NUMERICAL CALCULATION OF MEAN RADIANT TEMPERATURE AND TEMPERATURE ASYMMETRY FOR COMPLEX ROOM GEOMETRY

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1 Proceedings: Indoor Air 2 NUMERICAL CALCULATION OF MEAN RADIANT TEMPERATURE AND TEMPERATURE ASYMMETRY FOR COMPLEX ROOM GEOMETRY M Prek * University of Ljubljana, Faculty of Mechanical Engineering, Askerceva 6, 1 Ljubljana, ABSTRACT Radiative heat losses represent a substantial part of the total heat balance of the human body in a closed space. The thermal balance is strongly affected by surface temperatures and by the angle factor between a body and a wall surface. Since a room s internal surfaces are composed of various parts, the calculation of view factors becomes more complex. The proposed algorithm is based on the computation of view factors, thus allowing for the complex impact of these factors on thermal radiative heat exchange. The matrix based approach makes it possible to determine the influence of various parameters on mean radiant temperature, as well as on temperature asymmetry. This approach also enables the consideration of other parameters (air temperature gradient, air velocity...) whose impact on thermal sensation is expressed as the PMV value. INDEX TERMS Thermal comfort, Thermal environment, Radiant temperature. INTRODUCTION Thermal comfort is ensured by maintaining the declared values of the factors determining thermal comfort constant, while eliminating any disturbing influences causing local thermal discomfort. Regarding personal comfort, the human occupant has a subjective preference for a warmer building structure and cooler indoor air, indicating that radiative heat exchange should be favoured. The human body s thermal balance is strongly affected by radiative heat exchanges with surrounding surfaces, which are a function of the surface temperatures and angle at which the human body senses them. This means that in order to establish local thermal comfort or discomfort, view factors of people with respect to the envelope surfaces must first be assessed. Several methods for determining the mean radiant temperature has been proposed. The most comprehensive (ISO 7726:1998), involves a graphical or numerical procedure for calculating the view factors between the human body and a simple plane surface. In practice, this method becomes very complex, taking into account the influence of body posture and orientation with respect to composite room surfaces. Algorithms for computing these view factors for simple plane surfaces have already been determined (Nucara, Pietrafesa and Rizzo, 1999). However, in practical cases, the room s internal surfaces are composed of various parts, each possessing a specific thermal situation. In this case, the calculation of view factors becomes much more complex and requires a careful management of the thermal and geometric properties of the surfaces. A comprehensive algorithm is introduced here which allows for the computation of angle factors of human body with respect to the given complex situation. * Author matjaz.prek@fs.uni-lj.si 689

2 Proceedings: Indoor Air 2 ANGLE FACTOR BETWEEN HUMAN BODY AND ROOM SURFACE As presented in the work (Fanger, 197) and determined with standard (ISO 7726:1998), the angle factor between the human body and a rectangular surface F P A can be computed as: FP A 1 π x a = y c z b = y c = x z = = y y f P x z y y x z d d y y (1) where a and b are the width and height of the isothermal surface A, and f P is the projected area factor. In order to calculate the view factor, a method which avoids the direct calculation of the projected area factor has been established. The view factors are determined as solutions of equations and presented in the form of sets of graphs (Fanger, 197) as a function of dimensionless geometrical parameters a/c and b/c. Another step toward simplification was made (Rizzo, Cannistraro and Franzitta, 1992), where the view factors are given in the form of exponential equations dependent on geometrical parameters. This simplified method for calculating the view factors enables good agreement between the calculated and actual values determined from solutions of equations, as well as experimental data. EFFECTIVE RADIATION AREA OF HUMAN BODY The effective radiation area of a human body is defined as the surface area of a human body which directly contributes to the radiation exchange between the body and its surroundings. By calculating the angle factor F A P with projected area of human body A P on a plane perpendicular to the direction on the differential surface element da on the sphere, the effective radiation area can be derived from surface integration of projected area with spherical coordinate system as follows: α=π β= π 2 Aeff = AP cos β dβ dα (2) π α= β= To calculate the effective radiation area of a human body in Equation (2), the projected area A P of a human body irradiated by the parallel rays must be calculated. This projected area is equal to the surface area of the human body where parallel rays reach directly and projects on a plane perpendicular to the parallel rays (Ozeki, Konishi and Narita, ). ENERGY BALANCE ON THE HUMAN BODY A steady-state energy balance on the human body may be developed using environmental factors surrounding the body. We can set the rate of heat produced in the body equal to the total amount of heat rejected by the body to its environment. The clothing adds an additional thermal resistance that is in series with the convective and radiative resistances from the surface of clothing, as is shown in Figure1. 69

3 Proceedings: Indoor Air 2 surface of skin tsk R cl /A sk surface of clothing t cl 1/h C A cl t Q C Q sen 1/h R A cl t r Q R Figure 1. Thermal network of a clothed body. The clothing has a thermal resistance per unit inner surface (skin) area, R cl /A sk, that is in series with the outer convective and radiative resistances. The total sensible heat transfer Q sen can be determined by adding the convective and radiative components into single heat transfer coefficient h: Qsen ( t t ) Ask sk o = (3) A R sk cl + hacl where t o is the operative temperature. Placing the outer thermal resistance (1/h C A cl for convective and 1/h R A cl for radiative heat transfer) in series with the clothing resistance allows us to eliminate the clothing surface temperature. THE INFLUENCE OF COMPOSITE ROOM SURFACES The view factor for an arbitrary human position in the room can be determined by splitting the surface into four sub-surfaces and applying the additive property (Prek and Novak, 1). The division of the surface into four sub-surfaces and an application of the additive property requires the determination of the sign for certain sub-surfaces. According to the co-ordinate system, one can deduce a general algorithm: FP A = FP i sign( ai bi i= 1 When the view factors for individual sub-surfaces are determined, the view factor for the composed wall can be expressed as: F P wall orientation = FP total surface F orientation P sub surface (5) sub surfaces orientation and the mean radiant temperature T mrt for a given orientation in the room is then: ) () 691

