Modeling and Design

Chapter Three Modeling and design

In this chapter, We are in interested in developing a designing sequence with general parameters for most parts of monopropellant thruster. The design procedure includes : nozzle design, catalytic bed calculations, valve and regulator selection, tank calculations , and injectors calculations. As discussed in pervious section

-:Exhaust gas properties 3.1 I)Nitrous Oxide( ) properties:

The equation of reaction for nitrous oxide ( 3.1.1)

The mole fraction can be estimated by (3.1.2.a,b)

By using mass fraction one can get mole fraction (3.1.3.a,b) get gas constant

(3.1.4)

Then to determine chamber temperature(Tc), so apply balance of the energy equation as follows:

(3.1.5)

:Where for reactants (3.1.6)

:And for products of reaction (3.1.7)

30 by equating the above equations and integrating as shown (3.1.8)

(3.1.9)

From above equations , temperature of reaction in catalytic bed Tc is estimated , where Ts is standard or ambient atmosphere, the only unknown will be Tc other properties can be obtained now as follows: (3.1.10)

(3.1.11)

(3.1.12)

(3.1.13)

(3.1.14)

Similarly, hydrogen peroxide properties could be gotten but with a slight :change as follows

II)Hydrogen Peroxide properties:

(3.1.15)

(3.1.16)

(3.1.17.a,b)

The same procedure can be followed to find properties .from eqn.(3.1.4) gas constant can be calculated

(3.1.18) 31 (3.1.19)

(3.1.20)

(3.1.21)

(3.1.22)

After applying balance and getting chamber temperature the last properties can be estimated from equations (3.1.10,11,12,13,14).

3.2Nozzle Design:-

A convergent-divergent nozzle is used in thrusters to accelerate the exhaust gas thus producing the required thrust force. The nozzle is designed to produce certain thrust force using certain propellant at a known ambient condtions. Assuming the convergent and divergent nozzle angles, the nozzle length can be computed using the estimated throat and exit diameters. A theortical nozzle model based on a quasi-one- dimensional and steady flow is used to compute the nozzle parameters and geometry. The flow in the nozzle is assumed to be isentropic. The thrust chamber pressure (Pc) is chosen to have a certain value. The thruster is designed to produce certain thrust force F and the propellant type is chosen.

As the chemical properities of propellant and the products of dissociations are calculted in section (3.1). After computing the dissociation temperature (Tc), and the exhaust gas properties, the characteristic velocity of the exhaust gases is calculated using the following equation:

 1 R *T   1 2*( 1) (3.2.1) C*  c *    2 

The conditions at the nozzle throat can be calculated from the following :equations

32 2 T  T ( ) th c  1 . (3.2.2)   1  2  (3.2.3) Pth  Pc *    1

th  Pth /R*Tth 

(3.2.4) uth   RTth

(3.2.5)

Assuming the nozzle is expanding ideally to the back pressure (i.e :pe=pb). Hence the exit conditions are estimated as follows

 1   (3.2.6) 2  P   M  *  c  1 e     1  Pe   

33 Tc Te    1 2  (3.2.7) 1 * M e   2 

Pe e  (3.2.8) R *Te

ue  M e *  * R*Te (3.2.9)

Then, the nozzle expansion ratio and thrust coefficient can be obtained :from the following equations

 1 Ae 1 2  1 2 2* 1    * *1 *M e  M e  1 2 Ath (3.2.10)

 1  1 2  1 2  Pe   Pe Pa CF   *   * * 1     *  1  1  Pc  Pc

(3.2.11) By calculating thrust coefficient, nozzle dimensions can be estimated as :follows

F A  (3.2.12) th CF *Pc A th (3.2.13) Dth  2* 

34 :From equation (3.3.10), exit area is obtained as follows (3.2.10a) Ae  Ath * A ex (3.2.14) De  2*  Computing throat and exit diameters and assuming divergent angle (Theta2), the length of divergent part of nozzle is calculated as follows:-

Dex  Dth L  (3.2.15) div nozzle 2*tanTheta2 Then the nozzle mass flow rate could be estimated from the following :equation P *A m˙  c th (3.2.16) C After getting catalytic bed area which is nozzle inlet area from equations (3.3.3) , (3.3.11), the nozzle inlet diameter is estimated as in equations (3.3.4), (3.3.12), The length of convergent part of nozzle is calculated as :follows

Dc  Dth L  conv nozzle 2*tanTheta1 (3.2.17) The total length of nozzle equals:

(3.2.18)

3.3 Catalytic Bed Calculations:- No one can deny that most of the catalytic bed calculations and relations are empirical as each fuel or propellant has its special catalysts and its relations. So we can use two methods for geometry calculations one for hydrogen peroxide and another for nitrous oxide.

