CN103033430B - The ultrafast cold test device of a kind of simulation and test method - Google Patents

Abstract

The present invention's ultrafast cold test device of open a kind of simulation and test method, it is by vacuum tank, U-lag, chuck, jig, nozzle and support thereof, gas-holder, cold water storage cistern, water pump forms, concrete steps are as follows: the solenoid valve be installed on wedge is placed in opening, establishment and running test program, sample as resistance electrified regulation, after thermal deformation terminates, fast sample is cooled to a period of time before target temperature, open the gas trap of water pump and the flowing of control gas-holder gas respectively, chilled water is run in pipeline and refrigerating gas is flowed out by gas-holder, concrete time in advance and water pump, the sequencing that gas trap is opened, switch off the pump and control gas-holder gas flowing gas trap.The ultrafast cold analogue means design science of the present invention is reasonable, control simple and easy to do, improve cooling power by heat transfer, convection current and radiation three kinds of modes, reach the ultra-rapid cooling technological requirement of simulated production reality, for the applied research in actual production of ultrafast refrigeration technique provides effective means.

Description

The ultrafast cold test device of a kind of simulation and test method

Technical field

The invention belongs to thermal modeling test technical field, be specifically related to the ultrafast cold test device of a kind of simulation and test method.

Background technology

Ultra Fast Cooling provides on the austenitic basis of work hardening at the continuous rolling process in modern times, take ultra-rapid cooling as core, carrying out ultra-rapid cooling and stopping cooling at dynamic phase trasnsition point, carrying out comparatively Slow cooling subsequently and controlling rolling after-hardening austenite.The rolling process for cooling application of this technology, decrease micro alloying element, alloying element, avoid the large pressure rolling of low temperature, the ferrous materials obtained has the feature of function admirable, saving resource and the energy.

Gleeble-3800 hot modeling test machine GU Generic Unit is by the removable U-type groove in vacuum tank, sample, anvil head, fixture close contact composition thermal deformation device, be connected with hydraulic system again, form closed heating circuit and pressurizing loop, various heating cooling and the test of high temperature tension and compression deformation can be carried out, for simulating heat processing technique process testing, existing GU Generic Unit is not equipped with sample cooling appts, sample cooling can only be realized by the heat transfer of clamp assembly, cooling power is quite limited, the fastest cooling velocity of sample is 20 DEG C/s, the range of application of simulation test is restricted.Be difficult to realize the ultrafast cold analog functuion of cooling speed close to 100 DEG C/s.Existing clamp assembly is tungsten carbide anvil head unit, comprises base, muff, tungsten carbide anvil are first-class, and its overall thermal conductivity can be bad, can not realize the ultra-rapid cooling of sample.Patent " resistor-type thermal simulation the cartridge device " (patent No.: 03259544.1) propose to be provided with the by-pass water cooling system improving sample cooling effect on anvil headstock, chuck can be made to obtain water-cooled, improve the cooling effect of sample, increase sample cooling velocity.But water flowing in anvil headstock, can cause being clipped in the middle sample axial-temperature gradient of anvil head large, be out of shape uneven, make the hot deformation behavior distortion of sample, the precision of simulation test is not high, and the effect of its extraordinary simulation test is affected.

Patent " a kind of sampling emergent cooling device of the thermal simulation machine " (patent No.: be 200720067245.1) that the hydraulic wedge unit be equipped with based on thermal simulation machine proposes, one pipeline punishes into four tunnels at vacuum cover upper inlet, connected by four flexible pipes, another terminating nozzles of flexible pipe, be fixed on anvil headstock by telescoping shoring column, heat eliminating medium enters nozzle by pipeline and is sprayed on sample, realizes sample accelerating cooling.This device shortcoming is that the pressure flow of heat eliminating medium after shunting is limited, and sample cooling velocity increase rate is limited, again because it takes up room greatly, is not suitable for applying at GU Generic Unit.

The cartridge device that patent " a kind of chuck device of thermal simulation machine " (patent No.: ZL201120088251.1) proposes is the effect being realized fast cooling samples by heat radiation and heat transfer, but be difficult to realize slow cooling function at once after rapid cooling by means of only heat radiation and heat conducting mode, can not simulate ultrafast cold technological process, this patent does not propose cooling control method for the ultrafast refrigeration technique of simulation.

