Ben Sami
email: sami.ben-ali@inria.fr

Silviu-Ioan Ali

Olivier Filip

Sentieys

A Stochastic Rounding-Enabled Low-Precision Floating-Point MAC for DNN Training

Keywords: DNN, low-precision MAC unit, stochastic rounding

Training Deep Neural Networks (DNNs) can be computationally demanding, particularly when dealing with large models. Recent work has aimed to mitigate this computational challenge by introducing 8-bit floating-point (FP8) formats for multiplication. However, accumulations are still done in either half (16-bit) or single (32-bit) precision arithmetic. In this paper, we investigate lowering accumulator word length while maintaining the same model accuracy. We present a multiply-accumulate (MAC) unit with FP8 multiplier inputs and FP12 accumulations, which leverages an optimized stochastic rounding (SR) implementation to mitigate swamping errors that commonly arise during low precision accumulations. We investigate the hardware implications and accuracy impact associated with varying the number of random bits used for rounding operations. We additionally attempt to reduce MAC area and power by proposing a new scheme to support SR in floating-point MAC and by removing support for subnormal values. Our optimized eager SR unit significantly reduces delay and area when compared to a classic lazy SR design. Moreover, when compared to MACs utilizing single-or half-precision adders, our design showcases notable savings in all metrics. Furthermore, our approach consistently maintains near baseline accuracy across a diverse range of computer vision tasks, making it a promising alternative for low-precision DNN training.

I. INTRODUCTION

Deep Neural Networks (DNNs) have emerged as a cornerstone of modern artificial intelligence, enabling groundbreaking advancements in areas such as image recognition, natural language processing, and autonomous systems. The success of DNNs, however, comes at a significant cost. Training modern networks requires substantial computational resources, making it an arduous and resource-intensive process. As the scale and complexity of DNN models continue to grow, the demand for efficient training methods becomes increasingly urgent.

In response to the computational demands of DNN training, recent research has explored the use of lower-precision arithmetic. By reducing the bit-width of numerical representations, such as using 8-bit floating-point (FP8) formats, significant computational savings can be achieved. However, while much attention has been given to optimizing the precision of multiplicative operations, accumulative operations in DNNs, such as summations and weight updates, often remain in higherprecision formats like half-precision (16-bit) or single-precision (32-bit). This disparity in precision can lead to inefficiencies in hardware utilization and energy consumption.

In this paper, we attempt to reduce the precision of accumulations in forward (FWD) and backward (BWD) matrix multiply operations during training while preserving the model's accuracy. Our primary focus is the design of a dedicated multiplyaccumulate (MAC) unit optimized for efficiently handling FP8 multiplication inputs and FP12 accumulations. Recognizing that low-precision arithmetic can limit accuracy, we use stochastic rounding to mitigate the accuracy loss.

The paper is organized as follows. Section II presents some background on stochastic rounding (SR) as well as related work on low-precision DNN training and SR hardware. Section III details our main contribution on an SR-enabled floating-point MAC unit design. Finally, Section IV presents some results on low-precision training showing that our approach consistently maintains near baseline accuracy across a diverse range of image classification tasks.

II. BACKGROUND AND RELATED WORK

Rounding errors pose a prominent challenge within floatingpoint arithmetic. While DNN training can exhibit some resilience to these errors, their cumulative impact may ultimately result in substantial information loss. Various research efforts [START_REF] Wang | Training Deep Neural Networks with 8-bit Floating Point Numbers[END_REF], [START_REF] Zamirai | Revisiting BFloat16 Training[END_REF] study the potential benefits of employing stochastic rounding as a strategy to mitigate the effects of rounding errors in this context. SR is particularly effective against stagnation [START_REF] Blanchard | A class of fast and accurate summation algorithms[END_REF], a frequent occurance when computing the sum of a large number of terms with small magnitude and a large forward error is produced.

A. Stochastic Rounding

As opposed to deterministic rounding schemes, SR randomly maps a real value x to one of the two closest values in the number system being used. The probability of choosing either one is given by 1 minus its relative distance to x. More formally, for a finite set F ⊆ R, we denote the two rounding candidates ⌊x⌋ = max{y ∈ F|y ⩽ x} and ⌈x⌉ = min{y ∈ F|y ⩾ x}.

If x /

∈ F, we have that ("w.p" stands for "with probability") SR(x) = ⌈x⌉, w.p. q(x) = x-⌊x⌋ ⌈x⌉-⌊x⌋ , ⌊x⌋, w.p. 1 -q(x).

In case F is a normalized p binary digit precision floatingpoint system with maximum exponent e max and machine epsilon ε := 2 1-p , x ∈ F can be written as x = (-1) sx • 2 ex • m x , Fig. 1: Bit alignment in algorithms for sum-based stochastic rounding implementations. The random bits are added to the significand m x , followed by its truncation.

where s x ∈ {0, 1} is the sign bit, e x is an integer exponent between e min := 1 -e max and e max inclusive, and m x ∈ [0, 2) can be represented exactly using p binary digits, p -1 of which are stored explicitly. The case m x < 1 corresponds to subnormal values and only occurs when e x = e min .

If x ∈ R is written in normalized form as (-1) sx • 2 ex • m x , but this time with real-valued m x ∈ [0, 2), and we denote by tr(m x ) := sign(m)•2 1-p 2 p-1 |m x | its p-digit truncation, then SR(x) can be written as [4, Eq. (4.1)]

SR(x) = (-1) sx • 2 ex • tr(m x ), w.p. 1 -ε x , (-1) sx • 2 ex • (tr(m x ) + ε) , w.p. ε x , (1)
where [START_REF] Wang | Training Deep Neural Networks with 8-bit Floating Point Numbers[END_REF] and |x| is between the smallest and largest representable positive values in F.

ε x := (m x -tr(m x ))/ε ∈ [0,
From an implementation point of view, it makes more sense to consider the equivalent formulation [4, Eq. (4.4)] of ( 1)

SR(x) = (-1) sx • 2 ex • tr(m x ), X ⩾ ε x , (-1) sx • 2 ex • (tr(m x ) + ε) , X < ε x , (2)
where X ∼ U [0,1) is a random variable uniformly distributed over [0, 1). Definition (2) holds true in the discrete case as well [START_REF] Fasi | Algorithms for Stochastically Rounded Elementary Arithmetic Operations in IEEE 754 Floating-Point[END_REF]Sec. 4]: if X is generated on r bits, then x will be rounded up in 2 r ε x cases out of 2 r . What value of r is best in an SR implementation is an open question. It is certain that a lower value of r leads to a cheaper hardware implementation, but it will impact the accuracy benefit that SR potentially brings. Most hardware and software implementations of SR (see [START_REF] Croci | Stochastic rounding: implementation, error analysis and applications[END_REF]) do not directly use the comparison suggested by [START_REF] Zamirai | Revisiting BFloat16 Training[END_REF], but instead take the equivalent route of adding bits from a random source to the part of the number that will be truncated. If a carry out occurs after this addition, the rounded up value will be the result. A pictorial representation of this process is given in Fig. 1 (adapted from [5, Fig. 2]). Exceptional value handling can be done as suggested in [START_REF] Croci | Stochastic rounding: implementation, error analysis and applications[END_REF]Sec. 7.2].

B. Low Precision DNN Training

The FWD and BWD passes in deep learning models are frequently implemented as General Matrix Multiplications (GEMM), which constitute the most computationally demanding aspect of a training algorithm. Recent research has introduced the adoption of FP8 formats for multiplicative operations within GEMM procedures [START_REF] Micikevicius | FP8 formats for deep learning[END_REF]- [START_REF] Tatsumi | Mixing Low-Precision Formats in Multiply-Accumulate Units for DNN Training[END_REF]. Although quantizing inputs into 8-bit formats substantially reduces memory access time for transferring operands, it is essential to note that in most related work, accumulations are still conducted using 16-bit or 32-bit floating-point formats, jeopardizing overall gains. Moreover, quantizing the inputs leads to potential accuracy degradation due to the limited representation range of FP8 formats. To address this issue, some approaches employ loss scaling to mitigate accuracy loss [START_REF] Mellempudi | Mixed precision training with 8-bit floating point[END_REF]- [START_REF] Micikevicius | Mixed precision training[END_REF]. Others use distinct FP8 formats for the FWD and BWD passes, while introducing roundoff schemes to minimize memory access overhead [START_REF] Sun | Hybrid 8-bit floating point (HFP8) training and inference for deep neural networks[END_REF].

