In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the relative volume change of a fluid or solid as a response to a pressure (or mean stress) change. In its simple form, the compressibility
κ
{\displaystyle \kappa }
(denoted β in some fields) may be expressed as
κ
=
−
1
V
∂
V
∂
p
{\displaystyle \kappa =-{\frac {1}{V}}{\frac {\partial V}{\partial p}}}
,where V is volume and p is pressure. The choice to define compressibility as the negative of the fraction makes compressibility positive in the (usual) case that an increase in pressure induces a reduction in volume. The reciprocal of compressibility at fixed temperature is called the isothermal bulk modulus.
This situation is really confusing me. First of all, does accounting compressibility affect the measurements for CL? If yes, how does CL change as a function of the increasing mach number?
Back to the problem, above is the "in-flight" lift curve for a plane traveling at high speeds, so the CLs...
There's a huge volume of data on NIST database:
https://webbook.nist.gov/cgi/fluid.cgi?T=293.15&PLow=10&PHigh=1000&PInc=10&Applet=on&Digits=5&ID=C7727379&Action=Load&Type=IsoTherm&TUnit=K&PUnit=bar&DUnit=mol%2Fl&HUnit=kJ%2Fmol&WUnit=m%2Fs&VisUnit=uPa*s&STUnit=N%2Fm&RefState=DEF
I'm interested...
Starting from v(P,T),
dv=(dv/dp)_T dp + (dv/dT)_P dt
i implemented conditions when T and P are constant and ended up with
ln V = aT^3/P + constant and ln V = bT^3 /3P + constant
If i assume that the constant is 0, i can say that a/b = 1/3 but how do i justify this assumption?
Today, while studying about bulk Modulus, I encountered a doubt. Please consider this thought experiment.
I'm considering Caesium as an example as it seems to have a quite low Bulk Modulus (comparatively) of 1.6 GPa.
Let's say I apply a pressure of X GPa.
Volume change ratio can be given by...
Please help me with this problem I am facing, I am lacking notions of acoustics and I would be very grateful if someone could clarify them:
A tube has a revolution symmetry arounf the ##x## axis and has a section dependent of the value of the abscissa (x), so the profile ##S(x)## is known. The...
Homework Statement :
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A gas obeying the equation of state PV=RT undergoes a hypothetical reversible process
PV^\frac{5}{3} e^\frac{-PV}{E_0} = c_1 Can we prove that the thermal compressibility of the gas undergoing this process tends to a constant value at very high temperature? Here, E_0...
G'Day All,
This is my first post so please let me know if I have completed this form incorrectly, or missed a point of etiquette etc...
1. Homework Statement
The problem is to determine the pressurisation rate of a tank being filled by a pipe connected to a compressor.
Assumptions:
Pipe...
Homework Statement
The constant-volume heat capacity of a particular simple system is
c_v = AT^3
where A is a constant. In addition the equation of state is known to be of the form
(v-v_0)p = B(T)
where B(T) is an unspecified function of T. Evaluate the permissible functional form of B(T)...
Homework Statement
A cylinder is fitted with a piston, and the cylinder contains helium gas. The sides of the cylinder are adiabatic, impermeable, and rigid, but the bottom of the cylinder is thermally conductive, permeable to helium, and rigid. Through this permeable wall the system is in...
Hi everyone!
1. Homework Statement
Given is a function for the internal energy: ##U(T,V)=Vu(T)##
Asked is to derive the entropy balance equation. In order to do so i need to find the "isothermal and adiabatic compressibility": $$\kappa_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial...
Several questions:
What does compression actually means? In the case of isentropic calculations a changed in density (or specific volume) is included in the calculations (isentropic.jpg) so isentropic means that compressibility is included in those calculations or what?In a previous thread, a...
Homework Statement
Please consider ethylene at 152oF and 126 atm. Please determine the molar volume (ft3/lbmole) if Z is determined by Corresponding States Theory.
Homework Equations
Z=PVm/RmT
Vm= Molar volume
R=Rm/M
M= molecular weight
Rm=1545(ft*lbf)/(lbmol*oR)
Zc=(Pcvc)/(R*Tc)
Tc=283 K...
Homework Statement
Why is it that when considering the compressibility factor of a gas Z we see the value decrease at first and increase as value for Pr increases for an isotherm Tr=1?
Homework Equations
z=1+(Pb/RT)-(a/VRT)
The Attempt at a Solution
It is my understanding that as Pr is low...
I was trying to find the compressibility of water and compressibility of air to compare.
