A spherical pot contains 0.750.75 L of hot coffee (essentially water) at an initial temperature of 9595°C. The pot has an emissivity of 0.600.60, and the surroundings are at 20.0 20.0°C. Calculate the coffee's rate of heat loss by radiation.

Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Young & Freedman Calc 14th Edition

Ch 17: Temperature and Heat

문제 67

17장, 문제 67

A spherical pot contains $0.75$ L of hot coffee (essentially water) at an initial temperature of $95$ °C. The pot has an emissivity of $0.60$ , and the surroundings are at $20.0$ °C. Calculate the coffee's rate of heat loss by radiation.

검증된 단계별 안내

First, understand that the rate of heat loss by radiation can be calculated using the Stefan-Boltzmann Law, which is given by the formula:

P = e σ A (T^{4} - T^{4})

, where

P

is the power (rate of heat loss),

e

is the emissivity,

σ

is the Stefan-Boltzmann constant (

5.67 \times 10 ⁻⁸ W / m^{2} K^{4}

A

is the surface area, and

T

are the temperatures in Kelvin.

Convert the temperatures from Celsius to Kelvin. The initial temperature of the coffee is 95°C, which is

95 + 273.15 = 368.15

K. The surrounding temperature is 20°C, which is

20 + 273.15 = 293.15

Calculate the surface area of the spherical pot. The volume of the pot is given as 0.75 L, which is equivalent to 0.75 × 10⁻³ m³. Use the formula for the volume of a sphere:

V = \frac{4}{3} π r^{3}

. Solve for the radius

r

, and then use the formula for the surface area of a sphere:

A = 4 π r^{2}

Substitute the values into the Stefan-Boltzmann Law formula. Use the emissivity

0.60

, the Stefan-Boltzmann constant

5.67 \times 10 ⁻⁸

, the calculated surface area, and the temperatures in Kelvin.

Calculate the difference in the fourth powers of the temperatures:

{368.15}^{4} - {293.15}^{4}

. Multiply this result by the emissivity, Stefan-Boltzmann constant, and surface area to find the rate of heat loss by radiation.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law describes the power radiated from a black body in terms of its temperature. It states that the total energy radiated per unit surface area is proportional to the fourth power of the black body's temperature. For non-black bodies, emissivity is included to account for the material's efficiency in emitting radiation.

Emissivity

Emissivity is a measure of how effectively a surface emits thermal radiation compared to a perfect black body. It ranges from 0 to 1, where 1 represents a perfect emitter. In this problem, the pot's emissivity of 0.60 indicates it emits 60% of the radiation a black body would at the same temperature.

Temperature Difference

The rate of heat loss by radiation depends on the temperature difference between the object and its surroundings. The greater the difference, the higher the rate of heat transfer. Here, the coffee's initial temperature is 95°C, and the surroundings are at 20°C, creating a significant temperature gradient driving the heat loss.

관련 실천

교과서 질문

The blood plays an important role in removing heat from the body by bringing this energy directly to the surface where it can radiate away. Nevertheless, this heat must still travel through the skin before it can radiate away. Assume that the blood is brought to the bottom layer of skin at $37.0$ °C and that the outer surface of the skin is at $30.0$ °C. Skin varies in thickness from $0.50$ mm to a few millimeters on the palms and soles, so assume an average thickness of $0.75$ mm. A $165$ -lb, $6$ -ft-tall person has a surface area of about $2.0$ m² and loses heat at a net rate of $75$ W while resting. On the basis of our assumptions, what is the thermal conductivity of this person's skin?

1749

views

교과서 질문

An electric kitchen range has a total wall area of $1.40$ m² and is insulated with a layer of fiberglass $4.00$ cm thick. The inside surface of the fiberglass has a temperature of $175$ °C, and its outside surface is at $35.0$ °C. The fiberglass has a thermal conductivity of $0.040 W / m \cdot K$ . What is the heat current through the insulation, assuming it may be treated as a flat slab with an area of $1.40$ m² ?

4165

views

교과서 질문

The emissivity of tungsten is $0.350$ . A tungsten sphere with radius $1.50$ cm is suspended within a large evacuated enclosure whose walls are at $290.0$ K. What power input is required to maintain the sphere at $3000.0$ K if heat conduction along the supports is ignored?

2061

views

교과서 질문

A carpenter builds an exterior house wall with a layer of wood $3.0$ cm thick on the outside and a layer of Styrofoam insulation $2.2$ cm thick on the inside wall surface. The wood has $k=0.080\,W/m$\cdot$ K$ , and the Styrofoam has $k=0.027\,W/m$\cdot$ K$ . The interior surface temperature is $19.0$ °C, and the exterior surface temperature is $-10.0$ °C. What is the rate of heat flow per square meter through this wall?

845

views