·Table of Contents ·Methods and Instrumentation | Thermal effects influence upon penetrant testing sensitivityN.P.Migoun, P.P.Prokhorenko, A.B.GnusinInstitute of Applied Physics, National Academy of Sciences, Minsk, Belarus Contact |
(1) |
Fig 1: Filling of a dead-end cylindrical defect with a liquid. |
Fig 2: Filling of cylindrical dead-end capillaries
with ethanol (s = 2.18·10^{-2} N/m).
1, 3 - calculated curves for l_{¥}/l_{0}; 2, 4 - experimental data for l_{1}/l_{0}, where l_{1} - depth of capillary filling, reached by the meniscus 2 min after the immersion. 1, 2 - R = 7 mm, l_{0} = 8.9 mm; 3, 4 - R = 33 mm, l_{0} = 13.3 mm. |
(2) |
where l - current depth of penetration.
Thus, the pressure of compressed air at the stage of capillary filling changes under the law
(3) |
and reaches the maximum value at l = l_{¥} after a cooling of the air locked in a capillary to the temperature of a liquid T_{1}:
(4) |
where l_{¥} - maximum depth of capillary filling determined by the equality of pressures
(5) |
Expressing air mass m from (1) and taking into account (5), after simple transformations we find the expression for determination the maximum depth of capillary filling
(6) |
where factor k = T_{1}/T_{2} .
The graph of maximum depth of capillary filling depending on the temperature of a capillary pre-heating is shown in Fig. 2.
Experimental researches were conducted with a use of defect simulators - cylindrical dead-end capillaries. The capillary pre-heated in the oven to particular temperatures (40-60°C) was immersed in the dish with ethanol and then a kinetics of liquid penetration into capillary channel was observed with the help of the microscope (see curves 2, 4 in Fig. 2). The interval between the getting the capillary out of the oven and its immersion in the dish with a liquid was about 1 sec. Wherewith it is possible to explain some mismatch between theoretical and experimental curves in Fig. 2 at T > 45°C (since during this time the capillary can be cooled to certain temperature).
At the initial moment of immersion of a heated capillary into ethanol the temperature of a liquid in the area near the mouse of a capillary will change thereby changing its surface tension and, consequently, kinetics of filling. A mass of a liquid used in experiment, many times greater than a mass of a capillary. That is why capillary's temperature will quickly become equal to the temperature of surrounding liquid, that allows us not to take into account this change of temperature in further calculations.
For determination of a relation of method sensitivity with the temperature of a part's pre-heating we consider the process of penetrant extraction from a cylindrical defect.
Following the analysis conducted in [1], let us write the Washburn equation for a penetrant in defect's channel neglecting gravitational and inertial terms:
(7) |
where m^{N} - dynamic viscosity of penetrant, P_{p} - pressure of compressed air in defect's channel, P_{1}(t) - pressure in the region near defect's outlet, t - time.The pressure of compressed air P_{p} can be determined from the formulas (1), (2) as.
(8) |
The equation describing one-dimensional migration of penetrant in a developer above a cylindrical defect along a tested was formulated in [1]:
(9) |
where k_{p} - permeability of porous medium, - capillary pressure in a developer determined as P_{c}^{d}= 2s/ R_{e}, where R_{e} - effective radius of pores of a developer: R_{e} = R_{p} / cosq_{p}, R_{p} - average radius of pores of a developer, q_{p} - wetting angle of particles of a developer, r - coordinate of a front of migrating liquid.
Consider a case of full penetrant extraction from defect's cavity (at l = 0). Formulating the condition corresponding to this case, we write:or
(10) |
This inequality can be transformed to
(11) |
where s- surface tension of penetrant, q
- wetting angle of a capillary wall. Now we can state that under the condition (11) a penetrant will be completely extracted from a defect by the developer, thus forming an indication on its external surface.
