Thermal energy transfer

• Scientists, engineers, and inventors started working on steam engine technology in the late 18th and early 19th centuries.
• Examples of this technology include the massive steam-powered beam-engine pumps that are used to extract water from deep Cornish tin mines.
• A methodical and basic grasp of the properties of heat energy, as well as how it relates to the behavior of steam and the other components of the engines and the work that the engines do, was necessary for the development of the engines.
• Thermodynamics is the name given to this field of research, and Britain was the world leader in both the underlying physics of thermodynamics and the invention of new engines.
• The kinetic theory of matter, which examines the microscopic, particle-scale behavior of matter, complements thermodynamics, which focusses on the macroscopic (large-scale) behavior of a system.
• Certain elements of thermal physics, like the behavior of engines, are best understood in terms of macroscopic thermodynamics.
• On the other hand, other parts, like Brownian motion (the microscopic random motion of smoke or pollen particles), are best understood in terms of microscopic kinetic theory.

1. Internal Energy:

• Internal energy (U) is a fundamental concept in thermodynamics, representing the total energy of a system due to its internal molecular motions and interactions. It includes:
  1. Kinetic energy (KE) of molecules
  2. Potential energy (PE) of molecular interactions (e.g., chemical bonds, electrostatic forces)
  3. Potential energy of nuclear interactions (e.g., nuclear bonds)
• Internal energy is a state function, meaning that only the change in internal energy (ΔU) is meaningful, not the absolute value.
• [math] \Delta U = \sum (\text{kinetic energies}) + \sum (\text{potential energies}) [/math]
• Take a look at the water glass (Figure 1).

  
  Figure 1 Glass of a water showing
  internal energy

• The energy of the water particles is divided into two categories:
• Potential energy, which is related to any forces or interactions between the particles.
• Kinetic energy, which is related to how quickly the particles move, vibrate, or rotate (such as any electrostatic attraction or repulsion).
• The temperature of the particles affects their kinetic energy, and any intermolecular interactions between the particles affect their potential energies.
• The only factors that affect internal energy in perfect gases—which lack intermolecular interactions—are the kinetic energies.

2. The first law of thermodynamic:

• The First Law of Thermodynamics, also known as the Law of Energy Conservation, states:
• “Energy cannot be created or destroyed, only converted from one form to another.”
• A system’s growth in internal energy is equal to the heat it absorbs from outside sources less the work it does itself.
  This can be expressed symbolically as follows:
• [math] \Delta U = \Delta Q – \Delta W [/math]
• Where
  ΔQ is the thermal energy contributed to the system.
  ΔW is the work the system does.
  ΔU is the increase in internal energy of the system (often a gas).
• This means that the total energy of a closed system remains constant, but can be transformed between different forms, such as:
  – Kinetic energy (KE)
  – Potential energy (PE)
  – Thermal energy (Q)
  – Work (W)
  – Electrical energy
  – Chemical energy
• The First Law of Thermodynamics has far-reaching implications:
  1. Energy conservation: Energy cannot be created or destroyed, only converted.
  2. Universal applicability: Applies to all physical systems, from microscopic to macroscopic.
  3. Reversibility: Processes can be reversed, but with energy conversions.
  4. Causality: Cause-and-effect relationships between energy transformations.
• This fundamental principle has numerous applications across various fields, including:
  1. Engineering (mechanical, thermal, electrical)
  2. Physics (mechanics, thermodynamics, electromagnetism)
  3. Chemistry (thermodynamics, kinetics)
  4. Biology (metabolism, energy conversion)
  5. Environmental science (energy flows, conservation)
• The internal energy (U) of a system can be increased or decreased by two methods:
• 1. Heating: When energy is transferred to the system through heat (Q), the internal energy increases. This is represented by the equation:
• [math] \Delta U = Q [/math]
• 2. Work done: When work (W) is done on the system, the internal energy also increases. This is represented by the equation:
• [math] \Delta U = W [/math]
• Conversely, when energy is transferred out of the system through heat or work is done by the system, the internal energy decreases.

