Shear Matters: A Practical Guide to Newtonian and Non-Newtonian Flow and Pressure Drop in Sanitary Tubing

1. Introduction to Fluid Flow in Process Engineering

In fluid mechanics, understanding how fluids behave under various flow conditions is essential for designing efficient piping systems. Whether in petrochemical plants or food processing facilities, engineers must account for viscosity, flow regimes, piping size and material, and pressure drop to ensure proper design and optimal performance.

2. Viscosity and Fluid Classification

Viscosity is a measure of fluid’s resistance to deformation or flow and is defined as a ratio of shear stress to shear rate.

$$\mu =\frac{\tau }{\gamma }$$

Where:

• $\mu$  – viscosity
• $\tau$  – shear stress
• $\gamma$  – shear rate

Fluids are broadly classified into:

• Newtonian fluids: Constant viscosity regardless of shear rate (e.g., water, air). In a piping system, the viscosity of Newtonian fluids is constant regardless of how fast or slow the fluid moves through the piping.

• Non-Newtonian fluids: Viscosity changes with shear rate (e.g., ketchup, yogurt, cheese curds). In a piping system, the viscosity of non-Newtonian fluids vary and is directly dependent on how fast or slow the fluid moves through the piping.

While classifying fluids as Newtonian or non-Newtonian provides a foundational understanding of their behavior, it is only the first step in predicting how these fluids will perform in real-world piping systems. Accurate modeling—especially for non-Newtonian fluids common in cheese and dairy, food and pharmaceutical processing—requires consideration of several key fluid dynamic parameters.

Chief among these is the Reynolds number, a dimensionless value used to determine whether flow is laminar, transitional, or turbulent. For non-Newtonian fluids, however, the traditional Reynolds number must be modified to account for variable viscosity. Additionally, the Power-Law model describes the relationship between shear stress and shear rate, capturing how the fluid responds to deformation. The shear rate at the pipe wall, in particular, plays a critical role in determining the apparent viscosity of non-Newtonian fluids under flow conditions.

By integrating these parameters, engineers can estimate pressure drops, size pumps accurately, and design piping systems that preserve product integrity and maximize process efficiency. The following sections explore these concepts in greater detail and demonstrate their application in modeling flow within sanitary tubing systems.

3. Reynolds Number and Flow Regimes

The Reynolds number (Re) is a dimensionless quantity used to predict flow patterns in a pipe:

Reynolds Number Calculation:

$$Re=\frac{\rho VD}{\mu }$$

• $Re$  – Reynolds number
• $\rho$  – density
• $V$  – velocity
• $D$  – hydraulic diameter (pipe ID)
• $\mu$  – viscosity

Flow regimes:

• Laminar: Re < 2300 – smooth, orderly flow

• Transitional: 2300 < Re < 4000 – unstable, mixed flow

• Turbulent: Re > 4000 – chaotic, eddy-dominated flow

For non-Newtonian fluids, the Reynolds number must be determined to understand the flow characteristic in the pipe for a given flowrate and pipe size.

4. The Power-Law Model for Non-Newtonian Fluids

The Power-Law model is one model commonly used to describe shear-thinning or shear-thickening fluids:

Power-Law Fluid:

$$\tau =K{\gamma }^n$$

Where:

• $\tau$  – shear stress
• $\gamma$  – shear rate
• $K$  – consistency
• $n$  – Power-Law flow index

And:

• $n<1$  – shear-thinning (pseudoplastic)
• $n=1$  – Newtonian
• $n>1$  – shear-thickening (dilatant)

5. Shear Rate in Tubing

In circular pipes, the shear rate at the wall for a Newtonian fluid is:

$$\dot{\gamma }=\frac{4Q}{\pi R^3}$$

Where:

• $Q$  – volumetric flowrate (ft³/s)
• $R$  – pipe radius in (ft)

For non-Newtonian fluids, this relationship is more complex and depends on the flow behavior index n.

6. Determining the Power Law Flow Index (n) and Apparent Viscosity using Experimental Data

In practical applications, the Power Law Flow Index, n and the consistency index, K in the Power-Law model are often determined empirically using rheological data. One common method involves measuring the apparent viscosity of a fluid at various shear rates using a viscometer or rheometer. These data points can then be plotted to analyze the relationship between shear stress and shear rate.

To estimate K and n, engineers can plot apparent viscosity vs. shear rate and then using excel can fit a second-order polynomial to this point curve:

While this polynomial does not directly yield n, it can be used to approximate the behavior of the fluid and derive a local or average value of n over the measured shear rate range. 