Additional work explores the use of fixed-point accumulators in conjunction with FP8 multipliers to reduce the area overhead associated with FP16 accumulators [START_REF] Tatsumi | Mixing Low-Precision Formats in Multiply-Accumulate Units for DNN Training[END_REF]. Beyond FP8 formats, alternative formats for DNN training, such as BFloat16 [START_REF] Osorio | A BF16 FMA is all you need for DNN training[END_REF], [START_REF] Kalamkar | A study of BFLOAT16 for deep learning training[END_REF] and block floating-point formats [START_REF] Zhang | FAST: DNN Training Under Variable Precision Block Floating Point with Stochastic Rounding[END_REF], have also been investigated. Approaches like [START_REF] Sun | Ultralow precision 4-bit training of deep neural networks[END_REF] lower precision further by employing 4-bit integer arithmetic, complemented with gradient scaling to minimize accuracy loss, but for small DNN models.

Several studies utilize SR in low-precision DNN training. However, when it comes to hardware implementations, a predominant focus has been on integrating SR with integer arithmetic [START_REF] Zhang | FAST: DNN Training Under Variable Precision Block Floating Point with Stochastic Rounding[END_REF], [START_REF] Gupta | Deep learning with limited numerical precision[END_REF], [START_REF] Chang | ESRU: Extremely Low-Bit and Hardware-Efficient Stochastic Rounding Unit Design for Low-Bit DNN Training[END_REF] due to its simplicity and low hardware cost. While some work explores its usage in conjunction with floating-point arithmetic [START_REF] Wang | Training Deep Neural Networks with 8-bit Floating Point Numbers[END_REF], [START_REF] Mellempudi | Mixed precision training with 8-bit floating point[END_REF], the primary focus is evaluating accuracy gains rather than delving into hardware specifics.

More recent research focuses on optimizing hardware for stochastic rounding units. Within this context, an efficient stochastic rounding unit (SRU) was introduced, relying on a reduced bitstream for random values [START_REF] Chang | ESRU: Extremely Low-Bit and Hardware-Efficient Stochastic Rounding Unit Design for Low-Bit DNN Training[END_REF]. Furthermore, another work [START_REF] Yuan | You Already Have It: A Generator-Free Low-Precision DNN Training Framework Using Stochastic Rounding[END_REF] seeks to minimize the SRU's footprint by utilizing entropy generated during model training instead of hardwarebased random generators. While such hardware optimizations predominantly adhere to integer arithmetic, our work explores SR hardware implementation within floating-point MAC units, and its application to the training of DNN models.

III. SR-ENABLED FLOATING-POINT MAC UNIT DESIGN

Our MAC unit designs are composed of three major parts, as summarized in Figure 2.

a) Multiplier: This is an exact variant that computes the product of two p m -bit precision values with E m exponent bits as a p a := 2p m -bit precision result with E a := E m +1 exponent The first (a) does stochastic rounding in a lazy fashion late in the addition process, after normalization, whereas the second (b) starts the process early, with the caveat that a small rounding correction is required towards the end of the process. bits. Taking this full result eliminates the need for rounding that would otherwise consume extra logic. For example, our reference FP8 design with E5M2 multiplier inputs will output FP12 E6M5 results.

b) Accumulator: The multiplier feeds into a p a -bit precision adder, with its other input being the previous adder output. An in-depth presentation is left to Sec. III-A and Sec. III-B. c) Random Number Generator: The design is completed by a r-bit pseudo-random number generator (PRNG) that operates in parallel and asynchronously with the multiplier and whose output will be used in the SR operation of the adder. It is based on a Galois linear feedback shift register (LFSR).

A. Standard SR-Enabled Floating-Point Adder

The most involved part in the unit is the adder and the integration of SR. To highlight what happens, we first consider the case where round-to-nearest-even (RN) is performed and then present the changes needed to support SR. Our discussion partially follows [5, Sec. 7.3], using a dual-path floating-point adder design 1 . For simplicity, we assume s x = s y = 0 to avoid sign interactions, which might transform the addition into a subtraction. The analysis carries over to effective subtraction. 1 Our design is a variation on a standard dual-path floating-point adder [START_REF] Muller | Handbook of Floating-Point Arithmetic[END_REF]Fig. 8.11]. The main difference is that we use one integer adder/subtractor circuit for computing the (un-normalized) significand of the result, as opposed to the vanilla design that has a different circuit for each path. This saves some area at the expense of latency. The ideas for our SR-enabled units should be applicable to other adder designs, such as the aforementioned standard one.

Let x = 2 ex m x and y = 2 ey m y be the precision p normalized inputs in the adder (i.e., m x = tr(m x ) and m y = tr(m y )). The basic addition algorithm can be summarized as follows:

(i) reordering and swap: if e y > e x , swap x and y. (ii) significand alignment: compute m y 2 -(ex-ey) (i.e., shift m y to the right e x -e y places). (iii) significand addition: compute m t = m x + m y 2 -(ex-ey) .

The situation e x -e y > 1 is handled by the far (f ) path, whereas e x -e y ⩽ 1 is handled by the close (c) path. (iv) normalization: since m t ∈ [0, 2 p+1 ) we might have to shift m z to the right by one bit (if m t ⩾ 2 p ) and increment e z . A shift left is possible with effective subtraction, handled with a leading zero detector (LZD) and shifter. (v) rounding: the significand of the rounded sum m z is computed by rounding the normalized exact sum m t . In case of RN, we need bits at positions p + 1, p + 2 in the significand (the guard and round bits), and the sticky bit (logical OR of all the bits after the (p + 2)nd). The sticky bit can be computed during (ii)-(iii).

A straightforward implementation of SR in the floating-point adder case would simply amount to replacing the sticky bit computation (and that of the guard and round bits) with adding r randomly generated bits to the normalized result significand starting at position p + 1, like in our discussion from Sec. II-A. Figure 3a summarizes this, and it seems that most hardware SR

B. Reduced-Latency SR-Enabled Floating-Point Adder

The rounding is deferred until the result has been normalized. Such a lazy execution impacts the critical path, and one can think that latency can be improved if rounding is started earlier, in an eager fashion. This is the premise of our optimized eager SR-enabled floating-point adder whose schema is given in Figure 3b. The rounding operation starts immediately after the significand alignment (ii), where we tentatively add the r-2 least significant bits (LSBs) of the generated random bitstring to the r -2 bits from y's shifted significand (the Sticky Round block), starting at position p + 3. The p + 1 most significant bits (MSBs) of the shifted value of y are used in computing the addition, whereas the two most significant bits of the Sticky Round output, denoted with S ′ = S ′ 1 S ′ 2 are potentially used in a Round Correction stage that occurs after the normalization of the addition result. For our implementation, the Normalization block relies on a carry-dependent 1-bit left shift. In the event of a carry during addition, the new carry bit becomes the updated implicit bit while the exponent is incremented. Conversely, when no carry occurs, the result requires a left shift by one bit. Additionally, the implicit bit will no longer be needed. Therefore, the block outputs a denormalized value. 2 There is a patent from IBM on floating-point addition with SR [START_REF] Bradbury | Stochastic rounding floating-point add instruction using entropy from a register[END_REF], but it is not clear based on the given description if the rounding operation is performed after normalization or not.

