CN103471723B - A kind of new method predicting the day and night temperature of stratosphere balloon - Google Patents

Abstract

Predict a new method for the day and night temperature of stratosphere balloon, it has five large steps: the assumed condition of the new method of the day and night temperature of step one, proposition prediction stratosphere balloon; Step 2, according to the environment for use of stratosphere balloon and the ABC of Engineering Thermodynamics, determine the energy exchange mechanism of stratosphere balloon and each energy exchange item; Step 3, for step 2 propose each energy exchange item set up its thermal model respectively, obtain the Parameter Expression form that concrete each energy exchange item is concrete; Step 4, according to principle of energy balance, set up respectively stratosphere balloon by day with the energy-balance equation at night; Step 5, each energy exchange item step 3 obtained substitute into the energy-balance equation of step 4, the Simple iterative method in recycling method of value solving namely try to achieve stratosphere balloon by day with the temperature at night.The present invention only needs capsule cloth material parameter, hoverheight and time, just can obtain the day and night temperature of stratosphere balloon.

Description

A kind of new method predicting the day and night temperature of stratosphere balloon

Technical field

The invention provides a kind of new method predicting the day and night temperature of stratosphere balloon, belong to technical field of aerospace.

Background technology

Stratosphere balloon is a kind of low cost, high efficiency aircraft, and can perform reconnaissance mission for a long time, carry out goods input etc., have boundless application potential and prospect, therefore, correlation technique becomes the focus studied this year.The temperature of stratosphere balloon when suspending is one of very crucial input parameter for designer, especially most important to choosing of its capsule cloth material, and therefore, how to obtain effective temperature is all the problem that engineering staff needs to solve all the time.Can directly measure its temperature by stratosphere balloon flight test, but experimentation cost is higher, and the cycle of wasting time and energy is long, is also difficult to all situations all to be measured by test; Numerical simulation method needs to set up complicated finite element model, calculation of complex, and counting yield is low; More existing analytic methods, the factor that model is considered is comprehensive not, and theory solves also more complicated.Therefore, the present invention proposes a kind of analytic method predicting the day and night temperature of stratosphere balloon, the method is very simple and practical, only need the capsule cloth material parameter of a small amount of stratosphere balloon, hoverheight and time, just can be easy to the day and night temperature obtaining stratosphere balloon, visible the present invention has Important Academic meaning and engineer applied is worth.

Summary of the invention

The invention provides a kind of new method predicting the day and night temperature of stratosphere balloon, it is easy that the method has calculating, precision advantages of higher, and its technical scheme is as follows:

A kind of new method predicting the day and night temperature of stratosphere balloon of the present invention, the method concrete steps are as follows:

The assumed condition of the new method of the day and night temperature of step one, proposition prediction stratosphere balloon.

Assumed condition comprises:

(1) when stratosphere balloon stand under load, the actual similar water-drop-shaped of shape, in order to simplified model, is reduced to the circle that radius is R by stratosphere balloon.The characteristic dimension L of stratosphere balloon can represent with diameter;

(2) the capsule cloth due to stratosphere balloon is very thin, in micron dimension, therefore, supposes that the temperature of capsule cloth thickness direction is identical, does not consider the heat trnasfer between capsule cloth internal layer and skin;

(3) do not consider the loss of stratosphere balloon helium gas inside, suppose that the weight of stratosphere balloon is constant;

(4) due to earth radius R _efor 6371.23km, and the distance of the sun distance earth is 1.5 × 10 ⁸km, therefore, suppose sunshine arrive the earth overhead time be directional light;

(5) suppose that the earth is thermal equilibrium body;

(6) change of intensity of solar radiation with solar radiation angle and atmospheric extinction coefficient is not considered;

(7) impact of carbon dioxide and ozone infrared radiation is not considered.

Step 2, according to the environment for use of stratosphere balloon and the ABC of Engineering Thermodynamics, determine the energy exchange mechanism of stratosphere balloon and each energy exchange item.

Stratosphere balloon carries out energy exchange mainly through radiation and convection current, comprising: the direct solar radiation item Q that stratosphere balloon episphere absorbs ₁t solar scattered radiation item Q that (), stratosphere balloon absorb ₂t earth reflection radiation term Q that (), stratosphere balloon lower semisphere absorb ₃t earth infrared radiation item Q that (), stratosphere balloon lower semisphere absorb ₄t infrared radiation item Q that (), stratosphere balloon are external ₅t internal reflection infrared radiation item Q that (), stratosphere balloon absorb ₆(t), convection heat transfer item Q between stratosphere balloon and outside atmosphere ₇(t) and the convection heat transfer item Q between stratosphere balloon and internal gas ₈(t).

Step 3, for step 2 propose each energy exchange item set up its thermal model respectively, obtain the Parameter Expression form that concrete each energy exchange item is concrete.

The area of the infinitesimal of stratosphere balloon can be expressed as

In formula, dA is the unit dimension on stratosphere balloon, θ be the latitude of spherical coordinate system or solar radiation angle or sunshine and imfinitesimal method between angle, for the longitude of spherical coordinate system.

The direct solar radiation that infinitesimal on stratosphere balloon episphere absorbs is

dQ ₁(t)=α ₁τI ₀cosθdA(2)

In formula, α ₁for the capsule cloth of stratosphere balloon is to the absorptivity of direct solar radiation, τ is air coefficient of transparency, I ₀for solar constant.

The solar scattered radiation that infinitesimal on stratosphere balloon absorbs is

dQ ₂(t)=α ₁I ₁dA(3)

In formula, I ₁for solar scattered radiation intensity.

The earth reflection radiation that infinitesimal on stratosphere balloon lower semisphere absorbs is

$d{Q}_3\left(t\right)={\mathrm{\α}}_1{\mathrm{\ρ}}_g\mathrm{\τ}{I}_0{\left(\frac{R_e}{R_e+H}\right)}^2\cos \mathrm{\θ}\cos \mathrm{\φdA}---\left(4\right)$

In formula, ρ _gfor ground surface reflectance mean value, R _efor earth radius, the hoverheight of H stratosphere balloon, φ is sunshine direction and the angle between earth center and stratosphere balloon center line connecting direction.

