Figure 3. Schematic of a hydronic heating system radiator

CHAPTER 3: THE CONSUMER'S HEAT LOAD

SIMPLE MODEL FOR THE CONSUMER'S SPACE HEATING EQUIPMENT-continue - CR95_170029

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Figure 3. Schematic of a hydronic heating system radiator.

q2/q0 = [(Tml)1/(Tml)0]n1 [(Tml)2/ (Tml)1]n2

(3-1)

where q = heat output from the radiator (W)

Tml = logarithmic mean temperature difference (C)

n1, n2 = empirically determined coefficients (dimensionless).

and the subscripts denote the following operating conditions

0 = "design" condition for the radiators, usually the maximum load condition

at maximum supply temperature

1 = condition of actual supply temperature with the flow rate as determined

under the design condition

2 = any actual operating condition.

Equation 3-1 uses the logarithmic mean temperature difference Tml and exper-

imentally determined constants to predict heat exchanger performance. The log-

arithmic mean temperature difference is defined as

(Ts - Ta ) - (Tr - Ta ) =

Ts - Tr

Tm1 =

(3-2)

l n(Ts - Ta ) - ln(Tr - Ta ) l n[ (Ts - Ta )/(Tr - Ta ) ]

One problem that results from using the logarithmic mean temperature differ-

ence is that an explicit expression for either the supply temperature Ts, or the return

temperature Tr, cannot be obtained from the expression for the logarithmic mean

temperature. This limits the extent of closed form analysis and ultimately, when

calculations are required, it forces solution by iterative numerical methods. As a

solution to these problems, the use of the arithmetic mean as an approximation for

the logarithmic mean was proposed by Soumerai (1987). The arithmetic mean

temperature difference for this case is defined as

Tma= [(Ts Ta) + (Tr Ta)]/2 = (Ts + Tr 2Ta)/2.

(3-3)

The arithmetic mean temperature difference has the advantage that it can be used

to find a very simple explicit expression for either the supply temperature Ts or the

return temperature Tr given the value of the arithmetic mean temperature differ-

ence. The disadvantage of using the arithmetic mean temperature difference as an

approximation for the logarithmic mean temperature difference is the error induced

by this approximation. As Soumerai (1987) points out, within certain ranges of the

temperatures involved, the resultant errors are usually acceptable, given the other

uncertainties in heat transfer engineering. Soumerai (1987) recommends the use of

the arithmetic mean as an approximation for the logarithmic mean in cases where

the approach factor AF is equal to or greater than 0.5. The approach factor for this

type of heat exchanger is given by

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