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ASSUMPTIONS ANALYSIS: THERMOSIPHON MATHEMATICAL MODEL INTRODUCTION PARAMETERS OPERATING CONDITIONS INITIAL STATUS TANK INPUT TEMPERATURE TANK ANALYSIS PANEL INPUT FLOW & TEMPERATURE COLLECTOR HEAT LOSS COLLECTOR PANEL ANALYSIS THERMOSIPHON PRESSURE SYSTEM FLOW RATE MOODY FRICTION FACTOR MACRO COMPUTATIONAL FLOW Introduction The following document contains an in depth analysis of the assumptions within the Mathematical Model for the thermosiphon solar hot water heater analysis. Many of the assumptions are relatively minor, while others ought to be accounted for to obtain more accurate results or to increase the programs versatility. An overview of the assumptions will provide the user with an understanding of system limitations, and possible computational oversights. If so desired, the Mathematical Model can be improved by accounting for some of these assumptions through further research and model editing. The assumptions are broken up into their respective sub-analyses. Realize that each assumption may affect further calculations throughout the program. Parameters 1. The system is a thermosiphon system. Note that the system water input must be positioned near the exit of the tank siphon output. 2. All pipes are circular. Operating Conditions 1. Some operating conditions may require the assumption that a set of values are constant for a given time frame (i.e. the system input water temperature is constant) Initial Status 1. Assume that the inputted tank initial temperatures are correct. Tank Input Temperature 1. assumes pipe inner diameter of system input to panel input = pipe inner diameter of tank siphon output to system input for purposes of heat loss in pipe from tank siphon output to panel input 2. assumes water temp between Tank Siphon Output and System Input = temp between System Input and Panel Input for purposes of Heat loss in pipe from Tank Siphon Out to Panel Input 3. assumes pipe wall thickness of system input to panel input = pipe wall thickness of tank siphon output to system input for purposes of heat loss in pipe from tank siphon output to panel input 4. assumes pipe insulation thickness diameter of system input to panel input = pipe insulation thickness of tank siphon output to system input for purposes of heat loss in pipe from tank siphon output to panel input 5. assumes pipe thermal conductivity of system input to panel input = pipe thermal conductivity of tank siphon output to system input for purposes of heat loss in pipe from tank siphon output to panel input 6. assumes insulation thermal conductivity of system input to panel input = insulation thermal conductivity of tank siphon output to system input for purposes of heat loss in pipe from tank siphon output to panel input 7. If the demand flow rate is greater than twice the flow rate due to the thermosiphon effect, then the check valve will shut and no flow will pass through the tank siphon output. Otherwise, the panel input flow is equal to the sum of the thermosiphon effect plus half of the demand flow rate. We assume that the system input flow potential would be equal on both sides of the flanged-tee. 8. Assume the panel input flow temperature is the weighted average of the tank siphon output temperature and the system input temperature based on their respective flow rates. 9. If flow is laminar, the heat transfer equations assume fully developed flow (Incropera, 487: eq. 8.55). If flow is turbulent, the heat transfer equations assume fully developed flow, smooth pipes, and the temperature of the water in the pipe is greater than the ambient temperature (Incropera, 491: eq. 8.60). Tank Analysis 1. Assume the tank is a cylinder with a horizontal axis of symmetry. 2. The tank is modeled as six partitions of uniform temperature. 3. The flow is determined as a function of the ratio of output flow rates and partition geometries (values a, b, and c are empirical). 4. The tank input is positioned in the top sector (of the six sectors) of the tank. 5. The tank output is positioned in the top sector opposite the tank input. 6. The tank siphon output is positioned under the tank output in the bottom sector. 7. The volumetric thermal expansion coefficient of air (K^(-1)) assumes air is an ideal gas (Incropera, 538: eq. 9.9). 8. The outer tank temperature is isothermal, flow over the cylinder remains laminar, Rayleigh number is less than 1012, free convection occurs on the tank (Incropera, 554: eq. 9.34) as well as forced convection. 9. Characteristic length for the circular tank face is the average height. 