Theoretical and empirical investigation of round and rectangular cooling channels

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Scuola di Ingegneria

Corso di Laurea in Ingegneria Aerospaziale

Tesi di Laurea

Theoretical and empirical investigation of round

and rectangular cooling channels.

Relatore Candidato

Prof. Luca d’Agostino Sabatino Francesco Plastina

Co-relatore Prof. Oskar HAIDN

Dott. Ing. Simona Silvestri

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Abstract

The thermal phenomena in rocket engines involve interactions among different processes, including combustion in the thrust chamber, expansion of hot- gases through the nozzle, heat transfer from hot gases to the nozzle wall through pro-cess of convection and radiation, conduction in the wall and convection to the cooling channel. The complexity of the thermal analysis in rocket engines is due several factors : three- dimensional geometry, hot gas heat transfer coefficient and coolant heat transfer coefficient, dependence on the wall temperature, de-pendence on the coolant pressure drop and its properties, axial conduction of heat within the wall and radiative heat transfer between hot gases and surfaces of the combustion chamber.

The aim of this paper is to study the coupled problem of coolant flow and wall structure heat transfer in the cooling channels of a liquid rocket engine thrust chambers.

They will be analyzed several phenomena influencing the convective heat transfer coeﬃcient and derived thereby adjustments to the formulation of the coeﬃcient currently used.

It will be presented a 2-D model for the prediction of the local coolant temperature profile and of the wall temperature profile, coolant side.

In this dissertation the procedure to evaluate the simplified quasi-2-D model is based on:

• one-dimensional governing equation for coolant mass conservation • two-dimensional governing equation for coolant energy balance

The energy balance equation is coupled to the wall heat transfer balance in the radial direction.

Turbulent thermal conductivity(presented in Appendix A), fluid skin fric-tion, and coolant-wall heat transfer coeﬃcients are evaluated by a number of built-in correlations . This model allows fast prediction of both the coolant flow evolution and the temperature distribution along the whole cooling channel structure.

It was implemented a MATLAB®_{program that, known the data of the}

prob-lem and the coeﬃcient of convective heat exchange, allows the evaluation of the wall temperature profile and temperature profile in the channel resulting from the thermal stratification.

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Validation of the model is carried out by comparison with straight channel solutions obtained with a previous available data obtained from Thermtest. Results show good agreement with available data.

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List of Figures

1.1 Space Shuttle Main Engine (SSME). . . 5

2.1 Best configuration for regenerative cooling . . . 10

2.2 Heat transfer schematic for regenerative cooling . . . 11

2.3 Moody chart for the friction factor in circular tubes. . . 15

2.4 Division region of interest for eddy diﬀusivity variation . . . 20

3.1 Schematic of cooling channels geometry for two cases studied . . 23

3.2 Balance of heat flux converging in a slice channel dy and constant width b . . . 26

4.1 Mesh on a semi-infinite strip used for the solution of PDE equations 31 4.2 The Crank-Nicolson stencil for 2D problem . . . 33

4.3 Ghost node scheme . . . 37

5.1 Combustion chamber.Are marked water manifolds which consti-tutes the refrigerant to the experiment. . . 42

5.2 Setup of TCN-T38-01-2 chamber test composed by two short seg-ments and one long segment . . . 43

5.3 Sub-scale combustion chamber and pressure sensors location . . . 45

5.4 Position of the thermocouples in the chamber segments . . . 46

5.5 Picture of real TUM test bench . . . 46

5.6 Short segment chamber combustion . . . 47

6.1 Sketch of heat transfer model of Thermest . . . 50

6.2 Heat flux vs. axial coordinate calculated by Thermtest for cham-ber combustion with rectangular cooling channel . . . 51

6.3 Heat flux vs. axial coordinate calculated by Thermtest for cham-ber combustion with cylindrical cooling channel . . . 52

6.4 Geometry of combustion chamber for chamber combustion with rectangular cooling channel . . . 53

6.5 Geometry of combustion chamber for chamber combustion with cylindrical cooling channel . . . 53

6.6 Thermtest parameters . . . 54 vii

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7.1 Nusselt prediction by Kraussold correlation and operating points. 58 7.2 Comparison Nusselt prediction between Kraussold (green) and

Bhatti and Shah whit various values of roughness . . . 59 7.3 Comparison Nusselt prediction between Kraussold (green) Bhatti

and Shah (blue) and Hasan analysis (magenta) . . . 61 7.4 Operative point with three diﬀerent correlation; Red for

rectan-gular and yellow for cylindrical cooling channels. . . 62 7.5 Temperature profile from inlet section to exit section in

cylindri-cal cooling channels Kraussold heat transfer coeﬃcient. . . 63 7.6 Thermal stratification in cylindrical cooling channels with

Kraus-sold heat transfer coeﬃcient in three axial positions. . . 64 7.7 Temperature profile from inlet section to exit section in

rectan-gular cooling channels Kraussold heat transfer coeﬃcient. . . 65 7.8 Thermal stratification in rectangular cooling channels with

Kraus-sold heat transfer coeﬃcient in three axial positions. . . 66 7.9 Temperature profile from inlet section to exit section in

cylindri-cal cooling channels Bhatti and Shah heat transfer coeﬃcient. . 67 7.10 Thermal stratification in cylindrical cooling channels with Bhatti

and Shah heat transfer coeﬃcient in three axial positions. . . 68 7.11 Temperature profile from inlet section to exit section in

rectan-gular cooling channels Bhatti and Shah heat transfer coeﬃcient. 69 7.12 Thermal stratification in rectangular cooling channels with Bhatti

and Shah heat transfer coefficient in three axial positions. . . 70 7.13 Entrance effect for heat transfer coefficient with iterative method

for cylindrical cooling channel . . . 73 7.14 Entrance eﬀect for heat transfer coeﬃcient with iterative method

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Acknowledgments

Alla mia famiglia,ai miei amici a Giusy.

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List of symbol

Latin letters

Symbol Description Unit

A Area [m2_]

b Thickness of rib [m]

C Nusselt empirical factor [ ]

Cp Specific heat capacity [J_/_KgK_]

D Diameter [m]

f Friction factor [ ]

h0 Static enthalpy [J/Kg]

H0 Total enthalpy [J/Kg]

h Heat transfer coeﬃcient [W_/_m2_K_]

k Thermal conductivity [W_/_mK_] ˙ m Mass flux [Kg_/_s_] O_/_F Mixture ratio _{[ ]} P Pressure [P a] p Geometry factor [ ] ˙q Heat flux [W_/_m2_] r Radius [m] tw Tickness of wall m T Temperature K u Flow velocity [m_/_s_]

X1 Fluid kinematic constant for Hinze [ ]

x Axial coordinate [m]

y Radial coordinate [m]

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Greek letters

Symbol Description Unit

↵ kwtw [W/K] k2tw [W/K] ⇢ubCp [Js_/mK] ✏ Roughness [m] ✏m Eddy diﬀusivity [ ] ⌘ Combustion eﬃciency [ ]

⇠ Combustion empirical factor [ ]

⌫ Dynamic viscosity [Kg_/_ms_]

⇢ Density [Kg_/_m3_]

s Stefan-Boltzmann constant [W/m2K4]

⌧ Shear stress [M P a]

! Molecular thermal diﬀusivity ⇥m2_/_s⇤

Dimensionless number

Symbol Description Re Reynolds number P r Prandtl number N u Nusselt number St Stanton number

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Chapter 1

Liquid Propellant Rocket

Engine

Since World War II, rocket jet-propulsion has evolved from a raw science to a refined engineering art. Today, most number of aerospace companies in the world produce a large number of sophisticated liquid-propellant rocket-engine power plants capable of propelling scientific, commercial, and military vehicles and use as the primary propulsion systems in most launch vehicles and spacecraft since the initial conquest of space.

