IA: 1P1: Thermofluid Mechanics Flashcards
What is the technical definition of a fluid?
A fluid is a substance which when at rest, cannot sustain shear stress. Shear stresses can only be sustained in a fluid when relative motion between fluid particles takes place. Solids on the other hand, are always capable of sustaining shear stresses.
When do fluids and solids act the same and when do they act differently?
- They act the same way when subjected to a uniform pressure
- They act differently when subjected to shear forces
What is Pascal’s law?
For a fluid at rest, the absence of shear stresses means that:
* Pressure acts uniformly in all directions
* Pressure forces act normal to surfaces
* Pressure force = pressure x area
Prove Pascal’s law by using a small element of fluid at rest:
How does pressure vary with depth in a fluid of uniform density?
Linearly
Prove pressure varys linearly with depth for a fluid of uniform density
For a fluid of constant density, what is the pressure at a depth h?
p = pₐ + ρgh
What is true for all pointas at an equal depth in a fluid?
The pressure is the same. They all have the same relative difference in height to the free surface at atmospheric pressure, pₐ, and so they have the same pressure.
What is a manometer?
A U shaped column of fluid that is connected to an unknown pressure pₘ (in air) at one end and the other is connected to a known pressure (in the below example it is open to the atmosphere). Inside the tube there is a denser fluid (usually water or mercury). The difference in height, h, can be used to determine the pressure difference.
Derive the expression for pₘ?
pₘ = ρₗgh + pₐ
What is a gauge pressure?
The pressure relative to the atmosphere
What is a barometer?
A long tube which is closed at one end and open at the other. It is filled with a fluid (usually mercury) and then inverted. The height of the column indicates atmospheric pressure. pₐ = ρ(Hg)gh
What is archimedes’ principle?
A body wholly or partially submerged in fluid receives an upthrust equal to the weight of fluid it displaces
Prove archimedes’ principle:
How can you calculate the force acting on a submerged vertical plane?
Derive the expression
Force = average pressure x area
How can you calculate the force acting on a submerged plane surface?
Derive the expressions
The total force on a plate of general shape submerged in a fluid is calculated by integrating the forces along the surface of the plate.
For a submerged plane surface, what can be said about the horizontal component of the force acting on it?
It is the same as that on a vertical plane with the same projected area. Therefore for the horizontal component, the shape does not matter as long as the projected area is the same. However, the shape does matter for the vertical component.
For a submerged plane, how can you determine the point of action of the equivalent net force (the centre of pressure)?
The centre of pressure is found by integrating the moments acting on the surface to find the moment arm needed to balance the net force.
For a vertical submerged plane, where is the centre of pressure (where h is the length submerged below the surface)?
2h/3 below the surface
What is a streamline?
Streamlines are curves which are in line with the velocity vectors at each point in the flow, the velocity tangential the streamline (vₛ) is |v| and the velocity normal to the streamline (vₙ) is 0:
* No mass can cross a streamline
* Streamlines cannot cross
* Streamlines cannot randomly stop in a flow
* In general streamlines move closer together in regions of high flow velocity and further apart in regions of low velocity
Why can’t streamline cross?
Because at any given point, the flow cannot have two different velocities at the same time.
What is a stagnation point?
A point on the surface of an object where the flow velocity is zero:
* Flow velocity = 0
* Every flow with a body has at least one
What is flow separation?
Streamlines will usually follow the surface of an object. However, flows will separate from the surface when they cannot follow irs curvature or if the pressure gradients are too large.
What is a steady flow?
In steady flow, the trajectories of particles passing through each point in space do not change with time
What is unsteady flow?
In unsteady flow, the trajectory of a particle will vary with time.
When can streamlines be used?
For steady flow only. Streamlines don’t make sense in unsteady flow so we use the concept of particle paths or “streak-lines”
What is quasi-steady flow?
A quasi-steady flow is a type of unsteady flow where changes occur gradually over time, allowing the fluid to adjust smoothly. Because these variations happen slowly compared to the fluid’s response time, the flow can be approximated as a series of steady states at different moments.
Why do we often assume incompressible flow?
