How Does Heat Flux Affect Steel's Internal Temperature After 30 Seconds?

[parrim2]

1. A large block of steel [k=45 W/m°C, alpha=1.4x10^-5 m2/s] is initially at a uniform temperature of 35 C. The surface is suddenly exposed to a constant heat flux of 3.2x10^5 W/m2 Find the temperature at a depth of 2.5 cm after 30 seconds has elapsed.

 

[Mapes]

How have you tried to solve the problem?

 

[parrim2]

no i didn't know how to solve it without the convection coefficient

 

[Mapes]

Then I'd recommend checking your lecture notes, textbook, handouts, or a reference like Incropera and DeWitt's Fundamentals of Heat and Mass Transfer for an equation relating temperature to heat flux for semi-infinite media.

 

[DeeNos]

The transient conduction problem described in this scenario involves heat transfer within a large block of steel that is initially at a uniform temperature of 35°C and suddenly exposed to a constant heat flux of 3.2x10^5 W/m2. The goal is to determine the temperature at a depth of 2.5 cm after 30 seconds has elapsed.

To solve this problem, we can use the one-dimensional heat conduction equation, which relates the rate of heat transfer to the thermal conductivity, surface area, and temperature gradient within the material. In this case, we can assume that the block of steel is in a steady state, meaning that the temperature is not changing with time, and thus the time-dependent term in the equation can be ignored.

Plugging in the given values, we can solve for the temperature gradient, which is equal to the heat flux divided by the thermal conductivity:

Q/A = k (dT/dx)

(3.2x10^5 W/m2) / (45 W/m°C) = (dT/dx)

dT/dx = 7111.11 °C/m

Next, we can use the boundary condition that the temperature at the surface of the block is 35°C, and the depth is 0, to solve for the temperature at a depth of 2.5 cm (or 0.025 m):

T(x) = T0 + (Q/A) (x/k)

T(0.025 m) = 35 + (7111.11 °C/m) (0.025 m / 45 W/m°C)

T(0.025 m) = 35.39 °C

Therefore, after 30 seconds has elapsed, the temperature at a depth of 2.5 cm within the steel block will be approximately 35.39 °C.

It is important to note that this solution assumes that the block of steel is a homogeneous material, and that there are no other external factors affecting the temperature distribution within the block. In a real-world scenario, there may be other variables to consider, such as variations in thermal conductivity or heat transfer at the boundaries of the block. Further analysis and experimentation may be necessary to obtain a more accurate and comprehensive solution to this transient conduction problem.