Although the studies and economic commitments being made in the develop-ment of these engines, many aspects of modeling, analysis, and design of thrust chambers still present important challenges. The first reason for this challenge is the complexity of the problem; although the basic concepts are well established, many of the detailed physiochemical processes of liquid-propellant combustion remain unresolved. Another reason is the diﬃculty and significant expense of conducting research and development in the harsh and hazardous environments of liquid rocket thrust chambers.

1.1 Characteristics

The choice of a liquid propellent rocket engine has considerable advantages. Typical liquid propellant has a density similar to water, approximately equal to 0.7 to 1.4 and requires a little pressure to avoid evaporation. This combination of density and low pressure allows the use of tanks light.

The injection into the combustion chamber requires a pressure of the propel-lant exceeds that of the chamber near of the injectors. This pressure is typically generated with a pump: turbo-pumps are chosen for their strength and eﬃciency , even if in the past have been employed other types of pumps. Turbo-pumps have a weight of less than 1% of their thrust. For these reasons, most orbital launch vehicles use liquid propellants.

Liquid-fueled rockets have higher specific impulse than solid rockets and are 4

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capable of being throttled. Another advantage is the possibility of modulation of the thrust in real time and a good control of the rate of mixing. They can be switched oﬀ without too much trouble and restarted in a short time.

Figure 1.1: Space Shuttle Main Engine (SSME).

1.1.1 Basic Elements of a Liquid Propellant Rocket

En-gine

The structure of a liquid propellant engine is relatively more complex than that of the other propellant rocket motors. In them it is possible to distinguish seven major subsystems:[1]

• Thrust Chamber ; • Propellant feed system;

• Turbine-drive system (gas generator, pre-burner, etc.); • Propellant Control System

• Electrical and Pneumatic Controller Systems; • Thrust-Vector Control System (TVC) • Interconnect Components

The thrust-chamber assembly generates power by combusting liquid propellants led to the required combustion pressure by the propellant feed system. In the thrust chamber there is the main injector; it atomizes, ignites, and promotes the complete combustion of the liquid propellants. The combustion products are made to escape through a converging-diverging nozzle to achieve high gas velocities and thrust level. The propellant feed system essentially consist of propellant tanks, lines, and turbo-pumps; propellant turbo-pumps and lines; and in the case of a pressure-fed engine system, simple propellant tanks and lines. The turbines are (in most cases) driven by energetic high-temperature gases produced in gas generators, pre-burners, heat exchangers heated by the

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main thrust-chamber combustion products, or gases tapped directly from the main combustion chamber. In each of the two cases above, a set of propellant valves in the propellant-control system meters the two propellants flowing to the injector controlling both total amount and relative amount. Total amount is used in thrust control while the relative amount is used in control of propellant mixture ratio. The valves also have the task of starting and shutting down the engine through proper regulation of propellant flow rates. These final control elements (valves) are electrically or pneumatically controlled. The controllers receive command signals from the vehicle or previously stored electric commands in the controller memory to eﬀect valve control during engine operation (to yield desired thrust and mixture-ratio variation or propellant consumption during flight). The TVC system eﬀects directional changes of the vehicle. Depending on the method of propellant feed, there are two basic types of liquid-propellant rocket engines: those having propellants directly fed from pressurized tanks to the thrust chamber and those having the propellants fed to the thrust chamber by a set of turbo-pumps. In the first case, the tank pressures may typically be a few hundred pounds per square inch. In the second instance, the propellants are supplied from the tanks to the inlets of turbo-pumps at relatively low pressures, typically 30 to 100 psi. The turbo-pumps then raise these inlet pressures to high levels, which in modern engines like the SSME may reach 8,000 psig or more. In the case of the SSME, low-pressure propellant pumps provide an intermediate pumping stage between tank and main turbo-pumps. Because of tank weight considerations, pressure feed will be limited to propulsion-system sizes that deliver relatively low stage velocity increments. Many modern liquid-propellant rocket engines use turbo-pump feed because it lowers the total weight of inert parts per stage.

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Chapter 2

Heat transfer and

Regenerative Cooling

A liquid-rocket combustion chamber converts propellants into high-temperature, high-pressure gas by combustion, the propellan releases the chemical energy , resulting in an increase in internal energy of the stored gas. The liquid propel-lants are injected at the injection plate with a axial velocity which is assumed to be zero in gas flow calculations. The combustion dynamics proceeds through-out the length of the chamber and is expected to be completed at the nozzle entrance. Heat liberated between injection plate and nozzle inlet increases the specific volume of the gas. To satisfy the conditions of constant mass flow, the gas must be accelerated toward the nozzle inlet with some losses of pressure.

The temperatures of the combustion of the propellants of the rocket mo-tors are generally more high of the critical points of metals and metal alloys commonly used. Thrust-chamber cooling will be major design consideration to avoid that the high temperature decreases the strength characteristics of the material.[2]

For short-duration operation (up to a few seconds) uncooled chambers may sometimes be used; the heat will be absorbed by a heavy chamber wall acting as a heat sink, until the wall temperature approaches the failure level but for most long- duration applications a steady-state chamber cooling system must be employed.

There are two distinct classes of cooling systems :

1. Active method, cooling systems subtract heat to the walls by means of forced convection.

2. Passive method, cooling systems that the high temperatures of the walls is avoided due to the radiation or insulation (thermal protection).

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2.1 Thrust chamber cooling

The main cooling systems employed are the following:[3][24]

• Ablative cooling: the method uses a combustion gas-side wall material that is sacrificed, by melting, chemical changes ans vaporization, to dissipate heat. As a result, relatively cool gases flow over the wall surface, thus lowering the boundary-layer temperature and assisting the cooling process. Ablative cooling may be applied only to the nozzle throat or to the entire combustion chamber liner and also to solid rockets.

• Film cooling: the chamber materials are protected from excessive heat with a thin film of coolant (which usually is the propellant) which is in-jected through orifices near the injector or near the throat with special slots. This method has been with large extend used, in particular for high heat fluxes, alone or in combination with regenerative cooling.

• Transpiration cooling: it is accomplished by introducing a coolant (gaseous or liquid) through porous chamber walls at a rate suﬃcient to maintain the desired combustion gas side wall temperature.

• Radiative cooling: heat is radiated away from the surface of the outer chamber walls. This method is applied when thermal stresses are low, such as monopropellant rocket, gas generator or nozzle extension. In high performance liquid rockets, radiative cooling is used generally in nozzle extensions and regenerative cooling in the combustion chamber; low thrust liquid rockets are, instead, cooled with a combination of film and radiative cooling.

• Dump cooling: with this principle, a small percentage of the propellant, such as the hydrogen in a LO2/LH2 engine, is fed through passages in the thrust chamber wall for cooling and is subsequently dumped overboard through openings at the rear end of the nozzle skirt.

• Regenerative cooling: the most widely applied method, utilizes one or pos-sibly both of the propellants fed through passages in the thrust-chamber wall for cooling, before being injected into the combustion chamber.

2.2 Regenerative Cooling

The high heat flux and then the high temperatures reached in thrust chambers require an active cooling of the structure. Regenerative cooling is one of the most used cooling methods in rocket engines used throughout the length of the chamber, by the injectors to expansion nozzle. The cooling is accomplished by flowing high velocity coolant over the back side of the chamber hot gas wall to convectively cool the hot gas liner. The coolant, heated up by the hot liner, is then discharged into the injector and used as a propellant.

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The best regenerative cooling solution to date is the “channel wall” design where the hot gas wall cooling is accomplished by flowing coolant through rect-angular or cylindrical channels, which are machined or formed in the same combustion chamber. see Fig. 2.1

Figure 2.1: Best configuration for regenerative cooling

Regenerative cooling technique has many advantages and some disadvan-tages: the main advantages are no performance loss (thermal energy absorbed by the coolant is returned to the propellant ), no change in wall geometry as a function of time, indefinite firing duration, and relatively light weight con-struction. Disadvantages include limited throttling with most coolants, reduced reliability with some coolants (e.g., hydrazine), high pressure drops required at high-heat-flux levels, and thrust levels, mixture ratios, or nozzle area ratios possibly limited by maximum allowable coolant-temperature.[4]

The temperature of the coolant increases from the entry of the channel until it leaves the cooling passages, as a function of the heat absorbed and the coolant propriety and flow rate.

