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Technical Thermodynamics
Chapter 3p3: Energy Conservation, 1st LawEgon P. Hassel, University Rostock, Germany, Inst Technical Thermodynamics
February 14, 2011
Rostock Harbor, 2010
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Chapter 3: Energy Conservation, 1st Law
Section 3.4: First law for stationary flow processes (SFP)
Figure 3.4.1: first law of thermodynamics for stationary flow processes (SFP)
In figure 3.4.1 we see a stationary flow process (SFP). We see one inflow, m1 , one
outflow, m2, a heat flow into the system, Q
12, and a workflow into the system, W
t,12. In a
SFP all flows are constant with time and additionally the system itself does not change
with time. That is, we do not treat system on or off switch. Regard e.g. a pump. In the
morning the pump has room temperature and is switched on, the water starts flowing
and the pump heats up. After maybe 15 min, the water flow is constant and the pump
has reached its working temperature. Then we speak of a stationary flow process. Afterperhaps eight hours the pump is switched off, and the flow process and pump
temperature are no longer constant, then we do not have a SFP. So again, in an SFP
all flows and the system are constant with time. Typically in text books we only see one
inflow, one outflow, one heat flow and one work flow, which serves only as a simple
student example, technically and typically we have many in- and outflows, think e.g. of a
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gas turbine, where we have an air inflow, a fuel inflow, the exhaust gas outflow and heat
flows coming from several parts. Such SFP are technically very important, see e.g.
figure 3.4.2.
Figure 3.4.2: stationary flow process (SFP), sketch of a water mill
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Figure 3.4.3: stationary flow process (SFP), photo of a water mill wheel
Figure 3.4.4: stationary flow process (SFP), photo of a wind mill
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For modern people it is hard to understand how important the invention and the use
of those energy conversion machines was. We have electricity which can do anything,
anytime and anywhere, uh, uh, about. The historic people mostly got their mechanical
power from animals and humans, until mills came in much use, like for water pumping,
lifting of building material, threshing straw and grinding grain. Such a stationary flow
process (= SFP) is technically very important. We define the technical work as the work
coming out or going into a system with a SFP without the work which is necessary to
get the fluid into the system (input work for the mass flows) or out of the system (output
work) and without the volume work. As SFP we define a flow process in which there are
input mass flows into the system and output mass flows out of a system and maybe
input or output heat flows and maybe input or output work flows, and all quantities, that
is all flows and all state variables of the flows and the system must be independent of
time, that is stationary, see also figure 3.4.5.
Figure 3.4.5: verbal definition of stationary flow processes (SFP)
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Figure 3.4.6: SFP, all quantities are independent on time.
Figure 3.4.7: We should distinguish between inlet (1) and outlet (2) and before (I)
and after (II), which is often mixed in literature.
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In figure 3.4.6 we see again that all quantities are independent on time, including the
state and state variables of the system. In figure 3.4.7 it is shown that we should
distinguish between inlet (1) and outlet (2) and before (I) and after (II), which is often
mixed in literature.
Figure 3.4.8: In a SFP technical work crosses the system boundary as electrical
work or shaft work and specific technical work can be related to the mass flows.
In an SFP technical work crosses the system boundary as electrical work or shaft
work and specific technical work is often be related to the mass flows, see figure 3.4.8.,
the result is the specific technical work w t12. We should be careful if we have several
mass flows, then we should note on which mass flow which work flow or heat flow is
related.
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Figure 3.4.9: For a SFP also the volume is constant.
Figure 3.4.10: Stationary flow process (SFP) separated into two parts, a work
delivering turbine and a heat exchanger.
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For a stationary flow system also the volume is kept constant, see figure 3.4.9. A
scheme of a SFP is shown in figure 3.4.10. Here the system consists of a turbine which
delivers technical work flow and a heat exchanger with the incoming or outflowing heat
flux. We have one mass inflow with the velocity c1 and the height z1, this mass flow
naturally flows out of the system with the velocity c2 and the height z2. For this system
we want to write the first law that is the energy conservation, see figure 3.4.11, as apple
balance and energy balance.
Figure 3.4.11: Reminder of apple and energy balance.
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Figure 3.4.12: derivation of the First law for stationary flow processes (SFP).
