Space tanker

Delivery vehicle

Estimation of the mass of the heat exchange unit

Vehicle for the delivery of fuel components or raw materials for their production

A specialized vehicle for delivering liquid cargo to orbit has the following advantages:
- unification of mission parameters;
- non-critical dimension;
- reduced required reliability;
- load inertia to dynamic and static loads;
- high density and low cost of cargo;
- no prompt delivery required.

This allows you to:
- significantly reduce development costs;
- apply risky technologies;
- use unconventional fuel components;
- without the inherent inconvenience of using the first stage for initial acceleration with a high degree of reusability and low depreciation per start;
- reduce manufacturing costs by eliminating unnecessary duplication of components and systems;
- reduce the cost of operating reusable delivery vehicles by reasonably limiting inter-flight service;
- reduce the cost of launch preparation, as the operations for pre-flight preparation and payload control are very simple;
- unload the fuel tanks of the last stage, placing the load in the area of the tail compartment;
- optimize the launch path, reduce the required parallax when using an air launch;
- improve weight characteristics, including by significantly reducing the weight of the fairing and the payload adapter;
- refuse cargo insurance.

It is assumed that the use of specialized delivery vehicle instead of general -purpose vehicle will fundamentally reduce the cost of fuel delivery by half, at the same time, the cost of its development will be reduced by 3-4 times. The advanced development of a specialized delivery vehicle should provide half the cost of components loaded into the upper stage, compared to the cost of removing a load of the same mass using a multi-purpose launch vehicle.
The fact that a specialized delivery vehicle is significantly more effective than a multi-purpose one can be seen by comparing the OA RASS with similar launchers of different levels of specialization:

	Payload, t Н=200 (400) km	Payload bay volume, cub. m	Relative dimension
OA RASS, with crew	8.4 (7.0)	45	1
OA RASS, cargo	9.5 (8.1)	54	1
OA, coaxial to the tank	11 (9.5)	54	1
Water tanker	12.9 (11.4)	14	0.25-0.5
RST	15 (13.3)	18-25	0.2-0.5

The concept of a vehicle designed to deliver liquid (dense) cargo to a low orbit should include the following provisions:

1. The use of an air launch from a subsonic carrier aircraft or the use of a booster with a high degree of reusability and low depreciation for the launch , provided that the cost of its creation is moderate.
2. Single-stage output circuit.
3. Application of a three-component engine, using hydrogen as one of the components.
4. Axisymmetric design of disposable tanks, engine thrust center close to the axis.
5. Front position of the liquid hydrogen tank.
6. Placement of payload between the fuel tanks and the engine compartment.
7. Launching with the help of the main propulsion system on the trajectory, at the apogee, touching the orbit of the OA.
8. The use of low-boiling and possibly cryogenic components in the corrective engines.
9. The use of low-boiling and possibly cryogenic components in the corrective remote control system, the use of a simplified mooring and gripping device and passive gripping devices for mooring and holding the spacecraft as part of the orbital complex.
10. Reusable use of at least the most valuable structural elements.
11. The possibility of returning to Earth expensive elements of the rendezvous system, removed from the upper stages during their maintenance and refueling on the OA.

In the case of a drop-off fuel tank, the feasibility of claim 8 depends on the ratio of the masses of the various parts of the structure and the specific impulses of the main and corrective engines, claim 7 and claim 8 are competing.

The vehicle with an energy exchange engine has additional properties:
12. Use of the energy of condensate formation from the fuel components to increase the specific impulse of the engine at the final stage of removal.
13. The absence of a separate payload compartment and the use of an empty tank of the third component for the accumulation of condensate.
14. The ability to refuel the correction engine on the OA.

Further, such a device with an energy-exchange engine is called "reusable space tanker (RST)».

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Reusable space tanker

In general, the RST has in its composition:

- three-component energy-exchange rocket engine (EeRE);
- liquid hydrogen tank;
- liquid oxygen tank;
- a dual-use tank that contains hydrocarbon fuel at the beginning of the flight , and is a condensate storage tank at the final stage of removal.

