5.3.3. Pipes and Intermediate Heat Exchangers

5.3.3.1. Contributions to Flow and Pressure

In considering the flow through a pipe or through an intermediate heat exchanger, several factors are taken into consideration. The contribution of the element to \(a_{\text{o}}\) in Eq. (5.2-10) is taken as

(5.3-41)\[\Delta a_{\text{o}} = L/A\]

where

\(L\) = the length of the element

\(A\) = the flow area of the element

The pressure drop contribution of the element to \(a_1\) in Eq. (5.2-12) is composed of a number of terms. One of these terms is the frictional pressure drop, which is written as

(5.3-42)\[\Delta p_{\text{fr}} = f \frac{L}{D_{\text{h}}} \frac{\rho v \left| v \right|}{2} = f \frac{L}{D_{\text{h}}} \frac{w \left| w \right|}{2 \rho A^{2}}\]

where

\(\Delta p_{\text{fr}}\) = the frictional pressure drop

\(f\) = the Moody friction factor

\(L\) = the length of the element

\(D_{\text{h}}\) = the hydraulic diameter of the element

\(\rho\) = the density of the liquid

\(v\) = the liquid velocity

\(w\) = the liquid mass flow rate

\(A\) = the flow area of the element

The Moody friction factor \(f\) for turbulent flow in pipes [5-2] is taken as

(5.3-43)\[f = C_{1} \left\lbrack 1 + \left( C_{2} \frac{\varepsilon}{D_{\text{h}}} + \frac{C_{3}}{\text{Re}} \right)^{C_{4}} \right\rbrack\]

where

\(C_1 = 0.0055\)

\(C_2 = 20,000\)

\(C_3 = 1.0 \times 10^{6}\)

\(C_4 = 1/3\)

\(\varepsilon\) = the user-supplied roughness of the element

\(\text{Re}\) = the Reynolds number

For laminar flow, the friction factor \(f\) is taken as

(5.3-44)\[f = 64/\text{Re}\]

The Reynolds number in either case is

(5.3-45)\[Re = \frac{D_{\text{h}} \left| w \right|}{A \mu}\]

where \(\mu\) is the viscosity of the fluid.

A second term is the pressure drop caused by bends in the flow path, and these are modeled as an additional frictional drop term, written as

(5.3-46)\[\Delta p_b = f \frac{L_{\text{B}}}{D_{\text{B}}} N_{\text{B}} \frac{w \left| w \right|}{2\rho A^{2}}\]

where

\(L_{\text{B}}/D_{\text{B}}\) = a user-supplied input number for an effective length-to-diameter ratio per bend

\(N_{\text{B}}\) =the number of bends in the element

A third term is the pressure drop to account for baffles or restrictive orifices within the element which cause greater pressure drops than would be accounted for by friction or bends above. This term is taken as

(5.3-47)\[\Delta p_{G2} = G_{2}' \frac{w \left| w \right|}{2\rho A^{2}}\]

where \(G_2'\) is the minor loss coefficient, defined as

(5.3-48)\[\begin{split}G_{2}' = \begin{cases} G_{2} & \text{if }Re \geq 100 \\ G_{2} \left( \frac{100}{Re}\right) & \text{if }Re < 100 \end{cases}\end{split}\]

based on the user-provided nominal loss coefficient, \(G_2\) (G2PRDR).

A forth term is the minor pressure loss that occurs at the outlet of an element. This term is defined as

(5.3-49)\[ \Delta p_{K_{out}} = K_{out}\left(Re_{out}\right) \frac{w \left| w \right|}{2\rho A_{out}^{2}}\]

where \(K_{out}\left(Re_{out}\right)\) is the anisotropic Re-dependent loss coefficient at the outlet of the element, \(Re_{out}\) is the Re number at the outlet of the elment and \(A_{out}\) is the area at the outlet of the element. Anisotropic Re-dependent loss coefficients are described in Section 5.3.3.2.

