The Stirling Cycle*

Figure 1 illustrates the main processes of the Stirling Cycle and how it appears in a PV diagram. Figure 2 shows
schematically the Stirling engine. The device consists of two opposed pistons, the hot piston and cold piston,
which move back and forth within the hot and cold cylinders, respectively. The hot cylinder is thermally
interfaced through a heat exchanger with a thermal reservoir at TH that provides the high temperature heat
source that energizes the cycle. The cold cylinder is thermally interfaced with a thermal reservoir at TC that
provides the low temperature sink for heat rejection.

Figure 1: Stirling Cycle main processes.

Figure 2: Stirling Engine main components.

The Stirling engine is an externally heated engine (like the Rankine cycle) and therefore it is flexible
with respect to the source of heat. Stirling engines can be energized by an external combustion process (e.g., the
combustion of biofuels) or from other sources of heat (e.g., solar or nuclear energy). The hot and cold cylinders
are separated by a regenerator. Physically, the regenerator is a porous matrix of solid material that has a large
capacity to store thermal energy. The fluid flows through the flow channels in the regenerator, transferring heat to
and from the solid. Regenerators are generally simple devices. A container that is filled with lead shot or metal
screens can be used as a regenerator in a Stirling cycle. Thermodynamically, the regenerator acts as a thermal
storage medium. Heat is transferred from the gas to the solid during one part of the cycle and then from the solid
back to the gas during a later part of the cycle. During normal operation there is a temperature gradient in the
regenerator matrix so that the solid material adjacent to the hot cylinder is at (or near) the hot reservoir
temperature and the solid material adjacent to the cold cylinder is near the cold reservoir temperature. In a well-
designed regenerator, the heat capacity of the solid material (the matrix) will be much larger than the heat
capacity of the fluid. Therefore, the temperature of the matrix will change very little as it absorbs and releases
heat during each cycle. The recuperative heat exchanger operates continuously by transferring energy from gas
flowing in one portion of the steady flow cycle to gas flowing in a different portion of the cycle. The regenerator
shown in Figure 1 operates in a transient fashion by transferring energy from the gas at one time in the cycle to
the same gas but at a different time in the cycle. Figure 3 illustrates the processes that together make up a
complete cycle of an ideal Stirling engine.
 Figure 3: Thermodynamic ststes and processes of the Dtirling Cycle

State 1 is defined as the state of the working fluid immediately before the compression process, shown in Figure
3 (a). The hot cylinder volume is zero and the cold cylinder volume is at its maximum value. In the limit that the
regenerator has no void volume, the working fluid entirely resides in the cold cylinder at state 1. The fluid is
assumed to be isothermal and at the cold reservoir temperature. During the compression process, the cold
cylinder volume is decreased while the fluid is maintained at a constant temperature, TC, due to heat transfer to
the cold reservoir. The pressure of the fluid increases during this process.

State 2 is defined as the state of the working fluid immediately after the compression process. During the cold-
to-hot blow process, the cold cylinder volume is decreased while the hot cylinder volume is increased by an
equal amount. The cylinders move together, as shown in Figure 8-43(b). The total volume available for the fluid
in the system is ideally constant during this process. At the conclusion of the cold-to-hot blow process, the cold
cylinder volume is zero and therefore all of the fluid has been pushed through the regenerator and into the hot
cylinder. The fluid is warmed from TC to TH as it flows through the regenerator by heat transfer from the solid
matrix; ideally, the fluid enters the hot cylinder at TH. The pressure of the fluid rises during this process.

State 3 is defined as the state of the fluid immediately after the cold-to-hot blow process. All of the fluid is in the
hot cylinder at TH. During the expansion process shown in Figure 8-43(c), the hot cylinder moves to its
maximum volume and the gas is maintained at constant temperature, TH, due to heat transfer from the hot
reservoir. The pressure of the fluid decreases during this process.