It greatly simplifies the equations of motion
What is a viscous fluid?
Any object immersed in a flow will experience viscous friction. This is because the fluid very close to the wall actually interacts with the asperities on the surface and does not move with the bulk flow. This behaviour is referred to as the no-slip condition. Therefore there must be velocity gradients near the surface which cause shear forces in the fluid and friction forces at the wall. Some distance from the surface the flow speed is similar to the freestream velocity. The region of varying velocity near the surface is known as the boundary layer.
What is an inviscid fluid?
If the boundary layer is very small relative to the rest of the flow, its effects can be ignored. Therefore it can be treated as inviscid and so:
* There is no frictional drag on any surfaces in contact with the fluid
What is the Reynolds number of a fluid?
It is a dimensionless measure of the importance of inertial (pressure) forces relative to viscous shear forces. It is given by:
ρ = density of the fluid
V = free stream velocity
μ = the viscocity of the fluid
L = characteristic dimension
What happens as the Reynolds number increases?
The viscous effects on the flow (and so the relative size of the boundary layers) get progressively less important
When can we use a control volume and the steady flow momentum equation?
Valid for any flow as long as we can define steady conditions on the boundaries:
* Must be steady
* Does not matter if it is inviscid
* Does not matter if it is incompressible
When can we use the equations of motion for a fluid (Bernoulli’s etc)?
When deriving these equations we limit ourselces to the steady flow of fluids with constant density and we only consider the result of forces due to pressure and the acceleration due to gravity. Therefore:
* Must be steady
* Must be inviscid
* Must be incompressible
What is a system?
An arbitary geometrical portion of the universe with fixed or moveable boundaries which may contain matter or energy or both:
* No mass can cross a system boundary
* However, energy (i.e work or heat) can cross a system boundary
What is conservation of mass?
The mass within a system remains constant, even if other factors (such as the volume) are required to change.
What is conservation of momentum?
The momentum of a system remains constant as long as no net force is applied to it
What is newtons second law?
The rate of change of momentum of a system is equal to the sum of the forces applied to it.
What is a control volume?
A specified region in space of fixed volume with fixed boundaries which can be used to analyse fluid behaviour. It is similar to a system but mass is allowed to cross the control surface.
What is a mass flow rate?
The amount of mass which crosses a control volume boundary (control surface) in unit time.
Prove the mass flow rate into a control volume equals the mass flow rate out if the amount of mass is constant
When does the mass remain constant within a control volume?
The mass always remains constant when there is steady flow. Hence the mass flow rate in is equal to the mass flow rate out.
What is the continuity equation / steady flow mass equation (NON INTEGRAL FORM)?
The conservation of mass equation for the control volume when the flow is steady:
How do you determine the mass flow rate when there is a uniform velocity and density?
How do you determine the mass flow rate when there is a non-uniform velocity or density?
Integrate the product of the velocity and density over the inlet or outlet area
How do you determine the mass flow rate when the dlow direction inro or out of the control volume is not perpendicular to the control surface?
You only consider the normal component of the velocity
What is the general expression for mass flow rate (works in all scenarios regardless of varying magnitude and direction)?
What is the integral form for the continuity equation / steady flow mass equation?
Where the vector dA points out of the control volume with direction normal to the CS
The integral sign just means “A closed integral over the complete control surface CS enclosing a control volume”
What is the steady flow momentum equation?
In one direction (i.e just the x direction)
- Fₙₑₜ = net force acting on the control volume
- ΣFₓ = sum of non pressure forces acting ON the fluid
(It is essentially just the control volume version of the impulse-momentum equation for a system!)
Determine F
What is the equation for the momentum flow rate?
How do you deal with momentum flow rate in 2 dimensions (i.e. the flow has velocity components in both x and y directions rather than just one)
- The mass flow rate is defined as perpendicular to the surface, therefore we can treat it as a scalar quantity
- Momentum flow rate is a vector quantity due to the velocity
Therefore, to determine the momentum flow rate in either the x or y directions, you use the same mass flow rate but just resolve the velocity into x and y components.
What is the steady flow momentum equation in more than one direction?