There are two main cases to consider: (a) When no normalization takes place: In this case, the addition result remains unshifted, and employing the eager design produces an identical outcome to calculating the rounding carry bit c as with the lazy implementation. The sole distinction is that, in the eager approach, the rounding addition unfolds across two stages, as shown in Figure 4a. (b) When normalization takes place: In this case, the addition result undergoes a leftward shift by 1 bit, the normalized value consistently retains an LSB value of 0 instead of aligning with the G bit value (as it would have been shifted into that position). To address this inherent discrepancy, we take corrective action by substituting the LSB of the normalized result with the G bit, as shown in Figure 4b. As this normalization shift also influences the effect of the Sticky Round output, the carry from the first stage becomes S ′ 2 instead of S ′ 1 . Through this correction, we achieve results that mirror those of the classic implementation. To further validate the eager SR adder design, we also conduct brute-force testing using a vast array of 10000 input pairs covering all the possible execution traces in the adder architecture. For every combination of input values x and y, we employ 1000 random integers and we calculate the probability of rounding occurrence accurately. We verify that, for each input configuration, the calculated probability aligns with the stochastic rounding definition outlined in Sec. II-A. This testing process further confirms the precision and adherence of the implementation to the specified stochastic rounding criteria.

C. Hardware Synthesis Results

1) Experimental Setup:

We developed an RTL implementation for the various MAC units. The objective is to examine the tradeoffs concerning power, area, and delay under different configurations and bit widths. To this end, we conducted RTL synthesis using Synopsys Design Vision 2019.03, in conjunction with an FDSOI 28nm technology. We also performed FPGA synthesis using Vivado 2022.1, targeting the Virtex UltraScale+ VU9P chip from Xilinx. During our experiments, we relax timing constraints and optimize design area.

2) Synthesis results: The synthesis results are presented in Figure 5 and Table I, which report the Energy, Area and Delay of various MAC configurations (RN, SR lazy, SR eager, with and without subnormals support), and for various adder bit widths (FP32, FP16, BFloat16, E6M5). FP32 corresponds to E8M23, FP16 to E5M10, Bfloat16 to E8M7, and E6M5 is our proposed 12-bit low-precision accumulator format.

Under identical configurations and bit settings, our optimized (eager) SR MAC design consistently outperforms the classic (lazy) SR implementation across all test metrics. The area gain is mainly due to having larger LZD and Normalization blocks in the classic case (p + r versus p + 2). This trend carries over when excluding support for subnormal values. In these assessments we set r to p + 3. This selection is made to align with the IEEE-754 definition of RN, ensuring consistency in the number of bits retained after shifting. In the subsequent

IV. LOW-PRECISION TRAINING RESULTS

As previously stated, our aim is to deploy SR-enabled MAC units suited for DNN training scenarios and explore their impact in terms of hardware complexity and model accuracy. We focus in particular on MAC units with FP8 multipliers and FP12 accumulators, since this configuration seems best suited for the models tested and because FP8 arithmetic is gaining traction in terms of commercial hardware support [START_REF] Micikevicius | FP8 formats for deep learning[END_REF]. To this end, we investigate two critical factors: the number of random bits r used for the rounding and the impact of using subnormal encodings (which add some overhead to the hardware).

In the following, all GEMM operations during training (FWD and BWD passes) are performed using low-precision MAC units. To examine the effects of these precision choices, we use a PyTorch software-based bit-accurate emulation flow of the MAC units. It uses custom CUDA kernels to expedite the training simulation runs to GPUs.

A. Training settings

We use stochastic gradient descent with a momentum coefficient of 0.9 across all our experiments. The initial learning rates and weight decay were configured to 0.1/0.01 and 0.0001/0.0005 for training ResNet-20 and VGG16 models, respectively. We also use a batch size of 128 for both models. To modulate the learning rate throughout training, we employed a cosine annealing scheduler. The ResNet-20 and VGG16 models were trained for 165 and 200 epochs, respectively. ResNet-50 training tests were ran for 100 epochs with a 0.01 initial learning rate and batch size of 16. Additionally, a dynamic loss scaling technique [START_REF] Micikevicius | Mixed precision training[END_REF] was applied to all experiments, using an initial scaling factor of 1024.

B. Impact of subnormals and number of random bits

To investigate the impact of the number of random bits on training accuracy, we opted for the FP12 format (E6M5). We started with a ResNet20 model and the CIFAR10 dataset. As is visible in Table III, using only r = 9 bits for the random values yielded notably inferior accuracy results. However, as we progressively increase the number of random bits, accuracy consistently improves. With 18 bits, we achieved accuracy levels close to the baseline with and without subnormal support 3 .

We also ran tests using larger VGG16 and ResNet50 models. For VGG16, we utilized the CIFAR10 dataset, while the ResNet50 model was trained on a more challenging dataset, Imagewoof, a subset of ImageNet. As shown in Table IV, FP12 with 18 random bits and without subnormals outperforms the FP16 accumulator while incurring minimal accuracy degradation for both models. RN results with sub 16-bit formats and without subnormal support are not reported since the accuracy degradation becomes significant in these cases.

C. Hardware Overhead of Varying Random Bit Sizes

To study the hardware overhead of different numbers of random bits, we again use the FP12 format without subnormal support. The results, as presented in Table V, show that even with a larger value of r, our SR implementation yields significant reductions in area, energy consumption, and latency compared to MAC units using half-precision formats. Putting it all together, our 12-bit SR design without support for subnormals reduces the delay, area and energy of the MAC unit by ≈ 50% w.r.t. FP32, while maintaining near baseline accuracy during training. When compared to FP16, delay is reduced by more than 20%, and area and energy by ≈ 10%.

V. CONCLUSION

Using small number representations such as FP8 formats can significantly impact the computational demand of DNN training. However, in most of the state-of-the-art, accumulations are still done in either half (16-bit) or single (32-bit) precision arithmetic. While stochastic rounding in lower precision can be used to mitigate the loss of accuracy, it has traditionally come with hardware overhead and loss of performance.

In this paper we present a novel architecture for floatingpoint MAC units employing stochastic rounding. The proposed eager design outperforms the classic lazy one on all benchmarks, achieving up to 32.2% latency and 18.5% area savings. We also test our design in various DNN training settings, and by tuning the number of bits for the rounding unit we reach baseline-comparable results. Training results indicate that a configuration using 18 random bits and without subnormal support gives the best tradeoffs across all training benchmarks. Despite the hardware overhead due to more random bits, our design results in 23% and 9.2% savings in latency and area, respectively, w.r.t. an FP16 accumulator with RN support. Furthermore, the hardware advantages of our proposed eager design hold even greater potential within a systolic array-based accelerator, which we plan to cover in future work.

Fig. 2 :

 2 Fig. 2: Overview of our floating-point MAC unit designs with SR support. The multiplier results are exact, with rounding only being done in the adder part.

Fig. 3 :

 3 Fig.3: Two possible implementations of a SR-enabled floating-point adder. The first (a) does stochastic rounding in a lazy fashion late in the addition process, after normalization, whereas the second (b) starts the process early, with the caveat that a small rounding correction is required towards the end of the process.

Fig. 4 :

 4 Fig. 4: Detailed flow of eager SR Round Correction operation in the two possible normalization cases. implementations follow this approach [5, Sec. 7.3] 2 . Effective +/-(values of s x and s y ) is handled with the op flag.

Fig. 5 :

 5 Fig. 5: Hardware cost for different floating-point MAC configurations.

TABLE I :

 I Hardware cost for different FP adder configurations.