The cyclogram of the EeRE operation includes three modes:

Three-component (initial). It corresponds to the maximum fuel density and minimum specific thrust impulse. Fuel is consumed from all three tanks.
Two-component (intermediate). The engine operates in the oxygen-hydrogen LRE mode closed circuit, the fuel density is reduced, the specific thrust impulse is increased. Fuel is consumed from the oxygen and hydrogen tanks, the tank double use – empty.
Energy exchange (final). It corresponds to the minimum fuel density and the maximum specific thrust impulse. Fuel is consumed from oxygen and hydrogen fuel the resulting condensate is drained into a dual-use tank.

Various variants of the RST are possible - both fully reusable and using disposable tanks. Below is the most studied basic version, in which the device has a single-use two-section external fuel tank (EFT), containing tanks of liquid hydrogen and liquid oxygen.

   The RST is an orbiter that includes a dual -use tank and a three-component EeRE. It must have an aerodynamic quality and atmospheric flight controls that ensure a sufficiently comfortable return from orbit and landing in a given area with high accuracy. Presumably, an aircraft-type device will be used, and the landing will be carried out on the airfield runway. The RST must be adapted for frequent use – at least 20-50 launches per year.
   The uncritical dimension allows you to create a RST with a relatively small starting mass, which, in turn, makes it possible to use a booster with a high degree of reusability and low depreciation per launch. It is proposed to use a subsonic carrier aircraft (CA) for this purpose, which is considered a modification of the AN-124. In addition to special devices for attaching the RST to the upper part of the fuselage and launch support equipment, the CA must have a double vertical tail unit spaced at a safe distance from its plane of symmetry. The type of CA (AN-124) determines the maximum mass of RST+EFB-140 t and the mass of payload in the base orbit (H=200 km, i=51°) - 5-7 t . The start from the CA is carried out on the "hill" in the negative overload mode.
   However, if the large-scale effect of reducing efficiency is not excessively large, it makes sense to consider the RST+EFB rocket unit of half the mass – 70 tons, capable of removing up to 2.5-3 tons of payload. Then, in the mode of intensive operation, the RST will be able to provide water to the ORC, equipped with solar panels with a total capacity of 100 kW (85 tons of water and about 30 starts per year). For a block of such mass , it is advisable to develop a special CA, adapted for separating the RST in the cabriolet mode and providing a higher start height for 3-5 km. At the same time, for the CA, safety during separation will increase, and the impact from the RST engine jet will decrease. For the RST itself, this will lead to an increase in the payload mass for the following reasons:

- losses due to an increase in the trajectory angle will decrease;
- the losses due to overexpansion in the nozzle will be reduced, or a nozzle with a higher degree of expansion will be used;
- reduced aerodynamic losses and maximum speed head;
- the actual effect of a higher start will appear.

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Energy-exchange rocket engine

EeRE is a liquid rocket engine that has heat exchangers in the high-pressure fuel paths, in which the fuel components are heated by an oncoming flow of the working fluid consisting of gaseous products of chemical reactions between these components, cooled in the heat exchanger by the fuel components to the state of condensate. From the set of possible chemical components for the combustion chamber and the working fluid , the combination of liquid oxygen and liquid hydrogen is taken as the basis, as the most effective and having a sufficient cooling resource.
The working fluid of the heat exchanger is the combustion products of oxygen and hydrogen with a minimal excess of the latter, i.e., an overheated hydrogen -vapor mixture. In the heat exchanger, the combustion products are cooled, followed by water condensation. Water is supplied from the engine to the dual-use tank, which is an element of the orbiter. The condensate temperature must be lower than the boiling point of the water in the tank (at a pressure of no more than 2-3 atmospheres) during the entire operating time of the engine, with the exception of permissible emissions in transient modes. It is assumed that it will be possible to reach the average temperature of the condensate 50°С.