Where ΣFₓ and ΣFᵧ are the sum of all non pressure forces on the fluid in the x and y directions respectively
For a square control volume, when the fluid enters perpendicular to the first face, why must you be careful if the fluid exits out the top or bottom surfaces?
They can still have a contribution to the x component of momentum
How do you solve a question which requires a control volume and the use of the steady flow momentum equation?
- Choose a control volume and define a coordinate system. If unknown, label the force on the fluid in the control volume, or calculate the components if given.
- Evaluate the pressure forces acting on all of the boundaries (often easiest to use gauge pressure)
- Solve for any unknowns using the conservation of mass (continuity) equation or equations of motion
- Calculate the momentum fluxes in the x- and y- directions for all inflows and outflows
- Assemble the SFME in the component direcitons, being careful to keep the correct sign for all forces.
When can viscous forces be ignored?
- Free stream: Can be ignored
- Separations: Can NOT be ignored
- Mixing: Can NOT be ignored
What is the equation for the net force acting on a fluid element with pressure gradients across each side?
How does the intrinsic coordinate system for acceleration differ between mechanics and fluids?
- The acceleration in the s direction is the same
- The acceleration in the normal direction is not the same. For mechanics (and in the mechanics data book) the positive direction is taken towards the instantaneous centre of curvature. For fluids the positive direction is taken away from the instantaneous centre of curvature.
What are the expressions for the position, velocity, and acceleration of a particle in a streamline (in intrinsic coordinates)?
The same as that given in the mechanics databook except for acceleration in the normal direction where it is the negative of the one given in the databook (this is because in fluids we define the position direction outwards rather than inwards)
What is Bernoulli’s equation (where gravity is ignored)?
This can only be used if the flow is horizontal or the elevation differences are negligible (and so the gravity term is not included)
What does Bernoulli’s equation do?
It links the local flow pressure to the local flow velocity at any point along a streamline (the constant is only the same for 2 points along the same streamline)
What is Bernoulli’s equation?
What is a pitot tube and how does it work?
A Pitot tube is a tube with a 90-degree bend at one end, used to measure pressure in a flow. The pressure readings depend on the orientation of the tube:
* If the open end is sideways to the flow, the fluid continues to move past the opening, and the tube samples the static pressure (p).
* If the open end is facing directly into the flow, the fluid is brought to rest at the stagnation point (v = 0), and the tube measures the total (stagnation) pressure (p + ½ρv²)
Describe the signifigance of these 2 terms in the Bernoulli equation:
- p is the static pressure
- ½ρv² is the dynamic pressure
Together (p + ½ρv²) it gives the sum of the static and dynamic pressure which is known as the pitot-pressure (also known as the total or stagnation pressure)
What is the expression for the pitot pressure?
What is a pitot-static tube and how does it work?
A pitot-static tube measures the difference between the static pressure and the pitot pressure. This can be used to determine the free stream velocity.
What is the equation for the normal pressure gradient to a streamline?
What does the equation for the normal pressure gradient to a streamline indicate?
- Whenever there are curved streamlines, there must be a pressure gradient across them
- There will always be a lower pressure towards the centre of curvature
What is the coandă effect?
The tendency of a fluid to stick to a curved surface and follow its contours
Explain how the coandă effect works
The Coanda Effect occurs when a fluid, flows over a curved surface. Instead of following a straight path, the fluid adheres to the curvature of the surface. The streamline curvature equation (∂p/∂n = ρv²/R) implies that the pressure on the inside of the jet is lower than the ambient pressure, p∞. This pressure difference causes the jet of fluid to “stick” to the curved surface. As a result, the fluid can stay attached to the surface, even when it changes direction, maintaining the flow along the surface for a longer distance.
What is true about the pressure gradient for straight and parallel streamlines?
- There is no pressure gradient across them: ∂p/∂n = 0
- There is no pressure gradient along them: ∂p/∂s = 0
What is true about the pressure gradient for straight streamlines?
- There is no pressure gradient across them: ∂p/∂n = 0
- There may be a pressure gradient along them (if the stream is accelerating), therefore we can not say ∂p/∂s = 0 unless they are straight and parallel
When does ∂p/∂n = 0 for streamlines?
