1. Ideal gas – model assumptions:
• Volume of a gas molecule is much smaller than the gas volume
→ gas molecules are material points
• Range of forces between two interacting molecules is much smaller than an average distance between molecules
→ intermolecular interactions are negligible and molecules move in straight lines between collisions
• Collisions between molecules are perfectly elastic (i.e. no loss of energy)
Pressure p is a physical value equal to the force acting on a surface of the body along the normal direction to that surface: p = dFn / dS
where: Fn is the normal force acting on dS
Unit of p: 1 pascal; 1 Pa = 1N / 1m2
Temperature: Absolute temperature - defined as a value proportional to the average kinetic energy of particles: T=2/(3k)*(mv2/2)
where: k = 1.38·10 - 23 J / K is the Boltzmann constant
Unit: 1 kelvin (K)
Equation of state of an ideal gas
pV = NkT or: pV = nRT
where: n - number of moles of gas
R = 8.3145 J/(mol·K) – gas constant
NA = 6.023·10 23 mol-1 - Avogadro constant
(numebr of gas molecules in 1 mole)
R = kNA
Principle of thermal equilibrium
„0” thermodynamic law If the bodies 1 and 2 are in thermal equilibrium and bodies 2 and 3 are in thermal equilibrium thus bodies 1 and 3 are in the same thermal equilibrium _ mean kinetic energies of molecules of two gases in contact are equal
3. Principle of equipartition of energy Mean kinetic energy per one degree of freedom is the same for all molecules and equals to kT/2.
• Energy is distributed uniformly over degrees of freedom
Total kinetic energy of a molecule (treated as a rigid body) is: E=f/2*kT
For N-particle system – internal energy U: U=Nf/2*kT
4. First law of thermodynamics
Heat absorbed by the system is equal to the increase of internal energy of the system and the work performed by
the system on the surroundings.ΔQ = ΔU + ΔW
HEAT ΔQ – energy exchanged between the system and surroundings due to temperature difference
HEAT and WORK
Energy transfer – as a heat Q or work W (by means of the force acting on the system)
Q and W – not a property of the system (contrary to T, p and V)
Thermodynamic process: Initial state P (pp, Vp, Tp) → final state K (pk, Vk, Tk)
INTERNAL ENERGY Difference ΔQ – ΔW = ΔU is the same for all processes !
U – the function of state, contrary to Q and W
Internal energy of the system U
- increases when it takes energy in a form of heat Q
- decreases when it performs work W
Unit of U, Q, W: 1 joule (J)
5. Heat capacity of a system C – the quantity of heat required to rise the temperature of a body through
one degree 1K (1 0C)
Specific heat capacity c – heat capacity per unit mass:
dQ = M c dT
Molar heat capacity Cm – heat capacity of unit amount of substancje: Cm=1/n*dQ/dT
6. Isothermic process
p ⋅V = n ⋅ R⋅T
If T = const
p1*V1=p2*V2
Isobaric process
p ⋅V = n⋅ R⋅T
If: p = const
V1/T1=V2/T2
Isochoric process
If: V = const
p1/T1=p2/T2
7. Second law of thermodynamics
Question: is it possible to build a machine which takes heat and transform it fully into work?
1) Perpetum mobile of second type (self-acting machine) can not be constructed
2) When two bodies with different temperatures are in contact, then the heat flows from the body with higher temperature to that with lower temperature
→ directional process
3) Efficiency of every cyclic machine working between temperatures T1 and T2 is not higher than (T1 - T2) / T1.
ENTROPY
Entropy is a measure of the system disorder
→ The larger disorder of molecule position and velocity, the higher probability that the system will be in that state.
2nd law of thermodynamics –related to entropy
1st law of thermodynamics– related to internal energy
”0” law of thermodynamics– related to temperature
Carnot cycle
→ determines the limit of possibility of transformation of heat to work
(1) Gas state is p1, V1, T1 (point A). Cylinder is placed on the heat pump → thermal expansion of gas to state p2, V2, T1 (point B). Gasabsorbs heat Q1.
(2) Cylinder is placed on the insulation → adiabatic expansion of gas to state p3, V3, T2 (point C). Gas performs the work during moving the piston and temperature decreases to T2.
(3) Cylinder is placed on the refrigerator (T2) → isothermal compression of gas to state p4, V4, T2 (point D). Gas gives heat Q2 to the refrigerator.
(4) Cylinder is placed on the insulator → adiabatic compression of gas to state p1, V1, T1 (point A). External forces perform work and temperature
of gas increases to T1.
8. The mechanisms of HEAT TRANSFER
• Conduction• Convection• Radiation
Conduction – energy transfer from the more energetic particles of a substance to the adjacent less energetic ones → a result of interactions between the particles.
Convection - energy transfer between a solid surface and the adjacent liquid or gas which is in motion → it involves combined effects of conduction and fluid motion.
Radiation - energy emitted by matter in the form of electromagnetic waves (or photons) → a result of the changes in the electronic configurations of the atoms or molecules.
9. CONDUCTION
The rating problems deal with the determination of the heat transfer rate for an existing system at a specified temperature difference.Q= dQ/ dt (J/s = W)
Heat transfer rate
The sizing problems deal with the determination of the size of a system in order to transfer heat at a
specified rate for a specified temperature difference.
Fourier's law of conduction Qcond=- Λ A*dT/dx (W)
The thermal conductivity Λ of a material is the rate of heat transfer through a unit thickness of
the material per unit area and per unit temperature difference.