10. The tank circular faces can be represented as vertical plates (Incropera, 546: eq. 9.26). Panel Input Flow & Temperature 1. Assumes pipe inner diameter of system input to panel input = pipe inner diameter of tank siphon output to system input for purposes of heat loss in pipe from tank siphon output to panel input. 2. Assumes water temp between Tank Siphon Output and System Input = temp between System Input and Panel Input for purposes of Heat loss in pipe from Tank Siphon Out to Panel Input 3. assumes pipe wall thickness of system input to panel input = pipe wall thickness of tank siphon output to system input for purposes of heat loss in pipe from tank siphon output to panel input 4. assumes pipe insulation thickness diameter of system input to panel input = pipe insulation thickness of tank siphon output to system input for purposes of heat loss in pipe from tank siphon output to panel input 5. assumes pipe thermal conductivity of system input to panel input = pipe thermal conductivity of tank siphon output to system input for purposes of heat loss in pipe from tank siphon output to panel input 6. assumes insulation thermal conductivity of system input to panel input = insulation thermal conductivity of tank siphon output to system input for purposes of heat loss in pipe from tank siphon output to panel input 7. If the demand flow rate is greater than twice the flow rate due to the thermosiphon effect, then the check valve will shut and no flow will pass through the tank siphon output. Otherwise, the panel input flow is equal to the sum of the thermosiphon effect plus half of the demand flow rate. We assume that the system input flow potential would be equal on both sides of the flanged-tee. 8. Assume the panel input flow temperature is the weighted average of the tank siphon output temperature and the system input temperature based on their respective flow rates. 9. If flow is laminar, the heat transfer equations assume fully developed flow (Incropera, 487: eq. 8.55). If flow is turbulent, the heat transfer equations assume fully developed flow, smooth pipes, and the temperature of the water in the pipe is greater than the ambient temperature (Incropera, 491: eq. 8.60). Collector Heat Loss 1. ? Collector Panel Analysis 1. Negligible heat loss through collector walls 2. Panel heat transfer is reduced to the symmetrical and repeated unit of half of the distance between the array pipes to the water in the array pipe. 3. The heat transfer through the horizontal array pipes is taken into account by assuming the same correlations as the vertical array pipes, but heat transfer is only accounted for in half of the horizontal array (see effective pipe length for model). Otherwise a portion of the absorber plate would be accounted for twice. 4. Water properties are evaluated at the average water temperature in the array pipe. 5. Assume the validity of Incropera’s Figure 8.9 (Incropera, 490) for constant surface temperature for laminar flow for thermal and combined entry lengths. 6. Assume the temperatures along the length of the each of the array pipe elements are constant. 7. Assume that my equations representing Figure 8.9 (Incropera, 490) are accurate (see excel file: entry length approx2???????). 8. Assume that the heat transfer correlation eq. 8.43 (Incropera, 482) is valid for short intervals of pipe. 9. Assume the validity of the dimensionless parameter x+: for turbulent Eq. 5.12, Eqn. 5.84, for laminar eq. 5.29, pg 5.9 para 2, eq. 5.31 (Rohsenow) Thermosiphon Pressure 1. Assumes no heat loss in the pipe from the panel output to the tank input. 2. Assumes reverse flow will be halted by a check valve System Flow Rate 1. Assumes the minor loss coefficients are accurate. 2. Assumes the Darcy-Weisbach equations are accurate. (A prior System Flow Rate worksheet that proved to be less accurate, but was intended to be more universal uses the following equations and their respective assumptions. For turbulent flow Eq. 5.12, Eqn. 5.84, for laminar flow eq. 5.29, pg 5.9 para 2, eq. 5.31, assumes turbulent when Reynolds Number is above 5000 & linear interpolation between 2300 and 5000 [Rohsenow]. These equations required a correction factor of 1.35 and a Moody Friction Factor Safety Factor of 2) Moody Friction Factor 1. Assumes the Moody Friction Factor worksheet provides a valid interpretation of the Moody Diagram (Incropera, 471: Fig. 8.3) Macro Computational Flow 1. Assumes small enough time steps to validate the quasi-equilibrium status. 2. Assumes small enough convergence range. 3. Assumes that the use of some of the previous time step’s variables are acceptable for the current time step (small enough time steps or small enough changes in the variables’ values).