International Journal of Polymer Science

Volume 2015 (2015), Article ID 367962, 8 pages

http://dx.doi.org/10.1155/2015/367962

## A Facile Approach to Evaluate Thermal Insulation Performance of Paper Cups

State Key Laboratory of Pulp and Paper Engineering, South China University of Technology, Guangzhou 510640, China

Received 29 September 2015; Accepted 20 October 2015

Academic Editor: Matheus Poletto

Copyright © 2015 Yudi Kuang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Paper cups are ubiquitous in daily life for serving water, soup, coffee, tea, and milk due to their convenience, biodegradability, recyclability, and sustainability. The thermal insulation performance of paper cups is of significance because they are used to supply hot food or drinks. Using an effective thermal conductivity to accurately evaluate the thermal insulation performance of paper cups is complex due to the inclusion of complicated components and a multilayer structure. Moreover, an effective thermal conductivity is unsuitable for evaluating thermal insulation performance of paper cups in the case of fluctuating temperature. In this work, we propose a facile approach to precisely analyze the thermal insulation performance of paper cups in a particular range of temperature by using an evaluation model based on the MISO (Multiple-Input Single-Output) technical theory, which includes a characterization parameter (temperature factor) and a measurement apparatus. A series of experiments was conducted according to this evaluation model, and the results show that this evaluation model enables accurate characterization of the thermal insulation performance of paper cups and provides an efficient theoretical basis for selecting paper materials for paper cups.

#### 1. Introduction

The first modern paper cup was initially developed by Lawrence Luellen in 1907. After a hundred years of development, paper cups have become one of the life’s necessities for serving hot and cold drinks; they are widely used in fast-food restaurants, coffee shops, and offices, among others, because they are inexpensive, biodegradable, and renewable and because there are environmentally friendly properties of paper [1, 2]. The consumption of paper cups in China has increased sharply in recent years. More than 25 billion paper cups were consumed in 2011, and the number will continue to increase, with an estimated future annual growth rate of 12.84%. Paper cups are sometimes used to serve hot drinks or food, and the thermal insulation performance of paper should be considered because of the possibility of serious scalding accident. However, there is a lack of uniform standards, such as national standard or industrial standard for the convenient and accurate evaluation of the thermal insulation performance of paper cups.

Heat is transferred via conduction, convection, and radiation [3], and the impact of each mode of heat transfer depends on the application. The thermal insulation performance of paper cups is determined by the paper materials. Paper is considered to be a type of porous fibrous material with a porosity of 40%~70%. When the volume fraction of cellulose fibers is greater than 1%, heat transfer by convection is negligible because the air cells in the fiber system are too small to support convection or turbulence [4–6]. Heat transfer by radiation is a linear function of temperature cubed () according to the Rosseland approximation [7]. Hence, the radiative heat transfer is also inappreciable at the relatively low operating temperature of paper cups [8, 9]. Therefore, an effective thermal conductivity is generally used to characterize the conductive heat transfer and thermal insulation performance of paper materials.

Schuhmeister first proposed a model for the thermal conductivity of fibrous materials [10]. Many researchers have since worked on modifications of this model [11–13]. The most basic expression for defining an effective thermal conductivity in porous media was given by Bhattacharyya [14] and Bankvall [12]

The above equation assumes that the conductive heat travels through both the fibers and the interstitial fluid (often air). Therefore the effective thermal conductivity was based on a weighted average of the thermal conductivity values of the fibers and the interstitial fluid. The first term in the right-hand side of (1) overpredicts the conductivity of the solid phase. The second term is expected to predict the conductivity of the fluid phase. However, there are some questions regarding this equation. The heat transfer does not occur in a parallel mode through the paper because of the random fiber system, and the thermal barrier resistance should be considered when heat flows through a number of fiber-to-fiber contact areas [15]. The impact of each parameter on the thermal insulation performance of paper was studied by computer simulation. Through this simulation, we can predict the effective thermal conductivity of the whole media [15, 16].

However, a variety of parameters such as temperature and pressure [17], moisture [18, 19], the nature of the raw materials, and the macrostructure and microstructure of the paper [20] will affect the veracity of the effective thermal conductivity [21]. Another problem with an effective thermal conductivity is that the numerical value changes with temperature. Paper cups are always used at a relatively low temperature (below 100°C) that fluctuates. Therefore, an effective thermal conductivity is unsuitable for practical applications. In this study, we proposed a facile approach to precisely analyze the thermal insulation performance of paper cups by establishing an evaluation model. For the first time, we propose a new parameter named “temperature factor,” which is a simple and convenient parameter for characterizing the thermal insulation performance of a paper cup. Moreover, an apparatus for the experiment is also designed, as shown in Section 2.

#### 2. Model Establishment

On the basis of MISO (Multiple-Input Single-Output) technical theory [22], the equation for the effective thermal conductivity can be expressed as shown below

This equation denotes the effective thermal conductivity as a function of the factors that influence the thermal insulation performance of paper cups. When this approach is used, only the value of is required to appraise the thermal insulation performance of paper cups. However, Fourier’s law states that thermal conductivity is independent of the temperature gradient but necessarily of temperature itself. The thermal conductivity is a function of temperature. That means the effective thermal conductivity was suited for evaluating the thermal conductivity of materials at a single temperature [23]. For paper cups, evaluation of the thermal insulation performance in a particular temperature range is more suitable for practical applications. Therefore, for assessing the thermal insulation performance of paper cups, this study includes the proposal of the temperature factor and the design of a corresponding apparatus for measurement of this new parameter. The temperature factor represents the thermal insulation performance of paper cups within a certain temperature interval, and the apparatus can help to quickly determine the value of the parameter.

##### 2.1. Experimental Apparatus

To evaluate the thermal insulation performance of paper cups, an apparatus to test the temperature difference between the inside and outside of the cup wall was set up, and its structure is shown schematically in Figure 1. This apparatus was assembled based on the method used by Iioka [24], which was comprised of a heating section, a support frame, and a measuring part.