Irradiation-Energy Storage Heat Transfer

[chriskay301]

Homework Statement

An oak slab (ρ=545 kg/m3, k=0.17 W/(m-K), and c=2385 J/(kg-K)) of thickness 10 cm is exposed on one side to irradiation (1x10^5 W/m2) from a laser, which starts heating it up. Convection is negligible (i.e., h is small). The slab is initially at 23°C. (a) If the oak surface temperature reaches 260°C, it will start burning. How long can the slab be irradiated before it starts burning? (b) As an approximate first analysis, assume that once the slab surface reaches the burning temperature, it immediately vaporizes. If the energy required to vaporize oak is 14.5 kJ/kg, how long would it take to vaporize the oak slab to a depth of 2 mm?

Homework Equations

I'm not really sure. Energy Balance (Ein - Eout + Egeneration = Estored)

The Attempt at a Solution

I'm not sure where to start. I know multiplying the irradiation value by the area will give me the Ein term. And I'm pretty sure the irradiation travels through the slab as conduction. But I'm not really sure where to go from there.

 

[KataKoniK]

Hello,

Thank you for your post. Based on the information provided, we can start by using the energy balance equation that you mentioned: Ein - Eout + Egeneration = Estored. Let's break this down:

Ein: This is the energy input, which in this case is the irradiation from the laser. As you mentioned, we can calculate this by multiplying the irradiation value (1x10^5 W/m2) by the area of the slab.

Eout: This is the energy output, which in this case is the heat transfer from the slab to its surroundings. Since convection is negligible, we can assume that the only mode of heat transfer is conduction. The heat transfer equation for conduction is Q = kAΔT/L, where Q is the heat transfer rate, k is the thermal conductivity, A is the area, ΔT is the temperature difference, and L is the thickness of the slab. We can rearrange this equation to solve for Eout: Eout = kAΔT/L. We know the values for k, A, and L, but we need to find the value for ΔT. To do this, we can use the fact that the surface temperature of the slab reaches 260°C and the initial temperature is 23°C. This gives us a temperature difference of 237°C.

Egeneration: This is the energy generated within the slab due to the irradiation. Since the slab is initially at 23°C and the surface temperature reaches 260°C, we can assume that all of the energy input is converted into heat within the slab. This means that Egeneration is equal to Ein.

Estored: This is the energy stored within the slab. Since the slab is initially at 23°C, we can assume that there is no stored energy at the beginning. However, as the slab is exposed to the irradiation, it will start to store energy in the form of thermal energy.

Now, we can plug in the values into the energy balance equation and solve for the time it takes for the slab to reach 260°C:

Ein - Eout + Egeneration = Estored

(1x10^5 W/m2)(A) - (0.17 W/(m-K))(A)(237°C)/(0.1 m) + (1x10^5 W/m2)(A) = (2385 J/(kg-K))(ρ)(A)(ΔT