State 4 is defined as the state of the working fluid immediately after the expansion process. During the hot-to-
cold blow process, the hot cylinder volume is decreased while the cold cylinder volume is increased by an equal
amount as shown in Figure 8-43(d). At the conclusion of the hot-to-cold blow process, all of the fluid has been
pushed through the regenerator and into the cold cylinder. The fluid is cooled from TH to TC as it flows through
the regenerator by heat transfer to the solid matrix; the regenerator stores this energy to be released during the
subsequent cold-to-hot blow cycle. At the conclusion of the hot-to-cold blow cycle the system has been returned
to state 1 and it is ready to begin another cycle.
Notice that none of the processes described in the ideal Stirling cycle are associated with any entropy generation.
There is never heat transfer through a temperature gradient or flow through a pressure gradient for the ideal
Stirling cycle. Indeed, it is this characteristic of the ideal Stirling cycle that has drawn so much attention; in
theory, it can obtain the maximum possible efficiency for converting heat into work. However, the ideal Stirling
cycle is also not practical. The compression and expansion processes combine heat and work during a single
process. These processes would need to occur very slowly in order to approach a reversible limit. Notice that
none of the components or processes in the other power cycles (i.e., those that are actually used, the Rankine
cycle, gas turbine engine, and 4-stroke engine) have this characteristic. For example, in the Rankine cycle, work
is done with some components (the pump and turbine) and heat transfer with different components (the
condenser and boiler). There is a good reason for this separation: work and heat occur with very different time
scales. Heat takes a substantial amount of time to transfer and requires components with large amounts of
surface area. The ideal Stirling cycle is never realized in practice. Instead, the hot and cold cylinders are nearly
adiabatic and equipped with hot and cold heat exchangers that interface the working fluid thermally with the
appropriate reservoirs. During the compression process, the cold piston moves in and compresses the working
fluid nearly adiabatically, causing its temperature to rise above TC. When the gas is subsequently pushed through
the cold heat exchanger during the cold-to-hot blow process, it is cooled to TC before it enters the regenerator. A
similar process occurs in the hot cylinder and hot heat exchanger during the expansion and hot-to-cold blow
process. Clearly, the temperature elevation and subsequent heat transfer results in irreversible processes. When
these and other practical considerations are taken into account, the efficiency advantage associated with the
Stirling cycle becomes less clear.

*THERMODYNAMICS -S.A. Klein and G.F. Nellis; Cambridge University Press, 2011

EES – Code
"Stirling Cycle"

$UnitSystem SI Radian Mass J K Pa

$TabStops 0.2 3.5 in

T_H=converttemp(C,K,650 [C]) "temperature of hot reservoir"

T_C=converttemp(C,K,20 [C]) "temperature of cold reservoir"
{F$='He'} "working fluid"
F$='CO2' "working fluid"
CR=5 [-] "compression ratio"
P_charge=1 [MPa]*convert(MPa,Pa) "charge pressure"
Vol_max=990 [cm^3]*convert(cm^3,m^3) "maximum cylinder volume"
Vol_min=Vol_max/CR "minimum volume"
“………………………………..”
"State 1"
P[1]=P_charge "pressure"
T[1]=T_C "temperature"
Vol[1]=Vol_max "volume"
v[1]=volume(F$,T=T[1],P=P[1]) "specific volume"
u[1]=intenergy(F$,T=T[1]) "internal energy"
s[1]=entropy(F$,T=T[1],P=P[1]) "entropy"
m=Vol[1]/v[1] "mass"
“………………………………..”
"State 2"
T[2]=T_C "temperature"
Vol[2]=Vol_min "volume"
v[2]=Vol[2]/m "specific volume"
s[2]=entropy(F$,T=T[2],v=v[2]) "entropy"
u[2]=intenergy(F$,T=T[2]) "internal energy"
P[2]=pressure(F$,T=T[2],v=v[2]) "pressure"
“………………………………..”
0=Q_C_comp/T_C+m*(s[2]-s[1]) "entropy balance"
W_C_comp=Q_C_comp+m*(u[2]-u[1]) "energy balance"
“………………………………..”
"State 3"
T[3]=T_H "temperature"
Vol[3]=Vol_min "volume"
v[3]=Vol[3]/m "specific volume"
P[3]=pressure(F$,T=T[3],v=v[3]) "pressure"
s[3]=entropy(F$,T=T[3],v=v[3]) "entropy"
u[3]=intenergy(F$,T=T[3]) "internal energy"
“………………………………..”
"State 4"
T[4]=T_H "temperature"
Vol[4]=Vol_max "volume"
v[4]=Vol[4]/m "specific volume"
s[4]=entropy(F$,T=T[4],v=v[4]) "entropy"
u[4]=intenergy(F$,T=T[4]) "internal energy"
P[4]=pressure(F$,T=T[4],v=v[4]) "pressure"
“………………………………..”
Q_H_exp/T_H=m*(s[4]-s[3]) "entropy balance"
Q_H_exp=W_H_exp+m*(u[4]-u[3]) "energy balance"

