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Tank Blowdown Math

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by Prof. Dean R. Wheeler, Brigham Young University

EDITOR’S NOTE

This posting is reprinted from the original article written March 13, 2019 with permission from the author. This article was intended for chemical engineering students to size relief valves for pressure vessels, but it applies well to amateur liquid rocketry as many use a pressure fed system to deliver propellants to the engine.

The PDF of this white paper can be found below.

https://www.et.byu.edu/~wheeler/Tank_Blowdown_Math.pdf

The RRS has several members engaged with liquid rocket projects. An important part of analyzing the performance of those systems is the pressurization system that drives the propellant into the engine. The tank blowdown problem is useful to designing the system and estimating performance. This derivation goes through the thermodynamics of the general tank blowdown problem and should be a useful starting point for a pressure-fed liquid rocket project.

INTRODUCTION

This document provides a mathematical model for computing the rate of expelling gas through a small orifice or nozzle attached to a tank. Furthermore, two models are described for how fast the tank will depressurize. Related material on compressible flow can be found in fluid mechanics and thermodynamics textbooks and web pages.

Figure 1 shows the tank and associated nozzle. The narrowest diameter of the flow path in the orifice or nozzle is known as the throat region. The tank and throat regions are described with their own sets of equations.

Provided the tank is large and the throat is small, it will take many seconds to empty the tank and gas velocities in the main part of the tank will be much smaller than the speed of sound. This means that gas pressure, temperature, and density in the tank will be spatially uniform, though they will be changing in time. Thus, we describe the tank using a transient mass balance. One can compare this to a model in heat transfer known as lumped capacitance.

In the nozzle region however, gas velocity is large and there are large spatial variations in the gas properties. In addition, there is relatively little gas contained in the nozzle region. Thus, flowrate in the nozzle adjusts rapidly to match current conditions in the tank, making it seem as if the nozzle is operating at steady state. This approximation for the nozzle is known as quasi-steady state.

Figure 1: Schematic of a task with nozzle or orifice, allowing gas to exit. Italicized are variables that pertain to twokey regions. During blowdown every variable depends on time,

EQUATIONS OF STATE

The P, T, and rho variables in Figure 1 denote absolute pressure, absolute temperature, and density in the tank or the narrowest part of the nozzle or throat (denoted by an asterisk,*, subscript), respectively. Note that if tank pressure is given experimentally as a gauge quantity, it must be converted to absolute to be used in the equations below.

The first relationship between gas variables is given by an equation of state. The ideal gas law is a fairly accurate representation for air when pressure is less than around 10 atmospheres or 150 psia. It states that:

Figure 1: The ideal gas equation

where “V” is the volume of the gas, “n” is the number of moles, and “R” is the universal gas constant (8.31446 J/mol/K). With the introduction of the molecular weight, M (effectively 0.028964 kg/mol/K for air), and the substitution that density is mass over volume, rho = n M / V, the ideal gas law is changed to

Equation 2: Density calculated from the ideal gas equation

This equation could be applied separately to the tank variables or to the thrust variables.

TEMPERATURE AND PRESSURE DURING EXPANSION

The second important relationship comes from figuring out what happens when gas in the tank or nozzle expands. When a gas expands, its internal energy is used to perform work on the surroundings, and the gas therefore tends to cool off. If the gas expands slowly, there is time for itmto absorb hest from its warmer surroundings and the expansion is essentially isothermal, meaning the temperature stays at its initial value or that of the surroundings.

On the other hand, if a gas expands quickly its temperature will drop dramatically. This is called adiabatic expansion, where adiabatic means no noticeable heat transfer from the surroundings (i.e. the walls of the tank). In adiabatic expansion, the pressure drops more rapidly than it would for an isothermal (slow) expansion. Adiabatic expansion could haolen inside the tank if it is emptying rapidly, but this depends on the relative sizes of thr tank and nozzle. On the other hand, adiabatic expansion certainly occurs when a gas moves from the tank through the nozzle region. In other words, here the gas is moving quickly and therefore expanding quickly.

