MAE 570 Intermediate Thermodynamics Exam 2 Concept Review

Carnot Vapor Cycle

• Most efficient cycle operating between two specified temperature limits

• Isothermal heat addition in boiler: limited heat transfer processes to two-phase systems severely limits max temp in cycle

• Isentropic expansion in turbine: cannot handle steam with high moisture content

• Isentropic compression: not practical for two phases

Rankine Cycle

1 → 2 Isentropic compression in pump

2 → 3 Isobaric heat addition in boiler

3 → 4 Isentropic expansion in turbine

4 → 1 Isobaric heat rejection in condenser

Rankine Cycle Benefits

• Superheats steam in boiler, completely condensed in condenser

• Ideal for vapor power plants

• No internal irreversibilities

• Eliminates impracticalities associated with Carnot cycle

Efficiency of Ideal Rankine Cycle

η_Rankine = w_net/q_in = 1 - (q_out)/(q_in)

Isentropic Efficiency of Pump

η_p = w_s/w_a = (h2s - h1)/(h2a-h1)

Isentropic Efficiency of Turbine

η_t = w_a/w_s = (h3-h4a)/(h3 - h4s)

Ideal Rankine vs. with Irreversibilities

For the same system with isentropic efficiencies added: quality of the steam @ turbine exit increases, net work done per unit mass decreases, thermal efficiency decreases, and steam mass flow rate increases

Ways to Increase Rankine Cycle Efficiency

• Increase the average temperature at which heat is transferred to the working fluid in the boiler

• Decrease the average temperature at which heat is rejected form the working fluid in the condenser

• Lower the pressure of the condenser (increases moisture content of the steam in the turbine though)

• Superheating steam in the boiler (limited by metallurgical considerations)

• Increasing pressure of the boiler (increases moisture content of the steam in the turbine though)

  • Turbine moisture can be corrected by reheating steam

• Operate at supercritical pressure (P > 22.06 MPa)

Ideal Reheat Rankine cycle

• Superheat steam to very high temperatures (limited metallurgically)

• Expand the steam in the turbine in two stages with reheating in between

• η_Rankine = w_net/q_in = 1 - (q_out)/(q_in)

• Single reheat in a modern power plant can improve efficiency by 4-5%

Multi-Stage Reheat Rankine Cycle

• Increases average temperature during reheat process

• Impractical to use more than 2 reheat stages

• Theoretical efficiency from second reheat = ½ of that of single reheat

Ideal Regenerative Rankine Cycle

• Steam extracted from turbine at various points to heat feedwater

• Feedwater heater (FWH) or regenerator heats feedwater

• FWH = HX where heat is transferred from steam to feedwater by mixing two fluid streams (open FWH) or without mixing them (closed FWH)

Open Feedwater Heater (FWH)

• Mixing chamber that mixes steam extracted from the turbine with feedwater exiting the pump

• Ideally, mixture leaves the heater as a saturated liquid at the heater pressure

Open Feedwater Heater (FWH) Performance

• y = ṁ₆/ṁ₅

• q_in = h₅ - h₄

• q_out = (1-y)(h₇-h₁)

• w_t = (h₅ - h₆) + (1-y)(h₆ - h₇)

• w_p = [(1-y)(h₂ - h₁)]_{pump 1} + (h₄ - h₃)_{pump 2}

Closed Feedwater Heater (FWH)

• Heat transferred from extracted stream to feedwater with no mixing taking place

• Streams can be at different prssures, since they do not mix

Closed Feedwater Heater (FWH) Performance

• y = ṁ₇/ṁ₆

• q_out = h₆ - h₅

• q_out = (1-y)(h₈-h₁)

• w_t = (h₆ - h₇) + (1-y)(h₇ - h₈)

• w_p = [(1-y)(h₂ - h₁)]_{pump 1} + [(y)(h₄ - h₃)]_{pump 2}

Combining Open and Closed FWH

Process Heat

A type of heat input required by industries. Process heat usually supplied by steam at 5 to 7 atm and 150 to 200ºC. Energy typically transferred to steam by burning coal, oil, natural gas, or another fuel in a furnace.

