The soil temperature at distances from 0, 10, 20 and 50 cm from the earth’s surface at the end of the 10 h period.

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Answer 1

Question

Chapter 18, Problem 79P

To determine

The soil temperature at distances from 0, 10, 20 and 50 cm from the earth’s surface at the end of the 10 h period.

Expert Solution & Answer

Explanation of Solution

Given:

Initial uniform temperature of the furnace (Ti) is 10°C.

Temperature of the area (T∞) is −10°C.

Convection heat transfer coefficient (h) is 40 W/m2⋅K.

Time period (t) is 10 h.

Calculation:

Write the given properties of the soil.

k=0.9 W/m⋅Kα=1.6×10−5 m2/s

Calculate the one-dimensional transient temperature distribution in the ground.

T(x,t)−TiT∞−Ti=erfc(x2αt)−exp(hxk+h2αtk2)[erfc(x2αt+hαtk)]={erfc(x2αt)−exp(hxk+(40 W/m2⋅K)2(1.6×10−5 m2/s)(10×3600 s)(0.9 W/m⋅K)2)[erfc(x2αt+(40 W/m2⋅K)(1.6×10−5 m2/s)(10×3600 s)0.9 W/m⋅K)]}=erfc(x2αt)−[exp(hxk+1138)[erfc(x2αt+33.7)]]

It is concluded that the last term in the temperature distribution relation should be zero irrespective of distance x as the exponential term tends to infinity.

Write the reduced form of one-dimensional transient temperature distribution in the ground.

T(x,t)−TiT∞−Ti=erfc(x2αt)T(x,t)=Ti+(T∞−Ti)erfc(x2αt)

Calculate the soil temperature at distance from 0 cm from the earth’s surface.

T(0,10 h)=10°C+(−10°C−10°C)erfc(02(1.6×10−5 m2/s)(10×3600 s))=10°C+(−20°C)erfc(0)=10°C+(−20°C)(1)=−10°C

Thus, the soil temperature at distance from 0 cm from the earth’s surface is −10°C.

Calculate the soil temperature at distance from 10 cm from the earth’s surface.

T(0.1 m,10 h)=10°C+(−10°C−10°C)erfc(0.1 m2(1.6×10−5 m2/s)(10×3600 s))=10°C+(−20°C)erfc(0.066)=10°C+(−20°C)(0.9257)=−8.5°C

Thus, the soil temperature at distance from 10 cm from the earth’s surface is −8.5°C.

Calculate the soil temperature at distance from 20 cm from the earth’s surface.

T(0.2 m,10 h)=10°C+(−10°C−10°C)erfc(0.2 m2(1.6×10−5 m2/s)(10×3600 s))=10°C+(−20°C)erfc(0.132)=10°C+(−20°C)(0.8519)=−7.0°C

Thus, the soil temperature at distance from 20 cm from the earth’s surface is −7.0°C.

Calculate the soil temperature at distance from 50 cm from the earth’s surface.

T(0.5 m,10 h)=10°C+(−10°C−10°C)erfc(0.5 m2(1.6×10−5 m2/s)(10×3600 s))=10°C+(−20°C)erfc(0.329)=10°C+(−20°C)(0.6418)=−2.8°C

Thus, the soil temperature at distance from 50 cm from the earth’s surface is −2.8°C.

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