Engineering
Dittus–Boelter and Gnielinski: Pipe Nusselt Number and Heat Transfer
Compare two turbulent circular-tube correlations, validate their input ranges and compute h from unrounded Nusselt values.
State the flow assumptions before using a correlation
This example compares empirical single-phase internal-flow correlations for a smooth circular tube with developed flow. Re and Pr use consistent bulk fluid properties; conductivity k is in W/(m K), internal diameter D in metres and L/D is dimensionless. A numeric range check cannot verify smoothness, development or small property variation.
These formulas are estimates, not exact identities for every fluid or tube. Strong property changes, entrance effects, roughness, phase change and unusual geometries need a suitable model. Do not infer a laminar value solely from Re: thermal boundary conditions and development matter.
Dittus–Boelter with an explicit heating/cooling choice
E2 uses the revised coefficient 0.023 with Re^0.8 and Pr^n. B4 must be Heating (n=0.4) or Cooling (n=0.3), referring to the fluid. Unknown labels are rejected rather than silently treated as cooling. Leading/trailing ordinary spaces and letter case are normalized.
This exercise checks Re>=10000, 0.6<=Pr<=160 and L/D>=10, following the documented range of the cited implementation. Passing these checks does not independently establish the physical assumptions.
=IF(COUNT(B2:B3,B7)<>3,"Enter numeric Re, Pr and L/D",IF(OR(B2<10000,B3<0.6,B3>160,B7<10),"Outside Dittus-Boelter example range",IF(AND(LOWER(TRIM(B4))<>"heating",LOWER(TRIM(B4))<>"cooling"),"Choose Heating or Cooling",0.023*B2^0.8*B3^IF(LOWER(TRIM(B4))="heating",0.4,0.3))))Calculate h without bypassing the model checks
E3 is E2×k/D in W/(m² K). It uses the unrounded Nu and propagates any Dittus–Boelter input error. A Reynolds number outside the selected model therefore cannot leave an apparently valid coefficient displayed.
The initial Re=50000, Pr=4.5, k=0.62 and D=0.05 give heating Nu about 241.10 and h about 2989.6 W/(m² K). Cooling gives a different Nu because the exponent changes.
=IF(NOT(ISNUMBER(E2)),E2,IF(COUNT(B5:B6)<>2,"Enter numeric conductivity and diameter",IF(OR(B5<=0,B6<=0),"Require positive conductivity and diameter",E2*B5/B6)))Compare Gnielinski using a supplied Darcy friction factor
B8 contains a supplied Darcy friction factor f, not the Fanning factor. For the same convention and flow, Darcy f is four times Fanning f; mixing them changes the answer substantially. The initial 0.021 is an illustrative input, not an automatic friction prediction for every tube.
E4 calculates (f/8)(Re−1000)Pr / [1+12.7 sqrt(f/8)(Pr^(2/3)−1)]. E5 converts that Nu into h. The supplied f must be appropriate to the actual flow and geometry. This basic expression does not add entry-length or wall-property correction terms.
The worksheet uses 3000<=Re<=5000000, 0.5<Pr<=2000, L/D>=10 and 0<f<1. The latter two are explicit exercise restrictions, not universal guarantees of correlation validity. The transition region is sensitive to conditions; the worksheet does not automatically switch a laminar calculation to Gnielinski.
The Gnielinski comparison does not use the Heating/Cooling label: this version has no corresponding exponent switch. It still requires suitable fluid properties and friction data.
=IF(COUNT(B2:B3,B7:B8)<>4,"Enter numeric Re, Pr, L/D and Darcy f",IF(OR(B2<3000,B2>5000000,B3<=0.5,B3>2000,B7<10,B8<=0,B8>=1),"Outside Gnielinski example range",IF(1+12.7*SQRT(B8/8)*(B3^(2/3)-1)<=0,"Check Darcy factor and Pr",(B8/8)*(B2-1000)*B3/(1+12.7*SQRT(B8/8)*(B3^(2/3)-1)))))Explore controlled changes
With Re and Pr held fixed, doubling k doubles h and doubling D halves h. In a physical redesign, however, diameter may also change Re, so that experiment isolates only the conversion formula.
Try Re=5000: Dittus–Boelter and its h output report their range restriction, while the separate Gnielinski comparison can be evaluated under its stated assumptions. Clear f: only the Gnielinski calculation becomes incomplete.
Try the example
Use the original values or change them to test your understanding.