10. Convection Qconv=hAs(Ts-t∞)
h - convection heat transfer coefficient; unit: W/(m2K)
AS - surface area through which convection heat transfer takes place
TS - surface temperature
T∞ - temperature of the fluid sufficiently far from the surface.
11.RADIATION Qrad=εσAs(Ts4-Tsurr4)
where: - emissivity of the surface; 0 ≤ ε ≤ 1
AS - surface area
Ts- surface temperature
Tsurr- average surrounding surface temperature
σ = 5.67x10-8 (W / m2K4) Stefan-Boltzmann constant
12. 1D Fourier`s law of heat conduction
Qcond=- Λ A*dT/dx (W)
13. Thermal resistance
Convection heat transfer - from a solid surface of area AS and temperature TS to a fluid sufficiently far from the surface of temperature T∞ and a convection heat transfer coefficient h. Rconv=1/(hAs) (0C/W)
Convection heat transfer - from a solid surface of area AS and temperature TS to a fluid sufficiently far from the surface of temperature T∞ and a convection heat transfer coefficient h. Rrad=1/(hradAs) (0C/W) Convection and radiation → total heat transfer: hcombined = hconvection + hradiation
Thermal resistance network
1D heat flow through a plane wall of thickness L, area A, and thermal conductivity Λ , exposed to convection on both sides to fluids at temperatures T∞ 1 and T∞ 2with heat transfer coefficients h1 and h2, respectively
14. LUMPED SYSTEM: T =f(t), T (x,y,z)=const
h – heat transfer coefficient
Energy balance of the solid for the time interval dt:
hAs(T∞-T)dt=mCpdT
Lumped system – great convenience in heat transfer analysis
characteristic length: Lc=V/As
Biot number Bi=hLc/ Λ
Fourier number: τ=αt/L^2
15. In fluid – convection and conduction (in the absence of bulk motion) For the temperature at the contact point of two bodies – no-temperature-jump condition : qconv=Qconv
Nusselt number Nu= hLc/ Λ , dimensionless convection heat transfer coefficient
17. Laminar flow – highly ordered → smooth streamlines; typical cases – high-viscosity fluids (i.e. oils) at low velocities.
Turbulent flow – highly disordered → fluctuations; typical cases – low-viscosity fluids at high velocities.
18. Thermal radiation – energy transitions of molecules, atoms and electrons of a substance. The electromagnetic wave spectrum 1µm=10-6m.
19. A blackbody adsorbs all incident radiation and emits radiation energy uniformly in all directions per unit area normal to direction of emission → diffuse emiter
Spectral blackbody emissive power Ebλ – the amount of radiation energy emitted by a blackbody at an absolute T per unit time, per unit surface area, and per unit wavelength about the wavelength λ.
20. ABSORPTION and EMISSION
Assumption: A small body of surface area AS, emissivity ε, and absorptivity α at temperature T contained in a large isothermal enclosure. ε (T) =α (T)
The total emissivity of a surface at temperature T is equal to its total absorptivity for radiation coming from a blackbody at the same temperature.
Radiation heat transfer between a room and its window is proportional to the emissivity of the glass
surface.
21. The greenhouse effect
The spectral transmittivity of low-iron glass at room T for different thickness Glass transmits over 90% of radiation in the visible range and is nontransparent to radiation in the IR range (λ > 3 m).
22. Boiling and evaporation - the liquid-to-vapour phase change processes that occur at a solidliquid interface when the surface is heated above the saturation temperature Tsat of the liquid → convection heat transfer.
Evaporation occurs when the vapour pressure is less than the saturation pressure of the liquid at a given
temperature, and it involves no bubble formation or bubble motion.
Boiling occurs when a liquid is brought into contact with a surface maintained at a temperature TS
sufficiently above the saturation temperature Tsat of the liquid.
Condensation Temperature of a vapour - reduced below Tsat
23. Mass transfer requires
the presence of two regions at different chemical compositions → movement of chemical species from a high concentration region toward a lower concentration (nonhomogeneous medium).
Mass diffusion
Concentration difference is the driving force for mass transfer.
Mass flow rate ∝ Normal area A ⋅ Concentration gradient dC/dx
Moisture – influence on the performance and durability of building materials → importance of moisture transmission
Moisture – affects the effective thermal conductivity of porous building materials → linear increase of heat transfer
Diffusion in a moving medium
Mass transfer - due to both diffusion and convection
24. Waves transfer energy (kinetic and potential energy of medium particles).
Energy transfer – through matter due to displacement of disturbance not involving translational motion of matter.
25. Wave superposition rule
If disturbances are produced by two wave motions, the resultant (total) wave is the algebraic sum of the waves acting on their own.
Note: The covering waves do not influence each other.
Wave interference
Let`s consider two waves with the same frequencies and amplitudes but with a phase difference of ϕ : y1 = A sin (kx – ωt – ϕ) y2 = A sin (kx – ωt)
The total wave equation – from the superposition rule:
yw = y1 + y2 = A sin (kx – ωt – ϕ) + A sin (kx – ωt)
25. Sound wave intensity
Wave intensity I at a point is defined as the average velocity of incoming energy to (or passing through) a unit surface (power)
Doppler effect
Christian Doppler (1842 r) found that when the source and receiver of waves are in relative motion, the wave frequency, as measured by the receiver, is different from the source frequency.
Sound waves
The case of motion of the source and receiver along the connecting straight line.