So in figure 3.4.12 the first law is written for SFP, with a reminder of the first law of
the closed system. For these observations within the next couple of figures we have
temporal start of observation as (1) and end of observation as (2). For this open system,
because all system state parameters are constant, there is no change in internal energy
and neither a change in the external energies:
On the rhs we have then the inflow on heat Q12, of work W12, the inflow of energies
connected with the mass inflow m during the time step as the specific internal energy
u1, the specific kinetic energy c2
/2 and the specific potential energy rho*g*z1, and thesame for the outflowing mass during the time step:
U2!U
1+E
a2!E
a1= 0
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Equation 3.4.1: 1st law for SPF between time step (1) and (2)
This is complete and correct. The work W12 and heat Q12 contain all work and heat
to and from the system. Lets look at the heat Q12 first. There is heat flux along the inflow
and outflow pipes where the mass flow happens and heat flux across the other system
boundary. Mostly the heat flux in the inflow and outflow pipes is neglected, but could be
considered if necessary. The cause for this is that the temperature gradient in the inflow
and outflow is thought to be small thus inducing a small heat flux, see also figure 3.4.13,
3.4.14 and 3.4.15.
Figure 3.4.13: derivation of the 1st law stationary flow processes (SFP)
0 = Q12
+W12
+ !m u1+
c1
2
2+ g z
1
#$
&'(
!m u2 +c2
2
2+ g z2
"
#$%
&'
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Figure 3.4.14: SFP, the heat flux in the pipes usually is neglected.
Figure 3.4.15: SFP, the heat flux in the pipes usually is neglected.
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Figure 3.4.16: SFP, about the total work.
In the same sense W12 contains all work going into or out of the system, see figure
3.4.16. This consists of the work across the system boundary without the inflow and
outflow pipes and especially the work across the inflow and outflow pipes. Which work
goes across the inflow and outflow pipes? It is the input and output work, see figure
3.4.17. Typically we must do work to press the mass into the system, then the system
gains energy, and we get work when the mass flows out of the system, then the system
loses energy.
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Figure 3.4.17: SFP, about the total work and the input and output work.
Figure 3.4.18: Input and output work with an SFP I.
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In figure 3.4.18 and 3.4.19 we show a sketch of an SFP with the input and output
pipes enlarged. At the input we see the input pressure of the system as p1 and the input
pipe area as A, for a certain time step the amount of mass input is m, which fills a
volume V.
Figure 3.4.19: Input and output work with an SFP II.
This leads to the equation for the input or output work for an SFP, See also figure
3.4.20:
Equation 3.4.2: Input (or output) work for a system with an SFP.
Winput = F dsTI
TII
! = pAdxTI
TII
! = p V( )dVTI
TII
! =
= pvdmTI
TII
! = prhodm
TI
TII
!
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or in differential form:
Equation 3.4.3: Input or output work for a system with an SFP.
Figure 3.4.20: Input/output work SFP
See figure 3.4.20. We want to have the correct sign for the input and output work. So
we define input work as
and output work as
!Winput = Fds = pAdx = p V( )dV= pvdm
0)( 11 >!"#"#=!"# mvpVp
022
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and thus the total work done at the system or by the system is
Here we see that we distinguish between the input and output works at the inflow
and outflow pipes and the remaining work over the system boundary, which is called:
technical work Wt or here Wt12.
Figure 3.4.21: derivation of the 1st law SFP
When we now put all these equations together, see figure 3.4.21 and 3.4.22, we
come up with
)( 112212
22111212
vpvpmW
mvpmvpWW
t
t
!"!
="!"+=
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Keeping in mind that the enthalpy h is defined as h = u + pv and building the time
derivative we can rearrange the equations still to get the first principle for SFP in the
commonly used form as, see figure 3.4.22:
Equation 3.4.4: 1st law for SFP in commonly used form.
Figure 3.4.22: 1st law SFP in commonly used form
0 = !Q12 +!Wt12 + !m(u1 + p1v1 +
c1
2
2+ gz1)!
!m(u2+ p
2v2+c2
2
2
+ gz2)
0 = Q12 +Wt12 + m(h1 +c1
2
2+ gz1) ! m(h2 +
c22
2+ gz2)
" 0 = Q12 +Wt12 + m(h1 + ea1) ! m(h2 + ea2)
" 0 = Q12 +Wt12 + m(ht1 ! ht2)
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As conclusion we see the equations again in figure 3.4.23.
Figure 3.4.23: 1st law SFP in commonly used form
In between we defined for saving writing the total enthalpy as abbreviation of the
enthalpy plus the kinetic energy plus the potential energy as:
Equation 3.4.5: Definition of total enthalpy.
The 1st law can also be written in specific form as, see figure 3.4.24:
Htotal ! Ht ! H+mc
2
2+mgz or
htotal !Ht
m! ht ! h+
c2
2+ gz
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0 = !Q12+
!Wt12 + !m(h1 +c
1
2
2+ gz
1)! !m(h
2+
c2
2
2+ gz
2) /: !m
" 0 = !q12 + !wt12 + (h1 +c
1
2
2
+ gz1)! (h2 +
c2
2
2
+ gz2 )
Equation 3.4.6: 1st law SFP in specific form.
