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

MTA launch event, 2021-02-20

by Dave Nordling, Reaction Research Society


The Reaction Research Society held another launch event at the Mojave Test Area (MTA) on February 20, 2021. The weather was not cooperative for much of this day with wind gusts well beyond acceptable limits for launch (> 25 MPH). Our neighbor, Dave Crisalli and his Polaris Propulsion team, were using the Dosa Building as he had construction activities planned but were cancelled for that day. The RRS and Polaris Propulsion were glad to share the Dosa Building as we both made good use of the day.

The three planned objectives (weather permitting) for this MTA launch event were:

  • Build a new pit toilet restroom just north of the original site.
  • Conduct Solar Cat operations at the MTA
  • Conduct model rocket launches from Keith Yoerg’s new wire launcher array

THE ALL-SOLAR POWERED SOLAR CAT PROJECT

Bill Inman and his colleague, John Wells, made the long journey to the MTA from Nevada. Bill had made further improvements to the launching system and solar collector powering the Solar Cat steam rocket. He was able to and a remote tracking motor and drive system to further automate his solar concentrator, but several minor problems in setup prevented a launch that day.

Bill Inman and John Wells examine and prepare the solar collector system from the trailer at the east side of our MTA.
Photovoltaic panel mounted to the front of the collector to power the tracker.
Bill Inman and John Wells set up the latest iteration of the Solar Cat steam rocket from just west of the alpha and beta launch rails

Bill is striving to use an entirely solar powered system including a photovoltaic power system for his auxiliary functions. Because of the east to west passage of the sun through the sky, the steam rocket must be launched in a northerly direction. This is possible if done from the northern or western edges of our launch site.

Although the winds were excessive throughout most of the day, Bill could still conduct some assembly testing and even conduct steam rocket heating operations while keeping the rocket secure on the ground. Launch would only be attempted if the winds lowered in that time. Sadly, much of the day passed in correcting minor problems and system tests. The system proved ready but insufficient sunlight remained that day and launch would have to be conducted from the MTA at the next opportunity.

BUILDING A NEW PIT TOILET AT THE MTA

The society has been examining many improvements to our Mojave Test Area which has stood for over 65 years. The site has been improved over the many years but time has taken its toll and renovations are needed.

The top priority selected by our membership and visitors was the restroom facilities. Our short term plan was to build a second pit toilet while we work on plans for a more luxurious option in the longer term. This effort is viewed as a stopgap solution which will serve our society for at least a few years. Dmitri Timohovich and Wilbur Owens contributed greatly to this effort. With the many people we had at the site, we were able to start and complete the project with time to spare that day.

Our starting point for the project.
Wilbur operates the backhoe to get the trench dug for the sonotube. While Dmitri completes the wooden deck for the new pit toilet,
The precarious job of installing the sonotube once the pit is at the proper depth.
After getting sonotube vertical, the rest of the pit was filled with a few bucket loads of dirt and a few of us with shovels.
The new restroom deck gets placed and aligned with the new sonotube.
The toilet booth is removed from the original concrete platform. Our president, Osvaldo Tarditti, pauses a moment to consider how much crap our society has taken from our visitors and members alike,
The toilet booth is placed on the new platform, but first some further trimming of the sonotube must be done,
RRS secretary, Keith Yoerg, and RRS member, Dave Nordling, stand at the original concrete platform now filled flush to the surface with dirt. The task is nearly complete.
Once firmly affixed to the new platform, the toilet booth was fit checked by Dmitri Timohovich. He is signalling that our pit toilet is now ready for business.

The pit toilet project was a success thanks to both our members providing their physical and material labor and the careful planning and coordination that took place starting in this new year. This improvement project will be only one of several to come. We hope to make our remote testing site both more functional but also a bit more comfortable to all who visit us after many hours drive from the city.

LAUNCHING ROCKETS FROM A NEW MULTI-WIRE RAIL STRUCTURE

With the last hours of the day upon us, the winds had subsided to a more reasonable speed. Keith Yoerg had a few model rockets prepared for launch with commercial motors. He had also built a multi-wire launcher which is a convenient way to display and launch several small vehicles successively.

Max Timohovich (left) views the Baby Bertha and the Big Bertha rockets as they sit on the launch rail made from PVC pipe and fittings.

Second thing introduced at this MTA launch event was a four channel launch box built by Dmitri Timohovich. With a clean wood finish and a rugged latched case, this box proved its function well with the launch of three model rockets that day.

The new launch box was tested at the 2/20.2021 MTA launch event

After some glitches with the electric matches, Keith was able to launch and recover the Baby Bertha (A8-3) and the Big Bertha (A8-3) rockets. We got excellent footage of these classic model rocket types. The last of the three launches was the slightly larger Star Orbiter (E16-6) which left the rails cleanly and the recovery system deployed without issue. Although the winds had subsided sufficiently at ground level, the higher level winds carried the Star Orbiter for a long horizontal trek west well beyond the property line, After some searching, the Star Orbiter was lost to the desert hoping to be recovered

Baby Bertha leaps off the wire rail with its tiny A8-3 motor,
Big Bertha comes back under its parachute landing just to my left. This great video can be seen on the RRS Instagram account.
The last photo of the Star Orbiter as it sits on the pad before the wind carried it far to the west.

IN CLOSING

The team cleaned up the area and put away the gear at sunset. We talked about setting the next launch date in March 2021. We hope to have a new date set soon, likely after March 12th.