The earth infrared radiation that infinitesimal on stratosphere balloon lower semisphere absorbs is

$d{Q}_4\left(t\right)={\mathrm{\α}}_2\mathrm{\τ}{I}_0{R}_e^2\frac{1-{\mathrm{\ρ}}_g}{4{(R_e+H)}^2}\cos \mathrm{\θdA}---\left(5\right)$

In formula, α ₂for the capsule cloth of stratosphere balloon is to the absorptivity of infrared radiation.

The external infrared radiation of infinitesimal on stratosphere balloon is

$d{Q}_5\left(t\right)=2\mathrm{\ϵ\σ}{T}_1^4\mathrm{dA}---\left(6\right)$

In formula, ε be the capsule cloth of stratosphere balloon to the emissivity of infrared radiation, σ is Si Difen-Boltzmann constant, T ₁for the capsule Bu Wendu of stratosphere balloon.

The internal reflection infrared radiation that infinitesimal on stratosphere balloon absorbs is

$d{Q}_6\left(t\right)=\frac{{\mathrm{\α}}_2\mathrm{\ϵ\σ}{T}_1^4}{1-r}\mathrm{dA}---\left(7\right)$

In formula, r is the reflectivity of stratosphere balloon.

Infinitesimal on stratosphere balloon and the convection heat transfer between outside atmosphere are

dQ ₇(t)=h ₁(T ₁-T ₂)dA(8)

In formula, h ₁for the convection transfer rate between stratosphere balloon and outside atmosphere, T ₂for the temperature of outside atmosphere.

Infinitesimal on stratosphere balloon and the convection heat transfer between helium gas inside are

dQ ₈(t)=h ₂(T ₁-T ₃)dA(9)

In formula, h ₂for the convection transfer rate between stratosphere balloon and helium gas inside, T ₃for the temperature of helium gas inside.

Solar scattered radiation intensity I in formula (3) ₁can be expressed as

$I_1=\frac{1}{2}{I}_0\cos \mathrm{\θ}{\cos }^2\frac{\mathrm{\θ}}{2}---\left(10\right)$

Sunshine direction in formula (4) and the included angle between earth center and stratosphere balloon center line connecting direction can be expressed as

$\mathrm{\φ}=\frac{\mathrm{\π}|12-t|}{12},(6a.m.\≤t\≤18p.m.)---\left(11\right)$

In formula, t represents the time, and from 6:00 AM to afternoon, 18 is the cycle on a daytime.

H in formula (8) and formula (9) ₁and h ₂can be expressed as

$h_1=N{u}_1\frac{{\mathrm{\λ}}_1}{L}---\left(12\right)$

$h_2=N{u}_1\frac{{\mathrm{\λ}}_1}{L}---\left(13\right)$

In formula, Nu ₁for the Nusselt number of forced convection, λ ₁for the heat-conduction coefficient of forced convection, Nu ₂for the Nusselt number of free convection, λ ₂for the heat-conduction coefficient of free convection.

In the thermal analyses of stratosphere balloon, solar constant I ₀for 1367W/m ², air coefficient of transparency τ is 0.9, ground surface reflectance mean value ρ _gbe 0.3, Si Difen-Boltzmann constant σ be 5.6696 × 10 ^-8w (m ²k ⁴), the emissivity of capsule cloth to infrared radiation of stratosphere balloon is equal with absorptivity, i.e. ε=α ₂.

Respectively to formula (9), integration is carried out to formula (2), and substitutes into correlation parameter, the total amount of each energy exchange item can be obtained:

$Q_1\left(t\right)=\left\{\begin{array}{c}1230.3\mathrm{\π}{R}^2{\mathrm{\α}}_1,(6a.m.\≤t\≤18p.m.)\\ 0\end{array}\right.---\left(13\right)$

$Q_2\left(t\right)=\left\{\begin{array}{c}1139.2\mathrm{\π}{\mathrm{\α}}_1{R}^2,(6a.m.\≤t\≤18p.m.)\\ 0\end{array}\right.---\left(14\right)$

$Q_3\left(t\right)=\left\{\begin{array}{c}366.8\mathrm{\π}{R^2\mathrm{\α}}_1{\left(\frac{R_e}{R_e+H}\right)}^2\cos \frac{\mathrm{\π}|12-t|}{12},(6a.m.\≤t\≤18p.m.)\\ 0\end{array}\right.---\left(15\right)$

$Q_5\left(t\right)=\underset{A}{\∫}2\mathrm{\ϵ\σ}{T}_1^4\mathrm{dA}=2\mathrm{A\ϵA\σ}{T}_1^4=45.4\mathrm{\π}{R}^2{\mathrm{\α}}_2\×{10}^{-8}{T}_1^4---\left(17\right)$

$Q_6\left(t\right)=\underset{A}{\∫}\frac{{\mathrm{\α}}_2\mathrm{\ϵ\σ}{T}_1^4}{1-r}\mathrm{dA}=\frac{A{\mathrm{\α}}_2\mathrm{\ϵ\σ}{T}_1^4}{1-r}=\frac{22.7\mathrm{\π}{R}^2{\mathrm{\α}}_2\mathrm{\ϵ}\×{10}^{-8}{T}_1^4}{1-r}---\left(19\right)$

$Q_7\left(t\right)=\underset{A}{\∫}{h}_1(T_1-T_2)\mathrm{dA}=A{h}_1(T_1-T_2)=4\mathrm{\π}{R}^2{h}_1(T_1-T_2)---\left(20\right)$

$Q_8\left(t\right)=\underset{A}{\∫}{h}_2(T_1-T_3)\mathrm{dA}=A{h}_2(T_1-T_3)=4\mathrm{\π}{R}^2{h}_2(T_1-T_3)---\left(21\right)$

Wherein, the infinitesimal described in step 3 refers to the minute cells that stratosphere balloon intercepts, and area is infinitely small.

