## Combustion stoichiometry

Stoichiometry is the mathematical concept of relating the amount of products of a chemical reaction to the amount of reactants.

In combustion the four most relevant atoms are:

- Carbon C
- Hydrogen H
- Sulfur S
- Oxygen

The three basic stoichiometric concepts in combustion for these four atoms are

\(\hspace{2cm} C + 2O \rightarrow CO_2 \hspace{2cm}\) |
One atom of carbon C and two atoms of oxygen react to one molecule of carbon dioxide CO_{2} |

\(\hspace{2cm} 2H + O \rightarrow H_2O \hspace{2cm} \) |
Tow atoms of hydrogen and one atom of oxygen react to one molecule of water |

\(\hspace{2cm} S + 2O \rightarrow CO_2 \hspace{2cm} \) |
One atom of sulfur S and two atoms of oxygen react to one molecule of sulphur dioxide |

The three above reaction equations always apply whatever the fuel (e.g. pure CH

_{4}) or fuel mixture (e.g. 50% H

_{2} and 50% C

_{3}H6) is. It is also irrelevant how the oxygen is supplied (e.g. as pure O

_{2} or as the oxygen content of air or as oxygen contained in a fuel species like CH

_{2}OH).

It is important to keep in mind that stoichiometry does not involve any thermodynamics.

## Volume fraction

see

Mole fraction

## Mole fraction

The mole fraction of species \(\alpha\) is defined as the fraction of volumetric flow of species \(\alpha\) and the total volumetric flow
$$
y_{\alpha} = \frac{\dot{V}_{\alpha}}{\dot{V}_{total}} \hspace{2cm} [\frac{\frac{m^3}{s}}{\frac{m^3}{s}} = 1]
$$
The mole fraction can vary between zero and one and has no dimension.

Multiplying the mole fraction by 100 gives the volumetric percentage.

## Mass fraction

The mass fraction of species \(\alpha\) is defined as the fraction of massflow of species \(\alpha\) and the total massflow
$$
x_{\alpha} = \frac{\dot{m}_{\alpha}}{\dot{m}_{total}} \hspace{2cm} [\frac{\frac{kg}{s}}{\frac{kg}{s}} = 1]
$$
The mass fraction can vary between zero and one and has no dimension.

Multiplying the mass fraction by 100 gives the mass percentage.
## Air to fuel ratio (AFR)

The air to fuel ratio is defined as the ratio of massflows of air and fuel
$$
AFR = \frac{ \dot{m}_{air}}{\dot{m}_{fuel}} [\frac{kg}{kg}]
$$
The AFR has no dimension.

The AFR is the reciprocal value of the

fuel to air ratio. Mathematically there is no reason why to choose one over or the other since they express exactly the same thing. However, when dealing with fuel-lean combustion a value of the AFR of 40 is much more handy than a value of the FAR of 0.025. Otherwise it is often a question of tradition and practice in an engineering environment whether to use AFR or FAR.

## Thermal diffusivity

Thermal diffusivity a is a gas property defined as thermal conductivity \(\lambda\) divided by density \(\rho\) and specific heat capacity \(c_p\) at constant pressure
$$
a = \frac{\lambda}{\rho \cdot c_p} [\frac{m^2}{s}]
$$
Thermal diffusivity describes quantitatively the heat transport in the gas on a molecular level.

Thermal diffusivity is the diffusivity of heat in the heat conduction equation.
## Web application

A web application is a special form of computer program that runs in a web browser. It can be invoked over the internet. The main advantage of a web application is that it does not require to be installed locally on a computer. Code maintenance and deployment happens on one server whereas the code application can happen on many computers in many differenct locations at the same time.

Ideally a web application uses a responsive design.

## Viscosity

Viscosity \(\mu\) is a transport property of a fluid which quantitatively describes how momentum is transferred within the fluid on a molecular level. For a fluid with higher viscosity (honey) more momentum is exchanged between adjacent fluid regions than for a fluid with lower viscosity (water). Just imagine dragging a spoon through a pot of honey and a pot of water and how the spoon moves the fluid.

Viscosity has the unit of kg / m / s.

## Transport properties

Transport properties of a species are those properties that describe how momentum (viscosity) and heat (conductivity) are transported through molecular interaction.

