Sub-Cooled Liquid Region: Compressed Liquid Properties and Approximations

so let's look more closely at the sub-cooled liquid region sub-cooled liquids are often called

compressed liquids which is a bit of a misnomer because liquids are virtually incompressible

anyway this is the naming of this region the sub cooled or compressed liquid region

so i've drawn the tv diagram and i've shown the dome but i've really exaggerated this region

i mentioned to you before that it's it's virtually vertical because there's very little change in density in

this region so i've really kind of exaggerated this region to show

how the different properties kind of interact with each other so what we have to do in order to sort

out how to evaluate properties in this region is we have to go back to the

state postulate and i've indicated that the state postulate say recall the state

possible the state postulate says that for a simple compressible substance the state is completely defined by two

independent intensive properties so the first thing we have to do is understand which

properties are independent because if we take two independent properties we can use them to define the whole state

so to determine whether something is independent we draw lines of constant properties so

in this case lines of constant specific volume are vertical lines lines of constant temperature are

horizontal lines and lines of constant pressure from our experiment looks like this

so it's tipped this way in the subcooled liquid region and then straight across up so in this

region we're only worried about this line so how do we know if the properties are independent of one

another well what you do is you take a line that that's a constant of one parameter or

one variable and see if it fixes the other so if i take a line of

of constant temperature first of all does it fix the specific volume well no the specific volume can

literally be any any range of values along that line of constant temperature

so pinning temperature doesn't fit specific volume what about pressure well the pressure

if i take this line of constant temperature the pressure can be a whole bunch of different things it can be

that or that or that so fixing temperature doesn't fix pressure either and then what about the other

combination of specific volume and pressure well if i take a line of constant specific volume

there's all kinds of different pressure contours that strike that line so it means that fixing specific volume

doesn't fix pressure either so basically what this means is that all of the properties in the sub cooled

region are independent of one another and what that means according to the state postulate

is that any combination of two of them defines the state

we can also show this graphically so for instance take a value of pressure so this

pressure and this specific volume it defines one point

in this region take temperature and this pressure it defines one point

in this region those two lines only cross each other once and finally take specific volume and

temperature so let's take this line of specific volume this line of temperature

they cross one time in this region and that's how you observe the independence of all the properties so we

take any two of the variables they cross once they define one particular point which means the

state is fixed so moving on with sub-cooled liquid water um

we've established that all of the properties are independent of one another so specific volume temperature

pressure and the other properties we're going to talk about as well um

i've just i want to talk a little more about about some cold liquids not just water so-called liquid

water refrigerants and so on and because we're going to simplify a little bit of what we understand about water

using our insight of some of the properties so i've noted here that specific volume is a weak function of

pressure but a moderate function of temperature now what does that mean

it's a weak function of pressure because if you apply pressure to a liquid it doesn't change

its volume and if it doesn't change its volume then it doesn't change its density

so you can pressurize a liquid to a very very high pressure and and changes in density are almost

unobservable and this is why liquids are used in hydraulic systems because they can withstand

massive changes in pressures without changing their their density or their specific volume

when i say that specific volume is a moderate function of temperature what i mean is that

when you warm a liquid it does undergo small changes in density now you don't under small changes in

temperature you don't notice these changes in density the way you would notice it is if the the

mass grew in volume that would mean that the density is changing so you have to actually have a have a large temperature

range to actually see changes in the specific volume due to temperature but we still say that it's a

moderate function of t and you'll see some of this when we start looking at

the thermodynamic tables so speaking of the tables and referring back to the introduction

video for the appendices table a7 contains properties for compressed liquid water

and there is no analogous table for refrigerant 134 so this is why it's so important for us

to understand a little more about the properties because this allows us to have a different approach

for evaluating specific properties as a given state here's an example of looking up

properties for liquid water so the example is find specific volume for water at five

megapascals and 20 degrees c so we've got two independent intensive property so they do define a state we

should be able to look up the value of the specific volume so the table to go to

is table a7 and from table a7 you can just straight look up the value specific volume

is equal to 0.000995 cubic meters per kilogram that's a lot of accuracy but

this is the the accuracy to which the tables are evaluated so we use the full accuracy of the table

another way of looking up this property knowing what we know about liquid water the fact that it is

incompressible virtually incompressible so in this respect

if something is incompressible then specific volume as a function of p and t which is what we've been asked for

can be approximated as being a function of t only because the pressure has such a low impact

on the value of of the specific volume or anything else so in this case if we if we are

interested in v as a function of t we can just simply look up

the saturated liquid value of specific volume at the temperature so graphically what

this looks like is this so here's the saturated liquid line

okay so this this line denotes f and using table four which contains all of the values on

this line we can just go down the scale of temperature to 20 degrees

and look up the value of the specific volume at the temperature 20 degrees which is right there

and if we do that from table a4 we get that specific volume at 20 degrees c

is point zero zero one zero zero two and you can see there's a minuscule difference between

this value and this value even though the pressures are wildly different because the pressure you're

seeing in table a4 is the saturation pressure for water to boil at 20 degrees whereas

that's compared to 5 megapascals which is on the order of 50 atmospheres of pressure so you can see there's such a

minuscule difference in the specific volume over this huge range of pressures

so we understand a lot more about the specific volume in terms of its relationship with pressure and

temperature what about internal energy so i'm noting here that internal energy

is also a weak function of pressure but a strong function of temperature now how do we know this

well so what we have to observe is how this the internal energy changes when we change the temperature or when

we change the pressure now clearly when we change the temperature so we can take a liquid

water and we can add energy to the water by heating it

okay so we add q that makes the temperature go up and the internal energy is is basically

manifest as changes in temperature so we raise the temperature lower the temperature

the internal energy changes accordingly so definitely adding energy as heat affects the the temperature and

consequently the internal energy or vice versa now think about the other means of adding energy to the

system that would be by changing its volume by doing pdv type work on it

well we've said that for an incompressible liquid the reason we call it incompressible is that when you

pressurize it it doesn't change its shape so what that means is that

if we pressurize this liquid the volume doesn't change so the integral of pdv actually is zero so in other words it's

virtually impossible to add energy to a fluid to change its internal energy

by simply pressurizing it and this is why we say the internal energy is a very weak

function of the pressure because under very high pressurization the volume still doesn't change

so you're not actually changing the energy level of the liquid let's look at an example for for this

case so the example is now to find the internal energy for water at the same

pressure and temperature as we looked at for a specific volume so if we just go straight into table a7

which is for it's called the compressed liquid table it's basically for the sub cooled liquid

region we can just look straight up internal energy equals

83.61 kilojoules per kilogram if we take into consideration what we know about

the incompressible liquid the fact that pressure has no impact on energy we make the same approximation we

made for specific volume where we say that internal energy you know evaluated as a

function of p and t can be approximated as being a function of t

only in which case we can use table a7 as we did before and just look up the value of the internal energy on

the saturated liquid line at the temperature and that's what i've indicated here so we're approximating u

as a function of p and t as u on the saturated liquid line that's what the f means at the

temperature 20 degrees and we get a value 83.91 kilojoules per kilogram

compared to 83.61 kilojoules per kilogram by looking it up in the sub-gold liquid tables

now i keep doing this because generally speaking you're not going to be using the compressed liquid

tables very often the sub-cooled liquid you're going to find that it's a lot faster a lot more efficient

to simply use the the tables for the mixture region which we're going to talk about next

uh to look up properties for the subcooled liquid region