Mathematics of Measurable Parameters

P M V SubbaraoProfessor

Mechanical Engineering Department

A Pro-active Philosophy of Inventions….

Basic Steps in Development of Instruments

• Development of Mathematical Model for Identification of Parameters to be measured.

• Identification of characteristics to be possessed by a general Instruments.

• Qualitative and Quantitative models for determination of Instrument design details.

• Selection of geometrical and physical parameters.

Thermodynamic Property

• Any suitable characteristic whose value depends on the condition of a system and which is relevant to our thermodynamic study is known as a Thermodynamic Property.

– Intensive property -- Local property -- Independent of mass or size of the system.

– Extensive property -- Depends on the extent of the system -- Obey Colligative Law.

Specific Internal Specific Specific

Temp Pressure Volume Energy Enthalpy Entropy

C MPa m3/kg kJ/kg kJ/kg kJ/kg/K

1 669.2 1 0.4334 3420 3853 8.2

2 1197 10 0.06774 4438 5116 8.2

3 669.2 3.567 0.1204 3407 3837 7.6

– Primitive Property -- Directly observable -- With simple experiments -- No change in system.

– Derived Property -- Vigorous experiments -- To be calculated using observations.

Generalization of the Concept of Property

• Corollary 1: A change of state is fully described by means of the initial and final values of all the primitive properties of the system.– A change occurs when at least one of its primitive properties changes

value.

• Corollary 2: A process is required for the determination of a derived property.

• Corollary 3: The change in value of a property is fixed by the end states of a system undergoing a change of state and is independent of the path.

• Corollary 4: Any quantity which is fixed by the end states of a process is a property.

• Corollary 5: When a system goes through a cycle, the change in value of any property is zero.

• Corollary 6: Any quantity whose change in a cycle is zero is a property of a system. 00 dewq 00 ds

T

q

How to Get An Objective Method to Identify A Property…..

Need for A New Branch in Mathematics !!!

Functions of Several Variables

• Let the temperature T depend on variables x, y and z.

T = F(x,y,z)= Constant . This is called as pfaffian function.

• F(.) is called as Point Function.

• The rate of change of f with respect to x (holding y constant) is called the partial derivative of f with respect to x and is denoted by .

•Similarly, the rate of change of f with respect to y is called the partial derivative of f with respect to y and is denoted by .

x

zyxFzyxxF

x

Fx

),,(),,(lim

0

z

zyxFzzyxF

z

F

y

zyxFzyyxF

y

F

z

y

),,(),,(lim

&),,(),,(

lim

0

0

Maxwell’s Relations to Create Temperature

The Capability of A Substance:

A total change in Temperature is expressed as:

0,,,

dzz

Fdy

y

Fdx

x

FdF

yxxzzy

Define functions M,N & P such that:

0),,(),,(),,( dzzyxPdyzyxNdxzyxMdF

As F(x,y,z) is a point function, differentiation is independent of order.

yxzyzxyxzyzxz

P

x

P

y

P

z

N

x

N

y

M

,,,,,,

::

This is called as pfaffian differential equation.

yxzyzxyxzyzxz

P

x

P

y

P

z

N

x

N

y

M

,,,,,,

::

This is a necessary and sufficient condition for F(.) to be a Point function or state function.

Thermodynamic properties are so related that F(.) is constant.

Every substance is represented as F(.) in Mathematical (Caratheodory) Thermodynamics.

This is shows a surface connectivity of Property of a substance.

p-v-T Diagram of A Substance

p-V-T Diagram of solid Phase

Volume

Temperature

Pressure

Top View : V – T Diagram

Front View : p – V Diagram

Inaccessibility of Thermodynamic State

From an point on any (property) surface (of a substance) only on the same surface are accessible along that solution surface.

Any given substance, starting from an initial state on its property relation surface it can access only those states which lie on this surface.

Consider a analytic function G(x,y,z) such that

0),,(),,(),,( dzzyxPdyzyxNdxzyxMdG

This total differential may not satisfy

yxzyzxyxzyzxz

P

x

P

y

P

z

N

x

N

y

M

,,,,,,

::

Then G(x,y,z) is not a property function (surface) of any substance…

However, if another analytic function H(x,y) is such that:

0),(),( dyyxNdxyxMdG

Then

),(),(

),(yxf

yxN

yxM

dx

dy

),( yxfdx

dy Is an ordinary differential equation with

F(x,y) = constant

All pfaffian differential equations in two variables are Point functions.

Theorem of Caratheodory: If, in the neighborhood of an arbitrary point in the domain of the variables of a Pfaffian Differential Equation, there are points which are inaccessible to it along the solution surface of the equation, then this equation has an integrating denominator.

It is possible to multiply any differential equation by a function of independent variables and make pfaffian differential equation.

This function is called integrating denominator.

• The expression dz may be called the“total differential” of f(x, y, ...), and may also be denoted by df.

TESTING FOR EXACT DIFFERENTIALS

In general, an expression of the form,

df = P(x, y, ...)dx + Q(x, y, ...)dy + . . .,

will not be the total differential of a function f(x, y, ...), unless P(x, y, ...), Q(x, y, ...) etc. can be identified with f/x, f/y etc., respectively.

If this IS possible, then the expression is known as an “exact differential”.

The expression, df = P(x, y)dx + Q(x, y)dy, is an exact differential if and only if

x

Q

y

P

Integration of Exact Differential

• we move in the X - Y plane from an initial point (xi,yi) , to a final point (xf,yf) , then the corresponding change in is given by

f = ff - fi = df = P(x,y)dx + Q(x,y)dy• The difference (f) on the left-hand side depends only on the initial

and final points, the integral on the right-hand side can only depend on these points as well.

• the value of the integral is independent of the path taken in going from the initial to the final point.

• that the integral of an exact differential over a closed circuit is always zero

0df

Conclusions

• A System should be described using an Exact function.

• The change in this function can be calculated without doing any action.

• Instruments can be easily developed to measure these functions.

• Function tables can be prepared and marketed.

• Any industrial process can use these tables.

• Primary functions are in general directly measured.

• Derived functions are calculated using the measured primary functions.

• Special instruments can be developed to measure derived properties !

• This function is called as PROPERTY in Thermodynamics.

Basic Steps in Development of Instruments

• Development of Mathematical Model for Identification of Parameters to be measured.

• Identification of characteristics to be possessed by a general Instruments.

• Qualitative and Quantitative models for determination of Instrument design details.

• Selection of geometrical and physical parameters.