Impingement Heat Transfer Coefficients

Ref: ASME paper, "Streamwise Flow and Heat Transfer Distributions for Jet Array Impingement with Crossflow", by Florschuetz, Truman, Metzger (1981).

Introduction

Impingement heat transfer (heating or cooling) generates high convection coefficients, because the fluid boundary layer is extremely disrupted by the impinging jets. Impingement is one tool available when the heat transfer mechanisms must be taken to the edge of technology. This occurs often in gas turbine design, where higher firing temperatures lead to more efficent turbines, and therefore require high-technology cooling schemes (like impingement).

Impingement equations are empirically derived from test data, and many efforts have been made (and papers written) on what these equations should look like. The equations are all curve-fits to test data, so they implictly include all the "reality factors" that purely mathematical derivations often ignore.

The referenced paper is one of many, and is used here simply because I am most familiar with it and have used in for many years in designing gas turbine impingement cooling schemes.

Impingement is not well documented in text books, though there are some nice introductory write-ups in a few books.

The major consideration in the discussion below is the effect of post-impingement flow (aka "crossflow" or "spent flow") on downstream impingement jets. This is a serious effect and cannot be overlooked.

Assumptions / Simplifications

• All jets (holes) are round.
• Uniform hole diameters everywhere.
• Uniform hole spacing (X/D, Y/D, Z/D) everywhere.
• Constant flow channel width for spent flow (crossflow).
• All spent flow travels from row #1 to Row #n.
• Uniform mass flow rate through every hole.
• All spent flow travels from row to row down to the exit.
• Crossflow degradation effects, row-by-row, simplified by D. Leo for school use only.

Parameters

• mdot = mass flow rate per hole (kg/s)
• k = fluid thermal conductivity (W/m/K)
• mu = fluid viscosity (kg/m/s)
• D = hole diameter (input in mm, converted to meters internally)
• Cd = hole (orifice) discharge coefficient
• X/D = nondimensionalized hole spacing row-to-row (in direction of spent flow)
• Y/D = nondimensionalized hole spacing in transverse direction
• Z/D = nondimensionalized distance between surface and source of the jets.
• Pr = fluid Prandlt number
• Re(jet) = Jet Reynold's number
• Nu₁ = Nusselt number with no crossflow degradation effect
• Gc/Gj = local (row by row) cross flow degradation factor.
• Nu = Nusselt number with crossflow degradation effect
• H = convection heat transfer coefficient due to impinging jets
• pi = 3.14159

Note: the terms "spent flow" , "crossflow" and "post-impingement flow" all mean the same thing.

Jet Reynolds Number

Re(jet) = 4 * mdot / pi / D / mu

Uncorrected Nusselt Number

Don't miss the negative signs in some of the exponents!!

Nu₁ = .363 * (X/D)^-.554 * (Y/D)^-.423 * (Z/D)^.068 * Re(jet)^.727 * Pr ^1/3

Simplified Local (by Row) Crossflow Degradation Factor

Gc/GJ = (pi / 4) * (Row # - 1) / (Z/D) / (Y/D - 1)

Simplified Row-by-Row Corrected Nusselt Number

Nu = Nu₁ * (1 - [Gc/GJ]^.66)

Local (Row-by-Row) Impingement Heat Transfer Coefficient

H(imp) = Nu * k / D

A Hand Calc Example

If you want to use a small calculator, do it this way .....

Extended Discussion Regarding High Spent Flow (Crossflow)

This discussion is included here simply to complete the topic. It is beyond the scope of heat transfer course MECH594.

The amount of spent flow increases with each row of impingement holes, degrading the local impingement heat transfer coefficient, H(imp). However, the heat transfer coefficient of the spent flow, if considered as flow along a tube or a duct actually increases as more rows of holes add flow to it. At some point, the H(duct) may become greater than H(imp). It is wise to calculate H(duct) as well, row-by-row.

The local H(duct) is considered as turbulent flown through a pipe, duct or tube, including an inlet effect (similar to the boundary layer build-up on a flat plate).

One good equation for the local H(duct) is:

H(duct) = [ k / Dh ] * {1.0 + 1.2 * Dh / X_LE)} * .023 * [ Re_Dh ]^0.8 * Pr ⁿ

where:
X_LE is the distance (meters) from the leading edge of the impingement plate (along the direction of the spent flow).

Dh is now the hydraulic diameter of the spent flow passage, so, per hole
Dh = [4 * Area] / Perimeter = 4 * Y * Z / [2*(Y + Z)]

Dh = 2 * D * (Y/D) * (Z/D) / [ Y/D + Z/D ]

where D is still the hole diameter.

Area = Y * Z = D * (Y/D) * D * (Z/D)

The spent flow Reynold's number is Re_Dh = m' * Dh / Area / mu

where the local spent flow m' = mdot * (Row# -1)

mdot is still the mass flow rate through one hole.

The exponent, n, for the Prandlt number is 0.4 if T(wall) > T(fluid), and it is 0.3 if T(wall) < T(fluid).

The effect of this is shown in the curve labeled H(duct) in the figure above.

Exact Correlation

The exact (non-simplified) correlation for the crossflow effect on heat transfer is per the referenced ASME paper. It is beyond the scope of heat transfer course MECH594.

Nu = Nu₁ * {1 - C * [X/D]^nx * [Y/D]^ny * [Z/D]^nz * [Gc/GJ]^ng}

The values for the constants are in the table below, and depend on whether the impingement jet pattern is in-line or staggered with regard to the direction of the spent flow (crossflow).

And, if that isn't tough enough, the exact equation for Gc/Gj is ...

Gc/Gj = sinh [beta * (Row# - 0.5)] / 1.4142 / Cd / cosh [beta * Row#]

where

beta = Cd * 1.414 *(pi / 4) / (Y/D) / (X/D)

and sinh and cosh are hyperbolic sine and cosine.