	Configuration	E	M	r	Energy	Area	Delay
					(nW/M Hz)	(µm 2 )	(ns)
		8	23	0	1.17	1404.01	4.71
	RN W/ Sub	5 8	10 7	0 0	0.65 0.52	692.62 581.05	2.73 2.14
		6	5	0	0.42	479.81	1.88
		8	23	0	1.15	1337.42	4.69
	RN W/O Sub	5 8	10 7	0 0	0.64 0.52	662.43 562.44	2.75 2.28
		6	5	0	0.42	462.67	1.88
		8	23 27	1.62	1897.36	5.19
	SR lazy W/ Sub	5 8	10 14 7 11	0.89 0.66	938.73 833.84	2.99 2.77
		6	5	9	0.57	636.64	2.20
		8	23 27	1.48	1677.37	5.50
	SR lazy W/O Sub	5 8	10 14 7 11	0.81 0.64	839.34 751.74	3.18 2.83
		6	5	9	0.53	574.14	2.20
		8	23 27	1.37	1550.89	4.75
	SR eager W/ Sub	5 8	10 14 7 11	0.76 0.61	777.48 670.41	2.72 2.33
		6	5	9	0.50	549.49	1.86
		8	23 27	1.35	1497.52	4.73
	SR eager W/O Sub	5 8	10 14 7 11	0.70 0.61	718.41 661.54	2.63 2.50
		6	5	9	0.50	541.50	1.87

TABLE II :

 II FPGA implementation results for FP adder designs. delve deeper into the impact of the value of r on both hardware performance and model accuracy.The results from TableIIconfirm that the eager MAC design still offers area and timing gains within an FPGA setup.

	Configuration	E	M	r	LUT FF Delay (ns)
	RN W/ Sub	5	10	-	302	49	8.30
	RN W/O Sub	5	10	-	301	49	8.29
	SR lazy W/O Sub	6	5	18	344	59	8.76
	SR eager W/O Sub	6	5	18	251	59	8.04
	section, we

TABLE III :

 III Impact of number format (E, M) and random number bits r on accuracy when training ResNet20 on CIFAR10.

	Configuration	E	M	r	Accuracy (%)
	FP32 Baseline	8	23	-	91.47
	RN W/ Sub	5	10	-	91.1
	RN W/ Sub	8	7	-	88.79
	RN W/ Sub	6	5	-	83.03
	SR W/ Sub	6	5	9	43.11
	SR W/ Sub	6	5	12	89.34
	SR W/ Sub	6	5	16	90.7
	SR W/ Sub	6	5	18	91.39
	SR W/O Sub	6	5	16	90.67
	SR W/O Sub	6	5	18	91.39

TABLE IV :

 IV Impact of number format (E, M) and number of random bits r on accuracy for VGG16 and ResNet50 models.

	Model/Dataset	Configuration	E	M	r	Accuracy (%)
		FP32 Baseline	8	23	-	93.46
	VGG16/CIFAR10	RN W/ Sub	5	10	-	93.06
		SR W/O Sub	6	5	18	93.11
		FP32 Baseline	8	23	-	80.94
	ResNet50/Imagewoof	RN W/ Sub	5	10	-	80.3
		SR W/O Sub	6	5	18	80.33

TABLE V :

 V Impact of random bits r on hardware overhead.

	Configuration	E	M	r	Delay	Area	Energy
					(ns)	(µm 2 )	(µW/M Hz)
				9	1.87	541.50	0.50
				12	1.88	569.40	0.53
	SR eager W/O Sub	6	5	14	1.96	591.11	0.55
				16	2.04	610.86	0.58
				18	2.10	628.48	0.60
	RN W/ Sub (FP16)	5	10	-	2.73	692.62	0.65
	RN W/ Sub (FP32)	8	23	-	4.71	1404.01	1.17

When we do not have subnormal support, values in the subnormal range are treated as zero.


D Bestion

P Fillion

REVISITING THE PIRT AND SCALING ANALYSIS WITHIN THE FRAME OF 3D SYSTEM CODE MODELLING

Keywords: PIRT, Scaling, 3D modelling

A working Group of OECD-NEA-CSNI-WGAMA was created in 2016 to make a state of the art of existing 3D capabilities in TH system codes, to propose further activities to improve the modelling, and to complement the validation and verification of such tools. 3D modules use a two-fluid model in a porous body approach with a phase-time-space averaging procedure where terms related to diffusion and dispersion appear which were not present in the 1D models. This paper presents a contribution to this work focusing on the revisiting of the process identification and ranking table (PIRT) including local 3D processes and on the scaling analysis to identify the order of magnitude of some terms of the equations depending on the flow situation. It is shown how the various components and flow situations may have specific dominant processes and may require specific additional modelling and validation efforts. The presence of internal structures, the porosity of the component and the space scales of the flow have a strong impact on the relative importance of wall transfers and diffusiondispersion terms. Examples are given of flow situations in a pressurized water reactor (PWR) core in loss of coolant accidents (LOCAs), and in steam line breaks (SLB). This shows how such 3D tools used for pressure vessel require a specific analysis at the component level and for each transient phase.

INTRODUCTION

A working Group of OECD-NEA-CSNI-WGAMA was created in 2016 to make a state of the art of existing 3D capabilities in TH system codes, to propose further activities to improve the modelling, and to complement the validation and verification of such tools [START_REF] Herer | 3DSYSTH: an OECD/NEA activity on multi-dimensional capabilities of thermalhydraulic system codes[END_REF]. The present work was partly done in this framework. However, it represents the opinion of the authors and was not yet endorsed by the working group. The current generation of industrial system codes (CATHARE [START_REF] Emonot | CATHARE-3: A new system code for thermalhydraulics in the context of the NEPTUNE project[END_REF], RELAP-5 [START_REF]RELAP5/MOD3 Code Manual, Volume I : Code Structure, System Models and Solution Methods[END_REF], TRACE [START_REF]Theory Manual[END_REF], SPACE [START_REF] Kim | Development of Physical Models and Correlation Packages for the SPACE code[END_REF], ATHLET [START_REF] Teschendorff | Methodology, status and plans for development and assessment of ATHLET code[END_REF]) have 3D models in porous medium approach which were initially devoted to the prediction of very large scale 3D effects during large break loss of coolant accident (LBLOCA) and which were validated on the data of the 2D-3D experimental program performed in UPTF [START_REF] Emmerling | 2D/3D Program Upper Plenum Test Facility, UPTF: Program and System Description[END_REF][START_REF] Weiss | UPTF, a full-scale PWR loss-of-coolant accident experiment program[END_REF][START_REF] Glaeser | Down-comer and tie plate countercurrent flow in the Upper Plenum Test Facility (UPTF)[END_REF][START_REF] Glaeser | The contribution of UPTF experiments to resolve some scale-up uncertainties in countercurrent two phase flow[END_REF], SCTF [START_REF] Murao | Large-scale multi-dimensional phenomena found in CCTF and SCTF experiments[END_REF] and CCTF [START_REF] Iguchi | Summary of CCTF test resultsassessment of current safety evaluation analysis on reflood behavior during a LOCA in a PWR with cold-leg injection-type ECCS[END_REF] facilities. The PERICLES 2D program (see [START_REF] Deruaz | 2D effects in the core during the reflooding phase of a LOCA, Safety of Thermal Water Reactors[END_REF][START_REF] Hj | A study of 2-Dimensional Effects in the Core of a PWR during the Reflooding Phase of a LOCA. Analysis of Data of PERICLES Experiments with the COBRA-NC code[END_REF]) also investigated reflooding with radial power profile. The first applications used rather coarse 3D nodalization of the Pressure Vessel (PV) with only a few hundred meshes to keep a reasonable total CPU time. Today the computer power allows a much finer 3D nodalization and many other transients may also use a 3D modeling for improving the code prediction accuracy [START_REF] Prea | CATHARE-3: a first computation of a 3-inch break loss-ofcoolant accident using both cartesian and cylindrical 3D meshes modelling of a PWR vessel, NURETH-17, Xi'an[END_REF]. A core modelling with one mesh per assembly may become a standard practice in near future with the CATHARE code for many accidental transient simulations including LOCAs. This allows to look at much finer multi-D physical processes and may provide a better accuracy of predictions. Specific requirements are necessary for such finer simulations. First, a more detailed PIRT exercise is needed to identify dominant phenomena at a smaller scale than for previous nodalizations. Based on porous body 3D equations, one can identify a list of new phenomena (compared to 1D approach) to model and to validate on specific separate effect tests. An order of magnitude analysis allows to neglect some of these processes or simplify some models, depending on the sub-component (core, lower plenum, annular downcomer, upper plenum) and depending on the physical situation encountered in accidental transients. Attention will be drawn on LOCA situations in PWRs and on the core component of the Pressure Vessel. The need to revisit the PIRT and scaling analyses is first presented. Then the identification of local phenomena is made on the basis of porous-3D set of equations. Some ways to estimate the relative impact of each basic phenomenon are presented. Attention is drawn on all diffusive and mixing processes. Considering the lack of large scale test facilities, a method to validate the 3D modelling is proposed using a new approach called "multiscale validation" where small scale separate effect tests are used together with several modeling scales to validate the system scale 3D models.