Proper balance of these parameters, to maintain the chamber walls at tem-peratures below those at which failure might occur because of melting or stress, becomes one of the major requirements for the design of regeneratively cooled thrust chambers.

Design of a regeneratively cooled thrust chamber has need of an precision analysis and consideration of wall structure, hot gas side heat flux, coolant side heat transfer, and the eﬀects of temperature increases on coolant properties. Simple but eﬃcient method for describe the heat transfer in regenerative method is shown in figure 2.2 , is essentially, the analysis of the heat flow between two moving fluids, through a multilayer wall.

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Radial Distance from Center of Chamber Temperatur e Coolant Hot gas Chamber Inner Wall Gas size

boundary layer boundary layer Coolant size

Taw

Tco

Twg Twc

b

Figure 2.2: Heat transfer schematic for regenerative cooling

After the combustion, the heat to hot combusted gases transfer heat energy through a layer of stagnant gas along the wall,boundary layer, and after the heat passes to the wall. The general steady-state correlation of heat transfer for this complicated convective heat transfer from the combustion gases through the layers, which include the metal chamber walls, to the coolant can be expressed by the following equations:

˙qw= hg(Taw Twg) = kw ✓ Twg Twc b ◆ = hc(Twc Tco) (2.1) where:

Taw;Tco Temperature, hot gas and coolant liquid, K;

Twg;Twc Temperature wall surface,hot and cold sides, K;

hg;hc Heat transfer coeﬃcients, gas and coolant sides, W/(m2 K)

kw Thermal conductivity of wall, W/(m K);

b Chamber-wall thickness, m;

Assume a station in the thrust chamber with gas temperature T aw and coolant bulk temperature T co referring to (2.1), it will be seen that the heat flux ˙qw,

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which must be the same through all layers, is a function of the temperatures and of heat- transfer coeﬃcients hg, hcand kw.

H = ₁ 1 hg + 1 kw + 1 hc (2.2) H is a overall heat transfer coeﬃcient, the value of H is composed of the individual coeﬃcients for the boundary layers and the chamber metal wall (2.2). The smallerH is, the smaller ˙qw.

Since the temperature diﬀerentials are inversely proportional to the heat-transfer coeﬃcients of the heat-flow paths, the temperature drop will then be steepest between hot gas and inner chamber wall. Electrical analogy.[5]

The heat absorbed by the propellant used for regenerative cooling raises the temperature of the propellant, and thus the energy level, before it is injected into the combustion chamber but the eﬀect in the overall performance in light. On the other hand, this technique with big pressure losses requiring a high gas pressurization levels, imposes an overall performance penalty.

2.3 Heat transfer Coeﬃcient: semi-empirical

Nus-selt correlations

The convective heat transfer from the hot gas to the inner wall and from the wall to the coolant is modeled using Nusselt correlations as a common root.

What is the correlation of Nusselt, and why is it used for the heat transfer coeﬃcients predictions ?

In LPR’s rocket engine combustion chamber design, semi-empirical correla-tions are fundamental for first design.

The heat transfer depend on basic parameters: • Viscosity ⌫

• Thermal conductivity k

• Combustion chamber diameter/Cooling Channel geometry D • Flow velocity u

• Density ⇢ • Specific heat Cp

The combination of these dimensional parameters lead to 4 independent dimen-sionless parameters, usually:

1. Reynolds number :

Re =uD

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2. Prandtl number: P r =Cp⌫ k (2.4) 3. Nusselt number: N u =hD k (2.5) 4. Stanton number: St = h ⇢uCp (2.6)

Note that this identity subsists:

St = N u

ReP r (2.7)

Reynolds number is the ratio of inertia force ⇢ud. and viscous force (h). Prandtl number denotes the ratio between frictional dissipation Cp⌫. and

ther-mal conduction k. Nusselt number can be considered the ratio of actual heat transfer rate h and the heat transfer of a conduction process Cp/k. and finally

Stanton number is the ratio of heat transfer h and mass heat transfer ⇢uCp.

Now, i want to find if exists a relation that measures the heat transfer directly and i want to write this with dimensionless parameters.

Considering turbulent flow only because it is the actual flow regime in regen-erative cooling channels, the solution of the Blasius equation in incompressible flat plate flow for the friction coeﬃcient has been found:

cf 2 = 0.029 Re1/5 x (2.8) where x is the distance from the edge.

The general form for the friction factor is : f 2 = C Res x (2.9) where C and s depend only on the specific boundary conditions for the boundary layer.

In the same way is possible to derive, for an incompressible flow between a moving plate and a fixed surface a relation between the friction factor and Stanton number :

St = f/2

P rl (2.10)

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Now combining the last equations (2.9) and (2.10), the desired relation be-tween the dimensionless parameters, that provide the rate of heat transfer:

St = C

Res_{P r}l (2.11)

Using the (2.7) the most commonly used equation for heat transfer, the Nusselt correlation:

N u = C ResP rl (2.12)

From the definition of Nusselt number is possible to to determine the heat transfer coeﬃcient h. The coeﬃcient is directly connected to heat flux through equation (2.1).

Exist diﬀerent Nusselt type correlations. In every cases C is a constant which is determined for each application and the exponents l and s are calculated from boundary layer conditions. [6]

The model today implemented in Thermtest (see Chapter 6 ) for heat trans-fer coeﬃcient in hot gas side and coolant side are based on Cinjarew and Kraus-sold correlations, respectively for the hot gas side and coolant side.

The thesis focuses on the implementation of new models for the heat transfer coeﬃcient on the coolant side, to adapt and check the current predictions of MATLAB®_{code Thermtest, developed by the TUM.}

2.3.1 Cross section of cooling channel

This thesis focused on two types of cooling channels, whit the circular and rectangular geometry(high aspect ratio channel). For the last geometry has become customary to base the Reynolds and Nusselt number on a dimensional parameter which is called “hydraulic diameter” and which is defined by the equation:

Dh= 4A

Pw (2.13)

where A is the cross sectional area and Pw is the hydraulic perimeter.

With some caution, we may use Dh directly in place of the circular tube

diameter for calculation of turbulent heat transfer and skin friction coeﬃcients. The results obtained by substitutingDhfor D in turbulent circular tube formulas

are generally accurate within ±20% and are often within ±10%. [7]

2.3.2 Coolant side heat transfer coeﬃcient Kraussold

In Thermtest a model by Kraussold is implemented:[8]

hc= 0.023Re0.8P rn (2.14)

were n = 0.3 for water with 1 < P r < 14 and liquids with high Prandtl numbers.

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Prediction of heat transfer coefficient for the coolant side with Kraussold does not take into account of many factors can change cooling effectiveness and give coefficients which are in poor agreement with measured values. In addition to the Kraussold model already implemented in Thermtest, three other semi-empirical models will be investigated which allow to evaluate the effect of the roughness, the entrance effects and the turbulent Prandtl effect.

2.3.3 Roughness

Manufacturing and operating conditions are often far from ideal, leading to the duct walls that are rough.

An inside diﬃculty in investigations of the influence of surface roughness is caused by the fact that no acceptable geometric description of a rough surface has been found as yet. It is generally assumed that the parameter of design is the ratio of the average height of the roughness peaks to the tube diameter: "/D.

Nikuradse produced a defined roughness pattern, for tubes with rough sur-faces, by gluing sand of fairly uniform size to the tube surface to form a cover which was made as dense as possible. Friction factors determined in this way are plotted in Moody diagram with Reynolds number on the abscissa and with e/D as parameter.