4 Proceedings: Indoor Air 2 N Tmrt orientation = FP i orientation Ti orientation i= 1 where T i is the absolute temperature of the i-th internal surface and F P i is the view factor between the person (generic point in room) and i-th surface (or sub-surface). CASE STUDY The potential use of the introduced algorithm can easily be shown by means of an application aimed at the calculation of the mean radiant temperature for a wall which is composed of different elements. In Figure 2, an example of a wall composed of a door, window and heating panel is shown. For the sake of clarity, the influence of only one wall on mean radiant temperature is analysed, all other room walls are isothermal; geometry and temperatures are given in Table 1. The parameters are defined by the body posture and, thus, are chosen dependent on the geometry of the room and its elements. (6) Figure 2. Assumed composition of the room in the case study. Table 1. Parameters of the room surfaces Element Length (m) Height (m) Surface temperature ( o C) Outer wall 6,5 2,5 18 Outer door 1, 2,2 1 Window 1 2,5 1 1 Heating panel 1 2,2,5 5 Window 2 1,2 1,2 1 Heating panel 2 1,,8 5 Inner walls, floor, ceiling In Figure 3 two different positions of heating panel within the wall is presented. In Figure 3a, the heating panel is mounted in front of the wall and fully exposed; in Figure 3b, the same heating panel is mounted in niche and under the window still. The influence of this condition (mainly caused by practical or architectural demands) on mean radiant temperature (Figure ) could be easily determined with calculation of view factors. 692

5 Proceedings: Indoor Air 2 a b Figure 3. Geometrical conditions for heating panel within the wall. a b Temperature ( o C) Width (dm) 6 1 Length (dm) Width (dm) Length (dm) Figure : Calculated radiant temperature for observed wall with different conditions for heating panel within the wall. 6 5 Length (dm) 3 PMV Width (dm) Figure 5. Calculated PMV value for plane,6 m above the floor for room geometry, shown in Figure 2 with heating panels, mounted as shown in Figure 3b. 693

6 Proceedings: Indoor Air 2 In Figure 5, the calculated PMV value is presented. In these example, the height of the generic point is used as a variable. It was set at,6 m, which corresponds to seated posture. Other variables for determining the PMV value are: air temperature 21 o C, air velocity,1 m/s, relative humidity %, clothing 1, clo, metabolic rate 1,2 met, external work met. The distribution of PMV value within the room is clearly evident. The shape depends on the given surface temperatures as well as on the calculated view factors. Since the heating panels are mounted in miche, the influence of cold outer wall caused the extension of low PMV value in the interior of the room. Since the algorithm possesses a general structure, it is easily applied to different surface compositions, such as the position of the element within the wall, the arbitrary geometry of the element and the surface temperature of the element. The last possibility is especially interesting because it enables the simulation of the mean radiant temperature under different operating conditions, such as different outdoor temperatures, which create different surface temperatures of the outer wall elements and heating panels. CONCLUSIONS The proposed algorithm is based on the computation of angle factors using the additive property. This algorithm enables the calculation of mean radiant temperature for composite room surfaces, even allowing for the complex impact of body posture. Consideration of other parameters (air temperature gradient, air velocity...) whose impact on thermal sensation is expressed as the PMV value, is enabled. In order to achieve optimal thermal conditions, the thermal balance between heat sources and sinks must be established. This method enables the determination of the influence of heating source characteristics (temperature, geometrical parameters) on mean radiant temperature as a part of overall environmental conditions and the insight in the interactive influence between heating system and building structure (determined with geometrical, architectural and physical properties), thus achieving the maximal possible thermal comfort area. This method could, ideally, be incorporated into the architectural planning phase of buildings in order to determine the best relationship between the building structure and its heating system. REFERENCES Fanger PO Thermal Comfort. Copenhagen: Danish Technical Press. ISO 7726:1998. Thermal environments - Instruments and methods for measuring physical quantities. ISO 773:199. Moderate thermal environments - Determination of the PMV and PPD indices and specification of the conditions for thermal comfort. Nucara A, Pietrafesa M, Rizzo G, et al Human body view factors for composite plane surfaces, Proceedings of the 8 th International Conference on Indoor Air Quality and Climate Indoor Air 99, Vol 1, pp Edinburgh: Indoor Air 99. Ozeki Y, Konishi M, Narita C, et al.. Numerical calculation of angle factors between human body and rectangular planes. In Air Distribution in Rooms, Proceedings of the 7 th International Conference on Air Distribution in Rooms, Awbi HB, ed. Amsterdam: Elsevier, Vol 1, pp Reading: ROOMVENT. Prek M, and Novak P. 1. Simplified method for evaluating thermal radiant environment, Thermal protection of man under hot and hazardous conditions, pp Ljubljana: 2 nd International IIR Conference Rizzo G, Cannistraro G, Franzitta G, et al Algorithms for the calculation of the view factors between human body and rectangular surfaces in parallelepiped environments. Energy and Buildings. Vol. 19 (1), pp