How To Determine Shell-Side Heat Transfer Coefficient (Hs) In Heat Exchanger Experiments

The step-by-step methodology starts with accurately defining the shell-side flow geometry and ends with applying a correlation that ties fluid properties and flow regime to the heat transfer coefficient. To determine the shell-side coefficient ( h_s ) in a unit operations experiment, you will calculate the cross-flow area, compute the mass velocity and Reynolds number, find the heat transfer factor ( j_h ) from a standard curve, and then solve the Nusselt number equation using the fluid’s Prandtl number and a wall‑viscosity correction. The procedure is systematic, but its accuracy hinges on correct input measurements and an understanding of the underlying assumptions.

The core of the method is the Nusselt number relation ( Nu = j_h Re Pr^{0.33} (\mu/\mu_w)^{0.14} ). You first build the Reynolds number from geometry and flow, then look up ( j_h ) for your baffle cut, and finally back‑calculate ( h_s ). Every step relies on consistent units and a clear idea of the fluid properties at the bulk and wall temperatures.

The Complete 6‑Step Methodology

1. Calculate the Shell‑Side Cross‑Flow Area (( A_s ))

The shell’s free area for cross flow depends on tube layout and baffle geometry. For a typical segmental baffle, the area at the shell centerline is:

( A_s = \frac{(p_t - d_o) D_s l_B}{p_t} )

( p_t ) = tube pitch (center‑to‑center distance).
( d_o ) = tube outer diameter.
( D_s ) = shell inner diameter.
( l_B ) = baffle spacing (distance between consecutive baffles).

This formula assumes the flow is normal to the tube bank at the centerline and that the baffle window area does not dominate. Measure or verify these dimensions directly from the exchanger’s engineering drawings or by direct measurement in the pilot plant.

2. Determine Mass Velocity (( G_s )) and Linear Velocity (( u_s ))

With the cross‑flow area known, the shell‑side mass velocity is:

( G_s = \frac{W_s}{A_s} )

where ( W_s ) is the mass flow rate of the shell‑side fluid (kg/s or lb/h). The linear velocity, useful for physical intuition, is then ( u_s = G_s / \rho ), where ( \rho ) is the fluid density evaluated at the bulk mean temperature.

Always use average bulk temperature (arithmetic average of inlet and outlet) to evaluate fluid properties unless a more rigorous method is specified.

3. Compute the Equivalent Diameter (( d_e ))

The equivalent diameter represents the characteristic length for flow across the tube bundle. The formula depends on the tube pitch layout:

For square pitch:
( d_e = \frac{4(p_t^2 - \pi d_o^2/4)}{\pi d_o} )
For triangular pitch:
( d_e = \frac{4\left(\frac{\sqrt{3}}{2}p_t^2 - \frac{\pi d_o^2}{4}\right)}{\pi d_o} )

These expressions ensure that the Reynolds number uses a length scale representative of the flow passages between the tubes.

4. Calculate the Shell‑Side Reynolds Number (( Re ))

The Reynolds number quantifies the flow regime:

( Re = \frac{G_s d_e}{\mu} )

(\mu) is the dynamic viscosity of the shell‑side fluid at bulk temperature. A Reynolds number below 100 typically indicates laminar flow, while values above 10⁴ fall in fully turbulent territory. The ( j_h ) factor curve will reflect the transition.

5. Find the Heat Transfer Factor (( j_h ))

( j_h ) is an empirical factor that accounts for the heat transfer characteristics of the tube bundle for a given Reynolds number and baffle cut. To use it:

Identify the baffle cut (e.g., 25%, 35% of shell diameter). This defines which curve to use on the standard Kern‑type correlation chart.
Read ( j_h ) from the curve at the calculated ( Re ).

If you are performing the experiment with a digital data‑acquisition system, you may interpolate tabulated values. If the lab uses a printed chart, careful reading is essential—small errors in ( j_h ) directly translate to errors in ( h_s ).

6. Calculate the Heat Transfer Coefficient (( h_s )) from the Nusselt Number

The final step solves the Nusselt relation for ( h_s ):

( Nu = \frac{h_s d_e}{k_f} = j_h Re Pr^{0.33} \left( \frac{\mu}{\mu_w} \right)^{0.14} )

Rearrange to extract ( h_s ):

( h_s = \frac{k_f}{d_e} j_h Re Pr^{0.33} \left( \frac{\mu}{\mu_w} \right)^{0.14} )

where:

( k_f ) = thermal conductivity of the fluid at bulk temperature,
( Pr ) = Prandtl number (( c_p \mu / k_f )) at bulk temperature,
( \mu_w ) = viscosity evaluated at the tube wall temperature.

The wall temperature correction accounts for the difference between bulk and surface viscosity, which can be significant for viscous fluids. In practice, you may need to iterate: assume ( \mu_w \approx \mu ) initially, get an estimate for ( h_s ), compute a wall temperature via heat balance, then re‑evaluate ( \mu_w ) until convergence.

Understanding the Underlying Principles

Why the Equivalent Diameter Matters for Shell‑Side Flow

Unlike tube‑side flow where the inner diameter is the obvious choice, the shell side presents a complex arrangement. The equivalent diameter converts the actual flow geometry into a circle that would give the same pressure‑drop and heat transfer behavior for a given hydraulic radius. Using the wrong layout formula will skew the Reynolds number and make your ( h_s ) unreliable.

