CN112311279B - Thermoelectric power generation module for fluid waste heat recovery and structure optimization method thereof - Google Patents

Thermoelectric power generation module for fluid waste heat recovery and structure optimization method thereof Download PDF

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CN112311279B
CN112311279B CN202011119994.0A CN202011119994A CN112311279B CN 112311279 B CN112311279 B CN 112311279B CN 202011119994 A CN202011119994 A CN 202011119994A CN 112311279 B CN112311279 B CN 112311279B
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吴麟麟
胡迎香
汪若尘
罗丁
余未
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Abstract

The invention provides a thermoelectric power generation module for fluid waste heat recovery and a structure optimization method thereof, and belongs to the technical field of thermoelectric power generation. The length of the thermoelectric semiconductor is determined according to the temperature difference of each thermoelectric semiconductor, and when the output power gain value of the thermoelectric semiconductor is optimal, the optimal length of the thermoelectric semiconductor of the improved thermoelectric power generation module is determined, and the cross-sectional area of the thermoelectric semiconductor is optimized; by changing the cross-sectional area of the thermoelectric semiconductor, the output performance of the thermoelectric power generation module for fluid waste heat recovery is effectively improved, and the thermoelectric conversion efficiency of the thermoelectric power generation module is improved. Compared with the traditional thermoelectric power generation module structure, the thermoelectric power generation module determined by the structure optimization method can solve the problem of electric energy loss caused by the fact that the current generated by the thermoelectric power generation module is limited by the minimum current of the thermoelectric semiconductor.

Description

Thermoelectric power generation module for fluid waste heat recovery and structure optimization method thereof
Technical Field
The invention belongs to the technical field of thermoelectric generation, and particularly relates to a thermoelectric generation module for fluid waste heat recovery and a structure optimization method thereof.
Background
In recent years, as the demand for energy is increasing, the thermoelectric conversion technology has attracted much attention in research on renewable energy sources due to its advantages of no pollution, no moving parts, no maintenance cost, long service life, etc., as an alternative energy technology with great development prospects. More and more researchers have attempted to apply thermoelectric generators to the field of waste heat recovery, including waste heat generated by automobiles, airplanes and helicopters, ships, and industries.
The thermoelectric generation module is a core generation unit in the thermoelectric conversion technology and consists of an upper alumina ceramic plate, a thermoelectric semiconductor, a copper electrode plate and a lower alumina ceramic plate. However, the thermoelectric power generation module has low conversion efficiency and is far from meeting the requirements of commercial application. Therefore, the development of thermoelectric materials with higher performance and the thermoelectric generation module based on structural optimization can improve the conversion efficiency of the thermoelectric generation system to some extent. When the thermoelectric power generation module is used for recovering waste heat contained in fluid, the temperature distribution on the surface of the thermoelectric power generation module is uneven, so that output currents of different thermoelectric semiconductors are different, and the overall output current of the thermoelectric power generation module is limited by the minimum output current.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a thermoelectric power generation module for fluid waste heat recovery and a structure optimization method thereof, and solves the problem that when fluid waste heat is recovered, the current generated by the thermoelectric power generation module is limited by the minimum current of a thermoelectric semiconductor to cause electric energy loss.
The present invention achieves the above-described object by the following technical means.
A structure optimization method of a thermoelectric generation module for fluid waste heat recovery comprises the following steps:
step (1), i is 1, delta l is q × j, and the length of the ith thermoelectric semiconductor of the improved thermoelectric power generation module is made to be equal to
Figure BDA0002731678780000011
Judging whether i is larger than or equal to N, if so, executing the step (2); otherwise, let i equal to i +1, recalculate LiUntil i ═ N;
wherein: Δ l is the length increment of the thermoelectric semiconductor; the iteration number j is 1, 2, 3, … …, and the initial value is 1; constant coefficient
Figure BDA0002731678780000012
LoriginIs the original length of the thermoelectric semiconductor; delta TiIs the temperature difference, Δ T, of the ith thermoelectric semiconductormaxIs the maximum temperature difference, Δ T, of all thermoelectric semiconductorsminThe minimum temperature difference for all thermoelectric semiconductors; n is the total number of thermoelectric semiconductors;
step (2), calculating the output power P of the improved thermoelectric power generation moduleΔlAnd the output power P of the traditional thermoelectric power generation modulebase ofΔlComparing said output power P when j takes different valuesbase ofΔlWhether or not P is greater than or equal toΔlIf yes, ending; otherwise, j is made to be j +1, and the step (1) is returned;
step (3), determining the lengths L of all thermoelectric semiconductors of the thermoelectric power generation modulei-allAnd comparing the improved thermoelectric generation module with different j valuesGain of block output power relative to output power of traditional thermoelectric generation module
Figure BDA0002731678780000021
When the value delta' is maximum, j is an optimal value to obtain the optimal delta L, thereby determining the optimal length L of the thermoelectric semiconductori' optimizing the cross-sectional area of the thermoelectric semiconductor.
