CN110967368A - Device and method for measuring heterogeneous content in porous material - Google Patents

Device and method for measuring heterogeneous content in porous material Download PDF

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CN110967368A
CN110967368A CN201910957240.3A CN201910957240A CN110967368A CN 110967368 A CN110967368 A CN 110967368A CN 201910957240 A CN201910957240 A CN 201910957240A CN 110967368 A CN110967368 A CN 110967368A
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porous material
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张腾飞
唐可欣
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Tianjin University
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    • G01MEASURING; TESTING
    • G01NINVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N25/00Investigating or analyzing materials by the use of thermal means
    • G01N25/20Investigating or analyzing materials by the use of thermal means by investigating the development of heat, i.e. calorimetry, e.g. by measuring specific heat, by measuring thermal conductivity
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01NINVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N25/00Investigating or analyzing materials by the use of thermal means
    • G01N25/005Investigating or analyzing materials by the use of thermal means by investigating specific heat

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Abstract

The invention discloses a device and a method for measuring heterogeneous content in a porous material, which comprises a handheld piercing type thin sheet heterogeneous component detection sensor probe and a portable device host; the sensor probe comprises a heating sheet, a temperature sensor, a shell and a handle, and is connected with the equipment host through a cable; the portable equipment host is provided with a main controller, a temperature acquisition and voltage output module, a display screen, a control key, a sensor plug interface, a power supply circuit and other structures. The functions of controlling the start and stop of measurement, collecting and processing measurement data, and calculating and displaying measurement results can be realized. The probe is externally provided with a pointed sheet, and the composition structure of the probe is that a heating sheet is tightly attached to a temperature sensor and is wrapped by a thin hard shell. Compared with the prior double-needle thermal pulse method, the heating sheet and the temperature sensor have almost no distance, so that the heating amount of the heating body is greatly reduced in the measuring process, the distance between the heating body and the temperature sensor is not easy to change, and higher measuring precision can be ensured after long-term use.

Description

Device and method for measuring heterogeneous content in porous material
Technical Field
The invention belongs to the field of material detection and analysis, and relates to a device and a method for measuring heterogeneous content in a porous material based on a plane heat source unsteady state heat conduction principle.
Background
The invasion of heterogeneous components in the porous material can change the characteristics of the material, greatly reduce the heat preservation and noise elimination performance of the material, induce the phenomena of mildew, corrosion and the like, and further shorten the service life of the material. The measurement of heterogeneous content in porous materials is an extremely necessary means and technique. The measurement of heterogeneous content of materials, especially the measurement method of water content of materials and related research are more, but all have certain theoretical and use limitations. The existing measurement methods generally calculate the heterogeneous content in the measured material by measuring other physical quantities having a certain corresponding relationship with the content of the heterogeneous component in the porous material, and can be divided into an electrical method, a thermal method, a radiation method and the like according to different measurement principles. Thermal methods have been widely studied because of their characteristics such as simplicity, low cost, and insensitivity to the salt content in water. The thermal method calculates the heterogeneous content in the material to be measured according to the temperature dynamic response signal of a certain fixed position of a linear or columnar heat source with the measuring distance arranged in the material to be measured.
Most of the current researches are to measure the heterogeneous content of the porous material by using a thermal pulse method based on a thermal method, but the method has the following defects due to the mode of using a heating body and a temperature measuring point which are separately arranged: (1) in the measuring process, in order to collect the effective temperature rise which is easy to be detected at the temperature measuring point, the heating element needs to provide larger heating value. However, the large heat generation amount makes the phenomena of convection and radiation heat transfer in the material to be measured obvious, and the measurement accuracy is influenced. (2) The method is very sensitive to the distance between the heating element and the temperature measuring point, and the measuring precision is greatly reduced if the distance is changed in the installation and use process.
Disclosure of Invention
The invention aims to provide a device and a method for measuring heterogeneous content in a porous material, and solves the problems that heterogeneous measuring equipment in the prior art is susceptible in precision, large in heat productivity, not easy to carry and the like.