3. Work done by an expanding gas:

• A gas does work on its surroundings as it expands because it exerts a force on them that causes them to move.
• The work performed, W, by an expanding gas at constant temperature (also known as an isothermal change) may be calculated using the first rule of thermodynamics.
• As illustrated in Figure 2, imagine a gas contained within a cylinder by a frictionless piston.
• The cylinder’s walls are under pressure (p) from the gas with volume V. Consequently, the frictionless piston in area A experiences a force F.
• Where,
• [math] F = pA [/math]
• The loudness, ΔV, rises as a result.
• In order to ensure that the external force acting on the piston equals the force generated by the pressure (p) of the gas in the cylinder, we assume that ΔV is extremely tiny and that the force moves the piston at a slow but constant pace.
• As a result, the gas’s pressure during expansion is essentially kept constant. Since the gas functions, ΔW is positive.
• The piston is moved by force across a distance, Δx, in such a way that:
• ΔW = -FΔx

• Substituting

• ΔW= -pAΔx
• But AΔx = ΔV, the change in volume of the gas. So, 
• ΔW= pΔA

4. Heating up substances and changes of state:

• When a material is heated, thermal energy is transferred to its particles, raising their internal energy U and, consequently, their average kinetic energy.
• The temperature of the particles rises when their average kinetic energy increases. A number of macroscopic, observable criteria determine the size of temperature change, .
• The quantity of thermal heat energy supplied (Q).
• The mass of the substance, m; and its condition.
• Each substance has a unique specific heat capacity (c).
• The equation establishes a relationship between these variables:
•  Q = mcΔθ
• The units of specific heat capacity (c) are Jkg^-1K^-1 because the thermal energy Q is measured in joules (J), the mass m is measured in kilograms (kg), and the temperature change Δθ is measured in kelvin (K).
• One of a material’s basic characteristics is its specific heat capacity, which is crucial information for scientists and engineers building engines and insulation systems.
• A material’s ability to change temperature is determined by its specific heat capacity.
• When one kilogram of a material is heated by 1 K, it takes a significant amount of thermal energy to raise it to 4186Jkg^-1K^-1 (usually rounded to 4200Jkg^-1K-1), which is the specific heat capacity of water.
• In contrast, materials with relatively low specific heat capacities, like gold (Au=126Jkg^-1K¹), only need a small amount of thermal energy to raise the temperature of one kilogram of the material.
• We can determine a material’s change in temperature in response to a change in thermal energy thanks to its specific heat capacity.

Example

• Warming water:
  • 50 kg of tap water (18°C) is heated for a hot-water bottle in an aluminum saucepan using a 3.0kW electric burner for 4.0 minutes.
  • Determine the final temperature of the heated water by assuming that the water absorbs 60% of the electrical energy and that there are no further heat losses.
  • Water has a specific heat capacity of C_W =4186Jkg^-1K^-1.
• Solution
• Total electrical energy produced by the electric hob is
• [math] E = 3.0 \times 10^3 \times 4.0 \times 60 \, \text{s} \\
  E = 7.2 \times 10^5 \, \text{J} [/math]
• Thermal energy supplied to the water is
• [math] Q = \frac{60}{100} \times 7.2 \times 10^5 \, \text{J} \\
  Q = 4.32 \times 10^5 \, \text{J} [/math]
• But [math] Q = mc\Delta \theta [/math]  so
• [math] \Delta \theta = \frac{Q}{mc} \\ \Delta \theta = \frac{4.32 \times 10^5 \, \text{J}}{1.5 \, \text{kg} \times 4186 \, \text{J} \cdot \text{kg}^{-1} \cdot \text{K}^{-1}} \\
  \Delta \theta = 68.8 \, \text{K} \approx 69 \, \text{K} [/math]
• The water in the saucepan has finally reached a temperature of 18°C +69°C=87°C since a temperature change of 1K is equivalent to a temperature change of 1°C.

⇒ Specific heat capacity:

• Specific heat capacity (c) is the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). It’s a measure of a substance’s ability to absorb and release heat energy.
  c = Q / (m × ΔT)
•  where:
  – c is the specific heat capacity
  – Q is the heat energy added or removed
  – m is the mass of the substance
  – ΔT is the change in temperature
• Units: J/(kg·K) or J/(kg·°C)
• Specific heat capacity is an important property of materials, as it:
  1. Determines the amount of heat energy required to change a substance’s temperature
  2. Affects the temperature change of a substance when heat energy is added or removed
  3. Influences the thermal conductivity and diffusivity of materials
  4. Plays a crucial role in various engineering applications, such as:
     – Heat transfer and thermal management
     – Thermodynamic cycles and energy conversion
     – Materials science and selection
     – Climate and weather modeling
• Some common specific heat capacities:
  – Water: 4186 J/(kg·K)
  – Air: 1005 J/(kg·K)
  – Copper: 385 J/(kg·K)
  – Aluminum: 900 J/(kg·K)