Further, the output power and the temperature difference are obtained by establishing a flow thermoelectric multi-physical coupling model in ANSYS simulation.
Further, the original length L of the thermoelectric semiconductororiginIs equal to the width W of the traditional thermoelectric generation module, and
Figure BDA0002731678780000022
further, when the number of the thermoelectric semiconductors is the same, the height and the width of the thermoelectric semiconductor of the traditional thermoelectric power generation module are respectively equal to those of the thermoelectric semiconductor of the improved thermoelectric power generation module.
Furthermore, when the number of the thermoelectric semiconductors is the same, the total volume of all the thermoelectric semiconductors of the conventional thermoelectric generation module is equal to that of all the thermoelectric semiconductors of the improved thermoelectric generation module.
A thermoelectric power generation module for fluid waste heat recovery comprises an upper-end alumina ceramic plate, a copper electrode plate, a thermoelectric semiconductor and a lower-end alumina ceramic plate; the thermoelectric semiconductor comprises a p-type thermoelectric semiconductor and an n-type thermoelectric semiconductor, and the p-type thermoelectric semiconductor and the n-type thermoelectric semiconductor are connected in series through copper electrode plates and then are sandwiched between an upper alumina ceramic plate and a lower alumina ceramic plate; the length of the ith thermoelectric semiconductor is LiSaid L isiAnd determining that i is more than or equal to 1 and less than or equal to N according to the structure optimization method.
In the technical scheme, the number of the upper-end alumina ceramic plates and the lower-end alumina ceramic plates is 1, the total number of the thermoelectric semiconductors is N, and the total number of the copper electrode plates is N; the height of the upper alumina ceramic plate and the height of the lower alumina ceramic plate are both H1The width and the length are equal to each otherIs 2W +2Ws(ii) a All thermoelectric semiconductors are H in height3The width of the thermoelectric semiconductor is W, and the width distance between two adjacent thermoelectric semiconductors is WsThe distance between the peripheral thermoelectric semiconductor and the ceramic plate boundary is W s2; the height of the copper electrode sheet is H2And a width of 2W + WsThe length of the ith copper electrode sheet and the length L of the ith thermoelectric semiconductoriAre equal.
The invention has the beneficial effects that: the length of the thermoelectric semiconductor is determined according to the temperature difference of each thermoelectric semiconductor, and when the output power gain value of the thermoelectric semiconductor is optimal, the optimal length of the thermoelectric semiconductor of the improved thermoelectric power generation module is determined, and the cross-sectional area of the thermoelectric semiconductor is optimized; when the same number of thermoelectric semiconductors is used, the output power of the improved thermoelectric generation module is higher than that of the conventional thermoelectric generation module within a certain range. Compared with the traditional thermoelectric power generation module, the thermoelectric power generation module can solve the problem of electric energy loss caused by the fact that the current generated by the thermoelectric power generation module is limited by the minimum current of the thermoelectric semiconductor, effectively improves the output performance of the thermoelectric power generation module for fluid waste heat recovery by changing the cross section area of the thermoelectric semiconductor, and improves the thermoelectric conversion efficiency of the thermoelectric power generation module.
Drawings
FIG. 1 is a schematic diagram of a thermoelectric semiconductor structure;
FIG. 2 is a flow chart for structural optimization of the thermoelectric generation module of the present invention;
FIG. 3 is a schematic structural view of a thermoelectric power generation module according to the present invention;
fig. 4 is a top view of a thermoelectric generator;
FIG. 5 is a cross-sectional view of a heat exchanger;
FIG. 6 is a sectional view of the cooler;
fig. 7 is a graph of the relationship between the output powers of the improved thermoelectric generation module and the conventional thermoelectric generation module under different load resistances when Δ l is 0.01 mm;
in the figure: 1-upper alumina ceramic plate, 2-copper electrode plate, 3-p type thermoelectric semiconductor, 4-n type thermoelectric semiconductor and 5-lower alumina ceramic plate.
