Wicks Thermodynamically Ideal Engine Cycle

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The Wicks Cycle is a theoretical ideal thermodynamic cycle proposed by American Engineer Frank Wicks in 1991. Similar to Carnot cycle, it provides an upper limit on the efficiency that any classical thermodynamic engine can achieve during the conversion of heat into work, but based on the more realistic assumption of a heat source in the form of a fluid stream of finite heat capacity.

The Carnot cycle is commonly described as the ideal engine, with efficiency based on hot and cold side temperatures. It is shown as a rectangle on a Temperature-entropy diagram (T-S diagram). This means all heat is received from a constant high temperature source and rejected to a constant low temperature heat sink. However, a constant hot side temperature is usually a flawed assumption. The heat source for virtually all engines is a fluid stream of finite heat capacity, such as fuel combustion products. It may also be hot water from a geothermal well. Heat is released over the temperature range from maximum down to the ambient air and water. The Wicks Cycle is the thermodynamically ideal heat engine for these conditions.

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The seminal analysis and implementation of an ideal engine operating between these conditions was presented in a 1991 paper by Wicks.. It was originally called the Ideal Fuel Burning Engine. It is now also known as the Wicks Cycle.

The following sections :

1. Derive the efficiency of the Wicks Cycle as a function of the maximum hot side temperature and constant cold side temperature
2. Define and analyze a three process engine that uses air as the working fluid and achieves the ideal efficiency.
3. Discuss the theoretical and practical importance of understanding and correctly applying.

While the Wicks Cycle should be recognized as the ideal efficiency for most engines, the Carnot cycle remains the ideal for Heat pump and refrigeration cycles, that use work to move heat from a constant low temperature source to a constant high temperature heat sink.

Introduction

Heat engines are human inventions. Most rely upon fire. The cooling combustion products become the heat source. A portion of this thermal energy is converted to mechanical power or work by a sequence of cyclic processes. The remaining energy in the form of lower temperature heat is rejected to the environment at ambient temperature.

A fundamental question is what is the maximum portion of this thermal energy that can be converted to work. While the Carnot cycle and its efficiency is enshrined in engineering education, it does not answer the fuel burning engine question, since it assumes all high temperature heat is available at the constant temperature, whereas the combustion products release heat over the entire temperature range from maximum to ambient.

Wicks Thermodynamically Ideal Engine Cycle

The Wicks cycle can use air or any ideal gas for the working fluid and is comprised of three processes, whereas the Carnot cycle, that can also use air as the working fluid, requires four processes.

The efficiency of a Carnot Cycle depends upon the absolute temperatures ($T_{hot}$ and $T_{cold}$) corresponding to the assumed constant temperatures of the heat source and the ambient temperature.

$\eta _{\ Carnot}={\frac {(T_{hot}-T_{cold})}{T_{hot}}}$

The efficiency of the ideal fuel burning engine or Wicks cycle is derived in the next section in terms of the maximum absolute temperature ($T_{hot,max}$) and ambient temperature ($T_{cold}$).

$\eta _{\ Wicks}=1-T_{cold}\cdot {\frac {\ln {(T_{hot,max}/T_{cold})}}{(T_{hot,max}-T_{cold})}}$

This is also the maximum attainable efficiency of a power producing [[bottoming cycle that receives heat from the exhaust of a gas turbine topping cycle. It is also the maximum efficiency of power plant that uses a stream of geothermal hot water the heat source.

Derivation of Ideal Efficiency

The purpose of a Heat engine is to produce Mechanical work. It requires receiving heat from a high temperature source and rejection of low temperature heat to a cold reservoir. Heat is defined as random energy that crosses a boundary because of a temperature difference. In ideal machines the heat transfer temperature difference is virtually zero. Work is defined as a mechanical force acting through a distance or the ability to lift a weight by a defined height. The ideal engine also requires ideal or reversible adiabatic turbines and pumps or compressors. This requires no internal frictional losses. Thus, the working fluid will have no entropy change as it flows from the inlet to discharge.

