Explanation Given: The length of the cylindrical aluminum block is L = 30 cm . The diameter of the cylindrical aluminum block is D = 15 cm . The density of the cylindrical aluminum block is ρ = 2702 kg / m 3 . The specific heat of the cylindrical aluminum block is c p = 0.896 kJ / kg . The thermal conductivity of the cylindrical aluminum block is k = 236 W / m ⋅ K . The thermal diffusivity of the cylindrical aluminum block is α = 9.75 × 10 − 5 m 2 / s . The heat transfer coefficient of a cylindrical aluminum block is h = 80 W / m 2 ⋅ K . The temperature of the cylindrical aluminum block is T i = 20 ° C . The temperature of the furnace is T ∞ = 1200 ° C . The temperature at the center is T o = 300 ° C . Formula used: The Biot number is given as, B i = h L k Here, L c is the characteristic length, h is the heat transfer coefficient and k is the thermal conductivity. The interpolation for the coefficient of Biot number is given as y − y 1 = y 2 − y 1 x 2 − x 1 ( x − x 1 ) Here, y is the property that is to be calculated, x is the given value, x 1 is the nearest lower value to the given value, x 2 is the nearest highest value to the given value, y 1 is the property of lower value and y 2 is the property of the higher value. The Fourier number is given as, τ = α t L 2 Here, α is the thermal diffusivity, t is the characteristics time and L is the characteristic length. The one-term approximation for cylindrical block is given as, θ ( 0 , 0 , t ) block = θ ( 0 , t ) wall θ ( 0 , t ) cyl = ( A 1 e − λ 2 τ ) wall ( A 1 e − λ 2 τ ) cyl T o − T ∞ T i − T ∞ = ( A e − λ 2 τ ) wall ( A e − λ 2 τ ) cyl The mass of cylinder block is given as, m = ρ π r o 2 L The maximum amount of heat transfer is given as, Q max = m c p ( T ∞ − T i ) The dimensionless heat transfer for wall is given as, ( Q Q max ) wall = 1 − θ o,wall sin ( λ ) λ Calculation: The Biot number for long cylinder can be calculated as, B i = h L k = 80 W / m 2 ⋅ ° C × 0.15 m 236 W / m ⋅ ° C = 0.050 Refer table 4-2 “Coefficients used in the one term approximate solution” and obtain values for, B i 1 = 0.04 B i 2 = 0.06 λ 1 = 0.1987 λ 2 = 0.2425 Further, A 1 = 1.0066 A 2 = 1.0098 The interpolation for the coefficient of Biot number for cylinder length can be calculated as, y = y 2 − y 1 x 2 − x 1 ( x − x 1 ) + y 1 λ = λ 2 − λ 1 B i 2 − B i 1 ( B i − B i 1 ) + λ 1 = 0.2425 − 0.1987 0.06 − 0.044 ( 0.05 − 0.04 ) + 0.1987 = 0.2206 Solving for the coefficient A , A = A 2 − A 1 B i 2 − B i 1 ( B i − B i 1 ) + A 1 = 1.0098 − 1.0066 0.06 − 0.04 ( 0.05 − 0.02 ) + 1.0066 = 1.0082 The Biot number radial distance can be calculated as, B i r o = h r o k = 80 W / m 2 ⋅ ° C × 0.075 m 236 W / m 2 ⋅ ° C = 0.0254 Refer table 4-2 “Coefficients used in the one term approximate solution” and obtain values for, B i 1 = 0.02 B i 2 = 0.04 λ 1 = 0.1995 λ 2 = 0.2814 Further, A 1 = 1.0050 A 2 = 1.0099 The interpolation for the coefficient of Biot number for cylinder length can be calculated as y = y 2 − y 1 x 2 − x 1 ( x − x 1 ) + y 1 λ = λ 2 − λ 1 B i 2 − B i 1 ( B i − B i 1 ) + λ 1 = 0.2814 − 0.1995 0.04 − 0.02 ( 0.0254 − 0.02 ) + 0.1995 = 0.2216 Solving for the coefficient A , A = A 2 − A 1 B i 2 − B i 1 ( B i − B i 1 ) + A 1 = 1.0099 − 1.0050 0.04 − 0.02 ( 0.0254 − 0.02 ) + 1.0050 = 1.0063 The Fourier number is greater than the 0.2 in all directions. The one-term approximation for the cylindrical block can be calculated as, T o − T ∞ T i − T ∞ = ( A e − λ 2 τ ) wall ( A e − λ 2 τ ) cyl ( 300 ° C − 1200 ° C 20 ° C − 1200 ° C ) = ( ( 1

6th Edition

ISBN: 9781260699326

Author: CENGEL

Publisher: MCG

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Chapter 4, Problem 121P

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The amount of heat transfer from the aluminum block.