The Role of the ( j_h ) Factor

The ( j_h ) factor is a testament to the empirical nature of shell‑side correlations. It absorbs the effects of baffle cut, tube layout, and bypass streams that theoretical models rarely capture precisely. The standard curves are based on idealized tube bundles with no leakage. In a real heat exchanger, construction clearances and bypass streams reduce the effective heat transfer coefficient. Therefore, always treat the calculated ( h_s ) as an ideal value that a Wilson plot or direct measurement of overall coefficient may refine.

The Viscosity Correction: A Double‑Edged Sword

The term ( (\mu / \mu_w)^{0.14} ) is a semi‑empirical correction. For heating of a cold fluid, the wall is hotter, so ( \mu_w < \mu ), and the factor is greater than 1, increasing ( h_s ). For cooling, the effect is reversed. The challenge lies in accurately estimating the wall temperature. In student experiments, a common simplification is to assume the wall temperature equals the average of the two bulk fluid temperatures, which is an acceptable starting point but introduces small systematic errors for fluids with strongly temperature‑dependent viscosity.

Understanding the Trade‑offs

Precision vs. Practical Simplicity

The six‑step method strikes a balance between theoretical rigor and practical lab execution. The largest sources of uncertainty are:

Measuring mass flow rate ( W_s ): A small error in a rotameter or turbine meter reading cascades through ( G_s ) and ( Re ).
Reading the ( j_h ) chart: The logarithmic scales and curve families require careful interpolation.
Wall temperature estimation: Without direct thermocouples on the tube wall, the correction remains an approximation.

If your goal is to demonstrate the educational principle, the standard methodology is robust. For a design‑grade number, you would complement this with an experimental Wilson plot or computational fluid dynamics (CFD).

When to Iterate and When to Simplify

For low‑viscosity fluids like water, the viscosity correction is close to unity and can often be neglected without significant error. However, if the lab uses an organic solvent or light oil, the correction becomes meaningful. In such cases, a single iteration using a simple heat balance to estimate the wall temperature is usually sufficient.

Limitations of the Kern Method

The Kern correlation, on which this methodology is based, works well for typical shell‑side conditions but does not explicitly account for bypass streams between the tube bundle and the shell, or for the effects of baffle‑tube leakage. Consequently, experimentally determined overall ( U ) values may be lower than what you would back‑calculate from ( h_s ) and the tube‑side coefficient alone. This gap illustrates the importance of the fouling factors and additional resistances described in the supplementary references.

Common Pitfalls to Avoid

Mixing Unit Systems

The correlation is not unit agnostic. Ensure ( G_s ) (kg/m²·s) and ( \mu ) (Pa·s) yield a dimensionless ( Re ), and that ( k_f ) (W/m·K) and ( d_e ) (m) produce ( h_s ) in W/m²·K. An all‑too‑common student error is using diameter in millimeters while thermal conductivity is in W/m·K.

Evaluating Properties at the Wrong Temperature

Property evaluation at the bulk mean versus the film temperature changes the Prandtl number and viscosity. The method outlined uses bulk temperature; do not substitute film temperature unless you are working with a correlation that specifically calls for it (e.g., a condensation correlation).

Ignoring the Baffle Cut Influence

If the exchanger has a non‑standard baffle cut, blindly using the 25% cut curve will distort ( h_s ). Always confirm the actual baffle configuration before selecting the appropriate ( j_h ) curve.

Making the Right Choice for Your Experiment

The step‑by‑step methodology is flexible. Choose your depth based on the experimental goal:

If your primary focus is a rapid educational demonstration of shell‑side heat transfer: Use the six steps as given, assume a wall viscosity correction factor of 1.0, and read ( j_h ) from the 25% baffle cut curve if your exchanger matches that standard. This will give a good first‑pass ( h_s ) for water‑water or similar low‑viscosity systems.
If your primary focus is achieving the highest accuracy for an unknown viscous fluid: Include the full wall‑viscosity iteration. Use a precise digital lookup table for ( j_h ) instead of a printed chart, and take extra care in measuring the mass flow rate and inlet/outlet temperatures to minimize propagation of error.
If your primary focus is linking the individual coefficient to the overall heat transfer performance: Combine this method with the tube‑side ( h_i ) calculation and the wall conduction resistance to compute ( U_o ). Then compare with the experimentally determined ( U ) from the overall heat balance and LMTD. The discrepancy will vividly demonstrate the role of fouling factors and bypass streams.

By following a clear, step‑wise approach and recognizing its inherent assumptions, you can extract meaningful shell‑side film coefficients and deepen your understanding of heat exchanger behavior.

Summary Table:

Step	Action	Key Equation / Parameter
1	Calculate Cross-Flow Area ($A_s$)	$A_s = \frac{(p_t - d_o) D_s l_B}{p_t}$
2	Determine Mass Velocity ($G_s$)	$G_s = \frac{W_s}{A_s}$
3	Compute Equivalent Diameter ($d_e$)	Square or Triangular Pitch geometry
4	Calculate Reynolds Number ($Re$)	$Re = \frac{G_s d_e}{\mu}$
5	Find Heat Transfer Factor ($j_h$)	Empirical curves based on baffle cut and $Re$
6	Calculate Coefficient ($h_s$)	$h_s = \frac{k_f}{d_e} j_h Re Pr^{0.33} (\mu/\mu_w)^{0.14}$

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How to Determine Shell-Side Heat Transfer Coefficient (hs) in Heat Exchanger Experiments