MTA launch event, 2021-01-09

by the Reaction Research Society


The Reaction Research Society held its first launch event of the new year on Saturday, January 9, 2021.  Dave Nordling was the pyro-op in charge.  We had a couple rockets prepared and some maintenance work we wanted to continue.  Dmitri Timohovich brought his whole family to the event and we enjoyed grilled burgers there at the Mojave Test Area.  The January winds were light and cool that day.  It was a good day for launch.

Bill Inman enjoys a burger in the George Dosa Building
Everyone relaxing for a short lunch before getting to launch.

Wolfram Blume brought his two-stage rocket, the Gas Guzzler.  His ramjet upper stage was rebuilt from last year’s unfortunate breakage when dropped during loading on the launch rail last year.  3D-printed plastic parts can sometimes be very brittle and care must be taken.

The 1515 launch rail was put into position with some help from Bill Inman and a few others.  Bill Inman is still working on his solar tracker for his latest iteration of the Scalded Cat.  He made the trip from Carson City to the Mojave Test Area to help others with operations and we were very thankful.

Wolfram was able to mount his booster stage on the rails and carefully erect the launcher.  The booster uses a commercial solid motor, an Aerotech “K” motor.  

Wolfram cleans the 1515 rail at the RRS MTA in preparation for mounting the booster
Wolfram’s booster sits on the launch rail

The ramjet for this first flight was loaded with water to simulate the weight, but would not be fired.  The primary goal was to demonstrate the staging and recovery systems powered only by the booster.  Wolfram went back to the loading area to complete the preparations of the upper stage. During a system checkout, the parachute deployment charge fired.  After some careful examination, the source of the problem seemed to be related to errant software commands.  Wolfram aborted his launch attempt and returned with his rocket stages for further examination back in Los Angeles.  Although the charges could be reloaded, he could not be certain that an early parachute deployment would occur and wreck his vehicle during flight.

The Gas Guzzler upper stage ramjet

Dmitri Timohovich and Waldo Stakes worked on completing the welding of the new steel plate on the vertical test stand. This plate on the vertical test stand was damaged during a test failure many years back and late last year was finally cut out and the space grinded to fit a replacement plate.  Unfortunately, the stick welding system would require a different type of welder and a more powerful source to drop a reliable weld.  The welding of the plate will be reattempted at the next event.

With the grinding complete, the plate is fitted and ready to be welded in place

The second launch of the day would be Dave Nordling’s nitrous oxide hybrid rocket.  This 38mm H-sized commercial hybrid motor kit from Contrails Rocketry (H-222 model) had a modified igniter and was mounted in a new 4-inch body made by Larry Hoffing. The prior launch attempt had issues with severing the nylon plastic filling line so the ignition energy was increased with small bit of composite solid propellant ignited by an electric match.

Several minor problems occurred during launch preparations. The nitrous bottle and manifold filling system was working well but the electrical control box failed during tests.  After some discussion, the defective switch box was removed and we were able to fire and get a clean launch

Dave Nordling leans against the old blockhouse with the second build of the hybrid rocket waiting for launch
The hybrid sits on the 1010 launch rail

We repacked the motor before launch and adjusted the vent tube to be more visible. The filling of the rocket went quickly and smoothly, only about 20 seconds before the white stream of liquid could be seen. The filling was stopped and with a short five-count and the rocket was fired. The rocket came off the rails quickly and it seemed that the modified igniter worked. The big problem was the parachute recovery switch wasn’t turned on before launch. This simple oversight would mean a rebuild would be necessary.

The simplest error can lead to sad results.

The rocket was recovered on the north end of the MTA site. It only seemed to reach about 300 feet of altitude. Unfortunately the ballistic landing broke both stages and the internal motor mounts and a complete rebuild is necessary. The motor case was parts were in tact and so it was extracted and will be reloaded,

The hybrid motor seems to arc to the north against wind.

Beckie Timohovich recorded the hybrid flight on her phone, The rocket seemed to immediately curve to the north off the rails opposite of the wind. It seems that the nylon fill line might be still holding fast despite the added solid propellant charge. The 3/16-inch nylon plastic line being strong enough to hold back the 900 psi nitrous pressure, it also poses a challenge to cut cleanly from the ignition charge. A static firing of the motor will be done next to get a better look at how well the fill line severs and measure the thrust curve directly..

The remnant of the fill line from within the hybrid after firing. The end looks smoothly extruded.

After recovering the hybrid rocket and putting away the equipment, we flew a water rocket for Dmitri’s young son. Although very simple, these things are very fun.

Max Timohovich holds the water rocket fired several times at the end of the afternoon.

The event was a partial success and there is more work to be done on our facilities including adding a new toilet facility at our site and welding in the plate on the vertical test stand. The next hybrid rocket launch may be a couple months away, but Bill Inman may have his next design of his solar heated steam rocket ready for launch af fhe MTA in a few weeks. He had his first successful flight in the Nevada desert just before Christmas. He is getting ready for a flight from the RRS MTA.

Bill Inman has his first successful flight of his solar-powered steam rocket on 12/22/2020

Wolfram seemed confident that he too might be ready to try his first launch of the Gas Guzzler at about that same time. If the next launch event occurs before the next monthly meeting on February 12, 2021, the announcement will come through the society email list.

The sun sets at the RRS MTA after a good day.