A large body of lava from a volcano has stopped flowing and...

A large body of lava from a volcano has stopped flowing and is slowly cooling. The interior of the lava is at $1200^{\circ}{C}$ , its surface is at $450^{\circ}{C},$ and the surroundings are at $27.0^{\circ} {C}$ (a) Calculate the rate at which energy is transferred by radiation from 1.00 $\mathrm{m}^{2}$ of surface lava into the surroundings, assuming the emissivity is $1.00 .$ (b) Suppose heat conduction to the surface occurs at the same rate. What is the thickness of the lava between the $450^{\circ} \mathrm{C}$ surface and the $1200^{\circ} \mathrm{C}$ interior, assuming that the lava's conductivity is the same as that of brick?

A large body of lava from a volcano has stopped flowing and is slowly cooling. The interior of the lava is at $1200^{\circ}{C}$ , its surface is at $450^{\circ}{C},$ and the surroundings are at $27.0^{\circ} {C}$
(a) Calculate the rate at which energy is transferred by radiation from 1.00 $\mathrm{m}^{2}$ of surface lava into the surroundings, assuming the emissivity is $1.00 .$ (b) Suppose heat conduction to the surface occurs at the same rate. What is the thickness of the lava between the $450^{\circ} \mathrm{C}$ surface and the $1200^{\circ} \mathrm{C}$ interior, assuming that the lava's conductivity is the same as that of brick?

Key Concepts

Stefan-Boltzmann Law

This law relates the power radiated per unit area of a black body to the fourth power of its absolute temperature, multiplied by the Stefan-Boltzmann constant. It is fundamental in calculating how much thermal energy is emitted through radiation from surfaces, particularly for high-temperature objects.

Radiative Heat Transfer

Radiative heat transfer is the process by which energy is emitted by a surface in the form of electromagnetic waves. It is especially significant in high-temperature environments, where the amount of energy radiated increases steeply with temperature.

Fourier's Law of Thermal Conduction

Fourier's Law describes the rate at which heat is conducted through a material, stating that the heat transfer rate is proportional to the negative gradient of temperature and the material's thermal conductivity. This concept is essential for understanding how heat energy moves from a hotter region to a cooler region within a material.

Thermal Conductivity

Thermal conductivity is a material property that measures the ability of a substance to conduct heat. It plays a crucial role in conduction problems by determining how efficiently heat is transferred through a medium when subjected to a temperature gradient.

Transcript

00:01 So to solve this question, we are going to use stephen buzmanlaw.

00:04 So for the part a, we are given that surface temperature is 450 degrees celsius.

00:10 So we must convert it to kelvin.

00:12 Let's convert it 450 plus 273.

00:16 It's 723 kelvin.

00:21 And the temperature of surrounding is 27 degrees celsius.

00:27 That is 300 kelvin.

00:31 And we are asked the rate at which energy is transferred by radiation so we'll use stephen boltzmann law sigma epsilon a times temperature of the body or the lava in this case to the four minus temperature of surrounding to the four we are given stephen bozeman constant which is 5 .67 times 10 power negative 8.

01:06 Epsilon is one because temperature is very high.

01:14 Since it's not given to us, we can safely assume that area is given to us as one meter square, 723 to the 4 minus 300 to 4.

01:31 Let's do this difference first.

01:39 So the answer here is 265 billion or it isn't finished yet.

01:49 We have to multiply it by 5 .67 times 10 power negative 8.

02:00 So the answer will be 15 ,000 and 34 joules per seconds.

02:13 So let's convert it.

02:17 Let's write it in terms of proper significant digits.

02:20 That is three significant digits.

02:22 So answer here will be q.

02:24 Divided by t is equal to 1 .5 times 10 power 4 1 .50 times 10 power 4 joules per seconds or what? answer given below the question is wrong.

02:43 So let's solve for the part b...