Figure 2.3: Moody chart for the friction factor in circular tubes. It can be seen that in laminar flow and in turbulent flow with small Reynolds

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numbers, the roughness has no influence on friction factor f.This fact can be thought as a situation in which the roughness peaks are completely integrated in the laminar sublayer. Roughness on a pipe wall can disrupt the viscous and thermal sublayers only if it is suﬃciently large. When the Reynolds number in-creases, the viscous sublayer becomes thin and small and the peaks of roughness influence f .

The roughness Reynolds number is defined: [7] Re" = ⇢u⇤_✏ µ . . . = Re ✏ D r f 8 (2.15)

Experimental data shows that the smooth region (i.e., f depends on Re alone), transitional region (i.e., f depends on both Re and✏_/_D), and fully rough

region (i.e., f depends on✏_/_Dalone) seen in fig: 2.3 corresponds to the following

ranges of Re✏: 8 > < > : Re✏< 5 hydraulically smooth 5 Re✏ 70 transitionally smooth 70 Re✏ f ully rough

for rough pipes to evaluate the friction factor it use an interpolation of the Moody’s diagram, correlation of Colebrook:

1 p f =−2log  0.27✏ D + 2.51 Repf (2.16)

it valid also in turbulent range.

The Colebrook equation is usually solved numerically due to its implicit nature.[16] 1 p f =−2log  ✏ 3.71D 5.042 Re log ✓ 1 2.8257 ⇣ ✏ D ⌘1.1098◆ + 5.8506 Re0.8981 (2.17)

In the fully rough regime, can we use the correlation of Bhatti and Shah for the local Nusselt number:[9]

N u = (f /8)ReP r

1 +pf /8(4.5Re0.2

✏ P r0.5 8.48)

(2.18) that can be applied for Re > 104_{, 0.5  P r  10 and 0.002 }✏_/_D_{ 0.05.}

Finally the roughness has the eﬀect of increase heat transfer and skin friction, the heat transfer coeﬃcient on a rough wall can be several times that for a smooth wall at the same Reynolds and Prandtl number. The friction factor, and thus the pressure drop will also be higher.

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2.3.4 Turbulent flow (role of turbulent Prandtl number

for heat transfer coeﬃcient)

Despite many years of intensive research into turbulent diffusion, it is still poorly understood and can only be rather crudely predicted in many cases [Philip and Webester, 2003]. Because of highly complex turbulent flow mechanism, the prediction of the transport rates necessarily involves the formulation of conceptual models which embody many simplifying assumptions . Turbulent Prandtl number plays an important role in determining the value of heat transfer coefficient and the assumption of unity turbulent Prandtl number is very poor for fluids of P r > 1, i.e., the momentum eddy diffusivity is not equal to thermal eddy diffusivity.

Thermal transport mechanisms are related by a factor, the Prandtl number, combining the molecular and eddy viscosities one obtain the Boussinesq relation for shear stress and heat flux

⌧ ⇢= (⌫ + ✏m) du dy (2.19) ˙q ⇢Cp = (! + ✏h) dT dy (2.20)

where ! is molecular thermal diﬀusivity , ✏h, ✏m are respectively the eddy

diﬀusivity for heat transfer and heat diﬀusivity for momentum transfer and y is a distance from the wall.[10]

The turbulent Prandtl number is the ratio between the momentum and thermal eddy diﬀusivities and the equation (2.20) can be written as:

˙q

⇢Cp = (↵ + ✏h/P rt)

dT

dy (2.21)

introducing the dimensionless parameters: 1. u⇤₌q⌧w ⇢ ; 2. u+₌u_/_u⇤; 3. y+₌ yu⇤ ⌫ ; 4. R✏= q f 8Re

it can write (inserisci riferimento calcolo) the Nusselt number that regulates the heat exchange as:

N u = p f /2ReP r ´R✏ 0 dy+ 1_/P r+₍✏m_⌫ ₎/P rt (2.22) Hence , if the variation of m/ with y+ and P rt are known, the value of Nu

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The eddy diffusivity in the viscous sublayer, damped turbulence layer, and turbulent core affect greatly the rate of heat (or mass) transport between the wall and bulk. Previous studies(metti bibliografia) showed that the eddy diffusivity is function of many variables such as the distance from the wall (y+), Re, and Pr. Most studies showed that in the near wall region ✏m/⌫ proportional to y3.

The region of study is divided into three zones for momentum eddy diﬀusivity variation:

Sublayer:

✏m

⌫ = 0.0064Re

0.322_y3_{0 < y}+_{< y}1+ _(2.23)

where y1+_{is the distance beyond which the eddy diﬀusivity becomes linear}

with yrather than y3_{variation. From semi empirical data y}1+_{= 8.4Re}0.161.

Buﬀer Zone:

✏m

⌫ = 0.45y y

1+_{< y}+_{< y}2+ _(2.24)

Region with linear variation of eddy diﬀusivity where y2+_{= 0.156Re}

Turbulent Core:

✏m

⌫ = 0.07R✏ (2.25)

In this region the large eddies in the core region and the small variation in turbulent intensity in the central region makes the eddy viscosity constant.

The expression for P rt is obtained from experimental results of Friend and

Mitzner with an iterative procedure choosing for each time a diﬀerent value of Re and Pr.

with this correlation is determined the average turbulent Prandtl number. The following relation is obtained:[11]

P rt= 6.374Re 0.238P r 0.161 (2.26)

The formula shows that for liquids of P r > 1 the P rt is less than one

indi-cating that the thermal eddy diﬀusion is larger than momentum eddy diﬀusion. Also P rt decreases with increasing Re and P r indicating that the increase in

thermal eddy diffusion is larger than that in momentum eddy diffusion. The expressions of momentum eddy diffusivity (2.23); (2.24) ;(2.25) and turbulent Prandtl number (2.26) are substituted in (2.22) and obtain the Nusselt number and then the coefficient of heat exchange.

2.3.5 Entrance eﬀect ( M.F.Taylor semi-empirical

corre-lation for Nusselt and heat transfer coeﬃcient)

Kraussold and Bhatti and Shah give correlations often favorite because of their simplicity. for geometry of the coolant passage noncircular , high temperature or high heat fluxes and wall to coolant temperature ratios of the order of 2.5 to 1 the assumption of constant physical properties is no longer valid, and the

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predicted heat-transfer coeﬃcients can be in considerable error the order (of the order ofpTb/Tw.[12]

Within of the entrance region hydraulic and thermal boundary layers gen-erally develop simultaneously. The heat-transfer and flow characteristics in this entrance region diﬀer from those in a fully established turbulent boundary layer. In first segment of the chamber combustion,non fully developed flow occupy a considerable portion of the cooled passage length.

Taylor model gives a correlation which is valid over the complete length of a tube (including the entrance region), as well as for wide variations in the physical properties. At the classical formulation of the Nusselt number is added to a coeﬃcientx_/_Dwhich is the ratio of distance from entrance of the test section

to inside diameter of test section. The success demonstrated in predicting heat transfer coeﬃcient in straight with tubes for turbulent single phase hydrogen flow, encourages the use of this equation in predicting heat transfer coeﬃcients in the coolant passages of a rocket chamber combustion.

The Taylor model started with a classical formulation of Nusselt number:[13] N u = 0.023Re0.8b P r0.3b ✓ Tw Tb ◆ C2 (2.27) where C2 is a function ofx/D and Tw and Tb are respectively the wall and

the bulk temperatures.

To determine exponent C2 , at each x/D , the exponent of the wall to fluid

temperature ratio was computed so that the heat-transfer coeﬃcient calculated by equation (2.27) was equal to the experimental heat-transfer coeﬃcient ana-lyzed in 11 previous studies.

The computed exponents determined for all the experimental data were curve fit by using first-, second-, third-, and fourth-order polynomials in D_/_x.