Cold temperature is split in ultrafast refrigeration technique requirement and final cooling temperature accurately controls, be only the time period that 1-2s is very short cool time, and often require under a certain specified temp, to treat temperature in this control procedure after cooling fast, although above-mentioned sample cooling appts can realize the quick cooling of sample, but do not propose cooling and start the control with the precise time terminated, late effect when cooling often owing to controlling makes enforcement cooling action postpone, cause cooling not in time, when needing until temperature often again owing to terminating the delay of cooling action, produce cooling overshoot.But the realization of ultrafast refrigeration technique completes often in a short period of time, if still adopt conventional control method, the accuracy that experiment parameter controls will inevitably be had influence on, have influence on follow-up histological test work.

Summary of the invention

The present invention's ultrafast cold test device of open a kind of simulation and test method, to reproduce the application process of ultrafast refrigeration technique in production reality, application hot modeling test machine studies ultrafast refrigeration technique to obtain the desirable microstructure and property of ferrous materials.

The object of the present invention is achieved like this, simulate device that ultrafast cold test method adopts by vacuum tank, U-lag, chuck, jig, nozzle and support thereof, gas-holder, cold water storage cistern, water pump forms, U-lag is placed in vacuum tank, described chuck is made up of two parts: wedge and the anvil head matched with it, wedge clamping anvil head is positioned at U-lag, and held out against by jig, described wedge has 3 holes communicated, one is provided with solenoid valve, the outer connecting hose of all the other holes, the other end of flexible pipe leads to the outside of vacuum tank, be connected with water return outlet with the water delivering orifice of cold water storage cistern respectively, form closed-loop path, the circulation of the water in this closed-loop path provides power by water pump.

The present invention's mounting bracket on U-lag is used for fixed nozzle, and nozzle is connected with flexible pipe, and the other end of flexible pipe leads to the outside of vacuum tank.

Two cooling jet 30 ~ miter angles of the present invention are arranged, be beneficial to convection heat transfer, nozzle is taper.

Gas flowing in gas-holder of the present invention is by the solenoid control be installed on flexible pipe.

It is as follows that the present invention simulates ultrafast cold test method concrete steps:

The solenoid valve be installed on wedge is placed in opening by 1, carry out vacuum pumping, wedge endoporus and vacuum tank is made to have identical vacuum tightness, vacuum state now in wedge can reduce the capacity of heat transmission, when vacuum tightness reaches desired value, close this solenoid valve, pipeline can not be leaked because of water flowing.

2 establishment and running test programs, sample as resistance electrified regulation, because the wedge capacity of heat transmission is poor, sample temperature difference is in the axial direction reduced, and uniformity of temperature profile during sample heating and thermal insulation, to ensure the homogeneity of being out of shape subsequently.

3, after thermal deformation terminates, fast sample is cooled to a bit of time before target temperature temperature, open the gas trap of water pump and the flowing of control gas-holder gas respectively, chilled water is run in pipeline and refrigerating gas is flowed out by gas-holder, the sequencing that concrete time in advance and water pump, gas trap are opened, by the length of the flexible pipe connected, hydraulic pressure and air pressure determine.

Two sections of time concrete computation processes are as follows respectively:

1) cool cycles water lines horizontal component and vertical length are respectively h, H, then according to bernoulli principle, all real fluid movement particles be on same streamline, its functional value had is all identical, then have

$P+\frac{1}{2}\mathrm{\ρ}{v}^2+\mathrm{\ρgH}=C_1---\left(1\right)$

Wherein, P is the pressure of water, and ρ is the density of water, and v is the flow velocity of water, C ₁for constant (C ₁bernoulli equation according to fluid motion is determined)

Water is at pipeline vertical portion and horizontal component flowing time t ₁and t ₂calculated by following formula respectively:

t ₁=H/v （2）

t ₂=2h/v （3）

Formula (1) is substituted into formula (2), (3), the time that cooling circulating water arrives wedge can be obtained:

$t=t_1+t_2=(H+2h)\×\sqrt{\frac{\mathrm{\ρ}}{2(C_1-P-\mathrm{\ρgH})}}---\left(4\right)$

If the start-up time of water pump is t _b, then the unlatching water pump time is in advance:

