Convection Heat Transfer Calculator
Work out the convection coefficient over a flat plate from the flow speed, the fluid properties and the surface temperature.
What it does
Put a cold can in a freezer and it is the air moving past it that takes the heat away, not the metal. Convection is the name for that, and working out how much heat a moving fluid can carry is one of the most common calculations in engineering.
The engineering text behind this page starts with Newton's law of cooling, which says that the heat flux from a surface to the fluid is the convection coefficient times the difference between the surface temperature and the temperature of the fluid away from it. The convection coefficient is the number that says how well that particular fluid and that particular flow can carry heat away, and it is not a property of the fluid alone.
The way to get it is through the Nusselt number, which is the convection coefficient times the length of the surface divided by the thermal conductivity of the fluid. The University of Waterloo lecture notes behind this page give the correlations for flow over a flat plate: in the laminar range the local Nusselt number is 0.332 times the square root of the Reynolds number times the cube root of the Prandtl number, and the average over the whole plate is 0.664 with the same shape. In the turbulent range the average is 0.037 with the Reynolds number to the 0.8 power.
This page does that calculation for a plate in a stream of fluid. Give it the plate length, the flow speed, the density, viscosity and thermal conductivity of the fluid, the Prandtl number, the two temperatures and the area, and it returns the Reynolds number, the Nusselt number, the convection coefficient, the heat flux and the heat rate.
The limits are worth stating. The correlations are for an isothermal flat plate in a steady stream, and they assume the fluid properties do not change much across the boundary layer. The transition region between laminar and turbulent flow is smeared out over a range of Reynolds numbers rather than switching at one value, and the average correlations are for a plate that starts laminar at the leading edge.
How to use
- Choose metric or imperial for the lengths, the flow speed, the fluid properties and the answer.
- Enter the length of the plate measured along the direction of the flow, which is the characteristic length for this correlation.
- Enter the flow speed and the density and viscosity of the fluid, then its thermal conductivity and the Prandtl number.
- Check the Reynolds number. Below about 500,000 the plate is treated as laminar all the way, and above it the turbulent correlation is used.
- Enter the surface temperature and the temperature of the fluid away from the surface, and the area you care about.
- Read the convection coefficient, which is the number to use in a heat balance, and the heat flux and heat rate for the area.
Formula
- Re
- Reynolds number, the ratio of the inertia of the flow to its viscosity, dimensionless.
- Nu
- Nusselt number, the average convection coefficient times the length divided by the thermal conductivity.
- Pr
- Prandtl number, the ratio of the momentum diffusivity to the thermal diffusivity of the fluid.
- h
- Average convection coefficient, in watts per square metre per kelvin.
- k
- Thermal conductivity of the fluid, in watts per metre per kelvin.
The heat flux is the convection coefficient times the difference between the surface temperature and the fluid temperature, and the heat rate over an area is that flux times the area. The correlations are the average values over the whole plate, so they already include the growth of the boundary layer along the plate.
Worked example
Example: water at 20 degrees C flowing at 0.1 m per s over a 1 m plate
Defaults: metric units, a 1 m plate, a flow speed of 0.1 m per s, a density of 998 kg per m3, a viscosity of 0.001 Pa s, a conductivity of 0.6 W per m K, a Prandtl number of 7, a surface at 60 degrees C, a fluid at 20 degrees C and an area of 1 m2.
- The Reynolds number is 998 x 0.1 x 1 / 0.001 = 99,800.0, which is below 500,000, so the plate is treated as laminar over its whole length.
- The average Nusselt number is 0.664 x the square root of 99,800 times the cube root of 7, which is 0.664 x 315.92 x 1.9129 = 401.27.
- The convection coefficient is the Nusselt number times the thermal conductivity divided by the length: 401.27 x 0.6 / 1 = 240.76 W per m2 per K.
- The temperature difference between the surface and the fluid is 60 - 20 = 40 K.
- The heat flux is 240.76 x 40 = 9,630.39 W per m2.
- Over an area of 1 m2 the heat rate is the same number, 9,630.39 W.
- The tool shows 99,800.0, 401.27, 240.76 W per m2 per K, 9,630.39 W per m2 and 9,630.39 W.
