COMPENDIUMOF EQUATIONS

MAE 301 : ENGINEERING THERMODYNAMICS -I

-Mitesh Ghelani

1 Introduction

Thermo I (MAE 301) deals with the transfer and conversion of different forms of energy, e.g. heat and work in different devices (or systems) via different processes using different fluids. In addition to the conservation laws of energy and mass, entropy generation has to be checked as well. We only deal with simple compressible substances which undergo quasi-equilibrium processes

2 Balances for Systems (or Control Volumes)

2.1 Mass Balance: X X ∆m m˙ − m˙ = in out ∆t C.&v.

2.2 Energy Balance: X X ∆E  E˙ − E˙ = in out ∆t C.&v. or for observation time ∆t, X X E − E = ∆E| in out C.&v. 2.3 Entropy Balance: X X ∆S  S˙ − S˙ + S˙ = in out gen ∆t C.&v. or for observation time ∆t, X X S − S + S = ∆S| in out gen C.&v. Notes:

X ˙ X ˙ X ˙ X E = Q + W + mh˙ ; ∆E|C.v. = [∆U + ∆PE + ∆KE] ≈ ∆U|C.v. & C.&v. & in/out

Q X X X X Q˙ = ; W = W + W ; W = W 0 + W + W + ····· ∆t mech elec mech b dary spring shaft

Z 2 Z 2 ˙ Wb∆t = Wb0dary = p dV& whereas Wflow = ν dp 1 1

1 ˙ X ˙ X X Q S = ms˙ + ; ∆S|C.v. = [m2s2 − m1s1]C.v. = [Sfinal − Sinitial]C.v. Ts & & &

dh = du + d(pν) ; ∆H = ∆U + p∆V&, when p = constant

3 Identification of System (i.e., Device) , Fluid and Process

3.1 Type of System: Open System: A steady or transient system; for e.g. a turbine or tank with valve, respectively.

Closed System: A piston-cylinder device or tank.

Cycles: A Heat Engine, Heat Pump, Refrigerator, CARNOT Cycle, RANKINE Cycle, BRAYTON Cycle, DIESEL Cycle, OTTO Cycle, etc.

3.2 Type of Fluid:

Ideal Gases: Air, He, N2, CO2, etc. Water: We study water in three states, i.e., saturated liquid, mixture, or saturated steam.

R-134a: We study refrigeration fluid (R-134a) in three states: saturated liquid, mixture or vapor.

3.3 Type of Process: Expansion Vs. Compression: Change in system/fluid volume is referred as Expansion or Compres- sion. If the volume increases then it is called Expansion and if the volume decreases then it is called Compression. Either of these processes can be categorized as: Adiabatic process, Isother- mal process, Isochoric process, Isentropic process , Polytropic process or Isobaric process. We shall see each process individually in Section 7.

Heat Exchanger Vs. Stream Mixing: The process of transfer of thermal energy from one stream of matter (solid or fluid) to another by radiation, convection or conduction, to affect a change in temperature of the two streams. If heat is transfered without streams coming in direct contact, then the device is called a Heat Exchanger. If heat is transfered with streams coming in direct contact, then it is called Mixing Chamber. Also, this process may or may not see the phase change of the stream. We shall see Phase Changes in Section 6. Also, Heat Transfer, Q = mCp∆T , where typically Qout = Qin.

4 Data Resources

For solving problems, another aspect that is important once you have identified your system, fluid and process is to collect / calculate the unknown data using Property Tables and/or some relations.

2 4.1 Ideal Gases: 1. Equation of state for Ideal Gas (IG):

Simplest Form : pV& = mRT . An elaborate explanation of this is given in Section 6. 2. Polytropic Process Equations: Polytropic process is near realistic process. Refer to Section 7 for further information.

3. GIBBS Relations: GIBBS relations are used for calculating the change in entropy. A list of these equations are given in Section 5.

4. Data Tables for Air: Air is the most common IG in Thermodynamic Problems. Hence, there exist extensive Data Tables for different parameters of air which can be used directly for solving problems. Using data tables has its own advantages. First you need not calculate values using equations. Secondly, it gives you very precise values at given conditions.