A fifth item is the second term in Eq. (5.2-1), which is proportional to the square of the mass flow rate, and is due to the difference in the fluid densities at the inlet and outlet ends of the element. It has the form

(5.3-50)\[\Delta p_A = \frac{w^{2}}{A^{2}} \left( \frac{1}{\rho_{O}} - \frac{1}{\rho_{I}} \right)\]

where

\(\rho_{\text{O}}\) = the fluid density at the element outlet

\(\rho_{\text{I}}\) = the fluid density at the element inlet

A final term is the gravity-head term for the element, taken as

(5.3-51)\[\Delta p_{gr} = g\left( z_{\text{out}} - z_{\text{in}} \right) \overline{\rho}\]

where

\(z_{\text{out}}\) = the height of the element outlet

\(z_{\text{in}}\) = the height of the element inlet

\(g\) = the acceleration of gravity

\(\overline{\rho}\) = the average of the inlet and outlet fluid densities

Taking the above five terms together, the contribution from the element to \(a_1\) is

(5.3-52)\[\Delta a_{1} = - \Delta t \left\lbrack \Delta p_{fr} + \Delta p_b + \Delta p_{G2} + \Delta p_{K_{out}} + \Delta p_A + \Delta p_{gr} \right\rbrack\]

The contribution of the element to \(a_2\), which is the derivative of the pressure drops with respect to time, as shown in Eq. (5.2-15), is zero, since the friction factors and the geometry are assumed not to change independently with time. Hence

(5.3-53)\[\Delta a_{2} = 0\]

The contribution of the element to \(a_3\), however, which is the derivative of the pressure drops with respect to the mass flow rate, as shown in Eq. (5.2-17), is obtained by differentiation. For laminar flow, differentiating Eq. (5.3-44) gives

(5.3-54)\[\frac{\partial \text{f}}{\partial \text{w}} = - \frac{f}{w}\]

For turbulent flow, differentiating Eq. (5.3-43) gives

(5.3-55)\[\frac{\partial \text{f}}{\partial \text{w}} = - \frac{C_{1}C_{3}C_{4}}{\text{Re} \left| w \right|} \left( C_{2} \frac{\varepsilon}{D_{\text{h}}} + \frac{C_{3}}{\text{Re}} \right)^{C_{4} - 1}\]

Taking the derivative with respect to \(w\) of each term in Eq. (5.3-52), we have for the contribution to \(a_3\)

(5.3-56)\[\begin{split}\Delta a_{3} = - \Delta t \left\{ 2\frac{\Delta p_{fr} + \Delta p_b + \Delta p_{G2} + \Delta p_{K_{out}} + \Delta p_A}{w} \right. \\ \left. + \frac{\Delta p_{fr} + \Delta p_b}{f} \frac{\partial \text{f}}{\partial \text{w}} + \frac{\Delta p_{K_{out}}}{K_{out}(Re_{out})} \frac{\partial Re_{out}}{\partial \text{w}} \frac{\partial K_{out}}{\partial Re_{out}} \right\}\end{split}\]

5.3.3.2. Anisotropic Re-dependent Loss Coefficients

Anisotropic Re-dependent loss coefficients can be defined at element interfaces using Interface Loss Coefficient Tables. The definition of the loss coefficient tables is shown in Table 5.3.1, and examples are shown in Listing 5.3.1 and Listing 5.3.2. Loss coefficient tables are assigned to the outlet of an element using EllLossCoefTableID and to the inlet of a segment using SegLossCoefTableID,

By default, loss coefficients at element and segment interfaces are zero. Interfaces assigned to an interface loss coefficient table will overwrite the default interface loss coefficient. The pressure drop at element interface \(i\) caused by the loss coefficient is defined as

(5.3-57)\[ P_{i+} - P_{i-} = K_i(Re_i) \frac{w|w|}{2\rho_i A_i^2}\]

where

(5.3-58)\[ Re_i = \frac{Dh_i |w|}{A_i \mu_i}\]

and \(P_{i+}\)/\(P_{i-}\) are the pressure downstream and upstream the interface, respectively. In PRIMAR4, the pressure drop at an interface is included in the pressure drop of the element that defines the interface. In the case of a segment inlet interface, the pressure drop is included in the pressure drop of the first element of the segment.

