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).