For straight streamlines
When does ∂p/∂s = 0 for streamlines?
For straight and parallel streamlines
How can streamlines reveal areas of high and low pressure?
Streamline curvature reveals the pressure gradient normal to the curve (∂p/∂n = ρv²/R), therefore you can see the direction in which pressure increases and decreases. Pressure is higher on the “outside” of the curve, and lower on the “inside”
What is the Magnus effect?
The magnus effect occurs when a spinning object moves through a fluid. As the object spins it causes the streamlines to curve around it in the direction it is rotating. This is because on one side of the object the rotation moves in the same direction as the airflow so the air wraps round more on that side, and on the other side the spin moves against the airflow causing the air to wrap round the object less. This streamline curvature leads to a pressure gradient and hence a lift force acting on the object
How can you determine the direction of the force acting on the object from the magnus effect?
- Draw a diagram of the streamlines around the object, determine the direction of the pressure gradient’s normal to the curves ( ∂p/∂n = ρv²/R) throughout and hence determine the area’s of high and low pressure. From this you can determine the direction of the lift force.
- As a rough approximation (can be used to check) you can use a right hand rule. Thumb indicates the rotational axis (anticlockwise), index finger indicates the direction of motion **of the object ** and the middle finger indicates the direction of the force acting on the ball.
What is the pressure coefficient, Cₚ?
It is a dimensionless number that describes the relative velocities or pressures at a point on a body moving through a fluid:
p = Local static pressure
p₀ = Pitot pressure
p∞ = Free stream pressure
What does it mean when the pressure coefficient, Cₚ is equal to 1?
- The flow stagnates (v =0)
- The local static pressure equals the pitot pressure
What does it mean when the pressure coefficient, Cₚ is equal to 0?
- The magnitude of the velocity matches the freestream velocity (although not necessarily the direction)
- The local static pressure equals the free stream pressure
What does it mean when the pressure coefficient, Cₚ is less than zero?
- The fluid velocity is faster than the freestream
- Local static pressure is lower and the free stream pressure
Why is the pressure coefficient (Cₚ) used?
If two flows are geometrically similar, the effects of friction are small, and the pressure changes are small compared to the average pressure then:
* Regardless of the absolute dimensions or absolute velocity, the pressure coefficients will be the same
As it is independent of absolute dimensions or velocity, we can use it for easy comparisons between models and real systems, and it allows for clear visualization of high/low pressure regions around objects.
What is thermodynamics?
The study of the inter-relationships between heat, work, and energy
What is system analysis?
This considers processes applied to a fixed quantity of matter.
What is control volume analysis?
This considers flow processes with heat or work transfers
What is classical thermodynamics?
Classical thermodynamics treats matter as a continuum. We are only concerned with the properties of macroscopic systems and not with the behabiour of individual atoms or molecules
What is molecular thermodynamics?
Molecular thermodynamics describes the behaviour of matter at a molecular level using the techniques of kinetic theory and statistical mechanics.
What is an isolated system?
If there is not heat or work exchange between a closed system and its surroundings the system is isolated.
What is the difference between a system and a closed system?
A closed system strictly has a fixed quantity of matter. However, we follow the CUED tradition of using the word “system” to also mean “closed system”
What are thermodynamic properties?
Quantifiable characteristics of a system that depend only on the state of the system and not on how it arrived at that state.
How is the absolute Kelivin scale related to the celcius scale?
What is an extensive thermodynamic property?
A thermodynamic property that depends on the size or the extent of the system. A typical example is volume: if two identical systems are brought together to form a single system, the volume is doubled.
What is an intensive thermodynamic property?
A thermodynamic property that does not depend on the size of the system. Typical examples are pressure and temperature: if two identical systems are brought together, the pressure and temperature are unchanged.
What are specific properties of a thermodynamic system?
Specific properties are properties per unit mass and are thus a subset of intensive properties. They are represented by lower case letters.
i.e: Volume (V) is an extensive property, but the specific volume (v) is an intensive property
How can the specific volume be written?
As the reciprocal of density:
What is the two property rule?