The thermodynamic relationships for pressure and temperature for reversible adiabatic expansion of a constant heat capacity ideal gas are:

Equation 3A: Adiabatic pressure and density relationship
Equation 4A: Adiabatic temperature and density relationship

where the subscript, “o” indicates the initial state of the gas before the expansion started. This means if we know how the density is changing from an initial state to some later state, we can compute P and T as well. In the case of the nozzle, we apply the above equations as the gas travels between the tank and the throat. In the case, they become

Equation 3B: Adiabatic pressure and density relationship between tank and throat regions
Equation 4B: Adiabatic temperature and density relationship between tank and throat regions

The parameter, “gamma” , is the dimensionless ratio of specific heats ( gamma =. Cp / Cv ), and by statistical theory of gases, gamma = 7/5 = 1.4, for low temperature diatomic molecules, nitrogen (N2) and oxygen (O2) and so that value is used here.

CHOKED FLOW

Next, we need to determine the gas density in the nozzle when the tank is at a specified conditions. Recall that that the nozzle is treated as if it instantaneously responds to whatever state the tank is in. A fuller discussion of the nozzle flow equations can be found in other sources like textbooks that cover ideal compressible flow in nozzles.

Choked flow means that the flow is exactly at the speed of sound in the throat region. A higher speed cannot be achieved in the throat, regardless of upstream or downstream conditions. Thus, choked flow acts to limit how much gas flow can pass through a given size orifice, This is the reason why pressure relief valves on tanks must be properly sized to accommodate sufficient flow.

Choked flow happens for a large pressure drop across the nozzle or orifice, specifically if the upstream tank pressure meets the following condition relative to atmospheric pressure downstream from the nozzle:

Equation 5: Choked flow condition

Equation 5 is the origin of the rule of thumb or approximation that choked flow occurs for upstream pressure that is more than twice the value of downstream pressure (absolute). If the tank pressure drops below this limit, the speed of gas in the throat is subsonic, and less gas will flow than in the choked flow regime. The solution to subsonic flow in the nozzle is complicated and is less important to know because it is at the end of the tank’s discharge when pressure is low, and so will be neglected here.

The solution to choked flow in the throat region follows a simple relationship, derived from energy and mass balances:

Equation 6: Throat to tank density ratio

This can be substituted from Equation 3B and 4B to determine pressure and temperature in the throat in terms of tank conditions.

For choked flow the throat velocity is exactly the speed of sound, which is what makes it easier to analyze. For ideal gases, speed of sound, c, is determined solely by temperature. Thus, we can relate throat velocity to throat temperature, and in turn to tank temperature:

Equation 7: Speed of sound at the throat

For example, if T_tank = 294 Kelvins, then c_o = 314 m/sec for air.

MASS FLOW RATE

Now we can determine the mass flow rate, “m_dot”, through the nozzle or orifice. This comes from the following standard relationship, applied at the throat, because that is where conditions are known:

Equation 8: Mass flow,rate at the throat

where “A_*” is the throat cross-sectional area given by

Equation 9: Area of a circle

and where “d_*” is throat diameter.

Dimensionless parameter, Cd, in Equation 8 is the discharge coefficient, accounting for friction between fluid and walls and a phenomenon known as vena contracta. In essence, Cd, is needed in Equation 8 because the effective area for fluid at speed, v_o, is somewhat smaller than actual throat area. Cd would be equal to 1.0 for a perfect (frictionless or thermodynamically reversible) nozzle: in practice for a smoothly tapering nozzle it might be as high as 0.98, while for a sharp-edged orifice it might be as low as 0.60. Anything that causes separation of flow from the nozzle wall or increases frictional contact will decrease Cd.

Making the appropriate substitutions into Equation 8 leads to an equation for mass flow in terms of readily determined quantities:

Equation 10: Mass flow rate in terms of readily determined quantities

Frequently in industrial situations, mass flow rates are expressed instead as volumetric flow rates corresponding to a gas at a standard temperature and pressure (even though the gas is not actually at that temperature and pressure). For instance, a mass flow meter used for gases may express mass flow as standard liters per minute (SLPM) or standard cubic feet per minute (SCFM). In other words, even though m_dot (mass flow) is the key value being measured, it is expressed as

Equation 11: Standard volumetric flow and mass flow rate

which requires knowing what rho_std value is programmed by the manufacturer into the flow meter. This can be determined from the ideal gas law, given specified P_std and T_std values. As an example, the American manufacturer, Omega, assumes a standard temperature “T_std” of 70 degrees Fahrenheit (294.26 Kelvins) and a standard pressure “P_std” of 1 atmosphere which equals 14.696 psia (101,325 Pscals) thus by the ideal gas law, the standard density “rho_std” would equal 1.2 kg/m3 for air (molecular weight 28.97 g/mole).