5. Measuring the specific heat capacity of water using a continuous flow method:

• The specific heat capacity of a fluid can be measured using a continuous flow method (Figure 4.7).
• Where the fluid moves over an electric heater at a constant rate.
• It is assumed that the thermal energy transferred from the apparatus to the surroundings is constant.
• The experiment is carried out and then the flow rate of the fluid is changed, and a second set of readings is taken.
• The heat loss can then be eliminated from the calculations.
• 
  Figure 2 Measurement of specific heat capacity by the continuous flow method.
• As seen in Figure 2, a fluid passes through an insulated tube that holds an electric heating wire.
• The two electronic thermometers record the fluid’s temperature rise, which is computed using
• [math] \Delta \theta = T_2 – T_1 [/math]
• Using a timer and a beaker on a balance, the mass of the fluid that passes through the device in one minute is calculated, or m₁.
• Next, the fluid flow rate is adjusted to get a different number, m₃, and the heater settings are adjusted to yield a temperature differential of .
• The fluid’s specific heat capacity may therefore be estimated by supposing that for both flow rates, the thermal losses to the environment remain constant.
• For the first rate, the electrical energy supplied to the fluid in time t₁ is given by
• [math] I_1 V_1 t_1 = m_1 c \Delta \theta + E_{\text{lost}} \qquad (1)[/math]

• where [math]E_{lost}[/math] is the thermal energy lost to the environment, I₁ and V₁ are the initial current and p.d of the heater. Regarding the second rate of flow:

• [math] I_2 V_2 t_2 = m_2 c \Delta \theta + E_{\text{lost}} \qquad (2)[/math]

• Since [math]E_{lost}[/math] is a constant across experiments, subtracting equation (2) from equation (1) yields

• [math] I_1 V_1 t_1 – I_2 V_2 t_2 = m_1 c \Delta \theta – m_2 c \Delta \theta \\ I_1 V_1 t_1 – I_2 V_2 t_2 = c \Delta \theta (m_1 – m_2) [/math]
• If the experiments are both run for the same time t,then
• [math] c = \frac{(I_1 V_1 – I_2 V_2)t}{\Delta \theta (m_1 – m_2)} [/math]

6. Changing state:

• Thermal energy is utilized to raise the internal energy of liquid molecules when they are heated to their boiling point.
• This is quantified as a change in temperature.
• At the boiling point, on the other hand, the temperature changes no more and all of the thermal energy input is utilized to overcome the forces that hold the liquid’s particles together, turning it into a gas.
• The quantity of thermal energy, Q (in J), needed to alter a substance’s condition without causing a change in temperature is determined by
• Q = ml
• where l is the substance’s specific latent heat (‘latent’ meaning ‘hidden’), expressed in Jkg^-1, and m is the mass of the material, expressed in kg. All phase shifts associated with state transitions are covered by this equation.

  
  Figure 3 Kinetic theory graph

• As a result, water, for instance, has a specific latent heat of vaporization (I_v) which deals with the phase change from liquid to gas (and vice versa), and a specific latent heat of fusion, l_f, which deals with the phase change from solid to liquid (and vice versa).
• Figure 3 shows the connection between the idea of latent heat and the kinetic theory models of solids, liquids, and gases. The intermolecular bonds keeping the particles together are loosened by applying thermal energy to a material that is changing states (totally in the case of a liquid converting into a gas). Despite the substance receiving thermal energy during the change of state, the temperature does not change; this is why the thermal energy is known as latent heat.
• Table 1 displays the values of l_v and l_f for a few chosen materials. Once more, the high-water values indicate that a large amount of the planet Earth’s water is liquid and that the average temperature is maintained within a narrow range.

• Material	Specific latent heat of vaporization, [math]{l_v}/{\text{kJ}  \text{kg}^{-1}} [/math]	Specific latent heat of fusion, [math]{l_f}/{\text{kJ}  \text{kg}^{-1}} [/math]
  Water	2260	334
  Carbon dioxide	574	184
  Nitrogen	200	26
  Oxygen	213	14
  Lead	871	23