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Chapter 4, Problem 120P

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Chapter 4 Solutions

HEAT+MASS TRANSFER:FUND..(LL)-W/CONNECT

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Ch. 4 - Consider a sphere and a cylinder of equal volume...Ch. 4 - Obtain relations for the characteristic lengths of...Ch. 4 - Obtain a relation for the time required for a...Ch. 4 - A brick of 20310257mm in dimension is being burned...Ch. 4 - Prob. 15P Ch. 4 - Prob. 16P Ch. 4 - Prob. 17P Ch. 4 - Metal plates...Ch. 4 - Prob. 19P Ch. 4 - Prob. 20P Ch. 4 - Prob. 21P Ch. 4 - A 6-mm-thick stainless steel strip...Ch. 4 - After heat treatment, the 2-cm-thick metal plates...Ch. 4 - A long copper rod of diameter 2.0 cm is initially...Ch. 4 - Prob. 25P Ch. 4 - Steel rods...Ch. 4 - Prob. 27P Ch. 4 - Prob. 28P Ch. 4 - To warm up some milk for a baby, a mother pours...Ch. 4 - A person is found dead at 5 p.m. in a room whose...Ch. 4 - Prob. 31P Ch. 4 - In an experiment, the temperature of a hot gas...Ch. 4 - Prob. 33P Ch. 4 - Pulverized coal particles are used in oxy-fuel...Ch. 4 - Oxy-fuel combustion power plants use pulverized...Ch. 4 - Plasma spraying is a process used for coating a...Ch. 4 - Consider a spherical shell satellite with outer...Ch. 4 - Prob. 38P Ch. 4 - Prob. 39P Ch. 4 - Prob. 40EP Ch. 4 - Consider a sphere of diameter 5 cm, a cube of side...Ch. 4 - Prob. 42P Ch. 4 - An egg is to be cooked to a certain level of...Ch. 4 - What is an infinitely long cylinder? 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How is it...Ch. 4 - How is the product solution used to determine the...Ch. 4 - Prob. 109CP Ch. 4 - Consider a short cylinder whose top and bottom...Ch. 4 - Prob. 111P Ch. 4 - Prob. 112P Ch. 4 - Prob. 113P Ch. 4 - A hot dog can be considered to be a cylinder 5 in...Ch. 4 - Prob. 115P Ch. 4 - Prob. 116P Ch. 4 - A 2-cm-high cylindrical ice block...Ch. 4 - Prob. 118P Ch. 4 - Prob. 119P Ch. 4 - Prob. 120P Ch. 4 - Prob. 121P Ch. 4 - Prob. 122P Ch. 4 - Prob. 123P Ch. 4 - Prob. 124CP Ch. 4 - How does refrigeration prevent or delay the...Ch. 4 - What are the environmental factors that affect the...Ch. 4 - What is the effect of cooking on the...Ch. 4 - Prob. 128CP Ch. 4 - Prob. 129CP Ch. 4 - Prob. 130CP Ch. 4 - Prob. 131CP Ch. 4 - How does the rate of freezing affect the...Ch. 4 - Prob. 133CP Ch. 4 - Prob. 134CP Ch. 4 - Prob. 135CP Ch. 4 - Prob. 136CP Ch. 4 - Prob. 137CP Ch. 4 - Prob. 138P Ch. 4 - Chickens with an average mass of 2.2 kg and...Ch. 4 - Prob. 140EP Ch. 4 - Prob. 141P Ch. 4 - Prob. 142P Ch. 4 - A long roll of 2-m-wide and 0.5-cm-thick 1-Mn...Ch. 4 - Prob. 144P Ch. 4 - Prob. 145P Ch. 4 - Prob. 146P Ch. 4 - During a picnic on a hot summer day, the only...Ch. 4 - Two metal rods are being heated in an oven with...Ch. 4 - Stainless steel ball bearings...Ch. 4 - Prob. 150P Ch. 4 - Prob. 151P Ch. 4 - In Betty crockers Cookbook, it is stated that it...Ch. 4 - A watermelon initially at 35C is to be cooled by...Ch. 4 - Prob. 154P Ch. 4 - Prob. 155P Ch. 4 - Prob. 156P Ch. 4 - Prob. 157P Ch. 4 - Prob. 158P Ch. 4 - Prob. 159P Ch. 4 - Prob. 160P Ch. 4 - Prob. 161P Ch. 4 - Prob. 162P Ch. 4 - Prob. 163P Ch. 4 - Lumped system analysis of transient heat...Ch. 4 - Prob. 165P Ch. 4 - Prob. 166P Ch. 4 - An 18-cm-long, 16-cm-wide, and 12-cm-high hot iron...Ch. 4 - Prob. 168P Ch. 4 - Prob. 169P Ch. 4 - Prob. 170P Ch. 4 - Prob. 171P Ch. 4 - Prob. 172P Ch. 4 - A long 18-cm-diameter bar made of hardwood...Ch. 4 - Consider a 7.6-cm-long and 3-cm-diameter...Ch. 4 - Consider a 7.6-cm-diameter cylindrical lamb meat...Ch. 4 - Prob. 176P Ch. 4 - A small chicken (k=0.45W/m.K,=0.1510-6m2/s) and...Ch. 4 - A potato may be approximated as a 5.7-cm-diameter...Ch. 4 - When water, as in a pond or lake, is heated by...Ch. 4 - A large chunk of tissue at 35C with a thermal...Ch. 4 - Prob. 181P Ch. 4 - Citrus trees are very susceptible to cold weather,...

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