Combining Equations 10 and 11 and the ideal gas law leads to

Equation 12: Combining Equations 10 and 11 for standard volumetric flow rate

where “c_std” is the speed of sound at the standard temperature:

Equation 13: Standard volumetric rate and mass flow rate relationship

Makers of valves and orifices may provide an experimentally determined size parameter known as flow coefficient, Cv. For gases this dimensionless parameter can be converted to Cd*A_* by

Equation 14: Discharge area relationship4 to valve coefficient (metric units)

The key design principles resulting from the above analysis are, provided tank pressure is large enough to generate choked flow, that (1) mass flow rate of a gas through an orifice is proportional to throat area and tank pressure and (2) flow rate does not depend on downstream pressure.

TWO MODELS OF TANK BLOWDOWN

Equation 10 gives the rate of mass loss from a tank at a given gas density and temperature. To determine how long it will take to depressurize the tank, we must do a transient mass balance on the tank. The ordinary differential equation for this is:

Equation 15: Change of mass in time

where “m_dot” comes from Equation 10 and “m” is the mass of gas in the tank. This in turn is:

Equation 16: Mass in the tank

where V_tank is the fixed tank volume. With these substitutions we get for the governing equation

Equation 17: Mass flow rate from the tank

To make things more manageable, let us create a discharge time constant called “tau”

Equation 18: Time constant for blowdown of a tank

where “c_o” is the speed of sound at the initial temperature “T_o” (i.e. at the beginning of blowdown)

Equation 19: Speed of sound at initial conditions

With this new time constant, Equation 17 becomes:

Equation 20: Mass flow rate change in the tank

The last thing to do before solving this equation is figure out what to do with T_tank. We have two options:

ISOTHERMAL TANK ASSUMPTIONS

Assume gas temperature in the tank does not change in time, based on blowdown taking a long time so that heat can be readily absorbed from the walls. Thus, T_tank = T_o. This leads to Equation 20 becoming

Equation 21: Tank density change in time

which can be separated and integrated to give the solution

Equation 22: Tank density as a function of initial conditions

where “rho_o” is initial density in the tank. We then convert densities to pressure using the ideal gas equation.

Equation 23: Tank pressure as a function of initial conditions

The equation tells us how tank pressure varies with time, for an isothermal tank and choked exit flow.

ADIABATIC TANK ASSUMPTIONS

Assume the gas cools as it expands in the tank, due to no heat transfer from the walls, based on the blowdown taking a short time to complete. Thus, T_tank is given by Equation 4A. This leads to Equation 20 becoming

Equation 24: Mass flow rate from the tank

which can be separated and integrated to give a solution.

Equation 25: Density of the tank as a function of time

We then convert densities to pressures using Equation 3A for adiabatic expansion.

Equation 26: Tank pressure as a function of time

This equation tells us how tank pressure varies with time, for an adiabatic tank and choked exit flow. The tank temperature can likewise be predicted from Equation 4A.

Equation 27: Tank temperature as a finction of time

COMPARISON OF THE TWO MODEL ASSUMPTIONS

The isothermal and adiabatic models of tank blowdown can be considered two extremes, with the correct answer (i.e., with the true amount of heat transfer) lying somewhere in between them. Figure 2 shows an example of the respective blowdown curves (Equation 23 and 26). As noted previously, adiabatic tank conditions lead to more rapid pressure loss than do isothermal conditions.

The curves predict that the tank will have lost 80% of its original pressure at a time in the range of 1.3*tau < t < 1.6*tau. This shows the value of evaluating the variable, tau, to get an approximation of the time it takes to depressurize the tank.

