三维光热材料的制备及其太阳能脱盐性能

王伟; 田子谕; 桓茜; 孙佳瑞; 白师丽; 刘卫国

doi:10.6054/j.jscnun.2023073

三维光热材料的制备及其太阳能脱盐性能

王伟^{1, 2, ,},
田子谕³,
桓茜^{1, 2},
孙佳瑞²,
白师丽²,
刘卫国^1, ,

1.
西安工业大学光电工程学院，西安 710021
2.
陕西工业职业技术学院航空工程学院，咸阳 712000
3.
陕西科技大学机电工程学院，西安 710021

基金项目:

陕西省教育厅科研计划项目 22JK0263

详细信息

通讯作者:
王伟，Email: wangwei05@sxpi.edu.cn

刘卫国，Email: wgliu@163.com

中图分类号: P747, TK519
计量
- 文章访问数: 87
- HTML全文浏览量: 70
- PDF下载量: 77
出版历程
- 收稿日期: 2023-10-28
- 网络出版日期: 2024-02-26
- 刊出日期: 2023-12-24

Preparation and Solar Desalination Performance of 3D Photothermal Materials

WANG Wei^{1, 2, ,},
TIAN Ziyu³,
HUAN Xi^{1, 2},
SUN Jiarui²,
BAI Shuaili²,
LIU Weiguo^1, ,

1.
School of Optoelectronic Engineering, Xi'an Technological University, Xi'an 710021, China
2.
School of Aeronautical Engineering, Shaanxi Polytechnic Institute, Xianyang 712000, China
3.
College of Mechanical & Electrical Engineering, Shaanxi University of Science & Technology, Xi'an 710021, China

摘要

摘要:
光热多孔材料是太阳能海水淡化应用的关键所在，受生物质天然多孔互联结构的启发，通过冷冻干燥和高温碳化工艺制备具有三维互联孔道结构的生物质碳基蒸发器，研究其微结构特征、水输送和蒸发性能以及三维结构对材料蒸发性能的增强作用。结果表明：该蒸发器具有高度互联的孔道结构和高达95.5%太阳光吸收率，在太阳光下实现了1.71 kg/(m²·h)的蒸发速率和89%的光热转换效率，可以有效蒸发和淡化海水。此外，在三维结构特征下，实现了高达2.65 kg/(m²·h)的蒸发速率，三维结构有效提升了材料的蒸发性能，显示了优异的太阳能蒸发/脱盐效果。
- 界面式蒸发 /
- 太阳能脱盐 /
- 三维多孔结构 /
- 生物质材料
Abstract:
Photothermal porous materials are the key to solar desalination applications. Inspired by the natural porous interconnected structure of biomass, a carbon-based evaporator with a 3D interconnected pore structure was prepared by freeze-drying and high-temperature carbonization processes, and the microstructural features, water transport and evaporation properties, as well as the enhancement of the material's evaporation performance by the 3D structure, were investigated. The results showed that the evaporator had a highly interconnected pore structure with the optical absorption of 95.5% and achieved an evaporation rate of 1.71 kg/(m²·h) and a photo-thermal conversion efficiency of 89% under one solar, which can effectively evaporate and desalinate seawater. In addition, an evaporation rate of 2.65 kg/(m²·h) was realized under 3D structural features, and the 3D structure effectively enhanced the evaporation performance of the material, showing excellent solar evaporation/desalination effects.
- interfacial evaporation /
- solar desalination /
- 3D porous structure /
- biomass materials

HTML全文

设 $\frac{1}{p}+\frac{1}{q}=1 \quad(p>1)$ ，α, β ∈ $\mathbb{R}$ ，K(x, y)非负可测，若 $L_{p}^{\alpha}(0, +\infty)=\left\{f(x) \geqslant 0:\|f\|_{p, \alpha}=\left(\int_{0}^{+\infty} x^{\alpha} f^{p}(x) \mathrm{d} x\right)^{1 /p} < +\infty\right\}$ , 则称不等式

$\int_{0}^{+\infty} \int_{0}^{+\infty} K(x, y) f(x) g(y) \mathrm{d} x \mathrm{d} y \leqslant M\|f\|_{p, \alpha}\|g\|_{q, \beta}$

为Hilbert型积分不等式. 由于此类不等式与积分算子T：

$T(f)(y)=\int_{0}^{+\infty} K(x, y) f(x) \mathrm{d} x$

有密切的联系，故而Hilbert型积分不等式对于研究算子T的有界性与算子范数有重要意义.

1991年，XU和GAO^[1]首次提出了研究Hilbert型不等式的权系数方法. 该方法的核心是：引入2个搭配参数a、b，利用Hölder不等式，可得到如下形式的不等式：

$\begin{aligned} &\int_{0}^{+\infty} \int_{0}^{+\infty} K(x, y) f(x) g(y) \mathrm{d} x \mathrm{d} y \leqslant \\ &\quad W_{1}^{1 / p}(b, p) W_{2}^{1 / q}(a, q)\left(\int_{0}^{+\infty} x^{\alpha(a, b)} f^{p}(x) \mathrm{d} x\right)^{1 / p} \times \\ &\quad\left(\int_{0}^{+\infty} y^{\beta(a, b)} g^{q}(y) \mathrm{d} x\right)^{1 / q}. \end{aligned}$

(1)

一般地，随意选取的搭配参数a、b并不能使式(1)的常数因子W₁^1/p(b, p)W₂^1/q(a, q)最佳. 已有的相关研究^[2-13]基本上都是凭借丰富的经验和娴熟的分析技巧选取适当的搭配参数a、b，从而获得最佳的Hilbert型不等式.