$t_{s1}=t_1+t_2+t_b=(H+2h)\×\sqrt{\frac{\mathrm{\ρ}}{2(C_1-P-\mathrm{\ρgH})}}+t_b---\left(5\right)$

2) because compressed air quality is less, if the length of its flow line is l, then have

$P+\frac{1}{2}{\mathrm{\ρ}}_0{v_0}^2+{\mathrm{\ρ}}_0\mathrm{gl}=C_0---\left(6\right)$

Wherein, P is the pressure of air, ρ ₀for compressed-air actuated density, v ₀for the flow velocity of air, C ₀for constant (Bernoulli equation according to fluid motion is determined)

Pressurized air is at tube runs time t ₀calculated by following formula:

t ₀=l/v ₀（7）

(6), (7) formula simultaneous obtains:

$t_0=l/\sqrt{\frac{2\×(C_0-P-{\mathrm{\ρ}}_0\mathrm{gl})}{{\mathrm{\ρ}}_0}}---\left(8\right)$

If the response time of gas trap is t _d, then the unlatching gas trap time is in advance:

$t_{s2}=l/\sqrt{\frac{2\×(C_0-P-{\mathrm{\ρ}}_0\mathrm{gl})}{{\mathrm{\ρ}}_0}}+t_d---\left(9\right)$

4, switch off the pump and control gas-holder gas flowing gas trap.

Due to ultrafast refrigeration technique, after requiring to implement cooling fast, need to treat that temperature is at a certain specific temperature value, and when carrying out the experiment of this technical modelling, because chilled water and refrigerating gas all have certain pressure, there is certain inertia, switch off the pump and control gas-holder gas flowing gas trap time, due to the existence of this inertia, cause the cooling overshoot phenomenon of sample.In order to avoid this phenomenon, the in advance a bit of time is needed to carry out shutoff operation.

It is the starting mode of pump time that setting water pump shifts to an earlier date the shut-in time, because the cooling of wedge can not produce considerable influence to cooling overshoot, negligible.Then have

t _f1=t _b

Shut-in time is shifted to an earlier date for gas trap, can according to following formulae discovery:

$t_{f2}=l/2\×\sqrt{\frac{2\×(C_0-P-{\mathrm{\ρ}}_0\mathrm{gl})}{{\mathrm{\ρ}}_0}}+t_d---\left(10\right)$

Sample, under the effect that the thermal convection of the heat transfer of recirculated water and the pressure gas of cooling is common, can reach the cooling velocity 150 DEG C/s of expectation, even higher.

The ultrafast cold analogue means design science of the present invention is reasonable, control simple and easy to do, cooling power is improved by heat transfer, convection current and radiation three kinds of modes, the ultrafast cooling control method of this simulation can make cooling velocity control precisely, reach the ultra-rapid cooling technological requirement of simulated production reality, for the applied research in actual production of ultrafast refrigeration technique provides effective means.

Accompanying drawing explanation

Fig. 1 is ultra-fast cooling device overall construction drawing;

Fig. 2 is nozzle device structure figure;

Fig. 3 a-d switches off the pump and the cooling effect figure of gas trap time in advance;

Fig. 4 is the empirical curve of the specific embodiment of the invention.

In figure: 1 nozzle; 2 supports; 3 U-type groove; 4 wedges; 5 jigs; 6 anvil heads; 7 flexible pipes; 8 vacuum tanks; 9 solenoid valves; 10 cold water storage cisterns; 11 water pumps; 12 gas-holder; 13 samples; 14 gas traps.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in detail

The present invention simulates the device that ultrafast cold test method adopts and is made up of vacuum tank, U-lag, chuck, jig, nozzle and support thereof, cold water storage cistern, water pump and gas-holder as shown in Figure 1, U-lag 3 is placed in vacuum tank 8, wedge 4 clamps anvil head 6 and is positioned at U-lag 3, and held out against by jig 5, composition chuck component, this assembly uses in pairs, and sample 13 is clipped between two anvil heads 6.Described wedge 4 has 3 holes communicated, and one is provided with solenoid valve 9, the outer connecting hose 7 of all the other holes.The other end of flexible pipe 7 leads to the outside of vacuum tank 8, is connected respectively with the water delivering orifice of cold water storage cistern 10 with water return outlet, forms closed-loop path.The circulation of the water in this closed-loop path provides power by water pump 11.Temperature is controlled near 0 degree Celsius in the present embodiment, keep water to be liquid.On U-lag 3, mounting bracket 2 is for fixed nozzle 1, and nozzle 1 is connected with flexible pipe 7, and the other end of flexible pipe 7 leads to the outside of vacuum tank 8, connects with cold water storage cistern 10.Gas flowing in described gas-holder 12 is controlled by the gas trap 14 be installed on flexible pipe.