Figure 2: Comparison of isothermal and adiabatic blowdown curves.

MTA launch event, 2020-12-12

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by Dave Nordling, Reaction Research Society

The RRS held it’s last launch event of this difficult but eventful year, 2020. COVID-19 continues to pose a significant threat to the wellbeing of our members and the world at large. One of the advantages of our remote testing site is the ease that our members can socially distance themselves and with masks and proper planning of shared tasks the risk of contagion is easily mitigated. I was the pyro-op in charge for this event. We had three planned launches that would depend on good weather and a work task to repair our vertical test stand.

The winds were strong that day coming into the MTA site from the western route. A lot of sand swirling was a poor omen for the weather that day.

My late arrival found our participants waiting at the gate and with my apologies we entered and began our set up.

Wolfram Blume’s Gas Guzzler was to take it’s first flight today, but he decided to scrub for the day. The winds were a persistent nuisance and prevented launch operations for much of the day, but after 2pm calm winds prevailed. It is difficult to know when the weather will change except that it inevitably does. Wolfram’s first flight will have to wait for the new year,

The Gas Guzzler solid-motor booster stage to the left, the gasoline-fueled ramjet upper stage to the right.

REPAIR OF THE VERTICAL LAUNCH STRUCTURE

The vertical launch structure at the RRS MTA has had a bent panel from an explosion from a failed test over a decade or more ago. This stretched 1/4” steel panel was significantly bowed away from the others which made mounting very difficult. Replacement panels were made back in October, but today was the day the bent panel would be cut away and the sides grinded to fit the replacement panel.

Dimitri Timohovich was able to cut away the bent panel using a length of aluminum channel clamped to the side for careful alignment of the plasma cutting process.

Dimitri Timohovich cutting away the bent panel from the vertical test stand.
The bent panel removed.

Dmitri brings a lot of mechanical skills and the society is grateful he joined us in helping make improvements to our site. The winds were too high that day for the shield gas flow needed in the welding process. The edges were blended to allow the replacement panel to be fitted accurately within the vertical launch rail using two lengths of unistrut. With careful measurements and the right equipment, Dmitri or Waldo Stakes can stick-weld the replacememt panel in place and keep a reasonable horizontal and vertical positional accuracy with the hole patterns of the plates above and below,

The replacement panel was bolted and held in place for the weld operations to take place at a later date.

The last step after a successful welding of the replacement panel would be applying the spray-on galvanizing paint product to protect the metal from the caustic and harsh desert environment for years to come, The unistrut pieces looked to be handy for future projects so we decided to leave them in place.

THE SOLAR HEATED SIMPLE-CAT

Bill Inman and his colleague from Nevada arrived with a third design iteration to his parabolic solar collector heating system for his two-inch steam rocket, This design featured a larger area collector and a launch rail system for his 2-inch SimpleCat steam rocket prototype. The launch rails guide the 2-inch steam rocket vessel at the collector’s focus for heating. At the exhaust end of the rails is the steam release mechanism that was simplified from prior successful designs.

It was dubious if the launch would even be possible that day but Bill’s solar collector system could be deployed from his trailer on our site and at least collect heating data even if the steam rocket wouldn’t fly.

After correcting some initial fit problems, the larger parabolic mirror was deployed.
The new solar collector in work under the early high winds and poor winter sunlight.

Launching was not possible for much of the day so the two groups waited for an opportunity. Bill Inman reviewed with me his steam rocket design and it’s simplified nozzle plug release design, The steam rocket despite its conceptual simplicity has many dangers. The mechanism for controlled release of the 400 degree Fahrenheit pressurized water liquid must be stable, sturdy, reliable and safe to remotely operate, Keeping a safe distance during the planned 3 to 4 hour solar heating cycle is crucial and having the continuous ability to safely scram the system at a safe distance is an absolute must. Bill’s design has a relief valve to avoid vessel over-pressure and relies on defocusing the sun away from the vessel if an abort is necessary, Removing the heat source immediately allows the fluid and vessel to cool if left alone for an hour or more and will ultimately return to ambient temperature once the heat source is removed,

During deployment, the solar collector and mounting frame had several fit problems which were solved at the site. The sun wasn’t consistent that day, but the mechanism held sturdy in the periodic gusting winds. By the end of the day, the collector was not able to generate sufficient heating for the rocket, but the experience in the field was valuable. Bill will be returning in the new year to try again.