Using Excel to Simplify the Analysis

Microsoft Excel provides a user-friendly platform for performing this analysis:

• Input your shear rate and viscosity data into two columns and generate a scatter plot with shear rate along the x-axis and viscosity along the y-axis.  

• Create a scatter plot using the values for shear rate and viscosity.

On the scatter plot, apply a trendline.  Choose a power trendline and check the boxes by the following 2 options, Display Equation on Chart and Display R-squared value on chart.  See example scatter plot below in Fig. 1.

The displayed formula will look like the following:

$$y=K{x}^{n-1}$$

Where:

• $K$  – consistency index (Pa·s)
• $n$  – power-law flow index
  Note: the fitted exponent is n − 1. Add 1 to that slope to obtain n.
• $x$  – actual shear rate
• $y$  – apparent viscosity at that shear rate

The R² value (also called the coefficient of determination) on a scatter plot trendline indicates how well the trendline fits the data.   

• Range: It ranges from 0 to 1.

• Interpretation:

• R² = 1: The trendline perfectly fits the data — all points lie exactly on the line.

• R² = 0: The trendline does not explain any of the variability in the data.

• Higher R²: Indicates a better fit — the model explains more of the variation in the data.

As noted previously, this method allows process engineers to characterize non-Newtonian fluid flow behaviors, predict that flow behavior and estimate the dynamic viscosity across a range of flows for a given piping system.  Understanding this apparent viscosity at various flowrates in a piping system allows engineers to accurately estimate the pressure drop on non-Newtonian fluids in piping systems as further described in the subsequent section 7. 

7. Pressure Drop in Sanitary Tubing

Newtonian Fluids: Pressure drop is a critical factor in designing hygienic piping systems. For Newtonian fluids, the widely accepted Darcy-Weisbach is used:

∆P=f*LD*v22

$$\Delta P=f*\frac{L}{D}*\frac{\rho v^2}{2}$$

Where:

• $\Delta P$  – pressure drop (Pa)
• $h_f$  – head loss (m)
• $f$  – Darcy friction factor (dimensionless)
• $L$  – length of the pipe (m)
• $D$  – hydraulic diameter of the pipe (m)
• $\rho$  – fluid density (kg/m³)
• $v$  – average flow velocity (m/s)

It is probably worth noting the following:  

• The friction factor f is the same dimensionless value used in SI units and is determined based on the Reynolds number and relative roughness.

• For laminar flow (Re<2100Re<2100):

$$f=\frac{64}{Re}$$

• For turbulent flow, f is typically found using the Moody chart.

Non-Newtonian fluids:  The pressure drop calculations, for non-Newtonian fluids must incorporate the Power-Law parameters like the Power Law Flow index, n and Consistency Index, K as discussed above. 

The following equation is widely used and accepted for streamline laminar flow conditions for non-Newtonian fluids in piping systems.  

$$\Delta P=\frac{Q*v*L}{\mathrm{219,500}*D^4}$$

Where:

• $\Delta P$  – pressure drop (psi)
• $v$  – apparent viscosity (centipoise) at flow Q and pipe D
• $Q$  – flow rate (gallons per hour)
• $L$  – length of pipe (ft)
• $D$  – inside diameter of pipe (in)

8. Conclusion 

Accurately modeling the flow dynamics of non-Newtonian fluids in piping systems presents significant complexities. A solid understanding of Newtonian and non-Newtonian fluid behavior, laminar and turbulent flow regimes, shear effects on viscosity, and the Power-Law model for predicting pressure drops is essential for:

• Ensuring consistent and predictable flow rates and pressures

• Designing efficient and hygienic process piping systems

• Optimizing pump sizing and piping layouts

• Minimizing product degradation

• Reducing energy consumption

ESC Process Solutions, a division of ESC, Inc., is a leader in this specialized field. With over 30 years of experience in process engineering across the pharmaceutical, food, and dairy industries, we bring deep expertise in handling both Newtonian and non-Newtonian fluids.

Leveraging this experience, ESC has developed proprietary modeling tools specifically designed for both Newtonian and non-Newtonian fluid systems. These tools enable us to accurately predict and model complex piping networks, ensuring optimal flow conditions, precise pressure profiles, and compliance with stringent sanitary standards.

By combining deep industry knowledge with innovative engineering solutions, ESC Process Solutions delivers high-performance, customized process designs. Our commitment to understanding each client’s needs and product characteristics ensures that every system is engineered for both efficiency and integrity—hallmarks of a true industry expert.