Goulds Pumps -

Section E -- Properties of Liquids

E-3 Friction Loss of Pulp Suspensions in Pipe

APPENDIX E

The following are three examples which illustrate the method for determination of pipe friction loss in each of the three regions shown in Figure 3.

Example 1.
Determine the friction loss (per 100 ft of pipe) for 1000 U.S. GPM of 4.5% oven-dried unbeaten aspen sulfite stock, never dried, in 8 inch schedule 40 stainless steel pipe (pipe inside diameter = 7.981 in). Assume the pulp temperature to be 95^o F.

Solution:
a) The bulk velocity, V, is

and Q = flow = 1000 U.S. GPM.
D = pipe inside diameter 7.981 in.

0.4085(1000) = 6.41 ft/s.
V = 7.981²

b) It must be determined in which region (1, 2, or 3) this velocity falls. Therefore, the next step is to determine the velocity at the upper limit of the linear region, Vmax.

Vmax = K' C^a,
and K' = numerical coefficient = 0.85 (from Appen-dix B, Table I),

C = consistency = 4.5%,

a = exponent = 1.6 (from Appendix B, Table I).
V_max = 0.85 (4.5^1.6) = 9.43 ft/s.

c) Since V_max exceeds V, the friction loss, Δ H/L, falls within the linear region, Region 1. The friction loss is given by the correlation:

Δ H/L=F K V^a C^b D^y [2] and F = correction factor = F₁ F₂ F₃ F₄ F₅,

F₁ = correction factor for pulp temperature. Since the pulp temperature is 95^o F,

F₁ = 1.0,
F₂ = correction factor for pipe roughness. For stain-less steel pipe,

F₂ = 1.25 (from Appendix D),
F₃ = correction factor for pulp type. Numerical coefficients for this pulp are contained in Appendix C, Table II, and have already incor-porated this factor.

F₄ = correction factor for beating. No additional beating has taken place, therefore

F₄ = 1.0 (from Appendix D),
F₅ = design safety factor. This has been assumed to be unity.

F₅= 1.0.
F = (1.0) (1.25) (1.0) (1.0) (1.0) = 1.25,
K = numerical coefficient = 5.30 (from Appendix C, Table II),

a,b,y=exponents = 0.36,2.14, and -1.04, respectively (from Appendix C, Table II),

V, C, D have been evaluated previously.

Δ H/L= (1.25) (5.30) (6.41^0.36) (4.5^2.14) (7.981^-1.04)
= (1.25) (5.30) (1.952) (25.0) (0.1153)
=37.28 ft head loss/100 ft of pipe.

This is a rather substantial head loss, but may be acceptable for short piping runs. In a large system, the economics of initial piping costs versus power costs should be weighed, however, before using piping which gives a friction loss of this magnitude.

Example 2.
Determine the friction loss (per 100 ft of pipe) of 2500 U.S. GPM of 3% oven-dried bleached kraft pine, dried and reslurried, in 12 inch schedule 10 stainless steel pipe (pipe inside diameter = 12.39 in). Stock temperature is 125^oF.

Solution:
a) V, the bulk velocity, is

b) The velocity at the upper limit of the linear region, V_max, is V_max K' C^a and K' = 0.59 (from Appendix B, Table I),
= 1.45 (from Appendix B, Table I).
V_max = 0.59 (3.0^1.45) = 2.90 ft/s.
c) Region 1 (the linear region) has been eliminated, since the bulk velocity, V, exceeds V_max.

The next step requires calculation of V_w.
V_w = 4.00 C^{l .40}
= 4.00 (3.01.40) = 18.62 ft/s.
d) V exceeds V_max, but is less than V_w, indicating that it falls in Region 2. The friction loss in this region is calculated by substituting V_max into the equation for head loss, Equation 2.

Δ H/L=F K (V_max)^aC^bD^y,
andF=F₁ F₂ F₃F₄F₅; (iv)
F₁ = 1.528 - 0.00556T, and T = stock temperature = 1250 F
F₁ = 1.528 - 0.00556 (125) = 0.833,
F₂ = 1.25 (from Appendix D),
F₃ = F₄ = F₅ = 1.0,
F = 0.833(1.25) (1.0) = 1.041,
K = 8.80 (from Appendix C, Table II),
a,b, y= 0.31,1.81, and -1.34, respectively (from Appen-dix C, Table II),

V_max, C, and D have been defined previously. Δ H/L= 1.041(8.80) (2.900.31) (3.01.81) (12.39-1.34) = 1.041(8.80) (1.391) (7.304) (0.03430) = 3.19 ft head loss/100 ft of pipe.

Example 3.
Determine the friction loss (per 100 ft of pipe) for 2% oven-dried bleached kraft pine, dried and reslurried, through 6 inch schedule 40 stainless steel pipe (inside diameter = 6.065 in). The pulp temperature is 90^o F; the flow rate 1100 U.S. GPM. Solution:

a) The bulk velocity is

b) It must be determined in which region (1, 2 or 3) this velocity falls. To obtain an initial indication, determine V_max.
V_max = K' C^a,
and K' = 0.59 (from Appendix B, Table I),
a = 1.45 (from Appendix B, Table I).
V_max = 0.59(20^1.40) = 1.61 ft/sec.
c) Since V exceeds V_max, Region 1 (the linear region) is eliminated. To determine whether V lies in Region 2 or 3, the velocity at the onset of drag reduction, V_w, must be calculated.
V_w = 4.00 C^1.40,
= 4.00 (2.0^1.40) = 10.56 ft/sec.
d) V exceeds V_w, indicating that it falls in Region 3. The friction loss is calculated as that of water flowing at the same velocity
(Δ H/L)_w = 0.579 V^1.75 D^-1.25,
= 0.579 (12.22^1.75) (6.065^-1.25)
= 4.85 ft head loss/100 ft of pipe.
This will be a conservative estimate, as the actual friction loss curve for pulp suspensions under these conditions will be below the water curve.

REFERENCES (v)