Otto-von-Guericke University Magdeburg

Chair for PROCESS SYSTEMS ENGINEERING

Prof. K. Sundmacher

Winter term 2015/2016

Script 2. Momentum and energy balances

Contents 0.1 Momentum Balance .............................................................................................................1

0.2 Energy Balance ...................................................................................................................3

0.2.1 Balance of Total Energy ....................................................................................................3

0.2.2 Balance of Internal Energy ................................................................................................5

0.2.3 Balance of Enthalpy ..........................................................................................................6

0.2.4 Differential Equation of a Temperature Field .......................................................................7

0.1 Momentum Balance

Contrary to the material balances, for which various choices of quantity measures exist, the momentum

has only one measure, this is the mass multiplied by the velocity of center of gravity. Apart from this it is

possible to convert mass and velocity into other measures, e.g. volume or velocity of volumetric center.

The momentum balance is based on the Law of Newton, which says that the temporal change of

momentum of a body is equivalent to the sum of forces on a body:

f

f,j

jF

dt

mvd (1.1)

with

m Mass kg

vj Velocity of center of mass m·s-1

t Time s

Fj,f Force Vector N

Derivating the momentum balance is analogous to derivating a mass balance. The change of momentum

inside a volume is equivalent to the change of the integral taken over the volume of momentum density:

V

j

V

j

jdVv

tdVv

dt

d

dt

mvd (1. 2)

with

V Volume m3

Density kg·m-3

Momentum is transported convectively with in- and outflow through the surface of a volume. The

transported momentum density is written as vj, the velocity of the momentum entering the volume is

equivalent to the fluid velocity in opposite direction to the surface normal vector.

A

kkj dAnvv (1.3)

with

nk Surface normal vector 1

A Surface area of volume m2

Besides the convective momentum transport, the momentum in a volume can also be changed by forces,

e.g. forces which work at the surface of a volume. Surface forces are not only scalar, hydrostatic pressure

p, but also shear stresses jk in a fluid, which are caused by friction. Both pressures are summarized in a

pressure tensor Pjk=pjk+jk . Hence the non-convective term is

A

kjk dAnP (1.4)

with

Pjk Pressure tensor N·m-2

p Hydrostatic pressure fraction N·m-2

jk Unit tensor 1

jk Friction tensor N·m-2

Additionally to the surface forces there may be forces acting on the masses within the volume. These

volume forces are usually caused by fields like the gravitation field of the earth or a centrifugal force field.

Another source of a volume force may be an electric fields, if the fluid contains ions.

V

,j dVf (1.5)

with

fj, Unit-mass volume force on component N·kg-1

After application of the Gaussian Theorem we get, analogous to the derivation of the material balance,

the local form of the momentum balance:

,jjkkj

k

j fPvvz

vt

(1.6)

0.2 Energy Balance

0.2.1 Balance of Total Energy

Analogous to the material balance, an energy balance can be set up for various energy measures. In

classical physics, where engineering science can be found in most cases, we will find a law of

conservation only for total energy. Hence we derivate the balance of this energy at first.

The derivation of total energy in a respective volume is equal to the integral taken over the volume of the

change of energy density:

VV

dVet

edVdt

d

dt

dE (1.7)

with

E Total energy J

t Time s

Density kg·m-3

e Unit-mass total energy J·kg-1

V Volume m3

While material flows through the surface of a volume, energy is also transported convectively. This can be

expressed via the integral taken over the surface of the mass-flux density in opposite direction to the

surface normal vector, multiplied by the energy density of the mass flow

A

kk dAnev (1.8)

with

vk Velocity of center of gravity of a fluid m·s-1

nk Surface normal vector 1

A Surface area of volume m2

The change of energy within the respective volume may have two causes besides convective transport.

On the one hand this can be work done on the volume, and on the other hand heat transfer in or out of

the system.

Work performed on a volume may itself be composed of two different terms. On the one hand surface

forces, i.e. hydrostatic pressure or friction tension, can cause work on a volume. This is equivalent to the

pressure perpendicular to the surface, multiplied by the flow velocity in the same direction

A

kjjk dAnvP (1.9)

with

Pjk Pressure tensor N·m-2

Added to this is the work induced by volumetric forces. This is the unit -mass force on a component,

multiplied by the mass flow density of the component

V

,j,j dVvf

(1.10)

with

fj, Unit-mass volumetric force on component N·kg-1

v j, Velocity of component m·s-1

Heat put in or drawn out of the system can be subdivided into a heat flux which is not connected to

material transport and a heat flow connected to material transport. The first one is based on processes

like heat conduction or radiation of heat, while the second one is connected to diffusion of components

that have different specific enthalpy. Both heat fluxes are summarized to one expression:

,kk

'

k jhqq (1.11)

with

q´k Total heat-flux density W·m-2

qk Heat-flux density because of material transport W·m-2

h Partial specific enthalpy of component J·kg-1

The heat quantity introduced to the system is equivalent to the integral taken over the surface of the

volume of heat-flux density:

A

k

'

k dAnq (1.12)

As a result we get the total energy balance in local form

,j,j

'

kjjkk

k

vfqvPevz

et

(1.13)

0.2.2 Balance of Internal Energy

For engineering tasks often balancing internal energy is more useful than balancing total energy. The

difference between both energies lies in the kinetic energy. It holds

uve j 2

2

1 (1.14)

with

u Unit-mass internal energy J·kg-1

In multiplying density to each side of the equation and forming the partial derivation by time of it, we get

the following equation

2

2

1jv

te

tu

t

(1.15)

The second term on the right side is replaced with the help of the local momentum balance of equation

1.23 multiplied with vj. The resulting equation is the balance equation of kinetic energy.

j,j

k

jk

jkj

k

j vfz

Pvvv

zv

t

22

22

1 (1.16)

Using this relation plus the global continuity equation 1.15 and rearranging the equation we receive the

local balance of internal energy

k

j

jk,k,k

'

kk

k z

vPjfquv

zu

t

(1.17)

In the case of closed systems this is the balance of the first law of thermodynamics.

Important for understanding both elaborated energy balances is to realise the differences between them.

The work done by surface forces can be found within the total energy balance. It itself can be subdivided

into two parts:

k

j

jk

k

jk

jjjk

k z

vP

z

PvvP

z

(1.18)

The first part is also found in the balance of kinetic energy. It is equivalent to the work introduc ed to the

flow when flowing through a pressure gradient. Since the kinetic energy is not an internal energy, we do

not find this term here.

The second term is in the balance equation of internal energy. It poses the conversion of mechanical into

internal energy. This happens via compression, so pjk of the pressure tensor P jk, or via friction, that is the

term jk inside the pressure tensor P jk.

At this point we may ask ourselves where to find potential energy in the equation. Well, this energy is

incorporated in the work done by volume forces. In the case of gravitation on earth, the specific

volumetric force is the gravity gj, which is the gradient of the gravitation potential

j

jz

g

(1.19)

with

gj Gravity force N·kg-1

Gravitation potential J·kg-1

As a result we can write the term for the volumetric force as follows:

j

jjj,j,jz

vvgvf

(1.20)

In the case of a flow alongside equipotential lines, i.e. at constant height level, this term is irrelevant and

becomes zero. For movements alongside the gradient of a potential field, this term describes the increase

or decrease in potential energy.

0.2.3 Balance of Enthalpy

Derivating the enthalpy balance from the balance of internal energy is made via the definition of enthalpy

puh (1.21)

with

h Unit-mass enthalpy J·kg-1

Multiplying this with mass density and forming it into differential form results to

t

ph

tu

t

(1.22)

Putting this equation into the balance of internal energy and using the following equation

jk

k

j

j

j

z

vp

z

vp

(1.23)

as well as the continuity equation 1.15 results in the local enthalpy balance

k

j

jk,k,k

k

k

'

kk

k z

vjf

z

pvqhv

zt

ph

t

(1.24)

Comparing this balance to the balance of internal energy we can find a modified pressure term. In

enthalpy work via compression is not considered, but only the friction term.

0.2.4 Differential Equation of a Temperature Field

Finally we can derivate an enthalpy balance in temperature form out of the general enthalpy balance. This

form can directly be used to calculate temperature fields. We start with the total differential of the enthalpy

Dt

Dch

Dt

Dp

p

h

Dt

DTccpTh

Dt

Dp,, (1.25)

with

T Temperature K

c Molar concentration of component mol·m-3

This total differential is to be converted into its local form:

α j

αj

αα

j

j

j

pjp

j

jz

cv

t

ch

z

p

p

hv

t

p

p

h

z

Tcv

t

Tc

z

hv

t

h (1.26)

Using these equations as well as the mass balance for components we receive the following differential

equation for the temperature field in local form

k

j

jk

k

,k,k

j

jp

k

k

j

jp

z

v

z

hfj

z

Tvc

z

qh

z

pv

t

p

p

h

t

Tc 1

(1.27)

The energy source term h which can be found in the equation, represents the released reaction

enthalpy. In other energy balances the released reaction enthalpy is given only in its implicit form.