Mathematical Problems in Engineering

Volume 2016, Article ID 9071531, 7 pages

http://dx.doi.org/10.1155/2016/9071531

## Heat Transfer Analysis and Modification of Thermal Probe for Gas-Solid Measurement

National Key Laboratory of Science and Technology on Multispectral Information Processing, School of Automation, Huazhong University of Science and Technology, Wuhan 43004, China

Received 22 November 2015; Accepted 5 January 2016

Academic Editor: Yuan Fan

Copyright © 2016 Hong Zhang and Xiangying Qi. 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

The presented work aims to measure the gas-solid two-phase mass flow-rate in pneumatic conveyor, and a novel modified thermal probe is applied. A new analysis of the local heat transfer coefficients of thermal probe is presented, while traditional investigations focus on global coefficients. Thermal simulations are performed in Fluent 6.2 and temperature distributions of the probe are presented. The results indicate that the probe has obviously stable and unstable heat transfer areas. Based on understanding of probe characteristics, a modified probe structure is designed, which makes the probe output signal more stable and widens the measuring range. The experiments are carried out in a special designed laboratory scale pneumatic conveyor, and the modified probe shows an unambiguous improvement of the performance compared with the traditional one.

#### 1. Introduction

It is usual that the solid powder is transported in industrial plant by pneumatic conveyor. The particularly important examples are the transport of granular and pulverized coal for boiler and blast furnace. In those processes, on-line measurement of mass flow-rate in gas-solid two-phase flow is a common problem that has never been solved successfully [1].

Lots of methods were used to solve this problem, including static electricity, microwave, tomography, and pressure drop. Nowadays, the thermal method is also applied to the gas-solid measurement. As a long time used sensor which measures gas mass flow in industry, the thermal probe is extended to measure the gas-solid two-phase flow. The first use of thermal probe to measure gas-solid flow should be owed to Moriyama et al. [2] since 1985. The authors developed a noninvasive differential temperature sensing method for the mass flow rate 0–1,000 kg/h and mass flow ratio 0–47.2 kg/kg. The researchers including Zheng et al. [3, 4], Liu et al. [5], Yuan and Lu [6], and Zhang and Li [7] have focused on this area recently.

Our previous works include investigations on traditional heat balance method, thermal probe method [7, 8], and data fusion to achieve better performance [9]. By applying the thermal probe theory to measure the mass flow-rate of gas-solid two-phase flow, the variety of solids loads in pneumatic conveying system, but two shortages are observed including signal fluctuation and severe on-linearity causing narrow measurement range.

Commonly, the thermal probe has fast response time and high sensitivity and the issues of worse zero stability and signal fluctuation. To apply the thermal probe to detect the gas-solid flow, the heat transfer characteristics research in detail is needed to solve the problems of stability and range extension. Theoretical analysis and numerical simulation help us to study and find new solution to improve the probe performance in gas-solid.

While the details of heat transfer behavior in gas-solid flow are not yet well understood, the early experimental results have already supported the thermal probe’s application in solids mass flow-rate measurement [10–12]. Experimental investigations have proved that the wall Nusselt number is a function of the solid mass loading ratio. The Nusselt number will significantly increase in higher solids loading ratios because the heat transfer is enhanced caused by gas boundary layer influence due to solids. By using a four-way coupling Eulerian-Lagrangian model to simulate nonisothermal gas-solid flow in a vertical pipe, El-Behery et al. pointed out that the presence of solid particles reduces the boundary layer thickness and this reduction increases as the mass loading ratio increases [13]. With the recent improvement of the related mathematical models, the computational fluid dynamics (CFD) tools are strong enough to describe the heat transfer and dynamics behavior of gas-solid two-phase flow. The commercial software such as Fluent has been validated in many situations [14]. Fluent 6.2 was chosen as a simulation tool in analyzing the heat transfer of the thermal probe in the presented work.

#### 2. Analysis of Heat Transfer Equation of Thermal Probe in Single-Phase Flow

For a thin heated cylinder placed in fluid (namely, hot wire), the way of heat transfer between cylinder and fluid is mainly performed as forced convection. King concluded an empirical formula of convective heat transfer between “infinite” cylinder and fluid through experimental and theoretical studies in 1914:

In (1), and are constant. With regard to the specific fluid within identified temperature and pressure, fluid density , cylinder diameter , and fluid viscosity can be regarded as constant and integrated into and . According to the definition of Nusselt number and the relationship between Nusselt number and fluid flow of convective heat transfer, heat transfer coefficient of cylinder can be obtained aswhere is the fluid heat conductivity and is the cylinder diameter.