Figure 3.4.24: 1st law SFP in specific form.
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Figure 3.4.25: 1st law SFP, remarks
Note that q12 and w12 are related to the mass flow m-point, see figure 3.4.25. If there
are more than one inflow and one outflow one should be careful on what quantity the
specific values are related to. Typically these are related to the corresponding mass
flows but not to the system mass. In this law all quantities are taken at the system
boundary. The state changes within the system can be arbitrarily, even irreversible and
non quasi static.
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Figure 3.4.26: SFP with multiple inflows and outflows.
For multiple inflows and out flows, see figure 3.4.26, the 1st law for SFP looks like
Equation 3.4.7: SFP with multiple inflows and outflows.
0 = !Ql
l
! + !Wtkk
! + !mjinletj
! hj +cj2
2+gzj
"
#$$
%
&''( !m i
outleti
! h i +ci2
2+gzi
"
#$
%
&'
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Figure 3.4.27: 1st law SFP even shorter
To write this even shorter, see figure 3.4.27, with inflowing mass flows with plus sign
and outflowing mass flows with minus sign, we get:
0 = !Ql
l
! + !Wtkk
! + !mjinletj
! hj +cj2
2+gzj
"
#$$
%
&''( !m i
outleti
! hi +ci2
2+gzi
"
#$
%
&'
0 = !Qii
! + !Wtjj
! + !mkk
! htk
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Chapter 3: Energy Conservation, 1st Law
Section 3.5: Technical work
Figure 3.5.1: Technical work I.
The process which a mass element undergoes from inlet (1) to outlet (2) in an SFP
can often be approximated by a process like a polytropic process, e.g. an isothermal
process or else, figure 3.5.1 and 3.5.2. If we regard a certain fixed mass element then
this mass element is a closed system and we can work with the 1st law for closed
systems and the corresponding process formulas.
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Figure 3.5.2: SFP and technical work, sketch of a fixed mass flowing through a
system.
In figure 3.5.2 we see a fixed mass element flowing through a system in an SFP. The
following derivation can be made only if we are able to determine meaningful average
values for the state quantities for this mass element as it travels through the system.
This can be done often. If this can be done then further we can approximate the real
process with a polytropic process. However, we can apply the first principle for this
mass element in two ways. First we can treat the mass element as a closed system and
apply the 1st law of closed systems to this system with respect to the moving mass
element coordinate system. Then we do not have kinetic and potential energy changes.
Secondly we can write up the 1st law for SFP in a spatially fixed coordinate system. By
this we find out an alternative formulation for the technical work. Lets do this, see figure
3.5.3.
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Figure 3.5.3: SFP and technical work, fixed mass flowing through a system.
We write the first law for closed system to the moving mass element in the
coordinate system of this mass element, we do not have kinetic and potential energy
here:
the work done on this system is
if we put this together we get
du = !q +!w
!w = !pdv +!wdiss
du+ pdv = !wdiss +!q
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Figure 3.5.4: derivation of technical work SPF
then we use the definition of the enthalpy h, h = u + pv, see figure 3.5.4
and rearranged and put together
dh = d(u + pv) = du + d(pv) = du+ pdv + vdp
dh ! vdp = du + pdv = !wdiss +!q
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Figure 3.5.5: technical work in open systems, here in SFP.
If we now write the 1st law in a stationary fixed outside system for SFP we get, see
figure 3.5.5
and put in the dh from above we get
and this equation rearranged gives a formula for the determination of the technical
work as
Equation 3.5.1: equation for determination of technical work in open systems (SFP)
!q +!wt = dh + dea
q + wt = vdp + wdiss + q + dea
wt = vdp + wdiss + dea
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Figure 3.5.6: technical work in SFP and open systems.
In integral form we get for the technical work in open systems
with
which holds always.
wt12=
vdp+
1
2 (c22
c12
)+
g(z2z1)
+
wdiss,121
2
wdiss
0, wdiss,12
0
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Figure 3.5.7: Technical work in specific and absolute form.
Figure 3.5.8: technical work, open systems, like SFP.
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In figure 3.5.7 we see again the technical work in specific and absolute form. The
fluid can do work or work can be done at the fluid, see figure 3.5.8, if there is a pressure
change, or a change in kinetic energy, or a change in potential energy, or dissipation
work is done at the fluid.
Figure 3.5.9: qualitative example of technical work.