Step 4, according to principle of energy balance, set up respectively stratosphere balloon by day with the energy-balance equation at night.

For daytime, the gross energy of the capsule cloth of stratosphere balloon is

△Q _film(t)=Q ₁(t)+Q ₂(t)+Q ₃(t)+Q ₄(t)-Q ₅(t)+Q ₆(t)-Q ₇(t)-Q ₈(t)(22)

The gross energy of the helium gas inside of stratosphere balloon is

△Q _gas(t)=Q ₈(t)(23)

When the energy exchange of stratosphere balloon reaches equilibrium state, △ Q _film(t)=0, △ Q _gast ()=0, substitutes into formula (22) and formula (23), can obtain the capsule cloth about stratosphere balloon and helium gas inside energy-balance equation by day:

$\left\{\begin{array}{c}{Q}_1\left(t\right)+Q_2\left(t\right)+Q_3\left(t\right)+Q_4\left(t\right)-Q_5\left(t\right)+Q_6\left(t\right)-Q_7\left(t\right)-Q_8\left(t\right)=0\\ Q_8\left(t\right)=0\end{array}\right.---\left(24\right)$

For night, there is no direct solar radiation, solar scattered radiation and earth reflection radiation, i.e. Q ₁(t)=0, Q ₂(t)=0, Q ₃t ()=0, therefore, the capsule cloth of stratosphere balloon and helium gas inside at the energy-balance equation at night are

$\left\{\begin{array}{c}{Q}_4\left(t\right)-Q_5\left(t\right)+Q_6\left(t\right)-Q_7\left(t\right)-Q_8\left(t\right)=0\\ Q_8\left(t\right)=0\end{array}\right.---\left(25\right)$

Step 5, each energy exchange item step 3 obtained substitute into energy-balance equation (24), (25) of step 4, the Simple iterative method in recycling method of value solving can try to achieve stratosphere balloon by day with the temperature at night.

Formula (13) to formula (21) is updated in formula (24) and formula (25) respectively, can obtains

$\left\{\begin{array}{c}(\frac{22.7\mathrm{\π}{R}^2{\mathrm{\α}}_2\mathrm{\ϵ}\×{10}^{-8}}{1-r}-45.4\mathrm{\π}{R}^2{\mathrm{\α}}_2\×{10}^{-8})T_1^4-4\mathrm{\π}{R}^2{h}_1{T}_1+[366.8\mathrm{\π}{R}^2{\mathrm{\α}}_1{\left(\frac{R_e}{R_e+H}\right)}^2\cos \frac{\mathrm{\π}|12-t|}{12}\\ +2369.5\mathrm{\π}{R}^2{\mathrm{\α}}_1+214.0\mathrm{\π}{R}^2{\mathrm{\α}}_2{\left(\frac{R_e}{R_e+H}\right)}^2+4\mathrm{\π}{R}^2{h}_1{T}_2]=0\\ T_1-T_3=0\end{array}\right.$

(26)

$\left\{\begin{array}{c}(\frac{22.7\mathrm{\π}{R}^2{\mathrm{\α}}_2\mathrm{\ϵ}\×{10}^{-8}}{1-r}-45.4\mathrm{\π}{R}^2{\mathrm{\α}}_2\×{10}^{-8})T_1^4-4\mathrm{\π}{R}^2{h}_1{T}_1+[214.0\mathrm{\π}{R}^2{\mathrm{\α}}_2{\left(\frac{R_e}{R_e+H}\right)}^2+4\mathrm{\π}{R}^2{h}_1{T}_2]=0\\ T_1-T_3=0\end{array}\right.$

(27)

From (26) and formula (27), when the energy exchange of stratosphere balloon reaches equilibrium state, the capsule Bu Wendu of the stratosphere balloon of day and night is equal with helium gas inside temperature.By the Simple iterative method in method of value solving, equation (26) and (27) are solved respectively, can obtain stratosphere balloon by day with capsule Bu Wendu and the helium gas inside temperature at night.

A kind of new method predicting the day and night temperature of stratosphere balloon of the present invention, be characterized in very simple and practical, only need the capsule cloth material parameter of a small amount of stratosphere balloon, hoverheight and time, just can be easy to the day and night temperature obtaining stratosphere balloon.

Accompanying drawing explanation

True form schematic diagram when Fig. 1 a is stratosphere balloon stand under load.

Simplification schematic shapes when Fig. 1 b is stratosphere balloon stand under load.

Fig. 2 is stratosphere balloon energy exchange schematic diagram.

Fig. 3 is the stratosphere balloon surface infinitesimal schematic diagram by sunlight

Fig. 4 is the solar radiation angle schematic diagram of stratosphere balloon surface infinitesimal.

Fig. 5 is stratosphere balloon and the relative position schematic diagram between the earth and the sun.

Fig. 6 is be the FB(flow block) of the method for the invention.

In figure, symbol description is as follows:

R in Fig. 1 b is stratosphere balloon radius.

Q in Fig. 2 ₁t direct solar radiation item that () absorbs for stratosphere balloon episphere, Q ₂t solar scattered radiation item that () absorbs for stratosphere balloon, Q ₃t earth reflection radiation term that () absorbs for stratosphere balloon lower semisphere, Q ₄t earth infrared radiation item that () absorbs for stratosphere balloon lower semisphere, Q ₅t () is the external infrared radiation item of stratosphere balloon, Q ₆t internal reflection infrared radiation item that () absorbs for stratosphere balloon, Q ₇t () is the convection heat transfer item between stratosphere balloon and outside atmosphere, Q ₈t () is the convection heat transfer item between stratosphere balloon and internal gas.

DA in Fig. 3 is the unit dimension on stratosphere balloon, and n is the normal direction of the infinitesimal on stratosphere balloon, θ be the latitude of spherical coordinate system or solar radiation angle or sunshine and imfinitesimal method between angle, for the longitude of spherical coordinate system.

The latitude that θ in Fig. 4 is spherical coordinate system or solar radiation angle or sunshine and imfinitesimal method between angle;

φ in Fig. 5 is sunshine direction and the angle between earth center and stratosphere balloon center line connecting direction.

