fluid-mechanics

Fluid Mechanics Calculator

Computational skill for pipe flow, pump design, and flow measurement. Covers core ChBE 311 and MAE 244 topics including dimensional analysis, flow regimes, energy balance, and momentum transfer.

SI units throughout. (Pa, m, m/s, kg/m^3, Pa·s). Convert from field units as needed: 1 psi = 6894.76 Pa; 1 ft = 0.3048 m; 1 gal/min = 6.309e-5 m^3/s.

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Module 1 — Flow Regimes and Reynolds Number

def reynolds_number(rho_kg_m3, v_m_s, D_m, mu_Pa_s):
    """
    Reynolds number: Re = rho * v * D / mu
    rho:    fluid density (kg/m^3)
    v:      mean velocity (m/s)
    D:      pipe inner diameter (m)
    mu:     dynamic viscosity (Pa*s = kg/m/s)
    Returns: Re (dimensionless)
    Flow regime: Re < 2100 laminar; 2100-4000 transitional; > 4000 turbulent
    """
    return rho_kg_m3 * v_m_s * D_m / mu_Pa_s

def flow_regime(Re):
    """Classify flow regime by Reynolds number."""
    if Re < 2100:
        return {"regime": "laminar", "f_formula": "f = 64/Re"}
    elif Re < 4000:
        return {"regime": "transitional", "f_formula": "Use turbulent correlation; uncertain"}
    else:
        return {"regime": "turbulent", "f_formula": "Colebrook-White or Moody chart"}

def pipe_velocity(Q_m3_s, D_m):
    """Mean velocity from volumetric flow rate: v = Q / A"""
    import math
    A = math.pi / 4.0 * D_m**2
    return Q_m3_s / A

def hydraulic_diameter(A_m2, P_wetted_m):
    """
    Hydraulic diameter for non-circular ducts: D_h = 4*A / P_wetted
    Use D_h in place of D for Re and pressure drop in non-circular channels.
    """
    return 4.0 * A_m2 / P_wetted_m

---

Module 2 — Friction Factor and Pressure Drop

import math

def friction_factor_laminar(Re):
    """Darcy friction factor for laminar flow: f = 64/Re (Darcy-Weisbach)."""
    if Re <= 0:
        return None
    return 64.0 / Re

def blasius_turbulent(Re):
    """
    Blasius correlation for smooth pipes, turbulent flow (Re 4000–100,000):
    f = 0.316 * Re^(-0.25)  (Darcy friction factor)
    """
    if Re < 4000:
        return None
    return 0.316 * Re**(-0.25)

def colebrook_white(Re, eps_over_D, tol=1e-8, max_iter=50):
    """
    Colebrook-White equation (implicit, solved iteratively):
    1/sqrt(f) = -2.0 * log10(eps/(3.7*D) + 2.51/(Re*sqrt(f)))
    Re:         Reynolds number (turbulent, > 4000)
    eps_over_D: relative roughness (eps/D; steel pipe eps ≈ 0.046 mm)
    Returns: Darcy friction factor
    """
    if Re < 4000:
        return friction_factor_laminar(Re)
    # Initial guess: Swamee-Jain explicit approximation
    eps_D = eps_over_D
    f = (0.25 / (math.log10(eps_D/3.7 + 5.74/Re**0.9))**2)
    for _ in range(max_iter):
        rhs = -2.0 * math.log10(eps_D/3.7 + 2.51/(Re * math.sqrt(f)))
        f_new = 1.0 / rhs**2
        if abs(f_new - f) < tol:
            return f_new
        f = f_new
    return f

def darcy_weisbach_dP(f, rho_kg_m3, v_m_s, L_m, D_m):
    """
    Darcy-Weisbach pressure drop:
    dP = f * (L/D) * (rho*v^2/2)   [Pa]
    f:   Darcy friction factor (NOT Fanning; Darcy = 4 * Fanning)
    Returns: pressure drop (Pa), head loss (m)
    """
    dP = f * (L_m / D_m) * 0.5 * rho_kg_m3 * v_m_s**2
    h_f = dP / (rho_kg_m3 * 9.81)  # head loss in meters
    return {"dP_Pa": round(dP, 2), "dP_psi": round(dP / 6894.76, 4),
            "head_loss_m": round(h_f, 4)}

def minor_loss_dP(K, rho_kg_m3, v_m_s):
    """
    Minor (fitting) pressure loss: dP = K * (rho*v^2/2)
    Common K values:
      Gate valve fully open: K=0.1; half open: K=2.1
      Globe valve:           K=6-10
      90-deg elbow:          K=0.9 (std); K=0.3 (long-radius)
      Tee (flow-through):    K=0.4
      Tee (branch):          K=1.5
      Sudden expansion:      K=(1-A1/A2)^2
      Sudden contraction:    K≈0.5*(1-A2/A1)
    """
    return K * 0.5 * rho_kg_m3 * v_m_s**2

