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A deposit inside a heater passage can change more than the wall-to-fluid thermal resistance. By narrowing the flow path it can also reduce flow or increase the pressure needed to maintain it. Which response occurs depends on the hydraulic boundary. Compare pressure and flow together before attributing a changed heater temperature entirely to the deposit's conductivity or assuming an unchanged total flow protects every branch.
Key design decisions
- Define the remaining fluid radius and the length over which narrowing occurs.
- Distinguish pressure maintained across the affected passage from total flow maintained by the system.
- Measure branch distribution when parallel paths can hide local flow loss behind an unchanged total reading.
Identify the fluid passage affected by the deposit
Use a section view that identifies the wetted wall, deposit, remaining opening and the pressure-measurement locations. The simple model below describes a long straight circular passage with a uniform deposit around its circumference. A coating on an open plate, a short entrance obstruction and a deposit island in a wide chamber are different hydraulic geometries.
Keep the electrical heater construction separate from the fluid passage description. A thick-film element can heat through a wall without being the wetted material. The deposit radius in this calculation belongs to the flow opening, not the resistor artwork or an assumed ceramic hole. Confirm the actual assembly before translating an observed surface layer into a hydraulic change.
Use the fourth-power radius dependence within its limits
For steady, fully developed laminar flow of a Newtonian incompressible fluid through a straight circular passage, hydraulic resistance is 8 mu L divided by pi a to the fourth power. Here mu is dynamic viscosity, L the passage length and a the fluid radius. Pressure drop equals this resistance multiplied by volumetric flow.
The formula assumes a no-slip wall and approximately uniform relevant fluid properties. Entrance effects, abrupt constrictions, turbulence, boiling, gas pockets and non-Newtonian behavior require additional treatment. A real deposit can create several of these conditions, so the calculation is a bounded sensitivity model rather than a universal prediction for any dirty heater.
Rh = 8 mu L/(pi a^4); delta p = Rh Qv; Rh,fouled/Rh,clean = [a0/(a0 - t)]^4
- Rh is hydraulic resistance in Pa s/m³; mu is dynamic viscosity in Pa s.
- L and a are passage length and remaining fluid radius in metres.
- Qv is volumetric flow in m³/s; t is uniform radial deposit thickness; a0 is clean radius.
The compared passage retains the same length,fluid viscosity,laminar fully developed flow regime and circular uniform section. Deposit thickness is less than the clean radius.
Calculate pressure and flow alternatives from one narrowing
Assume a clean radius of 1.0 mm and a uniform radial deposit thickness of 0.1 mm, leaving 0.9 mm. The area remains 81 percent of the original, but laminar hydraulic resistance rises by 1/0.9 to the fourth power, approximately 1.5242. Area loss alone therefore does not describe the flow effect.
If pressure difference across the passage remains fixed, flow falls to 0.9 to the fourth power, or 65.61 percent of the clean value. If flow is held fixed, the passage pressure drop must instead rise to 152.42 percent. These are alternative controlled boundaries. They must not be combined into one operating result as though pressure and flow both remain unchanged.
| Quantity | Pressure across passage held fixed | Flow through passage held fixed |
|---|---|---|
| Remaining cross-sectional area | 81% of clean | 81% of clean |
| Hydraulic resistance | 152.42% of clean | 152.42% of clean |
| Volumetric flow | 65.61% of clean | 100% of clean |
| Passage pressure drop | 100% of clean | 152.42% of clean |
| Mean fluid velocity | 81% of clean | 123.46% of clean |
Retain viscosity, pressure locations and flow regime
For an assumed length of 0.20 m, viscosity of 0.001 Pa s and clean flow of 1.0 mL/s, the clean 1.0 mm-radius passage needs approximately 509.3 Pa pressure drop under the model. Maintaining that same flow after narrowing to 0.9 mm requires approximately 776.2 Pa. The values describe the straight passage alone, excluding fittings, valves and other circuit losses.
With an assumed density of 1,000 kg/m³, the clean mean velocity is approximately 0.318 m/s and Reynolds number is about 637. At fixed flow after narrowing, Reynolds number rises to about 707; at fixed passage pressure it falls to about 464. These checks support examining a laminar model for this numerical scenario but do not establish that a real rough, developing or obstructed passage meets every assumption.
Check branch flow even when the total is regulated
Consider two otherwise equal parallel passages sharing the same pressure difference, with only one narrowed as above. Its flow is 0.6561 times the clean branch's flow. If a controller maintains total flow at 2.0 mL/s, the clean branch carries approximately 1.208 mL/s and the narrowed branch approximately 0.792 mL/s. The total instrument still reports the intended two millilitres per second.
The controller raises the common pressure difference enough to recover total flow, but it does not restore equal distribution. A branch-local heater can consequently lose cooling while the clean branch receives more than before. A total flow switch or meter addresses the combined stream; branch protection and observability need the actual hydraulic architecture and thermal load.
Separate useful heat rate from a hotter outlet
For single-phase steady heating, useful fluid heat is mass flow times specific heat times temperature rise. Suppose 8 W enters a fluid with density 1,000 kg/m³ and assumed specific heat 4,000 J/(kg K). At 1.0 mL/s the fluid rise is 2 K. If fixed passage pressure reduces flow to 0.6561 mL/s while the same 8 W still reaches the fluid, the rise becomes approximately 3.048 K.
A hotter outlet in that comparison does not demonstrate improved heating throughput: the assumed useful heat rate remains 8 W while less fluid is processed. In the two-branch fixed-total-flow example, equal 8 W loads give different branch rises, approximately 1.656 K in the clean branch and 2.524 K in the narrowed branch. Complete mixing produces a 2 K combined rise and can conceal that difference. None of these fluid balances determines the local wall temperature.
Measure the hydraulic change before inferring deposit thickness
Record pressure on both sides of the defined passage, flow and fluid temperature during the comparison. Viscosity changes can alter the pressure-flow relationship without any geometric narrowing. A partially closed valve, trapped gas or changed external hose can also move the operating point. A higher system pressure drop alone does not identify the deposit location.
Do not invert one resistance ratio into a claimed layer thickness unless the geometry, viscosity and flow assumptions are independently justified. Patchy deposits and short restrictions can produce the same measured resistance through different mechanisms. Use inspection or an appropriate geometry measurement to connect the hydraulic result to the physical deposit rather than reporting a precise thickness inferred from an unverified ideal tube.
Keep the hydraulic and thermal conclusions together
Review the actual pump or pressure source characteristic, control range and pressure limits with the system owner. A constant-flow assumption ceases to apply when the source can no longer supply the required pressure. Do not compensate an unexplained restriction by raising pressure or heater input outside the agreed equipment envelope.
The completed comparison should retain deposit location, passage geometry, fluid state, pressure-flow data and individual thermal observations. Use the separate deposit thermal-resistance assessment to examine wall temperature and local heat transfer. Together the two analyses distinguish narrowed flow from an insulating surface layer without assigning an unsupported fouling rate, cleaning interval or deposit-resistant product claim.
Provide the deposited passage and hydraulic operating boundary
Send the actual fluid geometry and matched pressure-flow measurements before selecting a scale-related correction.
- Fluid-passage section, length, clean dimensions, deposit location and independently measured remaining openings.
- Fluid identity, temperature-dependent viscosity/density information and single-phase operating conditions.
- Pressure tap locations, clean/fouled flow readings and the pump or pressure-control characteristic.
- Parallel-path arrangement, branch flow visibility, heater power allocation and inlet/outlet temperature records.
- Equipment pressure limits, thermal protection, inspection evidence and the intended comparison conditions.
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