Building heat system sizing

May 6th, 203 MVK60 Heat and Mass Transfer Project report Building heat system sizing Arnaud BELTOISE Dept. of Energy Sciences, Faculty of Engineering, Lund University, Box 8, 2200 Lund, Sweden

. Introduction The project included in Heat and Mass Transfer course is the perfect occasion to put into practice what has been seen during the lectures. It is with this in mind one is interested here in understanding a central heating system performance in a house and calculating its power in the meantime. Heating efficiency is a really modern concern, in link with the energy price rise and more recently with the protection of environment. As much for public buildings as for private housing, it is no longer possible to design such a system without been sure it matches with the requirements needed. A preliminary study is of course compulsory, in order to take into account all the parameters playing a role in the heating process. In this short project, only a very simplified case will be analyzed, allowing to solve the problem in an analytic way. In reality for much more complicated issues one prefers to use special calculation software, well suited for such complex geometries and variables. An opening will be made on this topic at the end of this document, presenting alternative ways to design such systems. 2. Nomenclature 2.. Geometrical parameters A top wall, surface area of the top wall, m² A bottom, surface area of the bottom wall, m² A wall + window, surface area of one of the two identical vertical walls with both a window, m² A wall, surface area of one of the two last two identical remaining walls, m² L, length of the room, m l, width of the room, m H, height of the room, m L w, width of the window, m H w, height of the window, m V, speed of wind, m.s - e, outer wall thickness (thermal insulation), m e 2, inner wall thickness (concrete), m e glass, window thickness (glass), m

2.2. Convective parameters h forced, convective coefficient on the top wall, W.m -2.K - h out, convective coefficient on the outside of every vertical wall, W.m -2.K - h in, convective coefficient on all the inside walls, W.m -2.K - 2.3. Conductive parameters k, outer wall thermal conductivity (thermal insulation), W.m -.K - k 2, inner wall thermal conductivity (concrete), W.m -.K - k glass, window thermal conductivity (glass), W.m -.K - k air, air thermal conductivity, W.m -.K - 2.4. Radiation parameters σ, Sefan-Boltzmann constant, W.m -2.K -4 ε, window surface emissivity, dimensionless 2.5. Thermal parameters T int, temperature inside the room (order), K T ext, outdoor temperature, K, heat transfer through the top wall, W, heat transfer through the two vertical walls with windows, W, heat transfer through the two remaining walls, W, heat transfer through the floor, W P int, intern heating power (the unknown), W 3. Abstract 3.. Assumptions One considers here a private housing - a house with the simplest geometry possible. The surface control is basically a cube made of flat walls. All the openings are summarized at only two large identical windows, receiving the same constant amount of energy from the sun. The roof is approximate with a horizontal wall, windblown, with a large constant wind speed compared to the vertical sides. As for the bottom wall (the floor), it is taken adiabatic. All vertical walls obey convection laws on their outer side, as well their inner side with a different coefficient. The wall itself is composed of two successive lavers, with different conductive coefficients, k and k 2. Outer temperature is set constant along the time, and uniform around the control surface volume. One fixes the inner temperature the order to follow as constant too and uniform due to convection movements inside the control volume. Inside the volume, one puts the heating system (radiators) and the goal in this exercise is to guess the power value to give to the heat system to reach the order, the inner temperature. One also assumes the case is completely steady (this approximation could be done if the period of time is short compared to the day length.

3.2. Known numerical values Here are all the values given to solve the case: L 25 m l 0 m H 3 m L w 5 m H w m V 0 m.s - T int 20 C 293 K T ext 0 C 273 K h in 9.09 W.m -2.K - h out 6.67 W.m -2.K - σ 5.67.0-8 W.m -2.K 4 ε 0.07 µ air 2.0 * 0-5 kg.m -.s - ρ air.2 kg.m -3 Cp air 0 3 J.kg -.K - k air 0.026 W.m -.K - e 0.05 m k 0.04 W.m -.K - ρ 200 kg.m -3 e 2 0.20 m k 2 0.92 W.m -.K - ρ 2 2300 kg.m -3 e glass 0.006 m k glass.2 W.m -.K - ρ glass 2700 kg.m -3 4. Problem statement Using the fact one studies a steady case. 0, one can write: + + + The goal here to calculate the heat transfers to be able to estimate P int, the unknown, in order to dimension our heating system. 4.. Calculation of One has to remember that the top wall is blown by the wind as it is an exposed area. First one compares the Reynolds number at the edge of the roof with the critical Reynolds number, equal to 5 * 0-5.! "#$% & "#$ 6 * 0 6 plus grand que Re crit. One calculates ' () *+,$#- & "#$! "#$ % 0.833 0 and Prandtl number Pr & "#$ 3 "#$ 4 "#$ 0.769. 0 x 0.833 m laminar flow, 0.833 m x l turbulent flow. Laminar flow : Nu x 0.332 Re x 0.5 Pr 0.333 235.6 x 0.5 As h 4 "#$ 9:; 6.3 '.> W.m -.K -, one integrates on the domain to find the heat transfer.

