CN104866680A - Optimum spacing obtaining method for cooling pipelines on back side of mirror body in extreme ultraviolet collecting system - Google Patents

Abstract

The present invention discloses an optimum spacing obtaining method for cooling pipelines on the back side of a mirror body in an extreme ultraviolet collecting system. The optimum spacing obtaining method comprises the following steps of: S1, preliminarily determining an overall array layout scheme of the cooling pipelines on the back side of the mirror body according to distribution characteristics of a heat flux density of the mirror body; S2, building a physical heat transferring model which can reflect an actual situation; S3, building a numerical calculation model for heat conduction between the mirror body and the cooling pipelines, and preliminarily obtaining temperature distribution of the mirror body in the axial direction; S4, considering pipe-wall temperature difference caused by the situation that a water current absorbs heat and the temperature is risen, adding a group of energy conservation relational expressions in the calculation, and solving a more accurate mirror body temperature once again; and S5, on the basis of the steps of S1, S2, S3 and S4 of the technical scheme, writing an optimization procedure of an optimum spacing between the cooling pipelines, and adding a spacing between single pipes, which is as a final cyclic variable, into procedure operation so as to obtain the optimal solution of the spacing of the adjacent single pipes are obtained. According to the optimum spacing obtaining method disclosed by the present invention, the optimal solution of the spacing between the cooling pipelines can be accurately calculated, so that the uniform and stable temperature distribution of the whole mirror body in the extreme ultraviolet collecting system is realized, therefore, the array layout of the cooling pipelines is accurately designed in an optimizing manner.

Description

The superior distance acquiring method of mirror body dorsal part cooling duct in a kind of extreme ultraviolet collection system

Technical field

The present invention relates to a kind of in extreme ultraviolet photolithographic, the superior distance acquiring method of extreme ultraviolet collection system reflector body cooling duct alignment placement.

Background technology

Proximal pole ultraviolet photolithographic (EUVL) is considered to one of photoetching technique most with prospects of future generation.In the last few years, abroad the research and development institution of several extreme ultraviolet lithographies achieves some breakthrough progress, if light source power is progressively and improve steadily, collection system, illuminator, optical projection system are constantly optimized design, and the preparation technology of mask plate and photoresist constantly improves etc.

But, at the extreme ultraviolet photolithographic technical development initial stage, R&D work is the raising of emphasis light source power simply, and do not focus on the high heat that high power light source under working state of system brings and import problem, and the high temperature caused by intense radiation is fatal to the life-span of follow-up system and performance impact.

For extreme ultraviolet collection system, if the high heat absorbed can not discharge timely and effectively, then can within tens of second, mirror substrate temperature be made sharply to increase, the temperature distortion of collection system mirror body can be caused on the one hand, cause the light through catoptron reflection effectively cannot converge at intermediate focus place, collection system collection efficiency is declined to a great extent, intermediate focus place hot spot also can gross distortion, energy distribution is uneven, this is all extremely disadvantageous to follow-up illuminator and exposure system, finally can cause the of poor quality of photoetching chip, yield poorly, cannot meet the demands, on the other hand, after mirror absorption high heat, cause steep temperature rise, mirror substrate and rete life-span are seriously reduced.

Summary of the invention

The object of the invention is to the heat dissipation problem for collection system mirror body in extreme ultraviolet photolithographic, propose the superior distance acquiring method of mirror body dorsal part cooling duct in a kind of extreme ultraviolet collection system.The method, by carrying out theoretical analysis to the axial non-uniform Distribution of cooling duct, is requiring, under the condition that whole mirror surface temperature range is minimum, to utilize the thermal conduction study such as heat conduction, convection current equation to accurately calculate the optimum solution of cooling duct spacing.

To achieve these goals, following technical scheme is adopted:

S1: according to mirror body heat current density characteristic distributions, tentatively determines mirror body dorsal part cooling tube global alignment placement scheme, as shown in Figure 1.

S2: set up the heat transfer physical model that can reflect actual conditions, as shown in Figure 2.

S3: set up heat conducting mathematical calculation model between mirror body and cooling duct, as shown in Figure 3, and tentatively tries to achieve mirror body Temperature Distribution in axial direction.

S4: consider the pipe surface temperature otherness that current heat absorption intensification causes to add one group of energy conservation relation formula in interative computation, as shown in Figure 4, again solve more accurate mirror temperature.

