# Natural Convection Heat Transfer Coefficient Calculator Spreadsheet

## Where to Find a Natural Convection Heat Transfer Coefficient Calculator Spreadsheet

Convection heat transfer takes place between a solid surface and fluid that is at a different temperature and is in contact with the surface.  If the fluid is flowing past the surface due to an external driving force like a fan or pump, then the heat transfer is called forced convection.  When  fluid motion is due to density differences within the fluid (caused by temperature variation), then the heat transfer is called natural convection or free convection.

## Newton’s Law of Cooling for Natural Convection Heat Transfer Coefficient Calculator

Newton’s Law of Cooling [ Q = hA(Ts – Tf) ] is a simple expression used for the rate of convective heat transfer with either forced or natural convection.  The parameters in Newton’s Law of Cooling are:

• Q, the rate of forced convection heat transfer (Btu/hr – U.S. or W – S.I.)
• Ts, the solid temperature (oF – U.S. or oC – S.I.)
• Tf, the fluid temperature (oF – U.S. or oC – S.I.)
• A, the area of the surface that is in contact with the fluid (ft2 – U.S. or m2 – S.I.)
• h, the convective heat transfer coefficient (Btu/hr-ft2oF – U.S. or W/m2-K – S.I.)

## Dimensionless Nusselt, Rayleigh, Grashof, and Prandtl Numbers

A natural convection heat transfer coefficient calculator typically makes estimations using correlations of dimensionless numbers, specifically correlations of Nusselt number (Nu) with Prandtl number (Pr), Grashof number (Gr), and/or Rayleigh number (Ra), where Ra = GrPr.  The Nusselt, Grashof and Prandtl numbers are defined in the box at the left.

### Following is a list of the parameters that appear in these dimensionless numbers, with units are given for both the U.S engineering system and S.I. system of units:

• D, a characteristic length parameter (e.g. diameter for natural convection from a circular cylinder or a sphere or height of a vertical plate)  (ft for U.S.,  m for S.I.)
• ρ, the density of the fluid  (slugs/ft3 for U.S.,  Kg/m3 for S.I.)
• μ, the viscosity of the fluid  (lb-sec/ft2 for U.S.,  N-s/m2 for S.I.)
• k, the thermal conductivity of the fluid  (Btu/hr-ft-oF for U.S.,  W/m-K for S.I.)
• Cp, the heat capacity of the fluid  (Btu/lb-oF for U.S.,  J/kg-K for S.I.)
• g, the acceleration due to gravity (32.17 ft/sec2 for U.S.,  9.81 m/s2 for S.I.)
• β, the coefficient of volume expansion of the fluid  ( oR for U.S.,  K for S.I.)
• ΔT, the temperature difference between the solid surface and the fluid  ( oF for U.S., oC or K for S.I.)

The following sections provide equations for estimating the heat transfer coefficient for several common natural convection configurations.

## Natural Convection Heat Transfer Calculator for a Vertical Plane

The box at the right shows two correlations for convection heat transfer between a vertical plane and a fluid of different temperature in contact with it.  The first can be used for all values of Rayleigh number and the second is only for laminar flow, indicated by Ra < 109.  The screenshot image below shows an example of an Excel spreadsheet to use as a natural convection heat transfer coefficient calculator for a vertical plate using the two equations shown here.

## An Excel Spreadsheet as a Natural Convection Heat Transfer Calculator

For low cost, easy to use Excel spreadsheet packages to use as a natural convection heat transfer coefficient calculator for natural convection from a vertical plane, a horizontal plane, an inclined plane, a horizontal cylinder or a sphere in either U.S. or S.I. units (for only \$16.95),  click here to visit our spreadsheet store.

References

1. Incropera, F.P., DeWitt, D.P, Bergman, T.L., & Lavine, A.S., Fundamentals of Heat and Mass Transfer, 6th Ed., Hoboken, NJ, John Wiley & Sons, (2007).