In order to achieve the purpose, the technical scheme of the invention is as follows:
a device for measuring heterogeneous content in a porous material is characterized by comprising a handheld piercing type thin sheet heterogeneous component detection sensor probe and a portable device host;
the handheld piercing type thin sheet heterogeneous component detection sensor probe comprises a heating thin sheet, a temperature sensor, a shell, a lead, a handle and the like, wherein the heating thin sheet is tightly attached to the temperature sensor, the shell wraps the heating thin sheet and the temperature sensor, the lead of the heating thin sheet and the lead of the temperature sensor are arranged in the handle, and the heating thin sheet and the temperature sensor probe are connected with a portable equipment host through cables;
the portable equipment host is provided with a main controller, a temperature acquisition and voltage output module, a display screen, a control key, a sensor plug interface, a power supply circuit and other structures. The main controller is respectively and directly connected with the temperature acquisition and voltage output module, the display screen and the control button. The power supply circuit is connected with the display screen, the main controller and other components and supplies power to the components. The temperature acquisition and voltage output module is connected with the sensor plug interface.
The main controller is used for carrying out overall control and data processing on the host, and comprises display screen display control, measurement start-stop control, measurement process control, data acquisition sequential control and the like, converting data acquired by the sensor probe into temperature, and calculating and solving heterogeneous content.
The temperature acquisition and voltage output module outputs constant voltage according to the instruction of the main controller to supply power for the heating sheet of the sensor probe; meanwhile, the temperature signal of the sensor probe is collected and fed back to the main controller.
The display screen is used for displaying the measuring process and the measuring result. The control key is used for a user to control the start and stop measurement of the equipment host.
The overall appearance of the probe of the detection sensor is a pointed thin sheet.
The shell of the probe of the detection sensor is a hard shell.
The heating sheet of the probe of the detection sensor has the same shape as the probe and is in a sheet shape.
The temperature sensor of the probe of the detection sensor is a thermocouple or a thermal resistor. The temperature sensor is closely attached to the heat-generating sheet, and one or more temperature sensors may be disposed on the heat-generating sheet in order to obtain the average temperature of the surface of the heat-generating sheet.
The sensor plug interface is a connection interface of the probe of the detection sensor and the portable equipment host, and the connection interface of the equipment host can be multiple, so that the equipment host can be connected with multiple detection sensors, and multi-position monitoring of heterogeneous content is realized.
The method for measuring heterogeneous content by using the device comprises the following steps:
a. the probe of the handheld penetrating type sheet heterogeneous component detection sensor is penetrated into a material to be detected;
b. after the temperature field distribution in the material to be detected is stable, pressing a control button, and starting heating of the heating sheet;
c. the temperature sensor detects the surface temperature change of the heating sheet, and the temperature acquisition and direct current output module acquires data;
d. after the measurement is finished, the heating sheet stops heating, and the main controller processes the temperature measurement data;
e. the main controller solves the thermophysical property parameters according to a preset calculation program to obtain heterogeneous content;
f. and displaying the heterogeneous content calculation result on a display screen of the host.
The software equipment mainly comprises a heterogeneous content calculation program and host operating software. The manual intervention operation of the equipment host operation program is less, and the one-click operation under simple human-computer interaction is supported.
Compared with the existing method and device for measuring the heterogeneous content in the material by the unsteady thermal method, the method and the device have the beneficial effects that:
1. the sensor probe is in a sheet integrated arrangement mode, and effective temperature change can be detected by using a small heating value. The volume occupied by the power supply in the portable equipment host is reduced, and the overall portability of the portable equipment host is enhanced;
2. the probe of the puncture type sensor is held by a hand, and the shape of the pointed probe is matched with the handle, so that a user can conveniently arrange the probe in a tested material at random, and the measuring position is flexible;
3. the heating thin sheet is tightly attached to the temperature sensor, the whole probe is wrapped by the shell, and the probe is not easy to deform and is beneficial to reducing errors caused by distance change between the heating thin sheet and the temperature sensor or appearance change of the probe in the using process.
The invention applies the unsteady heat conduction principle of the plane heat source to the test of the heterogeneous content in the porous material, and the measuring probe in the integrated form of the sheet plane heat source with the tip and the temperature sensor is penetrated into the porous material to measure the heterogeneous content, thereby realizing the portable measurement of the heterogeneous content in the porous material, and having simple and rapid measuring method and small and portable measuring equipment.