Transport properties describe the interaction between molecules as a function of temperature.

They are calculated for a single species as a function of temperature and for a
fluid as a function of the participating species mass fractions.

## Thermal properties

Thermal properties of a species are specific enthalpy [kJ/kg] and heat capacity [kJ/kg/K].

Thermal properties describe how molecules change their internal state as a function of temperature.

They are calculated for a single species as a polynomial function of temperature and for a fluid as a function of the participating species mass fractions.

## Stanton number

duda
The Stanton number, named after the British aerodynamics pioneer T.E. Stanton (1865-1931), is a

(dimensionless number. It is defined as the ratio of the convective

heat transfer coefficient \( HTC \) to the product of the fluid's heat capacity \( c_p \), the fluid's bulk velocity \( U \) and the fluid's density \( \rho \).
$$
St = \frac{HTC}{c_p \cdot U \cdot \rho}
$$
The Stanton number (St) can be expressed using the

Reynolds number (Re), the

Nusselt number and the

Prandtl number as
$$
St = \frac{Nu}{Re \cdot Pr}
$$
The knowledge of the Stanton number for a

heat transfer problem allows to derive the

heat transfer coefficient and to perform a 1D heat transfer balance across the wall.
Heat transfer

correlations sometimes are expressed for the Stanton number rather than for the

Nusselt number.

## Species transport

Species transport means that in the flow network model individual chemical species like CH_{4} or O_{2} are modelled. This allows for more advanced combustion modelling like Chemical equilibrium, Perfectly stirred reactor (PSR) or Plug flow reactor (PFR).

Species transport is also a prerequisite for an enthalpy balance.

Species transport requires a species balance for flow elements and nodes

## Species

Species can be either atoms like C,O,H or molecules like O

_{2}, CH

_{4} or C

_{12}H

_{23}.

Characteristics:

- Molecular weight
- volumetric content of carbon (C), hydrogen (H), sulfur (S) which can be burnt
- volumetric content of oxygen (O)
- volumetric content of inerts
- transport properties
- thermal properties

Species are the constituents of

fluids and can be defined by the user.

## Fluid

A fluid is a gaseous mixture of one or more

species.

Examples (in mass percent)

Fluids can be defined by the user.

## Fuel to air ratio (FAR)

The fuel to air ratio is defined as the ratio of the massflow rates of fuel and air.
$$
FAR = \frac{ \dot{m}_{fuel}}{\dot{m}_{air}} [\frac{kg}{kg}]
$$
The FAR has no dimension.

FAR=0 means pure air whereas FAR=infinity means pure fuel. The stoichiometric FAR is a specific value for each fuel somewhere inbetween zero and infinity and means that the amount of oxygen in the air is just sufficient to burn all of the fuel. If the FAR is less than the stoichiometric value the case is fuel lean which means that we have more oxygen available than is needed to burn the fuel. Some oxygen remains after combustion.

If the FAR exceeds the stoichiometric value the case is fuel rich which means that there is not enough oxygen to burn the fuel, i.e. fuel, but not oxygen is left after combustion.

The FAR is the reciprocal value of the

air to fuel ratio (AFR)
## Glossary

A glossary lists relevant special terms of a specific field in alphabetical order and provides a short description for each term.

The aerocomb.com glossary lists terms which are relevant to 1d combustion and aerothermal modelling.

A term in the glossary may have a different meaning in another scientific or technical field. It is therfore important to keep the context of the glossary in mind.

## GUI

= Graphical User Interface.

Via a GUI the user can interact with a computer program using a mouse and a keyboard. The GUI can contain various types of standardised input fields
such as textfields, select lists or radio buttons as well as interactive graphics. Traditionally a GUI was a feature of stand-alone locally installed programs. With the
advances in internet technology a GUI can nowadays be realised entirely in a web browser without any local installation.

On a smart device the mouse mouse functions are replaced by touch gestures,

## Heat transfer

Heat flows from higher temperatures to lower temperatures. If the heat is not continuously supplied at the hot end and not continuously drained at the cold end, heat
transfer will eventually lead to a temperature equilibrium.

Heat can be transferred by conduction and by radiation. The first necessitates physical contact between the hot and the cold part whereas the latter necessitates
"sight contact" between the hot and the cold part. Please note that convection is a special case of conduction between a moving fluid and a solid.