2. Lienhard, J.H, IV and Lienhard, J.H. V, A Heat Transfer Textbook: A Free Electronic Textbook

3. Bengtson, Harlan HFundamentals of Heat Transfer, an online continuing education course for engineering PDH credit

4. Bengtson, Harlan H., Convection Heat Transfer Coefficient Estimation, an online continuing education course for PDH credit.

# An Excel Spreadsheet as a Rectangular Weir Flow Calculator

## Where to Find a Rectangular Weir Flow Calculator Spreadsheet

The following section, which gives background on sharp crested rectangular weirs in general, also appears in the companion article, “Suppressed Rectangular Weir Calculations with an Excel Spreadsheet

Background on Sharp Crested Rectangular Weirs in General

The picture at the left shows a rectangular weir measuring open channel flow rate in a natural channel.  The diagram below right shows a longitudinal cross-section of a sharp crested weir, with some of the terminology and parameters often used for sharp crested weirs included on the diagram.

The weir crest is the top of the weir. For a rectangular weir it is the straight, levelbottom of the rectangular opening through which water flows over the weir. The term nappe is used for the sheet of water flowing over the weir. The equations for calculating flow rate over a weir in this article require free flow, which takes place when there is air under the nappe. The drawdown is shown in the diagram as the decrease in water level going over the weir due to the acceleration of the water.  The head over the weir is shown as H in the diagram; the height of the weir crest is shown as P; and the open channel flow rate in the open channel (and over the weir) is shown as Q.

Image Credits:  Rectangular, Sharp-Crested Weir: flowmeterdirectory.co.uk

Sharp Crested Weir Parameters:  H. H. Bengtson, Ref #2

## The Francis Equation for a Rectangular Weir Flow Calculator

A contracted rectangular weir is one for which the weir extends across only part of the channel, so that the length of the weir, L, is different from as the width of the channel.  The picture at the left shows a contracteded rectangular weir being used to measure the flow of water in a triangular open channel.  The diagram below right shows some of the key parameters used in contracted rectangular weir flow rate calculations. Specifically, the height of the weir crest, P, the head over the weir, H, the weir length, L, and the channel width, B, are shown on the diagram of a contracted rectangular weir in a rectangular channel.  The U.S. Bureau of Reclamation, in their Water Measurement Manual (Ref #1 below), recommend the use of the Francis equation (shown below) for completely contracted rectangular weirs, subject to the condition that  H/L < 0.33,  B – L > 4 Hmax,  and > 2Hmax.

For U.S. units:  Q = 3.33(L – 0.2H)H3/2,  where

• Q is the water flow rate in ft3/sec,
• L is the length of the weir in ft,
• H is the head over the weir in ft,
• B is the width of the channel in ft, and
• Hmax is the maximum expected head over the weir in ft.

For S.I. units:  Q = 1.84(L – 0.2H)H3/2, where

• Q is the water flow rate in m3/sec,
• L is the length of the weir in m, and
• H is the head over the weir in m.
• B is the width of the channel in m, and
• Hmax is the maximum expected head over the weir in m.

Image Credits:  Contracted Rectangular Weir picture: Food and Agricultural Organization of the United Nations.

Contracted Rectangular Weir Diagram – Bengtson, Harlan H.

## The Kindsvater-Carter Formula for a Rectangular Weir Flow Calculator

If any of the three required conditions given in the previous section are not met, then the more general Kindsvater- Carter Equation, shown below should be used.

U.S. units: Q  =  Ce(2/3)[(2g)1/2](L + kb)(H + 0.003)3/2

S.I. units: Q  =  Ce(2/3)[(2g)1/2](L + kb)(H + 0.001)3/2

Ce is a function of L/B and H/P, while  kb is a function of L/B.  There are graphs, tables and equations available for obtaining values for Ce and kb for specified values of L/B and H/P. The equations given below were prepared from information in Reference #3 at the end of the article.