若选取的搭配参数a、b能够使式(1)的常数因子最佳，则称其为适配参数或适配数. 文献[14]曾讨论了齐次核的Hilbert型级数不等式的适配参数问题，本文将对拟齐次核的Hilbert型积分不等式讨论搭配参数a、b成为适配数的充分必要条件，并讨论其应用.

1. 预备知识

设G(u, v)是λ阶齐次函数，λ₁λ₂>0，则称K(x, y)=G(x^λ₁, y^λ₂)为拟齐次函数. 显然K(x, y)为拟齐次函数等价于：对t>0，有

$\begin{gathered} K(t x, y)=t^{\lambda_{1} \lambda} K\left(x, t^{-\lambda_{1} / \lambda_{2}} y\right), \\ K(x, t y)=t^{\lambda_{2} \lambda} K\left(t^{-\lambda_{2} / \lambda_{1}} x, y\right). \end{gathered}$

下面给出本文证明过程中所需的引理.

引理1 设1/p+1/q=1 (p>1)，a, b, λ∈ $\mathbb{R}$ ，λ₁λ₂>0，G(u, v)是λ阶齐次非负函数，K(x, y)=G(x^λ₁, y^λ₂)，aq/λ₁+bp/λ₂=1/λ₁+1/λ₂+ λ，记

$\begin{gathered} W_{1}(b, p)=\int_{0}^{+\infty} K(1, t) t^{-b p} \mathrm{~d} t, \\ W_{2}(a, q)=\int_{0}^{+\infty} K(t, 1) t^{-a q} \mathrm{~d} t, \end{gathered}$

则W₁(b, p)/λ₁=W₂(a, q)/λ₂，且

$\begin{aligned} &\omega_{1}(b, p, x)=\int_{0}^{+\infty} K(x, y) y^{-b p} \mathrm{~d} y=x^{\lambda_{1}\left(\lambda-b p / \lambda_{2}+1 / \lambda_{2}\right)} W_{1}(b, p), \\ &\omega_{2}(a, q, y)=\int_{0}^{+\infty} K(x, y) x^{-a q} \mathrm{~d} x=y^{\lambda_{2}\left(\lambda-a q / \lambda_{1}+1 / \lambda_{1}\right)} W_{2}(a, q) . \end{aligned}$

证明由aq/λ₁+bp/λ₂=1/λ₁+1/λ₂+ λ，可得- λ₁λ + λ₁bp/λ₂-λ₁/λ₂－1=－aq. 则有

$\begin{aligned} &W_{1}(b, p)=\int_{0}^{+\infty} t^{\lambda_{2} \lambda} K\left(t^{-\lambda_{2} / \lambda_{1}}, 1\right) t^{-b{p}} \mathrm{~d} t= \\ &\ \ \ \ \ \ \frac{\lambda_{1}}{\lambda_{2}} \int_{0}^{+\infty} K(u, 1) u^{-\lambda_{1} \lambda+\lambda_{1} b p / \lambda_{2}-\lambda_{1} / \lambda_{2}-1} \mathrm{~d} u= \\ &\ \ \ \ \ \ \frac{\lambda_{1}}{\lambda_{2}} \int_{0}^{+\infty} K(u, 1) u^{-a q} \mathrm{~d} u=\frac{\lambda_{1}}{\lambda_{2}} W_{2}(a, q), \end{aligned}$

故W₁(b, p)/λ₁=W₂(a, q)/λ₂.

作变换y=x^λ₁/λ₂t，有

$\begin{aligned} &\omega_{1}(b, p, x)=\int_{0}^{+\infty} x^{\lambda_{1} \lambda} K\left(1, x^{-\lambda_{1} / \lambda_{2}} y\right) y^{-b p} \mathrm{~d} y= \\ &\ \ \ \ \ \ x^{\lambda_{1}\left(\lambda-b p / \lambda_{2}+1 / \lambda_{2}\right)} \int_{0}^{+\infty} K(1, t) t^{-b p} \mathrm{~d} t=\\ &\ \ \ \ \ \ x^{\lambda_{1}\left(\lambda-b p / \lambda_{2}+1 / \lambda_{2}\right)} W_{1}(b, p). \end{aligned}$

同理可证ω₂(a, q, y)=y^{λ₂(λ－aq/λ₁+1/λ₁)}W₂(a, q). 证毕.