The objective of heat transfer calculations in combustion is to obtain wall temperatures and to layout the

cooling system. A secondary effect that can be taken into account is the transfer of
heat from one

flow element across a wall to another

flow element leading to a change in temperature in both elements. This can be of interest for emissions calculations.

## Mach number

The Mach number is the ratio of flow velocity U to the speed of sound a.
$$
Ma = \frac{U}{a}
$$
The Mach number is an indication as to whether the flow is compressible. With a Mach number of <=0.3 the flow can be treated as incompressible whereas a higher Mach
number requires the flow to be treated as compressible.

The evolution of the Mach number in a

flow element depends on the following factors:

## Mass balance

Mass balance applies both to a flow element and a network node.

For a flow element it means that the massflow entering a flow element equals the massflow leaving the flow element. This is true for all flow elements at any time during the iteration.

For a network node it means that the sum of massflows flowing into the node equals the sum of massflows leaving the node. In a converged solution all nodes will have a mass balance within a specified tolerance.

## Loss coefficient

The loss coefficient of a flow element is defined as the

pressure drop across the element divided by the dynamic pressure at the inlet of the element
$$
\zeta = \frac{\Delta p_t}{0.5 \cdot \rho U_1^2} [-]
$$
The total loss coefficient is the sum of of individual loss coefficients
$$
\zeta_{total} = \zeta_{friction} + \zeta_{discharge} + \zeta_{inlet} + \zeta_{combustion} + \zeta_{area} + \zeta_{bend} + \zeta_{mixing} + \zeta_{swirl}
$$

## Kinematic viscosity

Kinematic viscosity \(\nu\) is defined as

viscosity \(\mu\) divided by density \(\rho\)
$$
\nu = \frac{\mu}{\rho} [\frac{m^2}{s}]
$$
Kinematic viscosity is the diffusivity of momentum in the momentum equation (Navier-Stokes equation).

## Impingement cooling

Impingement cooling uses double walls. One impermeable and the other perforated. The impermeable wall adjacent to the hot combustion stream is cooled by impingement
jets that develop through the perforation holes of the outer wall.

With a growing number of impingement jets a crossflow accumulates between the two walls and gradually makes the jets less efficient.

Correlations provide heat transfer coefficients as a function of hole sizes, hole spacing and hole pattern as well as wall distances.

## Flow element

A 1d flow element is the ideal model of a real three dimensional geometry.

Characteristics:

- One inlet - one outlet
- One upstream node - one downstream node
- Streamwise variation of quantities between inlet and outlet
- No spanwise variation of quantities
- The cross section may vary from inlet to outlet
- The element's axis is either straight or bent

## Enthalpy polynomial

NASA provides polynomials with 8 coefficients \( a_1, a_2, a_3, a_4, a_5, a_6, a_7, b_1 \) for the enthalpy of a species

\[\begin{aligned}
\frac{H(T)}{R_{gen} \cdot T} = -a_1 \cdot T^2 + \frac{a_2}{T} ln(T) + a_3 + a_4 \cdot \frac{T}{2} + a_5 \cdot \frac{T^2}{3} + a_6 \cdot \frac{T^3}{4} + a_7 \cdot \frac{T^4}{5} + \frac{b_1}{T} [ \frac{\frac{kJ}{kmol}}{\frac{kJ}{kmol \cdot K} \cdot K} = 1 ]\\
\end{aligned} \]

\[\begin{aligned}
\frac{H(T)}{R_{gen} \cdot T} = -a_1 \cdot T^2 + \frac{a_2}{T} ln(T) + a_3 + a_4 \cdot \frac{T}{2} + a_5 \cdot \frac{T^2}{3} + a_6 \cdot \frac{T^3}{4} + a_7 \cdot \frac{T^4}{5} \\
+ \frac{b_1}{T} [ \frac{\frac{kJ}{kmol}}{\frac{kJ}{kmol \cdot K} \cdot K} = 1 ]\\
\end{aligned} \]