As shown in Figure 2, the cooling jet the adopted distribution in miter angle up and down, nozzle is taper.Such sample 13, under the effect that the thermal convection of the heat transfer of recirculated water and the pressure gas of cooling is common, reaches the cooling velocity 150 DEG C/s of expectation, even higher.

Said apparatus is adopted to simulate its concrete steps of ultrafast cold test to sample as follows:

1, the solenoid valve 9 be installed on wedge 4 is placed in opening, carry out vacuum pumping, make wedge 4 endoporus have identical vacuum tightness with vacuum tank 8, when vacuum tightness reaches desired value, close this solenoid valve 9, pipeline can not be leaked because of water flowing.

2, test routine is worked out with Quiksim software and Gleeble text editing lingware (GSL), sample 13 as resistance electrified regulation, because wedge 4 capacity of heat transmission when vacuum state is poor, sample 13 temperature difference is in the axial direction reduced, uniformity of temperature profile during sample 13 heating and thermal insulation, to ensure the homogeneity of being out of shape subsequently.

3, gas pressure intensity and chilled water pressure are increased to 7 atmospheric pressure, the hose length of logical refrigerating gas and chilled water selects 6 meters and 7 meters respectively.

If water is at pipeline vertical portion and horizontal component flowing time t ₁and t ₂, cool cycles water lines horizontal component and vertical length are respectively h, H,

According to bernoulli principle, all real fluid movement particles be on same streamline, its functional value had is all identical, then have

$P+\frac{1}{2}\mathrm{\ρ}{v}^2+\mathrm{\ρgH}=C_1---\left(1\right)$

Wherein, P is the pressure of water, and ρ is the density of water, and v is the flow velocity of water, C ₁for constant (C ₁bernoulli equation according to fluid motion is determined)

t ₁=H/v（2）

t ₂=2h/v（3）

Formula (1) is substituted into formula (2), (3) respectively, the time that cooling circulating water arrives wedge can be obtained:

$t=t_1+t_2=(H+2h)\×\sqrt{\frac{\mathrm{\ρ}}{2(C_1-P-\mathrm{\ρgH})}}---\left(4\right)$

If the start-up time of water pump is t _b, then the unlatching water pump time is in advance:

$t_{s1}=t_1+t_2+t_b=(H+2h)\×\sqrt{\frac{\mathrm{\ρ}}{2(C_1-P-\mathrm{\ρgH})}}+t_b---\left(5\right)$

2) because compressed air quality is less, if the length of its flow line is l, then have

$P+\frac{1}{2}{\mathrm{\ρ}}_0{v_0}^2+{\mathrm{\ρ}}_0\mathrm{gl}=C_0---\left(6\right)$

Wherein, P is the pressure of air, ρ ₀for compressed-air actuated density, v ₀for the flow velocity of air, C ₀for constant (Bernoulli equation according to fluid motion is determined)

Pressurized air is at tube runs time t ₀calculated by following formula:

t ₀=l/v ₀（7）

(6), (7) formula simultaneous obtains:

$t_0=l/\sqrt{\frac{2\×(C_0-P-{\mathrm{\ρ}}_0\mathrm{gl})}{{\mathrm{\ρ}}_0}}---\left(8\right)$

If the response time of gas trap is t _d, then the unlatching gas trap time is in advance:

$t_{s2}=l/\sqrt{\frac{2\×(C_0-P-{\mathrm{\ρ}}_0\mathrm{gl})}{{\mathrm{\ρ}}_0}}+t_d---\left(9\right)$

Chilled water arrives wedge 4 and refrigerating gas arrival nozzle 1 time used is respectively 0.8s and 0.5s, and therefore, 0.5s opens gas trap 14 first in advance, then opens water pump 11 through 0.3s.After thermal deformation terminates, when needing to implement ultra-rapid cooling to sample 13, according to above-mentioned interval time, open gas trap 14 and water pump 11 successively;

4, switch off the pump and control gas-holder gas flowing gas trap.