2. 适配参数的充分必要条件

定理1 设1/p+1/q=1 (p>1)，a, b, λ∈ $\mathbb{R}$ ，λ₁λ₂>0，G(u, v)是λ阶齐次非负可测函数，K(x, y)=G(x^λ₁, y^λ₂)，W₁(b, p)与W₂(a, q)如引理1所定义. 那么

(1) 若 $\alpha=\lambda_{1}\left[\lambda+\frac{1}{\lambda_{2}}+p\left(\frac{a}{\lambda_{1}}-\frac{b}{\lambda_{2}}\right)\right], \beta=\lambda_{2}\left[\lambda+\frac{1}{\lambda_{1}}+\right.$ $\left.p\left(\frac{b}{\lambda_{2}}-\frac{a}{\lambda_{1}}\right)\right]$ , 则有

$\begin{aligned} &\int_{0}^{+\infty} \int_{0}^{+\infty} K(x, y) f(x) g(y) \mathrm{d} x \mathrm{d} y \leqslant \\ &\quad W_{1}^{1 / p}(b, p) W_{2}^{1 / q}(a, q)\|f\|_{p, \alpha}\|g\|_{q, \beta}, \end{aligned}$

(2)

其中, $f(x) \in L_{p}^{\alpha}(0, +\infty), g(y) \in L_{q}^{\beta}(0, +\infty)$ .

(2) 式(2)中的常数因子W₁^1/p(b, p)W₂^1/q(a, q)是最佳的，当且仅当aq/λ₁+bp/λ₂=1/λ₁+1/λ₂+ λ，W₁(b, p)和W₂(a, q)都收敛. 当aq/λ₁+bp/λ₂=1/λ₁+1/λ₂+ λ时，式(2)化为

$\begin{aligned} &\int_{0}^{+\infty} \int_{0}^{+\infty} K(x, y) f(x) g(y) \mathrm{d} x \mathrm{d} y \leqslant \\ &\ \ \ \ \ \ \ \ \frac{W_{0}}{\left|\lambda_{1}\right|^{1 / q}\left|\lambda_{2}\right|^{1 / p}}\|f\|_{p, a p q-1}\|g\|_{q, b p q-1}, \end{aligned}$

(3)

其中, W₀=|λ₁|W₂(a, q)= |λ₂|W₁(b, p).

证明 (i)选择a、b为搭配参数. 根据Hölder不等式和引理1，利用权系数方法，有

$\begin{aligned} &\int_{0}^{+\infty} \int_{0}^{+\infty} K(x, y) f(x) g(y) \mathrm{d} x \mathrm{d} y= \\ &\qquad \int_{0}^{+\infty} \int_{0}^{+\infty}\left(\frac{x^{a}}{y^{b}} f(x)\right)\left(\frac{y^{b}}{x^{a}} g(y)\right) K(x, y) \mathrm{d} x \mathrm{d} y \leqslant \\ &\qquad \left(\int_{0}^{+\infty} \int_{0}^{+\infty} \frac{x^{a p}}{y^{b p}} f^{p}(x) K(x, y) \mathrm{d} x \mathrm{d} y\right)^{1 / p} \times \\ &\qquad \left(\int_{0}^{+\infty} \int_{0}^{+\infty} \frac{y^{b q}}{x^{a q}} g^{q}(y) K(x, y) \mathrm{d} x \mathrm{d} y\right)^{1 / q}=\\ &\qquad \left(\int_{0}^{+\infty} x^{a p} f^{p}(x) \omega_{1}(b, p, x) \mathrm{d} x\right)^{1 / p} \times \\ &\qquad \left(\int_{0}^{+\infty} y^{b q} g^{q}(y) \omega_{2}(a, q, y) \mathrm{d} y\right)^{1 / q}= \\ &\qquad W_{1}^{1 / p}(b, p) W_{2}^{1 / q}(a, q) \times \\ &\qquad \left(\int_{0}^{+\infty} x^{a p+\lambda_{1}\left(\lambda-b p / \lambda_{2}+1 / \lambda_{2}\right)} f^{p}(x) \mathrm{d} x\right)^{1 / p} \times \\ &\qquad \left(\int_{0}^{+\infty} y^{b q+\lambda_{2}\left(\lambda-a q / \lambda_{1}+1 / \lambda_{1}\right)} g^{q}(y) \mathrm{d} x\right)^{1 / q}= \\ &\qquad W_{1}^{1 / p}(b, p) W_{2}^{1 / q}(a, q)\|f\|_{p, \alpha}\|g\|_{q, \beta}, \end{aligned}$

故式(2)成立.

(ii) 充分性：设aq/λ₁+bp/λ₂=1/λ₁+1/λ₂+ λ，W₁(b, p)和W₂(a, q)收敛. 由引理1，有W₁(b, p)/λ₁=W₂(a, q)/λ₂，故

$W_{1}^{1 / p}(b, p) W_{2}^{1 / q}(a, q)=\left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{1 / q} W_{1}(b, p)=\frac{W_{0}}{\left|\lambda_{1}\right|^{1 / q}\left|\lambda_{2}\right|^{1 / p}},$

且α=apq－1，β=bpq－1，于是式(2)可化为式(3).

设式(3)的最佳常数因子为M₀，则M₀≤W₀/(|λ₁^1/q|λ₂|^1/p)，且用M₀取代式(3)中的常数因子后，式(3)仍然成立.