def total_pressure_drop(f, rho_kg_m3, v_m_s, L_m, D_m, K_total=0.0):
    """
    Total pressure drop: pipe friction + minor losses.
    K_total: sum of all fitting K values in the pipe run.
    """
    dP_pipe = f * (L_m / D_m) * 0.5 * rho_kg_m3 * v_m_s**2
    dP_minor = K_total * 0.5 * rho_kg_m3 * v_m_s**2
    return {"dP_pipe_Pa": round(dP_pipe, 2),
            "dP_minor_Pa": round(dP_minor, 2),
            "dP_total_Pa": round(dP_pipe + dP_minor, 2),
            "dP_total_psi": round((dP_pipe + dP_minor) / 6894.76, 4)}

---

Module 3 — Bernoulli and Mechanical Energy Balance

def bernoulli_exit_velocity(P1_Pa, rho1, v1, z1_m,
                              P2_Pa, rho2, z2_m, g=9.81):
    """
    Bernoulli equation (no friction, incompressible):
    P1/rho + v1^2/2 + g*z1 = P2/rho + v2^2/2 + g*z2
    Solve for v2 (exit velocity).
    Returns: v2 (m/s) or None if energy not sufficient
    """
    rho = 0.5 * (rho1 + rho2)  # approximate; use consistent density
    v2_sq = v1**2 + 2.0*((P1_Pa - P2_Pa)/rho + g*(z1_m - z2_m))
    if v2_sq < 0:
        return None  # No solution — check inputs
    return math.sqrt(v2_sq)

def mechanical_energy_balance(P1_Pa, rho, v1, z1_m,
                                P2_Pa, v2, z2_m,
                                eta_pump=0.75, g=9.81):
    """
    Mechanical energy balance with pump work and friction loss:
    W_pump (per unit mass) = (P2-P1)/rho + (v2^2-v1^2)/2 + g*(z2-z1) + h_f
    Returns: required pump work per unit mass (J/kg), pump head (m)
    """
    delta_p_term = (P2_Pa - P1_Pa) / rho
    delta_ke = 0.5 * (v2**2 - v1**2)
    delta_pe = g * (z2_m - z1_m)
    W_shaft = delta_p_term + delta_ke + delta_pe  # net fluid work (no friction yet)
    # Include friction if h_f (head loss in m) is provided separately
    return {"W_shaft_J_kg": round(W_shaft, 2),
            "pump_head_m": round(W_shaft / g, 3),
            "note": "Add friction loss h_f (m) to pump_head_m for actual pump head required"}

def pump_power_kW(Q_m3_s, delta_P_Pa, eta=0.75):
    """
    Pump shaft power: P_shaft = Q * delta_P / eta
    Q:       volumetric flow rate (m^3/s)
    delta_P: total pressure rise across pump (Pa)
    eta:     overall pump efficiency (0.60-0.80 typical)
    Returns: shaft power (kW), fluid power (kW)
    """
    P_fluid_kW = Q_m3_s * delta_P_Pa / 1000.0
    P_shaft_kW = P_fluid_kW / eta
    return {"P_fluid_kW": round(P_fluid_kW, 3),
            "P_shaft_kW": round(P_shaft_kW, 3),
            "P_shaft_hp": round(P_shaft_kW / 0.7457, 2)}

---

Module 4 — Flow Measurement

def venturi_flow_rate(Cd, A1_m2, A2_m2, delta_P_Pa, rho_kg_m3):
    """
    Venturi meter volumetric flow rate:
    Q = Cd * A2 * sqrt(2*delta_P / (rho*(1-(A2/A1)^2)))
    Cd: discharge coefficient (0.95-0.99 for well-designed venturis)
    A1: upstream pipe area (m^2)
    A2: throat area (m^2)
    delta_P: pressure differential (Pa)
    Returns: Q (m^3/s), velocity at throat (m/s)
    """
    beta_sq = (A2_m2 / A1_m2)**2
    Q = Cd * A2_m2 * math.sqrt(2.0 * delta_P_Pa / (rho_kg_m3 * (1.0 - beta_sq)))
    v_throat = Q / A2_m2
    return {"Q_m3_s": round(Q, 6), "Q_gpm": round(Q * 15850.3, 2),
            "v_throat_m_s": round(v_throat, 3)}

def orifice_flow_rate(Cd, A_orifice_m2, delta_P_Pa, rho_kg_m3):
    """
    Orifice meter flow rate: Q = Cd * A_o * sqrt(2*delta_P/rho)
    Cd: discharge coefficient (0.60-0.65 for standard sharp-edged orifice)
    Returns: Q (m^3/s)
    """
    Q = Cd * A_orifice_m2 * math.sqrt(2.0 * delta_P_Pa / rho_kg_m3)
    return {"Q_m3_s": round(Q, 6), "Q_gpm": round(Q * 15850.3, 2)}

