(Solution) - Using the temperature distribution of Problem 4 18 show that the -(2025 Original AI-Free Solution)

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Using the temperature distribution of Problem 4-18, show that the time lag between maximum points in the temperature wave at the surface and at a depth x is given by

Problem 4-18
When a sine-wave temperature distribution is impressed on the surface of a semi-infinite solid, the temperature distribution in the solid is given by
where
Tx,? = temperature at depth x and time ? after start of temperature wave at surface
Tm = mean surface temperature
n = frequency of wave, cycles per unit time
A = amplitude of temperature wave at surface
If a sine-wave temperature distribution is impressed on the surface of a large slab of concrete such that the temperature varies from 35 to 90ºC and a complete cycle is accomplished in 15 min, find the heat flow through a plane 5 cm from the surface 2 h after the start of the initial wave.