V. Pressure Drop Calculation in Two-Phase Flow

The mixture momentum equation is,

[eqn: - partialp over partialz = 1 over g_c left[ partialG over partialt + partial over partialz left( {G^2 A} over {<rho' >} right) right] + g over g_c < bar rho >sin theta][eqn: + {tau_w P_f} over A]                 (4.5.1)

acceleration          gravity      friction

[eqn: therefore ~ TRIANGLE p ~=~ TRIANGLE p_accel ~+~ TRIANGLE p_grav ~+~ TRIANGLE p_frict ~+~ TRIANGLE p_local]                 (4.5.2)

(1) acceleration pressure drop ([eqn: TRIANGLE p_accel] )            0

[eqn: TRIANGLE p_accel] = [eqn: 1 over g_c int from z=0 to L_H partial over partialz ( G^2 over {<rho' >} ) dz]   =  [eqn: int from 0 to lambda_0 d over dz ( G^2 over rho_f )dz ~+~ int from lambda_0 to L_H d over dz ( G^2 over {<rho_2phi >} )dz]

(4.5.3)

In homogeneous flow,

[eqn: <rho'> ~=] [eqn: <rho_H > ~=~ 1/(v_f +<x>v_fg )]                              (4.5.4)

∴ [eqn: TRIANGLE p_accel] = [eqn: G^2 over g_c left[ 1 over {<rho(L_H ) >} - 1 over {<rho( lambda_o )>} right]]  = [eqn: v_fg over g_c G^2 <x``(L_H )>]       (4.5.5)

In separate flow,

[eqn: 1 over <rho'> ~==~ left[ (1-<x>)^2 over {rho_l ``(1-<alpha>)} + <x>^2 over {rho_g <alpha>} right]]

(2) gravitational pressure drop ([eqn: TRIANGLE p_grav] )

[eqn: TRIANGLE p_grav] = [eqn: g over g_c int from z=0 to L_H {< bar rho >} dz]                                           (4.5.6)

i) single-phase region

[eqn: {TRIANGLE p^1phi}_grav] = [eqn: g over g_c int from z=0 to lambda_o {<rho_H >} dz ~=~ g over g_c rho_f`` lambda_o]                           (4.5.7)

ii) two-phase region

[eqn: {TRIANGLE p^2phi}_grav] = [eqn: g over g_c int from lambda_o to L_H {<rho' >} dz]                                        (4.5.8)

Heare, the axial density profile can be obtained from the energy balance by        using the void-quality relation and the definition of mixture density.

(3) frictional pressure drop ([eqn: TRIANGLE p_fric] )

[eqn: {TRIANGLE p^2phi}_fric ~>~ {TRIANGLE p^1phi}_fric]     where [eqn: {TRIANGLE p^1phi}_grav] = [eqn: K G^2 over {2g_c rho_l }]

= [eqn: {TRIANGLE p^1phi}_fric][eqn: phi_lo^2]                                     (4.5.9)

Here  [eqn: phi_lo^2] is called as the two-phase multiplier.

∴[eqn: {TRIANGLE p^2phi}_fric] = [eqn: K G^2 over {2g_c rho_f } phi_lo^2 (<x>, p, .....)]                       (4.5.10)

Thus, [eqn: phi_lo^2] [eqn: under TRIANGLE] [eqn: (dp/dz)_2phi / (dp/dz)_1phi]                                 (4.5.11)

To determine [eqn: phi_lo^2], we can write [eqn: TRIANGLE p_2phi] for two-phase flow

[eqn: {TRIANGLE p^2phi}_fric] = [eqn: K G^2 over {2g_c <rho_2phi > }]                                       (4.5.12)

1) For homogeneous flow, [eqn: <rho_2phi > ~=~ <rho_H >]

∴ [eqn: K G^2 over {2g_c rho_f } left( rho_f over {<rho_H >} right)]

By comparing Eqs.(4.5.10) and (4.5.12),

[eqn: phi_lo^2] = [eqn: rho_f / <rho_H >] = [eqn: (1~+~ v_fg over v_f <x>)]   from Eq.(4.5.4)         (4.5.13)