The probe with a sharp front end is matched with the handle, so that the probe can be conveniently penetrated into the porous material to be detected; the portable host has compact structure and small volume, does not need other auxiliary equipment, and can be suitable for various test conditions; the setting of sensor plug interface is convenient for the host computer to connect different sensor and measure. The invention can realize the miniaturization and accurate measurement of the heterogeneous content in the porous material, and has the advantages of low manufacturing cost, simple measurement method, rapidness and accuracy.
Drawings
FIG. 1 is a schematic structural view of a hand-held piercing sheet heterogeneous composition detection sensor probe of the present invention;
FIG. 2 is a front view of a hand-held piercing sheet heterogeneous composition detection sensor probe of the present invention;
in the figure; 1 is a heating sheet; 2 is a temperature sensor; 3 is a metal shell; 4 is a heating circuit; 5 is a temperature sensor signal line; 6 is a handle; 7 is a cable connected with the host;
FIG. 3 is a schematic diagram illustrating the internal operation of the host of the portable device according to the present invention;
FIG. 4 is a flowchart of a calculation procedure for measuring heterogeneous content in a porous material based on the unsteady heat conduction principle of a planar heat source according to the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings.
As shown in FIGS. 1-2, the device for measuring heterogeneous content in porous material of the present invention comprises a hand-held piercing type thin sheet heterogeneous component detection sensor probe and a portable device host;
the handheld piercing type sheet heterogeneous component detection sensor probe comprises a heating sheet 1, the heating sheet 1 is tightly attached to a temperature sensor 2, a shell 3 wraps the heating sheet 1 and the temperature sensor 2, a lead of the heating sheet 2 and a lead of the temperature sensor 3 are arranged in a handle 6, and the sensor probe is connected with a host through a cable 7;
the portable equipment host is provided with a main controller, a temperature acquisition and voltage output module, a display screen, a control key, a sensor plug interface, a power supply circuit and other structures.
The invention provides a method for determining heterogeneous content of a porous material based on a thermal method, which comprises the following steps:
a. the probe is inserted into the material to be tested by holding the handle, the probe is constructed as shown in figure 1,
the heating sheet 1 is closely attached to the temperature sensor 2, and the heating sheet 1 and the temperature sensor 2 are wrapped by the shell 3. The cable 7 is connected with the portable host, and comprises a voltage-stabilizing power supply circuit wire and a temperature sensor signal wire.
b. And standing the tested material containing the probe for a period of time, starting a control button after the temperature distribution is uniform and consistent, and starting the test. The heating sheet generates heat, and the main controller controls the temperature acquisition&The voltage output module collects voltage data V of a temperature measuring point of the temperature sensor at the moment iE,i
c. And after the measurement is finished, the heating sheet stops heating, the main controller processes the temperature measurement data, and the temperature data acquisition is finished. The voltage data is processed by the main controller to obtain heating sheet surface temperature rise data delta TE,i. And the main controller calculates and matches the temperature data with a temperature rise database stored in advance to obtain the thermophysical parameters lambda and rho c of the porous material.
(1) The method for establishing the temperature rise database comprises the following steps of:
Figure RE-GDA0002381642330000041
when x is equal to l, the total weight of the compound,
Figure RE-GDA0002381642330000042
when x is equal to 0, the total weight of the compound,
Figure RE-GDA0002381642330000043
when T is 0, T is T0(4)
Wherein T is the temperature of any point in the porous material to be detected, and K; t is0Is the initial temperature, K, of the porous material to be measured; lambda is the thermal conductivity coefficient of the porous material to be measured, Wm-1K-1(ii) a Rho is the density of the measured porous material, kgm-3(ii) a c is the specific heat capacity of the porous material to be measured, Jkg-1K-1(ii) a ρ c is the volumetric heat capacity of the porous material to be measured, Jm-3K-1
Figure RE-GDA0002381642330000044
The heat flow density of the single-side heat radiation of the sheet plane heat source under the condition of symmetrical heat transfer is W/m2(ii) a Q is the total input power of the plane heat source, W; a is the area of the planar heat source in sheet form, m2(ii) a l is the thickness of the porous material to be measured, m.