Cogeneration

• Produces more than one useful form of energy (i.e. process heat and electric power) from the same energy source

• Utilizes already-existing work potential to produce power (rather than wasting it)

• Produces electricity while meeting process-heat requirements of certain industrial processes

Utilization Factor

• ε_u = (Ẇ_net + Q̇_P)/Q̇_in

• For ideal steam-turbine cogeneration plant: 100%

• Actual plants have ε_u up to 80%

Cogeneration Energy Balance

• Q̇_in = ṁ₃(h₄ - h₃) (heat supplied in boiler)

• Q̇_out = ṁ₇(h₇ - h₁) (heat rejected in condenser)

• Q̇_P = ṁ₅h₅ + ṁ₆h₆ - ṁ₈h₈ (processed heat)

• Ẇ_net = (ṁ₄ - ṁ₅)(h₄ - h₆) + ṁ₇(h₆ - h₇) (work done by turbine)

Combined Gas-Vapor Cycle

• Combines Brayton and Rankine cycles, resulting in a higher thermal efficiency than either of the cycles individually

• Uses high-temperature exhaust gases from gas turbine as energy source for bottoming cycle

• Increases efficiency without significant increases in initiial cost

• Recent turbine developments made combine cycle much more economically attractive

Clausius Statement

• Heat cannot flow spontaneously from a colder body to a warmer body without external work

• Implication: a refrigerator cannot operate unless its compressor is driven by an external power source (i.e. electric motor)

Refrigerator

The transfer of heat from lower-temperature regions to higher temperature ones

Heat pump

Type of refrigerator that is used to heat a space by transferring heat from a cooler medium

Coefficient of Performance (COP)

• Quantifies performance of refrigerators and heat pumps

• COP_HP = COP_R + 1

Refrigerator COP

COP_R = (desired output)/(required input) = (cooling effect)/(work input) = Q_L / W_net

Heat Pump COP

COP_HP = (desired output)/(required input) = (heating effect)/(work input) = Q_H / W_net

Reversed Carnot Cycle

• Reverses order of regular Carnot cycle, including directions and heat/work interactions

• Counterclockwise on T-s diagram

• Most efficient refrigeration Cycle Operating between T_L and T_H, but not a suitable model in reality

• 2 → 3 involves the compression of a liquid-vapor mixture (two phase compressor needed)

• 4 → 1 involves the expansion of a high-moisture content refrigerant in a turbine

• COP of Carnot refrigerator: COP_R = (Q_L)/(Q_H - Q_L) = (T_L)/(T_H - T_L)

Ideal Vapor-Compression Refrigeration Cycle Traits

• Vaporizes refrigerant completely before compressing

• Replaces turbine with throttling device (i.e. expansion valve or capillary tube)

• Most widely used cycle for refrigerators, AC, and heat pumps

Ideal Vapor-Compression Refrigeration Cycle

1 → 2 Isentropic compression (compressor)

2 → 3 Isobaric heat rejection (condenser)

3 → 4 Isenthalpic Throttling (expansion device)

4 → 1 Isobaric heat absorption (evaporator)

COP of Vapor Compression Cycle

• q_L = h₁ - h₄

• q_H = h₂ - h₃

• w_net = h₂ - h₁

• COP_R = q_L/w_net = (h₁ - h₄)/(h₂ - h₁)

• COP_HP = q_H/w_net = (h₂ - h₃)/(h₂ - h₁)

Actual Vapor Compression Cycle (Irreversibilities)

• Non-isentropic compression

• Superheated vapor at evaporator exit

• Subcooled liquid at condenser exit

• Pressure drops in condenser and evaporator

Ideal Vapor Compression vs. with Irreversibilities

For the same system with isentropic efficiencies added: more heat absorbed in evaporator, more heat rejected in condenser, more net work done per unit mass of refrigerant, lower COP, (slightly) lower refrigerant mass flow rate

Selecting a Refrigerant

• Typically chlorofluorocarbons (CFCs), HFCs, HCFCs, ammonia, hydrocarbons, carbon dioxide, air, or even water

• Need to consider temperature of refrigerated space and environment

• Ammonia very low on global warming potential (GWP) and ozone depletion potential (ODP) but is very toxic