For two-phase friction factor, [eqn: f_2phi] ,

① [eqn: f_2phi ~=~ f_1phi]

② [eqn: f_2phi] has the same Re dependence as [eqn: f_1phi]

[eqn: f_2phi over f_1phi ~=~ {C_1 / Re_2phi^n } over {C_1 / Re_1phi^n } ~=~ left( mu_2phi over mu_1phi right)^n]                           (4.5.14)

★ appropriate correlations for two-phase viscosity

ⓐ McAdams et al.   :  [eqn: mu_2phi over mu_1phi ~=~ left[ 1+x`` left( mu_f over mu_g -1 right) right]^-1]

ⓑ Cichitti et al.      :  [eqn: mu_2phi over mu_1phi ~=~ left[ 1+x`` left( mu_g over mu_f -1 right) right]]

ⓒ Dukler et al.      :  [eqn: mu_2phi over mu_1phi ~=~ left[ 1+beta`` left( mu_g over mu_f -1 right) right]]

Thus, the modified [eqn: phi_lo^2]  considering the two-phase viscosity effect is,

[eqn: phi_lo^2] = [eqn: (1~+~ v_fg over v_f <x>)][eqn: left[mu_f over mu_g <x> + (1-<x>) right]^-n]               (4.5.15)

2) For separate flow,

By Chisholm, use momentum density against [eqn: <rho_2phi >] in Eq. (4.5.13)

[eqn: phi_lo^2] = [eqn: rho_f over <rho'>] = [eqn: left[ (1-<x>)^2 over (1-<alpha>) + rho_f over rho_g <x>^2 over <alpha> right]]                      (4.5.16)

3) other [eqn: phi_lo^2]

① Levy model for ideal annular flow

[eqn: phi_lo^2] = [eqn: left[ (1-<x>) over (1-<alpha>) right]^2-n]   ,where typical value of n is 0.25   (4.5.17)

② Martinelli-Nelson [eqn: phi_lo^2]

- steam-water data

- [eqn: phi_lo^2] ∝ <x> at any given pressure

[eqn: phi_lo^2] = [eqn: PROD (G,P) (1.2[(rho_f /rho_g )-1]<x>^0.824 ) +1.0]              (4.5.18)

where [eqn: PROD (G,p) = 1.36+0.0005p+0.1(G/10^6 ) -0.0007p(G/10^6 ),~~~(G/10^6 )<0.7]

[eqn: = 1.26-0.0004p+0.119(G/10^6 ) +0.00028p(G/10^6 ),~~~(G/10^6 )>0.7]

‡ negligible effects of G at a given quality

‡ not account for surface tension effect, important at high pressure            ③ Baroczy correlation

- correct for the influence of G on [eqn: phi_lo^2] for fluids other than water

[eqn: phi_lo^2](G) = Ω[eqn: phi_lo^2](Gref)

= [eqn: left[ 1+ C over X_tt + 1 over X_tt^2 right] (1-<x>)^2-n]                        (4.5.19)

where   [eqn: C ~=~ -2.0 + (28-0.8G^1/2 )``times][eqn: exp left[ - {(log_10 LAMBDA +2.5)^2} over 2.4-G/10^4 right]]

[eqn: LAMBDA ~=~ rho_g over rho_f left( mu_f over mu_g )^0.2],       n = 0.2

■ Models recommanded in 10CER50  Appendix k.

◦ modified Baroczy correlation

◦ Thom correlation ( > 250 psia )

◦ Martinelli-Nelson correlation ( < 250 psia )

(4) local pressure loss ([eqn: TRIANGLE p_local] )

For abrupt change of flow area in single-phase flow

[eqn: p_1 + G_1^2 over {2g_c rho_l} ~=~ p_2 + G_2^2 over {2g_c rho_l} ~+~K G^2 over {2g_c rho_l}]                        (20)

From momentum balance for uniform velocity profile,

[eqn: K ~=~ left(1- A_1 over A_2 right)^2 ~=~ (1- sigma``)^2]    ,where [eqn: sigma (flow~area~ratio) under TRIANGLE ~A_1 over A_2]

Thus,   [eqn: TRIANGLE p_exp] = [eqn: [(1- sigma^2 ) -K_exp ] G_1^2 over {2g_c rho_l}]                             (4.5.21)

[eqn: TRIANGLE p_cont] = [eqn: [(1- sigma^2 ) - sigma^2 K_cont ] G_1^2 over {2g_c rho_l}]                            (4.5.22)

[eqn: (1- sigma^2 ) G_1^2 over {2g_c rho_l}] is representative the reversible pressure change which can be        derived directly from the single phase Bernoulli equation without the               head-loss term.

