Fundamentals of Thermal-Fluid Sciences
Fundamentals of Thermal-Fluid Sciences
5th Edition
ISBN: 9780078027680
Author: Yunus A. Cengel Dr., Robert H. Turner, John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 18, Problem 77P
To determine

The temperatures of the stainless steel slab and copper slab from the surface.

Expert Solution & Answer
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Explanation of Solution

Given:

Initial uniform temperature (Ti) is 20°C.

Depth from the surface (x) is 1cm.

Uniform heat flux (q˙s) is 8kW/m2.

Total exposure time (t) is 60s.

Calculation:

Write the properties of the stainless steel.

  k=14.9W/mKα=3.95×106m2/s

Calculate the temperature of the stainless steel slab from the surface (T(x,t)).

  T(x,t)=Ti+q˙sk[4αtπexp(x24αt)xerfc(x2αt)]=20°C+8kW/m214.9W/mK[4(3.95×106m2/s)(60s)πexp((1cm)24(3.95×106m2/s)(60s))(1cm)erfc((1cm)2(3.95×106m2/s)(60s))]=20°C+8000W/m214.9W/mK[4(3.95×106m2/s)(60s)πexp((0.01m)24(3.95×106m2/s)(60s))(0.01m)erfc((0.01m)2(3.95×106m2/s)(60s))]=20°C+8000W/m214.9W/mK[4(3.95×106m2/s)(60s)πexp((0.01m)24(3.95×106m2/s)(60s))(0.01m)erfc(0.3248)]

Refer Table 18-4, “The complementary error function”, obtain the value corresponding to the error function of 0.3248.

  erfc(0.3248)=0.646

Calculate the temperature of the stainless steel slab from the surface (T(x,t)).

  T(x,t)=20°C+8000W/m214.9W/mK[4(3.95×106m2/s)(60s)πexp((0.01m)24(3.95×106m2/s)(60s))(0.01m)(0.646)]=20°C+4.9°C=24.9°C

Thus, the temperature of the stainless steel slab from the surface is 24.9°C.

Write the properties of the copper.

  k=401W/mKα=117×106m2/s

Calculate the temperature of the copper slab from the surface (T(x,t)).

  T(x,t)=Ti+q˙sk[4αtπexp(x24αt)xerfc(x2αt)]=20°C+8kW/m2401W/mK[4(117×106m2/s)(60s)πexp((1cm)24(117×106m2/s)(60s))(1cm)erfc((1cm)2(117×106m2/s)(60s))]=20°C+8000W/m2401W/mK[4(117×106m2/s)(60s)πexp((0.01m)24(117×106m2/s)(60s))(0.01m)erfc((0.01m)2(117×106m2/s)(60s))]=20°C+8000W/m2401W/mK[4(117×106m2/s)(60s)πexp((0.01m)24(117×106m2/s)(60s))(0.01m)erfc(0.05968)]

Refer Table 18-4, “The complementary error function”, obtain the value corresponding to the error function of 0.05968.

  erfc(0.05968)=0.9327

Calculate the temperature of the copper slab from the surface (T(x,t)).

  T(x,t)=20°C+8000W/m2401W/mK[4(117×106m2/s)(60s)πexp((0.01m)24(117×106m2/s)(60s))(0.01m)(0.9327)]=20°C+1.7°C=21.7°C

Thus, the temperature of the copper slab from the surface is 21.7°C.

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Chapter 18 Solutions

Fundamentals of Thermal-Fluid Sciences

Ch. 18 - Prob. 11PCh. 18 - Prob. 12PCh. 18 - Prob. 13PCh. 18 - Prob. 14PCh. 18 - Consider a 1000-W iron whose base plate is made of...Ch. 18 - Metal plates (k = 180 W/m·K, ρ = 2800 kg/m3, and...Ch. 18 - A 5-mm-thick stainless steel strip (k = 21 W/m·K,...Ch. 18 - A long copper rod of diameter 2.0 cm is initially...Ch. 18 - Prob. 21PCh. 18 - Steel rods (ρ = 7832 kg/m3, cp = 434 J/kg·K, and k...Ch. 18 - Prob. 23PCh. 18 - The temperature of a gas stream is to be measured...Ch. 18 - Prob. 25PCh. 18 - A thermocouple, with a spherical junction diameter...Ch. 18 - Prob. 27PCh. 18 - Prob. 28PCh. 18 - Carbon steel balls (ρ = 7833 kg/m3, k = 54 W/m·K,...Ch. 18 - Prob. 31PCh. 18 - Prob. 32PCh. 18 - Prob. 33PCh. 18 - Prob. 34PCh. 18 - Prob. 35PCh. 18 - Prob. 36PCh. 18 - Prob. 37PCh. 18 - Prob. 38PCh. 18 - Prob. 39PCh. 18 - A body at an initial temperature of Ti is brought...Ch. 18 - In a meat processing plant, 2-cm-thick steaks (k =...Ch. 18 - Prob. 42PCh. 18 - Prob. 43PCh. 18 - Prob. 45PCh. 18 - Prob. 46PCh. 18 - Prob. 47PCh. 18 - Prob. 48PCh. 18 - Prob. 49PCh. 18 - A long iron rod (ρ = 7870 kg/m3, cp = 447 J/kg·K,...Ch. 18 - Prob. 51PCh. 18 - A long 35-cm-diameter cylindrical shaft made of...Ch. 18 - Prob. 54PCh. 18 - Prob. 55PCh. 18 - Prob. 56PCh. 18 - A father and son conducted the following simple...Ch. 18 - Prob. 58PCh. 18 - Prob. 59PCh. 18 - Citrus fruits are very susceptible to cold...Ch. 18 - Prob. 61PCh. 18 - Prob. 63PCh. 18 - Prob. 64PCh. 18 - Prob. 65PCh. 18 - White potatoes (k = 0.50 W/m·K and α = 0.13 × 10−6...Ch. 18 - Prob. 67PCh. 18 - Prob. 68PCh. 18 - Prob. 69PCh. 18 - Consider a hot semi-infinite solid at an initial...Ch. 18 - Prob. 71PCh. 18 - Prob. 72PCh. 18 - Prob. 73PCh. 18 - Prob. 74PCh. 18 - Prob. 75PCh. 18 - Prob. 76PCh. 18 - Prob. 77PCh. 18 - Prob. 78PCh. 18 - Prob. 79PCh. 18 - Prob. 81PCh. 18 - Prob. 82PCh. 18 - Prob. 83PCh. 18 - Prob. 84PCh. 18 - Prob. 85PCh. 18 - Prob. 86PCh. 18 - Prob. 88PCh. 18 - Prob. 89PCh. 18 - A 2-cm-high cylindrical ice block (k = 2.22 W/m·K...Ch. 18 - Prob. 91PCh. 18 - Prob. 93PCh. 18 - Prob. 94RQCh. 18 - Large steel plates 1.0-cm in thickness are...Ch. 18 - Prob. 96RQCh. 18 - Prob. 97RQCh. 18 - Prob. 98RQCh. 18 - Prob. 99RQCh. 18 - Prob. 100RQCh. 18 - Prob. 101RQCh. 18 - Prob. 102RQCh. 18 - The water main in the cities must be placed at...Ch. 18 - Prob. 104RQCh. 18 - Prob. 105RQCh. 18 - Prob. 106RQCh. 18 - Prob. 107RQ
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