The fourth-order polynomial gave the best fit but was only slightly better than the more easily used first-order fit:

C2= 0.57 1.59x D

(2.28) Equation (2.28) and (2.27) are combined to give:

N u = 0.023Re0.8b P rb0.3 ✓ Tw Tb ◆ ✓0.57 1.59 x D ◆ (1+F1Dx) (2.29) The correlation approximates well the behavior of the thermal exchange coeﬃcient at all points of the channel with an entrance straight.

If the manifold have a angle bend entrance of 45° or 90° withx_/_D< 5:

N u = 0.023Re0.8b P rb0.3 ✓ Tw Tb ◆ ✓0.57 1.59 x D ◆ (1+F1Dx) (2.30) where F1 = 3.5 for the angle the 45° angle bend and F 1 = 5 for the 90°

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y

Turbulent Core

Buffer Zone

Sublayer

Figure 2.4: Division region of interest for eddy diﬀusivity variation Method developed by Taylor to compute a heat transfer coeﬃcient need of a profile of a local bulk temperature and a local profile for wall temperature.

To take into account the entrance effects we have to make an iterative cal-culation. The bulk temperature and the wall temperature profiles are directly taken from the analysis of the thermal stratification (see Chapter 3-4) wherein has been used the mean heat transfer coefficient from analysis of Kraussold. Hence will be calculated the new coefficient of local heat exchange with the formula proposed by Taylor . Will be in turn used these local coefficients for the calculation of the temperature at the wall and into the coolant according to the method developed in Chapter 3-4 for thermal stratification

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Chapter 3

Thermal stratification model

The objective of the present chapter is to developing a computational strategy that is able to describe the coupled hot-gas! wall!coolant environment that occurs in liquid rocket engines, and to provide a quick and reliable prediction of thermal stratification phenomena in circular and rectangular cooling channels.

This approach, which is a simplification of that presented by Pizzarelli [15], is still widely relying on empirical relationships. It allows to compute the radial stratification of both the wall and the coolant flow temperatures. This result is obtained by considering the one dimensional steady state evolution of the hot gas flow which was calculated in Thermtest and coupling the result, in terms of heat flow hot gas, with a “quasi 2-D” flow evolution through the cooling channels.

The approach for “quasi 2D” model is based on the idea to consider the one-dimensional approach for the coolant mass and momentum equation, while coolant and wall energy equations are developed with two-dimensional equa-tions. For simplicity it is assumed steady-state condition and thus hot-gas, wall, and coolant behaviors are coupled by heat balance from hot gas to wall and from wall to coolant. The model is developed for any fluid evolving through cooling channels, by considering any equation of state, and thus compressible gas, supercritical fluid and liquid can be considered as coolants.

3.1 Transmission of heat into the cooling channel

The problem of steady-state heat transferred into cooling channel follows the dynamics : the heat is transferred from the hot gas to the coolant via conduction in solid: internal wall, ribs and external wall.(See fig:3.1). At steady state conditions the heat flow must be equal to that leaving .

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Heat flux from hot gas goes to the coolant from the bottom wall and to the rib. The steady-state wall heat transfer balance can be written:

qw,hg= kw

Sw

(Tw,hg Tw,co) (3.1)

where kw and Sw are the wall thermal conductivity and thickness, Tw,co is the

coolant side wall temperature. For internal wall has been considered a one-dimensional radial heat transfer . Then, the heat transfer balance through the thickness is calculated by assuming again steady state operation:

@ @y ✓ kwtw@Tw @y ◆ = 2qw (3.2) and qw= hw,co(Tw T ) (3.3)

where Tw is the wall temperaturey is the radial direction,qwis the heat flux

from the fin to the coolant equal at the heat flux to the hot gas and tw is the

fin thickness.

Note that this equation assumes a non uniform fluid temperature and a fin thickness that is much smaller than its axial length and infinitely tall.

To close the problem are now required boundary conditions, respectively at the bottom of the rib y = 0 where the radial heat flux entering the rib is equal to that entering the wall from the hot-gas side, and at the top of the rib y = h where the external wall is assume adiabatic:

qw,hg= kw @Tw

@y _y=0

0 = kw

@Tw

@y _y=h

3.2 Equation for the coolant flow

The cooling channel flow model is developed by using the steady-state conser-vation laws of mass and energy, taking into account the eﬀects of heat transfer. The mass governing equation is written in a one-dimensional form, whereas a simplified arrangement of the 2-D energy equation is considered because the thermal stratification is strongly pronounced in the radial direction. Hence the fluid energy equation can be reduced to a 2-D balance in the stream wise direc-tion x and radial direcdirec-tion y. Moreover, the transverse velocity components v and w are neglected because they are much smaller than u.

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3.2.1 Mass equation

The steady-state integral mass conservation equation for the channel cross-sections is: d dx ¨ A ⇢udA = 0 (3.4)

where A is the cross section area, ⇢is the constant coolant density and xis axial direction in the channel.

Moreover ˙m (mass flow rate) is considered constant along the axial direction ˙

m = ¨

A

⇢udA = const (3.5)

Introducing a quasi 2-D version of the model at the mass equation, it is possible express a middle height velocity as:

¯ ⇢umA = ˙m (3.6) where ¯ ⇢ = 1 A ¨ A ⇢dA (3.7) um= m˙ ´h(x) 0 ⇢b(x, y)dy (3.8)

3.2.2 Energy equation

Integral equation steady state for energy in the channel cross section is: d dx ¨ A (⇢uH0)dA dx = ¨ Sw qwdSw (3.9) Where: ( H0= h +u 2 2 h = CpT (3.10) is the total enthalpy and qw is the heat flux entering in coolant through the

inner wall whit thickness Sw.

Thermal stratification depends on the heat flux through the coolant in radial direction and on the heat flux exchanged with the ribs walls.

The channel is split in radial direction in tiny slices of height dy they having the same abscissa x and the same valued of u . To solve for T (y), the balance equation (3.9) is modified . The equation is to be written for a slice of height dyrather than for the whole channel height.(See fig:3.2.2)

@

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dy qw dx b qc(y+dy) qw qc(y)

Figure 3.2: Balance of heat flux converging in a slice channel dy and constant width b

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If it is possible express qw(y)and qw(y)as a function of T (y), the equation

(3.11) becomes a diﬀerential equation for the unknown variable T .

Assumed the turbulent mixing conditions in according to Kanynski [17] qc(y)

is expressed by:

qc(y) = kt@T

@y (3.12)

Where kt is the turbulent conductivity in the non shear direction. The ratio

of kt to the fluid thermal conductivity can be obtained as a function of the

Re.(The relation is derived in the appendix A) kt k = 0.0053Re 0.9 D P r (6.374Re 0.238_{P r} 0.161₎1.5 (3.13)

The energy balance equation becomes: @ @x(⇢H0ub) = @ @y ✓ ktb@T @y ◆ + 2qw (3.14)

and the heat flux is related to the unknown coolant and wall temperature using:

qw= hw,co(T Tw) (3.15)

hw,co is the convection heat transfer coeﬃcient estimated by means of the

Nusselt number with the semi-empirical correlations it can be viewed in the Chapter 2.

To close the problem, the boundary conditions for the energy equation are the same used to find the thermal profile in the wall (3.2). The conditions state that the heat flux qw,hg enters at the channel bottom and the channel top is

adiabatic. This conditions are in accord whit the hypothesis of axisymmetric temperature distribution on internal and external walls.

qw,hg= kt @T

@y _y=0

0 = kt @T

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Chapter 4

Numerical method

The governing equations can be discretized considering a 2D grid: M nodes (i = 1, ..., M ) for the axial discretization axis x and N nodes (j = 1, ..., N) for the radial discretization axis y . The computations proceed started from the entrance of the coolant at the manifold and moving along the axial direction. The solution at each axial position is computed with a cross and iterative method from that at the previous one.

The governing equations are written for each value of i, assuming known the solution at the previous axial position (i 1 , or the channel inlet condition).

To solve the system of equations, the following computation strategy is used at each axial station:

1. From Thermtest are picked up the values of hot gas heat flux which con-stitute the values for y = 0 for each axial section.