S5: based on S1, S2, S3, S4 technical scheme steps, writes cooling tube optimal spacing optimizer.Under the prerequisite that cooling tube total arrangement is known, minimum temperature difference is reached for optimization aim with mirror body, by the related physical property parameter of input light source power, mirror dignity type, cooling tube size, water quench parameter and material, optimizer of the present invention can automatically perform the Optimum Operation of cooling tube spacing, after loop iteration, each adjacent cooling tube distance values after optimization can be provided.Afterwards, each single tube spacing is joined in sequential operation as last loop variable, try to achieve the optimum solution of adjacent single tube spacing.After optimizer of the present invention has been write, after tested, meet and be suitable for requirement, the result can referring to embodiment.

The concrete operation method of described step 2 is as follows: for certain layer of mirror body provided, determine the arrangement situation of its dorsal part cooling duct, G1, G2, G3 are same end to end coiled pipes, G4, G5 are the end to end coiled pipes of another root, can regard 5 single tubes as along mirror body axis direction, and each single tube is wound with half circumference of mirror body.S1, S2, S3, S4 are the arc length direction spacing between the adjacent cooling tube of G1, G2, G3, G4, G5 successively.

The concrete operation method of described step 3 is as follows: (θ is as computing intermediate variable for θ to utilize two angles, can numerical values recited be provided) axial plane from the next fillet body of mirror body cutting, this fillet body is stretched vertically, form the fillet shape rectangular parallelepiped that a length is S (S is former fillet body arc length vertically), this rectangular parallelepiped is divided into the little rectangular parallelepiped that length is p, obviously, p is less, and numerical stability is higher.If the width of little rectangular parallelepiped is u, thickness is the thickness of d, d also i.e. mirror body.Contact mirror body with cooling duct little rectangular parallelepiped (the little rectangular parallelepiped of the black in Fig. 3), is referred to as " camera bellows ", temperature be followed successively by above-mentioned in T1, T2, T3, T4, T5.

Each little rectangular parallelepiped be applied in mirror somascope face heat flow density and mirror body dorsal part heat flow density respectively on two surfaces taking radial direction as normal direction, because little cuboid dimensions is very little, the heat flow density on two little surfaces can regard constant value as; Each little rectangular parallelepiped carries out the transmission of hot-fluid by the bin contacted, the Inner Constitution of these bins and each little rectangular parallelepiped " passage " of Energy Transfer, the heat entering into each little rectangular parallelepiped is flow into " camera bellows " by this " passage ", and cooled water-band is walked, when whole diabatic process enters into stable state, if neglect the radiation between mirror body and environment, then according to energy conservation, within the identical time, the heat sum that each little rectangular parallelepiped absorbs equals the heat that in " camera bellows ", chilled water is taken away.

In order to the above-mentioned physical process of vivider description, each little rectangular parallelepiped is imagined as one by one with little " water tank " of water inlet, and mirror body dorsal part and cooling tube contact position (camera bellows) are equivalent to there be one " suction pump ", the water yield entering into each little " water tank " is all taken away in time, and whole system is in the process of a mobile equilibrium.Two adjacent water pumps draw water to the water tank between them, the pumpage of each water pump pair water tank close with it is strong, conclude thus, at two pump houses, a location point must be there is, make the current of each water tank be positioned on the left of this location point all flow to the water pump in left side, the current being positioned at each water tank on the right side of this location point all flow to the water pump on right side.

For convenience's sake, first the heat flow density of mirror body both sides is added summation, then is applied to one of them in two surfaces.Suppose total N number of little rectangular parallelepiped between adjacent two " camera bellows ", wherein have n1 the heat conduction " camera bellows " to the left that will absorb, have n2 the heat conduction " camera bellows " to the right that will absorb, have n1+n2=N.And these little rectangular parallelepipeds are pressed Ln1, L (n1-1) ... L3, L2, L1, R1, R2, R3 ... R (n2-1), Rn2 are numbered, the total border heat flow density of each little rectangular parallelepiped is fz (i), i=Rn1 ... Rn2, the heat that each little rectangular parallelepiped absorbs:

Q(i)＝fz(i)*u*p (16)

Q (i) when this little rectangular parallelepiped inside is conducted vertically, its heat flow density:

fx(i)＝Q(i)/(u*d)＝fz(i)*p/d (17)

The total axial heat flux density of i-th little rectangular parallelepiped is:

$q_x\left(i\right)=\underset{s=1}{\overset{i}{\mathrm{\Σ}}}\mathrm{fx}\left(s\right)=\frac{p}{d}\underset{s=1}{\overset{i}{\mathrm{\Σ}}}\mathrm{fz}\left(s\right)---\left(18\right)$

Obtained by Fourier heat equation:

$q_n=-k\frac{\∂T}{{\∂}_n}---\left(19\right)$

Wherein k is the thermal conductivity of material, q _nbeing direction vector is certain cross section on heat flow density, for the party's rate of temperature change upwards.