I)Hydrogen Peroxide:

35 This catalytic bed consist of screens which is packaged in casing of rocket and supported as shown in picture

Fig. 3.1 Catalytic bed screens.

This for silver catalyst activity=75 kg h2o2/ liter silver,minte. (3.3.1)

Liquid feed load=.014 kg/mm2,min (3.3.2)

It is required to calculate inlet nozzle area to get diameter of catalytic

(3.3.3) Note that mass flow rate is gotten from equation (3.2.16)

Now catalytic bed diameter is gotten

(3.3.4)

36 From activity eqution(3.3.1.1) and flow rate equation(3.2.16) volume of catalytic bed is calculated as follows:

(3.3.5)

Then bed length

(3.3.6)

To have no of possible screens

(3.3.7)

(3.3.8)

To get pressure difference or pressure drop across catalytic bed

(3.3.9)

II)Nitrous Oxide: From previous chapters we can say that catalyst bed efficiency and ability to decomposition depend on length. So catalyst bed length here is variable to obtain the most efficient catalytic bed .so A design parameter to get suitable initial length which is called characteristic length(L*) For nitrous oxide from previous experience it can be let that bed loading(G)=7 kg/sec.m2 (3.3.10) So bed area from

(3.3.11) Where Ac is A_inlet nozzle And diameter (3.3.12)

37 As mentioned previously the nitrous oxide catalysts are similar to hydrazine catalysts. So from others experiments it can be estimated that the characteristic length is (L*)=10.75 m. As the length is changed experimentally to ensure the required length, the calculated length is only initial length but the bed volume is obtained first using:

(3.3.13) Then initial length is obtained as follows:

(3.3.14)

3.4Tank Design and Calculations:- To design and manufacture pressure or propellant tank u should determine its geometry and make selection to a suitable and available materials .so we follow this simple method of design and calculation by using simple equation of state.

I)Hydrogen peroxide tank As mentioned in pervious chapter we should use pressurized tank for this fuel .then our tank has two things inside first liquid hydrogen peroxide down and inert gas which is nitrogen above liquid fuel. For initial state fuel still in tank

(3.4.1) From equation of state after finish of fueling (3.4.2)

For spherical tank (3.4.3) To determine thickness of tank wall we should determine diameter and make simple stress analysis for material selected. (3.4.4) For material density and (3.4.5)

From above equation we can get thickness(t)

(3.4.6)

38 Then outer diameter of tank

(3.4.7)

Then volume material of tank

( - ) (3.4.8)

Now we can get mass of material tank

(3.4.9)

So the total tank mass

(3.4.10)

If we assume that residual mass of propellant equal zero as no fuel accumulation occur then

System with nitrogen pressurization

As we have two cases initial and final case .for final case the tank volume is the nitrogen volume when fuel is finished so we can say

(3.4.11)

Where tank volume is constant in two cases then

(3.4.12)

Apply equation of state for fuel and nitrogen in two cases

(3.4.13.a,b,c)

From above five equations we can get five unknowns Then we design for minimum mass of nitrogen by plotting it with pressure. by plotting curve for pressure with tank mass we can choose design point which have minimum mass with suitable tank pressure which is design point .