Embodiment

Fig. 6 is the FB(flow block) of the method for the invention, and the present invention divides five steps to realize, and is specially:

The assumed condition of the new method of the day and night temperature of step one, proposition prediction stratosphere balloon.

Assumed condition comprises:

(1), during stratosphere balloon stand under load, the actual similar water-drop-shaped of shape (as shown in Figure 1a), in order to simplified model, is reduced to the circle (as shown in Figure 1 b) that radius is R by stratosphere balloon.The characteristic dimension L of stratosphere balloon can represent with diameter;

(2) the capsule cloth due to stratosphere balloon is very thin, in micron dimension, therefore, supposes that the temperature of capsule cloth thickness direction is identical, does not consider the heat trnasfer between capsule cloth internal layer and skin;

(3) do not consider the loss of stratosphere balloon helium gas inside, suppose that the weight of stratosphere balloon is constant;

(4) due to earth radius R _efor 6371.23km, and the distance of the sun distance earth is 1.5 × 10 ⁸km, therefore, suppose sunshine arrive the earth overhead time be directional light;

(5) suppose that the earth is thermal equilibrium body;

(6) change of intensity of solar radiation with solar radiation angle and atmospheric extinction coefficient is not considered;

(7) impact of carbon dioxide and ozone infrared radiation is not considered.

Step 2, according to the environment for use of stratosphere balloon and the ABC of Engineering Thermodynamics, be defined as the energy exchange mechanism of stratosphere balloon and each energy exchange item.

Stratosphere balloon carries out energy exchange mainly through radiation and convection current, comprising: the direct solar radiation item Q that stratosphere balloon episphere absorbs ₁t solar scattered radiation item Q that (), stratosphere balloon absorb ₂t earth reflection radiation term Q that (), stratosphere balloon lower semisphere absorb ₃t earth infrared radiation item Q that (), stratosphere balloon lower semisphere absorb ₄t infrared radiation item Q that (), stratosphere balloon are external ₅t internal reflection infrared radiation item Q that (), stratosphere balloon absorb ₆(t), convection heat transfer item Q between stratosphere balloon and outside atmosphere ₇(t) and the convection heat transfer item Q between stratosphere balloon and internal gas ₈(t).Give stratosphere balloon energy exchange schematic diagram in fig. 2.

Step 3, for step one propose each energy exchange item set up its thermal model respectively, obtain the Parameter Expression form that concrete each energy exchange item is concrete.

In figs. 2 and 3, give the infinitesimal of stratosphere balloon, its area can be expressed as

in formula, dA is the unit dimension on stratosphere balloon, θ be the latitude of spherical coordinate system or solar radiation angle or sunshine and imfinitesimal method between angle, for the longitude of spherical coordinate system.

The direct solar radiation that infinitesimal on stratosphere balloon episphere absorbs is

DQ ₁(t)=α ₁τ I ₀in cos θ dA (2) formula, α ₁for the capsule cloth of stratosphere balloon is to the absorptivity of direct solar radiation, τ is air coefficient of transparency, I ₀for solar constant.

The solar scattered radiation that infinitesimal on stratosphere balloon absorbs is

DQ ₂(t)=α ₁i ₁in dA (3) formula, I ₁for solar scattered radiation intensity.

The earth reflection radiation that infinitesimal on stratosphere balloon lower semisphere absorbs is

$d{Q}_3\left(t\right)={\mathrm{\α}}_1{\mathrm{\ρ}}_g\mathrm{\τ}{I}_0{\left(\frac{R_e}{R_e+H}\right)}^2\cos \mathrm{\θ}\cos \mathrm{\φdA}---\left(4\right)$

In formula, ρ _gfor ground surface reflectance mean value, R _efor earth radius, the hoverheight of H stratosphere balloon, φ is sunshine direction and the angle between earth center and stratosphere balloon center line connecting direction, and the earth, position relationship between stratosphere balloon and sun three are as shown in Figure 5.Fig. 4 is the solar radiation angle schematic diagram of stratosphere balloon surface infinitesimal.

The earth infrared radiation that infinitesimal on stratosphere balloon lower semisphere absorbs is

$d{Q}_4\left(t\right)={\mathrm{\α}}_2\mathrm{\τ}{I}_0{R}_e^2\frac{1-{\mathrm{\ρ}}_g}{4{(R_e+H)}^2}\cos \mathrm{\θdA}---\left(5\right)$

In formula, α ₂for the capsule cloth of stratosphere balloon is to the absorptivity of infrared radiation.

The external infrared radiation of infinitesimal on stratosphere balloon is

$d{Q}_5\left(t\right)=2\mathrm{\ϵ\σ}{T}_1^4\mathrm{dA}---\left(6\right)$

In formula, ε be the capsule cloth of stratosphere balloon to the emissivity of infrared radiation, σ is Si Difen-Boltzmann constant, T ₁for the capsule Bu Wendu of stratosphere balloon.

The internal reflection infrared radiation that infinitesimal on stratosphere balloon absorbs is

$d{Q}_6\left(t\right)=\frac{{\mathrm{\α}}_2\mathrm{\ϵ\σ}{T}_1^4}{1-r}\mathrm{dA}---\left(7\right)$

In formula, r is the reflectivity of stratosphere balloon.

Infinitesimal on stratosphere balloon and the convection heat transfer between outside atmosphere are

dQ ₇(t)=h ₁(T ₁-T ₂)dA(8)

In formula, h ₁for the convection transfer rate between stratosphere balloon and outside atmosphere, T ₂for the temperature of outside atmosphere.

Infinitesimal on stratosphere balloon and the convection heat transfer between helium gas inside are

dQ ₈(t)=h ₂(T ₁-T ₃)dA(9)

In formula, h ₂for the convection transfer rate between stratosphere balloon and helium gas inside, T ₃for the temperature of helium gas inside.