REVISITING PIRT AND SCALING ANALYSIS FOR A 3D MODELLING

The role of PIRT and scaling

In a reactor safety analysis the safety criteria are first defined which give limits to some parameters of interest or Figures of Merit (FoM), e.g. peak cladding temperature (PCT) in LOCAs, or liquid temperature field in a core in a main steam line break (MSLB), or boron concentration field in a core for a boron dilution transient.

Phenomena identification and ranking table (PIRT) and scaling analysis are used to investigate reactor accidental situations with the following objectives:

 Identify phenomena which have a dominant or significant effect on the FoMs and parameters of interest of the safety analysis.

 Identify all the phenomena which should be properly modelled and validated in the numerical tool to be used in the safety demonstration.

 Rank the phenomena to establish a hierarchy necessary to define scaling criteria for integral effect test (IET) facilities and separate effect test (SET) facilities which should be built and operated. The criteria are to preserve the dominant phenomena in the scaled experiment and to accept some distortion of the phenomena which have a lower influence on the parameters of interest.

 Establish a validation matrix which should be able to validate the numerical simulation tool on all dominant and influent phenomena in IETs and SETs.

 List and rank the phenomena and the related terms of the equations used in the simulation tool which should be included in the uncertainty analysis. The input uncertainty to be used for negligible terms may be neglected or better treated conservatively. The input uncertainty to be used for dominant terms must be treated with special care and possibly validated against SETs. The input uncertainty to be used for influent but not dominant phenomena should also be evaluated and preferably validated on SETs.

Why and when revisiting PIRT and scaling analysis for 3D modelling?

Usual scaling methods such as H2TS ("Hierarchical Two Tiered Scaling") [START_REF] Zuber | Appendix D: a hierarchical, two-tiered scaling analysis, an integrated structure and scaling methodology for severe accident technical issue resolution[END_REF] have two steps: a topdown, T-D, and a bottom -up (B-U). The T-D step uses non-dimensional form of mass (M), energy (E) and momentum (MM) conservation equations written at the system level to establish the scaling hierarchy i.e. what phenomena have priority in order to be scaled (e.g. in an IET), and to identify what phenomena must be included in the bottom-up analysis. The B-U step continues the analysis at the component or at the process level to scale SETs and both steps give requirements for the physical modelling in the simulation tool and of the validation of the tool. The scaling analysis is based on the PIRT but it can also help the PIRT by helping in the ranking of phenomena.

When 3D phenomena were identified in the LBLOCA (Large Break Loss of Coolant Accident) PIRT, they were investigated in specific large scale tests (UPTF, SCTF, CCTF). At that time, all the 3D local processes were not identified and that was the main reason for such very expensive tests. The downcomer refill at a reactor scale in UPTF was necessary since very complex countercurrent flow with high condensation rates and with azimuthal heterogeneity were encountered and no clear scaling criteria could be defined. In such complex cases the only way is to perform scale 1:1 tests in prototypical geometry and prototypical thermalhydraulic conditions. The 3D effects in the core during reflooding were also investigated with almost scale 1:1 tests in CCFT and SCTF. The vertical scale was preserved, the geometry of assemblies was prototypical and the radial transfers were well represented by using 8 full rod bundles (SCTF) with representative power differences between assemblies.

Reflooding models were first validated on 1D reflooding tests without radial power profile. When the models extrapolated from 1D were used in a 3D modelling of reflooding tests with radial power profile (SCTF, PERICLES-2D [START_REF] Morel | Validation of the CATHARE code against PERICLES 2D REFLOODING Tests[END_REF][START_REF] Dor | Core radial profile effect during reflooding, validation of CATHARE 2 3D module using SCTF tests[END_REF]) the predictions were rather satisfactory although no particular effort was spent to the specific 3D phenomena such as radial transfers which were observed in the experiments. This was possibly due to some compensating errors but since the tests were very close to reactor conditions, the same compensations are expected in reactor simulations. There was no need to investigate more deeply local 3D processes. Today 3D modeling is envisaged in other situations such as SLB, boron dilution, and SBLOCAS (Small Break LOCA) without any large scale test facility to validate the codes. Therefore, the only way is to identify each basic 3D phenomenon, to validate the model on existing small scale data in a separate effect way and to use existing IETs for a more global validation even if the relative weight of each process is not fully representative of the reactor geometry.

Identifying phenomena from basic 3D equations

The system of equations used in porous-3D two-fluid model is the basis for identification of all relevant 3D processes. The local instantaneous mass momentum and energy equations are time averaged and integrated over a domain of space characterized by a scale. For example, in a PWR core, the domain of integration can be a section of an assembly (assembly scale modelling) or a section of a sub-channel (sub-channel modelling). Equations can be written as follows (see Chandesris et al., [START_REF] Chandesris | A macroscopic turbulence model for flow in porous media suited for channel, pipe and rod bundle flows[END_REF][START_REF] Chandesris | Rod bundle thermalhydraulics mixing phenomena: 3D analysis with CATHARE 3 of various experiments[END_REF]):

𝜕𝜙𝛼 𝑘 𝜌 𝑘 𝜕𝑡 + ∇. (𝜙𝛼 𝑘 𝜌 𝑘 𝑉 𝑘 ) = 𝜙Γ 𝑘 (1
In these equations, 𝛼 𝑘 ,𝜌 𝑘 ,𝑉 𝑘 , 𝑒 𝑘 , ℎ 𝑘 are the volume fraction, the density, the velocity, the internal energy and the enthalpy for the phase k, 𝜙 is the porosity, P the pressure, Γ 𝑘 the interfacial mass exchange. 𝑝 𝑖 and 𝑓 𝑖 𝑇𝐷 are void dispersion terms due to space averaging of interfacial pressure forces, and time averaging of drag and added mass forces. They tend to homogenize void fraction. 𝜏 𝑖 𝑖𝑠 the interfacial friction force, 𝜏 𝑤𝑘 the wall friction force, q 𝑘𝑖 , q 𝑤𝑘 the interfacial and the wall to phase k heat transfer, 𝑆 𝑐 the heating surface, 𝜏 𝑘 𝑙+𝑡+𝑑 the stress tensor which accounts for laminar turbulent and dispersive effects, and 𝑞 𝑘 𝑙+𝑡+𝑑 the laminar turbulent and dispersive heat flux

Diffusion and dispersion terms

The momentum and energy dispersive and diffusive terms came out during the double (time and space) averaging process of the local convection terms:

< 𝑣𝑣 ̅̅̅ > 𝑓 =< 𝑣̅ > 𝑓 < 𝑣̅ > 𝑓 +< 𝑣′𝑣′ ̅̅̅̅̅ > 𝑓 +< 𝛿𝑣 ̅̅̅ 𝛿𝑣 ̅̅̅ > 𝑓 (4)
< 𝑣ℎ ̅̅̅̅ > 𝑓 =< 𝑣̅ > 𝑓 < ℎ ̅ > 𝑓 +< 𝑣 ′ ℎ′ ̅̅̅̅̅ > 𝑓 +< 𝛿𝑣 ̅̅̅ 𝛿ℎ ̅̅̅ > 𝑓 (5) 𝑥 ̅ is the time average of the quantity x and x' the deviation from this average: < 𝑥 > 𝑓 is the spatial average of the quantity x and 𝛿𝑥 the deviation from this average

The first rhs terms of equations ( 4) and ( 5) are the macroscopic convection of the mean velocity and enthalpy, the second rhs terms are the turbulent diffusion of momentum and energy, and the third rhs terms are momentum and energy dispersion terms (see Drouin et al, 2010, [21]).