For compressibility of water I found 46.4e-6
For compressibility of a gas... I am having a tough time finding anything. compressibility factor I can find, which is 1 for Hydrogen... but how does that relate...
How do you find a value for Z using the compressibility chart when the temperature line you're trying to use ends?
For example, if you look at the compressibility chart and try to find a value for z with P_r = .75 and T_r = .96, you will notice that the line for T_r = .95 ends well before the...
I am really stuck at this question.
I tried to get the equation of volume with independent variables P and T, but the equation itself does not give a nice form, and thus I cannot get the derivative of V with respect to P. What should I do?
> The isothermal compressibility $\kappa_t$ of a substance is defined as $$ \kappa_t = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right )_T $$ Obtain an expression for the isothermal compressibility of an ideal gas. (PV = RT) in terms of p.
I believe that the ideal gas law equation...
I am trying to figure if there is a calculation for working out how much water I would have to pump into a system to increase the pressure.
For example, I have a pipeline which is 2" and 6.5 km long. I worked the volume out using πr² × height so π×0.0254m²×6500m which is 13.17m³.
How do I now...
Hello!
I have some troubles finding a way to estimate how much temperature of liquid would rise when it is suddenly adiabatically pressurized to thousands of bars. In normal conditions liduids such as water are considered to be incompressible, but certainly not in 5000 bars. Because water (or...
Hi! I have the following problem:
pt + (c2u)x + (c2v)y = 0
ut + (u2+p)x + (uv)y = α(uxx+uyy)
vt + (uv)x + (v2+p)y = α(vxx+vyy)
It is a formulation of the incompressible Navier-Stokes equations.
I would like to know an exact solution.
Can anyone help me?
Thanks
I'm doing an independent research in aeronautics (realistically I'm just learning the material on my own and testing out of it due to unforeseen circumstances last semester). One area of study that my professor advised me to research in depth was just this simple phrase "compressibility is...
The pressure exerted on the walls of the container by a real gas is less compared to an ideal gas. This is due to the attractive forces of the gas pulling the molecules back towards the rest of the gas molecules. However, there is also a relationship whereby at lower temperatures, the z is even...
At low temperatures, z falls below 1 and the reason for that is because the intermolecular interactions cause the pressure exerted to be lesser than expected. PVm/RT=z and since P is less than expected z drops below 1.
However, as the pressure increases z increases to be above 1 because as P...
Homework Statement
1 kg of water is at room temperature and the pressure is isothermally increased on the system from 1 atmosphere to 1000 atmospheres. What is the work done? What is the change in heat? What would be the temperature change if this was done adiabatically? The volumetric...
Homework Statement
Show that the coefficient of volume expansion can be expressed as
β= -1÷ρ (∂ρ÷∂T) keeping P (pressure) constant
Where rho is the density
T is Temperature
Homework Equations
1/v =ρ
β= 1/v (∂v÷∂T) keeping P (pressure ) constant
The Attempt at a Solution
I started with...
Homework Statement
I was asked to find the vapour pressure and saturated liquid molar density of propane at 263.15K using a generalised compressibility chart.
(not allowed to use NIST or steam tables either, the chart i was given does not have reduced volume lines)
Homework Equations
Tr=T/Tc...
Homework Statement
Homework Equations
No idea...
The Attempt at a Solution
Does anyone know how to do question 6b)? Its a past paper question for an exam in 2 days and i have literally no idea where to start.
Homework Statement
Obtaining an equation of state from compressibility and expansivity.
States of superheated steam are observed to have an isothermal compressibility k=(rNT) / (VP^2)
and a volume expansivity B=(N/V)((r/P)+(am / T^m+1)).
r,m and a are constants.
a) Find dv in terms of dP...
Homework Statement
Please see attachment. The correct answer is supposed to be ##\alpha = .15 k_B/U_0##.
The Attempt at a Solution
The minimum of the potential is at ##x = a##. Expanded about this minimum we get ##U = -U_0 + 36\frac{U_0}{a^2}(x - a)^2 - 252\frac{U_0}{a^3}(x - a)^3 + ...##...
Homework Statement
Consider a solid of compressibility ##\kappa##. Assume that the atoms in this solid are arranged on a regular cubic lattice, the distance between their nearest neighbors being ##a##. Assume further that a restoring force ##-k_0 \Delta a## acts on a given atom when it is...
I'm a physics noob but this is something I was thinking about when I read in my textbook that fluids are generally incompressible.
Let's say I had 1000 L of water enclosed in a sphere several meters thick (say, 2 meters thick) made of an extremely hard, dense metal with 0 outlets or holes of...