\[\begin{aligned}
\frac{H(T)}{R_{gen} \cdot T} = -a_1 \cdot T^2 + \frac{a_2}{T} ln(T) + a_3 \\
+ a_4 \cdot \frac{T}{2} + a_5 \cdot \frac{T^2}{3} + a_6 \cdot \frac{T^3}{4} + a_7 \cdot \frac{T^4}{5} \\
+ \frac{b_1}{T} [ \frac{\frac{kJ}{kmol}}{\frac{kJ}{kmol \cdot K} \cdot K} = 1 ]\\
\end{aligned} \]

\[\begin{aligned}
\frac{H(T)}{R_{gen} \cdot T} = \\
-a_1 \cdot T^2 + \frac{a_2}{T} ln(T) + a_3 \\
+ a_4 \cdot \frac{T}{2} + a_5 \cdot \frac{T^2}{3} + a_6 \cdot \frac{T^3}{4} + a_7 \cdot \frac{T^4}{5} \\
+ \frac{b_1}{T} [ \frac{\frac{kJ}{kmol}}{\frac{kJ}{kmol \cdot K} \cdot K} = 1 ]\\
\end{aligned} \]

Where R is the general gas constant (R=8.314 J/kmol/K)

Specific enthalpy hhx for a species can be calculated from H(T)
\[\begin{aligned}
hhx(T) = \frac{H(T)}{M} \cdot R_{gen} \cdot T \\
[ \frac{1}{\frac{kg}{kmol}} \cdot \frac{kJ}{kmol \cdot K} = \frac{kJ}{kg} ]
\end{aligned} \]

## Reynolds number

The Reynolds number named after the British fluid dynamics pioneer O. Reynolds (1842 - 1912) is defined as velocity U times characteristic length L divided by kinematic viscosity \( \nu \). As a dimensionless number it has the dimension unity.

$$
Re = \frac{U \cdot L}{\nu} [ \frac {\frac{m}{s} \cdot m}{\frac{m^2}{s}} = 1 ]
$$
Two fluid systems with different dimensions, different velocities and different fluids will behave similar with respect to turbulence when their Reynolds numbers are similar.
## Correlation

A correlation in statistics describes the relationship between two variables. The existence of a correlation does not necessarily mean that one variable is the cause for the other.

In aerothermal modelling correlations are used to compute a desired quantity using known quantities. Correlations are mostly derived from experiments.

Example: flat plate correlation

$$
Nu = 0.023 \cdot Re^{0.8} \cdot Pr^{0.4}
$$
The correlation formulated 1930 by Dittus and Boelter relates the

Nusselt number (Nu) to the

Reynolds (Re) and

Prandtl (Pr) numbers of the

fluid. If we know the Reynolds and

Prandtl numbers of the

fluid flowing past a flat plate, we can work out the

Nusselt number and hence the

heat transfer coefficient.

## Film cooling

Film cooling is a means of reducing hot side heat transfer by injecting a film of cooling air alongside the hot side combustor wall.

Typically the cooling air enters from the combustor cold side through a number of holes and is deflected to form a cooling film that gradually mixes with the hot combustor flow.

Several stages of film cooling can be applied in series to achieve the desired wall temperature.

For heat transfer calculations there are correlations that allow the derivation of heat transfer coefficients from the film cooling geometry and the film mass flow.

## Effective area

Effective area (total to static) is a concept of describing the

loss coefficient of a

flow element. It has the unit of \( \sf mm^2 \) (area) and becomes zero for an infinitely big

loss coefficient
and equal to the geometric passage area for a

loss coefficient of zero.

It is defined as
$$
area_{eff} = \frac{\dot{m}}{\sqrt{2 \cdot \rho \cdot (p_{total,1} - p_{static,2})}}
$$

\( \dot{m}\)= massflow, \( p_{total,1} \)= total pressure upstream, \( p_{static,2}\): static pressure downstream, \( \rho \): density (assumed constant)

Effective area,

discharge coefficient and

loss coefficient can be converted into each other.

Effective area does not depend on the boundary conditions. It only depends on the geometry of the flow element.

The concept of effective area can also be applied to the entire flow network of a combustor.

## Combustion model

The combustion model is the part of the overall physical model that describes the chemical reaction of fuel and oxidant to combustion products.

Depending on the species mass fractions entering into the combustion process the combustion model derives the species mass fractions leaving the combustion process.

Applying an enthapy balance, the change in species mass fractions will lead to a change in temperature.