The following analysis uses the British thermal unit (Btu) for the unit of energy. It is the amount of heat required to raise the temperature of one pound of water one degree Fahrenheit (F) or Rankine (R). The work equivalent of one Btu of heat is 778 ft-lbs. Ideal engine analysis requires entropy that is based on Absolute Temperature. Zero on the Rankine scale is -460 degrees Fahrenheit.

Figure 1: Energy Flow Rate for Ideal Engine Operating Between Finite Heat Capacity Hot Stream In and Infinite Heat Capacity Heat Sink

The temperatures and energy flow is shown in Figure 1. The rectangle on the top represents heat transfer from the cooling hot stream. This becomes the heat input (Q_hot) for the engine. The product of flow rate (lb/hr) and hot stream heat capacity (Btu/lb.F) is the heat capacity rate (Btu/hr.F). The rate of heat extracted (Btu/hr) from the hot stream is the heat capacity rate times the difference between the initial temperature (T_hot,max) and final ambient temperature (T_cold). A heat engine is comprised of a sequence of repeated processes. The 1st Law of Thermodynamics is conservation of energy. It requires energy in to be equal to the energy out. Energy in is denoted as Q_{hot, and} Energy out : W_net and Q_cold.

Figure 2 presents the same ideal heat engine on an absolute Temperature-entropy diagram (T-S diagram). A maximum high temperature of 1040 F or 1500 R and constant cold temperature of 40 F or 500 R is assumed. The hot stream flow rate (${\dot {m}}$) is one lb per hour. It is assumed to be air with heat capacity C_p of .24 (Btu/lb.R)

Figure 2: Absolute Temperature vs Entropy

The total area under the upper curve is ${\dot {Q}}_{hot}$. The bottom rectangle is ${\dot {Q}}_{cold}$. The upper area defined by an exponential curve and vertical and horizontal lines. It is the Exergy or Availability. It is the rate of work that will be produced by the ideal engine. This ideal engine must be a perfect fit to Exergy. This indicates a three process cycle, as compared with a four processes in a Carnot cycle and most other familiar engines.

Equations

• The definition of efficiency ($\eta$) for a heat engine is equation (1)

$\eta ={\frac {{\dot {W}}_{net}}{{\dot {Q}}_{hot}}}$ (1)

• Conservation of energy requires equation (2) and that is rearranged as (2’).

${\dot {Q}}_{hot}={\dot {W}}_{net}+{\dot {Q}}_{cold}$ (2)

${\dot {W}}_{net}={\dot {Q}}_{hot}-{\dot {Q}}_{cold}$ (2')

• Substituting (2’) into (1) yields (3) for any ideal or non-ideal engine.

$\eta =1-{\frac {Q_{cold}}{Q_{hot}}}$ (3)

The 2^nd law of thermodynamics defines an ideal process or cycle. It requires reversibility. This means a zero change of total Entropy.

The entropy changes are related to the heat source and sink. The heat source releases heat to the cycle and thus loses entropy at a rate of $\Delta {\dot {S}}_{hot}$ (Btu/R.hr). The cold reservoir receives heat and increases in entropy at rate $\Delta {\dot {S}}_{cold}$ (Btu/R.hr). For an ideal engine total.

• In accordance with the 2^nd Law, the total entropy change for an ideal engine must be zero. This means the entropy change of the hot and cold reservoirs must be equal in magnitude and opposite in sign as shown in Figure 2 and equation (4).

$\Delta {\dot {S}}_{hot}=-\Delta {\dot {S}}_{cold}$ (4)

• Since the hot reservoir changes in temperature, finding the rate of entropy change requires defining and solving a differential equation (5) where $\delta {\dot {Q}}$ is an increment of heat.

$d{\dot {S}}={\frac {\delta {\dot {Q}}}{T}}$ (5)

• Equation (6) defines an increment of heat from the hot stream.