Following relation is had between adjacent two little rectangular parallelepipeds:

$q_x\left(i\right)=-k\frac{\mathrm{\ΔT}(i,i-1)}{p}---\left(20\right)$

$T(i-1)=\mathrm{\ΔT}(i,i-1)+T\left(i\right)=-\frac{p^2}{k*d}\underset{s=1}{\overset{i}{\mathrm{\Σ}}}\mathrm{fz}\left(s\right)+T\left(i\right)---\left(21\right)$

Because fz (i) and T (Ln1), T (Rn2) two boundary temperatures are known, only need circulate and bring the apportioning cost of n1 and n2 into, utilize above-mentioned iterative relation, when α=T (L1)-T (R1) < 0.001 (or more high precision), iteration terminates, at this moment think and the actual temperature having solved each little rectangular parallelepiped also obtain whole mirror body Temperature Distribution in axial direction immediately.

The concrete operation method of described step 4 is as follows: the heat setting every root single tube to take away as Qi, i=1,2,3,4,5, obtained by heat transfer equation:

Q _i＝h·A·ΔT _LM(22)

Obtained by the heat power equation of energy conservation and liquid coolant:

$Q_i=c_p\&CenterDot;\stackrel{.}{m}\&CenterDot;\mathrm{\&Delta;T}---\left(23\right)$

Wherein, A is inner surface of tube wall area, c _pfor the specific heat capacity of chilled water, m is the mass velocity of current, Δ T=T _out-T _infor the temperature rise of current, T _inwith T _outbe respectively the temperature of current at water inlet and water delivering orifice, in order to the convenience that engineering calculates, adopt geometric mean temperature to replace log-mean temperature difference:

ΔT _AM＝T _i-(T _in+T _out)/2 (24)

H is the convective heat-transfer coefficient of chilled water,

h＝λ*Nu/D

Nu＝0.012*(Re^0.87-280)*Pr^0.4

Re＝D*w/v (25)

Pr＝v/[λ/(ρ*c _p)]

Wherein, λ is the coefficient of heat conductivity of chilled water; Nu is nusselt number, and what provide in formula is rule-of-thumb relation when being in the zone of transition between laminar flow and turbulent flow; Re is Reynolds number; Pr is planck number, when being in zone of transition, and Pr=7; D is water pipe diameter, and w is the mean flow rate of current, and v is kinematic viscosity, and ρ is density.

Single tube pipe wall average temperature is obtained by equation (22), (23), (24), (25):

$T_i=T_{\mathrm{in}}+(\frac{1}{2c_p\stackrel{.}{m}}+\frac{1}{\mathrm{hA}})\&CenterDot;Q_i;(i=\mathrm{1,2},...5)---\left(26\right)$

At chilled water physical parameter, flow parameter, under cooling tube size and the given condition of five single tube spacing, Qi is only had to be the variable of loop iteration in formula (26), again according to energy conservation, through tube wall conduction, the heat Qi that takes away of chilled water convection current is from a part of region of the mirror body near this single tube.When giving five single tubes initial value respectively, formula (16) ~ (21) are utilized to obtain n1, n2, effective heat sink region of each single tube is carried out the curve surface integral of heat flow density successively, can be obtained the heat that each single tube absorbs after determining to these regions, mirror surface:

Q _i＝∫∫ _sfz(i)ds (27)

After Qi determines, according to formula (26), each new single tube temperature that one group is different from initial value can be obtained, this new temperature array is directly brought into or brings formula (16) ~ (21) again into after do certain algorithm process together with last initial value and carry out computing, circulation like this is gone down, until the pipe surface temperature solved converges to the precision met the demands.

Although above tried to achieve pipe surface temperature is correct, but not also the net result that we want, because for above-mentioned solution procedure, spacing between each single tube is assumed to be known at the very start, and give certain initial value, this may be optimum solution hardly, so, also spacing to be joined sequential operation as last loop variable.So far, the spacing optimum solution solution procedure between adjacent single tube is complete.