Solar scattered radiation intensity I in formula (3) ₁can be expressed as

$I_1=\frac{1}{2}{I}_0\cos \mathrm{\θ}{\cos }^2\frac{\mathrm{\θ}}{2}---\left(10\right)$

Sunshine direction in formula (4) and the included angle between earth center and stratosphere balloon center line connecting direction can be expressed as

$\mathrm{\φ}=\frac{\mathrm{\π}|12-t|}{12},(6a.m.\≤t\≤18p.m.)---\left(11\right)$

In formula, t represents the time, and from 6:00 AM to afternoon, 18 is the cycle on a daytime.

H in formula (8) and formula (9) ₁and h ₂can be expressed as

$h_1=N{u}_1\frac{{\mathrm{\λ}}_1}{L}---\left(12\right)$

$h_2=N{u}_1\frac{{\mathrm{\λ}}_1}{L}---\left(13\right)$

In formula, Nu ₁for the Nusselt number of forced convection, λ ₁for the heat-conduction coefficient of forced convection, Nu ₂for the Nusselt number of free convection, λ ₂for the heat-conduction coefficient of free convection.

The heat-conduction coefficient of forced convection can be expressed as

${\mathrm{\λ}}_1=0.0241{\left(\frac{T_2}{273.15}\right)}^{0.9}---\left(14\right)$

The Nusselt number of forced convection can be expressed as

Nu ₁=(0.037Re ^4/5-817.0)Pr ^1/3(15)

In formula, Re is Reynolds number, and Pr is the Prandtl number of air.

The Prandtl number of air can be expressed as

Pr=0.804-3.25×10 ^-4T ₂(16)

Reynolds number can be expressed as

$\mathrm{Re}=\frac{\mathrm{VL}}{v}---\left(17\right)$

In formula, V is the flowing velocity of stratosphere balloon ambient atmosphere, and L is the characteristic dimension of stratosphere balloon, and ν is the kinetic viscosity of Sudden warming in stratosphere.

The kinetic viscosity of Sudden warming in stratosphere can be expressed as

$v=\frac{1.46\×{10}^{-6}{T}_2^{1.5}}{{\mathrm{\ρ}}_1(T_2+110.4)}---\left(18\right)$

In formula, ρ ₁for the density of air.

The temperature of normal atmosphere and density T ₂and ρ ₁can be expressed as with stratosphere balloon hoverheight H

$\left\{\begin{array}{cc}{T}_2=288.15-6.5H& H\∈\left[\mathrm{0,11}\mathrm{km}\right)\\ T_2=216.65& H\∈[11\mathrm{km},20\mathrm{km})\\ T_2=216.65+(H-20)& H\∈[20\mathrm{km},32\mathrm{km})\end{array}\right.---\left(19\right)$

$\left\{\begin{array}{cc}{\mathrm{\ρ}}_1=1.225\×{\left(\frac{288.15-6.5H}{288.15}\right)}^{4.25588}& H\∈\left[\mathrm{0,11}\mathrm{km}\right)\\ {\mathrm{\ρ}}_1=0.364\×e^{-\frac{H-11}{6.34162}}& H\∈[11\mathrm{km},20\mathrm{km})\\ {\mathrm{\ρ}}_1=0.088\×{\left[\frac{216.65+(H-20)}{216.65}\right]}^{-35.1632}& H\∈[20\mathrm{km},32\mathrm{km})\end{array}\right.---\left(20\right)$

Due to stratosphere balloon reach energy equilibrium time, the free convection energy exchange of its helium gas inside and capsule cloth is 0, so do not need the expression providing the Nusselt number of free convection and the heat-conduction coefficient of free convection again.

In the thermal analyses of stratosphere balloon, solar constant I ₀for 1367W/m ², air coefficient of transparency τ is 0.9, ground surface reflectance mean value ρ _gbe 0.3, Si Difen-Boltzmann constant σ be 5.6696 × 10 ^-8w (m ²k ⁴), the emissivity of capsule cloth to infrared radiation of stratosphere balloon is equal with absorptivity, i.e. ε=α ₂.

Respectively to formula (9), integration is carried out to formula (2), and substitutes into correlation parameter, the total amount of each energy exchange item can be obtained:

$Q_1\left(t\right)=\left\{\begin{array}{c}1230.3\mathrm{\π}{R}^2{\mathrm{\α}}_1,(6a.m.\≤t\≤18p.m.)\\ 0\end{array}\right.---\left(21\right)$

$Q_2\left(t\right)=\left\{\begin{array}{c}1139.2\mathrm{\π}{\mathrm{\α}}_1{R}^2,(6a.m.\≤t\≤18p.m.)\\ 0\end{array}\right.---\left(22\right)$

$Q_3\left(t\right)=\left\{\begin{array}{c}366.8\mathrm{\π}{R^2\mathrm{\α}}_1{\left(\frac{R_e}{R_e+H}\right)}^2\cos \frac{\mathrm{\π}|12-t|}{12},(6a.m.\≤t\≤18p.m.)\\ 0\end{array}\right.---\left(23\right)$

$Q_5\left(t\right)=\underset{A}{\∫}2\mathrm{\ϵ\σ}{T}_1^4\mathrm{dA}=2\mathrm{A\ϵA\σ}{T}_1^4=45.4\mathrm{\π}{R}^2{\mathrm{\α}}_2\×{10}^{-8}{T}_1^4---\left(25\right)$

$Q_6\left(t\right)=\underset{A}{\∫}\frac{{\mathrm{\α}}_2\mathrm{\ϵ\σ}{T}_1^4}{1-r}\mathrm{dA}=\frac{A{\mathrm{\α}}_2\mathrm{\ϵ\σ}{T}_1^4}{1-r}=\frac{22.7\mathrm{\π}{R}^2{\mathrm{\α}}_2\mathrm{\ϵ}\×{10}^{-8}{T}_1^4}{1-r}---\left(26\right)$

$Q_7\left(t\right)=\underset{A}{\∫}{h}_1(T_1-T_2)\mathrm{dA}=A{h}_1(T_1-T_2)=4\mathrm{\π}{R}^2{h}_1(T_1-T_2)---\left(27\right)$

$Q_8\left(t\right)=\underset{A}{\∫}{h}_2(T_1-T_3)\mathrm{dA}=A{h}_2(T_1-T_3)=4\mathrm{\π}{R}^2{h}_2(T_1-T_3)---\left(28\right)$

Wherein, the infinitesimal described in step 3 refers to the minute cells that stratosphere balloon intercepts, and area is infinitely small.