39 -real equation model ( van der waal eqn. of state)

Van der waal model usually used for high pressures because of the effect of compressibility as the system of simple equation of state will have some errors. so this model can be used here . The eqn. of state

(3.4.14) Where

, (3.4.15)

Where Tc,Pc are data of critical point of fuel For van der waal model

C=0 ,d=0 , (3.4.16)

From eqn.(3.4.16) a ,b can be calculated in eqn.(3.4.15)

After applying eqn. (3.4.16) in eqn.(3.4.14) become

(3.4.17)

Eqn.(3.4.17) non linear eqn. so that it need to be simplified .after algebraic simplification the new eqn. become

(3.4.18)

Where A,B,C,D, constants depend on fuel and system properties Eqn(3.4.18) is quadratic eqn.in specific volume

II)-Nitrous oxide tank

40 The big advantage of nitrous oxide that it has natural self pressurization system .no need to built pressurization system so we design for it only. Similarly by using the same procedure of hydrogen peroxide we can find mass of tank hence minimum mass for point of design. The only difference for nitrogen system that we have cylindrical tank then volume will change to First we have

(3.4.19)

(3.4.20)

Solve above equation for inner diameter (Din).Then apply the same procedure to get curves and choosing design point

3.5 Regulator Selection:-

The two most important parameters to consider during regulator selection and operation are droop and supply pressure effect. Droop is the difference in delivery pressure between zero flow conditions and the regulator’s maximum flow capacity. Supply pressure effect is the variation in delivery pressure as supply pressure decreases while the cylinder empties. Single-stage and two-stage regulators have different droop characteristics and respond differently to changing supply pressure. The single-stage regulator shows little droop with varying flow rates but a relatively large supply pressure effect. Conversely, the two-stage regulator shows a steeper slope in droop but only small supply pressure effects.

The effect of these differences on performance can be illustrated with some examples. For instance, when a centralized gas delivery system is supplying a number of different chromatographs, flow rates are apt to be fairly constant. Supply pressure variations, however, may be abrupt, especially when automatic changeover manifolds are used. In this scenario, a two-stage regulator with a narrow accuracy envelope (supply pressure effect) and a relatively steep droop should be used to avoid a baseline shift on the chromatographs. On the other hand, if gas is being used for a short-duration instrument calibration, a singlestage regulator with a wide accuracy envelope (supply pressure effect) but a

41 comparatively flat droop should be chosen. This will eliminate the need to allow the gas to flow at a constant rate before the calibration can be done. Delivery Pressure Range - Determining an appropriate delivery pressure range for a regulator can be confusing but can be accomplished by following these steps:

1. Determine the gas pressure needed. 2. Determine the maximum pressure the system might require (this pressure and the gas pressure are often the same) 3. Select a delivery pressure range so that the required pressures are in the 25% to 90% range of the regulator’s delivery pressure (a regulator’s performance is at its best within this range).

Relieving/Non-Relieving - A relieving regulator has a hole in the center of the diaphragm. As long as the diaphragm is in contact with the poppet, the regulator does not relieve. When the pressure under the diaphragm increases as a result of back pressure from downstream, the diaphragm will rise, allowing the pressure to relieve through the opening in the diaphragm. While the internal gas is relieving through this opening, the surrounding atmosphere (i.e. air) is diffusing into the gas stream. Oxygen (a component of air) is a harmful contaminant, especially when a gas stream is intended to be oxygen-free. It is well documented that oxygen affects gas chromatographic results. Relieving regulators should not be used in specialty gas applications.

Accuracy Envelopes for Single and Two-Stage Regulators at Two Supply Pressures - The envelopes are bounded by inlet pressure curves of 2000 psig (138 bar) and 500 psig (35 bar). Each regulator was set to the shown delivery pressure with 2000 psig (138 bar) inlet pressure and zero flow. Once set, this delivery pressure was not manually changed during the evaluation. The above curves generated are the result of increasing flow through the regulator to its capacity, decreasing the flow rate through the regulator to zero.

Linked Poppet/Tied Diaphragm - The poppet and diaphragm are mechanically linked. An increase in pressure in the cavity below the diaphragm will cause the diaphragm to move upward, pulling the poppet to improve its seal against the seat. A tied diaphragm regulator is effective in corrosive gas service, especially in the event that corrosive particles form under the poppet or on the seat. Tied diaphragm or linked poppet are terms used by manufacturers to describe this regulator feature.