$\delta {\dot {Q}}={\dot {m}}\cdot C\cdot dT$ (6)

• Substituting (6) into (5) and integrating between the temperature limits results in equation (7). This the rate of entropy decrease from the heat source in (Btu/hr.R)

$\Delta {\dot {S}}_{hot}={\dot {m}}\cdot C\cdot \ln {\frac {T_{cold}}{T_{hot,max}}}$ (7)

• Equation (4) requires the change of entropy of the cold reservoir to be equal but opposite. Applying Logarithm rule $ln{\frac {A}{B}}=-ln{\frac {B}{A}}$ yields equation (8).

$\Delta {\dot {S}}_{cold}={\dot {m}}\cdot C\cdot \ln {\frac {T_{hot,max}}{T_{cold}}}$ (8)

• Multiplying rate of cold reservoir entropy change by the constant T_cold (R) results in the rate of heat rejection to the cold reservoir.

${\dot {Q}}_{cold}=T_{cold}\cdot \Delta {\dot {S}}_{cold}$ (9)

• Equation (10) is the rate of heat released by the hot stream.

${\dot {Q}}_{hot}={\dot {m}}\cdot C\cdot [T_{hot,max}-T_{cold}]$ (10)

• Substituting (8) into (9) and (9) and (10) into (3) yields engine efficiency (11).

$\eta _{ideal}=1-T_{cold}\cdot {\frac {\ln {(T_{hot,max}/T_{cold})}}{(T_{hot,max}-T_{cold})}}$ (11)

• $T_{hot,max}$= 1500 R and $T_{cold}$ = 500 R results in a 45.07 % ideal efficiency.

This confirms efficiency predicted by Figure 2. It shows ${\dot {W}}_{net,hot}$ = 108.17 (Btu/hr) and ${\dot {Q}}_{hot}$ = 240 (Btu/hr). Thus, the ratio or efficiency of 45.07 % per equation (1).

• In comparison, a Carnot cycle efficiency between constant hot ($T_{hot}$) and cold ( $T_{cold}$) temperatures is defined by equation (12).

$\eta _{Carnot}={\frac {T_{hot}-T_{cold}}{T_{hot}}}$ (12)
If a Carnot cycle equation is miss used, an ideal of efficiency of 66.67 % results. This provides a case in point. Users of thermodynamics equations, along with all equations, should understand the assumptions that produced the equation.

Specification and Analysis of an Ideal Engine

Figure 3:Ideal Engine Between Finite Heat Capacity Hot Stream and Constant Temperature Heat Sink

The prior analysis defines the ideal availability and efficiency. It does not define how to implement. It could be an infinite number of Carnot cycles with varying hot temperatures all with the same cold temperature. Another possibility is three process engine cycle shown in Figure 3. It is comprised of 3 three stages :

• (1-2) : Isothermal Compressor : Requires heat rejection at the rate of work input
• (2-3) : Ideal Heat exchanger : Process in which all heat from the hot stream is transferred at a zero temperature difference between the streams.
• (3-1) : Reversible Adiabatic Turbine : Converts a change in enthalpy into mechanical power.

Ambient air can be the working fluid. It is the intake to the Compressor at ambient pressure and temperature and exhaust of the Turbine is also reduced to ambient pressure and temperature.

Ambient air at point (1) is assumed to be an ideal gas with values of $C_{p}$= .24 (Btu/lb.F) and k= $C_{p}/C_{v}$ = 1.4. The corresponding ideal gas constant $R=C_{p}-C_{v}=(2/7)\cdot C_{p}$. Pressure is in atmospheres (atm). Enthalpy (Btu/lb) is the product of $C_{p}$ and the absolute temperature ($R$).

Entropy ($s-s_{1}$) as a function of temperature and volume ratios is relative to initial value per equation (13).