Detailed Description
The invention will be further described with reference to the following figures and specific examples, but the scope of the invention is not limited thereto.
If the thermoelectric generation module is used for recovering waste heat generated by automobile exhaust, a complete thermoelectric generator consists of a heat exchanger, a thermoelectric generation module and a cooler, wherein the thermoelectric generation module is arranged between the heat exchanger and the cooler.
As shown in fig. 1, a thermoelectric generation module for fluid waste heat recovery comprises an upper alumina ceramic plate 1, a copper electrode sheet 2, a thermoelectric semiconductor and a lower alumina ceramic plate 5; the thermoelectric semiconductor comprises a p-type thermoelectric semiconductor 3 and an n-type thermoelectric semiconductor 4, wherein the p-type thermoelectric semiconductor 3 and the n-type thermoelectric semiconductor 4 are connected in series through a copper electrode plate 2 and then are sandwiched between an upper alumina ceramic plate 1 and a lower alumina ceramic plate 5.
Referring to fig. 1, the number of the upper alumina ceramic plate and the lower alumina ceramic plate is 1, the total number of the thermoelectric semiconductors is N, and the total number of the copper electrode plates is N; the height of the upper alumina ceramic plate and the height of the lower alumina ceramic plate are both H1The width and the length are equal and are both 2W +2Ws(ii) a The length of the ith thermoelectric semiconductor (i is more than or equal to 1 and less than or equal to N) is LiThe length of the (i + 1) th thermoelectric semiconductor is Li+1And all the thermoelectric semiconductors have a height of H3The width of the thermoelectric semiconductor is W, and the width distance between two adjacent thermoelectric semiconductors is WsThe distance between the peripheral thermoelectric semiconductor and the ceramic plate boundary is W s2; the height of the copper electrode sheet is H2And a width of 2W + WsThe length of the ith copper electrode sheet and the length L of the ith thermoelectric semiconductoriAre equal.
As shown in fig. 2, a method for optimizing the structure of a thermoelectric power generation module for fluid waste heat recovery specifically includes the following steps:
step (1), calculating to obtain the hot end temperature Th of the ith thermoelectric semiconductor through a flow thermoelectric multi-physical coupling model of the thermoelectric power generation systemiCold end temperature TciThe hot end temperature of the (i + 1) Th thermoelectric semiconductor is Thi+1Cold end temperature Tci+1(ii) a Output power P of the thermoelectric power generation module and temperature difference Delta T of the ith thermoelectric semiconductori(ii) a Calculating the maximum temperature difference delta T according to the temperature difference of all the thermoelectric semiconductorsmaxAnd a minimum temperature difference Δ Tmin；
The thermoelectric semiconductor hot end temperature Th, cold end temperature Tc, temperature difference Delta T and output power P are obtained by establishing a flow thermoelectric multi-physical coupling model in ANSYS simulation, wherein the flow thermoelectric multi-physical coupling model is as follows:
1) for the flow field region, the mass, momentum and energy conservation law is followed:
Figure BDA0002731678780000041
Figure BDA0002731678780000042
Figure BDA0002731678780000043
in the formula, v is a fluid velocity, ρ is a fluid density, P' is a fluid pressure, μ is a dynamic viscosity, λ is a thermal conductivity coefficient, c is a specific heat capacity, T is a temperature of the thermoelectric semiconductor, and;
in addition, the fluid flow follows the k- ε turbulence model:
Figure BDA0002731678780000044
Figure BDA0002731678780000045
wherein: sigmak、σεIs the Brownian coefficient of k-epsilon turbulence, GkFor turbulent kinetic energy due to mean velocity gradients, GbFor turbulent kinetic energy due to buoyancy, YMContribution of wave expansion in compressible media to the overall dissipation rate, C1ε、C2ε、C3εAre all constants;
2) for the solid areas of the heat exchanger, cooler and ceramic plates, energy conservation is satisfied:
Figure BDA0002731678780000046