Usefulness of the present invention is, according to the overall principle of cooling tube topological design, by to the theoretical analysis of the axial non-uniform Distribution of cooling duct and program calculation, the optimum solution of cooling duct spacing can be calculated accurately, make whole mirror body in extreme ultraviolet collection system reach uniform and stable Temperature Distribution, thus being accurately optimized property of cooling duct alignment placement is designed.

Accompanying drawing explanation

Fig. 1 is mirror body dorsal part cooling duct general arrangement scheme;

Fig. 2 is mirror body dorsal part cooling duct arrangement sketch;

Fig. 3 is heat conducting mathematical calculation model between mirror body and cooling duct;

Fig. 4 is that adjacent single tube spacing optimal value solves process flow diagram.

Embodiment

Below in conjunction with accompanying drawing, technical scheme of the present invention is further described; but be not limited thereto; everyly technical solution of the present invention modified or equivalent to replace, and not departing from the spirit and scope of technical solution of the present invention, all should be encompassed in protection scope of the present invention.

In EUV light source collection system, given light source and individual layer mirror body, the total heat flow density of its minute surface is:

Hyperboloid portion: fz (x)=3557*exp (-0.02009*x)-814.7*exp (-0.2342*x);

Ellipsoid part: fz (x)=0.00687*x.^3-2.149*x.^2+217.8*x-5381;

(equation initial point is the intersection point of mirror body axis and mirror body front end face)

Cooling water pipe internal diameter 5mm, external diameter 8mm; Chilled water flow velocity 0.5889m/s, reynolds number Re=2944, water inlet temperature 20 DEG C.

Suppose total N number of little rectangular parallelepiped between adjacent two " camera bellows " (mirror body dorsal part and cooling tube contact position), wherein there is n1 the heat conduction " camera bellows " to the left that will absorb, there is n2 the heat conduction " camera bellows " to the right that will absorb, have n1+n2=N.And these little rectangular parallelepipeds are pressed Ln1, L (n1-1) ... L3, L2, L1, R1, R2, R3 ... R (n2-1), Rn2 are numbered, the total border heat flow density of each little rectangular parallelepiped is fz (i), i=Rn1 ... Rn2, the heat that each little rectangular parallelepiped absorbs:

Q(i)＝fz(i)*u*p (16)

Q (i) when this little rectangular parallelepiped inside is conducted vertically, its heat flow density:

fx(i)＝Q(i)/(u*d)＝fz(i)*p/d (17)

The total axial heat flux density of i-th little rectangular parallelepiped is:

$q_x\left(i\right)=\underset{s=1}{\overset{i}{\mathrm{\&Sigma;}}}\mathrm{fx}\left(s\right)=\frac{p}{d}\underset{s=1}{\overset{i}{\mathrm{\&Sigma;}}}\mathrm{fz}\left(s\right)---\left(18\right)$

Obtained by Fourier heat equation:

$q_n=-k\frac{\&PartialD;T}{{\&PartialD;}_n}---\left(19\right)$

Wherein k is the thermal conductivity of material, q _nbeing direction vector is certain cross section on heat flow density, for the party's rate of temperature change upwards.

Following relation is had between adjacent two little rectangular parallelepipeds:

$q_x\left(i\right)=-k\frac{\mathrm{\&Delta;T}(i,i-1)}{p}---\left(20\right)$

$T(i-1)=\mathrm{\&Delta;T}(i,i-1)+T\left(i\right)=-\frac{p^2}{k*d}\underset{s=1}{\overset{i}{\mathrm{\&Sigma;}}}\mathrm{fz}\left(s\right)+T\left(i\right)---\left(21\right)$

Because fz (i) and T (Ln1), T (Rn2) two boundary temperatures are known, only need circulate and bring the apportioning cost of n1 and n2 into, utilize above-mentioned iterative relation, as α=T (L1)-T (R1) < 0.001, iteration terminates, at this moment think and the actual temperature having solved each little rectangular parallelepiped also obtain whole mirror body Temperature Distribution in axial direction immediately.