42 Gauges - Generally single and two-stage regulators are equipped with two gauges - a cylinder or inlet pressure gauge and a delivery or outlet pressure gauge. The cylinder pressure gauge has the higher pressure range and is located adjacent to the inlet port. The delivery pressure gauge of the lower pressure range is located adjacent to the outlet port. The actual pressure gauge range is usually greater than the pressure range for which the regulator is rated. For example, a regulator that has a delivery pressure range of 1-50 psig (0.1-3 bar) will typically be supplied with a 0-60 psig (0-4 bar) delivery pressure gauge. This ensures that the rise in delivery pressure as a result of the regulator’s supply pressure effect will not exceed the gauge pressure range. Not all cylinder regulators have two gauges. A line regulator is typically provided with a single gauge that monitors the outlet pressure or reduced pressure. This gauge is usually situated in the 12 o’clock position. Regulators designed for liquefied gases may not have a cylinder pressure gauge because the cylinder pressure varies only with temperature as long as liquid is present in the cylinder.

Regulator Placement - Specialty gas regulator applications are divided into two types. The first is when the regulator is fastened to a gas cylinder using a CGA, DIN or BS fitting. The second application is when a regulator is located in a gas line, providing a means to further reduce the line pressure. A line regulator is identified by having the inlet and outlet opposite of each other and by a single gauge as discussed above. from above the selection of pressure regulator estimated by: 1. Calculation of Cv(flow ceoffient).

2. Amount of flow rate.

3. Minimum pressure difference (∆p).

4. Maximum inlet pressure and flow rate.

5. Type of flow to select material of regulator.

6. Chemical compatibility of material to match propellant.

The goal in designing is to have suitable pressure difference in pressure regulator.So it be considered that pressure drop vary between 10% and 20% of chamber pressure(Pc). ∆P=.17Pc (3.5.1) To calculate Cv we need to define volume flow rate(Q)

43 (3.5.2) Now we can determine regulator flow capacity(Cv)from (3.5.3)

Reading Flow Curves Flow Performance: The flow properties of a pressure regulator are illustrated by the flow curve. The vertical axis indicates the delivery pressure at which the regulator is set and the horizontal axis indicates the gas flow that the regulator passes. The curves are made by setting the delivery pressure while there is no gas flow and then slowly opening the outlet valve downstream while measuring both the flow and the delivery pressure. Typically, as flow increases, delivery pressure drops. The portion of the curve to the far left is fairly flat and it is in this range that the regulator demonstrates a stable pressure regulation even though the flow is changing. For example, increasing the flow from point “A’’ to point “B’’ shows only a slight decrease in pressure. The portion of the curve to the right shows a rapid drop in pressure with increasing flow rate, indicating that the regulator valve seat is almost wide open. If flow is increased from point “B’’ to point “C’’, there is a large drop in pressure that is typical for all regulators.

Fig 3.2/ regulator flow curve

44 3.6 Valve Sizing and Selection: Sizing flow valves is a science with many rules of thumb that few people agree on. In this article I'll try to define a more standard procedure for sizing a valve as well as helping to select the appropriate type of valve. **Please note that the correlation within this article are for turbulent flow

STEP #1: Define the system Key Variables: Total pressure drop, design flow, operating flow, minimum flow, pipe diameter, specific gravity

STEP #2: Define a maximum allowable pressure drop for the valve The usual rule of thumb is that a valve should be designed to use 10-15% of the total pressure drop or 10 psi, whichever is greater.

STEP #3: Calculate the valve characteristic

At this point, some people would be tempted to go to the valve charts or characteristic curves and select a valve. Don't make this mistake, instead, proceed to Step #4!

STEP #4: Preliminary valve selection Don't make the mistake of trying to match a valve with your calculated Cv value. The Cv value should be used as a guide in the valve selection, not a hard and fast rule. Some other considerations are: a. Never use a valve that is less than half the pipe size b. Avoid using the lower 10% and upper 20% of the valve stroke. The valve is much easier to control in the 10-80% stroke range.

45 Before a valve can be selected, you have to decide what type of valve will be used (See the list of valve types later in this article). For our case, we'll assume we're using an equal percentage, globe valve (equal percentage will be explained later). The valve chart for this type of valve is shown below. This is a typical chart that will be supplied by the manufacturer (as a matter of fact, it was!)

