Fluid Mechanics Calculator

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)

Learning Resources

Textbooks

Free Online Resources

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

Roughness Reference

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

Error Handling

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.