h?)(+ @AB0CDBEFG 6.3 I '.> ().>.9 W.m -.K - Turbulent flow : Nu x 0.0296 Re x 0.8 Pr 0.333 37.07 x 0.8 As h 4 "#$ 9:; 29.56 '.L W.m -.K -, one integrates on the domain to find the heat transfer. h?)(+ @MNEONADMFG h H' 29.56 I.P QA.P '.P () S20.24 W.m -.K - Average h forced on the top wall: h?)(+,$#- h H',$#-,$#- h?)(+ @AB0CDBEF+ @,$#- F h?)(+ @MNEONADMF85.4 W.m -.K - Let s do the heat balance on the upper wall: h U @V V F+ W L L U @V V + F+ W I I U @V + V + F Hence, + h?)(+ U @V + V + F X h + L W L + I W I + U @V V + F Y h?)(+ 359 Z 4.2. Calculation of [\]]^ One will notice this is the same case as before, same convective coefficient inside, same conductive coefficient in the double layer wall. Only the convective coefficient on the outer side of the wall changes. Be careful, there are two identical walls, so think to multiply the result by two! 2 _h U @V V F+ W L L U @V V + F+ W I I U @V + V + F Hence, + h ` U @V + V + Fa 2 X h + L W L + I W I + U @V V + F Y h ` 733 Z

4.3. Calculation of [\]]^ [bcd[^ These two walls have one window each. One must consequently separate what happens through the window and through the rest of the wall (like previous case, only the area to take into account changes because of the window on the wall). 2 Q + S Let s beginning to calculate. The area to take into account is (L H L w H w ) X h + L W L + I W I + h ` Y @L H Lw HwF @V V + F 733 W Now, let s concentrate on. One must add the radiation term in the heat balance equation, only considering the L w H w area: h @Lw HwF@V V F+ 4 i @Lw HwF@V + V + F+ i W I @Lw HwF@V + V + F+ h ` @Lw HwF@V + V + F +j k @Lw HwFlV m V m + n I Hence, X h + o W o + h ` Y @Lw HwF @V V + F +j k @Lw HwFlV m V m + n 722 Z As Therefore, 2 Q + S 490 Z 4.4. Calculation of pq As it was said in the early hypothesis in 3.., one obviously guess the value of as the floor is considered as adiabatic, the heat flux is zero: 0 Z 4.5. Calculation of bc Using the heat balance equation, one must find the final value to provide to the system in order it can be able to regulate the temperature inside the building:

+ + + So that, 359+ 733+490+ 08802 Z 5. Literature survey The design of a heating system must be performed accurately. Indeed, the choice of a device too much powerful significantly increases the installation cost without bringing savings and consumption and the risk of short operation cycles is higher, which is bad for the device. The choice of a low power device also induced very long periods in action. It is therefore imperative to perform a first accurate calculation of heat losses first. Never too little, nor too much! After having gathered several sources, it seems one can differentiate at least 5 methods to dimension one s heat system: Method : Manual J Load Calculation (see point 4. of the study) This is the proper and scientific method for calculating furnace or boiler size taught to technicians and recommended for use by professionals.it consists of taking information about your home's construction materials, insulation levels, number of windows, sizes of rooms, etc, and making a calculation based on those factors to determine the appropriate heating and cooling requirements needed. Method 2: Use an equipment sizing estimator This is an online tool that will give you a rough estimation. It won't be exact, but when used in conjunction with other information it can provide a fairly close approximation. Method 3: Comparison between your home to similar homes in your area If one of the neighbor house is as big as yours, why not to ask his heating system size in order you can set something working well at your place. Method 4: Ask a contractor Most professionals give free in-home estimates for installing new heating systems, during which they will recommend a unit size. A contractor familiar with the homes in your neighborhood will likely be able to give you an idea over the phone of what size you might need. Method 5: Replace an existing unit by a similar one If the unit you have now is the correct size for your home, replace it with the same size? Look at the name plate on the furnace, usually located somewhere inside the unit. Remember, furnaces are rated by input power but you will want to determine what the actual output power is when selecting the right size replacement unit.

6. Conclusions After this short study, one manages to find easily the value of 8800 W. It corresponds to nearly the power provided by 9 radiators of kw each, which seems plausible to heat a 750 m 3 room, by winter weather (only 0 C outside compared to the 20 C inside the room for a good comfort of users). This way, one can assume the assumptions that one made were not that bad as one obtain something realistic at least. The other big approximation concerns the numerical values, which can depend on the sources, the building process, material aging but also some effects due the quality of materials themselves. In this problem come a lot of errors from everywhere, and one should be able to detect them. In the industrial world, experts use specific softwares, much more suited for this kind of heat transfer problem. In our study, we have made the assumption to be in a steady case, which is clearly wrong if we consider time length about one day or more. The sun radiations will change during the day, the floor temperature also, and the user himself can decide to lower the room temperature to 8 C or will activate the air ventilation, way enough when the office is empty for instance Moreover, lots of alternatives exist, even if they don t have the same scientific and physical aspects. 7. References Chadia Zayane, January th 20, «Identifying a model of thermal behavior building from its load curve, Mines ParisTech (in French) http://cg.ensmp.fr/bibliotheque/public/zayane_these_0289.pdf Aurélie KAEMMERLEN, October 29 th 2009, Heat transfer through the thermal insulation of the building, Henri-Poincaré University, Nancy http://www.scd.uhp-nancy.fr/docnum/scd_t_2009_002_kaemmerlen.pdf Jan L.M. Hensen, June 7 th 99, On the thermal interaction of the building structure and heating and ventilation system, Technische Universiteit Eindhoven http://www.bwk.tue.nl/bps/hensen/publications/9_dissertation.pdf Stefan Thor Smith, June 2009, Modelling thermal loads for a non-domestic building stock. Associating a priori probability with building form and construction - using building control laws and regulations, University of Nottingham http://etheses.nottingham.ac.uk/895//thesis.pdf http://www.climatechange.ie/pdf/radiatorssizeposition.pdf http://nesa.uni-siegen.de/wwwextern/idea/keytopic/3.htm