def rotameter_correction(Q_cal, rho_cal, rho_fluid, mu_cal, mu_fluid):
    """
    Correct rotameter reading for different fluid properties.
    Q_actual = Q_cal * sqrt(rho_cal/rho_fluid) * (mu_fluid/mu_cal)^n
    n ≈ 0 for turbulent float; n ≈ 0.5 for laminar (viscous) float
    Conservative: use n=0 (turbulent) for most process rotameters.
    """
    return Q_cal * math.sqrt(rho_cal / rho_fluid)

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Learning Resources

Textbooks

Author(s)	Title	Edition	ISBN
Bird, Stewart & Lightfoot	Transport Phenomena	2nd ed. (2007)	978-0-470-11539-8
Fox, McDonald & Pritchard	Introduction to Fluid Mechanics	9th ed. (2015)	978-1-118-01297-1
White, F.M.	Fluid Mechanics	8th ed. (2015)	978-0-07-339827-3
Munson, Okiishi, Huebsch & Rothmayer	Fundamentals of Fluid Mechanics	8th ed. (2016)	978-1-119-08070-1

Free Online Resources

Resource	URL	Notes
LearnChemE — Fluid Mechanics YouTube	youtube.com/playlist?list=PL4xAk5aclnUi83lMy9KFvYrSbqU8QopBO	Prof. Liberatore, CSM; 60+ videos ChBE 311 level
MIT OCW 1.060	ocw.mit.edu/courses/1-060-engineering-mechanics-ii	Fluid statics, pipe flow, open channel
NPTEL Fluid Mechanics	nptel.ac.in (search "Chemical Engineering Fluid Mechanics")	IIT Kharagpur; Chakraborty
HyperPhysics Fluids	hyperphysics.phy-astr.gsu.edu/hbase/fluid.html	Quick reference
Engineering Toolbox	engineeringtoolbox.com	Pipe roughness, viscosity tables

WVU Courses

• ChBE 311 Fluid Mechanics — core ChBE curriculum
• MAE 244 Fluid Dynamics — mechanical engineering perspective
• PNGE 361 Production Engineering — pipe flow applied to tubing and surface lines

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Roughness Reference

Material	Roughness eps (mm)	eps/D for 2" pipe
Drawn tubing (smooth)	0.0015	0.00003
Commercial steel	0.046	0.0009
Galvanized iron	0.15	0.003
Cast iron	0.26	0.005
Concrete	0.3–3	0.006–0.06

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Output Format

## Pipe Flow Analysis — [System Description]

### Input Summary
| Parameter | Value | Units |
|-----------|-------|-------|
| Flow rate (Q) | | m^3/s or gpm |
| Pipe ID (D) | | m or in |
| Fluid density | | kg/m^3 |
| Fluid viscosity | | Pa*s or cP |
| Pipe length | | m or ft |
| Pipe roughness (eps/D) | | dimensionless |

### Flow Regime
| Parameter | Value |
|-----------|-------|
| Mean velocity | m/s |
| Reynolds number | |
| Flow regime | Laminar / Turbulent |
| Friction factor (f, Darcy) | |
| Method | Laminar / Blasius / Colebrook-White |

### Pressure Drop
| Component | dP (Pa) | dP (psi) |
|-----------|---------|---------|
| Pipe friction | | |
| Minor losses | | |
| **Total** | | |

**Reasonableness:** Compare dP to available pump head; check v < 3 m/s for most liquids

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Error Handling

Condition	Cause	Action
Re in transition zone (2100-4000)	Slug/intermittent flow	Use turbulent correlation; note uncertainty
v > 5 m/s in liquid line	Excessive velocity, erosion risk	Increase pipe diameter
Colebrook-White not converging	Very low Re or extreme eps/D	Check inputs; use explicit Swamee-Jain
Negative v2 from Bernoulli	Insufficient energy at inlet	Check pressures and elevations; system needs pump

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Caveats

• Darcy vs. Fanning: The Darcy friction factor (f_D = 4 * f_F) is used here and in the Darcy-Weisbach equation. Many ChBE textbooks (including BSL) use the Fanning friction factor. Verify which convention your textbook uses before substituting.
• Incompressible flow assumed. For gas flow, use compressible flow equations or break into segments with average properties. Valid for liquids and low-velocity gas (Ma < 0.3).
• Colebrook-White is implicit. Must be solved iteratively. The Swamee-Jain explicit approximation (used as initial guess above) is accurate to ±2%.
• Minor loss K values are approximations. Use manufacturer data for control valves and other precision-critical fittings.