What is the correct type of pressure (static or total) used in the compressibility factor, KP, when calculating fan power? Howden's Fan Engineering book seems to indicate total pressures should be used, but I also have a PDF from Howden that indicates static pressures should be used. Online...
Homework Statement
Find the volume of 2 kg of ethylene at 270 K, 2500 kPa using Z
Homework Equations
Method to Solve for Z, Using Tr and Pr
PV = ZnRT
The Attempt at a Solution
Tr = 0.9561
Pr = 0.496
Z found to be approx. 0.75
R given on a table at 0.2964 kJ/kg K
From...
β is expansibility , (1/V)(∂V/∂T) at constant temperature
κ is isothermal compressibility , (-1/V)(∂V/∂P) at constant pressure
How to prove (∂P/∂T) at constant V = (β/κ)
Thank you
Homework Statement
Determine the volume, in m^3, occupied by 20 kg of hydrogen (H2) at 1170 kPa, 2220°C.
Homework Equations
Z=pv/rt, Pr=P/Pc, Tr=T/Tc, and for hydrogen M = 2.016 (kg/kmol) Tc = 33.2 (K) Pc = 13.0 bar Zc=pc*vc/(RTc)
The Attempt at a Solution
I know if I find Z then...
Homework Statement
Variables: N (number of particles), μ (chemical potential), P (pression), V (volume).
k is Boltzmann's constant. I often use β=1/kT.
The (isothermal) compressibility is given by
\kappa_{T} = -\frac{1}{V}\left (\frac{\partial V}{\partial P}\right )_{N,T}
The...
Homework Statement
Assume that the distance across a microscopic cell is larger than the correlation length
of the liquid, so whatever is happening in one cell is statistically uncorrelated with what
is happening in an adjacent cell. Further, assume that each cell has two distinct possible...
Correct me if I am wrong, but I think that compressibility, generally tells how difficult it is to compress an object. Meaning, the higher the compressibility, the easier it will be to compress an object with a given pressure.
I was asked in my University to describe an system of infinite...
Water is compressible, but it takes a lot of pressure to increase the density by 10 or 20%
Deep Ocean pressure measurements
http://nctr.pmel.noaa.gov/Dart/Pdf/Eble_J_atmo_91.pdf
http://seagrant.gso.uri.edu/factsheets/abyss.html
http://www.windows2universe.org/earth/Water/density.html...
Homework Statement
To a very good approximation, ammonia obeys the Bertholet equation of state,
which readsPV=nRT+\frac{9}{128}(\frac{nRTc}{Pc})(1-6\frac{Tc^2}{T^2})Pa)Suppose we have 500 grams of ammonia under a pressure of P=3.04 atm
and at T=323K. Calculate the volume of ammonia according...
I was running some structural relaxations under constant pressure, and found out that my system has compressibility which increases when pressure is applied. Of course, this is only up to a point, since at pressure high enough, phase transition will occur and compressibility will drop sharply...
Homework Statement
Sketch the following compressibility isotherms(fancy word for graphs) (compressibility factor, PV/nRT versus pressure
:) for oxygen at a temperature just above the critical temperature
::) for oxygen at a temperature well above the critical temperature eg. 200 degrees...
I have a question on the quantity -(dV/dP)T,N where V = volume, P = pressure, T = temp, N = number of moles and T, N are held constant. I see in textbooks that this quantity is always positive at equilibrium. It makes intuitive sense, as if it were negative, it would be unphysical. I've been...
I am going to be running some experiments with refrigerants. They will be at various temperatures and pressures. I will then be doing some modeling of certain phenomena. I would like to be able to quantify whether the ideal gas model is reasonable during certain stages of the experiment. My...
Homework Statement
Calcualte the volumes in cubic feet of the gases in the attached chart, assuming the gases are ideal
Homework Equations
Ideal gas equation PV=nRT
The Attempt at a Solution
Please see the attached sheet also, this has my attempt, if i use my method and then use a...
Homework Statement
A certain metal whose thermal expansion coefficient \beta is 5,0 × 10^-5 °C^-1 and whose isothermal compressibility \kappa_T is 1,2 × 10^-6 atm^-1 is at an initial pressure of 1 atm and an initial temperature of 20°C. A thick layer of Invar is thermally insulating the...
I have been asked to show that if an ideal gas is compressed isothermally its compressibility is 1/P whereas if the same gas is compressed adiabatically its compressibility is 1/yP
Where y is gamma
I have managed to do the first bit about isothermal compressibility, but cannot work out...