2. The partial diﬀerential equation (3.2) is solving considering T from j 1 position.

3. The value of Tw calculated with step 2 is used for solved equation (3.14)

4. The new value of T is used for step 2 and the procedure is repeated until to complete the mesh.

4.1 Finite Diﬀerence Method

Numerical methods are useful for solving fluid dynamics, heat and mass transfer problems, and other partial-diﬀerential equations of mathematical physics when such problems cannot be handled by the exact analysis because of nonlinearities, complex geometries, and complicated boundary conditions.

The finite difference method is one of several techniques for obtaining nu-merical solutions to partial differential equation (PDE).The method consist in replace the partial differential equation with a discrete approximation. “Dis-crete” means that the numerical solution is known only at a finite number of

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points in the physical domain. The number of those points can be selected by the user of the numerical method. In general, increasing the number of points not only increases the resolution (i.e., detail), but also the accuracy of the numerical solution.

Finite-diﬀerence methods are simple to formulate, can be readily extended to two or three-dimensional problems, and are very easy to learn and apply to the solution of partial-diﬀerential equations encountered in the modeling of engineering problems.[18]

The mesh is the set of locations where the discrete solution is calculated. These points are called nodes, and if one were to draw lines between adjacent nodes in the domain the resulting image would resemble a net or mesh. Two are parameters of the mesh x, the local distance between adjacent points inx direction, and y, the local distance between adjacent points iny direction. For the simplicity x and y are uniform throughout the mesh.

The core idea of the finite-diﬀerence method is to replace continuous deriva-tives with so-called diﬀerence formulas that involve only the discrete values associated with positions on the mesh.

In fig:4.1 the mesh used for computed the solutions of equations of the prob-lem; the open squares indicate the location of the (known) initial values. The solid squares indicate the location of the (known) boundary values. The open circles indicate the position of the interior points where the finite diﬀerence approximation is computed.

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x=0 y=0 j j-1 j+1 i i-1 i+1 1 N M y=h x=L

Figure 4.1: Mesh on a semi-infinite strip used for the solution of PDE equations The finite diﬀerence method for the generic function (x) using discrete approximations is:

@ @x t

i+1 i

x (4.1)

where the quantities on the right hand side are defined on the finite difference mesh. Approximations to the governing differential equation are obtained by replacing all continuous derivatives by discrete formulas such as those in (4.1). Exist various types of finite difference approximations for first order or sec-ond order; scheme forward, backward o central. To compute the different prob-lems these scheme are assembled into a convergent discrete approximation. Both

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the space derivatives are replaced by finite diﬀerences. Doing so requires spec-ification both spatial locations of the function values in the finite diﬀerence formulas.

4.2 Crank Nicolson method

Up to now we know that the finite diﬀerence method are a means of obtain-ing numerical solutions to partial diﬀerential equations which describe physical phenomena.[18]

The most common finite diﬀerence methods for the solution of partial dif-ferential equations are:

• Explicit method • Implicit method • Crank Nicolson method

These methods are closely related but diﬀer in stability, accuracy and execution speed. In the formulation of a partial diﬀerential equation problem, there are three components to be considered:

• The partial diﬀerential equation

• The grid on which the partial diﬀerential equation is required to be satis-fied.

• The boundary conditions and initial conditions to be met.

Crank Nicolson method is used for solving parabolic partial diﬀerential equa-tions, was developed by John Crank and Phyllis Nicolson in the middle 20th century used for numerically solving the heat equation and similar partial dif-ferential equations. For diﬀusion equations and other similar equations, the Crank–Nicolson method is unconditionally stable.[25]

Crank-Nicolson scheme is based on the trapezoidal rule, is a combination of the explicit Euler scheme and implicit Euler scheme.

uj+1_i un i x = F n i ✓ u, y, x,@u @y, @2_u @x2 ◆ (f orward Euler) uj+1_i un i x = F n i ✓ u, y, x,@u @y, @2_u @x2 ◆ (backward Euler) uj+1_i un i x = 1 2  Fin+1 ✓ u, y, x,@u @y, @2_u @x2 ◆ + Fin ✓ u, y, x,@u @y, @2_u @x2 ◆ (Crank N icolson)

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j j+1

i i+1

i-1

Figure 4.2: The Crank-Nicolson stencil for 2D problem

In this case the equation which determine the thermal stratification are: @ @y ✓ kwtw@Tw @y ◆

= 2qw Equation f or rib temprerature prof ile

@ @x(⇢ubcpT ) = @ @y ✓ ktb @T @y ◆

+ 2qw Equation f or coolant temperature prof ile

Now replace the partial derivative occurring in the partial diﬀerential equa-tions by approximaequa-tions based on Taylor series expansions of function near the point or points of interest.

After applying these equations to all the nodes of grid, shall obtain a system with tridiagonal coeﬃcient matrix.

↵ " Ti+1 wj+1 2T i+1 wj + T i+1 wj 1 2 y2 ! + T i wj+1 2T i wj + T i wj 1 2 y2 !# 2hw(Twij T i j) = 0 (4.2) Tji+1 Tji x ! = "

Tj+1i+1 2Tji+1+ Tj 1i+1

2 y2 ! + T i j+1 2Tji+ Tj 1i 2 y2 !# +2hw(Tji Twij) (4.3)

where ↵ = kwtw; = ⇢ubCp and = ktb

The above equations can be written in a matrix system equations as: (

A1Tw = B1c 2hwT

A2T = B2d + 2hwTw

(4.4) where c ; d are vectors of known terms, that is, the values of T and Tw at

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4.3 Crank Nicolson method for cylindrical

cool-ing channel

The application of the Crank-Nicolson method for finite diﬀerencing in cylin-drical symmetry for round cooling channel is now illustrated.[18]

The constituent equations of physical problem are illustrated in algorithm 4.3

Where p = 1 for the cylindrical symmetry and i = 2 . . . M 1 are internal node of mesh.

4.3.1 Stability of Crank-Nicolson method

To convergence of this method is the condition [27]: x kw

y2 <

1

2 (4.5)

1_/₂descends to Von Neumann stability analysis.

However, that the ratio of spatial step x times the thermal diﬀusivity to the square of space step, y2_{, is large.}

The Crank Nicolson Method is unconditionally stable and has higher order of accuracy. The price of solving a tridiagonal system at each step is worth paying since this method allows large step sizes. [26]

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Algor itmo 4.1 Tr id ia go na lm at ri x fo r th er m al st ra ti fic at io n di ﬀe re nc e m et ho d sc he m e 2 6 6 6 6 6 6 4 2↵↵ 0 .. . 0 ↵ 2↵↵ .. . 0 0 ↵ 2 ↵ . .. 0 . . . . . . . .. . .. ↵ 000 ↵ 2↵ 32 7 7 7 7 7 7 56 6 6 6 6 4 T i+1 w1 T i+1 w2 T i+1 w3 . . . T i+1 wn 3 7 7 7 7 7 5 = 2 6 6 6 6 6 6 4 2↵ +2 hw ↵ 0 .. . 0 ↵ 2↵ +2 hw ↵ .. . 0 0 ↵ 2↵ +2 hw . .. 0 . . . . . . . .. . .. ↵ 000 ↵ 2 ↵ +2 hw 32 7 7 7 7 7 7 56 6 6 6 6 4 T i w1 T i w2 T i w3 . . . T i wn 3 7 7 7 7 7 5 2 6 6 6 6 6 6 4 +2 0 .. . 0 +2 .. . 0 0 +2 . .. 0 . . . . . . . .. . .. 000 +2 32 7 7 7 7 7 7 56 6 6 6 6 4 T i+1 1 T i+1 2 T i+1 3 . . . T i+1 n 3 7 7 7 7 7 5= 2 6 6 6 6 6 6 4 2 +2 hw 0 .. . 0 2 +2 hw .. . 0 0 2 +2 hw . .. 0 . . . . . . . .. . .. 00 0 +2 32 7 7 7 7 7 7 56 6 6 6 6 4 T i 1 T i 2 T i 3 _{. . . T}i n