Examples of combustion models:

## Chemical equilibrium

Chemical equilibrium means that for a reversible chemical reaction the rates of the forward reaction and the backward reaction are equal. The effect is that there are
no net changes in the concentrations of reactants and products although both reactions still take place.

In order to reach chemical equilibrium a sufficient residence time time in the combustor is required.

Increasing pressure shifts the equilibrium towards a state with less volume and vice versa.

Increasing temperature in an exothermal reaction shifts the equilibrium towards the products.

**Try the aerocomb.com combustion tool online**

and make chemical equilbrium calculations yourself.
## Adiabatic combustion temperature

is the temperature that results if a combustible mixture is completely burned. The combustion process changes the species composition of the mixture, but not its enthalpy. In order to preserve enthalpy the temperature of the mixture after combustion must change.

Adiabatic combustion temperature depends on the type of fuel, the fuel to air ratio and the temperature of the mixture prior to combustion.

**Try the aerocomb.com combustion tool online **

and compute the adiabatic combustion temperature yourself.
## Boundary conditions

Boundary conditions in an aerothermal network model are imposed via flow elements with no upstream node. These elements do not receive information from the network, but supply boundary conditions into the network.

Boundaries can be of pressure type or massflow/pressure type. In the first case the pressure in the network at the position of the boundary element determines the massflow into or out of the network through the boundary element.
In the latter case the boundary fluid and temperature are also specified.

## Heat transfer coefficient

The heat transfer coefficient (HTC) relates the heat flux between fluid and wall to the temperature difference between wall surface temperature and the flow bulk temperature
$$
HTC = \frac{\dot{q}}{T_{wall,surface}-T_{flow,bulk}} [\frac{W}{m^2 \dot K}]
$$
Heat transfer coefficients can be calculated using

correlations for different geometries and

cooling arrangements. The knowledge of the HTC and the fluid bulk temperatures for both sides of a wall allows us to perform a 1D heat transfer calculation across the wall and to derive the wall temperature profile.

## Cooling

In order to prevent overheating and mechanical disintegration the combustor walls must be cooled. The heat transferred from the combustion process to the walls
must be compensated.

Typical cooling technologies for gas turbine combustors are

In a 1d combustor network model each of the above cooling schemes requires special correlations that take the geometry and the various
fluid streams into account and deliver heat transfer coefficients and wall temperatures.

## Air excess factor

Air excess factor (AEF) is defined as the ratio of the actual

air to fuel to air to the stoichiometric

air to fuel ratio. Often the Greek letter λ is used for AEF.

**AEF < 1** |
fuel rich |

**AEF = 1** |
stoichiometric |

**AEF > 1** |
fuel lean |

AEF is the reciprocal value of

equivalence ratio (EQR). Whether AEF or EQR is used, often depends on the industry and its engineering environment.

## Aerothermal network

see

Flow network
## Node

A node is a logical element with no geometric properties like a node in an electrical circuit. Flow elements can be connected upstream or downstream of a node.

At a node the first law of thermodynamics applies:

## Convective Cooling

Convective cooling is the most basic type of cooling. A stream of hot combustion products flows along one side of the combustor
wall and the stream of cooling air flows along the other side of the wall.

In a steady state process with constant fluid temperatures and constant flow rates on both sides of the wall a heat transfer
equilibrium will be reached. The resulting wall temperature profile not only depends on fluid temperatures and flow rates, but
also on the wall material(s).

**Try the aerocomb.com heat transfer tool online **

and compute convective heat transfer yourself.
## Complete combustion

Complete combustion means, sufficient oxygen provided, that all of the fuel carbon is transformed into CO_{2}, all of the fuel hydrogen is transformed into H_{2}O and
all of the fuel sulfur is transformed into SO_{2}.

If there is not enough oxygen available a part of the carbon, hydrogen and sulfur in the fuel will remain unburnt.

Complete combustion cannot deal with intermediate species such as CO, intermediate hydrocarbons or radicals such as OH.

Complete combustion will result in the adiabatic combustion temperature.

**Try the aerocomb.com combustion tool online**

and make complete combustion calculations yourself.
## Perfectly stirred reactor (PSR)

The perfectly stirred reactor is a special

flow element which can take chemical kinetics into account.

