Tank Blowdown Math

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_o by

Equation 14: Cd * A_o = Cv * 16.2 mm^2

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: dm / dt = – m_dot

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

Equation 16: m = rho_tank * V_tank

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

Equation 17:

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

Equation 18:

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

Equation 19:

With this new time constant, Equation 17 becomes:

Equation 20:

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:

which can be separated and integrated to give the solution

Equation 22: rho_tank = rho_o * exp (- t / tau )

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

Equation 23: P_tank = P_o * exp (- t / tau )

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:

which can be separated and integrated to give a solution.

Equation 25:

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

Equation 26:

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:

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.

FURTHER NOTES FROM THE EDITOR

This paper uses SI (metric) units of measure. It is important to note that the speed of sound calculation if done in English units requires the use of a gravitational constant factor to have the right answer. For most work done in the United States, English units are still in common use.

There are some other unit conversions included to aid of measure

November 2020 virtual meeting

by the Reaction Research Society


The RRS held its monthly meeting on November 13th. The teleconference brought several of us together again from different parts of the city and around the country.

The November 2020 monthly meeting by teleconference

PAST EVENTS

The society discussed the prior launch event at the MTA held on November 7, 2020. Details of the event are in the firing report posted this month. More RRS members are buidling more rockets and this is a good trend, We were not able to work on site improvements at the event, but these tasks can be resumed at the next event.

Bent panel on the vertical thrust stand to be removed and replaced.

The 2020 Constitution Committee is almost ready to release the recommended draft with updated policy listings according to RRS vice-president Frank Miuccio who is the single executive council representative on the committee. Release of the draft will go out in December 2020. If ratified by the voting membership, this new Constitution will clarify RRS rules and policies and supersede all past revisions.

FACILITY UPGRADES AT THE MOJAVE TEST AREA

The society discussed the current progress of facility upgrades approved by the executive council. At the top of the list is a new restroom facility. A couple contractor bids are forthcoming with one being delivered just a few hours before the meeting. Cost is an important consideration and other designs besides a block and concrete design are being considered. Containerized, portable systems may offer an effective solution with fewer complications.

The council approved the construction of a similar pit toilet like the one being used now at the MTA. The second toilet is meant to replace the first as it will not be usable for much longer. A few RRS members are already working to put in this stop-gap measure until the better facility can be selected and funded.

Second pit toilet at the MTA approved as stop-gap measure.

Other site upgrades discussed at the meeting was replacement of the roof structure on the existing blockhouse which would be a temporary but safer solution until a new structure can br funded and built. Another was removal and replacement of the bent panel on the vertical thrust stand which may be done at the next launch event. Still another improvement option is purchasing some of the existing containers from our tenant for more storage space at the site and a water storage tank.

The blockhou at the RRS mTA. Replacement of the roof would enhance the safety of this shelter.

Other improvements include locked and secured areas for high-pressure gas bottles and cryogenic liquid cylindeers. Towable water-spraying trailers have also been discussed to augment fire-fighting at the MTA. A larger launch rail system for heavier liquid rockets at the RRS MTA is also being considered.

DONATE TO THE RRS THROUGH AMAZON SMILE

Amazon Smile is a program that allows Amazon to donate a small percentage of eligible purchases to a non-profit group, such as the Reaction Research Society.

go to Amazon Smile and see for yourself

You first enter: smile.amazon.com

then select the Reaction Research Society as your chosen recipient. There is no extra cost to you and you allow the RRS to recieve quarterly donations from Amazon when sufficient purchase amounts have been made in that quarter. With the holiday season approaching, this is a simple way to benefit the RRS and the many activities we support.

For more details and assistance, contact Chris Lujan or just go to Amazon Smile.

NOMINATIONS FOR EXECUTIVE COUNCIL

As per the Constitution, the society holds nominations at the November monthly meeting for the four elected executive council positions. Terms will start in the new calendar year, First, an election chairman is appointed by the council. This person can not run for office in that election cycle. Dave Nordling was selected as election chairman.

Only administrative members in active status can be nominated and hold office. The nominees will be listed on the ballots going out to the eligible voting membership. Ballots will be sent by email which underscores the importance of all members keeping their contact information up to date with the RRS treasurer. Also, one of the requirements of active membership is being current with dues payments also tracked by the RRS treasurer.

treasurer@rrs.org

ANNUAL MEMBERSHIP DUES

RRS policy is that all annual dues are to be paid on January 1 of each calendar year. New members joining the society throughout the year can pay a pro-rated dues for their first partial calendar year. Membership dues are $40 per year and student membership is $20 per year.

The executive council voted to keep dues at their current levels for the new calendar year, 2021. However, dues will increase to $50 per year starting in 2022 (January 1). All payments are made to the society to the RRS treasurer or the RRS president.

president@rrs.org

NEXT MTA EVENT

The next launch event at the Mojave Test Area will be held on December 12th. The nitrous oxide hybrid will be launched with the upgraded igniter and new airframe. Other rocket launches are planned.

IN CLOSING

The next monthly meeting will be held on December 11 by teleconference. Anyone with questions or suggestions should contact the RRS secretary.

secretary@rrs.org


October 2020 virtual meeting

by the Reaction Research Society


The RRS held it’s monthly meeting by teleconference on October 9th at our usual starting time of 7:30pm. We had a few members calling in from out-of-state. We had a few new topics to cover.

