119.1 = = 2 2 i D,i i i h Nu k D 119.1 0.668W m K 0.022m 3616W m K = = × = Estimating h o : For external flow, characterize the flow with o o D,o 6 2 o V D 20m s 0.027m Re 31,124 17.35 10 m s ν × = = = × evaluating thermophysical properties at T f,o = (T s,o + T o )/2 when the surface temperature is determined from the thermal circuit analysis result, ( ) ( )
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