取充分小的ε>0及δ>0，令

$\begin{aligned} f(x) =&\left\{\begin{array}{l} x^{\left(-a p q-\left|\lambda_{1}\right| \varepsilon\right) / p}\quad (x \geqslant 1), \\ 0 \quad (0<x<1); \end{array} \right.\\ g(y) = &\begin{cases}y^{\left(-b p q-\left|\lambda_{2}\right| \varepsilon\right) / q}\quad (y \geqslant \delta) ,\\ 0 \quad (0<y<\delta).\end{cases} \end{aligned}$

则

$\begin{array}{c} \begin{aligned} &\|f\|_{p, a p q-1}\|g\|_{q, b p q-1}= \\ &\quad\left(\int_{1}^{+\infty} x^{-1-\left|\lambda_{1}\right| \varepsilon} \mathrm{d} x\right)^{1 / p}\left(\int_{\delta}^{+\infty} y^{-1-\left|\lambda_{2}\right| \varepsilon} \mathrm{d} y\right)^{1 / q}= \\ &\quad\left(\frac{1}{\left|\lambda_{1} \varepsilon\right|}\right)^{1 / p}\left(\frac{1}{\left|\lambda_{2}\right| \varepsilon} \delta^{-\left|\lambda_{2}\right| \varepsilon}\right)^{1 / q}= \\ &\quad\frac{1}{\varepsilon\left|\lambda_{1}\right|^{1 / p}\left|\lambda_{2}\right|^{1 / q}} \delta^{-\left|\lambda_{2}\right| \varepsilon / q}, \end{aligned}\\ \begin{aligned} &\int_{0}^{+\infty} \int_{0}^{+\infty} K(x, y) f(x) g(y) \mathrm{d} x \mathrm{d} y= \\ &\quad\int_{1}^{+\infty} x^{-a q-\left|\lambda_{1}\right| \varepsilon / p}\left(\int_{\delta}^{+\infty} y^{-b p-\left|\lambda_{2}\right| \varepsilon / q} K(x, y) \mathrm{d} y\right) \mathrm{d} x= \\ &\quad\int_{1}^{+\infty} x^{-a q-\left|\lambda_{1}\right| \varepsilon / p+\lambda \lambda_{1}}\left(\int_{\delta}^{+\infty} y^{-b p-\left|\lambda_{2}\right| \varepsilon / q} K\left(1, x^{-\lambda_{1} / \lambda_{2}} y\right) \mathrm{d} y\right) \mathrm{d} x= \\ &\quad\int_{1}^{+\infty} x^{-1-\left|\lambda_{1}\right| \varepsilon}\left(\int_{x^{-\lambda_{1} / \lambda_{2}} \delta}^{+\infty} t^{-b p-\left|\lambda_{2}\right| \varepsilon / q} K(1, t) \mathrm{d} t\right) \mathrm{d} x \geqslant \\ &\quad\int_{1}^{+\infty} x^{-1-\left|\lambda_{1}\right| \varepsilon}\left(\int_{\delta}^{+\infty} t^{-b p-\left|\lambda_{2}\right| \varepsilon / q} K(1, t) \mathrm{d} t\right) \mathrm{d} x= \\ &\quad\frac{1}{\left|\lambda_{1}\right| \varepsilon} \int_{\delta}^{+\infty} t^{-b p-\left|\lambda_{2}\right| \varepsilon / q} K(1, t) \mathrm{d} t. \end{aligned} \end{array}$

于是

$\frac{1}{\left|\lambda_{1}\right|} \int_{\delta}^{+\infty} t^{-b p-\left|\lambda_{2}\right| \varepsilon / q} K(1, t) \mathrm{d} t \leqslant \frac{M_{0}}{\left|\lambda_{1}\right|^{1 / p}\left|\lambda_{2}\right|^{1 / q}} \delta^{-\left|\lambda_{2}\right| \varepsilon / q}.$

先令ε→0⁺，再令δ→0⁺，得

$W_{1}(b, p)=\int_{0}^{+\infty} t^{-b p} K(1, t) \mathrm{d} t \leqslant \frac{M_{0}}{\left|\lambda_{1}\right|^{1 / p}\left|\lambda_{2}\right|^{1 / q}}.$

再根据引理1，可得到W₀/(|λ₁|^1/q|λ₂|^1/p)≤M₀. 所以式(3)的最佳常数因子M₀=W₀/(|λ₁|^1/q|λ₂|^1/p).

必要性：设式(2)的常数因子W₁^1/p(b, p)W₂^1/q(a, q)是最佳的，则W₁(b, p)和W₂(a, q)是收敛的. 下证aq/λ₁+bp/λ₂=1/λ₁+1/λ₂+ λ.

记 $\frac{1}{\lambda_{1}} a q+\frac{1}{\lambda_{2}} b p-\left(\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}+\lambda\right)=c, a_{1}=a-\frac{\lambda_{1} c}{p q}, b_{1}=$ $b-\frac{\lambda_{2} c}{p q}$ ，则

$\begin{gathered} \alpha=\lambda_{1}\left[\lambda+\frac{1}{\lambda_{2}}+p\left(\frac{a_{1}}{\lambda_{1}}-\frac{b_{1}}{\lambda_{2}}\right)\right]=\alpha_{1}, \\ \beta=\lambda_{2}\left[\lambda+\frac{1}{\lambda_{1}}+p\left(\frac{b_{1}}{\lambda_{2}}-\frac{a_{1}}{\lambda_{1}}\right)\right]=\beta_{1}, \\ W_{2}(a, q)=\int_{0}^{+\infty} K(t, 1) t^{-a q} \mathrm{~d} t=\frac{\lambda_{2}}{\lambda_{1}} \int_{0}^{+\infty} K(1, t) t^{-b p+\lambda_{2} c} \mathrm{~d} t . \end{gathered}$

