Chemical to Mechanical Energy
WORKING PRINCIPLES
A) HEAT ENERGY
i)Elevation in boiling point (Raoult’s Law application) Raoult’s Law for a solution containing a non volatile solute states that the relative lowering of vapour pressure of a liquid is equal to the mole fraction of the solute.
( P10-P1 )/ P10 = X2
where P10 is the Vapour Pressure of pure solvent
P1 is the Vapour Pressure of the solution with non volatile solute
X2 is the mole fraction (concentration of solute).
We also know that boiling point of a liquid is the temperature at which Vapour Pressure of the liquid is equal to the atmospheric pressure.Hence, due to the presence of the non volatile solutes in our solution (impurities), more heat / higher temperature is required to make the V.P. of the solution= atmospheric pressure.
Figure -2
ΔTb=Kb m
where ΔTb is the elevation in boiling point of pure solution
Kb is the Molal Elevation Constant (Ebbullioscopic constant)
m is the molality of the solution.
ii) Change in physical state of water
Current temperature of solution (water) = 299.15 K
Elevated boiling point of water = Normal Boiling Point + ΔTb
= 373.15 K + 2K = 375.15 K
Difference b/w current temperature and elevated boiling point,
dT = 375.15 K - 299.15 K = 76 K
Now, Specific Heat Capacity (S) is defined as the amount of heat absorbed / evolved per unit mass to change its temperature by one unit. 4
Specific Heat Capacity for water ( Sw ) = 4.186 kJ Kg-1 K-1
And, Latent Heat of vaporisation ( also known as Hidden Heat) is the heat absorbed by a liquid to convert into gaseous state without increasing the temperature. [4]
Finally, the mass of water = Volume occupied x Density of water
M = πr2h x 1000 Kg m-3
M = [3.14 ( .0762)2 ( .0762) ] m3 x 1000 Kg m-3
= 1.38 Kg
where r = radius of container
h = height of water column
Here, the total heat absorbed Qin to convert water at K to steam at 373.15 K is given by
Qin = MSw dT + MHv = M(Sw dT + Hv )
where M is the mass of solution (water)
Sw is the Specific Heat Capacity of water (4.186 kJ kg-1 K-1)
dT is Difference b/w current temperature and elevated boiling point.
Hv is the Latent Heat of vaporisation (2257 kJ/kg).
B) MECHANICAL ENERGY
iii) Rotational analogues of motion.
As the steam flows through the nozzle its pressure falls from inlet pressure to the exit pressure (atmospheric pressure, or more usually, the condenser vacuum). Due to this high ratio of expansion of steam, the steam leaves the nozzle with a very high velocity.
Now, we will calculate the work done by the torque generated by the steam.
Calculating the number of revolutions of rotor blades= 30
Circumference of the path covered by the metal fins= 2π (7.62) , where r is the radius of the path.
Angular displacement(θ , the rotational analogue of linear displacement). In radians θ =(Arc length/radius) = (n 2πr/ r)
Angular acceleration (α) is the rotational analogue of linear acceleration.It is given by the rate of change of angular velocity or the second derivative of angular displacement with respect to time
which is the average angular acceleration for the entire rotation.
where ω, is angular velocity (rotational analogue of linear velocity) and θ is the angular displacement.
Torque (τ) is the rotational analogue of linear force and is given by the product of moment of inertia (rotational analogue of substance mass) and the angular acceleration. Just like rate of change of linear momentum gives us the applied force, rate of change of angular momentum gives us angular force (Torque applied)
where I is the moment of inertia calculated by
I = mL2/4,
where L is the diameter of the path covered by the fins.
Work done is defined as the scalar product of displacement and the component of force in the direction of displacement.Since the work done here is due to the torque generated by the steam discharge, total work done is given by
iv) Work energy theorem:
This work done is the obtained due to the change in kinetic energy of the steam between its initial position (point of discharge) and final position.
Well just a short proof , for the curious minds, is given by-
In our case Vf is the velocity with which the steam is impinged on the rotor blades and V0 is its discharge velocity (the velocity with which it leaves the tin nozzle).
v) According to Law of conservation of energy :
energy can neither be created nor be destroyed but can be converted from one form to another. This theorem can also be restated in the form of the First law of thermodynamics.
Hence the energy used to mechanically rotate the rotor fins can be taken as the transformed mechanical output to the heat energy input.
vi) Calculating efficiency of the energy converter:
The efficiency of the heat engine is defined as the ratio of energy output to the energy output
η = (Energy output) / (Energy input)
= (Work Done) / ( Heat Supplied)
= ( W / Q in ) x 100
vii) Carnot efficiency theorem:
No engine can give an efficiency greater than that of the carnot engine and the efficiency of the carnot engine is independent of the nature of substance. Therefore, in our project, the efficiency of our energy converter will always be less than 1 ( or 100%) . i.e. η < 1
Conclusion:
This project was primarily an effort to apply my theoretical knowledge attained from school classrooms to a small working model created solely to spread awareness about physical sciences among the juniors with some scientific entertainment.
Our primary conclusion is that the chemical energy of the fuel is converted into heat energy which in turn is converted into mechanical energy; also, the efficiency of our energy converter is always less than 100%.
We have practically observed a rise in the boiling point of water due to its non volatile impurities (a direct consequence of Raoult’s law). We then went on to heat the water from the initial temperature to its elevated boiling point. Consequentially we calculated the heat energy supplied during the process of vaporisation.
At appropriate temperature, water was converted into steam which was then saturated by restriction on its discharge. We use the rotational analogues of motion in their full glow once the discharged steam impinged on the rotor blades and generated torque which performedthe mechanical work. In explanation of this process of energy conversion we also reviewed energy conservation law ( First Law of thermodynamics) and work-energy theorem. Finally we concluded by calculating the efficiency of our energy converter which was in agreement with Carnot’s theorem for efficiency of heat engines.
References:
1. Definition retrieved from - https://en.wikipedia.org/wiki/Energy
2. Referred from chapter “Solutions” in the grade 12 NCERT textbook for chemistry (part 1).
3. Figure-2 retrieved from - http://chemistry.tutorvista.com/physical-chemistry/colligative- properties.html
4. Definitions of Specific heat and latent heat of vaporization retrieved from– chapter “Heat” in the class 11 NCERT textbook for physics (part 2).
5. Referred from 11 NCERT textbook for physics (part 1).
6. Definition retrieved from chapter “Work, energy and power” in the grade 11 NCERT textbook for Physics (Part 1).
7. Figure-3 retrieved from-
http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Work/WorkEngergyTheorem.html
8. Definition retrieved from chapter “Thermodynamics” in the grade 11 NCERT textbook for Physics (Part 2).
9. Referred from chapter “Thermodynamics” in the grade 11 NCERT textbook for physics (Part 2).