LIST OF PHENOMENA FROM 3D EQUATIONS

List of phenomena

One may establish a list of phenomena based on these equations and see which phenomenon exists in 1D models and which are specific to 3D (see Table 1). From this list there are several phenomena which are simply extrapolated from 1D models without any specific effort to take into account possible effects of the 3D geometry of the flow in the respective components of a pressure vessel, the downcomer, the lower plenum, the core, the upper plenum, the upper head. This is true for interfacial heat and mass transfers, interfacial friction, and wall friction. Depending on the local geometry there may be significant differences with typical 1D flows in pipes or axial flow in rod bundles. For example the impact of spacer grids in a PWR core is known to increase the turbulence intensity by a factor 4 or 5 and to improve wall heat transfers. New phenomena appear that were not present in 1D models such as laminar and turbulent diffusion of momentum and energy and dispersion of momentum and energy. The relative impact of each phenomenon depends on the component, the accidental sequence and the phase of the transient. Various modelling and validation needs are reported in the table but in many cases the absence of information is not dramatic since the phenomenon has a low or very low impact. Therefore some guidance is provided here below to help the evaluation of the order of magnitude of each term.

Ranking phenomena related to momentum

Let's consider first the relations between pressure and flowrate which may be described by the mixture momentum equation. One may consider one privileged flow direction (vertical upward in a core, vertical downward in a downcomer and consider mixture momentum balance in this direction integrated over a distance z.

In steady state, it reduces to: The r.h.s terms are 1: axial acceleration 2: contribution from radial flow, 3: gravity, 4: wall friction,5: form losses, 6: viscous and turbulent momentum diffusion, and momentum The gravity wall friction and form loss terms are very often dominant terms. The form pressure losses (5 th ) is related to singular geometries such as spacer grid, perorated plated, ….. Only the first third and fourth r.h.s terms exist in a 1D model. The axial acceleration term may be due to variations of the porosity or to density differences. Density differences are likely to occur in case of heating, e.g. in a core. It is relatively small in single phase liquid since 𝜕𝜌 𝑙 𝜕ℎ 𝑙 ⁄ is rather small but may be significant in two phase with boiling and in single phase vapour since 𝜕𝜌 𝑣 𝜕ℎ 𝑣 ⁄ is rather high. The radial acceleration term exists in case of radial flow and may result from several reasons. In the case of the core geometry, both buoyancy driven crossflows and friction driven crossflows were already identified [START_REF] Bestion | Scaling considerations about LWR core thermalhydraulics[END_REF][START_REF] Bestion | 3D Core thermalhydraulic phenomena in PWR SBLOCAs and IBLOCAs[END_REF][START_REF] Alku | Modelling of turbulent effects in LOCA conditions with CATHARE-3[END_REF]. A radial power profile creates radial density gradients which decrease the axial gravity term (3 rd term) and increase the axial acceleration (1 st term) and the friction (4 th term) and form losses (5 th term).

Δ𝑃 Δ𝑧 = ( Δ𝑃 Δ𝑧
A first analysis [START_REF] Bestion | Scaling considerations about LWR core thermalhydraulics[END_REF] proposed a non-dimensional number 𝐹 1 , to characterize the limit between both types of crossflows. From the simplified axial momentum equation:

𝜕𝑃 𝜕𝑧 = 𝜌𝑔 -𝜌 ( 𝐾 𝑆𝐺𝑧 Δ𝑧 𝑆𝐺 + 4 𝐷 ℎ𝑧 𝐶 𝑓𝑧 ) 𝑉 2
2 𝐾 𝑆𝐺𝑧 is the form loss coefficient of a spacer grid, Δ𝑧 𝑆𝐺 is the distance between grids, and 𝐶 𝑓𝑧 the friction coefficient along rods. One can see if it increases or decreases with heating and density change:

( 𝜕 𝜕𝑃 𝜕𝑧 𝜕𝜌 ) 𝐺 = -𝑔 + ( 𝐾 𝑆𝐺𝑧 Δ𝑧 𝑆𝐺 + 4 𝐷 ℎ𝑧 𝐶 𝑓𝑧 ) 𝑉 2 2
For a given mass flux G, the pressure gradient decreases with heating (decreasing ) for 𝐹 1 <1 and there will be crossflows from low power (high ) to higher power (high ) creating the chimney effect. For 𝐹 1 >1 there will be crossflows from high power to low power. A low 𝐹 1 creates buoyancy driven crossflow and a high 𝐹 1 creates friction driven crossflow.

𝐹 1 = ( 𝐾 𝑆𝐺𝑧 Δ𝑧 𝑆𝐺 + 4 𝐷 ℎ𝑧 𝐶 𝑓𝑧 ) 𝑉 2

2𝑔 ;

It was found that the momentum diffusion and dispersion effects related to the radial gradient of axial velocity (due to radial density gradients) were small and negligible compared to gravity and friction.in a pure vapour flow. More recent investigations have found that in vapour flow, the acceleration may not be negligible in the balance and that a modified number 𝐹′ 1 should be considered:

𝐹′ 1 = [ 𝐾 𝑒𝑓𝑓𝑧 𝐿 𝑍 + 8𝛼𝜑 𝐺𝐶𝑝 𝐷 ℎ ]𝑉 2 2𝑔 with 𝐾 𝑒𝑓𝑓𝑧 𝐿 𝑍 = 𝐾 𝑆𝐺𝑧 Δ𝑧 𝑆𝐺 + 4 𝐷 ℎ𝑧 𝐶 𝑓𝑧 𝛼 = - 1 𝜌 𝜕𝜌 𝜕𝑇

Thermal expansion coefficient

: wall heat flux

Laminar and turbulent momentum diffusion and momentum dispersion were modeled and validated for a PWR core geometry on the basis of small rod bundle data [START_REF] Chandesris | Rod bundle thermalhydraulics mixing phenomena: 3D analysis with CATHARE 3 of various experiments[END_REF]. It was found that radial dispersion is usually larger than radial diffusion and depends significantly on the type and geometry of spacer grids. There may be a factor 30 between simple spacer grids and mixing grids. such order of magnitude analyses on may conclude that for a core geometry:  High velocity liquid flow in a core with radial power differences does not create big density and velocity differences between high power and low power regions due to low thermal expansion and induce small friction driven crossflows. In such case, radial momentum diffusion and dispersion will also be small. This is a situation encountered in high power high flowrate SLB  Low velocity liquid flow in a core with radial power differences create higher density differences -than in previous case-between high power and low power regions and may induce buoyancy driven crossflows. In such case, radial momentum diffusion and dispersion may play a second order role. This is a situation encountered in low power low flowrate SLB.  High velocity boiling flow in a core with radial power differences create density differences -due to boiling-between high power and low power regions and induces some friction driven crossflows.