The reactants are assumed to have perfectly mixed upon entering the PSR. The reaction into products follows a chemical reaction mechanism and is primarily dependent on the reactor

residence time. The longer the

residence time the more the reaction will be completed.

## Plug flow reactor (PFR

The plug flow reactor is a special

flow element which can take chemical kinetics into account.

The flow is assumed to move like a piston through the reactor. Species concentrations, velocity, enthalpy and temperature are assumed homogeneous normal to the flow direction and can vary in flow direction. The reaction into products follows a chemical reaction mechanism.

## Prandtl number

The Prandtl number, named after the German aerodynamics pioneer L. Prandtl (1875 - 1953), is a

dimensionless number. It is defined as the ratio of kinematic viscosity to thermal diffusivity
$$
Pr = \frac{\nu}{\alpha}
$$
The Prandtl number only depends on the fluid as it only relates fluid properties to each other. It is a measure of the ratio of momentum transport ( \(\nu\) ) to thermal transport ( \( \alpha \) ) in the fluid.

## Premixed flame

A premixed flame will develop when a gaseous mixture of fuel and oxidant is ignited. Since fuel and oxidant are already mixed, premixed flames are more dominated by chemical kinetics than by fluid mechanics and diffusion processes.

Any intermediate between a pure

diffusion flame and a pure premixed flame is also possible.

A premixed flame is more sensitive to the stoichiometry than a

diffusion flame. A premixture may be too fuel lean to burn, whereas a

diffusion flame with the same stoichiometric ratio between fuel and oxidant streams may still burn.

## Enthalpy balance

Enthalpy balance for a node means that the total enthalpy entering the node equals the total enthalpy leaving the node.

$$
h_{t,in} = h_{t,out} \space \space \space [\frac{kJ}{kg \cdot s}]
$$

For a flow element the total enthalpy at the outlet equals the total enthalpy at the inlet plus/minus the energy gained or lost by heat transfer across the element's walls.

$$
h_{t,outlet} = h_{t,inlet} + \dot{q} \space \space \space [\frac{kJ}{kg \cdot s}]
$$

## Flow split

The distribution of mass fluxes in the branches of a flow network. As massflow must be conserved the total massflow into a network node must be equal to the total
massflow leaving the network node.

80 + 20 = 40 + 60

The flow split is a function of the resistance of the flow elements. In general terms high resistance in a flow element results in a high pressure drop along the element
and accordingly in a low massflow through the element.

## Effusion cooling

Effusion cooling is a method of liner cooling where the liner wall is perforated with a large number of small holes. Cooling air flows through these holes from the
cold side of the combustor to the hot side. The cooling effect is twofold. There is convective cooling of the wall by the cooling air flowing through the holes, but
there is also a film of cooling air protecting the hot side of the combustor, similar to the situation in conventional film cooling.
## Equivalence ratio

Equivalence ratio (EQR) is defined as the ratio of the actual

fuel to air ratio to the stoichiometric

fuel to air ratio.

**EQR < 1** |
fuel lean |

**EQR = 1** |
stoichiometric |

**EQR > 1** |
fuel rich |

The advantage of using EQR instead of

FAR is that the above table is true for any fuel. The value of the stoichiometric fuel to air ratio varies from fuel to fuel, but EQR=1 always means stoichiometric conditions.

## Diffusion flame

A diffusion flame as oposed to a

premixed flame is characterised by separate fresh streams of fuel and oxidant prior to combustion. Fuel and oxidant must mix on a molecular level until a surface of stoichiometric conditions is achieved where a flame can establish. This is also true for liquid fuel (spray flame) where the fuel droplets must evaporate before the mixing and combustion.

Diffusion flames are to a greater extent determined by fluid mechanics, molecular diffusion and flow turbulence than by chemical kinetics. The simplified approach "mixed equals burnt" for diffusion flames basically means that the time scale of fluid mechanics is much larger than the chemical timescale.

## Discharge coefficient

The discharge coefficient of a

flow element is defined as the ratio of its

effective area to its geometric area (flow passage) at the inlet.
$$
c_d = \frac{a_{effective}}{a_{geometric,1}}
$$

## Dimensionless number

A dimensionless number in aerothermal modelling is a number derived from other quantities such that its resulting physical dimension is exactly unity.