STEP #5: Check the Cv and stroke percentage at the minimum flow If the stroke percentage falls below 10% at our minimum flow, a smaller valve may have to be used in some cases. Judgements plays role in many cases.

STEP #6: Check the gain across applicable flow rates

Gain is defined as:

OTHER NOTES:

Another valve characteristic that can be examined is called the choked flow. The relation uses the FL value found on the valve chart. I recommend checking the choked flow for vastly different maximum and minimum flow rates. For example if the difference between the maximum and minimum flows is above 90% of the maximum flow, you may want to check the choked flow. Usually, the rule of thumb for determining the maximum pressure drop across the valve also helps to avoid choking flow.

46 3.7 Injectors design :-

For our system we use the showerhead injector may be designed as following by using hydrazine monopropellant model of design . The number of orifices is calculated from:

(3.7.1)

The injector pressure drop is calculated from:

(3.7.2)

Now we can get orifice diameter from:

(3.7.3)

For a good and safe design we should have space between orifices which is calculated from the equation

(3.8.4)

Therefore ,the diameter of each orifice row is:

(3.8.5)

The number of orifice in each row is calculated from:

47 (3.8.6)

Valve selection procedure for gas 3.8 :(application(N2O

Cv:flow discharge coefficient

Q:volume flow rat in ft/m3

G: specific gravity

T: gas temperature in F0 P1: maximum inlet pressure in psi

P2:maximum outlet pressure in psi m.:mass flow rate ibm/s

ρ:gas density in kg/m3 if o.53 p1 < p2

(Cv=(Q/1349)*(√(460+T)*G/∆P*P2)…………………(3.8.1

If o.53 p1 >p2

(Cv=(Q/704 P1)*(√(460+T)*G)………………………(3.8.2

P=pressure drop∆

:Givens m.=0.015

T=68 F0 ρ air=1.225 kg/m3

48 ρ n2o=1.775 kg/m p1=15*14.5 psi p2=13*14.5 psi

P=2*14.5 psi∆

:Calculations

:From equation of state (ρ =P1/R T……………………(3.8.3 also To calculate Cv we need to define volume flow rate(Q)

Q= m./ ρ……………..(3.8.4) :gas constant calculated from (R = Ru/Mw……………………………(3.8.5 8341/44= then R=188.96 J/kg—K

(G = ρ n2o/ ρ air …………(3.8.6 1.45=1.775/1.225= from equ 3 after substituting about T ,R

ρ =27.092 kg/ m3 from equ 4 by substituting about m., ρ then Q=(0.015/27.092)*(127132.8002)=70.389

Q=70.389 ft3/hr

From equ 1 ((Cv=(70.389/1349)* (√ (460+68)*1.45)/(29*13*14.5

Cv=0.02

49 Valve selection procedure for liquid application 3.9 :H2O2

:Givens m.=0.015 T=68 F0 ρ water=1000 kg/m3 ρ h2o2=1.391*1000 kg/m3 p1=15*14.5 psi p2=13*14.5 psi P=2*14.5 psi∆

Calculations :specific gravity calculated from G = ρ h2o2/ ρ water

G=1.391/1=1.391 :volumetric flow rate calculated from

Q= (m./ ρ h2o2 )*(1/6.3088*10^(-3)) in gal/min

((3-)^10*1/6.3088 )*(1000*0.015/1.391)=

Q=1.7092956 gal/min

:flow discharge coefficient calculated from

(Cv=Q*(√ G/ ∆P

(1.391/29)*1.7092956=

Cv =0.004

50 . HEATING SYSTEM 3.10

Required Evaluation the time required to heating the catalytic bed up to 523 Kelvin The available power(electrical power ) used to this process taken as : Ẅ which related :with temperature in the following relation

(Ẅ=m' cp ∆T ……………..(1 (W/t……………………(2= :Given Ẅ=4000 watt Dcb: catalytic bed diameter =0.048225 m Dc:coil of heater diameter =0.008 m

;Where W:heat transfer from heater coil to the catalytic bed t: total time used in heating ;Also (W=m cp ∆T……………...(3 .(m cp [Tf-Ti ]…………(4 = m: mass of catalytic bed cp: specific heat of catalytic bed material ρ: density of catalytic bed material V: volume of catalytic bed (m=ρV……………..(5 :FOR SHELL 405 -1 (iron..36 wt,γ_Al2 o3)

