General resistance in a thermal liquid branch

**Library:**Thermal Liquid / Elements

The Flow Resistance (TL) block models a general pressure drop in a thermal-liquid network branch. The pressure drop is proportional to the square of the mass flow rate. The constant of proportionality is determined from a nominal operating condition specified in the block dialog box.

Use this block when the only data available for a component is its pressure drop as a function of its mass flow rate. Combine the block with others to create a custom component that more accurately captures the pressure drop that it induces—for example, a heat exchanger based on a chamber block.

The volume of fluid inside the flow resistance is assumed to be negligible. The mass flow rate in through one port must then exactly equal the mass flow rate out through the other port:

$${\dot{m}}_{\text{A}}+{\dot{m}}_{\text{B}}=0,$$

Energy can enter and leave the flow resistance through the thermal-liquid ports only. No heat exchange occurs between the wall and the environment. In addition, no work is done on or by the fluid. The energy flow rate in through one port must then exactly equal the energy flow rate out through the other port:

$${\varphi}_{\text{A}}+{\varphi}_{\text{B}}=0,$$

The relevant external forces on the fluid include those due to pressure at the
ports and those due to viscous friction at the component walls. Gravity is ignored
as are other body forces. Expressing the frictional forces in terms of a loss factor
*ξ* yields the semi-empirical expression:

$$\Delta p=\xi \frac{{\dot{m}}^{2}}{2\rho {S}^{2}},$$

*Δp*is the pressure drop from port**A**to port**B**—that is,*p*_{A}+*p*_{B}.*ξ*is the loss factor.*ρ*is the fluid density.*S*is the flow area.

The pressure drop equation is implemented with two modifications. First, to allow for a change in sign upon reversal of flow direction, it is rewritten:

$$\Delta p=\xi \frac{\dot{m}\left|\dot{m}\right|}{2\rho {S}^{2}},$$

$$\Delta p=\xi \frac{\dot{m}\sqrt{{\dot{m}}^{2}+\text{}{\dot{m}}_{\text{Th}}^{2}}}{2\rho {S}^{2}},$$

Above $${\dot{m}}_{Th}$$, the pressure drop approximates that expressed in the original equation (curve

**II**) and it varies with $${\dot{m}}^{2}$$. This dependence is commensurate with that observed in turbulent flows.Below $${\dot{m}}_{Th}$$, the pressure drop approximates a straight line with slope partly dependent on $${\dot{m}}_{Th}$$ (curve

**III**) and it varies with $$\dot{m}$$. This dependence is commensurate with that observed in laminar flows.

For ease of modeling, the loss factor *ξ* is not required as a
block parameter. Instead, it is automatically computed from the nominal condition
specified in the block dialog box:

$$\frac{\xi}{2{S}^{2}}=\frac{{\rho}_{*}\Delta {p}_{*}}{{\dot{m}}_{*}^{2}},$$

$$\Delta p=\frac{{\rho}_{*}\Delta {p}_{*}}{\rho {\dot{m}}_{*}^{2}}\left(\dot{m}\sqrt{{\dot{m}}^{2}+{\dot{m}}_{\text{Th}}^{2}}\right).$$

$$\Delta p=\frac{C\dot{m}}{\rho}\sqrt{{\dot{m}}^{2}+{\dot{m}}_{\text{Th}}^{2}},$$

$$C=\frac{{\rho}_{*}\Delta {p}_{*}}{{\dot{m}}_{*}^{2}}.$$

If the fluid mass density is treated as invariant, then its nominal and actual
values must always be equal. This is the assumption made in the block. The ratio of
the two is then always `1`

and the fraction *C*/*ρ* reduces to:

$$K=\frac{C}{\rho}=\frac{\Delta {p}_{*}}{{\dot{m}}_{*}^{2}}.$$

$$\Delta p=K\dot{m}\sqrt{{\dot{m}}^{2}+{\dot{m}}_{\text{Th}}^{2}}.$$

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