3) for the p-type thermoelectric semiconductor and the n-type thermoelectric semiconductor, the following are satisfied:
Figure BDA0002731678780000051
Figure BDA0002731678780000052
in the formula, λp(T) is the thermal conductivity of the p-type thermoelectric semiconductor, λn(T) is the thermal conductivity of the n-type thermoelectric semiconductor,
Figure BDA0002731678780000053
is the resistivity of the p-type thermoelectric semiconductor,
Figure BDA0002731678780000054
is the resistivity of the n-type thermoelectric semiconductor,
Figure BDA0002731678780000055
the seebeck coefficient of a p-type thermoelectric semiconductor,
Figure BDA0002731678780000056
the seebeck coefficient of an n-type thermoelectric semiconductor,
Figure BDA0002731678780000057
as a vector of current density, TpIs the temperature, T, of a p-type thermoelectric semiconductornIs the temperature of the n-type thermoelectric semiconductor;
4) for the copper electrode sheet, satisfy:
Figure BDA0002731678780000058
in the formula, λcoIs the heat conductivity of the copper material,
Figure BDA0002731678780000059
the resistivity of the copper material;
5) in addition, the electric field conservation equation is:
Figure BDA00027316787800000510
Figure BDA00027316787800000511
Figure BDA00027316787800000512
in the formula (I), the compound is shown in the specification,
Figure BDA00027316787800000513
is the electric field density vector, phi is the electromotive force,
Figure BDA00027316787800000514
is the seebeck electromotive force.
Setting a boundary condition: the natural convection heat transfer coefficient, the ambient temperature, the inlet mass flow of air and water, the inlet temperature of air and water, and the outlet boundary of air and water are set as pressure outlets.
Step (2), the original length and width of all thermoelectric semiconductors of the traditional thermoelectric power generation module are equal, namely LoriginW; let i equal to 1,
Figure BDA00027316787800000515
The length of the ith thermoelectric semiconductor of the improved thermoelectric power generation moduleThe degree is as follows:
Figure BDA00027316787800000516
judging whether i is greater than or equal to N, if so, carrying out the next step; if not, let i equal i +1, recalculate LiUp to
Figure BDA00027316787800000517
Wherein Δ l is the length increment of the thermoelectric semiconductor; the iteration number j is 1, 2, 3, … …, and the initial value is 1; constant coefficient
Figure BDA00027316787800000518
(q holds a significant digit).
Step (3) calculating the output power P of the improved thermoelectric power generation moduleΔlAnd the output power P of the traditional thermoelectric power generation modulebase ofΔl(ii) a Wherein, the same quantity of thermoelectric semiconductor, the total volume of all thermoelectric semiconductor of traditional thermoelectric generation module equals the total volume of all thermoelectric semiconductor of improved thermoelectric generation module, the height, the width of traditional thermoelectric generation module thermoelectric semiconductor and the height, the width of improved thermoelectric generation module thermoelectric semiconductor are equal respectively, H promptly3′＝H3W ═ W; length L of all thermoelectric semiconductors of conventional thermoelectric power generation moduleoriginAre all equal to each other and are
Figure BDA0002731678780000061
Step (4), when j takes different values, comparing the output power P of the traditional thermoelectric generation modulebase ofΔlWhether the output power P of the improved thermoelectric generation module is more than or equal toΔlIf yes, stopping circulation; if not, the calculation is returned to the step (2) to re-calculate, i is 1, and j is j + 1.
Step (5), determining the lengths L of all thermoelectric semiconductors of the improved thermoelectric power generation modulei-allComparing the gain of the output power of the improved thermoelectric generation module under different j values compared with the output power of the traditional thermoelectric generation module, namely comparing
Figure BDA0002731678780000062
J is an optimal value when delta' is taken as the maximum value, the optimal delta L is obtained through calculation, and the optimal length L of the thermoelectric semiconductor is determinedi' optimizing the cross-sectional area of the thermoelectric semiconductor.