4.2 Two-Phase Fluids For most substances, the relationships among the thermodynamic properties are too complex to be expressed by simple equations. Therefore, properties are frequently presented in the form of tables. Some thermodynamic properties can be measured easily, but others cannot and are calculated by us- ing the relations between them and measurable properties. The results of these measurements and calculations are presented in tables in a convenient format. In this class, we will mostly deal with Water and R-134a . Following are the two tables which we will use extensively:

1. Steam tables for H20 2. Property tables for R-134a

4.3 Phase Diagrams for Two-Phase Fluids: Here we will see the T-v , p-v and T-s diagrams of water. T-v diagram of water is shown below:

3 dia water.jpg

Figure 1: p -v diagram of Water

dia water.jpg

Figure 2: T -v diagram of Water

4 T-V dia water1.jpg Figure 3: T -v or p -v diagram of Water showing specific mixture volume 1

of p-V T-V dia water2.jpg Figure 4: T -v or p -v diagram of Water showing specific mixture volume 2

5 Following is T-s diagram for Water which shows the Isobars and Isochors:

5 Special Cases / Applications

5.1 Closed System (First Law):

Qnet − Wnet ≈ ∆Usystem ; msystem = constant

5.2 Conservation of Mass for Steady Flow: X X v V m˙ = m˙ ;m ˙ = ρAv = A ; ν = & i e ν m inlets exits

5.3 First Law for Steady Open System with Uniform Flow: X  v2  X  v2  W˙ + Q˙ + m˙ h + + gz = m˙ h + + gz net net i 2 e 2 inlets i exits e

5.4 General Relations: dh = du + d(pν); du = T ds − pdν ; dh = T ds + νdp

 ∂u   du   ∂h   dh  Cv = = ; Cp = = ∂T v dT IG ∂T p dT IG

5.5 Ideal Gas Relations:

p2V p1V & 2 & 1 Cp k R pV = mRT ; = ; k = ; Cp = R ; Cv = T2 T1 Cv k − 1 k − 1

∆H = mCp∆T ; ∆U = mCv∆T

" !# Z T2 C (T ) p  Z T2 C (T ) V ∆S = m p dT − R ln 2 = m v dT + R ln & 2 T p1 T V T1 T1 & 1

5.6 Efficiency(η) or Coefficient of Perfomance (COP): 1. Gerneral Definition: Desired Output η or COP = Required Input 2. Heat Engine (HE): • 1st Law for a Cycle: Wnet = QH − QL

6 Following is T-s diagram for Water which shows the three major regions:

dia water.jpg

Figure 5: T -s diagram of Water

dia water 02.jpg

Figure 6: T -s diagram of Water with Isobars and Isochors

7 • Thermal Efficiency:

Wnet,out QH − QL QL ηHE,th = = = 1 − Qin QH QH • Isentropic Efficiency:

isentr. Wactual isentr. Wisentr. ηT = ; ηC = Wisentr. Wactual 3. Refrigerator (R):

QL QL 1 COPR,th = = = Q Wnet,in QH − QL H − 1 QL 4. Heat Pump (HP):

QH QH 1 COPHP,th = = = Q Wnet,in QH − QL 1 − L QH 5. CARNOT or Reversible Heat Engine (HE):

Wnet,out QH − QL QL TL ηHE,th,rev = = = 1 − = 1 − Qin QH QH TH 6. CARNOT or Reversible Refrigerator (R):

QL QL 1 1 COPR,th,rev = = = Q = T Wnet,in QH − QL H − 1 H − 1 QL TL 7. CARNOT or Reversible Heat Pump (HP):

QH QH 1 1 COPHP,th,rev = = = Q = T Wnet,in QH − QL 1 − L 1 − L QH TH

8. Relation between COPR and COPHP :

COPHP = COPR + 1

6 Pure Substances and Phase Changes

6.1 Definitions (see also Section 4.3): Pure Substance: A substance that has a fixed chemical composition throughout is called a pure sub- stance. Phase Change: A pure substance exists in different phases (viz Solid, Liquid or Gaseous) depending on its energy level. Transition from one phase to another either by giving away energy or by accepting energy is known as ”Phase Change”. Compressed Liquid: In the liquid phase, a substance that is not about to vaporize is called a com- pressed liquid.

8 Superheated Vapor: In the gas phase, a substance that is not about to condense is called a super- heated vapor.

Saturation Temperature and Pressure: At a given pressure a substance changes phase at a fixed temperature is called Saturation Temperature. Likewise, at a given temperature a substance changes phase at a fixed pressure, is called Saturation Pressure.

Saturated Liquid and Vapor: During a boiling process, both the liquid and the vapor coexist in equi- librium, and under this condition liquid is called Saturated Liquid and vapor is called Saturated Vapor.

Quality: In a saturated vapor-liquid mixture, the mass fraction of vapor is called the quality and it is given by

m x = vapor mtotal . Quality ranges from 0 (saturated liquid) →1 (saturated vapor). Quality is only significant in the saturated mixture region and it is used to express average values of any intensive property y as given below:

y = yf + xyfg

where f stands for saturated fluid(liquid) and g for saturated gas(vapor), while y could be specific volume(v), internal energy(u), enthalpy(h) or entropy(s).