In described step 4, if the heat that every root single tube is taken away is Qi, i=1,2,3,4,5, obtained by heat transfer equation:

Q _i＝h·A·ΔT _LM(22)

Obtained by the heat power equation of energy conservation and liquid coolant:

$Q_i=c_p\&CenterDot;\stackrel{.}{m}\&CenterDot;\mathrm{\&Delta;T}---\left(23\right)$

Wherein, A is inner surface of tube wall area, c _pfor the specific heat capacity of chilled water, m is the mass velocity of current, Δ T=T _out-T _infor the temperature rise of current, T _inwith T _outbe respectively the temperature of current at water inlet and water delivering orifice, in order to the convenience that engineering calculates, adopt geometric mean temperature to replace log-mean temperature difference:

ΔT _AM＝T _i-(T _in+T _out)/2 (24)

H is the convective heat-transfer coefficient of chilled water,

h＝λ*Nu/D

Nu＝0.012*(Re^0.87-280)*Pr^0.4

Re＝D*w/v (25)

Pr＝v/[λ/(ρ*c _p)]

Wherein, λ is the coefficient of heat conductivity of chilled water; Nu is nusselt number, and what provide in formula is rule-of-thumb relation when being in the zone of transition between laminar flow and turbulent flow; Re is Reynolds number; Pr is planck number, when being in zone of transition, and Pr=7; D is water pipe diameter, and w is the mean flow rate of current, and v is kinematic viscosity, and ρ is density.

Single tube pipe wall average temperature is obtained by equation (22), (23), (24), (25):

$T_i=T_{\mathrm{in}}+(\frac{1}{2c_p\stackrel{.}{m}}+\frac{1}{\mathrm{hA}})\&CenterDot;Q_i;(i=\mathrm{1,2},...5)---\left(26\right)$

At chilled water physical parameter, flow parameter, under cooling tube size and the given condition of five single tube spacing, Qi is only had to be the variable of loop iteration in formula (26), again according to energy conservation, through tube wall conduction, the heat Qi that takes away of chilled water convection current is from a part of region of the mirror body near this single tube.When giving five single tubes initial value respectively, formula (16) ~ (21) are utilized to obtain n1, n2, effective heat sink region of each single tube is carried out the curve surface integral of heat flow density successively, can be obtained the heat that each single tube absorbs after determining to these regions, mirror surface:

Q _i＝∫∫ _sfz(i)ds (27)

After Qi determines, according to formula (26), each new single tube temperature that one group is different from initial value can be obtained, this new temperature array is directly brought into or brings formula (16) ~ (21) again into after do certain algorithm process together with last initial value and carry out computing, circulation like this is gone down, until the pipe surface temperature solved converges to the precision met the demands.

Finally, based on S1, S2, S3, S4 technical scheme steps, write cooling tube optimal spacing optimizer, and spacing is joined sequential operation as last loop variable.So far, the spacing optimum solution solution procedure between adjacent single tube is complete.

Through optimizing, G1, G2, G3, G4, G5 single tube axial coordinate value is respectively:

8.00 19.64 31.28 50.68 73.96 105 [mm]

The bulk temperature distribution of mirror surface is realistic, and temperature value difference is less than 1 DEG C.

Claims (1)

1. the superior distance acquiring method of mirror body dorsal part cooling duct in extreme ultraviolet collection system, is characterized in that described method step is as follows:

S1: according to mirror body heat current density characteristic distributions, tentatively determines mirror body dorsal part cooling tube global alignment placement scheme;

S2: set up the heat transfer physical model that can reflect actual conditions, concrete operation method is as follows:

For certain layer of mirror body provided, determine the arrangement situation of its dorsal part cooling duct, G1, G2, G3 are same end to end coiled pipes, G4, G5 are the end to end coiled pipes of another root, 5 single tubes are regarded as along mirror body axis direction, each single tube is wound around half circumference of mirror body, and S1, S2, S3, S4 are the arc length direction spacing between the adjacent cooling tube of G1, G2, G3, G4, G5 successively;

S3: set up heat conducting mathematical calculation model between mirror body and cooling duct, and tentatively try to achieve mirror body Temperature Distribution in axial direction, concrete operation method is as follows:

Utilize axial plane that two angles are θ from the next fillet body of mirror body cutting, this fillet body is stretched vertically, form the fillet shape rectangular parallelepiped that a length is S, this rectangular parallelepiped is divided into the little rectangular parallelepiped that length is p, width is u, and thickness is the region of d, mirror body and each cooling tube joint, be referred to as " camera bellows ", temperature is followed successively by T1, T2, T3, T4, T5;