As shown in fig. 3, the thermoelectric material used in the module for medium temperature difference power generation in this embodiment is Bi2Te3The relevant dimensional parameters are: 16, H1＝0.8mm，H3＝1mm，W＝1.4mm，Ws＝1.1mm，H20.35mm (see table 1); setting a boundary condition: the natural convection heat transfer coefficient is 15W/(m)2K), ambient temperature 300K, inlet mass flow rates of air and water set to 40g/s and 166.7g/s, respectively, inlet temperatures of air and water set to 500K and 300K, respectively, outlet boundaries of air and water set as pressure outlets, and outlet pressure set to standard atmospheric pressure. As shown in fig. 4, a thermoelectric power generation module formed by 16 thermoelectric semiconductors, 16 copper electrode sheets and a ceramic plate is placed between a heat exchanger and a cooler to form a complete thermoelectric power generator; FIG. 5 is a cross-sectional view of a heat exchanger, in which the diameter of the inlet and outlet pipes is 20mm, the length of the inlet and outlet pipes is 50mm, the length of the heat collection box is 85mm, the width and the height of the heat collection box are both 60mm, automobile exhaust enters the heat collection box from the inlet of the heat exchanger, the container contains a shunting fin, and the heat exchanger sufficiently absorbs heat through the shunting fin to recover the waste heat of the automobile exhaust with higher efficiency; FIG. 6 is a sectional view of a cooler, the length and width of which are 40mm, the height of which is 12mm, the diameter of a cooler pipe is 2.75mm, and the length of which is 149.5mm, the cooler is cooled by a water cooling method, and cooling water flows in from an inlet of the cooler, is cooled by an inner pipe of the cooler, and then flows out from an outlet of the cooler.
TABLE 1 improved thermoelectric Generation Module dimensional parameters
Figure BDA0002731678780000063
Figure BDA0002731678780000071
The materials used for the thermoelectric generator and the corresponding material properties are listed in table 2.
TABLE 2 materials and corresponding Material Properties for thermoelectric generators
Figure BDA0002731678780000072
Figure BDA0002731678780000081
The results of the temperature difference Δ T and the length increment Δ l of the 16 thermoelectric semiconductors of the improved thermoelectric power generation module according to the optimization method are listed in table 3.
TABLE 3 temperature Difference DeltaT and Length increment Deltal for 16 thermoelectric semiconductors of improved thermoelectric Power Module
Figure BDA0002731678780000082
In addition, when the load resistance is 0.3 Ω, the output power relationship of the improved thermoelectric generation module and the conventional thermoelectric generation module for different Δ l values is shown in table 4.
TABLE 4 relationship of output power of improved thermoelectric generation module and traditional thermoelectric generation module with different delta l values when load resistance is 0.3 omega
Figure BDA0002731678780000091
As shown in table 4, when the load resistance is 0.3 Ω, the output power results of the improved thermoelectric generation module and the conventional thermoelectric generation module having different Δ l values indicate that the output power P of the thermoelectric generation module increases as the length increment Δ l of the thermoelectric semiconductor increases; when delta l is less than or equal to 0.03mm, the output power of the improved temperature difference power generation module is higher than that of the corresponding traditional temperature difference power generation module, and the gain value delta' of the output power of the improved temperature difference power generation module is reduced along with the increase of the delta l; when delta l is 0.01mm, the gain value delta' of the improved thermoelectric generation module is maximum compared with that of the traditional thermoelectric generation module; when delta l is larger than or equal to 0.04mm, the output power of the improved thermoelectric generation module is lower than that of the corresponding traditional thermoelectric generation module, and the gain value delta' of the output power of the improved thermoelectric generation module is negative.
As shown in fig. 7, under different load resistances, the output power results of the improved thermoelectric generation module and the conventional thermoelectric generation module when Δ l is 0.01mm show that: the output power of the improved thermoelectric generation module is obviously greater than that of the traditional thermoelectric generation module, and the optimization effect of the thermoelectric generation module is verified. If a larger number of thermoelectric semiconductors are arranged in the improved thermoelectric power generation module, the optimization effect is more obvious.
The present invention is not limited to the above-described embodiments, and any obvious improvements, substitutions or modifications can be made by those skilled in the art without departing from the spirit of the present invention.