Two functional forms are currently available to capture the dependence of the loss coefficient on the Reynolds number:

(5.3-59)\[\begin{split} K_i(Re_i) = \begin{cases} C_1 + C_2 Re_i^{C_3} & \text{Kind} = 1 \\ C_1 + C_2 exp\left(Re_i{C_3}\right) & \text{Kind} = 2 \\ \end{cases}\end{split}\]

(5.3-60)\[ \begin{align}\begin{aligned}\begin{split} \begin{split} C_{1} & \geq 0 \\ C_{2} & \geq 0 \\ C_{3} & \ge -2 \end{split} & \text{Kind} = 1 \\\end{split}\\\begin{split} \begin{split} C_{1} & \geq 0 \\ C_{1} + C_{2} & \geq 0 \end{split} & \text{Kind} = 2\end{split}\end{aligned}\end{align} \]

Two sets of coefficients may be defined for a given interface: one set for forward flow and one set for reverse flow. Additionally, an interface area, \(A_i\), and hydraulic diameter, \(Dh_i\), may be defined for the evaluation of the interface pressure drop and interface Re. If an interface area and interface hydraulic diameter are not provided or are defined as zero, the area and hydraulic diameter of the element upstream or downstream the interface are used. If the upstream element has a smaller or equal area than the downstream element, the upstream element’s area and hydraulic diameter are used for the interface. If the upstream element has a bigger area than the downstream element, the downstream element’s area and hydraulic diameter are used for the interface.

At very low flow rates, the derivative of the loss coefficients may approach infinity. This introduces numerical difficulty when SAS attempts to reverse flow through an element. In order to improve the robustness of the semi-implicit solution scheme, an optional lower limit has been introduced for Re-dependent loss coefficients, \(ReL\). When the Reynold number drops below \(ReL\), Eq. (5.3-59) is evaluated at \(ReL\). Additionally, the derivative of \(K_i(Re_i)\) is assumed to be zero.

Table 5.3.1 Input in a Interface Loss Coefficient Table

Column Label	Description	Range
iEll (Use only with element outlet table)	Index of the element outlet where the coefficient will be defined.	1 \(\leq\) iEll \(\leq\) NELEMT
iSGL (Use only with segment inlet table)	Index of the segment inlet where the coefficient will be defined.	1 \(\leq\) iSGL \(\leq\) NSEGLP + NSEGLS
Kind	Kind of functional form for calculating the anisotropic loss coefficient. See Eq. (3.9-24)	1 or 2
C1f, C2f, C3f	Function coefficients for forward flow	See Eq. (3.9-25)
C1b, C2b, C3b	Function coefficients for backward flow	See Eq. (3.9-25)
A (optional)	Area used to evaluate the Reynolds number and minor pressure loss at the interface. Optional area and hydraulic diameter must be provided as a pair.	\(A \geq 0\)
Dh (optional)	Hydraulic Diameter used to evaluate the Reynolds number at the interface. Optional area and hydraulic diameter must be provided as a pair.	\(Dh \geq 0\)
ReL (optional)	Lower Reynolds number limit for evaluating the functional form. For Reynolds numbers below the lower limit, the functional form is evaluated at ReL and the derivative is assumed to be 0.0.	\(ReL \geq 0\)

Listing 5.3.1 Example of Element Outlet Anisotropic Loss Coefficient Input

TABLE <id> <name>
iEll  Kind  C1f    C2f       C3f   C1b    C2b     C3b
2     1     0.5 0.0001         1   1.5 100000      -1
3     2     1       10 -0.000001     2 0.0001 0.00001
4     1     1   100000        -1   2.2  20000      -1
END

Listing 5.3.2 Example of Segment Inlet Anisotropic Loss Coefficient Input

TABLE <id> <name>
iSGL  Kind  C1f    C2f       C3f   C1b    C2b     C3b   A            Dh  ReL
1     1     0.5 100000        -1   1.5      2       0  2.2          6.5  0.0
4     1     1   100000        -1   2.2  20000      -1 0.002 0.003183099  1.0
END