Some of the attendees to the October 2020 monthly meeting

SUMMARY OF RECENT MTA EVENT

We held a work event at the Mojave Test Area on October 3rd which was very successful despite higher temperatures for that autumn Saturday. We cleared a lot of tumbleweeds, mended the barbed wire fence at our front gate, painted the metal window gratings on the front of the Dosa Building and even cleared off the decks of the large vertical test stand. We had a lot of great help and we hope to continue making these site improvements to make our facility more attractive and useful.

The large vertical test stand at the RRS MTA after some clearing.

We agreed to meet at the MTA again on November 7th which will coincide with a static fire test of USC’s Rocket Propulsion Lab latest multi-grain solid motor design. We will also attempt to remove and replace a bent panel on the vertical thrust stand. The nitrous oxide hybrid rocket by Dave Nordling, Larry Hoffing and Osvaldo Tarditti is also ready for launching. If there are other member projects that are ready we will add those to the event and notify the pyro-op in charge.

LIQUID ROCKET PROJECTS

Liquid rocket projects have become more popular recently and some have started within the RRS.

Loyola-Marymount University (LMU) in Marina del Rey started a capstone project for their upper classmen in their undergraduate aerospace engineering program to design and build a large liquid rocket. The LMU Lion project was inspired by the FAR-MARS competition. Dave Nordling has been supporting the early design work on behalf of the RRS starting early this calendar year prior to the pandemic restrictions ending in-person meetings. LMU has restarted the project with the new academic year with a series of specialized coursework and short presentations on topics from experts around the industry. Dave was glad to present the history and capabilities of the RRS. The presentation was well received and LMU has looked at using our vertical test stand when they get their first liquid rockets systems ready for test.

Loyola Marymount Aerospace Research Society

The Compton Comet project at Tomorrow’s Aeronautical Museum has been restarted with RRS members, Waldo Stakes, Kent Schwitkis and Dave Nordling. The Compton Comet is a liquid rocket to be built, tested and flown at the RRS. It is a larger vehicle design which has several parts built and will use a surplus XLR11 single-chamber fueled by LOX and a 75% ethanol-water blend, It is a very ambitious project for the Compton College STEM students but it will provide an excellent means of learning practical skills.

Engine section built from a surplus tailcone and ethanol-LOX engine sits on the workbench

UCLA is continuing Project Ares for this next academic year. Last year’s liquid rocket vehicle design was in its final preparations for a Spring 2020 launch at FAR until the pandemic closed campuses around the country including the UCLA Lab. UCLA invited a few RRS members to attend their preliminary design review by teleconference. They are proceeding with several design improvements from last year’s vehicle design and when their laboratory access is restored under carefully regulated conditions, they hope to have another static fire at the RRS and flight from FAR next spring.

Thr Rocket Project at UCLA

Richard Garcia had started a design for a small liquid fueled rocket that would be easier to build and serve as the basis for a common or standard design for society members wanting to test and fly a liquid rocket at reasonable cost. Propellants are ethanol and liquid oxygen. The design has features proven from past successful liquid motor testing at the RRS MTA. The first prototype of the small 125 lbf motor is in build now. After successful demonstrations of the motor in hot-fire, the vehicle will be built.

Illustration of the RRS standard liquid rocket concept

There is a rocket hangar space opening up at the Compton/Woodley Airport which RRS members will soon have access. It has been a goal to have a work space within the city centrally located for most of our members. Operations at the rocket hangar will be limited to construction activities and small-scale pressure tests and cold flow operations, but it will offer our members a greater convenience for those with limited working space in their homes. Contact Wilbur Owens and Xavier Marshall for details. Social distancing and mask protocols would apply.

MTA FIREFIGHTING MEASURES

Fires are one of the greatest risks that come with amateur rocketry. At the behest of several members we have been discussing way of better preparing to fight fires from our site. The roughly one dozen pressurized water containers we have in our storage container are filled and made ready at every event. These have been useful for containing any fires starting at the pads. The RRS is looking at storing large quantities of water at the MTA. We’re also looking at trailer mounted water tanks that could be pulled by a small all-terrain vehicle (ATV) with a motorized pump spray system. These are commonly found in agricultural locations and would be an excellent addition to help limit the propagation of downrange fires until county resources can arrive.

An example of a 200-300 gallon water tank with a motorized pump system

MTA FACILITY UPGRADES

New restroom facility designs have been discussed over this summer. Concepts have been discussed with contractors and firm cost proposals are being prepared. Issues like cost and permits are important concerns. The society last year approved this facility upgrade project as the top priority.

One of several concepts for a new restroom facility at the RRS MTA under discussion

ANNUAL ELECTIONS FOR EXECUTIVE COUNCIL

Next meeting teleconference will be held on November 13th. After appointing an election chairman, we will be holding nominations for executive council positions at the meeting. Administrative members of society are encouraged to participate as we select our next year’s leadership. Active membership is also required so be certain to pay your dues if you haven’t all year.

If there are any questions or comments, please contact me RRS secretary. You can also follow the RRS on Facebook and on Instagram.