While the hot wire conducts convective heat transfer with fluid, it is heated by an external electrical source. Then the hot wire heat transfer equation derived by King’s equations can be expressed aswhere is the resistance of hot wire when it is at a certain temperature and is the resistance of hot wire when it is at fluid temperature. The resistances of hot wire and are functions of temperature, and is heating power in the thermal equilibrium state; moreover, () has a linear relationship with the temperature difference and is characterized as the temperature difference between hot wire and fluid. and can be determined by calibration.

Considering that (1) is based on an infinite hot wire, the rate of length and diameter of hot wire is large enough (the rate of length and diameter of typical hot wire is more than 300, which are, resp., 1.25 mm and 4 *μ*m). If cylinder length/diameter rate is taken into account, the index on velocity , which is available for calibration, should be changed. Therefore, for the cylinder with finite-length, (3) can be rewritten as

Notably, and here contain fluid parameters such as fluid density, viscosity, thermal conductivity, and specific heat, and they are assumed to be constant during the measurement. However, these parameters may vary along with the state of temperature, fluid component, and so on. The related investigations belong to temperature and components compensation of thermal probe and are not discussed in this paper.

##### 2.1. Fluid Density Is Not a Constant

As the range of temperature and pressure is rather large, fluid density is bound to fluctuate. The following heat transfer equation of thermal probe reveals that heat transfer of thermal probe is related with local mass flow of fluid , so the fluid density can be expressed in the output of thermal probe and then thermal probe can be used to directly measure the mass flow of fluid:

This equation is called heat transfer equations of thermal mass flow meter. In other words, the local mass flow flowing around thermal probe has a determined nonlinear relationship with temperature of speed probe (which is characterized by ), temperature of fluid (which is characterized by ), and the heating current. The local mass flow of thermal probe can be obtained when the heat transfer probe works in constant power mode or constant temperature differential mode.

##### 2.2. Prandtl Number Is Not a Constant

At high Reynolds number, Prandtl number should not simply be seen as a constant in more general equations of fluid heat transfer because there was a certain degree of error in (1). Researches at home and abroad on heat transfer criteria equation of flowing around cylinder show that a more precise heat transfer expression is needed in a larger measurement range.

In different flow states, different heat transfer criteria equations can be used to analyze coefficients and . Here, criteria equation obtained from Kramers experiment is adopted and can be described as

It is appropriate at and range.

Meanwhile, constants and in (4) should be

Among them, the temperature coefficient of heating resistance, the resistance at the reference temperature, the length of heat cylinder, and the diameter are inherent constants of the probe; the thermal conductivity of fluid, the specific heat capacity , and the dynamic viscosity are related to the temperature. When determining the specific values of these parameters, the film temperature is applied, which is defined as the average values of the temperature of probe and the temperature of fluid; namely,

Leaving aside the small differences of specific coefficients in expressions of and , due to utilizing different heat transfer criteria equations, equations could be used to qualitatively analyze and research calibration constants for thermal mass flow probe.

The heat transfer criteria equations listed above are empirical relationship obtained by considering the overall heat transfer coefficient of the probe. Actually, the distribution of the local heat transfer coefficient has a big difference. The next section will specifically discuss the local heat transfer coefficient of the probe to help to improve the design of the probe in the later chapters.

#### 3. Analysis of Local Heat Transfer Coefficient of Thermal Probe

It can be seen from the whole heat transfer theory described above that the thermal probe measures the velocity of flow and mass flow was actually achieved by measuring the heat transfer characteristics of the thermal probe. So it is necessary to further study the local heat transfer coefficient of thermal probe. The heat transfer coefficient is closely related to the fluid mechanics behavior while fluid flows around the thermal probe.

##### 3.1. Analysis of Flow Patterns While Fluid Flows around the Probe

When the fluid passes along a perpendicular direction to the axis of the cylinder, the fluid has characteristics with boundary layer. The fluid flowing across the probe may cause emerging inverse flow, whirlpool, and vortex beam. When the Reynolds number is very low, the boundary layer separation does not occur and the fluid flowing around cylinder appears as a status of climb stream. With the increase of the Reynolds number, a certain frequency of vortex will be started up.

The angle starting at the windward stagnation can be used to represent the position of the boundary layer separation point (Figure 1). When the Reynolds number , the boundary layer separation point occurs in the range of ; when , the boundary layer is turned into a turbulent flow before separating, and the occurrence of separation is pushed back to the point at .