Step 4, according to principle of energy balance, set up respectively stratosphere balloon by day with the energy-balance equation at night.

For daytime, the gross energy of the capsule cloth of stratosphere balloon is

△Q _film(t)=Q ₁(t)+Q ₂(t)+Q ₃(t)+Q ₄(t)-Q ₅(t)+Q ₆(t)-Q ₇(t)-Q ₈(t)(29)

The gross energy of the helium gas inside of stratosphere balloon is

△Q _gas(t)=Q ₈(t)(30)

When the energy exchange of stratosphere balloon reaches equilibrium state, △ Q _film(t)=0, △ Q _gast ()=0, substitutes into formula (29) and formula (30), can obtain the capsule cloth about stratosphere balloon and helium gas inside energy-balance equation by day:

$\left\{\begin{array}{c}{Q}_1\left(t\right)+Q_2\left(t\right)+Q_3\left(t\right)+Q_4\left(t\right)-Q_5\left(t\right)+Q_6\left(t\right)-Q_7\left(t\right)-Q_8\left(t\right)=0\\ Q_8\left(t\right)=0\end{array}\right.---\left(31\right)$

For night, there is no direct solar radiation, solar scattered radiation and earth reflection radiation, i.e. Q ₁(t)=0, Q ₂(t)=0, Q ₃t ()=0, therefore, the capsule cloth of stratosphere balloon and helium gas inside at the energy-balance equation at night are

$\left\{\begin{array}{c}{Q}_4\left(t\right)-Q_5\left(t\right)+Q_6\left(t\right)-Q_7\left(t\right)-Q_8\left(t\right)=0\\ Q_8\left(t\right)=0\end{array}\right.---\left(32\right)$

Step 5, each energy exchange item step 3 obtained substitute into the energy-balance equation of step 4, the Simple iterative method in recycling method of value solving can try to achieve stratosphere balloon by day with the temperature at night.

Formula (21) to formula (28) is updated in formula (31) and formula (32) respectively, can obtains

$\left\{\begin{array}{c}(\frac{22.7\mathrm{\π}{R}^2{\mathrm{\α}}_2\mathrm{\ϵ}\×{10}^{-8}}{1-r}-45.4\mathrm{\π}{R}^2{\mathrm{\α}}_2\×{10}^{-8})T_1^4-4\mathrm{\π}{R}^2{h}_1{T}_1+[366.8\mathrm{\π}{R}^2{\mathrm{\α}}_1{\left(\frac{R_e}{R_e+H}\right)}^2\cos \frac{\mathrm{\π}|12-t|}{12}\\ +2369.5\mathrm{\π}{R}^2{\mathrm{\α}}_1+214.0\mathrm{\π}{R}^2{\mathrm{\α}}_2{\left(\frac{R_e}{R_e+H}\right)}^2+4\mathrm{\π}{R}^2{h}_1{T}_2]=0\\ T_1-T_3=0\end{array}\right.$

(33)

$\left\{\begin{array}{c}(\frac{22.7\mathrm{\π}{R}^2{\mathrm{\α}}_2\mathrm{\ϵ}\×{10}^{-8}}{1-r}-45.4\mathrm{\π}{R}^2{\mathrm{\α}}_2\×{10}^{-8})T_1^4-4\mathrm{\π}{R}^2{h}_1{T}_1+[214.0\mathrm{\π}{R}^2{\mathrm{\α}}_2{\left(\frac{R_e}{R_e+H}\right)}^2+4\mathrm{\π}{R}^2{h}_1{T}_2]=0\\ T_1-T_3=0\end{array}\right.$

(34)

From (33) and formula (34), when the energy exchange of stratosphere balloon reaches equilibrium state, the capsule Bu Wendu of the stratosphere balloon of day and night is equal with helium gas inside temperature.By the Simple iterative method in method of value solving, equation (33) and (34) are solved respectively, can obtain stratosphere balloon by day with capsule Bu Wendu and the helium gas inside temperature at night.

Claims (1)

1. predict a method for the day and night temperature of stratosphere balloon, it is characterized in that: the method concrete steps are as follows:

The assumed condition of the method for the day and night temperature of step one, proposition prediction stratosphere balloon;

Assumed condition comprises:

(1) when stratosphere balloon stand under load, the actual similar water-drop-shaped of shape, in order to simplified model, stratosphere balloon is reduced to the circle that radius is R, the characteristic dimension L of stratosphere balloon represents with diameter;

(2) the capsule cloth due to stratosphere balloon is very thin, in micron dimension, therefore, if the temperature of capsule cloth thickness direction is identical, does not consider the heat trnasfer between capsule cloth internal layer and skin;

(3) loss of stratosphere balloon helium gas inside is not considered, if the weight of stratosphere balloon is constant;

(4) due to earth radius R _efor 6371.23km, and the distance of the sun distance earth is 1.5 × 10 ⁸km, therefore, if be directional light during sunshine arrival earth overhead;

(5) set the earth as thermal equilibrium body;

(6) change of intensity of solar radiation with solar radiation angle and atmospheric extinction coefficient is not considered;

(7) impact of carbon dioxide and ozone infrared radiation is not considered;

Step 2, according to the environment for use of stratosphere balloon and the ABC of Engineering Thermodynamics, determine the energy exchange mechanism of stratosphere balloon and each energy exchange item;