于是可知式(2)等价于

$\begin{aligned} &\int_{0}^{+\infty} \int_{0}^{+\infty} K(x, y) f(x) g(y) \mathrm{d} x \mathrm{d} y \leqslant \\ &\quad W_{1}^{1 / p}(b, p)\left(\frac{\lambda_{2}}{\lambda_{1}} \int_{0}^{+\infty} K(1, t) t^{-b p+\lambda_{2} c} \mathrm{~d} t\right)^{1 / q}\|f\|_{p, \alpha_{1}}\|g\|_{q, \beta_{1}} . \end{aligned}$

又经计算有a₁q/λ₁+b₁p/λ₂=1/λ₁+1/λ₂+ λ，α₁=a₁pq－1，β₁=b₁pq－1，故式(2)进一步等价于

$\begin{aligned} &\int_{0}^{+\infty} \int_{0}^{+\infty} K(x, y) f(x) g(y) \mathrm{d} x \mathrm{d} y \leqslant \\ &\qquad W_{1}^{1 / p}(b, p)\left(\frac{\lambda_{2}}{\lambda_{1}} \int_{0}^{+\infty} K(1, t) t^{-b p+\lambda_{2} c} \mathrm{~d} t\right)^{1 / q} \times \\ &\qquad\|f\|_{p, a_{1} p q-1}\|g\|_{q, b_{1} p q-1} . \end{aligned}$

(4)

根据假设，式(4)的最佳常数因子是W₁^1/p(b, p)× ${\left({\frac{{{\lambda _2}}}{{{\lambda _1}}}\int_0^{ + \infty } K (1, t){t^{ - bp + {\lambda _2}c}}{\rm{d}}t} \right)^{1/q}}$ . 又由 $\frac{1}{{{\lambda _1}}}{a_1}q + \frac{1}{{{\lambda _2}}}{b_1}p = \frac{1}{{{\lambda _1}}} + \frac{1}{{{\lambda _2}}} + \lambda$ 及充分性的证明，可知式(4)的最佳常数因子为

$\begin{gathered} \frac{1}{\left|\lambda_{1}\right|^{1 / q}\left|\lambda_{2}\right|^{1 / p}}\left(\left|\lambda_{2}\right| \int_{0}^{+\infty} K(1, t) t^{-b_{1} p} \mathrm{~d} t\right)= \\ \left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{1 / q} \int_{0}^{+\infty} K(1, t) t^{-b p+\lambda_{2} c / q} \mathrm{~d} t \end{gathered}$

于是得到

$\begin{aligned} &\int_{0}^{+\infty} K(1, t) t^{-b p+\lambda_{2} c / q} \mathrm{~d} t= \\ &\quad W_{1}^{1 / p}(b, p)\left(\int_{0}^{+\infty} K(1, t) t^{-b p+\lambda_{2} c} \mathrm{~d} t\right)^{1 / q}. \end{aligned}$

(5)

对于1和t^λ₂c/q，应用Hölder不等式，有

$\begin{aligned} &\int_{0}^{+\infty} K(1, t) t^{-b p+\lambda_{2} c / q} \mathrm{~d} t=\int_{0}^{+\infty} t^{\lambda_{2} c / q} K(1, t) t^{-b p} \mathrm{~d} t \leqslant \\ &\qquad\left(\int_{0}^{+\infty} 1^{p} K(1, t) t^{-b p} \mathrm{~d} t\right)^{1 / p}\left(\int_{0}^{+\infty} t^{\lambda_{2} c} K(1, t) t^{-b p} \mathrm{~d} t\right)^{1 / q}= \\ &\qquad W_{1}^{1 / p}(b, p)\left(\int_{0}^{+\infty} K(1, t) t^{-b p+\lambda_{2} c} \mathrm{~d} t\right)^{1 / q}. \end{aligned}$

(6)

根据式(5)，可知式(6)取等号. 又根据Hölder不等式取等号的条件，可得t^λ₂c/q=常数，故c=0，即aq/λ₁+bp/λ₂=1/λ₁+1/λ₂+ λ₁. 证毕.

注1 定理1表明: 当且仅当aq/λ₁+bp/λ₂=1/λ₁+1/λ₂+ λ时，搭配参数a、b是适配参数. 因此，只要选取a、b满足aq/λ₁+bp/λ₂=1/λ₁+1/λ₂+ λ，就可以得到各种各样的具有最佳常数因子的Hilbert型积分不等式.

推论1 设1/p+1/q=1 (p>1)，λ₁λ₂>0，λ >0，1/r+1/s=1 (r>1)，α=p(1－ λλ₁/r)－1，β=q(1－ λλ₂/s)－1，则

$\begin{aligned} &\int_{0}^{+\infty} \int_{0}^{+\infty} \frac{f(x) g(y)}{\left(x^{\lambda_{1}}+y^{\lambda_{2}}\right)^{\lambda}} \mathrm{d} x \mathrm{d} y \leqslant \\ &\qquad\frac{1}{\left|\lambda_{1}\right|^{1 / q}\left|\lambda_{2}\right|^{1 / p}} \mathrm{~B}\left(\frac{\lambda}{r}, \frac{\lambda}{s}\right)\|f\|_{p, \alpha}\|g\|_{q, \beta}, \end{aligned}$

(7)

其中的常数因子是最佳的，f(x)∈L_p^α(0, +∞)，g(y)∈L_q^β(0, +∞).

