. Un astronauta que realiza una actividad fuera del vehículo...

. Un astronauta que realiza una actividad fuera del vehículo (paseo espacial) a la sombra del Sol lleva un traje espacial que puede aproximarse como perfectamente blanco $(e=0)$ excepto un parche de $5 \mathrm{~cm} \times 8 \mathrm{~cm}$ con la forma de la bandera nacional del astronauta. El parche tiene una emisividad de 0,300 . El traje espacial debajo del parche tiene un grosor de $0,500 \mathrm{~cm}$, con una conductividad térmica $k=0,0600 \mathrm{~W} / \mathrm{m}^{\circ} \mathrm{C}$, y su superficie interior está a una temperatura de $20,0^{\circ} \mathrm{C}$. ¿Cuál es la temperatura del parche y cuál es la tasa de pérdida de calor a través de él? Suponga que el parche es tan fino que su superficie exterior está a la misma temperatura que la superficie exterior del traje espacial que tiene debajo. Asuma también que la temperatura del espacio exterior es de $0 \mathrm{~K}$. Obtendrá una ecuación muy difícil de resolver en forma cerrada, por lo que puede resolverla numéricamente con una calculadora gráfica, con un software o incluso por ensayo y error con una calculadora.

. Un astronauta que realiza una actividad fuera del vehículo (paseo espacial) a la sombra del Sol lleva un traje espacial que puede aproximarse como perfectamente blanco $(e=0)$ excepto un parche de $5 \mathrm{~cm} \times 8 \mathrm{~cm}$ con la forma de la bandera nacional del astronauta. El parche tiene una emisividad de 0,300 . El traje espacial debajo del parche tiene un grosor de $0,500 \mathrm{~cm}$, con una conductividad térmica $k=0,0600 \mathrm{~W} / \mathrm{m}^{\circ} \mathrm{C}$, y su superficie interior está a una temperatura de $20,0^{\circ} \mathrm{C}$. ¿Cuál es la temperatura del parche y cuál es la tasa de pérdida de calor a través de él?
Suponga que el parche es tan fino que su superficie exterior está a la misma temperatura que la superficie exterior del traje espacial que tiene debajo. Asuma también que la temperatura del espacio exterior es de $0 \mathrm{~K}$. Obtendrá una ecuación muy difícil de resolver en forma cerrada, por lo que puede resolverla numéricamente con una calculadora gráfica, con un software o incluso por ensayo y error con una calculadora.

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann law governs the radiative power emitted by a blackbody, stating that it is proportional to the fourth power of the absolute temperature. When modified by the emissivity factor, this law allows us to calculate the radiative heat loss from surfaces that are not perfect blackbodies. It is essential for determining how objects in space lose heat through radiation.

Emissivity

Emissivity is a measure of a surface's efficiency in emitting thermal radiation compared to that of a perfect blackbody. Ranging between 0 and 1, a higher emissivity indicates greater radiative heat loss. It is a key parameter when considering radiative energy exchange in environments where conduction and convection may be minimal or absent, such as in space.

Fourier’s Law of Thermal Conduction

This concept describes the transfer of heat through a material due to a temperature gradient. It states that the heat flow rate is proportional to the material's thermal conductivity, the cross?sectional area through which heat is flowing, and the temperature difference between its two faces, while inversely proportional to the thickness of the material. This law is crucial for analyzing heat transport through layers such as a suit or any solid medium.

Thermal Equilibrium

Thermal equilibrium refers to the state in which the energy entering a system (or part of a system) is equal to the energy leaving it, resulting in a stable temperature. In problems involving radiative and conductive heat transfer, setting the conduction heat flow equal to the radiative heat loss is essential to solve for a steady-state temperature.

Transcript

00:01 The setup for this question is to look at how heat is being conducted through the space suit, right, so an astronaut, out to a patch, and then off into space.

00:17 Now, there are two different transfers of heat, right, method of transfer of heat that we are discussing over here.

00:27 The first would be via conduction.

00:30 We have the suit, the astronaut having a temperature of about 20 degrees celsius that is inside the suit.

00:45 And then this amount of heat is transferred over to the patch through the sickness of the suit via conduction.

00:56 Well, the second part of this question is the patch itself transferring heat to the space.

01:08 Now the issue with that is that space itself is considered to be kind of like a vacuum.

01:17 So there's very little air molecules that is around to conduct heat away.

01:26 So this vacuum over here so that is very little conduction.

01:33 Mostly the heat will be lost through radiation.

01:40 So radiation is it would be the main source of heat transfer from the patch to the environment.

01:55 So what determines the rates at which the patch emits this radiation? well, that's the staphael -boltzman's equation.

02:06 P equals to e, the emissativity times the staphael -botsman's constant, times the area times the temperature to the power 4.

02:28 So the vacuum itself, because its temperature of the vacuum is 0 kelvin's, so we do not expect it to actually absorb any radiation from the surroundings.

02:48 So the incoming radiation using the same equation with t equals to 0, be just zero.

02:59 So there's no incoming radiation that we need to be aware of.

03:06 So now let let us start off with the conduction part.

03:09 This conduction part would actually determine what is the inner, what is the temperature of our patch.

03:16 Because by conduction we know that the rates of heat transfer be given as the thermal conductivity multiplied by the area, multiplied by the difference in temperature.

03:38 I'm going to assume that the temperature is smaller than 20 degrees.

03:43 So we have 20 minus t.

03:57 And divided by the thickness of the spacesuit.

04:04 Now we assume that the patch is almost infinitely thin, such that it only has one temperature.

04:19 Now with this amount of energy that is being transferred from the astronaut itself to the patch.

04:32 We expect that at thermal equilibrium, this will also be equal to the amount of radiation that is being emitted by our patch.

04:47 So this is equals to e times sigma, a t to about 4 divided by...

04:59 So now with this equation, we are ready to solve.

05:11 I try and solve for this.

05:16 So for t.

05:17 So we get rid of a is the common factor.

05:25 And then we try to bring all the term with t to one side...