Ce is dimensionless, so the equation for Ce is as a function of L/B and H/P is the same for both S.I. and U.S. units and is as follows:

Ce = α(H/P) + β, where  β = 0.58382 + 0.016218(L/B), and

α = [-0.0015931 + 0.010283(L/B)]/[1 – 1.76542(L/B) + 0.870017(L/B)2]

The equation for kb as a function of L/B has different constants for S.I. and U.S. units.  The two versions of the equation for kb are as follows:

U.S. units: for 0 < L/B < 0.35:   kb = 0.007539 + 0.001575(L/B)  – (kb is in ft)

for 0.35 < L/B < 1.0:  kb = -0.34806(L/B)4 + 0.63057(L/B)3 – 0.37457(L/B)2 + 0.09246(L/B) – 0.000197 (kb is in ft)

S.I. units: for 0 < L/B < 0.35:   kb = 0.002298 + 0.00048(L/B) (kb is in m)

for 0.35 < L/B < 1.0:  kb = -0.10609(L/B)4 + 0.1922(L/B)3 – 0.11417(L/B)2 + 0.028182(L/B) – 0.00006 – (kb is in m)

Note that if H/L < 0.33,  B – L > 4 Hmax,  and P > 2Hmax, then the Francis Equation and the Kindsvater-Carter Equation will give nearly the same value for Q.  As conditions diverge more and more from the requirements, the calculations from the two equations will diverge more and more.  In these cases the value calculated by the Kindsvater-Carter formula should be used.

## An Excel Spreadsheet as a Contracted Rectangular Weir Flow Calculator

The Excel spreadsheet template shown below can be used as a contracted rectangular weir flow calculator, using both the Francis equation and the Kindsvater-Carter equation.  Only four input values are needed.  They are the height of the weir crest above the channel invert, P; the width of the channel, B; the weir length L; and the measured head over the weir, H. With these four input values, the Excel formulas will calculate the parameters needed and check on whether the conditions required for use of the Francis equation are met. If the conditions are all met, then the value of Q calculated with the Francis equation can be used.  If any of the conditions aren’t met, then the value of Q calculated with the Kindsvater-Carter formula should be chosen.  This Excel spreadsheet and others for suppressed and contracted rectangular weir calculations are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

References

1. U.S. Dept. of the Interior, Bureau of Reclamation, 2001 revised, 1997, 3rd ed,  Water Measurement Manual

2. Bengtson, H.H., Sharp Crested Weirs for Open Channel Flow Measurement, an Amazon Kindle ebook.

3. Bengtson, H.H., Open Channel Flow Measurement – Weirs and Flumes, An online continuing education course for PDH credit for Professional Engineers

4. Bengtson, H. H., Sharp-Crested Weirs for Open Channel Flow Measurement, An online continuing education course for PDH credit for Professional Engineers.

5. Merkley, Gary P., Weirs for Flow Measurement Open Course Ware, Utah State University.

# Suppressed Rectangular Weir Calculations with Excel Spreadsheets

Introduction to Suppressed Rectangular Weir Calculations

As shown in the diagrams and pictures below, the rectangular refers the the shape of the water cross-section as it goes over a sharp crested rectangular weir, which consists of a plate placed in an open channel so that the water is forced to flow through the rectangular open in the weir plate.  It can be used for open channel flow rate measurement, by measuring the height of water above the weir crest (the straight, level top of the weir opening), which can then be used to calculate the water flow rate over the weir.

Background on Sharp Crested Rectangular Weir Calculations in General

The picture at the left shows a rectangular weir measuring open channel flow rate in a natural channel.  The diagram below right shows a longitudinal cross-section of a sharp crested weir, with some of the terminology and parameters often used for sharp crested weirs included on the diagram.

The weir crest is the top of the weir. For a rectangular weir it is the straight, level bottom of the rectangular opening through which water flows over the weir. The term nappe is used for the sheet of water flowing over the weir. The equations for calculating flow rate over a weir in this article require free flow, which takes place when there is air under the nappe. The drawdown is shown in the diagram as the decrease in water level going over the weir due to the acceleration of the water.  The head over the weir is shown as H in the diagram; the height of the weir crest is shown as P; and the open channel flow rate in the open channel (and over the weir) is shown as Q.