The figure shows the simplest (demonstration) scheme of an energy-exchange rocket propulsion system that has a heat exchanger-condenser only in the fuel path. The numbers indicate: 1 - fuel tank; 2 - pump fuel high pressure; 3 - highway fuel high pressure; 4 - oxidizer tank; 5 - oxidizer pump high pressure; 6 - the pipeline oxidizer high-pressure; 7 - heat exchanger unit; 8 - turbine; 9 - combustion chamber; 10 - cooling jacket of the camera combustion and nozzle; 11 - chemical reactor (gasifier); 12 - highway condensate; 13 – tank condensate; 14 - discharge pipe; 15 - lowering the throttle on the highway condensate; 16 - jet stream; 17 – the pipeline pressurizes the hydrogen tank; 18 - dryer.

Liquid hydrogen from the fuel tank 1 enters the fuel high pressure pump 2, after which it is fed to the high-pressure fuel line 3. At the same time, liquid oxygen from the oxidizer tank 4 enters the high-pressure pump of the oxidizer 5, after which it is fed to the high-pressure oxidizer line 6. The fuel is fed through the high-pressure path 3 to the heat exchange unit 7, in which it is heated. The heated component rotates the turbine 8 TNA, located on a common shaft with pumps 2 and 5, after which it is discharged into the combustion chamber 9. Part fuel from the path of high pressure 3 passes through the cooling jacket of the camera combustion and nozzle 10 and enters the gas generator 11. A portion of the oxidizer from the main line of high pressure of 6 is fed into the combustion chamber 9, and the other (smaller) part of the gas generator 11. Formed in the last superheated steam passes through the heat exchanger-condenser heat exchange unit 7, where it is cooled and condenseries. The resulting water is fed through the condensate line 12 to the condensate storage tank 13. Excess steam, condensate, and gas impurities from the heat exchanger-condenser of the heat exchange unit 7 are discharged into the fuel line 3 after the turbine 8, which contributes to the stabilization of the heat exchange process. The high pressure in the cooled circuit of the heat exchange unit 7 is maintained by a step-down choke 15 installed on the condensate line 12. The hydrogen released from the condensate during the throttling process is mixed with the boost gas and is displaced by water from the storage tank along the main line 17 to the liquid hydrogen tank 1, maintaining the boost pressure in it. Along the way, this mixture passes through the desiccant 18, which avoids the formation of solid ice crystals in the liquid hydrogen.

In reality, the EeRE can contain both more than one turbopump and more than one heat exchange unit, as well as have a different fuel line scheme. In addition, it is possible to use the heat exchange unit in a supercritical mode (at a water vapor pressure above the critical one), in which there will be no condensation phenomenon, and water will be formed from the cooled products of the gas generator after their decompression.

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Estimation of the mass of the heat exchange unit

Before taking up the development of the EeRE, you should try to determine whether it is possible in principle to create a heat exchanger with such a low mass that does not devalue the idea of the RST. To do this, using the similarity theory [ link ] the mass of the heat exchange cells was estimated. The same theory is used to calculate the cooled combustion chambers of the LRE, in both cases, the heat transfer processes are not fundamentally different.

Since the choice of the material of the walls of the heat exchanger is a difficult task, for simplification, a conditional material with the properties of iron (k-t thermal conductivity 29 J/(s^.m^.degree), ultimate strength 20 kgf/mm² =2^.10⁸ N/m² by t=800°С). The heat exchanger was considered thin-walled (characteristic transverse size of the channels >> the thickness of the walls), and the heating of hydrogen occurred from the liquid state. The results of the evaluation are shown below:

Initial conditions for the hydrogen contour:

density of hydrogen at the tube inlet - 20 kg/m³;
pressure - 250 atm.;
gas inlet temperature - 300°К;
high-speed pressure - 0,4 MPa;
flow rate - 200 m/s.

Initial parameters of the heat exchanger:

wall thickness h=0,4 mm;
wall temperature - 500°К;
characteristic transverse cell size - 2 mm;
working time - 563 s.

