MM$313$ NUMERICAL ANALYSIS TERM PROJECT The thermal analysis of an apartment unit will be conducted using numerical methods. A top view of the room with dimensions in cm is provided in Figure $1.$ As shown in the figure, one of the walls of the room is parabolic. The parabolic wall of the room and the wall at y $=0$ cm within the range $0$ cm $$ x $500$ cm are the exterior walls. The wall at x $=0$ cm within the range $0$ cm $$ y $484$ cm faces the building$$s interior void, and its temperature is assumed to be constant at $15$C$.$ The ceiling of the room marks the start of the roof and is assumed to have a constant temperature of $8$C$.$ The floor of the room contains an underfloor heating system, which consists of several pipes. Hot water flowing through there pipes heats the room. Additionally, the occupants of the room, various electronic devices, and lighting fixtures emit heat. Based on this information, a $3$D thermal analysis of the room will be carried out. Furthermore, there is an electrical outlet on the wall at y $=484$ cm within the range $0$ cm $$ x $522$ cm$,$ and it has been observed that the outlet is experiencing a problem. The issue is suspected to be caused by overheating of the electrical wires, as heating has been observed in certain sections of the wall. Therefore, the region of maximum wall temperature will be calculated, and the issue will be examined. The dimensions of this wall are schematically presented in Figure $2.$ Figure $1.$ Figure $2.$ a$)$ The underfloor heating system consists of $10$ pipes, each with a diameter of $4$ cm$.$ One of these pipes is schematically shown in Figure $3.$ It is desired fort he flow in these pipes to remain laminar, and the maximum possible heat transfer to the room through the pipes is required. If the water enters the pipes at $50$C and exits at $45$C$,$ calculate the heat transfer to the room from the pipes by root$-$finding methods. $($water density, $=1000$ kg$/$m$3,$ specific heat capacity of water, Cp $=4\mathrm{.}18$ kJ$/$kgK$,$ dynamic viscosity of water, $=0\mathrm{.}000466$ kg$/$ms$.)$ Figure $3.$ b$)$ Given that the heat generation from people, lighting, and electronic devices in the room is approximately $6\mathrm{.}62$ W$/$m$3,$ calculate the total heat generation in watts $($W$).$ c$)$ It is assumed that the variation of temperature along the y$-$axis for the wall shown in Figure $2$ can be determined using the following formula: $()=22(122)+212()+1+22$ Here, egen is the heat generation per unit volume, with a value of approximately $75$ W$/$m$3.$ L is the thickness of the wall, with a value of $20$ cm$.$ kw is the thermal conductivity of the wall, with a value of $0\mathrm{.}5$ W$/$mK$.$ Additionally, T$2$ is the temperature of the wall$$s surface facing the other room, which is $18$C$,$ and T$1$ is the temperature of the wall$$s surface facing the room under investigation, which is $22$C$.$ Determine the region in the wall where the maximum temperature occurs. d$)$ Determine the boundary conditions for the room where the $3$D thermal analysis will be conducted. e$)$ Divide the room into cells with lengths not exceeding $45$ cm along the x and y axes and $1$ cm along the z axis. Calculate the temperatures at the interior points using the differential equation below. $($External air convection coefficient, h $=0\mathrm{.}5$ W$/$m$2$K; thermal conductivity of the air inside the room, k $=0\mathrm{.}1$ W$/$mK; External air temperature, T$=15$C$.)22+22+22+=0$ Please solve the above questions using only numerical methods, utilizing the attached information and the formula set below. Use four decimal places in your calculations. Additionally, indicate the nodes to which the temperatures belong. Useful Formulas: $==24=2=,=,=||,=|=()=()0$ Re is the Reynolds number, $$ is the average velocity, D is the diameter of each pipe inside the channels of the underfloor heating system, $$ is the volumetric flow rate of water inside a single channel of the underfloor heating system, $$ is the heat transfer from a single channel of underfloor heating system to room, qwall is the heat flux through wall. Reminder for Boundary Conditions: $1-$ T$($x$,$y$,$$)$ or T$($x$,$$,$z$)$ or T$($$,$y$,$z$)=$ $($ and are any numbers$)2-$ Tx$($$,$y$,$z$)$ or Ty$($y$,$$,$z$)$ or Tz$($x$,$y$,$$)=$ $($ and are any numbers$)3-$ Tx$($$,$y$,$z$)$ or Ty$($y$,$$,$z$)$ or Tz$($x$,$y$,$$)=$ h$($$)($Twall $$ Tfluid$)/$k $($ is any number, h is the convective heat coefficient, $$ is the length of any grid and k is the thermal conductivity.