证明记K(x, y)=G(x^λ₁, y^λ₂)=1/(x^λ₁+y^λ₂)^λ，则G(u, v)是－λ阶齐次非负函数. 选取搭配参数a= $\frac{1}{q}\left(1-\frac{\lambda \lambda_{1}}{r}\right), b=\frac{1}{p}\left(1-\frac{\lambda \lambda_{2}}{s}\right)$ , 可得

$\frac{1}{\lambda_{1}} a q+\frac{1}{\lambda_{2}} b p=\frac{1}{\lambda_{1}}\left(1-\frac{\lambda \lambda_{1}}{r}\right)+\frac{1}{\lambda_{2}}\left(1-\frac{\lambda \lambda_{2}}{s}\right)=\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}-\lambda,$

故a、b是适配参数. 又因为apq－1=p(1－ λλ₁/r)－1=α，bpq－1=q(1－ λλ₂/s)－1=β，且

$\begin{aligned} W_{0}=&\left|\lambda_{2}\right| W_{1}(b, p)=\left|\lambda_{2}\right| \int_{0}^{+\infty} \frac{1}{\left(1+t^{\lambda_{2}}\right)^{\lambda}} t^{\lambda \lambda_{2} / s-1} \mathrm{~d} t=\\ & \int_{0}^{+\infty} \frac{1}{(1+u)^{\lambda}} u^{\lambda / s-1} \mathrm{~d} u=\mathrm{B}\left(\frac{\lambda}{s}, \lambda-\frac{\lambda}{s}\right)=\mathrm{B}\left(\frac{\lambda}{r}, \frac{\lambda}{s}\right). \end{aligned}$

根据定理1，式(7)成立，且其常数因子是最佳的. 证毕.

3. 在求积分算子范数中的应用

根据Hilbert型不等式与相应积分算子的关系理论，由定理1可得如下定理.

定理2 设1/p+1/q=1 (p>1)，a, b, λ ∈ $\mathbb{R}$ ，λ₁λ₂>0，α=apq－1，β=bpq－1，G(u, v)是λ阶齐次非负可测函数，K(x, y)=G(x^λ₁, y^λ₂)，且

$\begin{aligned} &W_{1}(b, p)=\int_{0}^{+\infty} K(1, t) t^{-b p} \mathrm{~d} t<+\infty, \\ &W_{2}(a, q)=\int_{0}^{+\infty} K(t, 1) t^{-a q} \mathrm{~d} t<+\infty, \end{aligned}$

则当aq/λ₁+bp/λ₂=1/λ₁+1/λ₂+ λ时，积分算子T：

$T(f)(y)=\int_{0}^{+\infty} K(x, y) f(x) \mathrm{d} x, f(x) \in L_{p}^{\alpha}(0,+\infty)$

是从L_p^α(0, +∞)到L_p^β(1－p)(0, +∞)的有界算子，且T的算子范数为

$\|T\|=\frac{\left|\lambda_{2}\right| W_{1}(b, p)}{\left|\lambda_{1}\right|^{1 / q}\left|\lambda_{2}\right|^{1 / p}}=\left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{1 / q} \int_{0}^{+\infty} K(1, t) t^{-b p} \mathrm{~d} t.$

推论2 设1/p+1/q=1 (p>1)，λ₁λ₂>0，－1 < λ < min{1±4/λ₁, 1±4/λ₂}，α=p[1+ λ₁(λ －1)/2]－1，β=p[1+ λ₂(λ －1)/2]－1，则积分算子T：

$T(f)(y)=\int_{0}^{+\infty} \frac{\left|x^{\lambda_{1}}-y^{\lambda_{2}}\right|^{\lambda}}{\max \left\{x^{\lambda_{1}}, y^{\lambda_{2}}\right\}} f(x) \mathrm{d} x, f(x) \in L_{p}^{\alpha}(0,+\infty)$

是从L_p^α(0, +∞)到L_p^β(1－p)(0, +∞)的有界算子，且T的算子范数为

$\begin{aligned} &\|T\|=\frac{1}{\left|\lambda_{1}\right|^{1 / q}\left|\lambda_{2}\right|^{1 / p}}\left[\mathrm{~B}\left(\lambda+1, \frac{1-\lambda}{2}-\frac{2}{\lambda_{2}}\right)+\right. \\ &\ \ \ \ \ \ \left.\mathrm{B}\left(\lambda+1, \frac{1-\lambda}{2}+\frac{2}{\lambda_{2}}\right)\right] . \end{aligned}$

证明记K(x, y)=G(x^λ₁, y^λ₂)= |x^λ₁－y^λ₂|^λ/max{x^λ₁, y^λ₂}，则G(u, v)是λ －1阶齐次函数. 取a= $\frac{1}{q}\left[1+\frac{\lambda_{1}}{2}(\lambda-1)\right], b=\frac{1}{p}\left[1+\frac{\lambda_{2}}{2}(\lambda-1)\right]$ ，则