For a simple compressible system at rest, two independent intensive properties and the mass are sufficient to determine the state, when the system is in equilibrium.
What are the important characteristics of a simple compressible system?
- Intensive properties are uniform throughout
- Effects of electricity, capillartiy, gravity, and magnetism can be ignored
What does the two property rule mean by two independent intensive properties?
The two properties chosen cannot influence each other directly. i.e. specific volume and density are not independent so cannot be chosen
Why does the two property rule exist?
Because there are only 2 ways to change the state of a simple compressible system:
* By heat interactions with the surroundings
* By work interactions with the surroundings
What is thermodynamic equilibrium?
A thermodynamic system is in equilibrium when none of its thermodynamic properties are changing in time at a measurable rate.
For what states can thermodynamics tell us about?
Thermodynamics can only furnish relationships connecting equilibrium states. It can relate the two end equilibrium states but cannot tell us much about the intervening process. It is not possible to plot the process on the below p-v diagram (for a process carried out quickly) as the pressures and densities may be non uniform in the system during the process.
When is it possible to plot the process linking these two equilibrium states?
If the process is carried out very slowly, the departures from equilibrium may be kept very small and so the process effectively passes through a series of equilibrium states. This is known as a quasi-equilibrium or quasi-static process. This allows us to plot it on a p-v diagram:
What is absolute pressure?
The pressure p measured relative to the pressure in a vacuum
What is an Adiabatic process?
A process in which the heat transfer between the system and the surroundings is zero, Q = 0. A reversible and isentropic process is adiabatic. An adiabatic process is an idealised concept, requiring perfectly insulating walls or an infinitely short duration
What is the carnot cycle?
The Carnot cycle is a theoretical model for the maximum efficiency a cyclic heat engine can achieve, it has a cycle efficiency given by:
What is the Clausius inequality?
What is the Coefficient of Performance for a refrigerator and a heat pump?
Q₁ = Qₕₒₜ
Q₂ = Q𝒸ₒₗ𝒹
What is a compressor?
A steady flow device for raising the pressure of a gas. They are usually adiabatic
What is a thermodynamic cycle?
A process where the thermodynamic state of the system is the same as at the beginning
What is the cycle efficiency of a heat engine?
- Qhot = Q1
- Qcold = Q2
What is displacement work (p dV work)?
The mechanical work done by a system due to the fully resisted expansion of a gas (changes volume during a quasi-equilibrium process). It is equal to the area under the curve on a pressure-volume diagram.
What is enthalpy?
A property H used in the steady flow energy equation. It has the same units as energy but it is NOT energy, it is merely a shorthand for U + pV because this combination appears so frequently
What is entropy?
A property S associated with the second law of thermodynamics:
What is entropy creation?
What is fully resisted expansion?
A slow expansion process in which mechanical and thus thermodynamic equilibrium is maintained. A fully resisted expansion does work and is reversible. It is a special case of a quasi-equilibrium process.
(pressure is uniform through the system during the process)
What is heat?
Energy Q that is transferred across the boundary of a system by virtue of a temperature difference. It is the energy transferred that is not work.
What is a heat engine?
A cyclic device for converting heat into work. According to the Kelvin-Planck statement, complete conversion is not possible and so some heat input must be rejected.
What is the ideal gas equation in terms of the mass of the gas?
m = Mass of gas = nM (number of moles x molar mass of gas)
R = specific gas constant
What is internal energy?
The property U that represenents the thermal and intermolecular potential energy within a system.
What is an irreversible process?
A process associated with departure from equilibrium and entropy creation. It is a process where the system and surroundedings cannoty be restored to their original state.
What is an Isenthalpic process?
A process in which the enthalpy is constant.
What is an Isentropic process?
A process in which the entropy is constant. Reversible and adiabatic processes are also isentropic processes.
What is an isobaric process?
A process in which the pressure remains constant
What is an isochoric process?
A process in which the specific volume (or density) remains the same
What is an isothermal process?
A process in which the temperature remains constant, another relation which remains constant: pV = constant
What is a nozzle?
A steady flow device in which the flow is accelerated. They are usually adiabatic and reversible
What is partially resisted expansion?