Calculation results:

heat flow - 16,4 MW/m²;
surface area of the heat exchanger per 1 ton of condensate - 1,63 m²;
mass of cells per 1 t of condensate - 5,07 kg;
temperature difference on the inner and outer surfaces - 113°.

If the cells are arranged in the form of round tubes, then:

gas flow through one tube - 0,0126 kg/s;
tube length - 3,51 m;
tube mass - 21,5 g;
pressure loss ~ 47 atm.;
safety margin at t=800°С - 400%.

Parameters of the heat exchanger at payload weight 7,5 t:

condensate production rate - 13,32 kg/s;
energy output per kilogram of water produced - 15 MJ;
total heat flow through the heat exchanger - 200 MW;
general flow section for hydrogen - 0,0032 m²;
total number of tubes - 1004 pc.;
total mass of tubes - 38 kg;
the total flow section for the heat carrier - no more than 0,001 m²;
solid steel shell mass – 15 kg;
total mass without end elements - 53 kg;
total weight per 1 ton of payload without end elements– 7,1 kg.

In the calculations of the parameters of the RST, the total increase in the mass of the engine due to the presence of a heat exchange unit was taken into account 33,3 kg per ton of condensate, which is at the mass of payload 7.5 t is 250 kg. When the ratio of components in the combustion chamber is 5 , the increase in the enthalpy of the fuel will be 2.645 MJ/kg, and the temperature of the heated hydrogen will be about 1100 °K.

The assessment showed that there are no fundamental obstacles to the creation of an EeRE. There may be comments on the evaluation conditions. Many factors acting both in the direction of increasing and decreasing the mass of the heat exchanger are not taken into account. But the calculated increase in the mass of the engine is chosen with a sufficient margin to have confidence in the successful solution of the problem. In general, the creation of a heat exchange unit with acceptable mass and energy characteristics is a complex engineering and technological task.

more detailed

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Three-component EeRE

If compared with a similar launch vehicle, equipped with a "traditional" LRE and delivering water, the use of EeRE in its pure form can give no more than 12-15% increase in the mass of payload, and this is not enough to be associated with the new technology. For practical use, a three-component EeRE is interesting. The use of the third, dense component will significantly compensate for the decrease in fuel density corresponding to the optimal energy exchange mode. Compensation is achieved by increasing the proportion of the three-component section in the engine cycle diagram compared to the optimal proportion for a simple three-component LRE. The three-component EeRE generally has three consecutive operating modes:

Three-component mode.
Mode oxygen-hydrogen LRE.
Energy exchange mode.

When switching to each next mode, the fuel density decreases and the specific impulse increases. In addition, it will be possible to use the empty tank of the third component for collecting condensate. The three-component EeRE is able to provide 25% or more of the payload mass gain.

When developing an EeRE circuit, a three-component engine should be chosen as an analog RD-704, developed as part of the Multi-Purpose Aerospace System project (MASS). This LRE has two modes of operation:
- three-component (kerosene, hydrogen(l) and oxygen(l) are used));
- oxygen-hydrogen.

There is no doubt that in all types of three-component engines (including the third component will not be used to cool the combustion chamber. Then it is possible to further increase the efficiency of the product by replacing kerosene with a more energy-saturated component. Due to security reasons, this method is not acceptable for a multi-purpose system, but it is suitable for a specialized launcher that delivers low-cost cargo. Looking at chemical reactions, we can see that some of them give a large energy yield, but, nevertheless, do not they are used in rocket technology. For example, there is a cumulative reaction that competes with the combustion reaction in terms of the increase in enthalpy:

С₂H₂+ 3H₂ = 2СH₄+ 11760 kJ/kg

Compare with:

С₂H₂ + 4О₂= 2СО₂ + H₂О + 11760 kJ/kg

A very important circumstance is that in the first reaction there is no obvious oxidizer, and its product is an excellent fuel. Of course, the real effect of the interaction of acetylene with hydrogen will be greatly distorted by by-products-hydrocarbons of different structures, but the reaction of stoichiometric combustion of these substances will give an unambiguous gorenje output. For liquid components:

С₂H₂+ 3H₂ + 4О₂= 2СО₂ + 4H₂О + 11685 kJ/kg, ρ=0,67 g/sm³

С₂H₂+ 3H₂ + 3О₂= 2СО + 4H₂О + 10880 kJ/kg, ρ=0,60 g/sm³

This Gorenje is not much less than in the combustion reaction of oxygen-hydrogen fuel (~12600 kJ/kg, non-stoichiometric ratio of components), but the fuel density is twice as high. Unfortunately, pure liquid acetylene is unstable, explodes, and therefore cannot be used in LRE. Apparently, such a noticeable effect as acetylene could give is really unattainable, but it is not hopeless to search for other components or mixtures with acceptable performance properties.

There are many energy-saturated hydrocarbon compounds, the applicability of each of them depends not only on their physical and chemical properties, but also on the engine circuit and its specific design implementation. The table shows data on some chemical compounds that can be used in a three-component LRE, in pure form or in a mixture with more inert compounds.

Component	Chemical formula	Melting Point,°К	Boiling point, °К	Heat of formation, kJ/kg	Energy of full hydrogenation products, kJ / kg
Methane	СH₄	90,7	111,7	-4666	0
Acetylene	С₂H₂	192,4	-	8708	11760
Ethylene	С₂H₄	103,7	169,4	1864	6295
Cyclopropane	С₃H₆	273,5	283,0	1266	5772
Methylacetylene	С₃H₄	170,4	249,9	4629	8519
Propadiene	С₃H₄	136,9	238,6	4799	8660
Cyclobutane	С₄H₈	182,8	286,1	506	5108
Dimethylacetylene	С₄H₆	240,9	300,4	2707	6966
1,2-butadiene	С₄H₆	137,0	284,0	3001	7195
1,3-butadiene	С₄H₆	164,2	268,7	2037	6383
Vinyl Acetylene	С₄H₄	135,2	278,7	5853	9422
Butadiine	С₄H₂	236,8	283,4	9451	12039
Benzene	С₆H₆	278,7	353,3	1062	5542
Kerosene	СH_1,843	210	420	-1728	3171

   As a rule, the most energy-saturated hydrocarbons have a smaller proportion of hydrogen in their composition, they correspond to a lower density of three-component fuel. Compounds such as cyclopropane and cyclobutane are exotic and require special production, some of the others are intermediates of chemical production and are available. Benzene, methylacetylene, and dimethylacetylene were once studied as rocket fuels. Benzene gives stable mixtures with four-base hydrocarbons, and can be added to them to stabilize and improve performance properties.
   All of these compounds, with the exception of methane, are significantly more expensive than kerosene. The cost of both them, and the technology of handling them, can reduce the efficiency of their use - one kilogram of weight gain of PN will require from 20 to 100 kg of hydrocarbons.
   Energy-saturated fuels can also be successfully used in cargo vehicles that do not use EeRE. However, the optimal fuel used as the third component in this case may be different. This will depend on the volume of the tank of the third component – it has a different effect on the efficiency of the device with the EeRE and the device of the usual scheme.

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Comparison of the RST with a device using a three-component LRE

The table shows the results of evaluating the characteristics of 4 specialized vehicles for delivering water to low orbit, one of them, taken as an analog, carries water from Earth in a separate tank. All devices use three-component fuel l.oxygen- l.hydrogen-kerosene, the sequence of the tanks with the components is the same. All devices have an orbital reusable block and a disposable EFT. The mass gain coefficient of the engine equipped with a heat exchange unit is 0.033 kg/kg of condensate, the mass coefficient of the structure and the remaining components is 40.8 kg/m³. The temperature of the hydrogen at the outlet of the heat exchanger will be about 1100 °C. The table shows:

Analogue - the device using a three-component rocket engine of the RD-701 type.
RST(1). Corresponds to the maximum mass of PG, has an extended two-component (intermediate) section of the EeRE work.
RST(2). By increasing the three-component section, the two-component section is reduced to the technological minimum, while the payload mass decreases slightly, and the volume of EFT decreases significantly – by more than 8%.
RST(3). The three-component section increases and displaces the energy exchange section so much so that the payload mass is equalized with the analog. The RST has the minimum dimensions, the advantage of the EeRE is realized in increasing the weight of the structure, what contributes to the improvement of operational properties, increasing the resource reusable elements and reduce the cost of creating the device.

