modified Low Third Derivative (LTDm) methods. Among the reconstruction algorithms available today,. SBFM and LTDm are among the most promising.
International Journal of Ventilation ISSN 1473-3315 Volume 4 No 4 ________________________________________________________________________________________________________________________
Computed Tomography for Indoor Applications M. Cehlin1 and M. Sandberg2 1
Divison of Energy and Mechanics, Department of Technology and Built Environment, University of Gävle, Sweden. 2 Division of Indoor Environment, Institute of Technology and Built Environment, University of Gävle, KTH Research School, Sweden.
Abstract This paper deals with tomographic techniques for two-dimensional spatially resolved concentration measurements indoors. This represents a significant advance over the traditional point measuring method for mapping tracer gas and pollutants. Methods for recording of data are stressed as well as different types of tomographic reconstruction algorithms such as the Smooth Basis Function Minimization (SBFM) and the modified Low Third Derivative (LTDm) methods. Among the reconstruction algorithms available today, SBFM and LTDm are among the most promising. These algorithms show potential for reconstruction of gas concentration in rooms, since they are regularized to converge towards smooth concentration distributions. Using the LTD method and ‘snapshot’ configuration enables the examination and real-time monitoring of transient flows. Key words: computed tomography, tomographic reconstruction algorithms, pollution distribution monitoring, laser scanning. 1. Introduction Today most sampling techniques of physical variables of importance for both the indoor and outdoor environment are based on point sampling techniques. These point-samplers are usually integrated over a long period and placed at fixed locations in a test-region. Thus, the result gives only an approximate estimation about e.g. the gas concentration distribution because the spatial resolution is limited to the discrete location of the sampling devices. These point-measuring techniques are very time demanding and information about short-term fluctuation is lost, because concentrations are integrated over a long time. The traditional techniques of today are also intrusive, and might disturb the airflow and the concentration distribution in the test-region. These problems and limitations, however, can be reduced by using a whole field measuring technique based on computed tomography and optical sensing.
Tomography is best known for its use in medical Xray absorption imaging where it is an established diagnostic technique (Hounsfield 1973, Cormack 1973, Brooks et al 1976). However, the mathematical description of the process in absorption/scattering tomography, applies both in medical applications and in indoor testing. While the concepts are similar, some of the problems involved with reconstructing pollutant concentrations in air over large areas are different from those that arise in medical imaging. Principally, medical imaging is aimed at measuring the location of objects in space, whereas gas tomography is aimed at measuring both the location and the magnitude. Pollutant concentrations fluctuate in both time and space, which makes sampling and reconstruction in air much harder. Another important distinction is that the images should normally convey quantitative information rather than relative changes, for example absolute values for the gas concentration distribution.
Radon (1917) provided a basis for the description of scalar tomographic projections. Since then, images produced by computer tomography have been formed by computer processing of information from many individual path integrated measurements obtained non-destructively through the object or volume.
Normally, images produced by computer tomography require far fewer measurements than would be required using traditional point measuring techniques to obtain the same level of detail. Therefore computed tomography, in combination with optical sensing, provides an excellent tool for the investigation of pollutant dispersion indoors.
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appropriated to the specific species. They have been widely used for direct concentration measurements of airborne gases outdoors where the ray-integrated measurement is converted to a path average molar concentration by dividing the measurement by the known path length. Tunable lasers permit matching to an absorption/transmission window. However, they are limited to the sensing of one or a few gases. The scattering of light is a simple and more costeffective technique than the absorption of light. Using smoke as a tracer enables the use of small laser modules (module diameter under 10 mm) and detectors operating at visible wavelengths, which makes them suitable for measurements over small regions. This optical approach has the advantage of being relatively inexpensive (under US $10 for a laser module).
The rapid rise of computer fluid dynamics models for a growing number of applications must be accompanied by independent validation of each model against experimental results. In this context, tomography offers a suitable complement to point measurements for model verification. Tomographic techniques have an excellent spatial range and can be used for imaging in two and three dimensions. This paper deals with tomographic techniques for two-dimensional spatially resolved concentration measurements indoors. Methods for the recording of data are stressed as well as different suitable tomographic reconstruction algorithms. 2. Fundamental Concept of Computed Tomography
The degree of attenuation of light is dependent on the density profile and the size of the object being penetrated. For parallel beams of light sent through a medium in cross-section, the ratio of the light intensity traversing the object, I, to that incident on the medium; I0, is given by the Lambert-Beer law:
The basic components of any tomographic system are the sensing system and the tomographic reconstruction algorithm. The basis of any measurement is to exploit differences in the properties of the process being examined, e.g. chemical or particle concentrations. The performance of reconstructions is critically dependent on the correct choice and design of sensors, and an understanding of their limitations.
