6.3. 4-Way Heat exchanger

Fig. 6.3.1 Schematized 4-way heat exchanger

Fall type

type label	description	active
Heat exchanger	Heat exchanger with primary and secondary connection points. The fluid on both sides is assumed to be the same.	No

The four-way heat exchanger can be operated in co- and counter current flow. The heat transfer is determined by the temperature difference between the primary and secondary side and the heat transfer coefficient. The head loss is determined by a loss coefficient.

6.3.1. Mathematical model

Fig. 6.3.2 Overview of positive flow directions and numbering of connection points

Fig. 6.3.2 shows an overview of the positive flow direction and heat transfer direction.

The head loss on both the primary and secondary side is defined as:

(6.3.1)\[\Delta H_{i} = C_{h,i}Q_{i}\left| Q_{i} \right|\]

with:

variable	Description	Units
\(\Delta H_{i}\)	Head loss over the i-th side of the heat exchanger (i.e., primary (p) or secondary (s))	m
\(C_{h,i}\)	Hydraulic loss coefficient of the i-th side of the heat exchanger	s2/m5
\(Q_{i}\)	Flow rate on the i-th side of the heat exchanger	m3/s

This can be rewritten in terms of pressure and mass flow as follows (subscripts for primary and secondary side have been dropped for convenience):

(6.3.2)\[\Delta P = \rho g\left( C\frac{{\dot{m}}^{2}}{\rho^{2}} - \Delta z \right)\]

with:

variable	Description	Units
\(\Delta p\)	pressure drop	N/m2
\(\rho\)	density of the fluid at the upstream side	kg/m3
g	gravitational acceleration	m/s2
\(\dot{m}\)	mass flow rate	kg/s
\(\Delta z\)	height difference between connection points	m

The continuity equation applies for both sides, as:

(6.3.3)\[\dot{m}_{i,1} = \dot{m}_{i,2}\]

The heat transfer can be modelled in four different ways [4]:

• ε-NTU method

• P-NTU method

• LMTD method (Log Mean Temperature Difference)

• ψ-P method

The ε-NTU method is applied as it is the most practical method for computer programs. The ε-NTU method uses the effectiveness (ε) of the heat exchangers, defined as:

(6.3.4)\[\epsilon = \frac{\dot{Q}}{{\dot{Q}}_{\max}} = \frac{C_{t,Cold}\left( T_{C,out} - T_{\text{gen}} - T_{C,in} \right)}{C_{t,min}\left( T_{H,in} - T_{C,in} \right)}\]

with:

variable	Description	Units
\(\dot{Q}\)	Heat transfer from primary to secondary side.	W
\(\dot{Q}_{max}\)	Maximum heat transfer from primary to secondary side.	W
\(C_{t,cold}\)	Thermal capacity on the cold side	J/K/s
\(C_{t,min}\)	Minimum thermal capacity of both the cold and hot side	J/K/s
\(T_{C, out}\)	Temperature at the outlet of the cold side	K
\(T_{C, in}\)	Temperature at the inlet of the cold side	K
\(T_{H, in}\)	Temperature at the inlet of the hot side	K
\(T_{gen, in}\)	Temperature increase due to friction	K

The cold side is the side with the lowest temperature, this can be either the primary or secondary side

The flow thermal capacity for both the hot and cold side is calculated with:

(6.3.5)\[C_{t} = c_{p}\dot{m}\]

with:

variable	Description	Units
\(c_p\)	Specific heat of the fluid	J/kg/K

The temperature increase generated by friction is calculated with:

(6.3.6)\[T_{gen} = \frac{Q_{gen}}{c_p}\]

with

variable	Description	Units
\(Q_{gen}\)	The heat generated by friction	W

The generated heat flux is calculated from:

(6.3.7)\[Q_{\text{gen}} = C_{h}g\frac{{\dot{m}}^{3}}{\rho^{2}}\]

For different configurations of heat exchangers, formulas are available to calculate the effectiveness of the heat exchanger. We have implemented the following equation for the heat exchanger in co-current flow:

(6.3.8)\[\epsilon = \frac{1 - e^{- \text{NTU}*(1 + RC)}}{1 + RC}\]

and for counter-current flow:

(6.3.9)\[\epsilon = \frac{1 - e^{- \text{NTU}*(1 - RC)}}{1 - RC e^{- \text{NTU}*(1 - RC)}}\]

with:

variable	Description	Units
\(NTU\)	The number of transfer units	-
\(RC\)	The heat capacity ratio	-

The number of transfer units is given defined as:

(6.3.10)\[NTU = \frac{h}{C_{min}}\]

with:

variable	Description	Units
\(h\)	Overall heat exchange coefficient	W/K
\(C_{min}\)	Minimum thermal capacity of both the hot and cold side	W/K

The heat capacity ratio is defined as:

(6.3.11)\[RC = \frac{C_{min}}{C_{max}}\]

variable	Description	Units
\(C_{max}\)	Maximum thermal capacity of both the hot and cold side	W/K

An additional equation is required to define the temperature on all connection points. This additional equation defines the energy balance between the primary (1-2) and secondary (3-4) side as:

(6.3.12)\[\dot{m}_3 c_{p3} T_{3} - \dot{m}_4 c_{p4} T_{4} + fr Q_{gen, s} + \dot{m}_1 c_{p1} T_{1} - \dot{m}_2 c_{p2} T_{2} + fr Q_{gen, p} = 0\]

with:

variable	Description	Units
\(fr\)	Fraction of the generated heat supplied to the fluid	-

6.3.2. Properties

6.3.2.1. Hydraulic specifications

description	Input	unit	range	default	remarks
Loss coefficient primary side	Real	[s2/m5]	[0-200]
Loss coefficient secondary side	Real	[s2/m5]	[0-200]
Heat transfer coefficient	Real	[W/K]	[0-1e8]
Correction factor	Real	[-]	[0-1]
Fraction gen. heat to fluid	Real	[-]	[0-1]	0

See also “Mathematical model” (Mathematical model).

Remark

6.3.2.2. Component specific output

Heat transfer [W]

Generated heat flux [W]

6.3.2.3. H-actions

None

6.3.2.4. Component messages

Message	Type	Explanation