Bill Inman at the end of an unsuccessful test at the MTA on 12-12-2020

Bill was disappointed in the results from that day’s activities as this month would have marked the 20th anniversary of his original successful flight of the Scalded Cat from the RRS MTA. I told him to take comfort in the fact that he has come a long way in a short time building three prototype devices in the same number of months. Bill is prolific and dedicated to his goal of being successful. Time should prove the value of patience and persistence.

ABORTED ATTEMPT TO LAUNCH THE HYBRID

My patience with the weather was ultimately rewarded as the winds subsided just after 2PM that day. I decided to make the next attempt to launch the larger 3-inch rocket that Larry Hoffing built that is adapted to fit the 16-inch Contrails Rocketry hybrid 38mm motor. A more energetic ignition system able to simultaneously sever the nylon fill line and ignite the combustion of the hybrid solid propellant grain was added and ready since the past July 2020 event. With the cooler temperatures, the solenoid filling valve would likely open according to the pressure gauge on our red supply bottle from Nitrous Supply Inc. in Huntington Beach,

The nitrous oxide hybrid rocket sits on the table in the Dosa Building at the MTA waiting for the winds to subside.

Dmitri Timohovich helped me set up the nitrous oxide bottle and manifold. The two-channel filling and firing circuit needed some labelling to clarify the proper wiring. A new lead-acid 12-volt lawnmower battery was acquired for the society as the previous one finally had to be retired and recycled. The new battery was ready for a launch that would ultimately not happen that day.

The rocket’s recovery system passed checkout as the original internal 9-volt battery installed months earlier was still healthy, The venting tube needed to be realigned with the exit hole to allow the white jet of liquid to be visually indicated when the nitrous volume is fully filled. It is important to detect this at a distance from the blockhouse as it is not safe to examine it more closely.

Fill, drain and firing circuit for a Contrails hybrid rocket motor

The launch would not take place due to a missing push-to-connect fitting to join the fluid filling tube from the rocket back to the nitrous manifold. The schematic above shows the key parts of the system minus the separate vent solenoid that failed on the original manifold, It is always frustrating to be missing one critical item despite days of preparation. After a lot of searching in vain for a single fitting, the container of materials will be better organized in the future and extra 3/16” push-to-connect Prestolok fittings will be ordered to arrive in time for the next launch event.

It was all the more painful to stand at the MTA under nearly calm winds and have to wait for another day. These are the trials and tribulations of rocketry,

Bill Inman and John Krell next to the old blockhouse at the RRS MTA.

At the end of the day, we gathered to discuss the progress or lack thereof that day. We made plans for the next launch event which seems to be best held on January 9th. We were glad for each other’s company and stayed at a safe distance throughout, We’ll return again to the MTA soon.

December 2020 virtual meeting

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By the Reaction Research Society

The Reaction Research Society held its last monthly meeting of this difficult and eventful year 2020 on Friday, the 11th. The teleconference was well attended and included some students from Loyola Marymount University. We began the meeting with the treasurer’s report and moved to the first order of business.

Our meetings by teleconference will continue into the new year.

ELECTION RESULTS FOR 2021 EXECUTIVE COUNCIL

The results of the election were presented by this year’s appointed election chairman, Dave Nordling. The full slate of nominees were elected from the nominations at the November meeting.

President = Osvaldo Tarditti

Vice President = Frank Miuccio

Secretary = Keith Yoerg

Treasurer = Larry Hoffing

The society thanks our outgoing secretary and treasurer, Drew Cortopassi and Chris Lujan respectively, for their service in this year, 2020. We welcome our new executive council members as they start their annual term on January 1, 2021

We appreciate the many voting members who responded this year. The society was sad to learn of the passing of two of our lifetime members, Thomas McGaffey and Mike Gottlieb.