Image Credits:  Rectangular, Sharp-Crested Weir: flowmeterdirectory.co.uk

Sharp Crested Weir Parameters:  H. H. Bengtson, Ref #2

The Francis Equation for Suppressed Rectangular Weir Calculations

A suppressed rectangular weir is one for which the weir extends across the entire channel, so that the length of the weir, L, is the same as the width of the channel, B.  The picture at the left shows a suppressed rectangular weir being used to measure the flow of water in an open channel.  The diagram below right shows some of the key parameters used in suppressed rectangular weir flow rate calculations.  Specifically, the height of the weir crest, P, the head over the weir, H, and the weir length, L (equal to channel width, B) are shown on the diagram.  The U.S. Bureau of Reclamation, in their Water Measurement Manual (Ref #1 below), recommend the use of the Francis equation (shown below) for suppressed rectangular weirs, subject to the condition that  H/P < 0.33 and H/B < 0.33:

For U.S. units: Q = 3.33 B H3/2, where

• Q is the water flow rate in ft3/sec,
• B is the length of the weir (and the channel width) in ft, and
• H is the head over the weir in ft.

For S.I. units:  Q = 1.84 B H3/2, where

• Q is the water flow rate in m3/sec,
• B is the length of the weir (and the channel width) in m, and
• H is the head over the weir in m.

The same condition for H/P and H/B apply.

Image Credits:  Suppressed Rectangular Weir Picture – U.S. Dept. of the Interior, Bureau of Reclamation, 2001 revised, 1997 third edition, Water Measurement Manual.

Suppressed Rectangular Weir Diagram – Bengtson, Harlan H.

The Kindsvater-Carter Formula for Suppressed Rectangular Weir Calculations

If either of the requirements in the previous section (H/P < 0.33 and H/B < 0.33) are not met the the more general Kindsvater- Carter Equation, shown below should be used.

U.S. units: Q = [0.075(H/P) + 0.602](2/3)[(2g)1/2](L – 0.003)(H + 0.003)3/2

S.I. units: Q = [0.075(H/P) + 0.602](2/3)[(2g)1/2](L – 0.001)(H + 0.001)3/2

Note that if H/P < 0.33 and H/B < 0.33, then the Francis Equation and the Kindsvater-Carter Equation will give nearly the same value for Q.  As H/P and/or H/B increase more and more above the 0.33 limit the calculations from the two equations will diverge more and more.  In these cases the value calculated by the Kindsvater-Carter formula should be used.

An Excel Spreadsheet for Suppressed Rectangular Weir Calculations

The Excel spreadsheet template shown below can be used for suppressed rectangular weir calculations, to calculate the water flow rate over a suppressed rectangular weir, using both the Francis equation and the Kindsvater-Carter equation.  Only three input values are needed.  They are the height of the weir crest above the channel invert, P; the width of the channel, B (which equals the weir length L); and the measured head over the weir, H. With these three input values, the Excel formulas will calculate H/P and H/B. If both of these are less than 0.33, then the value of Q calculated with the Francis equation can be used.  If either of the conditions aren’t met, then the value of Q calculated with the Kindsvater-Carter formula should be chosen.  This Excel spreadsheet and others for suppressed and contracted rectangular weir calculations are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

References

1. U.S. Dept. of the Interior, Bureau of Reclamation, 2001 revised, 1997, 3rd ed,  Water Measurement Manual

2. Bengtson, H.H., Sharp Crested Weirs for Open Channel Flow Measurement, an Amazon Kindle ebook.

3. Bengtson, H.H., Open Channel Flow Measurement – Weirs and Flumes, An online continuing education course for PDH credit for Professional Engineers

4. Bengtson, H. H., Sharp-Crested Weirs for Open Channel Flow Measurement, An online continuing education course for PDH credit for Professional Engineers.

5. Bengtson, H.H., “A Sharp Crested Rectangular Weir Equations Spreadsheet,” an online blog article.

6. Merkley, Gary P., Weirs for Flow Measurement Open Course Ware, Utah State University.