In figure 3.5.9 we see the signs in the equation of the technical work if a fluid is
doing work within a machine in a SFP. The delivered work is negative, the dissipation
work is always positive, no exception, and thus the sum of the work caused by pressure
change or velocity change or height change must be negative too.
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Subsection 3.5.1: Example of technical work at a water dam for production ofelectricity
Figure 3.5.1.1: SFP, water power plant, numerical example for technical work and
dissipation etc.
As an example for the work done in an electrical power plant with a water dam we
study figure 3.5.1.1. On the lhs of the figure we see the water sea. The water in entering
the tube, falls down, spins the turbine blades within the water turbine and leaves the
pipe on the right hand side of the figure. We state that the height difference is 120 m, z1
- z2 = 120 m. The height difference between the water level on the inlet and the axis of
the inlet pipe should be the same as on the outlet side the height difference between the
water level and the pipe axis. We assume further the the whole system is adiabatic. The
environmental atmospheric pressure should be taken as constant, p1 = p2. If we draw
the inlet and outlet boundaries far from the pipe inlet and outlet, then we can neglect the
inlet and outlet velocities at the respective boundaries, c1 = 0 m/s, c2 = 0 m/s. Our last
assumption is that the water can be regarded as incompressible, meaning the density is
constant.
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Figure 3.5.1.2: water dam example I, 1st law SFP.
As to be seen in figure 3.5.1.2 the first law for SFP in this case is:
with the assumptions we made follows
and the enthalpy put in, h = u + pv:
and with p1 = p2 and v1 = v2
wt12 = (h2 + gz2 )! (h1 + gz1)
( )1211122212 zzgvpuvpuwt!+!!+=
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Figure 3.5.1.3: water dam example I, 1st law SFP
The work of the system wt12 < 0 comes from the reduction in potential energy of the
water, z1 > z2, see figure 3.5.1.3. The change of internal energy u2 - u1 comes from
dissipation. To see this, imagine the case with wt12 = 0 J/kg. In this case the whole
potential energy of the water of an equivalent of 120 m fall height is converted into
internal energy when the water reaches the lower height with a certain velocity and
mixes itself into the flow. This is a very nice example for dissipation. Potential energy is
transformed into internal energy.
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Figure 3.5.1.4: water dam example II, 1st law SFP
If we now define reversible technical work as, see figure 3.5.1.4:
we come up with a meaningful definition of the efficiency of the water turbine system
as:
that is the quotient of the real work divided by the reversible work, which always is
larger than the real work. Thus this efficiency is larger than zero and smaller than one.
wt12, rev
= vdp +1
2(c
2
2 c1
2)+ g(z
2 z
1)
1
2
1
,12
12!=
revt
t
w
w
"
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Figure 3.5.1.5: water dam example III, 1st law SFP
If we put these equations together and rearrange, see figures 3.5.1.5 and 3.5.1.6,
we get
and with the specific caloric state equation for incompressible fluids, here for water,
with cv,water = 4.19 kJ/(kg*K), we get
! =g z2
"
z1( ) + u2 " u1( )g z
2" z
1( )= 1"
u2"
u1
g z1" z
2( )
Technical Thermodynamics
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Figure 3.5.1.6: water dam example IV, 1st law SFP
Figure 3.5.1.7: water dam example V, 1st law SFP
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With an assumed efficiency of 85 %, see figure 3.5.1.7, we get a temperature
increase of 0.042 K. That means, if we let water fall 120 m * 0.85 that is 102 m, about
100 m, the potential energy converted into internal energy increases the temperature
0.042 K. On the next slide, figure 3.5.1.8, we see that if the efficiency of the turbine
system is zero the total potential energy gets converted to internal energy: u2 - u1 = g*
(z2 - z1).
Figure 3.5.1.8: water dam example VI, 1st law SFP
This example could also lead to some speculation, see figures 3.5.1.9 and 3.5.1.10.
There seem to exist energy types which can be converted freely into any other form of
energy, here in this example, the potential energy can be transformed into electricity,
and before that, into mechanical energy as kinetic energy, and into internal energy. But
the increase of the internal energy is directly and unconsciously considered by us as
waste. So we wonder, exist two kinds of energy, one from which can be freely converted
into any other kind of energy, and one form which is more or less waste energy. Later in
the chapter about exergy aka availability and anergy we will see that that this is correct.
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Figure 3.5.1.9: water dam example VII, 1st law SFP
Figure 3.5.1.10: water dam example VIII, 1st law SFP
- end of chapter 3 part 3 -
Technical Thermodynamics
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In France, 2009
"
Technical Thermodynamics
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