Due to ultrafast refrigeration technique, after requiring to implement cooling fast, need to treat that temperature is at a certain specific temperature value, and when carrying out the experiment of this technical modelling, because chilled water and refrigerating gas all have certain pressure, there is certain inertia, switch off the pump and control gas-holder gas flowing gas trap time, due to the existence of this inertia, cause the cooling overshoot phenomenon of sample.In order to avoid this phenomenon, in advance a period of time is needed to carry out shutoff operation.

Because the cooling of wedge can not produce considerable influence to cooling overshoot, therefore water pump shifts to an earlier date the shut-in time and can be set as starting mode of pump time t _f1=t _b

Shut-in time is shifted to an earlier date for gas trap, can according to following formulae discovery:

$t_{f2}=l/2\×\sqrt{\frac{2\×(C_0-P-{\mathrm{\ρ}}_0\mathrm{gl})}{{\mathrm{\ρ}}_0}}+t_d---\left(10\right)$

Substitute into formula and obtain the gas trap time for (0.5s)

Therefore, 0.5s closes gas trap 14 first in advance, then switches off the pump 11 through 0.3s.

List the cooling effect that the different shut-in time in advance reaches in accompanying drawing 3a-d, the curve that Fig. 3 a is pre-set time when being zero, can find out that cooling has overshoot phenomenon, therefore, need switch off the pump in advance and gas trap; The curve that Fig. 3 b is pre-set time when being 0.3s, still have overshoot, but overshoot reduces to some extent; Fig. 3 c is for switching off the pump in advance further and gas trap, time advance 0.7s can be found out that cooling curve does not have overshoot phenomenon, but cooling power is not enough to some extent, Fig. 3 d is the curve of open and close water pump and the gas trap drawn according to formula, there is no overshoot and not enough phenomenon, considering the cooling curve situation in Fig. 3 a-d, is accurately by switching off the pump of drawing of formula (10) and gas trap time as seen.

Based on the parameter that above-mentioned fast cooling device and fast cooling method adopt, carried out following simulated experiment, the empirical curve obtained as shown in Figure 4, can be found out and reach 120 in cooling velocity---during 150 DEG C/s, actual value is consistent completely with setting value.

Simulated experiment is 1.: under vacuum conditions, sample is raised to 1150 DEG C with the speed of 20 DEG C/S, after being incubated 4 minutes, drop to 1000 DEG C with 2 DEG C/S, with 5/S speed, sample compresses by 30% deflection, then 850 DEG C are dropped to 5/S speed with 2 DEG C/S, sample compresses by 60% deflection, drops to 600 DEG C of insulations 2 seconds, then be as cold as room temperature with 5 DEG C/S with 120 DEG C/S.

Simulated experiment is 2.: under vacuum conditions, sample is raised to 1150 DEG C with the speed of 20 DEG C/S, after being incubated 5 minutes, drop to 1000 DEG C with 5 DEG C/S, with 5/S speed, sample compresses by 30% deflection, then 850 DEG C are dropped to 5/S speed with 2 DEG C/S, sample compresses by 60% deflection, drops to 650 DEG C of insulations 2 seconds, then be as cold as room temperature with 20 DEG C/S with 150 DEG C/S.

Claims (6)

1. the ultrafast cold test device of simulation, it is characterized in that, it is by vacuum tank, U-lag, chuck, jig, nozzle and support thereof, gas-holder, cold water storage cistern, water pump forms, U-lag is placed in vacuum tank, described chuck is made up of two parts: wedge and the anvil head matched with it, wedge clamping anvil head is positioned at U-lag, and held out against by jig, described wedge has 3 holes communicated, one is provided with solenoid valve, the outer connecting hose of all the other holes, the other end of flexible pipe leads to the outside of vacuum tank, be connected with water return outlet with the water delivering orifice of cold water storage cistern respectively, form closed-loop path, the circulation of the water in this closed-loop path provides power by water pump.

2. one simulates ultrafast cold test device according to claim 1, it is characterized in that, on described U-lag, mounting bracket is used for fixed nozzle, and nozzle is connected with flexible pipe, and the other end of flexible pipe leads to the outside of vacuum tank, connects with cold water storage cistern.