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Algor itmo 4.2 Cy lin dr ic al Cr an k N ic ol so n eq ua ti on s ↵ 2 y 2 h⇣ T i+1 wj +1 h 1 p _2i i 2T i+1 wj + T i+1 wj 1 h 1+ p _2i i⌘ + ⇣ T i wj +1 h 1+ p 2i i 2 T i wj + T i wj 1 h 1 p _2i i⌘i 2 hw (T i wj T i j)= 0 T i+1 j T i j x ! = 2 y 2 h⇣ T i+1 j+1 h 1 p 2i i 2T i+1 j + T i+1 j 1 h 1+ p _2i i⌘ + ⇣ T i j+1 h 1+ p 2i i 2 T i +j T i j 1 h 1 p ₂i i⌘i +2 hw (T i wj T i j)

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4.3.2 Boundary Condition implementations for Crank

Nicol-son

The general form of the boundary conditions for T and T w are Neumann conditions:[19] 8 < : ˙qw,hg= kt @T_@y y=0 0 = kt @T_@y y=h f or T (4.6) 8 < : ˙qw,hg= kw @T_@yw y=0 0 = kw @T_w_@y y=h f or Tw (4.7)

The simple and immediate implementation is to replace the derivate in equa-tion (4.6) and (4.7) with a first order diﬀerences:

8 < : T2i+1 T i+1 1 y = q˙w,hg/k2 T2i+1 T i+1 1 y = 0 (4.8) 8 < : Ti+1 w2 T i+1 w1 y =q˙w,hg/kw Tw2i+1 T i+1 w1 y = 0 (4.9) This choice is wrong! In fact the first order diﬀerences approximation has a spatial accuracy of O( x).

We must introduce a ghost node, image a node ˜T0or ˜Tw0 that are outside of

the grid of domain.This valued not explicitly appear in the numerical scheme.

j

j+1

Ghost node

∆y

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Use a central diﬀerence approximation at j = 0 to impose the boundary conditions. Ti 2 T0i 2 y = ˙qw,hg k2 (4.10) Hence: ( Ti 0= T2i+ 2 y ˙ qw,hg k2 Ti w0= Tw2i + 2 y ˙ qw,hg k2 at y = 0 (4.11) ( Ti N = TNi 1 Ti wN = TwNi 1 at y = h (4.12)

The finite diﬀerence equation for first node and the last node are respectively for Tw and T :

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Algor itmo 4.3 Eq ua ti on fo r the fir st and la st no de resp ec ti vel y fo r Tw an d T 8 <h : 2 ↵ 2 ↵ 0 .. . 0 i T i+1 w1 = h 2 ↵ +2 hw 2 ↵ 0 .. . 0 i T i w1 2 hw T i 1 2 y kw ↵ ˙qw, h g (i ) +˙ qw, h g (i +1 ) h 0 .. . 02 ↵ 2↵ i T i+1 wN = h 0 .. . 0 2↵ 2↵ +2 hw i T i wN 2 hw T i N 8 < : h +2 2 0 .. . 0 i T i+1 1 = h +2 hw 2 2 0 .. . 0 i T i w1 2 hw T i w1 +2 y kw ↵ ˙qw, h g (i ) +˙ qw, h g (i +1 ) h 0 .. . 0 2 +2 i T i+1 N = h 0 .. . 0 2 +2 hw 2 i T i N 2h w T i wN

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Chapter 5

Workbench combustion

chambers

At Lehrstuhl für Flugantriebe of TUM (Institute for Flight Propulsion ) are located two high pressure combustion chamber test facility TCN-T38-01-2 and RF_TP1BBC. The first combustion chamber is composed of three segments characterized by cylindrical cooling channels while the second combustion cham-ber is formed by four segments, the first is the longest is equipped whit rectan-gular cooling channels like a typical rocket engine designs in order to provide a good comparability to full scale applications. The other three segments are characterized once again by circular cooling channels . Now will be presented a general description of chamber component with some geometric parameters and operating conditions. After will be presented the main equipment used for data collection.

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Water manifolds

Figure 5.1: Combustion chamber.Are marked water manifolds which constitutes the refrigerant to the experiment.

Both the combustion chambers present a modular setup, that consist in segments of diﬀerent length, which allows for an easy change of single modules. By the combination of short and long chamber segments, diﬀerent values of axial and characteristic length can be realized.

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Figure 5.2: Setup of TCN-T38-01-2 chamber test composed by two short seg-ments and one long segment

The cooled combustion chamber and nozzle segments are fabricated from oxygen-free copper, which has a certified yield strength of 289.0 MPa; the man-ifolds are made of high-temperature stainless steel.

Now are shown the main geometric parameters of the combustion chambers and cooling channels TCN-T38-01-2 and RF_TP1BBC and an overview over the nominal operating points and the maximum

operating conditions.

Description Symbol Value Unit

Chamber diameter Dcc 37 [mm]

Throat diameter Dth 16.53 [mm]

Segment Number [ ] 3 [ ]

Chamber length Lcc 443 [mm]

Segment length LLshort1short2 94.599.5 [mm][mm]

Llong 190 [mm]

Cooling channel diameter dcc 4 [mm]

Cooling channel number [ ] 36 [ ]

Minimum distance hot gases Sw 1 [mm]

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Description Symbol Value Unit Chamber diameter Dcc 30 [mm] Throat diameter Dth 19 [mm] Segment Number [ ] 4 [ ] Chamber length Lcc 356.5 [mm] Segment length Lshort1 77 [mm] Lshort2 77 [mm] Lshort3 42 [mm] Llong 160.5 [mm]

Cooling channel rectangular dimension wcc 1.5 [mm]

hcc 3.5 [mm]

Cooling channel rectangular number [ ] 34 [ ]

Minimum distance hot gases Sw 1.5 [mm]

Table 5.2: Combustion chamber RF_TP1BBC geometric main features

Chamber pressure Pcc,nom 8.0 [M P a]

Pcc,max 10 [M P a]

Maximum oxidizer mass flow (GOX) mOX˙ 1.00

h

kg s

i

Maximum fuel mass flow m˙F 0.6

h kg s i Mixture ratio O_/_F _1.4 _3.5 hkg s i Combustion temperature Tcc 2300 3800 [K]

Table 5.3: Combustion chamber TCN-T38-01-2 main operating conditions

Chamber pressure Pcc,nom 8.0 [M P a]

Pcc,max 10 [M P a]

Maximum oxidizer mass flow (GOX) m˙OX 0.932

h

kg s

i

Maximum fuel mass flow m˙F 0.274

h kg s i Mixture ratio O_/_F _3.4 hkg s i Combustion temperature Tcc 3600 [K]

Table 5.4: Combustion chamber RF_TP1BBC main operating conditions

5.1 Measurements

To describe the injection conditions,the temperature and pressure field in com-bustion chamber segments the experimental apparatus are equipped with a var-ious type of sensors.

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A regular positioning of pressure transducers provides measurement of the wall pressure along the chamber axis gives information about the acceleration of the hot gas hence on the heat release.

Static pressure measurement are provided with the sensors that having a measurement range from 0 to 100 bar, an over pressure limit of 200 bar and operated at a data acquisition rate of 1500Hz.

pressure transducers

Figure 5.3: Sub-scale combustion chamber and pressure sensors location An other type of dynamic pressure transducers are installed in the oxidizer feed line,in the fuel injector manifold and in the combustion chamber wall.

To determine the temperature field,the chamber segments are equipped with 90 thermocouples spring mounted in the chamber wall. The spring loading of the thermocouples will provide a constant force to ensure a continuous contact between the thermocouples tup and the base of the hole.

In the first segment the thermocouples are mounted on four azimuthal po-sitions to monitor the behavior of the adjacent external injectors an the one positioned at the centre of the chamber. A regular pattern along the chamber axis provides information about the combustion.