ANNUAL MEMBERSHIP DUES

The RRS treasurer reminds our society membership that annual dues are to be paid on January 1st of each year. This is a continuation of RRS policy set forth by the executive council in 2019. In year 2021, membership dues have increased for student membership while full membership dues remain the same. As always, student membership in the RRS is valid all year regardless of how many events we hold.

Full membership = $40 USD / year

Student membership = $30 USD / year

Annual membership dues are an important source of revenue to support the society’s operational costs. Associate, administrative, student and corresponding members are required to have their dues be fully paid to maintain active membership status.

For questions regarding dues payments, contact the RRS treasurer:

treasurer@rrs.org

There are two common means of dues payment:

(1) Click the “Donate” button on the RRS website which links to Paypal. This is the easiest method for the society to receive and confirm payment. When using this method of payment, please make a note that you are paying ‘annual dues’ and the name of the person it is for. The Paypal website gives you the option to share your address with the society, Please do so as this marks who the payment is from. Contact the RRS treasurer if you have questions.

(2) Payment by check can be submitted to our post office box in Los Angeles. As the executive council checks the post office box only periodically, it may take some time for your payment to register. With all mailed correspondence, please email or call the RRS treasurer to let them know it is coming.

Reaction Research Society
P. O. Box 90933
Los Angeles, California, 90009-0933

When making your dues payment, it is also important to update your contact information with the RRS treasurer. It is the responsibility of the RRS treasurer to maintain the membership roster and record payment of membership dues. It is the sole responsibility of every member, past and present, to keep their email and other contact information up to date with the society.

NEW PYROTECHNIC OPERATOR LICENSING

Keith Yoerg announced he has made significant progress in his application to becoming a licensed pyrotechnic operator. The RRS has been supportive of increasing our roster of pyro-op’s. Keith is one of several RRS members in this process. The RRS and the Friends of Amateur Rocketry organization have been working with CALFIRE since last year on improving definitions for state regulations on amateur rocketry in California. Our two organizations have been supportive of each other’s members’ desire for training and instruction in the course of becoming licensed pyro-op’s. It is to our mutual benefit to have more people knowledgeable about safe operations in our hobby.

SITE IMPROVEMENTS AT THE MTA

The RRS continues to evaluate its options for an improved restroom facility at the MTA. The council has put this as the top concern and is in the process of evaluating bids. The society would like to proceed with a replacement during this winter season. More storage space, replacing the roof on the old blockhouse and finding a towable fire-wagon with a water pump are also on our list of improvements.

MTA LAUNCH EVENT

The RRS has a scheduled launch event on December 12th. Dave Nordling will be the pyro-op in charge for this event. Three launches are planned including Wolfram Blume’s first systems flight test of his two-stage Gas Guzzler ramjet and Dave Nordling’s and Larry Hoffing’s 3-inch rocket with an improved nitrous oxide hybrid motor. Bill Inman also plans to be at the MTA to test his third design iteration of his solar collector which will now include his launch rail and his new SimpleCat 2-inch steam rocket.

LOYOLA MARS TEAM PRESENTATION

The society was happy to welcome upperclassmen students from Loyola Marymount University to our December meeting teleconference. Loyola MARS is a recurring senior capstone project to incrementally design, build, test and fly liquid rockets leading to a final design capable of reaching the von Karman line in under 10 years. It is an ambitious project that was inspired by the former Base11 competition. The RRS has supported the Loyola MARS team since its start and was impressed by their initial systems design. The society looks forward to supporting their first fluid systems tests and static hot-firing at the MTA in the coming new year.

As an educational non-profit group, the RRS provides assistance to several local universities who are building rockets for class projects. We welcome student groups to indicate their interest in attending our meetings by contacting the RRS executive council.

IN CLOSING

2020 Constitutional Committee is overdue in presenting the new draft and policy statements to our active membership for review. This will be delivered at next month’s meeting by the appointed committee. Once our membership has had the chance to offer it’s feedback and suggestions, the new Constitution will be sent to our voting membership for ratification.

The RRS treasurer’s report on membership status will also be deferred to next month’s meeting.

The RRS wishes everyone to be safe in this holiday season and take appropriate precautions in this COVID-19 pandemic.

The next monthly meeting will be on Friday, January 8, 2021.