An expansion process that does some work but is irreversible
What is a perfect gas?
An ideal gas with constant specific heat capacities, cᵥ and cₚ
What is a polytropic process?
A quasi equilibrium process in which:
What is a thermodynamic process?
When the state of a system changes. The succession of states the system passes through is the process path
What is a reversible process?
A process is reversible if the system and its surroundings can be returned to their initial state. Reversibility is synonymous with quasi-equilibrium and therefore take place very slowly. All reversible processes do not involve any substantial departures from equilibrium.
What is a semi-perfect gas?
An ideal gas with specific heat capacities that are functions of temperature only, cᵥ(T) and cₚ(T)
What is shaft work?
Work Wₓ that passes through the fsurface of a control volume or a system boundary by rotating a shaft
What are specific heat capacities?
The heat required to raise the temperature of 1kg of a substance by 1K:
What is the steady flow energy equation?
What is the Zeroth law of thermodynamics?
If systems B and C are each separately in thermal equilibrium with system A, then they would be in thermal equilibrium with each other if brought into thermal contact.
What is a thermal reservoir?
An infinitely large body which does not change temperature when heat is transferred to or from it
What is a throttle?
A steady flow device for expanding flow that is usually adiabatic and irreversible. In the absence of changes in kinetic and potential energy, a throttle is isenthalpic.
What is a turbine?
A steady flow device for producing power from the flow of a fluid, they are usually adiabatic.
What is unresisted expansion?
Unresisted (unrestrained) expansion is not fully resisted and is not a quasi-equilibrium process. There is no external pressure resisting the expansion, and although the gas expands, the system does not perform boundary work, since there’s no force acting through a boundary displacement. As a result, no work is done, and the process is highly irreversible. As you can see below, the system boundary does not move, therefore ΔV = 0, so W = 0
What is a valve?
A steady flow device for regulating flow that is usually adiabatic and irreversible
What is the equation for the first law of thermodynamics in a closed system?
What is the formal definition of work done by a system?
Work is done by a system on its surroundings if the sole effect of everything external to the system could have been the raising of a weight.
What does it mean if the work done by a system is greater or lesser than zero?
- W > 0: Work done BY the system
- W < 0: Work done ON the system
When can the we evaluate displacement/pdV work?
When the process is fully resisted (pressure is uniform through the system during the process)
What are the equations for shaft work and shaft power?
What is the equation for the work done to stretch an elastic wire?
Why is the work done stretching an elastic wire and compressing a spring negative?
Because you are doing work ON the system, therefore the work done BY the system is negative
What are the two main mechanisms of heat transfer across a system boundary?
- Conduction: Occurs in solid’s, liquid’s and gases. May be thought of as energy transfer from more energetic to less enegetic particles due to interactions between them, or (in gases and liquids) the diffusion of high energy parrticles to regions where they are less concentrated.
- Radiation: Radiation is emitted by matter due to changes in electronic configurations of the atoms or molecules. The energy is radiated as electromagnetic waves and can pass through a vacuum.
What is the equation for the first law of thermodynamics in a closed system where the energy term has been expanded?
Energy is divided into three components:
* ΔKE = Change in kinetic energy of the system
* ΔPE = Change in potential energy of the system
* ΔU = Change in the internal energy of the system
These three energy terms are extensive properties
For a system with no changes in gravitational or kinetic energy, how can the first law of thermodynamic be written in terms of intensive properties?
Are Q and W properties?
NO, they are quantities that depend on the process that takes place
How do we know when a system is adiabatic?
If the question specified the system is insulated we can assume that it is adiabatic
What is the first law of thermodynamics for a cyclic process?
Since E is a property (a function of state) the change in E for a cyclic process is zero, hence:
The net work output thus equals the net heat input for a cyclic process
What is the ideal gas equation of state?
What is the universal gas constant?
What is the specific gas constant?
M = molar mass of gas
What is the equation for enthalpy, H?
equation for both extensive and intensive properties.
For a constant volume process (with no shaft work, or motion etc), what does the first law of thermodynamics become?
For a constant pressure process, what does the first law of thermodynamics become?
