*2.1. Heat Accounting for Radiators*

The heat of the hot and cold fluids is exchanged through the solid walls. First, the hot fluid conducts the heat to the solid walls; next, it is conducted via one side of the solid walls toward the other side, and then the heat is transferred to the cold fluid via the walls. Overall, the heat exchange can be distinguished as a heat giving–heat conducting–heat giving process acting in tandem.

If a hot fluid flows in a metal foam or radiator, a cold fluid passes through the metal foam or radiator, and convection occurs to dissipate the heat. Based on the heat balance principle, it is clear that when the ducts are well insulated, then the amount of heat discharged per unit time of the hot flow and the amount of heat absorbed per unit time of the cold flow are equal.

For a radiator with hot and cold fluids flowing against each other, the heat balance equation reads as follows:

$$Q = q\_{m,\varepsilon}(H\_{\mathfrak{c},2} - H\_{\mathfrak{c},1}) = q\_{m,h}(H\_{h,1} - H\_{h,2}).\tag{1}$$

Here, *qm*,*<sup>c</sup>* and *qmhh* indicate the mass flow rate of the cold and hot fluids, respectively; *Hc* and *Hh* correspond to the enthalpy per unit of cold and hot fluid (the subscripts *c* for cold and *h* for hot) and the subscripts "1" and "2" correspond to the inlet and outlet side of each unit.

If there is no phase change in either the hot or cold fluid inside the radiator, and the specific heat coefficient at constant pressure.of the fluid, *cp*, does not change due to a change in the temperature, then Equation (1) can be expressed as follows:

$$Q = q\_{m,c}c\_{p,c}(t''{}\_1 - t'{}\_1) = q\_{m,h}c\_{p,h}(t'{}\_2 - t''{}\_2) \tag{2}$$

For overall heat transfer, the heat transfer equation required is as follows:

$$Q = kA\Delta t\_m.\tag{3}$$

Here, A is the contact area; Δ*tm* represents the average temperature difference between the two ends where the fluid is located, i.e.,

$$
\Delta t\_m = \frac{\Delta t\_{\rm li} - \Delta t\_{\rm c}}{\ln \left( \Delta t\_{\rm c} \Delta t\_{\rm h} \right)} \left( \Delta t\_{\rm c} = t''\_1 - t'\_1; \Delta t\_{\rm h} = t'\_2 - t''\_2 \right) \tag{4}
$$

While *K* is the heat transfer coefficient, the introduction of Equation (3) into Equation (1) and the association of Equations (1) and (2) lead to the following:

$$k = \frac{q\_{m,c}c\_{p,c}(t''\varepsilon - t'\_c)}{A\Delta t\_m} = \frac{q\_{m,h}c\_{p,h}(t'\_h - t''\_h)}{A\Delta t\_m}.\tag{5}$$

According to this Equation, it is clear that determining the inlet and outlet temperatures of the two fluids and the flow rate of the radiator suffices to obtain the total heat transfer coefficient, *K*.

During actual operation, the heat dissipation as well as the heat absorption achieved by the hot and cold fluids are not in an absolutely equal state and need to be based on the heat balance theory to obtain their relative error in heat balance Δ*Q* as follows:

$$
\Delta Q = \frac{(Q\_h - Q\_c)}{Q\_c} \times 100\%. \tag{6}
$$

Here, *Qh* denotes fluid heat release and *Qc* denotes fluid heat absorption; in the case of Δ*Q* < 5%, it can be assumed that the system is in a state of thermal equilibrium.

Forced convection is applied to the fluid in the radiator. The factors that will have an effect on the heat transfer coefficient correspond to the internal diameter of the radiator channel D, the fluid flow rate *v*, the fluid viscosity *μ*, the fluid density *ρ*, the fluid heat capacity *cp*, the fluid thermal conductivity *k*, etc. The expression is as follows:

$$h = f\left(l, D, \rho, \mu, c\_{\mathbb{P}}, k, v\right). \tag{7}$$

Implementing the measure analysis based on Buckingham's π theorem results in the following:

$$h = f\left(l, D, \rho, \mu, c\_p, k, v\right),\tag{8}$$

$$
\pi\_2 = \frac{Dv\rho}{\mu} = \text{Re}\_\prime \tag{9}
$$

$$
\pi\_3 = \frac{\mu c\_p}{k} = Pr.\tag{10}
$$

Therefore, Equation (7) can be related as follows:

$$N\mu = f\_1(\text{Re}, Pr). \tag{11}$$

From heat transfer and fluid mechanics, the Nusselt and Prandtl numbers for the flow of a work mass in an electronic heat sink component are known.

Where *de* is the equivalent diameter of the flow channel or the so-called hydraulic diameter, *h* denotes the convection heat transfer coefficient and *λ* denotes the thermal conductivity of water. *ρ* and *μ* denotes fluid density and viscosity; *cp* denotes constant pressure heat capacity.
