Explanation Given: The dimensions of the Fireclay brick is V = 0.06 m × 0 .09 m × 0 .20 m The initial temperature of the brick is T i = 1600 K . The temperature of the air is T ∞ = 40 °C . The convection heat transfer coefficient for the cooling process is h = 50 W / m 2 ⋅ K . Formula used: The expression for the Biot number is given as, B i = h L K Here, B i is the Biot number, h is the convective heat transfer coefficient, K is the thermal conductivity and L is the characteristic length. The expression for the Fourier number is given as, F o = α t L 2 Here, F o is the Fourier number, α is the thermal diffusivity, and t is the time. The expression for the temperature at the center of the brick is given as, T ( 0 , 0 , 0 , t ) − T ∞ T i − T ∞ = P 1 ( 0 , t ) × P 2 ( 0 , t ) × P 3 ( 0 , t ) Here, P ( 0 , t ) = T ( 0 , t ) − T ∞ T i − T ∞ The expression for the temperature at the corner of the brick is given as, T ( L 1 , L 2 , L 3 , t ) − T ∞ T i − T ∞ = P 1 ( L 1 , t ) × P 2 ( L 2 , t ) × P 3 ( L 3 , t ) Here, P ( L , t ) = θ ( L , t ) θ o P ( 0 , t ) Calculation: The below figure shows the schematic diagram of the brick. Figure-01 Refer to table A-3, “thermo physical properties of fire clay brick”. ρ = 2050 kg / m 3 K = 1.0 W / m ⋅ K c p = 960 J / kg ⋅ K α = 0.51 × 10 − 6 m 2 / s The Biot number for the dimension L 1 can be calculated as, B i 1 = h L 1 K = ( 50 W / m 2 ⋅ K ) × ( 0.03 m ) ( 1.0 W / m ⋅ K ) = 1.5 The Biot number for the dimension L 2 can be calculated as, B i 2 = h L 2 K = ( 50 W / m 2 ⋅ K ) × ( 0.045 m ) ( 1.0 W / m ⋅ K ) = 2.25 The Biot number for the dimension L 3 can be calculated as, B i 3 = h L 3 K = ( 50 W / m 2 ⋅ K ) × ( 0.1 m ) ( 1.0 W / m ⋅ K ) = 5 The Fourier number for the dimension L 1 can be calculated as, F o 1 = α t L 1 2 = ( 0.51 × 10 − 6 m 2 / s ) × ( ( 50 min ) × 60 sec ) ( 0.03 m ) 2 = 1.7 The Fourier number for the dimension L 2 can be calculated as, F o 2 = α t L 2 2 = ( 0.51 × 10 − 6 m 2 / s ) × ( ( 50 min ) × 60 sec ) ( 0

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ISBN: 9780470917855

Author: Bergman, Theodore L./

Publisher: John Wiley & Sons Inc

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Chapter 5, Problem 5S.10P

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The temperature at the center of the brick after 50 minutes of cooling.

The temperature at the corners of the brick after 50 minutes of cooling.

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Chapter 5, Problem 5S.9P

Chapter 5, Problem 5S.11P

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The temperature at the center of the brick after 50 minutes of cooling. The temperature at the corners of the brick after 50 minutes of cooling.

Solution Summary: The author analyzes the temperature at the center of the brick after 50 minutes of cooling. The convection heat transfer coefficient for the cooling process is h=50.

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The temperature at the center of the brick after 50 minutes of cooling. The temperature at the corners of the brick after 50 minutes of cooling.

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The temperature at the center of the brick after 50 minutes of cooling.

The temperature at the corners of the brick after 50 minutes of cooling.