[eqn: TRIANGLE p_static = TRIANGLE p_rev - TRIANGLE p_irrev]

By Vennard,

[eqn: sigma]	0 0.2 0.4 0.5 0.6 0.8 1.0
Kcont	0.385 0.34 0.266 0.219 0.164 0.053 0

o Kexp = [eqn: (1- sigma)^2]

o Kcont

For two-phase flow,

[eqn: TRIANGLE p_2phi ~=~ K G^2 over {2g_c <rho_2phi >} ~=~ K G^2 over {2g_c rho_f} PHI]                          (4.5.23)

[eqn: TRIANGLE p_exp] = [eqn: {G_1^2 sigma^2} over {g_c rho_l} left( [ rho_l over rho_g x^2 ( 1 over {alpha_1 sigma } - 1 over alpha_2 ) ] ~+~ (1-x)^2 [ 1 over {sigma (1-alpha_1 )} - 1 over {(1-alpha_2 )} ] right)]

[eqn: TRIANGLE p_cont] = [eqn: G_1^2 over {2 sigma^2 g_c} left[ left( 1 over C -1 right)^2 + (1- sigma)^2 right] left[ x over rho_g - (1-x) over rho_l right]]

where

[eqn: alpha_1, alpha_2] ; void fractions upstream and downstream

C   ; vena contracta area ratio

Thus, for sudden expansion with no phase change,

[eqn: phi_lo^2] = [eqn: rho_f over <rho'>] = [eqn: left[ (1-<x>)^2 over (1-<alpha>) + rho_f over rho_g <x>^2 over <alpha> right]]                    (4.5.26)

However, the experimental results for the grid-type spacer showed,

[eqn: PHI] = [eqn: (1+ v_fg over v_f <x>)]

Thus,   [eqn: TRIANGLE p_spacer = K_sp G^2 over {2g_c rho_f} (1+ v_fg over v_f <x>)]                              (4.5.27)

● Core pressure drop

① pressure loss along the bundle of bare rods

② pressure loss across spacers or grids

③ pressure loss at core entrance and exit

☞ pressure loss along the bundle of bare rods  ;  depend on P/D

f = C Re-n  + B

Authors	C	n	B	조 건
Miller et al.	0.296	0.2	0	37-rod bundle, □ 1.47 P/D, 0.625" O.D.
Wantland	1.76	0.39	0	100-rod bundle, □ 1.106 P/D, 3/16" O.D. Re=103 - 104, Pr=3 -6
	90.0	1.0	0.0082	102-rod bundle, ▽ 1.19 P/D, Re=2x103-104
Trupp and Azad	*1	*2	0	▽ 1.2<P/D<1.5, Re=104-105,

*1 ; [eqn: 0.287[(2sqrt3 / pi )(P/D)^2 - 1.30]]

*2 ; 0.368(P/D)-1.358

☞ pressure drop across the grid or spacer

I) By De Stordeur(1961)

[eqn: TRIANGLE p_spacer = rho C_s V_s^2 s / 2g_c A]

where A, Vs, s are unrestricted area, velocity in the spacer region               and projected area of the spacer.

Cs(Drag Coefficient) [eqn: sim] 1.65 for strap-type grids at Re([eqn: sim]105)

2) By Rehme(1973)

[eqn: TRIANGLE p_spacer = C_v (rho V_s^2 /2g_c ) (s/A)^2]

where Cv ; modified drag coefficient = 9.5 and 6,5 for Re=104 and 105

VB ; average bundle fluid velocity

☞ pressure drop across core or reactor vessel

[eqn: TRIANGLE p] = [eqn: K G^2 over {2g_c rho_l }]