Claims (7)

1. A structure optimization method of a thermoelectric generation module for fluid waste heat recovery is characterized by comprising the following steps:
step (1), i is 1, delta l is q × j, and the length of the ith thermoelectric semiconductor of the improved thermoelectric power generation module is made to be equal to
Figure FDA0003100301770000011
Judging whether i is larger than or equal to N, if so, executing the step (2); otherwise, let i equal to i +1, recalculate LiUntil i ═ N;
wherein: Δ l is the length increment of the thermoelectric semiconductor; the iteration number j is 1, 2, 3, … …, and the initial value is 1; constant coefficient
Figure FDA0003100301770000012
LoriginIs the original length of the thermoelectric semiconductor; delta TiIs the temperature difference, Δ T, of the ith thermoelectric semiconductormaxIs the maximum temperature difference, Δ T, of all thermoelectric semiconductorsminThe minimum temperature difference for all thermoelectric semiconductors; n is the total number of thermoelectric semiconductors;
step (2), calculating the output power P of the improved thermoelectric power generation moduleΔlAnd the output power P of the traditional thermoelectric power generation modulebase ofΔlComparing said output power P when j takes different valuesbase ofΔlWhether or not P is greater than or equal toΔlIf yes, ending; otherwise, j is made to be j +1, and the step (1) is returned;
step (3), determining the lengths L of all thermoelectric semiconductors of the improved thermoelectric power generation modulei-allAnd comparing the gain of the output power of the improved thermoelectric generation module relative to the output power of the traditional thermoelectric generation module when different j values are obtained
Figure FDA0003100301770000013
When the value delta' is maximum, j is an optimal value to obtain the optimal delta L, thereby determining the optimal length L of the thermoelectric semiconductori' optimizing the cross-sectional area of the thermoelectric semiconductor.
2. The method for optimizing the structure of the thermoelectric power generation module for fluid waste heat recovery according to claim 1, wherein the output power and the temperature difference are obtained by establishing a flow thermoelectric multi-physical coupling model in ANSYS simulation.
3. The method for structural optimization of a thermoelectric generation module for fluid waste heat recovery of claim 1, wherein the original length L of the thermoelectric semiconductor isoriginIs equal to the width W of the traditional thermoelectric generation module, and
Figure FDA0003100301770000014
4. the method for optimizing the structure of the thermoelectric generation module for fluid waste heat recovery according to claim 1, wherein the thermoelectric semiconductors of the conventional thermoelectric generation module have the same number of thermoelectric semiconductors, and the thermoelectric semiconductors of the conventional thermoelectric generation module have the same height and width as those of the thermoelectric semiconductors of the improved thermoelectric generation module.
5. The method for optimizing the structure of a thermoelectric generation module for fluid waste heat recovery according to claim 4, wherein the total volume of all thermoelectric semiconductors of a conventional thermoelectric generation module is equal to the total volume of all thermoelectric semiconductors of an improved thermoelectric generation module when the number of thermoelectric semiconductors is the same.
6. A thermoelectric power generation module for fluid waste heat recovery is characterized by comprising an upper-end alumina ceramic plate (1), a copper electrode plate (2), a thermoelectric semiconductor and a lower-end alumina ceramic plate (5); the thermoelectric semiconductor comprises a p-type thermoelectric semiconductor (3) and an n-type thermoelectric semiconductor (4), wherein the p-type thermoelectric semiconductor (3) and the n-type thermoelectric semiconductor (4) are connected in series through a copper electrode plate (2) and then are clamped between an upper alumina ceramic plate (1) and a lower alumina ceramic plate (5); the length of the ith thermoelectric semiconductor is LiSaid L isiThe structural optimization method of the thermoelectric power generation module for fluid waste heat recovery according to any one of claims 1 to 5 determines that i is greater than or equal to 1 and less than or equal to N.
7. The thermoelectric power generation module for fluid waste heat recovery according to claim 6, wherein the number of the upper alumina ceramic plate (1) and the lower alumina ceramic plate (5) is 1, the total number of thermoelectric semiconductors is N, and the total number of copper electrode sheets (2) is N; the height of the upper alumina ceramic plate (1) and the height of the lower alumina ceramic plate (5) are both H1The width and the length are equal and are both 2W +2Ws(ii) a All thermoelectric semiconductors are H in height3The width of the thermoelectric semiconductor is W, and the width distance between two adjacent thermoelectric semiconductors is WsThe distance between the peripheral thermoelectric semiconductor and the ceramic plate boundary is Ws2; the height of the copper electrode slice (2) is H2And a width of 2W + WsThe length of the ith copper electrode sheet and the length L of the ith thermoelectric semiconductoriAre equal.
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