Stratosphere balloon carries out energy exchange by radiation and convection current, comprising: the direct solar radiation item Q that stratosphere balloon episphere absorbs ₁t solar scattered radiation item Q that (), stratosphere balloon absorb ₂t earth reflection radiation term Q that (), stratosphere balloon lower semisphere absorb ₃t earth infrared radiation item Q that (), stratosphere balloon lower semisphere absorb ₄t infrared radiation item Q that (), stratosphere balloon are external ₅t internal reflection infrared radiation item Q that (), stratosphere balloon absorb ₆(t), convection heat transfer item Q between stratosphere balloon and outside atmosphere ₇(t) and the convection heat transfer item Q between stratosphere balloon and internal gas ₈(t);

Step 3, for step 2 propose each energy exchange item set up its thermal model respectively, obtain the Parameter Expression form that concrete each energy exchange item is concrete;

The cartographic represenation of area of the infinitesimal of stratosphere balloon is

In formula, dA is the unit dimension on stratosphere balloon, θ be the latitude of spherical coordinate system or solar radiation angle or sunshine and imfinitesimal method between angle, for the longitude of spherical coordinate system; D θ be the latitude of spherical coordinate system or solar radiation angle or sunshine and imfinitesimal method between the infinitesimal of angle; for the infinitesimal of the longitude of spherical coordinate system;

The direct solar radiation that infinitesimal on stratosphere balloon episphere absorbs is

dQ ₁(t)＝α ₁τI ₀cosθdA(2)

In formula, α ₁for the capsule cloth of stratosphere balloon is to the absorptivity of direct solar radiation, τ is air coefficient of transparency, I ₀for solar constant;

The solar scattered radiation that infinitesimal on stratosphere balloon absorbs is

dQ ₂(t)＝α ₁I ₁dA(3)

In formula, I ₁for solar scattered radiation intensity;

The earth reflection radiation that infinitesimal on stratosphere balloon lower semisphere absorbs is

${\mathrm{dQ}}_3\left(t\right)={\mathrm{\α}}_1{\mathrm{\ρ}}_g{\mathrm{\τI}}_0{\left(\frac{R_e}{R_e+H}\right)}^{2}cos\mathrm{\θ}cos\mathrm{\φ}dA---\left(4\right)$

In formula, ρ _gfor ground surface reflectance mean value, R _efor earth radius, the hoverheight of H stratosphere balloon, φ is sunshine direction and the angle between earth center and stratosphere balloon center line connecting direction;

The earth infrared radiation that infinitesimal on stratosphere balloon lower semisphere absorbs is

${\mathrm{dQ}}_4\left(t\right)={\mathrm{\α}}_2{\mathrm{\τI}}_0{R}_e^2\frac{1-{\mathrm{\ρ}}_g}{4{(R_e+H)}^2}cos\mathrm{\θ}dA---\left(5\right)$

In formula, α ₂for the capsule cloth of stratosphere balloon is to the absorptivity of infrared radiation;

The external infrared radiation of infinitesimal on stratosphere balloon is

dQ ₅(t)＝2εσT ₁ ⁴dA(6)

In formula, ε be the capsule cloth of stratosphere balloon to the emissivity of infrared radiation, σ is Si Difen-Boltzmann constant, T ₁for the capsule Bu Wendu of stratosphere balloon;

The internal reflection infrared radiation that infinitesimal on stratosphere balloon absorbs is

${\mathrm{dQ}}_6\left(t\right)=\frac{{\mathrm{\α}}_2{\mathrm{\ϵ\σT}}_1^4}{1-r}dA---\left(7\right)$

In formula, r is the reflectivity of stratosphere balloon;

Infinitesimal on stratosphere balloon and the convection heat transfer between outside atmosphere are

dQ ₇(t)＝h ₁(T ₁-T ₂)dA(8)

In formula, h ₁for the convection transfer rate between stratosphere balloon and outside atmosphere, T ₂for the temperature of outside atmosphere;

Infinitesimal on stratosphere balloon and the convection heat transfer between helium gas inside are

dQ ₈(t)＝h ₂(T ₁-T ₃)dA(9)

In formula, h ₂for the convection transfer rate between stratosphere balloon and helium gas inside, T ₃for the temperature of helium gas inside;

Solar scattered radiation intensity I in formula (3) ₁be expressed as

$I_1=\frac{1}{2}{I}_0{\mathrm{cos\θcos}}^2\frac{\mathrm{\θ}}{2}---\left(10\right)$

Sunshine direction in formula (4) and the included angle between earth center and stratosphere balloon center line connecting direction are expressed as

$\begin{array}{cc}\mathrm{\φ}=\frac{\mathrm{\π}\left|12-t\right|}{12}& 6a.m.\≤t\≤18p.m.---\left(11\right)\end{array}$

In formula, t represents the time, and from 6:00 AM to afternoon, 18 is the cycle on a daytime;

H in formula (8) and formula (9) ₁and h ₂be expressed as

$h_1={\mathrm{Nu}}_1\frac{{\mathrm{\λ}}_1}{L}---\left(12\right)$

$h_2={\mathrm{Nu}}_2\frac{{\mathrm{\λ}}_2}{L}---\left(13\right)$

In formula, Nu ₁for the Nusselt number of forced convection, λ ₁for the heat-conduction coefficient of forced convection, Nu ₂for the Nusselt number of free convection, λ ₂for the heat-conduction coefficient of free convection;

In the thermal analyses of stratosphere balloon, solar constant I ₀for 1367W/m ², air coefficient of transparency τ is 0.9, ground surface reflectance mean value ρ _gbe 0.3, Si Difen-Boltzmann constant σ be 5.6696 × 10 ^-8w/ (m ²k ⁴), the emissivity of capsule cloth to infrared radiation of stratosphere balloon is equal with absorptivity, i.e. ε=α ₂;

Respectively to formula (9), integration is carried out to formula (2), and substitutes into correlation parameter, namely obtain the total amount of each energy exchange item:

$Q_1\left(t\right)=\{\begin{array}{cc}1230.3{\mathrm{\πR}}^2{\mathrm{\α}}_1& 6a.m.\≤t\≤18p.m.\\ 0& \end{array}---\left(14\right)$