The expression allowing determining the value of capillary penetration depth of defects with provision for parts' pre-heating was obtained above. Now we find the expression for calculation the optimal depth of diffusion penetration of defect allowing during further development process completely extraction of all penetrant from a defect's cavity.
Suppose that a developer is applied at the moment when the meniscus reach the depth _{.} Developer application causes the drawing of a penetrant from defect's channel. Thereby a pressure of a gas locked in a channel is decreasing. For simplicity we neglect degassing of a liquid with a pressure decrease. Now by analogy with (2) we use the ideal gas law formulated for a moment of application a layer of powder developer
(12) |
Now we formulate the same equations for the moment, when development process can be considered as completely accomplished:
(13) |
where P_{2} - pressure of gas in a defect, l_{ex} - the length of extracted column of penetrant. Whence the value of pressure P_{2} is determined by expression
(14) |
Extraction of a penetrant by a powder developer from defect's channel takes place only on condition (10), where for pressure P_{p} we use the pressure P_{2}
(15) |
During the penetrant extraction process the pressure P_{2 }of a gas in its channel decreases and the value P^{d}_{c} + P_{2} can be also smaller than the sum of capillary and atmospheric pressures. The sign of equality in (15) corresponds to the moment of accomplishment of development process, when the value of the pressure P_{2} is not enough for further penetrant extraction from defect's channel.
Now we solve the equation (15) with a use of (14) and the condition l_{ex} = l_{¥} + l_{d}, which means, that in case of equality in (15) the penetrant is completely extracted from a channel. It results in the formula for evaluating the maximum value of diffusive penetration depth allowing the extraction of all penetrant from a defect by a developer with given R_{e}:
(16) |
Using the formulas (6), (16) we can calculate the values of capillary and optimal diffusive depths of defect's filling with penetrants both without provision for parts' pre-heating and taking account of it. (Hereinafter in the calculations of a threshold sensitivity, indication diameter and diffusive penetration depth we use the initial temperature of inspected parts equal to 20 °C). The results of such calculations are presented in Table 1 (parameters of the capillary and developer layer are following: depth of a defect l_{0} = 5 mm, effective radius of developer pores R_{e} = R/4).
Let us calculate a time necessary for defect's filling to the depth l_{d} due to diffusion and dissolution of a gas locked in defect's channel after reaching the meniscus the depth l_{¥} [1]:
(17) |
where y = 1 - P_{a} k / (P_{c} + P_{a}); k_{H} - Henry constant of solubility; D - diffusion coefficient of a gas in a liquid; T - temperature of gas.
R = 2 mm | R = 4 mm | R = 7 mm | ||
Without part's pre-heating | 1_{¥} | 8,9.10^{-4} | 4,9.10^{-4} | 2,9.10^{-4} |
2,7.10^{-3} | 1,5.10^{-3} | 8,8.10^{-4} | ||
With provision for part's pre-heating to temperature 60°C | 1_{¥} | 1,4.10^{-3} | 1,0.10^{-3} | 8,6.10^{-4} |
2,2.10^{-3} | 9,3.10^{-4} | 3,1.10^{-4} | ||
Table 1: |
The calculations under the formula (17) shows, that to fill a defects with above-mentioned parameters to a depth
it is necessary the time about 57 minutes. In practice not always there is a possibility to provide such time of penetration. Therefore it is reasonable to use a parts' pre-heating that will reduce a duration of diffusion penetration. For example in considered case it takes 16 minutes to reach the optimal depth
for the defect with radius R = 2mm and for the defect with R = 7mm - about 50 second.
Determine the expression for calculation the threshold sensitivity of a method with provision for part's pre-heating at a stage of penetrant application.
We neglect the evaporation of a penetrant during the development process and suppose that the sum of penetrant's volume in developer layer V_{1} and penetrant's volume in capillary channel V_{2} is equal to the initial volume
(18) |
where n is the factor describing residual depth of capillary filling after excess penetrant removal .