3. one simulates ultrafast cold test device according to claim 1, it is characterized in that, the cooling jet distribution in 30 ~ miter angle up and down, nozzle is the needle-like of band pin hole.

4. one simulates ultrafast cold test device according to claim 1, and it is characterized in that, gas-holder is connected with solenoid valve through flexible pipe.

5. adopt a kind of test method simulating ultrafast cold test device as described in Claims 1-4 any one, it is characterized in that, concrete steps are as follows:

1) solenoid valve be installed on wedge is placed in opening, carry out vacuum pumping, wedge endoporus and vacuum tank is made to have identical vacuum tightness, vacuum state now in wedge can reduce the capacity of heat transmission, when vacuum tightness reaches desired value, close this solenoid valve, pipeline can not be leaked because of water flowing;

2) establishment and running test program, be used as sample as resistance electrified regulation, because the wedge capacity of heat transmission is poor, sample temperature difference in the axial direction reduced, uniformity of temperature profile during sample heating and thermal insulation;

3) after thermal deformation terminates, fast sample is cooled to a period of time before target temperature, open the gas trap of water pump and the flowing of control gas-holder gas respectively, chilled water is run in pipeline and refrigerating gas is flowed out by gas-holder, the sequencing that concrete time in advance and water pump, gas trap are opened, by the length of the flexible pipe connected, hydraulic pressure and air pressure determine;

4) switch off the pump and control gas-holder gas flowing gas trap;

It opens water pump in advance, gas trap two time period concrete computation processes are as follows respectively:

(1) cool cycles water lines horizontal component and vertical length are respectively h, H, then according to bernoulli principle, all real fluid movement particles be on same streamline, its functional value had is all identical, then have

$P+\frac{1}{2}{\mathrm{\ρv}}^2+\mathrm{\ρ}gH=C_1---\left(1\right)$

Wherein, P is the pressure of water, and ρ is the density of water, and v is the flow velocity of water, C ₁for constant, C ₁bernoulli equation according to fluid motion is determined;

Water is at pipeline vertical portion and horizontal component flowing time t ₁and t ₂calculated by following formula respectively:

t ₁＝H/v (2)

t ₂＝h/v (3)

Formula (1) is substituted into formula (2), (3), the time that cooling circulating water arrives wedge can be obtained:

$t=t_1+t_2=(H+2h)\×\sqrt{\frac{\mathrm{\ρ}}{2(C_1-P-\mathrm{\ρ}gH)}}---\left(4\right)$

If the start-up time of water pump is t _b, then the unlatching water pump time is in advance:

$t_{s1}=t_1+t_2+t_b=(H+2h)\×\sqrt{\frac{\mathrm{\ρ}}{2(C_1-P-\mathrm{\ρ}gH)}}+t_b---\left(5\right)$

(2) because compressed air quality is less, if the length of its flow line is l, then have

$P+\frac{1}{2}{\mathrm{\ρ}}_0{v_0}^2+{\mathrm{\ρ}}_{0}gl=C_0---\left(6\right)$

Wherein, P is the pressure of air, ρ ₀for compressed-air actuated density, v ₀for the flow velocity of air, C ₀for constant

Pressurized air is at tube runs time t ₀calculated by following formula:

t ₀＝l/v ₀(7)

(6), (7) formula simultaneous obtains:

$t=l/\sqrt{\frac{2\×(C_0-P-{\mathrm{\ρ}}_{0}gl)}{{\mathrm{\ρ}}_0}}---\left(8\right)$

If the response time of gas trap is t _d, then the unlatching gas trap time is in advance:

$t_{s2}=l/\sqrt{\frac{2\×(C_0-P-{\mathrm{\ρ}}_{0}gl)}{{\mathrm{\ρ}}_0}}+t_d---\left(9\right).$

6. the ultrafast cold test device experiment method of one simulation according to claim 5, is characterized in that,

Switch off the pump in advance, the time concrete computation process of gas trap be respectively as follows:

It is starting mode of pump time, i.e. t that setting water pump shifts to an earlier date the shut-in time _f1=t _b

Gas trap shifts to an earlier date the shut-in time, according to following formulae discovery:

$t_{f2}=l/2\×\sqrt{\frac{2\×(C_0-P-{\mathrm{\ρ}}_{0}gl)}{{\mathrm{\ρ}}_0}}+t_d---\left(10\right)$