H = U + pV
What is the constant volume (isochoric) specific heat capacity?
cᵥ relates to changes in internal energy
Where v remains constant
What is the constant pressure (isobaric) specific heat capacity?
cₚ relates to changes in enthalpy
Where p remains constant
What is an ideal gas?
An ideal gas is a gas where both the enthalpy and internal energy are functions of the temperature ONLY. pv = RT is true for all ideal gases
What are the 2 types of ideal gases?
- Semi-perfect gas
- Perfect gas
What is the equation linking cᵥ and cₚ for ideal gases?
cₚ = R + cᵥ
R = specific gas constant = Universal gas constant / Molar mass
For a semi-perfect gas, how can you roughly determine the values of cᵥ and cₚ for a temperature range from tabulated data?
What is γ (gamma)?
cₚ / cᵥ
What is the rough value of γ (cₚ / cᵥ) for a monatomic perfect gas?
1.67
What is the rough value of γ (cₚ / cᵥ) for a diatomic perfect gas?
1.40
For a perfect gas what is the equation for specific internal energy (3 forms)?
This is because cᵥ and cₚ are constant so these equations can be formed
For a perfect gas what is the equation for specific enthalpy (3 forms)?
This is because cᵥ and cₚ are constant so these equations can be formed
What are the 4 forms of the ideal gas equation?
For an isobaric process (constant pressure), where only pdV work is done, how can the first law of thermodynamic ber rewritten?
- Q = m(R + cᵥ)(T₂ - T₁)
- Q = mcₚ(T₂ - T₁)
- Q = mΔh = ΔH
For an isochoric process (constant volume), where only pdV work is done, how can the first law of thermodynamics be rewritten?
- Q = mΔu
For an isothermal process (constant temperature), where only pdV work is done, how can the first law of thermodynamics be rewritten?
Q = W = mRT Ln(V₂ - V₁)
How can we know if the question has a perfect gas in it?
If it states cᵥ or cₚ as a constant. Therefore we can use:
* Δu = cᵥ(T₂ - T₁)
* Δh = cₚ(T₂ - T₁)
What are the equations for the change in specific internal energy and change in specific enthalpy for a perfect gas?
- Δu = cᵥ(T₂ - T₁)
- Δh = cₚ(T₂ - T₁)
What 3 relations are constant for an isentropic process?
Isentropic: adiabatic, a perfect gas, and is in quasi-equilibrium (reversible)
What is a process called if it is adiabatic, invovles a perfect gas, and is in equilbirium (reversible)?
Isentropic
For a polytropic process what happens when the polytropic index (n) is:
* n = γ
* n = 1
* n = 0
* n = ∞
- n = γ: Isentropic process
- n = 1: Isothermal process
- n = 0: Isobaric process
- n = ∞: Isochoric process
1 < n < γ usually (not always) represents a process with heat transfer
Note: n will usually be in the range 1 to γ
How can you calculate the work done on or by a system during a polytropic process?
What are the 2 ways in which the second law of thermodynamics is expressed?
- Kelvin-Planck Statement
- Clausius Statement
What is an irreversible process?
An irreversible process is associated with departures from equilibrium. They can be viewed as “thermodynamically irresponsible” since they are a lost opportunity for doing useful work. After an irreversible process, it can be restored to its original state but this requires suitable flows of heat and work from the surroundings which leave the surroundings permanently changed.
What are examples of irreversible processes?
- Unrestrained or partially resisted expansion of a gas
- Heat transfer across a finite temperature difference
- Processes involving friction
- A rapid chemical reaction (like an explosion)
- Mixing of different gases and liquids
- Mixing of two streams of fluids at different velocities or temperatures
Show how unrestrained expansion of an ideal gas in a container with insulating walls is irreversible:
DONT ACTUALLY SHOW JUST READ THE PROOF
What type of heat transfers are reversible and which are irreversible?
- Heat transfers across finite temperature differences are irreversible (as you would not expect a spontaneous flow of heat back from the cooler block to the hotter block)
- Heat transfers across an infinitesimal temperature differences is reversible
What is the Kelvin-Planck Statement?
