Technical Thermodynamics Chapter 3p3: Energy Conservation, 1st Law

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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


Egon P Hassel http://web.me.com/egon.hassel University Rostock, Germany, page 1
http://web.me.com/egon.hasselhttp://web.me.com/egon.hasselhttp://web.me.com/egon.hassel


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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( )




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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 -




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In France, 2009

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Technical Thermodynamics Chapter 3p3: Energy Conservation, 1st Law

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