In such case, radial momentum diffusion and dispersion may play a lower role. This is encountered in a blowdown phase of a LBLOCA.  Low velocity boiling flow in a core with radial power differences create high density differencesdue to boiling-between high power and low power regions and induces very high buoyancy driven crossflows. In such case, radial momentum diffusion and dispersion may play a relatively minor role. This is encountered in a core uncovery (SBLOCA, see figure 1, right) or in a LBLOCA reflooding situations (Figure 1, left) below the swell level or below the quench front.  High velocity vapour flow in a core with radial power differences creates big density and velocity differences between high power and low power regions due to high thermal expansion and induce buoyancy driven crossflows. In such case, radial momentum diffusion and dispersion will may play a relatively minor role. This is a situation encountered in low pressure core uncovery (SBLOCA, Figure 2, right).  Low velocity vapour flow in a core with radial power differences create higher density differences -than in previous case-between high power and low power regions and induces high buoyancy driven crossflows. In such case, radial momentum diffusion and dispersion may play a second order role. This is a situation encountered in high pressure core uncovery (SBLOCA). The first term is the axial convective term and the 2 first r.h.s. terms are the radial convection terms corresponding to crossflows. The third term due to wall heating is clearly dominant in a core and the laminar diffusion, turbulent diffusion and dispersion terms can play a significant role only when crossflows are very small. Crossflows are due to either buoyancy or friction and depend mainly on the friction and form losses for a radial flow. There is lack of information on such radial pressure losses and this is a weakness of current 3D core models. New experimental programs should provide information on wall friction for non-axial flow in a core (see the METERO program [25,26])

Ranking phenomena related to phase repartition

The core void fraction repartition may play a dominant role on the cooling capabilities particularly in the reactor core. In a SBLOCA when there is an uncovered zone, the swell level controls the PCT. It is mainly determined by the axial interfacial friction. Current models are satisfactory and extensively validated. However when 3D phenomena exist due to core radial power differences, (see figures 1 and 2). The interfacial friction for non-axial flow is very poorly known. It may affect to what extent the vapour axial flowrate is homogenized or not at the swell level. If the radial mixing below a swell level is perfect the void fraction and the vapour mass flux is homogeneous at the swell level and the highest power rods do not benefit from the large vapour production below the level. If the mixing is not perfect, the situation may be better for the high power rods.

The void dispersion term can also play a significant role in some conditions as shown by Valette in PSBT analysis [START_REF] Valette | PSBT simulations with Cathare 3[END_REF]. From sensitivity tests, it appears that it can be dominant only in high speed two-phase flow and it is much lower than the buoyancy driven crossflow.

New experimental programs should provide information on interfacial friction for non-axial flow and on void dispersion force in a core (see the METERO program [25,26])

Ranking phenomena for PWR core thermalhydraulics

One can then propose a ranking of core 3D phenomena based on the impact on a good prediction of flowrate and temperature or steam quality repartition which have an impact on cooling capabilities and PCT. Only phenomena which require models are considered. Crossflows depend on gravity and affect convection or acceleration terms which are not listed here, since they do not require any models but they are influenced by friction and pressure losses which are listed. Ranking uses 3 categories: dominant (D), influent (I) and negligible (N). The favorable (f) or unfavorable (uf) effect on cooling are mentioned, and the availability of validation data (v) or the lack of data (nv) are also mentioned. The wall transfers are necessarily dominant in all conditions since they may include the onset of nucleate boiling (ONB) and net vapor generation point (NVG) which creates void and affects reactivity (SLB), CHF occurrence which decrease dramatically heat transfer coefficients (LBLOCA blowdown), quenching which recovers good cooling (reflooding), film boiling which controls post-CHF overheating in LOCA blowdown phase, and vapor convective heat transfer which controls PCT in core uncover (SBLOCA). 6 main conditions are considered characterized by either single phase liquid single phase vapor or two-phase, and by low or high velocity. High velocity here corresponds to 𝐹′ 1 > 1 where friction crossflows may take place and low velocity corresponds to 𝐹′ 1 < 1 where buoyancy driven chimney effects may occur. This ranking is made on the basis of some quantitative evaluations, on some sensitivity tests, on some experimental data analyses, and on some expert judgement. Further checking may be necessary.

D v D v D v D v D v D v Energy diffusion I. f nv N I f nv N I f nv N Energy dispersion D f v I f v I f nv N f nv I nv N nv

Impact of porosity and geometrical scales on mixing processes

The core geometry has a rather low porosity and a small hydraulic diameter compared to other components of a PV. Therefore, friction terms often play a dominant role and dispersion may be much larger than turbulent diffusion. Therefore a turbulence model is not required and dispersion may be modelled without any additional transport equation. In a more open medium like an annular downcomer or upper and lower plena, due to much lower wall friction, one may expect buoyancy driven rather than friction driven crossflows. If turbulent mixing is expected to play a dominant role (e.g. in downcomer and lower plenum during SLB or boron dilution transients) a turbulence model is needed and the 3D modules of system codes may not be well adapted to these components. A coupling of the system code with a CFD in open medium may be necessary. Also wall friction and wall heat transfers which are extrapolated from 1D to porous 3D models are not adapted to open media with small meshes, for which a wall function approach is needed. In such case a CFD simulation with fine meshing may be used as a reference for a coarse 3D modelling of system code to tune some models and/or evaluate the uncertainty (see [START_REF] Bestion | The concept of multiscale validation[END_REF]).

USING PIRT AND SCALING FOR 3D MODELLING AND VALIDATION

It was shown in chapter 2 and 3 how revisiting the PIRT and listing all local 3D effects could identify the modelling weaknesses and the lack of validation on some influent or dominant phenomena. The analysis was illustrated by the PWR core 3D. After the identification and ranking presented in table 2 the analysis may go farther by identifying more precisely the sensitivity of each process on some specific core situation, the influence of geometrical characteristics and the availability of validation data. This was made for SBLOCA and already presented [START_REF] Bestion | Specific requirements for BEPU methods using System Thermalhydraulic codes with 3D-Pressure Vessel Modelling[END_REF]. Appendix 1 shows another Table which was used to determine the experimental needs. The METERO experimental program was defined to answer many of the identified needs which is also briefly presented in Appendix 1 by a Figure and a test matrix, more detail information being available in two recent papers [25,[START_REF] Bestion | Improved PWR LOCA Simulations Through Refined Core 3D Simulations -An Advanced 3D Modelling and the Associated METERO Validation Program[END_REF]. A large scale test facility in prototypical steam-water conditions with full power full pressure is not available and would be extremely expensive without having measurement techniques able to provide the necessary information on each basic process. Therefore separate effect validation experiments may be used to provide information on mixing processes. Since some available data are performed in small rod bundles (5x5 to 8x8), they cannot investigate transfers between assemblies of different power. However, they can be used to model radial transfers at the sub-channel modelling scale and then may provide useful information for validation of a core modelling at the assembly scale. This uses the new concept of multiscale validation also recently presented in [START_REF] Bestion | The concept of multiscale validation[END_REF]. A validation of core 3D effects in SBLOCA may use IET which have radial power differences (Rabe & Prea, 2019, [START_REF] Rabe | Chimney and diverging effects in core PWR: analysis and experimental characterization for predictive behavior during loss of coolant accident[END_REF]) although the limited radial core size

CONCLUSIONS

The paper has shown how to analyze 3D processes occurring in a LWR PV at a basic level in view of providing a better modeling and identifying the validation needs. This is based on the porous 3D equations and on an order of magnitude analysis in order to rank phenomena. The paper illustrated the method on core thermalhydraulics for a range of situations including LOCAs, SLBs.