(ρ av= ( ρ iron*.36+ ρ Al2o3*.64 )/2…………………………(6 ;Where ρ iron =7.86 g/cm3 ρ Al2o3=3.96 g/cm3 after substituting in Equ…(6) then (ρ av= 2595.6 kg/m3 ……………………………………..(ans 1 (Cp av=( Cp iron*.36+ Cp Al2o3*.64)/2………………………….(7 ;Where 51 Cp iron=448 J/kg-k Cp Al2o3=880 J/kg-k after substituting in Equ…(7) then

(Cp av=362.24 J/kg-k…………………………………….(ans 2

Now we can calculate the volume from the following formula (V =Л/4(D2)*L…………………………………………..(8 (D=Dcb- Dc………………………………………………(9 ;Where Dcb: catalytic bed diameter Dc: coil of heater diameter

Also after substituting in Equ…(5,8,9 ) then

(D:net catalytic bed diameter =0.040225 m…………….(ans3 (V=1.051315*10^(-4) m3…………………… ………(ans4 .. .… (m= 0.27288 kg ………………………………………….(ans5

; (Now we can calculate the( W )from the Equ (3

(W=22240.8 J/kg—k ……………………………………(ans6

Finally and from Equ 2 the required time for heating is

;where

(t=W/ Ẅ ……………………………………………..(equ 2a

(t=111.204 sec……………………………………………(ans7 required a battery of life time =5.56*1000 (L_t=1.5444 hr………………………..(ans 8

:FOR IRON—AL-2 (iron..95 wt, AL0.05 wt)

(ρ av= ( ρ iron*.95+ ρ Al2o3*.05 )/2…………………………(6--a

;Where ρ iron =7.86 g/cm3

52 ρ Al2o3=3.96 g/cm3 after substituting in Equ…(6--a) then

(ρ av= 4409kg/m3 ……………………………………..(ans 9 (Cp av=( Cp iron*.95+ Cp Al2o3*.05)/2………………………….(7—a ;Where Cp iron=448 J/kg-k Cp Al2o3=880 J/kg-k after substituting in Equ…(7--a) then

(Cp av=437.8J/kg-k…………………………………….(ans 10

Now we can calculate the volume from the following formula (V =Л/4(D2)*L…………………………………………..(8 (D=Dcb- Dc………………………………………………(9 ;Where Dcb: catalytic bed diameter Dc: coil of heater diameter

Also after substituting in Equ…(5,8,9 ) then

D:net catalytic bed diameter =0.040225 m…………….(ans3) (V=1.051315*10^(-4) m3…………………… ………(ans4 .… (m= 0.4659kg ………………………………………….(ans5

(W=45659.5 J/kg—k ……………………………………(ans6

Finally and from Equ 2 the required time for heating is

;where

(t=W/ Ẅ ……………………………………………..(equ 2a

(t=11.41sec……………………………………………(ans7 required a battery of life time =11.41*1000 (L_t=3.169444 hr………………………..(ans 8

53 (FOR IRON—Ag) (IRON 0.95.0.05Ag-3 The same steps of calculations so these final results

;Where ρ iron =7.86 g/cm3 ρ Al2o3=3.96 g/cm3 after substituting in Equ…(6--b) then

… ρ av= 6381kg/m3 also Cp iron=448 J/kg-k Cp Al2o3=880 J/kg-k after substituting in Equ…(7--b) then

.……………………………………Cp av=270.3/kg-k Also after substituting in Equ…(5,8,9 ) then m= 0.670841kg

W=22240.8 J/kg—k Finally and from Equ 2 the required time for heating is

;where

(t=W/ Ẅ ……………………………………………..(equ 2a

(t=10.19976 sec……………………………………………(ans7 required a battery of life time =10.19976*1000 (L_t=2.833266hr………………………..(ans 8

(FOR IRON—Mg o) (IRON0.95. Mg o0.05) -4

ρ av= 46285kg/m

Cp av=432 .05 J/kg-k m= 0.48660 kg

54 W=4730 .105J/kg—k

…t=11.824sec

L_t=3.28444 hr