Ozone Depletion Potential (ODP)

Relative amount of degradation to the ozone layer that it can cause, R-11 fixed at an ODP of 1.0

Global Warming Potential (GWP)

Measure of how much heat a greenhouse gas traps in the atmosphere over a specific time period, relative to carbon dioxide (CO₂)

Heat Pump Systems

• Energy source usually atmospheric air (air-to-air system)

• Higher COP, but more expensive to install: Water-source systems (well water) or ground-source (geothermal)

• Capacity and efficiency of heat pumps falls significantly at low temperatures, so usually a supplementary heating system rewuired

• Heat pumps most competitive in areas that have a large cooling load during the cooling season and a relatively small heating load during the heating season (i.e. Arizona)

Cascade Refrigeration System

• Moderately low temperatures required for some industrial applications, temp range may be too large for a simple vapor-compression cycle

• Improves COP of refrigeration system

• Up to 3-4 systems of cascading

Multistage Compression System

• When the fluid throughout the cascade refrigeration system is the same

• Heat exchanger replaced by mixing chamber (or flash chamber)

• Better heat transfer characteristics

Energy Balance in Cascading Systems

• Q̇_L= ṁ_B(h₁ - h₄) (heat absorbed in evaporator)

• Q̇_H = ṁ_A(h₆ - h₇) (heat rejected in condenser)

• Q̇_tr = ṁ_B(h₂ - h₃) = ṁ_A(h₅ - h₈) (heat transferred in HX)

• Ẇ_net = ṁ_B(h₂ - h₁) + ṁ_A(h₆ - h₅) (total work input)

• COP_R = Q̇_L/Ẇ_net

Energy Balance in Multistage Compression System

• Q̇_L= (1 - y)(h₁ - h₈) (heat absorbed in evaporator)

• Q̇_H = (h₄ - h₅) (heat rejected in condenser)

• h₆ = y(h₃) + (1 - y)h₇

• h₉ = y(h₃) + (1 - y)h₂

• Ẇ_net = (1 - y)(h₂ - h₁) + (h₄ - h₉) (total work input)

• COP_R = Q̇_L/Ẇ_net

Multipurpose Refrigeration System

• Allows for refrigeration at more than one temperature

• Routes all exit streams from evaporators to a single compressor to let it handle the compression process for the entire system

Energy Balance in Multipurpose Refrigeration System

• Q̇_L= Q̇_L,R + Q̇_L,F = (h₅ - h₄) + (h₁ - h₆) (heat absorbed in refrigerator and freezer)

• Q̇_H = h₂ - h₃ (heat rejected in condenser)

• Ẇ_net = h₂ - h₁ (total work input)

• COP_R = Q̇_L/Ẇ_net

Gas Refrigeration Cycle

• Also known as reversed Brayton cycle

• Lower COPs relative to the vapor compression refrigeration cycles or reversed Carnot cycle

• Involve simpler, lighter components, suitable for aircraft cooling

• Can also incorporate regeneration

• COP_R = q_L/w_net = (h₁ - h₄)/[(h₂ - h₁) - (h₃ - h₄)]

Gas Refrigeration Cycle with Regeneration

• Can decrease the lowest temperature of the cycle, achieving extremely low temperatures

• Further cools high-pressure gas to T₄ before expanding in the turbine

Absorption Refrigeration

• Economic when there is a source of inexpensive thermal energy at a temp of 100-200ºC

• Can use geothermal energy, solar energy, and waste heat from cogeneration or process

• Involve the absorption of a refrigerant by a transport medium (commonly ammonia and water)

• Liquid is compressed instead of a vapor, resulting in negligible work input for the cycle

• Complicated, large, much less efficient, more difficult to service

• Only considered for large industrial/commercial applications where the unit cost of thermal energy is low relative to electricity

• COP_max = (1 - T₀/T_s)*[T_L/(T₀ - T_L)]

Compressed Liquid Approximation

For compressed liquid @ pressure P & temperature T:

h ≈ h_f @ T

Formula for Compressed Liquid Enthalpy Value Through a Pump

h₂ = h₁ + v₂(P₂ - P₁)