【问题的提出】近年来北方城镇建筑物的窗户有些是三层的，即窗户上装三层玻璃，且中间留有一定空隙。他与两层玻璃窗相比，其保温效果如何呢 ？本文通过模型来描述热量通过窗户的传导过程，并将三层玻璃窗与两层玻璃窗的热量传导进行对比，对三层玻璃窗能减少多少热量损失给出定量分析结果。
【关键字】三层玻璃窗 量损失
【模型假设】
1、热量的传递过程只有传导，没有对流，即假定窗户的密封性能很好，三层玻璃之间的空气是不流动的。
2、室内温度与室外温度保持不变，热传导过程已处于稳定状态，即沿热传导方向，单位时间通过单位面积的热量是常数。
3、室内温度![](https://wkrtcs.bdimg.com/rtcs/image?w=23&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"23","h":"23","c":"a5749ec33f2c95fe8c19d702d76d4968.png"})
【符号说明】
| 单位时间单位面积双层玻璃窗传导的热量 |
| 单位时间单位面积三层玻璃窗传导的热量 |
| 室内温度 |
| 室外温度 |
d | 玻璃厚度 |
l | 夹层空气的厚度 |
| 内层玻璃窗的外侧温度 |
| 中层玻璃窗的内侧温度 |
| 中层玻璃窗的外侧温度 |
| 外层玻璃窗的内侧温度 |
| 玻璃窗的热传导系数 |
| 空气的热传导系数 |
【模型的建立与求解】
在本文假设中，热传导过程应满足如下的物理定律：
厚度为![](https://wkrtcs.bdimg.com/rtcs/image?w=14&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"14","h":"20","c":"8277e0910d750195b448797616e091ad.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=72&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"72","h":"43","c":"6a87d8d06c6bd72cda82455fd8ac2a99.png"})
其中![](https://wkrtcs.bdimg.com/rtcs/image?w=14&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"14","h":"20","c":"8ce4b16b22b58894aa86c421e8759df3.png"})
对于厚度为
![](https://wkrtcs.bdimg.com/rtcs/image?w=138&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"138","h":"47","c":"d255d6bc099ecb0b8dbc3dc7aeac6bbc.png"})
对于三层玻璃窗，设其内层玻璃的外侧温度为![](https://wkrtcs.bdimg.com/rtcs/image?w=23&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"23","h":"23","c":"73dbbadc957c0a6b4dad293229e4bca1.png"})
由（3）式消去
![](https://wkrtcs.bdimg.com/rtcs/image?w=138&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"138","h":"47","c":"0c24839b04f7671717e893569a63093f.png"})
将（4）式比上（2）式可得：
![](https://wkrtcs.bdimg.com/rtcs/image?w=94&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"94","h":"57","c":"591cf95354354051b49c450e071f9a7b.png"})
由于S大于0，显然, ![](https://wkrtcs.bdimg.com/rtcs/image?w=61&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"61","h":"27","c":"aa35110fcffed875b182510f0e8174e2.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=24&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"24","h":"49","c":"3b34adb09e5e6bbd50ab6d26f5d77b04.png"})
在分析三层玻璃比双层玻璃可减少多少热量损失时，我们做最保守的估计，取
【模型反思】不难发现，虽然三层玻璃窗比双层玻璃窗节约热量50%左右，,但双层玻璃窗节约热量已经达到了97%，三层玻璃实际的节约热量比双层的没有多多少。而且三层玻璃窗的制作工艺更加复杂，会增加一些费用。从经济适用的角度分析，三层玻璃窗不如双层的适用。
【参考文献】数学建模（第四版）姜启源 谢金星 叶俊 编
题目：三层玻璃窗的功效
姓名：刘强
年级：2009
学号：20
本文来源：https://www.2haoxitong.net/k/doc/2a34e4842f3f5727a5e9856a561252d380eb209a.html
文档为doc格式
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![](https://wkrtcs.bdimg.com/rtcs/image?w=24&md5sum=8f7b1807fe35d71a9556cd07fffccea0&sign=ef285124a3&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=454&ipr={"t":"img","w":"24","h":"23","c":"cf3aed12b96d168a3cb957b1a5c38875.png"})
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