Worksheet4-Heat Transfer-2023

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Dec 6, 2023

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Physics 100 Worksheet #4 Sections 13.1, 15.1, 14.1, 14.4, 14.5: Thermodynamics Instructions: 1. Bring this worksheet with you to class until completed. 2. Form small groups of 2 to 4 students. Discuss all answers with your group. Ask your TA or lecturer for hints if required. 3. Once completed, keep it to help you study for exams. ____________________________________________________________________________________ Introduction to the First Law of Thermodynamics So far we have looked at mechanical work and mechanical energy . We have then expanded our ideas to energy transformation processes and efficiency. In many of these cases, some fraction of energy is transformed into thermal energy and then lost from the system as heat . Now we will look at heat transfer and find out how thermal energy is ‘lost’ to the environment. Let’s start with a general principle: work and heat change the internal energy of a system . Both work and heat are energy transfers and have the unit joule. Positive and negative signs indicate that energy is transferred into ( + ) or out of ( − ) the system. Most of the macroscopic (describing big objects) physical quantities that we are going to use in this module can be understood also from the microscopic point of view, if you recall that all objects are made of myriads of atoms, that are in perpetual motion (gases and liquids) or vibration (solids). Q4.1 In your own words, define what the following physics quantities mean, and state their units: Internal energy : Thermal energy : Heat : Temperature :

Heat Capacity and Latent Heat Transferring heat into a system will, in general, change its temperature – unless the system is undergoing a phase change (melting or evaporating). The heat ( ? ) transferred to a substance effects a temperature change ( Δ? ) that depends on the substance ’s specific heat ( ? ) and mass ( 𝑚 ). The expression relating a substance’s temperature change to the amount of heat it absorbs is: ? = 𝑚?Δ? = 𝑚?(? f − ? i ) If a system undergoes a phase transition, the heat required to complete it is: ? = 𝑚𝐿 where 𝑚 is the mass of the substance, and 𝐿 is the latent heat of this transition. Q4.2 Time it takes for water in domestic water heater to warm up. Your roommate just took a very long shower, and now cold water is coming out of the hot tap. Channeling your frustration into something more productive, you set to investigate how long you will need to wait until you can take a shower. Looking at the specifications on your water heater, you see that it has a 40-L capacity and a power rating of 1500 W. You also determine that the heat capacity of water is 4.2 kJ/(kg K) and that cold water is about 10 o C, whereas water for comfortably showering is at least 35 o C. Before you start, how long do you expect it will take for the water to heat up?

For the following we need two ideas from thermodynamics: 1. Thermal equilibrium : Two bodies in contact have the same temperature in thermal equilibrium. If they are initially at different temperatures, heat transfer from the hotter to the colder object will occur. 2. The First law of thermodynamics, stating how the internal energy ( ? ) of a system changes due to the net heat transferred into the system and the net work done by the system : Δ𝐸 = Δ𝐸 ?ℎ + Δ𝐸 𝑐ℎ𝑒? = ? ?𝑒? + 𝑊 ?𝑒? Here 𝐸 ?ℎ is the thermal energy of the system, which is proportional to the temperature ? , and 𝐸 𝑐ℎ𝑒? is the chemical energy of the system, for example fat stored in our bodies. Important note about notation: here we defined the heat transferred to the system ( ? 𝑖? ; for example, heating water on a stove), and the work done on the system ( 𝑊 𝑖? ; for example, compressing a gas) as positive quantities, and the heat removed from the system ( ? ??? ; for example, coffee cup that is cooling down), and the work done by the system on the environment ( 𝑊 ??? ; for example, letting a gas expand and push on a piston) as negative quantities. This allows us to highlight similarities between heat and work. Note that your textbook defines positive work as work done by the system ( 𝑊 ??? ) and negative work as the work done on the system ( 𝑊 𝑖? ), so that they have instead: Δ? = ? ?𝑒? − 𝑊 ?𝑒? . This is confusing, so we will stick with the definition that heat or work that increases the system energy (e.g. done on the system) is positive, and that which decreases the system energy (e.g. done by the system) is negative, and use the notation 𝚫𝑬 for internal energy to stress that. In fact, in the following we will deal with the situations when no mechanical work is involved and there is no change in the chemical energy of a system. Then the first law simplifies to: Δ𝐸 ?ℎ = ? ?𝑒? = ? 𝑖? − ? ??? . Q4.3 You are in your room in the winter time and the heater is running. If the room is kept at constant temperature, what can you say about the amount of heat into the room and out of the room? Since thermal energy is proportional to temperature, constant temperature implies ? 𝒊? − ? ??? = ? . Humans, animals, and houses have a similar energy balance: • Internal source for generating thermal energy. • Extra heat from sun/environment. • Insulation for slowing down heat conduction: Fur, fat, sweaters, and fibreglass. • Heat transfer mechanisms ( Conduction , Convection , Radiation ) are the same. In our discussion, we will neglect evaporation (sweating) and discuss convection only conceptually . The heat coming from the heater into the room must be _________________________________________ ( greater than/equal to/less than ) the heat leaking out of the room. Why?

Heat Conduction Our sandbox will be a simple model of a one-room cabin. We will add features step by step to make the example more and more realistic. To make sense of this, let us first understand our simplified heat conduction physics model: As the textbook explains, contact between objects is required for heat conduction to occur. Heat is conducted through the walls of your room because the molecules inside the wall interact with each other. Hi gher temperature means that “warm” molecules have more energy than “cold” molecules and direct contact leads to an energy transfer. If the system (e.g. the wall) cannot reach thermal equilibrium because heat is constantly added to the room, the energy transfer from hot to cold is maintained and energy is flowing out of the room into the environment at a constant rate. Note that this model assumes that you have two thermal reservoirs that are at fixed temperatures, ? 1 and ? 2 , at all times. The equation governing heat conduction can be written as: ? Δ𝑡 = 𝑘𝐴 ? (? 2 − ? 1 ) ≡ 𝐻 (Sometimes we might use 𝐻 to denote ?/Δ𝑡 , the rate of conductive heat transfer – also referred to as rate of heat flow.)

Q4.4 Think about the outside wall of a room. What do the quantities in the equation above mean? Q4.5 Compared to the heat flow (?/𝚫?) going into the wall on the inside at 𝒙 = ? , the heat flow at 𝒙 = 𝒅/? in the middle of the wall is: We will need the thermal conductivities ? of a few materials: Substance Thermal conductivity ? (J/s ⋅ m ⋅ 0 C) Wood 0.12 (average) Glass 0.84 Glass wool 0.042 Concrete brick 0.84 Down feathers 0.025 Air 0.023 ? : ? : ( answer : thermal conductivity of the wall material e.g. concrete or wood) 𝑨 : 𝑻 ? : 𝑻 ? : 𝒅 : (Hint: Assume that you could measure the rate of heat flow inside a wall of thickness d at different depths.) Explain your answer: 0 x = x d = A. half of the value at x = 0. B. the same as at x = 0. C. more than half. D. zero.

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