Suppose total N number of little rectangular parallelepiped between adjacent two " camera bellows ", wherein there is n1 the heat conduction " camera bellows " to the left that will absorb, there is n2 the heat conduction " camera bellows " to the right that will absorb, there is n1+n2=N, and these little rectangular parallelepipeds are pressed Ln1, L (n1-1) ... L3, L2, L1, R1, R2, R3......R (n2-1), Rn2 is numbered, the total border heat flow density of each little rectangular parallelepiped is fz (i), i=Rn1......Rn2, the heat that each little rectangular parallelepiped absorbs is:

Q(i)＝fz(i)*u*p (16)；

Q (i) is when this little rectangular parallelepiped inside is conducted vertically, and its heat flow density is:

fx(i)＝Q(i)/(u*d)＝fz(i)*p/d (17)；

The total axial heat flux density of i-th little rectangular parallelepiped is:

$q_x\left(i\right)=\underset{s=1}{\overset{i}{\mathrm{\&Sigma;}}}\mathrm{fx}\left(s\right)=\frac{p}{d}\underset{s=1}{\overset{i}{\mathrm{\&Sigma;}}}\mathrm{fz}\left(s\right)---\left(18\right);$

Obtained by Fourier heat equation:

$q_n=-k\frac{\&PartialD;T}{{\&PartialD;}_n}---\left(19\right);$

Following relation is had between adjacent two little rectangular parallelepipeds:

$q_x\left(i\right)=-k\frac{\mathrm{\&Delta;T}(i,i-1)}{p}---\left(20\right);$

$T(i-1)=\mathrm{\&Delta;T}(i,i-1)+T\left(i\right)=-\frac{p^2}{k*d}\underset{s=1}{\overset{i}{\mathrm{\&Sigma;}}}\mathrm{fz}\left(s\right)+T\left(i\right)---\left(21\right);$

Because fz (i) and T (Ln1), T (Rn2) two boundary temperatures are known, only need the apportioning cost of tape loop people n1 and n2, utilize above-mentioned iterative relation, as enough hour of α=T (L1)-T (R1), iteration terminates, at this moment think and the actual temperature having solved each little rectangular parallelepiped also obtain whole mirror body Temperature Distribution in axial direction immediately;

S4: consider the pipe surface temperature otherness that current heat absorption intensification causes to add one group of energy conservation relation formula in interative computation, again solve more accurate mirror temperature, concrete operation method is as follows:

If the heat that every root single tube is taken away is Qi, i=1,2,3,4,5, obtained by heat transfer equation:

Q _i＝h·A·ΔT _LM(22)；

Obtained by the heat power equation of energy conservation and liquid coolant:

$Q_i=c_p\&CenterDot;\stackrel{\&CenterDot;}{m}\&CenterDot;\mathrm{\&Delta;T}---\left(23\right);$

Geometric mean temperature is adopted to replace log-mean temperature difference:

ΔT _AM＝T _i-(T _in+T _out)/2 (24)；

H is the convective heat-transfer coefficient of chilled water, then have:

h＝λ*Nu/D

Nu＝0.012*(Re^0.87-280)*Pr^0.4

Re＝D*w/v (25)；

Pr＝v/[λ/(ρ*c _p)]

Single tube pipe wall average temperature is obtained by equation (22), (23), (24), (25):

$T_i=T_{\mathrm{in}}+(\frac{1}{2c_p\stackrel{\&CenterDot;}{m}}+\frac{1}{\mathrm{hA}})\&CenterDot;Q_i;(i=\mathrm{1,2},...5)---\left(26\right);$

When giving five single tubes initial value respectively, formula (16) ~ (21) are utilized to obtain n1, n2, effective heat sink region of each single tube is carried out the curve surface integral of heat flow density successively, can be obtained the heat that each single tube absorbs after determining to these regions, mirror surface:

Q _i＝∫∫ _sfz(i)ds (27)；

After Qi determines, according to formula (26), each new single tube temperature that one group is different from initial value can be obtained, this new temperature array is directly brought into or brings formula (16) ~ (21) again into after do certain algorithm process together with last initial value and carry out computing, circulation like this is gone down, until the pipe surface temperature solved converges to the precision met the demands;

S5: based on S1, S2, S3, S4 step, writes cooling tube optimal spacing optimizer, and joins in sequential operation using each single tube spacing as last loop variable, tries to achieve the optimum solution of adjacent single tube spacing.