The partial differential equation set is solved by referring to the method in H.S. Carlslaw, J.C. Jaeger.connection of Heat in solids.2 d edition.Oxford Clarendon Press,1986:89-112, and the temperature rise of the boundary of the tested porous material x ═ l is resolved as follows:
Figure RE-GDA0002381642330000045
wherein α is the thermal diffusivity, m2S; only the thermophysical parameters λ and ρ c are unknown in the above equation.
Substituting the assumed thermal conductivity coefficient lambda, the volume heat capacity rho c and the heat flux density of the porous material to be detected
Figure RE-GDA0002381642330000046
Can calculate the theoretical temperature rise delta T of the temperature measuring point corresponding to the moment i under the unit heat flux densityq=1,iEnumerating all possible tested material thermophysical parameters lambda and rho c in matching combination as assumed thermophysical parameters to perform temperature rise calculation, thereby obtaining a group of temperature rise databases under unit heat flux density; it is composed ofThe assumed heat conductivity value of the porous material to be detected is that the heat conductivity value of the porous material and the heat conductivity value of the pure heterogeneous component are used as an upper limit and a lower limit during drying; the volumetric heat capacity of the measured porous material has the same physical value.
(2) The measured temperature rise data is matched with the temperature rise database in the following way,
the actual heating power of the heating sheet in the measuring process is compared with delta Tq=1,iMultiplying to obtain theoretical calculated temperature rise delta TM,i(ii) a The temperature rise delta T is calculated theoretically under the combination of certain assumed thermal conductivity and volumetric heat capacityM,iAnd the measured temperature rise delta TE,iAnd (3) comparing, and obtaining a root mean square difference value of the theoretical calculation temperature rise and the measurement temperature rise by utilizing a formula (6):
Figure RE-GDA0002381642330000051
wherein D is the root mean square difference value DEG C between the calculated temperature rise and the measured temperature rise, Delta TM,iThe temperature rise value (DEG C) at the moment i is calculated by a formula, and delta TE,iThe temperature rise value (DEG C) at the moment i is obtained through measurement, and n is the total sampling number of the temperature sensors in single measurement; and for each pair of enumerated thermophysical parameters lambda and rho c, calculating a corresponding time-by-time average difference value D of the theoretical calculation temperature rise and the measured temperature rise, wherein when the value D is minimum, the corresponding rho c value is the volumetric heat capacity rho c of the measured porous material.
d. And the main controller calculates the volume fraction of the heterogeneous content according to the one-to-one correspondence relationship between the volume heat capacity rho c and the heterogeneous content, and displays the calculation result on a display screen of the portable host.
The volumetric heat capacity is related to the heterogeneous content as follows:
ρdrycdrywcwxw=ρc (7)
where ρ isdryThe dry density (kgm) of the tested porous material before heterogeneous invasion-3),cdryThe dry specific heat capacity (Jkg) of the tested porous material before heterogeneous invasion-1K-1),ρdrycdryFor the porous material to be measuredVolumetric heat capacity (Jm) at drying before mass invasion-3·K-1),ρwIs the density (kgm) of pure heterogeneous components-3),cwSpecific heat capacity (Jkg) of pure heterogeneous composition-1K-1),ρwcwVolumetric heat capacity (Jm) as a purely heterogeneous component-3·K-1),xwVolume fraction of pure heterogeneous component, ρ c is the volumetric heat capacity (Jm) of the porous material to be tested-3·K-1);
According to the difference of the volumetric heat capacities before and after the intrusion of the heterogeneous material of the porous material to be detected, the volume fraction of the content of the pure heterogeneous component is solved:
Figure RE-GDA0002381642330000052
wherein x is heterogeneous content volume fraction.
After the temperature sensor is tightly attached to the heating sheet, a metal shell with a relatively thin thickness is wrapped outside the temperature sensor to form a sensor probe, and a lead of the heating sheet and a lead of the temperature sensor are arranged in the handle. The probe is in a pointed shape as a whole, the shell framework has certain rigidity, and the probe can be conveniently inserted into a tested material by holding the handle in actual operation. After the heating sheet is electrified and heated, the temperature rise response of the surface of the heating sheet is measured by the temperature sensor, and the portable equipment host deduces the thermophysical property parameter change of the material before and after the intrusion of the heterogeneous material according to the temperature rise response, so as to calculate the content of the heterogeneous material contained in the material. Because the heating body is tightly attached to the temperature measuring point, the influence on the measurement precision caused by the change of the distance between the heating body and the temperature measuring point does not exist; the heating element reduces the heating value, and can reduce the influence of convection heat transfer, radiation heat transfer or phase change of heterogeneous components on the measurement precision. The design of portable equipment host computer, sensor plug interface through the plug probe, can make the host computer connect different sensor probes respectively, realizes the convenient monitoring of multiposition of heterogeneous content.