I − σ ⋅ds =e ∫ I0
(1)
where
In chemical and particle concentration applications, tomographic technology involves acquisition of measurement signals from detectors, located mostly around the boundaries of an investigated region, revealing information about the concentration distribution within the region. Media, such as aerosol particles or chemical gases, illuminated by a beam of light, scatter and absorb some of that light, thereby diminishing the intensity of the beam along its axis. This process is often called attenuation or extinction and the total attenuation of light intensity is the sum of the attenuation due to scattering and absorption.
σ =σs +σa
(2)
σ is the total extinction coefficient, σs extinction coefficient due to scattering, σa extinction coefficient due to absorption and s is the path length of the light beam through the medium. Simply stated, the law claims that when a sample is placed in the beam, there is a direct and linear relationship between the amount (concentration) of its constituent(s) and the amount of energy it absorbs and scatters. For gases, the Beer-Lambert law also provides a relationship between absorption attenuation coefficient and gas concentration, c:
Laser sensing systems, such as tunable lasers and Fourier transform infrared (FTIR) instruments have been used extensively outdoors as well in industries to measure a variety of gases (Murray 1977, Herget et al 1980, Levine et al. 1985). They are capable of detecting a variety of contaminants at concentrations as low as a few parts per million over path lengths of a metre (Levine et al. 1989).
σa= cε
(3)
where ε is absorptivity. The scattering attenuation coefficient for monodispersive particles (Mie scattering) can be expressed as:
In FTIR, spectral analysis is used to measure several chemical species along the path of a light beam, by selecting the different absorption spectra which are
σs =
π 4
NQsπ d 2
(4)
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International Journal of Ventilation ISSN 1473-3315 Volume 4 No 4 ________________________________________________________________________________________________________________________
Figure 1. Light extinction measurements.
Optical path-integrated measurements provide only one-dimensional attenuation values, with no information about the distribution of σ along the beam path (the LIDAR technique is an exception, see Woods et al 1978). Therefore, to pinpoint the spatial distribution of σ, a network of overlapping path measurements must be obtained. Then these 1dimensional data can be converted into 2dimensional information with a tomographic reconstruction method. Computed tomography, based on optical sensing, was first applied for outdoor air pollution measurement on an urban scale (Byer et al 1979; Wolfe et al 1979, 1982). Optical remote sensing in combination with computed tomography for monitoring of air contaminant concentrations indoors was proposed initially by Todd et al (1990) and thereafter has been evaluated theoretically/numerically (Bhattacharyya et al 1997; Todd et al 1994; Samanta et al 2000) and experimentally (Yost et al. 1994; Drescher et al. 1996; Samanta et al 1996; Price 1999; Price et al. 2000; Fischer 2000; Todd et al. 2001a; Todd et al. 2001b; Cehlin et al 2002, Cehlin 2005).
and for polydispersive σs =
π
∞
∑ N i Q s iπ d i 2 4
(5)
i
where Qsi is the scattering coefficient for particles with size di and Ni is the number of particles with size di. For non-absorbing aerosol particles the intensity loss at a certain frequency due to Mie scattering is directly proportional to the number of particles per unit volume. When considering scattering tomography measurements one has to use lenses to ensure that only the attenuated parallel light reach the detectors, see Figure 1. Failure to include lenses at the detectors allows forward-scattered light from particles to reach the detectors and, for that case, the Lambert-Beer law does not hold. The discussion of extinction (Beer-Lambert law) holds for single scattering. If the particle concentration is sufficiently dense, the scattering behaviour changes relative to that for an isolated sphere and the particles may no longer scatter independently. This effect is known as multiple scattering and can be avoided only by diluting the particles. The light scattered by each particle will illuminate other particles in a direction that is not parallel to the incident light and a portion of the light can be scattered back into the original beam. This process allows multiple scattered light to reach the detector. Extinction measurements performed by Hodkinson (1962) using 1.8 µm polystyrene spheres showed that the results obeyed the Lambert-Beer law for the transmittance range 0.37≤T≤1 per centimetre. His experiment was only conducted down to T=0.37. However, the validity of the law may be extended even down to T