NOMENCLATURE

List of Acronyms

)

Figure 1 :Figure 2 :

 12 Figure 1: Radial transfers due to power profile. A high power assembly with two lower power neighbours. Left: low P LBLOCA reflooding; Right: high P SBLOCA core uncovering

  ; M: medium; L: low

Figure 3 :

 3 Figure 3: The METERO test facility with unbalanced feeding of two assemblies



  Mixing of a passive scalar in 1-phase flow  EV-US No spacers Scalar dispersion  dk D Mixing of a passive scalar in 1-phase flow  P losses and wall & interfacial friction in axial flow  P losses and wall & interfacial friction with transverse flows  P losses and wall & interfacial friction with buoyancy driven crossflows  Void dispersion tests  UV-UX-equal 𝜕𝑃 𝜕𝑧 UV: unequal velocity; EV equal velocity; ES equal scalar (tracer concentration); US unequal scalar concentration; UT: unequal temperature; UX: unequal quality; equal 𝜕𝑃 𝜕𝑧 means that velocity differences and temperature or quality difference are such that the axial P gradient are equal to avoid crossflows and to measure scalar or temperature or void diffusion-dispersion

)

  𝛼 𝑘 𝜌 𝑘 ( 𝜕𝑉 𝑘 𝜕𝑡 + 𝑉 𝑘 𝛻. 𝑉 𝑘 ) +𝛼 𝑘 𝛻𝑃 = (𝑝 𝑖 + 𝑓 𝑖 𝑇𝐷 ) 𝛻𝛼 𝑘 ∓ 𝜏 𝑖 + 𝛼 𝑘 𝜌 𝑘 𝑔 + 𝜏 𝑤𝑘 +

	1 𝜙 𝛻. (𝛼 𝑘 𝜌 𝑘 𝜏 𝑘 𝑙+𝑡+𝑑 ) (2)

𝜕𝜙𝛼 𝑘 𝜌 𝑘 𝑒 𝑘 𝜕𝑡 + ∇. (𝜙𝛼 𝑘 𝜌 𝑘 ℎ 𝑘 𝑉 𝑘 ) = 𝜙q 𝑘𝑖 + 𝑆 𝑐 q 𝑤𝑘 + 𝜙Γ 𝑘 ℎ 𝑘 + ∇. (𝛼 𝑘 𝑞 𝑘 𝑙+𝑡+𝑑 )

Table I . List of phenomena in a porous 3D model (M=mass equation, MM= momentum equation, E=energy equation)

 I

	Phenomenon		Equation	Present in	Validation	Models	Validation	Comments
		(M, MM, E)	1D models	in 1D	for 3D	in 3D
			Term
	Interfacial		M	yes	yes	Taken from	In some
	mass transfer		𝜙Γ 𝑘			1D	cases
	Space acceleration	MM 𝛼 𝑘 𝜌 𝑘 𝑉 𝑘 𝛻. 𝑉 𝑘	yes	No need	No need	No need	Depends on porosity & density
								variations
	Void		MM	yes	No	Yes for core	Some
	dispersion	(𝑝 𝑖 + 𝑓 𝑖 𝑇𝐷 ) 𝛻𝛼 𝑘			geometry	data
	Interfacial friction		MM 𝜏 𝑖	yes	yes	Taken from 1D	No data	No model in complex geometry
	Wall friction & P losses		MM 𝛼 𝑘 𝜌 𝑘 𝑔	yes	yes	No	No	Needs for non-axial flow
	Momentum diffusion (laminar)	𝜙 1	MM 𝛻. (𝛼 𝑘 𝜌 𝑘 𝜏 𝑘 𝑙 )	No	No need	yes	No	Very often negligible
	Momentum		MM	No	No need	Yes	Some data
	diffusion (turbulent)	1 𝜙	𝛻. (𝛼 𝑘 𝜌 𝑘 𝜏 𝑘 𝑡 )			for core	for core
	Momentum		MM	No	No need	Yes	Some data	geometry
	dispersion	1 𝜙	𝛻. (𝛼 𝑘 𝜌 𝑘 𝜏 𝑘 𝑑 )			for core	for core	dependent
	Energy		E	Yes	No need	Yes	No need
	transport	∇. (𝜙𝛼 𝑘 𝜌 𝑘 ℎ 𝑘 𝑉 𝑘 )
	Interfacial heat		E	Yes	Yes	Taken from	In some
	transfers		𝜙q 𝑘𝑖			1D	cases
	Wall		E	Yes	Yes	Taken from	In some
	Heat transfers		𝑆 𝑐 q 𝑤𝑘			1D	cases
	Energy		E	Yes	No need	Taken from	No need	Derived
	interfacial transfer		𝜙Γ 𝑘 ℎ 𝑘			1D		from 𝜙Γ 𝑘
	Energy		E	No	No need	yes	No	Very often
	Diffusion (laminar)		∇. (𝛼 𝑘 𝑞 𝑘 𝑙 )					negligible
	Energy		E	No	No need	Yes	Some data
	Diffusion (turbulent)		∇. (𝛼 𝑘 𝑞 𝑘 𝑡 )			for core	for core
	Energy		E	No	No need	Yes	Some data	geometry
	Dispersion		∇. (𝛼 𝑘 𝑞 𝑘 𝑑 )			for core	for core	dependent

Table 2 : Ranking of PWR core 3D phenomena with respect to cooling capabilities

 2

	PHENOMENON		THERMALHYDRAULIC CONDITIONS
		High V	Low V	High V	Low V	high V	Low V
		liquid	liquid	vapor	vapor	2-phase	2-phase
	Wall friction
	axial	D..v	N..v	D..v	N..v	I..v	N..v
	P losses axial	D u v	I.. u v	D.. u v	I.. u v	I.. u v	I.. u v
	P losses radial	I uf nv	D f nv	I uf nv	D f nv	I uf nv	D nv
	Void dispersion	-				D f v	I nv
	Interfacial
	friction axial					I v	D uf v
	Interfacial
	friction radial					I, uf nv	I uf nv
	Momentum	I f nv	N	I f nv	N	I	N
	diffusion
	Momentum	D	I	I	N	D	I
	dispersion
	Wall Heat
	transfers

Table 3 : PIRT analysis specific to a 3D approach for an uncovered core in a SBLOCA 3D EFFECTS IN AN UNCOVERED PWR CORE OF A SBLOCA

 3

	Process	Sensitivity on FoM (H,M,L)	Model Uncertainty (H, M, L)	Geometry effect	SET or global validation
	Flow regime identification	Possibly H	L in axial flow	possible effect of grids	No
	Interfacial friction	H	M	probably low	Axial flow:OK Radial flow : no
	Wall friction & form loss	M	axial flow: L radial flow: H	Spacer grids	Axial flow:OK Radial flow : no
	Void dispersion	L	H	?	No
	Interfacial H&M	L (pure vapor in
	transfers	dry zone)
	Wall HT regime identification	H	L		Axial flow:OK
	Convection to liquid	L	L	Spacer grids	Axial flow:OK
	Nucleate boiling	L	M		Axial flow:OK
	CHF (DNB or dry-out)	L to H	M	Spacer grids	OK
	Convection to vapor	H	L	Spacer grids	Axial flow:OK
	Heat diffusion-dispersion in liquid	L	M	Spacer grids	PSBT

Table 4 : Test series in METERO test facility

 4

	Process	Term of	Test series	Test conditions
		equation
	Wall friction	𝜏 𝑤𝑘 ⃗⃗⃗⃗⃗⃗	 P losses in axial flow	 Equal BC
			 P losses in non-axial flow	 UV
	Momentum diffusion-dispersion	  tk	 Pressure losses with transverse flow	𝜕𝑃 𝜕𝑧  With & without  UV-UT-equal
				spacers
	Scalar turbulent
	diffusion

ACKNOWLEDGMENTS

This study has been carried out in the framework of the CATHARE code developed by the CEA and supported by EDF, FRAMATOME and IRSN.

APPENDIX 1: ADDITIONAL INFORMATION ON CORE 3D VALIDATION NEEDS

The following Table 3 presents additional analysis made for a 3D approach for an uncovered core in a SBLOCA. for more information. This Table was used to define the METERO test program. The Figure 3 below shows the METERO test facility. Then the table 4 presents the test matrix and how each test series may answer some needs. It is not possible here to comment in detail these tables and the figure 3, the reader may go to reference [START_REF] Bestion | Specific requirements for BEPU methods using System Thermalhydraulic codes with 3D-Pressure Vessel Modelling[END_REF] for more information.