$s-s_{1}=C_{v}\cdot \ln {\frac {T}{T_{1}}}+R\cdot \ln {\frac {V}{V_{1}}}$ (Btu/lb.R)

Table I : Property Table for Ideal Engine Using Air as Ideal Gas for Working Fluid

Points	T (F)	T (R)	p (atm)	${\frac {V}{V_{1}}}$	h (Btu/lb)	($s-s_{1}$) (Btu/lb.R)
1	40	500	1	1.0	120	0
2	40	500	26.763	0.037365	120	-.26366
3	1040	1500	26.763	0.112095	360	0

Table II : 1^st Law Energy Transfer Rates (Btu/hr) Each Process and Full Cycle for ${\dot {m}}=1$ (lb/hr)

${\dot {Q}}={\dot {m}}*(h_{out}-h_{in})+{\dot {W}}$
Process	${\dot {Q}}$	${\dot {m}}*(h_{out}-h_{in})$	${\dot {W}}$
1 -2 (Isothermal Compressor)	-131.83	0	-131.83
2-3 (Heat from hot stream)	240	240	0
3-1 (Reversible adiabatic turbine)	0	-240	240
1-2-3-1 (Full Cycle or Net)	108.17	0	108.17

Calculating efficiency from Table II

The calculation from table II confirms the efficiency found by $T_{max}=1500$ R and $T_{cold}=500$ R in equation (11)

$\eta _{ideal,engine}={\frac {{\dot {W}}_{net,cycle}}{{\dot {Q}}_{hot}}}={\frac {108.17}{240}}=45.07\%$

Applications in Engineering Education and Practice

Practical fuel burning engines were developed in the early 1700s starting with the Thomas Newcomen steam engine and dramatic improvements by James Watt. These engines powered the Industrial Revolution but the scientific theory to describe performance was not yet understood. Coal consumption was a measure of efficiency.

Engine efficiency would later be defined as the ratio of net work output and the high temperature heat input.

Sadi Carnot in 1824 suggested that the maximum efficiency depended upon temperatures. Subsequent understanding of absolute temperature, reversibility and entropy provided the tools to derive the Carnot cycle efficiency as a function of the temperatures of the heat source and sink, and a four process cycle that achieves this efficiency. A century of students learned the Carnot efficiency equation and cycle, while not being challenged to derive it, and not realizing the assumed constant temperature of the source was not true for fuel burning engines.

Determining the maximum efficiency of fuel burning engines that receive heat over a temperature range required calculus as shown in the prior section. The derivation of this efficiency and identification and analysis was first presented in the 1991 paper by Wicks.

Text books now describe the ideal fuel burning engine as the Wicks cycle. It is also called the Wicks cycle in sources of solved problems, online lectures and engineering practice literature and journals

Wicks cycle considerations in engineering analysis and practice

1. Practical problems with the Wicks cycle and also with the Carnot cycle is the required pressure that depends upon the temperature ratio is unrealistically high. However, the pressure ratio for the bottom cycles in combined cycle engines is compatible with existing equipment. Thus, most of the engineering practice interest of the Wicks cycle is for bottoming cycles or power production from moderated temperature streams.
   
   
2. The Wicks cycle requires isothermal compression. This can be approached with a conventional adiabatic compressor with many Intercooler. A future possibility for isothermal compression is novel technology that entrains the air in water for compression and then separation.
   
   
3. An important recognition is that the maximum attainable performance should be known, but not pursued, because of the economic law of diminishing returns. Heat engines require Heat exchangers and Turbines. An ideal heat exchanger would recover all available heat with no lowering of temperature but require infinite surface area and cost. An ideal turbine would produce ideal work, but t an infinite cost. Thus, the practice goal should be to determine the economic optimum while understanding the ideal.
   
   
4. While the ideal Carnot and Wicks cycles are unachievable for practical reasons, the derivation and presentation is justified as knowledge for the pure sake of knowledge without regard for direct application.
   
   
5. The Carnot cycle efficiency remains valid for Refrigeration and Heat pumps that use cyclic processes to move heat from a constant lower temperature source to a constant high temperature sink.
   
   
6. Heating systems such as a furnace or boiler are thermodynamic processes. Higher efficiency can be achieved with cycles. The ideal heating system would be an ideal fuel burning engine or Wicks cycle driving and ideal heat pump or Carnot cycle. While this is not realistic, it provides the realization that heating systems can be made much more efficient by combing available engines and heat pumps. It provides a basis for comparing the actual to ideal performance that has been described as thermodynamics 2^nd Law Analysis.
   

References

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