It is impossible to construct a cyclic device whose sole effect is to produce positive work whilst receiving heat from a single thermal reservoir. In other words, we cannot convert all the heat from a heat source into useful work, there is a limit on the efficiency of this process. Some of the heat must be rejected to a cooler reservoir.
What is the Clausius Statement?
It is impossible to construct a cyclic device whose sole effect is the transfer of heat from a cooler to a hotter body. To transfer heat from a cooler body to a hotter body there must be a work input.
What are the stages of the Carnot cycle?
- Isothermal Expansion (A → B)
- Adiabatic Expansion (B → C)
- Isothermal Compression (C → D)
- Adiabatic Compression (D → A)
Explain the 4 stages of the Carnot cycle:
- Isothermal Expansion (A → B): The gas is brought into contact with a heat source at temperature T₁ and expands isothermally
- Adiabatic Expansion (B → C): The heat source is removed and the expansion continues adiabatically until the temperature is T₂
- Isothermal Compression (C → D): The gas is brought into contact with a heat sink at temperature T₂ and is compressed isothermally
- Adiabatic Compression (D → A): The heat sink is removed and the gas is returned to its initial state by adiabatic compression
What happens to the carnot efficiency as the ratio of the temperature between the hot and cold sink increases?
If the temperature of the hot sink (T₁) is far greater than the temperature of the cold sink (T₂) then the T₂/T₁ gets very small and so the carnot efficiency increases
T₁ = Tₕ
T₂ = T𝒸
What are the 2 theorems concerning the efficiency of a cyclic heat engine?
- The maximum efficiency of a cyclic heat engine operating between two thermal reservoirs is attained when the cycle is reversible.
- All reversible heat engines operating between the same thermal reservoirs are equally efficient.
ηₘₐₓ = reversible
What is the maximum efficiency of a cyclic heat engine and when is this obtained?
It is the same as the carnot efficiency and it is attained when the cycle is reversible.
What is the relationship between Q and T for a reversible cyclic heat engine / refrigerator / heat pump operating between T₁ and T₂
Q₁/T₁ = Q₂/T₂
This equation is valid for heat engines, heat pumps, and refrigerators
What is a refrigerator/heat pump?
A device that extracts heat from a cold thermal reservoir and delivering a greater quantity of heat to a hotter thermal reservoir. According to the Clausius statement there must be a work input.
What is the difference between a refrigerator and a heat pump?
- A refrigerator aims to remove as much heat as possible from a cold space.
- A heat pump aims to deliver as much heat as possible to a hot space.
What does the COP for a refrigerator and heat pump tend to be?
Coefficient of performance:
* Refrigerator: Usually > 1
* Heat pump: Always > 1
When is the best coefficient of performance achieved for a refrigerator or a heat pump?
When the cycles are reversible
How do you establish an empirical temperature scale?
- Select a thermometric substance for which some property (the thermometric property) varies with termperature in a “well-behaved” fashion. i.e. the length of a mercury column in a glass capillary tube, or the electrical resistance of a fine platinum wire.
- Specifying the values of temperature at two fixed, reproducible points. For example, centrigrade scales are set by fixing the ice and steam point temperatures to 0°C and 100°C respectively.
Intermediate readings will depend on the selected thermometric property and may not vary linearly
What is the ideal gas temperature scale?
An empirical temperature scale based on the (absolute) pressure of a sample of gas maintained at a constant volume.
What is the triple point of water?
The point at which ice, water, and steam coexist in equilibrium (0.01°C or 273.16K, 611.657 Pa)
What are the 4 temperature scales?
- Empirical
- Ideal gas
- Thermodynamic
- Practical temperature measurement
What is the thermodynamic temperature scale?
Kelvin created a temperature scale based on the second law which is independent of any thermometric substance. It states that for a reversible heat engine, Q₁/Q₂ = fn (θ₁, θ₂) → Q₁/Q₂ = θ₁ / θ₂. Therefore it can be said that Q₁/Q₂ = T₁ / T₂ = θ₁ / θ₂ and so the thermodynamic temperature scale is idenical to the ideal gas temperature scale.
What are 5 practical methods of temperature measurement?