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Figure 5.4: Position of the thermocouples in the chamber segments In the second segment the thermocouples are installed on six azimuthal positions to have a complete map of the external injectors behavior .

The water manifolds are equipped with temperature and pressure sensors. Moreover,in order to characterize the behavior of the injectors, the temperature measurement are displaced on six azimuthal positions to map all the injectors and a further sensor is positioned between the two injectors.

Below are some pictures of the test bench and a short segment of the com-bustion chamber

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Chapter 6

Heat transfer

analysis(Thermtest)

TUM developed a computational program Thermtest in order to predict and analyze the heat transfer. Thermtest is the in house MATLAB®_{written code}

that predicts the heat transfer from hot combustion gases in the water coolant to the external wall of the combustion chambers, of TUM test facility. This program is mainly used for test preparation and analysis.

The preparation of the tests on a new cooled combustion chamber requires the capability of prevision the expected thermal loads on the chamber . The convective heat transfer is calculated by semi-empirical Nusselt correlations. The advantage of this approach is a immediately solution and prediction of the most important parameters compared to more sophisticated CFD codes.

The heat transfer within the cooled structure of the combustion chamber can be subdivided into these subproblems :[20]

• prediction of fluid properties, gas composition ,pressure and temperature of the combustion and the products combustion (hot gas);

• evaluation of the heat transfer; thereafter the heat transfer coeﬃcient at the hot side chamber and at the cold wall. Considering also into account fractions driven by radiation as well as convection;

• heat conduction in the chamber walls. Eﬀects caused by curvature of walls, fins, holes and cavities, diﬀerent chamber materials and local heat have been considered;

• heat transfer into the coolant. The knowledge of the fluid behavior espe-cially in the case of cryogenic,unstable coolants are be of great importance; • heat transfer from the chamber coating to the external ambient.

Thermtest features an unsteady thermal transfer model using a FV or FD-like formulation on an arbitrary structured mesh, a fully-3D description of the

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Natural Convection Radiation

Coolant heat transfer Kraussold model

Convective heat transfer Cjnirawev modified Radiateve heat transfer

Shack Formula

Hot gas fluid properties Gordon-McBridge equation External ambient Chamber wall Cooling channel Combustion chamber

Figure 6.1: Sketch of heat transfer model of Thermest

rocket chamber, which might consist of an arbitrary set of metallic and ceramic materials, and an implicit discretization in time and space.

Thermtest utilizes one-dimensional hot gas properties acquired from NASA computer program CEA2 of S. Gordon and B. McBride [22]. The temperature of the flow and the ideal characteristic velocity are calculated using the built-in rocket problem. In CEA2 the evolution along z-axis is taken into account, but there are not reaction kinetics.

Thermal features of the fluid near the wall are calculated assuming an equi-librium composition for the temperature-pressure problem. The convective heat flux from the hot gas to the inner wall and from the wall to the coolant is mod-eled using Nusselt correlations. The hot wall convective coeﬃcient is calculated from a formulation proposed by Cinjarew:

N u = 0.0162(ReP r)0.82 ✓ Thg Tw ◆0.35 (6.1) hhg = 0.01975 ⌫0.18₍_{m Cp)}_˙ 0.82 D1.82 ✓ Thg Tw ◆0.35 (6.2)

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Figure 6.2: Heat flux vs. axial coordinate calculated by Thermtest for chamber combustion with rectangular cooling channel

The temperature of the hot gas is the so called recovery temperature that takes into account imperfect combustion and incomplete heat recovery:

Thg= Thg,static+ ⇠(Tth,combustion⌘2c⇤ Thg,static) (6.3)

⇠is the empirical factor depending on the Prandtl number and varying from 0.7 to 0.9 and by boundary layer. Thermtest used 0.8 which well approximates the real behavior.

To take into account the heat flux by radiation Thermtest assuming that the eﬀective outer diameter of the imaginary cylinder filled up with hot gas is virtually equal to the inner chamber diameter . The heat transfer coeﬃcient for radiation can be defined in this way:

hhg,rad= ⇣ s 1 ✏w + 1 ✏hg 1 ⌘ (Tw+ Thg) Tw2+ Thg2 (6.4) s= 5.67· 10 8 ⇥ _W m2_K4 ⇤

Thermtest for the emissivity of the hot gas used a the empirical formulation by Shack [21]. For water and carbon dioxide following equations are given for the thermal heat flux:

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Figure 6.3: Heat flux vs. axial coordinate calculated by Thermtest for chamber combustion with cylindrical cooling channel

˙qH2O= 5.74 (Pccrcc) 0.3✓Tgas 100 ◆3.5 (6.5) ˙qCO2= 4 (µCO2P cc rcc) 0.3✓Tgas 100 ◆3.5 (6.6) To calculate heat transfer from the chamber wall to the cooling channel Thermtest used a correlation by Kraussold, which can be viewed in the above chapters.

6.1 Geometry

Thermtest implemented a three-dimensional geometry for combustion chamber while the cooling mesh is two-dimensional. The mesh is created by a dedicated script highly modifiable.

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Figure 6.4: Geometry of combustion chamber for chamber combustion with rectangular cooling channel

The entire geometry includes the manifolds and the water cooling system (the red lines in fig:6.4; 6.5). Diﬀerent materials properties are implemented in library of Thermtest.

Figure 6.5: Geometry of combustion chamber for chamber combustion with cylindrical cooling channel

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6.2 Solving

Thermtest solves the various diﬀerential equations with approach of finite dif-ference method creating a sparse coeﬃcients matrix that is solved with an MATLAB®_method.

The input data for Thermtest are summarized in fig:6.6. The parameters are divided in inputs fixed and variable. The fixed parameters, common to several simulations, are:

• combustion chamber geometry and materials; • cooling channels geometry;

• coolant properties (water for the regenerative cycle);

• free-stream fluid properties from the solution of the RAK-problem with CEA2;

• hot wall fluid properties, calculated with the CEA2 TP-problem[23]. The variable input parameters are defined by test has to be simulated, they could be diﬀerent between each test and are:

• combustion chamber pressure Pc;

• mixture ratioO_/_F;

• combustion eﬃciency ⌘c;

• duration of the simulation.

Solid properties Cooling channels

geometry Combustion chamber

geometry Hot fluid properties

at wall Propellant fluid properties Fluid properties at freestream THERMTEST

FIXED PARAMETERS VARIABLE PARAMETERS

Mixture ratio Combustion chamber pressure Times Combustion efficiency Coolant properties

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By library present in Thermtest will be loading all the initial parameters like mesh file, gas, coolant and solid property ,the solving loop start. After that it begins the discretization , the first result will be the free-stream tempera-ture. Then the recovery temperature and the static hot gas temperature . After choosing the semi empirical models available , the hot wall heat transfer coef-ficient is calculated. Now Thermtest considering free convection and radiation, calculates the cold wall and using the Kraussold model the heat transfer coeﬃ-cient between solid and cooling channels is obtained. Finally will be generated and solved a matrix of dependencies and prepared itself for the next time-step.

The Thermtest output parameters are:

• combustion chamber temperature, function of time, radial, angular and axial coordinates,

• cooling fluid temperature, function of time and axial coordinate, • cooling fluid pressure,

• hot and cold wall heat flux, function of time and axial coordinate, • maximum wall temperature.

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Chapter 7

Results and Discussion

The three heat transfer models viewed in Chapter 2 will be analyzed and com-pared with model used by Thermtest. After that each correction will be applied to the analyzed load points to check its behavior and finally find out which diﬀerences exist with the model now used by the program.

Valuation of heat transfer we viewed is based on parameters that characterize the flow conditions and the flow molecular properties.

The relations used are a combination of dimensionless parameters like a Prandtl molecular number that characterize the intrinsic properties and the Reynolds number which provides an estimate of the operating conditions.