"cold to hot blow process"

h_C=enthalpy(F$,T=T_C) "enthalpy of fluid entering/leaving cold cylinder"
h_H=enthalpy(F$,T=T_H) "enthalpy of fluid entering/leaving hot cylinder"
Q_r_CTHB=m*(h_H-h_C) "heat transfer to fluid from regenerator during cold-to-hot
blow process"

{nd=0.5 [-]} "nondimensional piston positions"

Vol_C_CTHB=Vol_min*(1-nd) "volume of cold cylinder"
Vol_H_CTHB=Vol_min*nd "volume of hot cylinder"

{P_CTHB=P[2]} "guess for the pressure"

P_CTHB_MPa=P_CTHB*convert(Pa,MPa) "in MPa"
“………………………………..”
v_C_CTHB=volume(F$,T=T_C,P=P_CTHB) "specific volume in cold cylinder"
v_H_CTHB=volume(F$,T=T_H,P=P_CTHB) "specific volume in hot cylinder"
m_C_CTHB=Vol_C_CTHB/v_C_CTHB "mass in cold cylinder"
m_H_CTHB=Vol_H_CTHB/v_H_CTHB "mass in hot cylinder"
m_C_CTHB+m_H_CTHB=m "total mass balance"
“………………………………..”
W_C_CTHB=integral(P_CTHB*Vol_min,nd,0,1) "work done by cold piston during cold to hot blow"
W_H_CTHB=W_C_CTHB "work done to hot piston during cold to hot blow"
W_C_CTHB=m*h_C+Q_C_CTHB-m*u[2] "heat transfer to cold reservoir during cold to hot blow"
Q_H_CTHB+m*h_H=W_H_CTHB+m*u[3] "heat transfer from hot reservoir during cold to hot blow"

"hot to cold blow process"

Q_r_HTCB=m*(h_H-h_C) "heat transfer from fluid to regenerator during hot-to-cold

“blow process"

Vol_C_HTCB=Vol_max*nd "volume of cold cylinder"

Vol_H_HTCB=Vol_max*(1-nd) "volume of hot cylinder"

v_C_HTCB=volume(F$,T=T_C,P=P_HTCB) "specific volume in cold cylinder"

v_H_HTCB=volume(F$,T=T_H,P=P_HTCB) "specific volume in hot cylinder"
m_C_HTCB=Vol_C_HTCB/v_C_HTCB "mass in cold cylinder"
m_H_HTCB=Vol_H_HTCB/v_H_HTCB "mass in hot cylinder"
m_C_HTCB+m_H_HTCB=m "total mass balance"
“………………………………..”
W_C_HTCB=-integral(P_HTCB*Vol_max,nd,0,1) "work done by cold piston during hot to cold blow"
W_H_HTCB=W_C_HTCB "work done to hot piston during cold to hot blow"
“………………………………..”
W_C_HTCB+m*h_C=Q_C_HTCB+m*u[1] "heat transfer to cold reservoir during cold to hot blow"
Q_H_HTCB=W_H_HTCB+m*h_H-m*u[4] "heat transfer from hot reservoir during cold to hot blow"
“………………………………..”
W_net=W_H_exp-W_C_comp "net work per cycle"
Q_H_net=Q_H_exp+Q_H_CTHB+Q_H_HTCB "net heat transfer from hot reservoir per cycle"
eta=W_net/Q_H_net "efficiency"
eta_max=1-T_C/T_H

“END CODE………………………………..”