	Analogue	RST(1)	RST(2)	RST(3)
Type of LRE	RD-704	EeRE	EeRE	EeRE
Starting point mass, t	137,5	137,5	137,5	137,5
Payload mass, т	5,50	6,78	6,70	5,50
Specific impulse, N·s/kg:
1st mode	4067	4067	4067	4067
2nd mode	4508	4508	4508	4508
3rd mode	-	4874	4874	4874
Parameters 3rd mode:
ratio of combustion products	-	4,544	4,544	4,544
ratio of components for the combustion chamber	-	4,50	4,50	4,50
Fuel mass, t	118,2	123,3	123,7	122,9
including:
kerosene	10,3	9,4	11,6	12,4
of hydrogen	9,8	11,7	10,2	9,4
of oxygen	98,1	102,2	101,9	101,1
Fuel volume in EFT, cubic meters	236,3	254,1	232,9	221,5
Cargo tank volume, cubic meters	5,5	11,4	14,2	15,2
Fuel consumption, tons, total	118,2	116,5	117,0	117,4
1st mode	75,1	68,2	84,6	90,4
2nd mode	43,1	17,5	2,0	2,0
3rd mode	-	30,8	30,4	25,0
Construction mass, t	13,800	14,200	13,800	14,600
including:
EFT mass, t	5,790	5,790	5,300	5,040
cargo tank mass, t	0,135	0,280	0.345	0,370
LRE mass gain, t	-	0,226	0,223	0,183
remaining OA mass, t	7,875	7,904	7,932	9,007

   The table clearly shows that the use of the third component allows you to significantly compensate for the increase in tank volume due to a decrease in fuel density when the engine is running in energy exchange mode.
   EFT RST with an equal volume will be easier and cheaper to manufacture than its counterpart, since it consists of two fuel tanks instead of three, and therefore:
- it has a smaller surface area of the tanks;
- one inter-tank compartment instead of two;
- there is no fuel line of the third component with a connector, the total length of the fuel lines is less;
- the center of mass of the fuel is shifted back, as a result of which the average load on the tanks is reduced.
   The use of a device equipped with an EeRE makes it possible to increase the weight of the delivered water by a maximum of 23%. At the same time, the weight of the structure of the returned device increases (the weight gain gives an internal tank of a larger volume and a heavier LRE). It may seem that the increase in the remaining mass of the RST(1) and RST(2) is not sufficient to provide thermal protection for the increase in the mass of the engine and the internal tank. But here it should be taken into account that the RST has less total and dry weight corrective-brake remote control due to the possibility of refueling it on the OS, which does not make sense for an analog. These tables correspond to the output on the LEO, but the RST has a feature of the output trajectory. If the analog is output to the LEO and then makes a two-pulse interorbital flight to the OS, then it is more advantageous for the RST to stretch the final section of the output so that the apogee of the transition orbit touches the orbit of the OS. If we recalculate the characteristics of the devices on the OS, then the RST will not only not have a mass deficit, but will also provide an even greater mass of payload relative to the analog.

