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The paper presents a neutron thermalization unit, “a source of ultracold neutrons”, designed for basic research in a beam research hull reactor. For new-generation experiments in the fields of weak-interaction physics and astrophysics, statistical accuracy associated with high density of ultracold neutrons is necessary. To achieve high density for helium-4 in the source chamber, which is used as a converter of cold neutrons into the ultracold ones, it should be at the temperature of about 1 K. In case of applying vacuum pumping of helium-4 vapors in ultracold neutron sources, it has not yet succeeded to obtain a temperature below 1.4 K. To achieve lower temperatures, the required saturated vapor pressure should be less than 50 Pa, which is impossible due to hydraulic losses. It is proposed to use a heat exchanger where helium-4 will be cooled by helium-3. The reason is that the temperature of helium-3 is more efficiently maintained by vacuum pumping since its saturated vapor pressure is an order of magnitude higher than that of helium-4. However, between two heliums the temperature drop occurs due to Kapitsa jump and thermal bridge between the helium capsule and heat exchanger. To solve this problem, we proposed optimization using numerical simulation on the basis of a mathematical model of thermal processes in a chamber with superfluid helium. The model takes into account the contact thermal resistance of Khalatnikov acoustic mismatch model with a correction coefficient. An example of such optimization is presented for the ultracold neutron source located in Gatchina. The mathematical model was implemented in the general solver based on the finite element method. A heat exchanger design geometry was proposed with the temperature drop equal to 0.2 K; the temperature of helium-4 was achieved by vacuum pumping of helium-3 vapors at the pressure of 850 Pa. The temperature fall from 1.4 K to 1 K will increase the density of ultracold neutrons by almost an order of magnitude, and increase statistical accuracy of experiments with ultracold neutrons carried out in a non-beam research reactor.