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7.21.1 Radiator Equations



Modeling the Pressure Loss Through a Radiator


A radiator is considered to be infinitely thin, and the pressure drop through the radiator is assumed to be proportional to the dynamic head of the fluid, with an empirically determined loss coefficient which you supply. That is, the pressure drop, $\Delta p$, varies with the normal component of velocity through the radiator, $v$, as follows:


 \Delta p = k_L \frac{1}{2} \rho v^2 (7.21-1)

where $\rho$ is the fluid density, and $k_L$ is the non-dimensional loss coefficient, which can be specified as a constant or as a polynomial, piecewise-linear, or piecewise-polynomial function.

In the case of a polynomial, the relationship is of the form


 k_L = \sum_{n=1}^{N}{ r_n v^{n-1} } (7.21-2)

where $r_n$ are polynomial coefficients and $v$ is the magnitude of the local fluid velocity normal to the radiator.



Modeling the Heat Transfer Through a Radiator


The heat flux from the radiator to the surrounding fluid is given as


 q = h ( T_{{\rm air},d} - T_{\rm ext} ) (7.21-3)

where $q$ is the heat flux, $T_{{\rm air},d}$ is the temperature downstream of the heat exchanger (radiator), and $T_{\rm ext}$ is the reference temperature for the liquid. The convective heat transfer coefficient, $h$, can be specified as a constant or as a polynomial, piecewise-linear, or piecewise-polynomial function.

For a polynomial, the relationship is of the form


 h = \sum_{n=0}^{N}{ h_n v^n } ; 0 \leq N \leq 7 (7.21-4)

where $h_n$ are polynomial coefficients and $v$ is the magnitude of the local fluid velocity normal to the radiator in m/s.

Either the actual heat flux $(q)$ or the heat transfer coefficient and radiator temperature $(h, T_{\rm ext})$ may be specified. $q$ (either the entered value or the value calculated using Equation  7.21-3) is integrated over the radiator surface area.

Calculating the Heat Transfer Coefficient

To model the thermal behavior of the radiator, you must supply an expression for the heat transfer coefficient, $h$, as a function of the fluid velocity through the radiator, $v$. To obtain this expression, consider the heat balance equation:


 q = \frac{\dot{m}c_p \Delta T}{A} = h(T_{{\rm air},d} - T_{\rm ext}) (7.21-5)

where


$q$ = heat flux (W/m $^2$)
$\dot{m}$ = fluid mass flow rate (kg/s)
$c_p$ = specific heat capacity of fluid (J/kg-K)
$h$ = empirical heat transfer coefficient (W/m $^2$-K)
$T_{\rm ext}$ = external temperature (reference temperature for the liquid) (K)
$T_{{\rm air}, d}$ = temperature downstream from the heat exchanger (K)
$A$ = heat exchanger frontal area (m $^2$)

Equation  7.21-5 can be rewritten as


 q = \frac{\dot{m} c_p (T_{{\rm air},u} - T_{{\rm air},d})}{A} = h(T_{{\rm air},d} - T_{\rm ext})%%\label{eq6.8.6} (7.21-6)

where $T_{{\rm air},u}$ is the upstream air temperature. The heat transfer coefficient, $h$, can therefore be computed as


 h = \frac{\dot{m} c_p ( T_{{\rm air},u} - T_{{\rm air},d})} {A (T_{{\rm air},d}- T_{\rm ext})} %%\label{eq6.8.7} (7.21-7)

or, in terms of the fluid velocity,


 h = \frac{\rho v c_p ( T_{{\rm air},u} - T_{{\rm air},d})} {T_{{\rm air},d}- T_{\rm ext}} %%\label{eq6.8.8} (7.21-8)


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