Claims (9)

1. A device for measuring heterogeneous content in a porous material is characterized by comprising a handheld piercing type thin sheet heterogeneous component detection sensor probe and a portable device host;
the handheld piercing type thin sheet heterogeneous component detection sensor probe comprises a heating thin sheet, a temperature sensor, a shell, a lead, a handle and the like, wherein the heating thin sheet is tightly attached to the temperature sensor, the shell wraps the heating thin sheet and the temperature sensor, the lead of the heating thin sheet and the lead of the temperature sensor are arranged in the handle, and the heating thin sheet and the temperature sensor probe are connected with a portable equipment host through cables;
the portable equipment host is provided with a main controller, a temperature acquisition and voltage output module, a display screen, a control key, a sensor plug interface and a power supply circuit structure; the main controller is directly connected with the temperature acquisition and voltage output module, the display screen and the control button respectively; the power supply circuit is connected with the display screen, the main controller and other components and supplies power to the components; the temperature acquisition and voltage output module is connected with the sensor plug interface;
the main controller is used for carrying out overall control and data processing on the host, including display screen display control, measurement start-stop control, measurement process control, data acquisition sequential control and the like, converting data acquired by the sensor probe into temperature, and calculating and solving heterogeneous content;
the temperature acquisition and voltage output module outputs constant voltage according to the instruction of the main controller to supply power for the heating sheet of the sensor probe; meanwhile, the temperature signal of the sensor probe is collected and fed back to the main controller.
The display screen is used for displaying the measurement process and the measurement result; the control key is used for a user to control the start and stop measurement of the equipment host.
2. The apparatus for measuring heterogeneous content in porous material according to claim 1, wherein: the overall appearance of the probe of the detection sensor is a pointed thin sheet.
3. The apparatus for measuring heterogeneous content in porous material according to claim 1, wherein: the shell of the probe of the detection sensor is a hard shell.
4. The apparatus for measuring heterogeneous content in porous material according to claim 1, wherein: the heating sheet of the probe of the detection sensor has the same shape as the probe and is in a sheet shape.
5. The apparatus for measuring heterogeneous content in porous material according to claim 1, wherein: the temperature sensor of the probe of the detection sensor is a thermocouple or a thermal resistor; the temperature sensor is closely attached to the heat-generating sheet, and one or more temperature sensors may be disposed on the heat-generating sheet in order to obtain the average temperature of the surface of the heat-generating sheet.
6. The apparatus for measuring heterogeneous content in porous material according to claim 1, wherein: the sensor plug interface is a connection interface of the probe of the detection sensor and the portable equipment host, and the connection interface of the equipment host can be multiple, so that the equipment host can be connected with multiple detection sensors, and multi-position monitoring of heterogeneous content is realized.
7. A method for measuring heterogeneous content in a porous material, wherein the heterogeneous content is measured by using the apparatus of claim 1, comprising the steps of:
a. the probe of the handheld penetrating type sheet heterogeneous component detection sensor is penetrated into a material to be detected;
b. after the temperature field distribution in the material to be detected is stable, pressing a control button, and starting heating of the heating sheet;
c. the temperature sensor detects the surface temperature change of the heating sheet, and the temperature acquisition and direct current output module acquires data;
d. after the measurement is finished, the heating sheet stops heating, and the main controller processes the temperature measurement data;
e. the main controller solves the thermophysical property parameters according to a preset calculation program to obtain heterogeneous content;
f. and displaying the heterogeneous content calculation result on a display screen of the host.