Reference table
| Flow speed | Reynolds number | Nusselt number | Convection coefficient |
|---|---|---|---|
| 0.05 m per s | 49,900.0 | 283.74 | 170.24 W per m2 per K |
| 0.1 m per s | 99,800.0 | 401.27 | 240.76 W per m2 per K |
| 0.5 m per s | 499,000.0 | 897.26 | 538.36 W per m2 per K |
| 1 m per s | 998,000.0 | 4,458.67 | 2,675.20 W per m2 per K |
Limits
- The correlations are for a flat plate with a uniform surface temperature in a steady stream, and real surfaces are rarely that tidy.
- The switch between the laminar and the turbulent formula is at a Reynolds number of 500,000, but the transition is gradual and the true answer in the transition region sits between the two.
- The fluid properties are taken as constant, when in reality they change across the boundary layer, and they should really be evaluated at the film temperature.
- Natural convection, a phase change on the surface and any roughness on the plate all need different correlations.
Common mistakes
- Using the plate width as the characteristic length. For flow over a flat plate it is the length along the flow.
- Forgetting the cube root of the Prandtl number, which is a large factor for liquids and a small one for gases.
- Using the local correlation, 0.332, when you want the average over the plate, which is 0.664 and twice as big.
- Mixing up the units of the convection coefficient. It is watts per square metre per kelvin, and so is the answer you get in BTU per hour per square foot per degree F.
- Applying the laminar formula above the transition Reynolds number, which understates the heat transfer badly.
Try it yourself
Quick checks. Answers sit tucked under each question.
What happens if the flow speed is doubled to 0.2 m per s?
The Reynolds number doubles to 199,600.0 and, in the laminar range, the Nusselt number rises with the square root of it, so the convection coefficient rises by the square root of 2 to 340.49 W per m2 per K and the heat flux rises to 13,619.43 W per m2.
How much heat does a 1 m2 plate pull out of the water at this flow speed?
9,630.39 W with the surface 40 K above the fluid. That is the rate at which the water has to be cooled to hold the surface temperature, and it is the number that sizes a cooling coil or a radiator.
What changes if the flow turns turbulent?
The correlation changes from a square root of the Reynolds number to a 0.8 power, so the coefficient climbs faster. At 1 m per s the Reynolds number is 998,000.0, the Nusselt number is 4,458.67 and the convection coefficient is 2,675.20 W per m2 per K, well above the 240.76 of the laminar case at 0.1 m per s.
Questions
What is the Nusselt number?
The ratio of the convection heat transfer to the conduction heat transfer over the same area and temperature difference. The MIT lecture notes behind this page give it as h L over k, and the University of Toronto teaching notes describe it as a ratio that shows how strongly convection dominates in the flow.
Why does the flow speed matter so much?
Because the Reynolds number depends on it, and so does the Nusselt number. In the laminar range the coefficient grows with the square root of the speed and in the turbulent range with the speed to the 0.8 power, so moving air faster always moves more heat.
What is the Prandtl number?
The ratio of the momentum diffusivity to the thermal diffusivity of the fluid. It is about 7 for water and about 0.71 for air, and it appears in the correlation as a cube root, so it matters more for liquids than for gases.
What is the critical Reynolds number?
About 500,000 for a flat plate, which is where the flow changes from laminar to turbulent. The University of Waterloo notes use the same figure for the laminar and the turbulent formulations.
Does the surface temperature change the coefficient?
Not in these correlations, but it changes the fluid properties, so it changes the answer indirectly. A hot surface changes the viscosity and the density of the fluid next to it, and in a careful calculation the properties are taken at the film temperature, the mean of the surface and the fluid.
How is this used in a heat balance?
The convection coefficient is the number you need on the surface side of a balance. The Engineering LibreTexts chapter behind this page writes the convection resistance as one over h A, and it notes that finding the right coefficient is one of the main practical challenges of doing heat transfer work.
Sources
- University of Waterloo ECE 309 lecture notes: convection heat transfer over flat plates
- Engineering LibreTexts, 7.7: heat transfer revisited, with Newton's law of cooling and the convection coefficient