$\begin{aligned} &\frac{1}{\lambda_{1}} a q+\frac{1}{\lambda_{2}} b p=\frac{1}{\lambda_{1}}\left[1+\frac{\lambda_{1}}{2}(\lambda-1)\right]+\frac{1}{\lambda_{2}}\left[1+\frac{\lambda_{2}}{2}(\lambda-1)\right]= \\ &\ \ \ \ \ \ \frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}+\lambda-1, \end{aligned}$

故a、b是适配参数. 又 $a p q-1=p\left[1+\frac{\lambda_{1}}{2}(\lambda-1)\right]-1=\alpha$ ， $b p q-1=q\left[1+\frac{\lambda_{2}}{2}(\lambda-1)\right]-1=\beta$ . 则

$\begin{aligned} &\left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{1 / q} \int_{0}^{+\infty} K(1, t) t^{-b p} \mathrm{~d} t= \\ &\qquad\left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{1 / q} \int_{0}^{+\infty} \frac{\left|1-t^{\lambda_{2}}\right|^{\lambda}}{\max \left\{1, t^{\lambda_{2}}\right\}} t^{-\left[1+\lambda_{2}(\lambda-1) / 2\right]} \mathrm{d} t= \\ &\qquad\frac{1}{\left|\lambda_{1}\right|^{1 / q}\left|\lambda_{2}\right|^{1 / p}}\left[\mathrm{~B}\left(\lambda+1, \frac{1-\lambda}{2}-\frac{2}{\lambda_{2}}\right)+\right. \\ &\qquad\left.\mathrm{B}\left(\lambda+1, \frac{1-\lambda}{2}+\frac{2}{\lambda_{2}}\right)\right]<+\infty . \end{aligned}$

根据定理2，知推论2成立. 证毕.

推论3 设1/p+1/q=1 (p>1)，1/r+1/s=1 (r>1)，λ₁λ₂>0，α=p(1－ λ₁/r)－1，β=q(1－ λ₂/s)－1. 则积分算子T：

$T(f)(y)=\int_{0}^{+\infty} \frac{\ln \left(x^{\lambda_{1}} / y^{\lambda_{2}}\right)}{x^{\lambda_{1}}-y^{\lambda_{2}}} f(x) \mathrm{d} x, f(x) \in L_{p}^{\alpha}(0,+\infty)$

是从L_p^α(0, +∞)到L_p^β(1－p)(0, +∞)的有界算子，且T的算子范数为

$\|T\|=\frac{1}{\left|\lambda_{1}\right|^{1 / q}\left|\lambda_{2}\right|^{1 / p}}\left[\zeta\left(2, \frac{1}{r}\right)+\zeta\left(2, \frac{1}{s}\right)\right],$

其中ζ(t, a)是Riemann函数.

证明记

$K(x, y)=G\left(x^{\lambda_{1}}, y^{\lambda_{2}}\right)=\frac{\ln \left(x^{\lambda_{1}} / y^{\lambda_{2}}\right)}{x^{\lambda_{1}}-y^{\lambda_{2}}},$

则G(u, v)是－1阶齐次非负函数.

取搭配参数 $a=\frac{1}{q}\left(1-\frac{\lambda_{1}}{r}\right), b=\frac{1}{p}\left(1-\frac{\lambda_{2}}{s}\right)$ ，则

$\frac{1}{\lambda_{1}} a q+\frac{1}{\lambda_{2}} b p=\frac{1}{\lambda_{1}}\left(1-\frac{\lambda_{1}}{r}\right)+\frac{1}{\lambda_{2}}\left(1-\frac{\lambda_{2}}{s}\right)=\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}-1,$

故a、b是适配参数. 又apq－1=p(1－ λ₁/r)－1=α，bpq－1=q(1－ λ₂/s)－1=β，且

$\begin{gathered} \left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{1 / q} \int_{0}^{+\infty} K(1, t) t^{-b p} \mathrm{~d} t=\left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{1 / q} \int_{0}^{+\infty} \frac{\ln \left(t^{-\lambda_{2}}\right)}{1-t^{\lambda_{2}}} t^{\lambda_{2} / s-1} \mathrm{~d} t= \\ \frac{1}{\left|\lambda_{1}\right|^{1 / q}\left|\lambda_{2}\right|^{1 / p}}\left[\zeta\left(2, \frac{1}{r}\right)+\zeta\left(2, \frac{1}{s}\right)\right]<+\infty \end{gathered}$

根据定理2，知推论3成立. 证毕.

图 1 不同蒸发器的实物照片

Figure 1. Digital photos of different evaporators

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图 2 界面蒸发平台及户外蒸发测试

Figure 2. Interfacial evaporation platform and outdoor evaporation test

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图 3 三维CBG蒸发器的顶表面和侧表面的SEM形貌

Figure 3. SEM morphologies of the top and side surfaces of 3D CBG evaporator

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图 4 CBG-400蒸发器的空隙分布与XPS光谱

Figure 4. Pore size distribution and XPS spectra of CBG-400 evaporator

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图 5 不同碳化温度和高度下CBG的水吸收性能

Figure 5. Water absorption properties of CBG at different carbonization temperatures and altitudes

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图 6 不同碳化温度下CBG的热行为和CBG-400的光学吸收性能

Figure 6. Thermal behavior of CBG with different carbonization temperatures and the light absorption property of CBG-400

