In this chapter, we propose a method for the analytical description of the porous powder materials’ (PPMs) pore distribution based on the pore structure data obtained by mercury porosimetry. The mercury porosimetry method is mostly informative and reliable when speaking about the recurrence of results as compared with other methods of pore distribution investigation. In this chapter, we present a calculation method of correcting experimental data of mercury porosimetry, based on the presentation of a porous body by a statistical model of a serial type.
- porous powder material
- pore volume distribution on size
- average hydraulic pore size
- mercury porosimetry
- statistical model of porous body
Consistent with the multiple functions performed by porous powder materials (PPMs) in various technical devices, a variety of computational methods were developed to assess the effectiveness of the PPMs’ varying pore structure. The relevant calculations use characteristics of the pore structure of the PPMs determined experimentally. The pore volume distribution on size and the average hydraulic pore size are considered as main, most common, characteristics of the pore structure. In this chapter, we propose a method for the analytical description of the PPMs’ pore distribution based on the pore structure data obtained by mercury porosimetry. The mercury porosimetry method is mostly informative and reliable when speaking about recurrence of results as compared with other methods of pore distribution investigation. However, a pore distribution function provided by this method has a distorted character. It increases the volume of small pores that is provided by the narrowing and widening of pore channels on the way of mercury travel. In this chapter, we present a calculation method of correcting experimental data of mercury porosimetry, based on the presentation of a porous body by a statistical model of a serial type.
2. Analytical description of the pore structure
Mercury porosimetry is the most accurate and informative method of studying the pore volume distribution on size. The essence of this method consists of measuring the quantity of mercury pressed in the pre-evacuated porous material, depending on the applied external pressure .
Mercury porosimeter operates as follows. The test sample is placed in a sealed cell which is evacuated; simultaneously the sample is degassed. Then, mercury is introduced into the cell so that mercury completely closes the sample. The mercury is automatically subjected to a predetermined pressure, which is left for a certain time so that the mercury fills all the pores that have the size larger than the critical value. At each table pressure value, the volume of mercury, which went down in the pores of the sample, is measured with a permittance method. According to the experimental data, the integral
and the differential
functions of pore volume distribution on size are calculated. Here
For processing the experimental data, the following technique was developed. Because usually the minimum and maximum pore sizes of the PPMs differ by 1–2 orders of magnitude; the logarithmically uniform pressure table is pre-assigned that corresponds to the logarithmically uniform sequence of pore size values
However, because the automatically applied pressure is not exactly equal to the table value, and may differ from it by 1.5%, the real critical pore sizes
According to the obtained values of the volume of mercury which went down into the sample
the volume values are calculated in the points
In the last expression in the first approximation, as it was said,
and successive approximation of the volume values in the points
A satisfactory accuracy of the approximation of the experimental results (deviation less than 1%) is usually achieved when the number of iterations is
It is easy to obtain the expression for the approximating differential function by differentiating the last expression:
Figures 1 and 2 show the processed, accordingly described, technique data on the experimental study of pore volume distribution on size of the PPMs obtained by sintering a freely poured copper powder PMS-N with a particle size from −315 to 200 μm.
The average hydraulic pore size characterizes the transport and evaporative capacity of the PPMs at full saturation of its pore space with working fluid. Experimental determination of the average hydraulic pore size is based on the use of Laplace’s law. The test sample in the form of a tablet is placed in the sleeve so that the rubber gasket is tightly compressed on the side surface of the sample (Figure 3). At the bottom of the sleeve is a socket, connected to a hose of sufficient length, filled with a liquid which completely wets the sample. The lower end of the hose is placed in a vessel containing the same liquid. A slow rise of the sample is produced. At the moment of separation of the liquid in the hose from the sample, the height of the sample over the liquid level in the vessel
where is the surface tension, is the density of the liquid, and is the acceleration of free fall.
In case of partial draining of the pore space (e.g., with intense evaporation of the liquid inside the PPMs, the action of the mass forces, etc.), the pore size distribution becomes significant. The question arises about the relationship between the function of pore distribution and average pore size of PPMs. Special experiments and subsequent calculations showed that for the PPMs, fabricated with the same technology from different fractions of one powder, such a relationship exists. If the integral function of pore distribution of PPMs with the average hydraulic pore size of
respectively, for the differential pore distribution function:
Express provision is illustrated in Figure 4, where the experimental data for a porous bronze material BrOF10-1 depicts in the conventional coordinates (
3. Method to correct the data of mercury porosimetry
It is known  that pore distribution function, derived from the method of mercury porosimetry, is of a distorted character. It raises the volume of small pores that is caused by narrowing and widening porous channels on the way of mercury travel. Therefore, to use the data of mercury porosimetry in calculations of operational properties of porous materials, a correction of this data is necessary.
The developed method of correction of mercury porosimetry data is based on using a statistical model of a porous body of a serial type [3, 4]. In this model, a porous body is presented as a block of parallel capillaries, each of which consists of a number of successively disposed cylindrical elements. The diameter
Let us consider a process of mercury pressing into a model porous body. Let the mercury be on the left from the plane
Let the function of the mercury capillary part length be
Thereby, we derived an equation to find a function of the length
Thereafter we shall consider a model, in which all the elements have the same length
Justifying such a simplification is based on a smooth-changing a pore diameter. When the value of
giving a step-by-step solution:
The derived solution for
Using the solution (27), we may make a calculation:
A dependence exists between the functions
where we use the designation,
where there is marked
which forms the Cauchy problem together with a boundary condition
to determine a true function of pore quantity distribution on sizes
with which it is possible to obtain:
In Figure 5 the results of calculating the functions of pore volume on size distribution from Eq. (36) and—for comparison—directly from the experimental data are given. It is seen that as a result of data correction of mercury porosimetry, the curves of pore distribution displace considerably in the direction of large pores.
The relationship between the function of pore distribution and average hydraulic pore size, eliminating the need for a time-consuming set of experiments to determine the function of pore distribution of porous powder material, allowing to calculate the pore distribution function of porous powder material with any hydraulic average pore size from the known pore distribution function of the reference porous powder material with the fixed average hydraulic pore size, is explained.
The true function of pore distribution, obtained as a result of correcting mercury porosimetry data, enables to improve considerably the accuracy of calculations of processes and facilities parameters, where porous powder materials are used.