HVAC SYSTEMS - HFT Stuttgart

Page - 21 -
CHAPTER 02
Since neither the load Boolean switch δload nor the additional heating Qh & are
continuous functions of the time, it is not possible to solve the differential
equation analytically. Therefore, a simple forward differences method is used,
which enables the calculation of the storage temperature for the actual time
step n from the values of the previous time step (n-1) and equation (2.2.1-16)
can be written as:
( t ) T ( t )
⎛δ
⎜ col ⋅ m&
col
Δt
+ ⎜−
δloadm&
ms
c s ⎜
⎝
− U eff A ⋅
( )
( )
( ) ⎟ ⎟⎟⎟
⋅ c
⎞
col ⋅ T col out−T
s + Q&
,
h
⋅ c load ⋅ T − s Tload
ret
T −T
T s n = s n −1
load
,
s
amb
⎠
(2.2.1-17)
For the storage tank with stratification (Fig. 2-3) the equation becomes more
complex since for each layer an energy balance is required and additional
effects like convective heat transfer and heat conduction between the layers
needs to be considered. Since the exact mathematical model development is
difficult for convective heat flows, the convection and conduction between the
layers are approximated by a common effective heat conductivity λeff. According
to Eicker (2001) for good constructed hot water storage tanks without internal
heat exchangers this effective heat conductivity is in the region of the heat
conductivity of water and can be approximated by a value of λeff = 0.644 W m -1
K -1 . For heat storages with internal heat exchangers effective heat conductivity
is in the region of λeff = 1.0 - 1.5 W m -1 . With this the heat flow between layer i-1
and 1 or between i and i-1 is calculated according to the Fourier equation and
the heat flow between one layer and the two adjacent layers caused by heat
convection and heat conduction can be written as:
Q&
cc , i
= Q&
cc , i −1→i
λeff
= −As
z
λeff
= As
z
−Q&
⎛ eff
( T −T
) − ⎜−
A ( T −T
)
s , i
( T − T + T )
s , i + 1
cc , i →i
+ 1
s , i −1
⎝
2 s , i s , i −1
s
λ
z
s , i + 1
s , i
⎞
⎟
⎠
(2.2.1-18)