8. The method for measuring heterogeneous content in a porous material according to claim 7, wherein the heterogeneous content calculation method adopted in the step e simplifies the heat transfer process of the measured medium into a one-dimensional unsteady pure solid heat transfer process, and the adopted heat transfer differential equation, boundary conditions and initial conditions are as follows:
Figure FDA0002227745600000021
when x is equal to l, the total weight of the compound,
Figure FDA0002227745600000022
when x is equal to 0, the total weight of the compound,
Figure FDA0002227745600000023
when T is 0, T is T0(4)
Wherein T is the temperature of any point in the porous material to be detected, and K; t is0Is the initial temperature, K, of the porous material to be measured; lambda is the thermal conductivity coefficient of the porous material to be measured, Wm-1K-1(ii) a Rho is the density of the measured porous material, kgm-3(ii) a c is the specific heat capacity of the porous material to be measured, Jkg-1K-1(ii) a ρ c is the volumetric heat capacity of the porous material to be measured, Jm-3K-1
Figure FDA0002227745600000024
The heat flow density of the single-side heat radiation of the sheet plane heat source under the condition of symmetrical heat transfer is W/m2(ii) a Q is the total input power of the plane heat source, W; a is the area of the planar heat source in sheet form, m2(ii) a l is the thickness of the porous material to be measured, m.
9. The method of claim 8, wherein the temperature rise at the x ═ l boundary of the porous material being measured is used to resolve the solution:
Figure FDA0002227745600000025
wherein α is the thermal diffusivity, m2S; only the thermophysical parameters λ and ρ c are unknown in the above equation.
Substituting the assumed thermal conductivity coefficient lambda, the volume heat capacity rho c and the heat flux density of the porous material to be detected
Figure FDA0002227745600000026
Can calculate the theoretical temperature rise delta T of the temperature measuring point corresponding to the moment i under the unit heat flux densityq=1,iEnumerating all possible tested material thermophysical parameters lambda and rho c in matching combination as assumed thermophysical parameters to perform temperature rise calculation, thereby obtaining a group of temperature rise databases under unit heat flux density; wherein the assumed heat conductivity value of the porous material to be tested takes the heat conductivity value of the porous material and the heat conductivity value of the pure heterogeneous component in drying as upper and lower limits; the volumetric heat capacity of the measured porous material has the same physical value; the actual heating power of the heating sheet in the measuring process is compared with delta Tq=1,iMultiplying to obtain theoretical calculated temperature rise delta TM,i
The temperature rise delta T is calculated theoretically under the combination of certain assumed thermal conductivity and volumetric heat capacityM,iAnd the measured temperature rise delta TE,iAnd (3) comparing, and obtaining a root mean square difference value of the theoretical calculation temperature rise and the measurement temperature rise by utilizing a formula (6):
Figure FDA0002227745600000031
wherein D is the root mean square difference value DEG C between the calculated temperature rise and the measured temperature rise, Delta TM,iThe temperature rise value (DEG C) at the moment i is calculated by a formula, and delta TE,iThe temperature rise value (DEG C) at the moment i is obtained through measurement, and n is the total sampling number of the temperature sensors in single measurement; and for each pair of enumerated thermophysical parameters lambda and rho c, calculating a corresponding time-by-time average difference value D of the theoretical calculation temperature rise and the measured temperature rise, wherein when the value D is minimum, the corresponding rho c value is the volumetric heat capacity rho c of the measured porous material.
The volumetric heat capacity is related to the heterogeneous content as follows:
ρdrycdrywcwxw=ρc (7)
where ρ isdryThe dry density (kgm) of the tested porous material before heterogeneous invasion-3),cdryThe dry specific heat capacity (Jkg) of the tested porous material before heterogeneous invasion-1K-1),ρdrycdryThe volume heat capacity (Jm) of the porous material to be tested when dried before the invasion of the heterogeneous material-3·K-1),ρwIs the density (kgm) of pure heterogeneous components-3),cwSpecific heat capacity (Jkg) of pure heterogeneous composition-1K-1),ρwcwVolumetric heat capacity (Jm) as a purely heterogeneous component-3·K-1),xwVolume fraction of pure heterogeneous component, ρ c is the volumetric heat capacity (Jm) of the porous material to be tested-3·K-1);
According to the difference of the volumetric heat capacities before and after the intrusion of the heterogeneous material of the porous material to be detected, the volume fraction of the content of the pure heterogeneous component is solved:
Figure FDA0002227745600000032
wherein x is heterogeneous content volume fraction.
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