It is easy to see that
(19) |
where P- porosity of a developer, h - its thickness, r - radius of area on an external surface of a developer layer impregnated with penetrant (radius of an indication). Therefore
(20) |
Let us assume that D is a minimum diameter of a visualised indication. Then in order to detect the defect of radius R and depth l_{0} it is necessary, that
(21) |
Solving an inequality (21) with respect to R, it is possible to find the threshold value of defect radius R^{*}, which can be revealed with a use of powder developer with effective radius of pores consistent with the conditions (11) and at a given values of variables included in (21).
Deriving from (21) the expression for evaluation the indication diameter of cylindrical defect we obtain
(22) |
The experimental results of the influence of parts' heating before application of a penetrant upon the area of an indication with a use of control panels are shown in Fig. 3. For comparison the results of theoretical and experimental researches the curves of growth of indication width H_{ind} for a defect such as a crack with parallel walls are represented in Fig. 4.
Fig 3: The pictures of artificially cracked control panels.
Colour contrast penetrant and pre-heating before penetrant application were used: a, b -defect's width H = 3 mm; c, d - defect's width H = 4 mm ; a, c - processing of a part according to traditional technology (without a pre-heating):temperature of the panels T_{1} = 14°C, 20°C; h = 52 mm, 37 mm respectively; b, d - processing of a part with a use of its pre-heating:temperature of a pre-heating T_{2} = 38°C, 47°C; h = 51 mm, 39 mm respectively. |
Fig 4: Width of crack's indication as function of part's pre-heating temperature:
s = 2.18·10^{-2} N/m, n = 0.8; P
= 0.5; l_{0 }= 5 mm, h = 30 mm, H = 5 mm, L = 10 mm (1); l_{0 }= 5 mm, h = 30 mm, H = 10 mm, L = 10 mm (2); l_{0 }= 3 mm, h = 50 mm, H = 5 mm, L = 4 mm (3). |
(23) |
where T_{1}^{d} - the temperature of an air locked in a defect's channel, l_{p} - penetration depth.
Suppose that due to the heating of a part during the whole development process the temperature of an air in defect's channel was increased up to . The change of a solubility of air in penetrant during this process can be neglected.
Now we formulate the same equations for the moment, when development process can be considered as completely accomplished:
(24) |
where P_{2} - pressure of gas in a defect, l_{min} - residual depth of defect's filling. Whence the value of pressure P_{2} is determined by expression
(25) |
where k_{1} = T_{1}^{d} /T_{2}^{d} - relation of part's temperatures corresponding to respectively initial and final moments of a development process.
Now we can obtain from (15) the expression for determination a residual depth of defect's filling l_{min} after fixation of pressures equilibrium
(26) |
The negative values l_{min} in (26) correspond to the case when the whole penetrant is completely drawn from a defect's cavity, namely l_{min} = 0.
For calculation of threshold sensitivity we used the formula (21), modified taking into account the above-mentioned calculations:
(27) |
For example the results of numerical solution of the formula (27) with respect to R for the case R_{e} = R are presented in Fig. 5. One can see from this figure that the part's pre-heating before penetrant application to temperature 60°C in combination with its subsequent heating to the same temperature during development process allows to increase considerably the method's sensitivity. In the case corresponding to Fig. 5 a minimal radius R^{*} of detected defect is changing from 21.2 mm to 7.3 mm.
Fig 5: Threshold sensitivity of a method as function of the depth of defect penetration with/without use of part's pre-heating at two stages of the process (1- without heating, 2 - with use of heating to temperature 60°C at both stages). s = 2.18 · 10^{-2}N/m, n = 0.8, P=0.5 l_{0 }= 5 mm, h = 20 m m, R_{e} = R. |
Similar calculations for a case, when R_{e} = R/4 (h = 40 mm, l_{0} = 3 mm) shows, that it is possible to decrease the threshold sensitivity from 24.5mm to 13.5 mm and to reduce the optimal duration of penetrant application necessary for reaching such sensitivity from 20.5 minutes to 40 seconds.
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