”‘In the absence of compressed liquid data, a general approximation is to treat a com- pressed liquid as a saturated liquid at the given temperature”’.

∼ y = yf@T

Critical Point: The state beyond which there is no disticnt vaporization process is called the critical point.

Triple Line: All the three phases of a substance coexist in equilibrium at states along the triple line characterized by triple line temperature and pressure.

6.2 Ideal Gas Law (Equation of State): R pν = RT and R = u M where R is the gas constant, Ru is the Universal Gas Constant and M is the Molar mass or Molecular weight of the gas. Since ideal gas is a fictitious substance, we define the Compressibility Factor (Z) as: ν pν Z = actual or Z = νideal RT

9 The Z factor is approximately the same for all gases at the same reduced temperature and reduced pressure which are defined below: T p TR = and pR = Tcr pcr

6.3 van der Waals Equation of State:  a  p + (ν − b) = RT ν2 where, 27R2T 2 RT a = cr and b = cr 64pcr 8pcr Now that we have seen the Energy Balance and Ideal Gas Law, we will move to different processes and work formulation.

7 Processes

Isobaric Process: A process that occurs at constant pressure. An example would be an ideal Piston- Cylinder arrangement (p-c device), where only the piston weight exerts the constant pressure.

p = C

Isochoric Process: A process that occurs at constant volume, which means that boundary work done by the system is zero e.g. a rigid tank. V& = C Isothermal Process: A process that occurs at constant temperature, typically achieved by device/fluid cooling or heating. T = C

Adiabatic/Isocaloric Process: A process that occurs with no heat transfer/exchange because of per- fect insulation. k Q = 0 and pV& = 0 n Polytropic Process: Any process that obeys P V& = C is called a Polytropic Process. Here n is called the polytropic index and is always a real number.

n n−1 n−1 p v  T v  T p  n 2 = 1 2 = 1 2 = 2 p1 v2 T1 v2 T1 p1

0 1. if n = 0, then pV& = p = C and it is an isobaric process (constant pressure) 2. if n = 1, then for an ideal gas pV& = C and it is an isothermal process (constant temperature) 3. if n = k = Cp , then, for an ideal gas, it is an adiabatic process (no heat transferred) Cv Note that 1 < k < 2, since k = Cp = Cv+R = 1 + R > 1 Cv Cv Cv 4. if n =∞, then it is an isochoric process (constant volume)

Note that using above condition, resulting P-V-T relations are very useful.

10 8 Work Formulations

In general, δW = F ds

1. Shaft Work Ws = 2πNτ

2. Electrical Work We = VeI∆t

3. Spring Work(Ks = C) 1 W = K (x2 − x2) spring 2 s 2 1 4. Work for compressible substance Z V & 2 W = pextdV V & & 1 5. Work for a compressible substance undergoing quasi-static processes (e.g., p-c device)

Z V & 2 W = pdV V & & 1

6. Work for compressible substance undergoing isothermal quasi-static process

V p W = mRT ln & 2 and W = mRT ln 1 V p2 & 1 7. Work for compressible substance undergoing isobaric quasi-static process

W = p(V − V ) & 2 & 1 8. Work for quasi-static adiabatic process

W = −(U2 − U1)

9. Work for quasi-static adiabatic process for ideal gas

W ≡ ∆U = mcv(T2 − T1)

10. Work for quasi-static polytropic process

p2V − p1V W = & 2 & 1 for n 6= 1 1 − n V W = pV ln & 2 for n = 1 V & 1

11 11. Work for Ideal Gas undergoing an isentropic process with constant specific heats

mR(T − T ) W = 2 1 for n 6= 1 1 − n V W = mRT ln & 2 for n = 1 V & 1

12 9 Nomenclature:

Cp Specific heat at constant pressure

Cv Specific heat at constant volume E Energy

E˙ Energy Rate

H Enthalpy h Mass Specific Enthalpy

U Internal Energy u Mass Specific Internal Energy

S Entropy s Mass Specific Entropy

S˙ Entropy Rate

Q Heat

Q˙ Heat Flow Rate

V& Voulume v Velocity

Ve Voltage ν Specific volume

I Current

τ Torque

Ks Spring Constant N Revolutions per minute (RPM)

W Work(J)

W˙ Work Flow m Mass m˙ Mass Flow t Time

T Temperature p Pressure

R Gas Constant

13 Ru Universal Gas Constant ρ Density k Adiabatic Index

η Efficiency

COP Coefficient of Performance n Polytropic Index

M Molecular Weight

14