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Evaluation characteristics of the heat exchanger

We will evaluate the parameters of the heat exchanger elements using the similarity theory [напр., Кудрявцев В.М. и др. Основы теории и расчёта жидкостных ракетных двигателей. М.:Высш.шк.,1993]. The heat transfer from the wall to the coolant is determined by the formula:

q_ж=a_ж(Т_ст.ж-Т_ж),

where q_ж - perceived convective specific heat flux, a_ж - k-t of heat transfer. The latter is included in the number Nusselt, which is a similarity criterion:

Nu=a_жd/l,

where l - k-t of the thermal conductivity of the liquid, d - characteristic linear size, in a tubular heat exchanger d=d_г=4F/P - hydraulic diameter, where F - cross-sectional area of the channel, P - full wetted perimeter; I. e. d - inner tube diameter.

Nu is a function of the other two similarity criteria:

Re=rWd/m=md/Fm and Pr=mC_p/l,

where r, m, C_p - density, viscosity, heat capacity of a liquid or gas; m - second mass flow rate, W - flow rate.

For liquids, the experimentally obtained Nusselt-Kraussold dependence is more often used:

Nu=0.023Re^0.8Pr^0.4

However , in the case under consideration, the predominant part of the energy is taken by hydrogen in the gaseous state, and the parameters of the gas-dynamic flow significantly depend on the temperature. A formula that takes into account the influence of temperature is closer to reality:

Nu=0.023Re^0.8Pr^0.4(Т_г/Т_ст)^0.55

Substituting the formulas of dimensionless parameters into the equation, we find from it the k-t of heat transfer from the wall to the working fluid:

a_г=0.023(rW)^0.8K/d^0.2(Т_г/Т_ст)^0.55

Then the specific heat flow will be equal to:

q_г=0.023(rW)^0.8(K/d^0.2)(Т_ст-Т_г)(Т_г/Т_ст)^0.55,

q_г=0.023(2pr)^0.4(K/d^0.2)(Т_ст-Т_г)(Т_г/Т_ст)^0.55,

where K=C_p^0.4l^0.6/m^0.4, p=rW²/2 - high-speed pressure. For hydrogen gas K=1792.

Heat flow through the wall:

Q=2S∆Тl_ст/h,

where S, l_ст and h - surface area, thermal conductivity coefficient of the material and wall thickness, ∆Т - temperature difference on the wall.

The pressure loss on the pipe wall is determined by the formula:

∆P=l_трpL/d,

where l_тр - k-t of friction, l_тр=0.0032+0.221/Re^0.237, if Re=10⁵.. .10⁸, L - pipe length.

Of the launch vehicles that have ever been developed, the OA MASS (NPO Molniya) is the closest to the RST, so we will choose it as an analog. Then with equal starting masses The RST will be able to deliver up to 15 or more tons of water to the LEO. Since our calculations are indicative, then consider the case of a high-performance heat exchanger. To do this, select the cyclogram, according to which the RST for 563 seconds of flight in the condensate accumulation mode are gaining 15000 kg of water, while the rate of its production is 26.64 kg/s. Lowest heat flow through the heat exchanger Q=400 MW is realized in the case of water production from liquid components, with an energy output of 15 MJ per kilogram of water produced.

Let the basis of the heat exchanger consist of tubes of circular cross-section made of a material having the properties of iron (l_ст=29 Дж/(с^.m^.град), ultimate strength s₀=20 kgf/mm² =2^.10⁸ N/m² by t=800°С), in which the working fluid - hydrogen-is heated. Choose the inner diameter of the tubes d=2 mm and we will consider them thin-walled (d»h). The density of hydrogen at the entrance to the tubes is assumed to be 20 kg/m³, what corresponds to the pressure 250 atm at 300°К. The temperature of the gas at the inlet to the tubes is assumed Т_г=300°К, wall temperature Т_ст=500°К. High-speed pressure p accept it 0.4 MPa, what corresponds to W=200 m/s.

At these values, a four-fold safety margin of the tubes is provided at t=800°С, and the heat flow will be q=16.4 MW/m².

Then the surface area of the heat exchanger S=Q/q=24.4 m². When the wall thickness is h=0.4 mm the total weight of the tubes will be 76 kg, while the temperature difference on their surfaces-internal and external-will reach 113°.