$Q_2\left(t\right)=\{\begin{array}{cc}1139.2{\mathrm{\π\α}}_1{R}^2& 6a.m.\≤t\≤18p.m.\\ 0& \end{array}---\left(15\right)$

$Q_3\left(t\right)=\left\{\begin{array}{cc}366.8{\mathrm{\πR}}^2{\mathrm{\α}}_1{\left(\frac{R_e}{R_e+H}\right)}^{2}cos\frac{\mathrm{\π}\left|12-t\right|}{12}& 6a.m.\≤t\≤18p.m.\\ 0& \end{array}\right.---\left(16\right)$

$Q_5\left(t\right)=\underset{A}{\∫}2{{\mathrm{\ϵ\σT}}_1}^{4}dA=2{{\mathrm{A\ϵA\σT}}_1}^4=45.4{\mathrm{\πR}}^2{\mathrm{\α}}_2\×{10}^{-8}{T_1}^4---\left(18\right)$

$Q_6\left(t\right)=\underset{A}{\∫}\frac{{\mathrm{\α}}_2{{\mathrm{\ϵ\σT}}_1}^4}{1-r}dA=\frac{{\mathrm{A\α}}_2{{\mathrm{\ϵ\σT}}_1}^4}{1-r}=\frac{22.7{\mathrm{\πR}}^2{\mathrm{\α}}_2\mathrm{\ϵ}\×{10}^{-8}{T_1}^4}{1-r}---\left(19\right)$

$Q_7\left(t\right)=\underset{A}{\∫}{h}_1\left(T_1-T_2\right)dA={\mathrm{Ah}}_1(T_1-T_2)=4{\mathrm{\πR}}^2{h}_1(T_1-T_2)---\left(20\right)$

$Q_8\left(t\right)=\underset{A}{\∫}{h}_2\left(T_1-T_3\right)dA={\mathrm{Ah}}_2(T_1-T_3)=4{\mathrm{\πR}}^2{h}_2(T_1-T_3)---\left(21\right)$

Wherein, the infinitesimal described in step 3 refers to the minute cells that stratosphere balloon intercepts, and area is infinitely small;

Step 4, according to principle of energy balance, set up respectively stratosphere balloon by day with the energy-balance equation at night;

For daytime, the gross energy of the capsule cloth of stratosphere balloon is

ΔQ _film(t)＝Q ₁(t)+Q ₂(t)+Q ₃(t)+Q ₄(t)-Q ₅(t)+Q ₆(t)-Q ₇(t)-Q ₈(t)(22)

The gross energy of the helium gas inside of stratosphere balloon is

ΔQ _gas(t)＝Q ₈(t)(23)

When the energy exchange of stratosphere balloon reaches equilibrium state, Δ Q _film(t)=0, Δ Q _gast ()=0, substitutes into formula (22) and formula (23), obtains the capsule cloth about stratosphere balloon and helium gas inside energy-balance equation by day:

$\{\begin{array}{c}{Q}_1\left(t\right)+Q_2\left(t\right)+Q_3\left(t\right)+Q_4\left(t\right)-Q_5\left(t\right)+Q_6\left(t\right)-Q_7\left(t\right)-Q_8\left(t\right)=0\\ Q_8\left(t\right)=0\end{array}---\left(24\right)$

For night, there is no direct solar radiation, solar scattered radiation and earth reflection radiation, i.e. Q ₁(t)=0, Q ₂(t)=0, Q ₃t ()=0, therefore, the capsule cloth of stratosphere balloon and helium gas inside at the energy-balance equation at night are

$\left\{\begin{array}{c}{Q}_4\left(t\right)-Q_5\left(t\right)+Q_6\left(t\right)-Q_7\left(t\right)-Q_8\left(t\right)=0\\ Q_8\left(t\right)=0\end{array}\right.---\left(25\right)$

Step 5, each energy exchange item step 3 obtained substitute into the energy-balance equation of step 4, the Simple iterative method in recycling method of value solving namely try to achieve stratosphere balloon by day with the temperature at night;

Formula (14) to formula (21) is updated to respectively in formula (24) and formula (25),

$\left\{\begin{array}{c}\left(\frac{22.7{\mathrm{\πR}}^2{\mathrm{\α}}_2\mathrm{\ϵ}\×{10}^{-8}}{1-r}-45.4{\mathrm{\πR}}^2{\mathrm{\α}}_2\×{10}^{-8}\right){T_1}^4-4{\mathrm{\πR}}^2{h}_1{T}_1+\[366.8{\mathrm{\πR}}^2{\mathrm{\α}}_1{\left(\frac{R_e}{R_e+H}\right)}^2\cos \frac{\mathrm{\π}\left|12-t\right|}{12}\]\\ +2369.5{\mathrm{\πR}}^2{\mathrm{\α}}_1+214.0{\mathrm{\πR}}^2{\mathrm{\α}}_2{\left(\frac{R_e}{R_e+H}\right)}^2+4{\mathrm{\πR}}^2{h}_1{T}_2\]=0\\ T_1-T_3=0\end{array}\right.---\left(26\right)$

$\left\{\begin{array}{c}\left(\frac{22.7{\mathrm{\πR}}^2{\mathrm{\α}}_2\mathrm{\ϵ}\×{10}^{-8}}{1-r}-45.4{\mathrm{\πR}}^2{\mathrm{\α}}_2\×{10}^{-8}\right){T_1}^4-4{\mathrm{\πR}}^2{h}_1{T}_1+\[214.0{\mathrm{\πR}}^2{\mathrm{\α}}_2{\left(\frac{R_e}{R_e+H}\right)}^2+4{\mathrm{\πR}}^2{h}_1{T}_2\]=0\\ T_1-T_3=0\end{array}\right.---\left(27\right)$

From (26) and formula (27), when the energy exchange of stratosphere balloon reaches equilibrium state, the capsule Bu Wendu of the stratosphere balloon of day and night is equal with helium gas inside temperature; By the Simple iterative method in method of value solving, equation (26) and (27) are solved respectively, namely obtain stratosphere balloon by day with capsule Bu Wendu and the helium gas inside temperature at night.