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图 7 不同碳化温度下CBG的蒸发性能测试

Figure 7. Evaporation properties of CBG at different carbonization temperatures

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图 8 三维结构CBG-400的蒸发性能

Figure 8. Evaporation properties of CBG-400 with 3D structure

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图 9 3D-CBG在模拟太阳光照射下的温度变化

Figure 9. Temperature changes of 3D-CBG under simulated sunlight irradiation

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图 10 3D-CBG-400-6的户外测试环境条件、表面温度和蒸发性能

注：A图中光密度为0.964 kW/m²，环境温度为23 ℃；B图中蒸发率最高点的环境温度为45%。

Figure 10. Ambient conditions, surface temperatures and evaporation performance for outdoor testing of 3D-CBG-400-6

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参考文献(19)

[1]	杨地, 史潇凡, 张冀杰, 等. 光热材料在海水淡化领域的近期研究进展与展望[J]. 化学学报, 2023, 81(08): 1052-1063. YANG D, SHI X F, ZHANG J J, et al. Recent research progress and prospect of photothermal materials in seawater desalination[J]. Acta Chimica Sinica, 2023, 81(8): 1052-1063.
[2]	赵春波, 赵嵘, 戚剑飞, 等. 面向双碳目标的水淡技术: 生物质碳用于界面太阳能光蒸汽转化技术的研究进展[J]. 材料导报, 2023, 37(12): 5-17. ZHAO C B, ZHAO R, QI J F, et al. Utilizing biochars in interfacial solar-vapor conversion and seawater desalination: potential value for 'Double Carbon' goals and state of the art[J]. Materials Reports, 2023, 37(12): 5-17.
[3]	ZHOU L, LI X, NI G W, et al. The revival of thermal utilization from the sun: interfacial solar vapor generation[J]. National Science Review, 2019, 6(3): 562-578. doi: 10.1093/nsr/nwz030
[4]	GUO Y, ZHAO F, ZHOU X, et al. Tailoring nanoscale surface topography of hydrogel for efficient solar vapor generation[J]. Nano Letters, 2019, 19(4): 2530-2536. doi: 10.1021/acs.nanolett.9b00252
[5]	LIU H, ZHANG X, HONG Z, et al. A bioinspired capillary driven pump for solar vapor generation[J]. Nano Energy, 2017, 42: 115-121. doi: 10.1016/j.nanoen.2017.10.039
[6]	XU N, HU X, XU, et al. Mushrooms as efficient solar steam-generation devices[J]. Advanced Materials, 2017, 29(28): 1606762/1-5.
[7]	ZHU M, YU J, MA C, et al. Carbonized daikon for high efficient solar steam generation[J]. Solar Energy Materials and Solar Cells, 2019, 191: 83-90. doi: 10.1016/j.solmat.2018.11.015
[8]	LI Z, WANG C, LEI T, et al. Arched bamboo charcoal as interfacial solar steam generation integrative device with enhanced water purification capacity[J]. Advanced Sustainable Systems, 2019, 3(4): 1800144/1-10.
[9]	SUN P, ZHANG W, ZADA I, et al. 3D-structured carbonized sunflower heads for improved energy efficiency in solar steam generation[J]. ACS Applied Materials & Interfaces, 2019, 12(2): 2171-2179.
[10]	XU K, WANG C, LI Z, et al. Salt mitigation strategies of solar-driven interfacial desalination[J]. Advanced Functional Materials, 2021, 31(8): 2007855/1-9.
[11]	KUANG Y, CHEN C, HE S, et al. A high-performance self-regenerating solar evaporator for continuous water desalination[J]. Advanced materials, 2019, 31(23): 1900498/1-8.
[12]	ZHOU J, GU Y, LIU P, et al. Development and evolution of the system structure for highly efficient solar steam generation from zero to three dimensions[J]. Advanced Functional Materials, 2019, 29(50): 1903255/1-20.
[13]	LI X, LI J, LU J, et al. Enhancement of interfacial solar vapor generation by environmental energy[J]. Joule, 2018, 2(7): 1331-1338. doi: 10.1016/j.joule.2018.04.004
[14]	WANG Y, WU X, GAO T, et al. Same materials, bigger output: a reversibly transformable 2D-3D photothermal evaporator for highly efficient solar steam generation[J]. Nano Energy, 2021, 79: 105477/1-9.
[15]	WU X, WU Z, WANG Y, et al. All-cold evaporation under one sun with zero energy loss by using a heatsink inspired solar evaporator[J]. Advanced Science, 2021, 8(7): 2002501/1-12.
[16]	WANG Y, WU X, YANG X, et al. Reversing heat conduction loss: extracting energy from bulk water to enhance solar steam generation[J]. Nano Energy, 2020, 78: 105269/1-10.
[17]	MA M, CAO X, XU K, et al. Engineering a superhydrophilic TiC/C absorber with multiscale pore network for stable and efficient solar evaporation of high-salinity brine[J]. Materials Today Energy, 2022, 26: 101009/1-11.
[18]	WANG W, TIAN Z, HUAN X, et al. Carbonized sweet potato-based evaporator for highly efficient solar vapor generation[J]. Energy Technology, 2023: 2300671/1-8.
[19]	ZHOU L, TAN Y, WANG J, et al. 3D self-assembly of aluminium nanoparticles for plasmon-enhanced solar desalination[J]. Nature Photonics, 2016, 10(6): 393-398. doi: 10.1038/nphoton.2016.75