The gas consumption through one tube will be m=rWF=0.0126 kg/s, total consumption 25.2 kg/s, what corresponds to the increase in the enthalpy of the fuel 2.645 MJ/kg with a component ratio of 5 (one of the calculated variants of the EeRE). Then the total number of tubes will be N=2008 pcs, the length of each of them L=3.51 m, mass - 21.5 g, and their total cross-section F_sum=0.0063 m². When the gas flows in the tube, it is realized Re=4^.10⁶, and the pressure loss on the heat exchanger in the hydrogen path will be ~ 47 atm, what is acceptable.

The heat carrier flowing in the opposite direction has a flow rate of ~ 1.06 kg / kg of hydrogen, which corresponds to 1 mole of water per 8.5 moles of hydrogen. It is clear that the total cross-section of the heat exchanger F_т will not significantly exceed F_sum. By accepting F_t=0.01 m², find the diameter D=11.3 sm. The pressure of hydrogen at the inlet of the tube is 250 atm, at the outlet-about 200 atm. It is advisable to make the pressure of the coolant less, then in case of destruction of the tube, its fragments and the coolant will not get into the fuel path. To calculate the shell, we take P_т=2^.10⁷ Pa (~200 atm). Then, taking into account the five-fold margin of safety, the thickness of the shell wall H must make up H=5^.P_тD/(2s₀), and its mass - m_об=p/4((D+H)²-D²)Lr_ст, where r_ст - wall material density. Since there are no serious obstacles to the organization of cooling of the outer shell of the heat exchanger, it is possible to use strong steel with r_st=7800 kg/m³ and s₀=100 kgf/mm² =10⁹ N/m². Then H=0.57 sm, m_об=28.2 kg, and the total mass of the heat exchanger m_t≈105 kg, which is far from excessive. It can be expected that with such weight characteristics, the main contribution to the weight gain of the EeRE will not be given by the heat exchanger cells themselves, but by the auxiliary elements of the structure. Further, when evaluating the efficiency of the MCT, an estimated engine weight gain of 500 kg was selected for the considered capacitor power, or 33.3 kg per ton of condensate.

The above calculations, apparently, cannot be directly transferred to a real product, since the nonlinear temperature dependence of the thermodynamic parameters will affect a large range of temperature changes. On the other hand, there are additional opportunities to improve the mass perfection of the heat exchanger. Thus, the forced intensification of heat exchange will reduce the length of the tubes by up to two times, and the low-temperature part of the heat exchanger can have thinner walls be made using a material with higher thermal conductivity, such as copper or silver.

It should be noted that when the EeRE dimension changes, the heat exchanger structure will not scale. The geometric parameters of the tubes - the length, diameter of the passage section and wall thickness will remain, only their number will change.

The evaluation of the parameters of the heat exchanger showed that the question of its mass perfection is primarily a matter of engineering and technological art, consisting mainly in the selection or development of materials with the best properties and the creation of the smallest cell structure. The overall task will be to find a compromise between the desire to increase the power of the heat exchange unit and the growth of its mass. The intensification of heat exchange will increase the enthalpy of the products in the combustion chamber and significantly increase the specific gravity pulse, which, in turn, will reduce the ratio of the initial and final mass of the device in the condensate accumulation area. It follows from the Tsiolkovsky formula that this trend should lead to an increase in the total mass of the spacecraft in orbit. On the other hand, when the gas heating temperature increases the mass of the heat exchange unit increases significantly non-linearly. The reason is that as the temperature increases, the thermal conductivity and strength of metals decreases, which requires an increase in the heat exchange surface area and wall thickness. The problem of heating hydrogen to ~800 °К it is solved easily and painlessly for the mass but this is not enough for the device, the main purpose of which is to deliver water to orbit. Heating the hydrogen to ~1100 °К and oxygen up to ~800 °К the rocket engine industry has experience working with materials in the specified environments at the appropriate temperatures. When the temperature of the heated hydrogen increases to ~1400°K and more for the high-temperature part of the heat exchanger, heavy refractory metals and special alloys will be required.

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