Different configurations can be compared with respect to certain physical effects by looking at their appropriate dimensionless numbers.

Example:

Reynolds number
## Species balance

Species balance for a node means that the sum of mass fractions x of species \( \alpha \) entering the node (branches,in) equals the sum of mass fractions of species \( \alpha \) leaving the node (branches out).

$$
\sum_{branches,in} x_{\alpha} = \sum_{branches,out} x_{\alpha}
$$

Along a flow element, however, chemical reaction may occur. Species like O_{2} or CH_{4} may be produced or consumed. Hence there is only a conservation of atoms like N,H,O,C,S.

For an atom \( \beta \) we look into all

species and sum up the mass content of atom \( \beta \) in the

species
$$
\sum_{\alpha,inlet} x_{\alpha} \cdot b_{\alpha\beta} = \sum_{\alpha,outlet} x_{\alpha} \cdot b_{\alpha\beta}.
$$
\( \alpha \) are

species like O

_{2} or CH

_{4} and \( \beta \) are atoms like N,H,O,C,S. \(b_{\alpha\beta}\) is the mass content of atom \( \beta \) in

species \( \alpha \), e.g. 12/16 for C in CH

_{4}.
\( x_{\alpha} \) is the mass fraction of

species \( \alpha \).

## Responsive design

Responsive design is a special design of a web application that recognizes the screen size of the client device. An ordinary non-responsive web page that is wider than the client screen will hide some of its content using horizontal scroll bars. The non-responsive web page is mostly not easily usable on a smart device.

A responsive web page or responsive web application will adapt to the client device making it user-friendly.

You can try it just here by reducing your browser window. As the page is responsive, the page content gets constantly reformatted according to the window width.
## Péclet number

The Péclet number (Pe) named after the French physicist J.C.E. Péclet (1793-1857) is a dimensionless number defined as the ratio of convection to diffusion of heat.
$$
Pe = \frac{U \cdot L}{\alpha} [\frac{\frac{m}{s} \cdot m}{\frac{m^2}{s}} = 1]
$$
where U is velocity, L is a chararcteristic length and \(\alpha\) is the

thermal diffusivity.
The Péclet number can be written as the product of

Reynolds number and

Prandtl number:
$$
Pe = Re \cdot Pr
$$

## Nusselt number

The Nusselt number (Nu), named after the German heat transfer pioneer W. Nusselt (1882-1957), relates the convective

heat transfer coefficient (HTC) to the fluid conductivity \( \lambda \) divided by a characteristic length L.
$$
Nu = \frac{HTC \cdot L}{\lambda} [-]
$$
The Nusselt number is a

dimensionless number and can be calculated from empirical

correlations. The derivation of the Nusselt number and consequently the

heat transfer coefficient is the starting point for 1D

heat transfer calculations.

## Pressure drop

Pressure drop is the difference in total pressure across a single

flow element or across an entire

flow network.
$$
\Delta p = p_{t,outlet} - p_{t , inlet}
$$
Physical processes creating a pressure drop are:

- Friction
- Area change
- Flow curvature
- Mixing
- Swirl
- Combustion.

A pressure drop causes a change in

Mach number.

## Residence time

Residence time ($t_{res}$) for a flow element is a function of element volume (Vol), massflow through the element ($\dot{m}$) and the mean density ($\rho$)
$$
t_{res} = \frac{Vol \cdot \rho}{\dot{m}}
$$
## Riblet cooling

Riblet cooling is a means of enhancing the heat transfer on the cold side of the combustor wall with turbulators.

The turbulators enhance flow turbulence and hence heat transfer and cooling.

The turbulators increase the pressure drop as compared to a smooth wall.

Correlations allow the derivation of heat transfer coefficients as a function of the riblet geometry

## Flow network

A flow network consists of one or more 1d

flow elements (circles) interconnected via

nodes (squares). Both bifurcations and unifications of the flow path are possible.

Boundary conditions are imposed via elements without an upstream

node.

The basic aerodynamics (

flowsplit,

pressure drop,

enthalpy balance) of combustion chambers like those of gas turbines can be modelled using a flow network. The effort
for setting up a model and the computational effort for solving it are appreciably lower than with a full 3d CFD model.